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Brief Relational Mathematics Counseling
as an Approach to Mathematics Academic Support
of College Students taking Introductory Courses
A DISSERTATION
submitted by
Jillian M. Knowles
In partial fiilfillment of the requirements
for degree of
Doctor of Philosophy
LESLEY UNIVERSITY
May 24
2004
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS Hi
ABSTRACT v
LIST OF TABLES vi
LIST OF FIGURES viii
CHAPTER
I THE NEED FOR A NEW MATHEMATICS COUNSELING
APPROACH TO SUPPORT COLLEGE STUDENTS 1
II THE THEORETICAL CONTEXT FOR RELATIONAL
MATHEMATICS COUNSELING 15
III A NEW APPROACH: BRIEF RELATIONAL
MATHEMATICS COUNSELING 60
IV METHODOLOGY 103
V AN ACCOUNT OF THE SUMMER 2000 PS YC/STAT 1 04
CLASS 139
VI UNCOVERING MATHEMATICS RELATIONAL PATTERNS:
THREE PSYCH/STAT 104 COUNSELING CASE STUDIES 196
VII DEVELOPING THEORY: STUDENT CATEGORIES AND
WAYS OF COUNSELING 307
VIII REFLECTIONS ON BRIEF RELATIONAL MATHEMATICS
COUNSELING: EVALUATION OF THE PILOT STUDY AND
RECOMMENDATIONS FOR FURTHER RESEARCH 341
APPENDICES:
A: Proposed Brief Mathematics Counseling Approach Chart 374
B: Individual Assessment Instruments 375
C: Class Assessment and Observation Instruments 388
D: Informed Consent Forms etc 446
E: Coding and Analysis 451
F: ClassSeating 459
G: PS YC/STAT 1 04 Instructor Handouts and Syllabus 462
H: Class Data 471
I: Class Calendar of Events 474
J: Mathematics Counselor Tutoring Handouts 479
K: Karen's Data 486
L: Jamie's Data 491
M: Mulder's Data 498
REERENCES 503
Ill
ACKNOWLEDGEMENTS
First I thank the instructor and the students of the statistics in psychology summer
2000 class who so generously allowed me to pilot with them my new approach to
mathematics support. While their names must remain anonymous I hope that this
document reveals each one's unique learning history and approach to mathematics and
the courageous changes each made to improve in that approach.
My deepest gratitude and love go to my husband Robert, and daughters Christina,
Andrea, and Margaret who have gone beyond all expectations in supporting my efforts
and believing in my ability to complete this project. I could not have done it without them
and can only guess at the impact of the sacrifices each has made to make it possible.
A sincere thank you to my senior advisor and first reader Rebecca Corwin for her
ongoing efforts in reading draft after draft of this manuscript often under considerable
time pressure. Special thanks go to Cornelia Tiemey, my second reader, for her extra
work in reading drafts and for her incisive critiques. Paul Crowley also deserves my
thanks for his work as third reader, especially for helping me hone my counseling
insights and skills.
I am deeply gratefiil to Sarah Madsen Hardy and to Caroline Heller and the
Lesley Ph.D. Fall 2003 writing group for their wonderfiil support in helping me fmd and
develop my writing self.
My thanks go to those with whom I work, especially to Margaret Pobywajlo and
Roberta Kieronski but also to other colleagues, peer tutors, and students who have not
only supported me in my long writing process but also have continued to teach me and
deepen my insights into ways the learnmg process is embedded in our relational realities.
IV
Finally my deep thankfulness goes to Jesus, my extended family and friends, and
my community of faith who have prayed for and supported me through this long process
and have accepted me with the sometimes startling changes I have made.
I would like to dedicate this manuscript to the memory of my parents Molly and
Geoff Fraser. My mother inspired me with the vision to see potential in those who could
not see it in themselves. My father challenged me to move beyond sentiment to expect
and demand realistic effort and careful thinking of myself and of those with whom I
work.
Abstract
Traditional approaches to college mathematics support focus on cognitive aspects
of the student's approach and only incidentally address affective problems such as
mathematics and testing anxiety. Because such affective conditions may be symptoms of
underlying relational problems rooted in a student's learning history, I proposed a brief
relational mathematics counseling approach that integrates cognitive constructivist
mathematics tutoring into a brief relational conflict counseling framework (Mitchell,
1988; Windschitl, 2002).
I hypothesized that using this approach, professional tutors who also took on a
role of mathematics counselor, could help underachieving students, during an
introductory level mathematics college course, improve their approach to mathematics
and avoid failure. This pilot study was conducted with an introductory statistics class at a
small urban commuter university in the Northeast United States. Ten of the 13 students in
the class volunteered for counseling; each participant had an average of 5 sessions. The
counselor helped students explore their relationships with their mathematics selves,
internalized presences, and interpersonal attachments while sessions focused on
mathematics and course management. This new approach to mathematics support helped
the counselor and the student become aware of mathematics relational patterns impeding
the success of the student, and allowed both to develop constructive ways to change
counterproductive behaviors.
VI
List of Tables
Table 4.1
Table 4.2
Table 4.3
Table 5.1
Profile Summary of Students taking PSYC/STAT 104,
Summer 2000 114
PSYC/STAT 104, Summer 2000 Class and Research
Schedule 116
Conventions Used in Presentation of Transcripts 129
Grades Throughout the Course of all Individuals in
PSYC/STAT 104, Summer 2000 183
Table 5.2
Number of Individual Utterances during Lecture Portion
of Classes and Final Grade 185
Table 6.1
Table 6.2
Table 7.1
Table 7.2
Table 7.3
Table Al
Table El
Table E2
Table E3
Table E4
Focus Participants' Levels of Understanding of the
Variable on the Algebra Test 300
Focus Participants' Understanding of Arithmetic on the
Arithmetic for Statistics assessment 301
Criteria for Determining Level of Mathematical
Preparedness of PSYC/STAT 104 Participants 308
Emerging Categories of Mathematics Self Development
310
Criteria for determining Malleability of PSYC/STAT 104
Participants 312
Proposed Brief Relational Mathematics Counseling
Summary 374
Analysis scheme for Counseling Session Data: Student's
Mathematical Relational Matrix 463
Analysis of Lecture Session Student Exchanges with
Instructor 464
Analysis of Student's Problem Working Session
Behaviors 465
Protocol for Analysis of Exam Question Solutions 466
Table HI
Students' Expectations & Hopes in Relation to Effort,
Grades and Scores, Summer 2000 461
Vll
Table H2 Student Tier (Tobias) and Category (Knowles) in
Relation to Class Rank after Exam #1 and Pre- and Post-
Statistical Reasoning Assessment (SUA) scores 462
Table H3 Students' Pre and Post Positions on Feelings and Beliefs
with Net Number of Changes 463
Table Kl Karen's Individual PS YC/STAT 1 04, Summer 2000
Participation Profile 494
Table K2 Karen's Progress in Tests in Relation to Mathematics
Counseling Interventions 495
Table K3 Numerical Scores and Averages of Karen's JMK
Mathematics Affect Scales Responses 498
Table LI Jamie's Individual PS YC/STAT 104, Summer 2000
Participation Profile 499
Table L2 Jamie's Progress in Tests in Relation to Mathematics
Counseling Interventions 500
Table L3 Numerical Scores and Averages of Jamie's JMK Scale
responses 505
Table Ml Mulder's Individual PSYC/STAT 1 04, Summer 2000
Participation Profile 506
Table M2 Mulder's Progress in Tests in Relation to Mathematics
Counseling Interventions 507
Table M3 Numerical Scores and Averages of Mulder's JMK Affect
Scale Responses 510
VIU
List of Figures
Figure 5.1 Room and furniture configuration for PSYC/STAT104
class, second floor, Riverside Center, Brookwood State
University, Summer 2000 141
Figure 5.2 Individual's scores on Test #1, with each student's score
broken down into his Conceptual( multiple-choice Part I)
score, out of 40; his Symbol score (on Part I) out of 8;
and his Computational score (Part II) out of 52, total
possible 100 172
Figure 7. 1 Mathematics Self Development Categories of
PSYC/ST AT 104 participants. 331
Figure Fl Jillian's seating positions for the PSYC/STATS 1 04
course, second floor. Riverside Center, Brookwood State
University, Summer 2000 467
Figure F2 The most usual seating choices of students for
PSYC/STAT104 course, second floor. Riverside Center,
Brookwood State University, Summer 200 468
Figure KI Karen's pre and post scores on the pre Mathematics
Feelings Survey in relation to class extreme scores. 496
Figure K2 Karen's pre- and post-summary scores on the
Mathematics Beliefs Survey in relation to class range
scores. 496
Figure K3 Karen's responses on the JMK Mathematics Affect Scales
(in Mathematics Coimselmg Sessions 2 through 5) 497
Figure LI Jamie's responses on Questions 1 1 and 12 on Exam #1 .
Note her self-corrections in question 1 1 . 501
Figure L2 Jamie's responses on Questions 13 and 14 on Exam #1.
Note her self-corrections in questionl3. 501
Figure L3 Jamie's solution to question 7 (d) Find the area of the
figure. Algebra Test (Sokolowski ( 1997); see Appendix
D). Note Jamie's uiitial error that she scratched out and
replaced with the correct properly coordinated area
solution. 502
IX
Figure L4 Jamie's mathematics pre- and post-feelings and beliefs
Survey Profile Summary in relation to class range pre-
and post-scores. 503
Figure L5 Jamie's responses on the JMK Mathematics Affect
Scales, Sessions 1-5 504
Figure Ml Mulder's responses to the pre- and post-Mathematics
Feelings and Mathematics Beliefs surveys in relation to
class extreme scores. 508
Figure M2 Mulder' s responses on the JMK Mathematics Affect
Scales for Sessions 1,3, and 5. 509
CHAPTER I
THE NEED FOR A NEW MATHEMATICS COUNSELING APPROACH
TO SUPPORT COLLEGE STUDENTS
This research grew out of my experience as a mathematics learning specialist in
academic support centers at two- and four-year colleges and universities in the Northeast
United States. Over the years, certain students who came to me for help puzzled me. Some
whose skills seemed inadequate or whose experience of mathematics seemed too damaging
went on to succeed, while others succeeded at the introductory level but could not continue
and eventually changed to a major that had fewer mathematics requirements or none at all.
Too many of my students — ^typical students with no diagnosable learning disabilities —
withdrew or failed. Again and again I noticed that academic proficiency alone could not
explain my students' success or failure. I became convinced that there was another way of
understanding how college students learn mathematics — one that my training and
experience did not give me tools to address at that point.
A young woman I will call Janet was one such puzzling student. A freshman who
was taking a business precalculus course, Janet was in the practice of coming once or twice
a week to the Mathematics Support Center during drop-in time to work on her homework.
The peer tutors or I would check on her and sit down with her if she was struggling. One
morning there were few students and no peer tutors present, just me, the professional tutor.
Janet was sitting close to the table with her notebook and mathematics text on the table in
front of her. As I went over to check on how she was doing, she pulled her hands out from
under the table. Without thinking, I blurted out, "What were you doing?" Shamefaced, Janet
replied, "I was counting on my fmgers." She was working on factoring a quadratic equation
and was trying to work out what factors of 24 summed to II by tapping her fingers on the
underside of the table. I asked why under the table — ^and not above, where she could see her
fingers — and she told me that when she was six years old, in first grade at a parochial
school, her teacher had rapped students' knuckles whenever she caught them using their
fingers to help with arithmetic problems. The teacher had forbidden the use of fingers or
other counting materials, taking the need for them to mean a student had not done her
homework or memorized her addition facts. So Janet had learned to keep her counting
hidden and had never committed her addition facts reliably to memory.
Janet grasped quadratic equations — clearly she had the aptitude to memorize these
arithmetic facts. The question was why she hadn't. At the age of 19, Janet was stuck in
mathematical behavior that was now neither appropriate nor necessary, although it was
sensible in the early grades. She was managing in her course, although it always seemed to
be a quiet struggle and she never seemed confident of her outcomes. I began to wonder
about how Janet's first grade teacher's treatment and her ongoing need to rely on hidden
counting had affected Janet's view of herself as a mathematics learner. I wondered whether
she was now aware that it is considered developmentally appropriate for first graders to
count usmg physical objects. Or did she still believe, as she seemed to have then, that the
teacher was justified in her knuckle rapping and that she was bad at mathematics, as the
teacher implied?
Janet was confident and doing well in her other courses; it was only in mathematics
that she was struggling. Even within mathematics there seemed to be discrepancies in her
confidence and achievement. She grasped difficult precalculus concepts such as the
composition of fiinctions, but her tentative grasp of underlying arithmetic facts often seemed
to undermine her confidence in her understanding of such advanced concepts. Despite her
difficulties and unrealistic underconfidence, Janet did persist and earned a B~ in her
course — not as high a grade as I felt she could have earned, however. Her persistence also
puzzled me. I saw other low-confidence students, with no early trauma and with a sound
grasp of the underlying arithmetic facts and concepts, who came intermittently to the
mathematics support center, seemed to feel helpless to change their gloomy expected
outcome, and ended up withdrawing or failing.
Janet was the kind of student whom I often found myself wanting to help but not
knowing how. I could recommend the upcoming mathematics anxiety workshop. Although
anxiety did not seem to be at the core, Janet did have affective problems with mathematics
that included anxiety. Likewise, I could help her master her precalculus content. But from
past experience I knew that these interventions were unlikely to affect Janet's overall
approach to herself learning mathematics. And yes, I could (and did) tell her about research
and good practice in elementary education that showed that her teacher had been wrong and
that Janet's use of fingers in first grade did not mean that she was bad at mathematics. But I
suspected that simply communicating this information would not be enough to convince her
that she had always been and was now able to understand and master mathematics. Her
progress continued to be achieved at what appeared to be considerable emotional cost and
little sense of personal ownership; she attributed her relative success to the peer tutors and
me.
The struggling students who inspired me to undertake the research described in
this dissertation are in many ways typical of American college students. Regardless of
their major, in most U.S. colleges or universities, students are required to take a
mathematics course at an introductory college level for a liberal arts degree; some must
go further for their major. More students withdraw jB-om or fail these courses than any
other college courses (Dembner, 1996a, 1996b). Students who do not llilfill their
college's mathematics requirements often abandon or change their academic and career
goals. Students like Janet may persist in mathematics for their major, but fail to develop
the confidence to apply it to related courses or in the workforce.
Colleges and universities have attempted to address the problem of failing and
withdrawing mathematics students and the alarming attrition rate' of students from
mathematics and mathematics-related majors in college (Madison, 2001; National
Research Council, 1991). The most prevalent assumption on the part of colleges and
universities is that this failure and attrition can be attributed to students' deficient high
school mathematics backgrounds. Perhaps they are taking classes for which they do not
have the prerequisite knowledge.
Increasingly since the mid-1980s, academic institutions have attempted to support
struggling students and to encourage those who are more confident to continue studying
mathematics by instituting placement testing; developing short courses to teach
prerequisite mathematics and study skills; establishing learning resource centers that
usually provide peer and professional tutoring for individuals and groups, instituting
behavioral or cognitive counseling programs, and offering workshops focusing on study
skills or testing anxiety (Boylan, 1999; Hadwin & Winne, 1996). Many also offer pre-
college level developmental mathematics courses." All of these are efforts by the
institution to reduce failure and enhance retention. Much of it may be seen to fall under
the umbrella of what used to be called remedial but is currently called developmental
education. '"
From the perspective of the individual student, the outcomes of such efforts are
uncertain, however. In my position as a mathematics specialist in the learning support
center, a central piece of the university's failure reduction and retention effort, I see that
while students who make strategic use of such resources can achieve at a higher level in
mathematics coursework, many who need help are not strategic in accessing it. Furthermore,
the predominantly cognitive, skills-based approaches that the learning support center offers
are ineffective for understanding and addressing the problems faced by underconfident,
anxious, or avoidant students. The measures currently in place fail to fiilly address the
problems of college students like Janet — ^those with what I have come to understand as poor
mathematics mental health" The number of students I encounter who need help of a kind
not provided by current approaches leads me to believe that poor mathematics mental health
may be central to our national failings in mathematics.
The Learning Support Center Context
The research that is described in this dissertation grows directly out of the
questions that plagued me over the course of a 15-year career as a learning center
mathematics specialist. My objective was to find new approaches to helping students
struggling with mathematics in the specific context of a college or university learning
support center. I believe that academic support center personnel are well positioned to
apply a new, more holistic approach to helping students struggling with poor
mathematics mental health because of the opportunities for professional tutors to work
one-on-one with students, the separation from regular classroom dynamics, and the semi-
autonomy of typical learning support centers that makes organizational changes and the
piloting of innovative approaches easier. However, this study was designed with a lively
awareness of the practical challenges of working with students in this setting.
In academic support centers like the one where I work, mathematics learning
specialists overwhelmingly focus on mathematical skills and concepts. The pressures of
everyday practice in an academic support center, the urgency the students feel because of
the limited time available, and the importance of mathematics as an academic gatekeeper
combine to create among academic support personnel a tendency toward unreflective
pragmatism (Lundell & Collins, 1999). This pragmatism is characterized by only
incidental assessment of affective issues as well as limited mathematics assessments. This
short-term view leads to a default tendency to focus only on the course mathematics,
especially on procedures and skills rather than understanding. Mathematics tutors are
under great time pressure: Although students may make ongoing weekly appointments,
we see the typical student only when he" chooses to come in. The incidental nature of our
contact with the student exacerbates the problem. For too many students, this approach is
not working adequately.
The problem is not that mathematics learning specialists do not know about
cognitive and affective factors significant for achievement, but rather that we know them
abstractly and as separate factors, and lack an approach for gaining, prioritizing, and
using this knowledge effectively. In addition, 1 increasingly had the troublesome sense
that the cognitive and affective expressions that we see (e.g., Janet's fmger counting and
underconfidence) may be symptoms rather than causes of a student's real difficulties. We
deal daily with the interaction of these overt and hidden factors and their meaning for a
particular student, but this meaning often eludes us. The quick diagnosis of a student's
central problem, whether overtly cognitive or affective, underlying, or an interaction of
these, is a particular challenge in the learning center context.
For all of these reasons, it is unusual in mathematics academic support to find out
about students' mathematics learning histories and the understanding, beliefs, attitudes, and
habits that they developed as a result. My frustration with my limited ability to help Janet
and others like her grew at the same time as I began to recognize clues to the puzzle. More
and more, I became certain that the cognitive approach predominant in my field was not
enough to understand how and why college students undertake to learn mathematics. How
could I address the root source of Janet's arithmetic problem, her approach to coping with
the problem over many years, and her evidently low mathematics self-esteem? I began to
see that I needed a fiiller understanding of Janet's history and its effects on her present
mathematics experience in order to understand what I, as a mathematics learning specialist,
could do to help her change her mind about herself as a mathematics learner.
I began to wonder if affective issues and learning histories might be important
determiners of achievement in mathematics among typical college students. I suspected that
cognitive outcomes were related not only to academic preparation, but to relational
dynamics and affective experiences: an elementary school teacher who humiliated a student
for asking a question or a parent who told a student that she inherited the family "we-cannot-
do-math gene." Experiences like these may lead to otherwise inexplicable gaps in basic
number facts and number sense or hazy understanding of the algebraic variable. Why did
certain students fmd themselves unable to think, interact, or connect with the instructor?
Could it be that students with otherwise adequate mathematics skills and aptitudes are
limited by unconscious forces Imked to earlier mathematics learning experiences that cause
8
them to repeat counterproductive practices? Might one defining negative experience with a
teacher in grade school or in high school affect a student's lifelong learning of mathematics?
How might poor preparation interact with a student's mathematics identity to affect his
approach in the current course?
I wanted to fmd out what would happen if a mathematics learning specialist did
have the opportunity to delve into these questions in the learning center context. What
would be the result on a student's mathematics achievement in the current and future
semesters if I were able to offer support based on a more holistic picture of that student as
a mathematics learner? If mathematics learning specialists could fmd ways to understand
and help the whole person rather than dealing with his parts — ways to address the
mathematics mental health of their students — ^we might be able resolve these problems
and more effectively and reUably help him go on to achieve long-term mathematics goals.
The Study
In my capacity both as a mathematics learning specialist and a doctoral student, I
have searched for ways to understand students' mathematics mental health, diagnose their
difficulties, and help them holistically and effectively. Through the study that is described in
this dissertation, I have sought to create and test a more holistic approach of academic
support that would help the many students I encountered with academic mathematics
problems rooted deeply in relational conflict and other traumas that thwarted the
development of their mathematical identities.
This research was based on the hypothesis that an adequate knowledge of the student
as a whole person doing mathematics may be a pivotal part of academic support personnel's
plan for understanding and supporting him through his mathematics course. This hypothesis
led to four research questions that are mformed by the set of challenges particular to an
academic support center setting:
1 . What does a mathematics learning specialist need to know about a student in
order to understand him as a whole person doing mathematics?
2. What processes can be used to gain this understanding quickly while he is taking
a mathematics course?
3. How can a mathematics learning specialist use this fialler understanding of the
student to help him in the mathematics course he is taking?
4. What does a mathematics learning specialist need to understand about himself as
a counselor and tutor in order to help the student succeed?
The search to answer these questions and thus understand and effectively intervene
in each student's complex interactions between his mathematics affect and cognition led me
outside the narrow boundaries of the field of mathematics academic support. The field of
counseling psychology — in particular, relational psychotherapy — emerged as providing the
most perceptive ways to understand the effects of students' mathematics learning histories
on their current learning challenges. In chapter 2, 1 discuss the work of scholars I have
drawn from. By adapting theories and practices of relational psychotherapy to mathematics
learning, and then combining these new methods with the cognitive approaches that I had
been practicing for years, I arrived at a brief relational mathematics counseling approach. I
describe the development of this approach in chapter 3. To investigate the approach, I
piloted it with students taking a summer introductory- level statistics course taught at a small,
urban, commuter state university in the Northeast. The remainder of the dissertation
describes and discusses the study itself — ^the use of case study methodology and the criteria I
10
used in my choice of particular cases to present in chapter 4; the presentation of the class as
the case that creates the context for the individual cases in chapter 5; the individual cases in
chapter 6; analysis of results and developing theory in chapter 7; and evaluation, limitations
and implications of the study, as well as recommendations for further research in chapter 8.
The goal of this study was to develop, pilot, and evaluate a mathematics
counseling approach based on brief relational therapy approaches (with cognitive therapy
and developmental psychology contributions) designed to help individuals attain sound
mathematics mental health and success in reaching their own mathematics goals. This
involved identifying, adapting, and developing instruments and approaches that explore
students' mathematics learning; their history, beliefs, and attitudes about learning; and
their relational patterns as they participate in an introductory level college mathematics
course. Students engaged in a brief course of mathematics relational counseling with me
as the mathematics counselor using these instruments and approaches.
This study contributes an approach to the field of mathematics academic support
that combines aspects of mathematics and personal therapy approaches drawn from
cognitive, affective, and relational theory. It is designed to help college academic support
staff understand and help the student as a whole person doing mathematics. It combines
what are typically considered to be quite unrelated, disparate elements of mathematics
learners and those who help them, that is, mathematics cognition and affect expressed in
distinctive relational patterns (his, mine, and ours). The results provide some prelimmary
data to establish groundwork for the development and use of this individual counseling
approach to improve students' mathematics mental health and success in required college
mathematics courses.
11
My goals can be further summarized thus:
1 . To identify, adapt, and develop instruments and approaches that explore students'
mathematics learning, their history, feelings, attitudes, and beliefs about learning,
and their relational patterns as they may affect progress in an introductory-level
college mathematics course;
2. To pilot a mathematics counseling approach based on brief relational therapy
approaches (with cognitive therapy and developmental psychology contributions)
with the goal of helping individuals attain good-enough" mathematics mental
health and success; and
3. To evaluate assessment and treatment instruments and approaches, and more
importantly, the brief relational mathematics counseling approach itself
Over the past few years, my colleagues have looked at me quizzically when 1 tell
them that my research explores how relational therapy that is rooted in Freud can help
college students achieve in mathematics. Admittedly, my approach is quite
unconventional. On the surface, the teaching and learning of mathematics seem to have
little to do with the murky realm of unconscious motivations and relational conflicts. But
when 1 observed my students' behavior, addressed their achievement problems as
symptoms, and asked them to talk to me about how they felt about their teachers, their
peers, themselves, and the subject of mathematics itself, the results were rife with
conscious and unconscious motivations that were often in conflict, and counterproductive
relational patterns in which students seemed stuck.
While there are many tools to assess how affect effects achievement in
mathematics and cognitive and behavioral treatments to address problems, to my
12
knowledge the only practitioner who has attempted to understand how mathematics
issues can be addressed using a holistic individual approach based in Freudian
psychotherapy is Lusiane Weyl-Kailey (1985), a Parisian psychotherapist who had been a
mathematics teacher. Her work was conducted in a clinical setting with school children
whose psychological and emotional disturbances she found to be connected with their
mathematics learning problems. She used psychopedagogy — an integration of Freudian
therapeutic and pedagogical approaches — to understand the psychological effects of
mathematics on her clients in order to improve both their mathematics learning and their
psychological health (Tahta, 1993; Weyl-Kailey, 1985). While she is a psychotherapist
who brings her understanding of mathematics pedagogy into her therapy with disturbed
students who had mathematics learning issues, I am a mathematics educator who
proposes to bring Freudian-related relational conflict therapy as a new approach into the
learning support of average mathematics students who have affective and relational
barriers to their mathematics learning.
In this study I show that close psychological attention to unconscious motivations
and conflicts is applicable not only for those whose mathematics learning problems may
be related to personal emotional disturbances but more generally for ordinary college
students whose psychological functioning is within the range of "normal," and this
counseling approach may be appropriately delivered in the educational setting. In the
following pages, I will define a mathematics selfihaX we all have, no matter how deeply
neglected, damaged, or denied. I will explain how a teacher or tutor can be like a parent
in the psychological development of this mathematical self It is my hope that the theories
13
I have explored and the approach I have piloted will open the door to a new way of
thinking about academic support that nurtures and heals students' mathematics selves.
When I begin to describe my work and my dissertation project, many people
(university colleagues, students, friends, acquaintances, fellow partygoers or fellow
church members) want to tell me their mathematics story. Each one wished that when
they were struggling with the mathematics course that ended or changed their career
aspirations, they had had someone knowledgeable in mathematics, mathematics
pedagogical research fmdings, and relational counseling approaches who had been able to
help them understand and get over their fears and low confidence so that they could
proceed with their mathematics learning. For others the topic is so painful that they have
to change the subject or walk away. And there are some who have a story of struggle and
triumph and a few who never or rarely struggled, almost always "getting it" and
succeeding. It is for the many who, for want of someone who could listen knowingly and
intervene strategically, performed poorly or avoided or failed in the mathematics they
needed, that I pursued this dissertation research.
14
' Of the 3.6 million U.S. mathematics students in ninth grade in 1972, only 294,000 persisted to al-Ievel
mathematics courses as freshmen in college in 1976. Only 1 1,000 continued to graduate with a bachelor's
degree in mathematical sciences in 1980, and 2,700 succeeded in graduating with a master's degree in
1982. See National Research Council, 1991, p. 19, Figure 5. These figures are relatively dated but the
current progression appears to be similar.
" Currently, abnost all community colleges and more than 60 percent of other colleges and universities in
the United States offer developmental courses in mathematics, writing, study skills, and in some cases
reading (Bibb, 1999;Dembner, 1996), mathematics developmental courses being the ones most enrolled in
by freshmen, however (Phipps, 1998; Madison, 1990).
"' In recent discussion of the evolution of developmental education in colleges and universities, Payne and
Lyman note that the preference for the term "developmental" over "remedial and developmental" was
formalized in 1976 when the name of the professional journal was changed to reflect that. They point out,
however, that the field has been known by many other names in its long history (Payne, 1996). Higbee
(1996) sees the essential difference between "remedial" and "developmental" as the difference between "to
correct a previous wrong" and "to promote the growth of students to their highest potential" (p. 63), that is,
the difference between a deficit and a growth orientation.
™ Sheila Tobias (1993) uses the term "math mental health" to refer to a person's "willingness to learn the
mathematics [he] needs when [he] needs it" (p. 12), using it as the criteria to assess a student's mathematics
ftmctioning beyond the cognitive, fri contrast, in adopting her term I include under it all aspects of a
student's mathematics ftmctioning including cognitive factors.
" In odd numbered chapters I use the masculine, "he," "him," and "his" for the third person singular generic
pronoun. In even numbered chapters I use the feminine, "she," "her," and "hers."
" I have adapted the use of Winnicott's (1965) term "good-enough" for this study. A fiill discussion of his
use of it and my adaptation comes in chapter 2.
15
CHAPTER II
THE THEORETICAL CONTEXT FOR A RELATIONAL COUNSELING APPROACH
I identified in chapter 1 the central problem that learning specialists face when we
try to help students achieve their potential in college-level mathematics. We focus
narrowly on course-related mathematics skills and concepts; we may help the student
improve her grade but fail to understand and help her' as a whole person doing
mathematics. The focus is so much on helping her pass her course that we do not stop
long enough to Usten and understand what is really preventing the success she aspires to.
What if I had the opportunity to hear her story and understand how certain people or
experiences might have affected how she is doing mathematics now? What if I knew how
to help her unravel herself from beliefs and behaviors that seemed to be standing in the
way of her success, beliefs and behaviors that had developed over the years as the result
of those people and experiences? I determined that if there were a way to use an
individual counseling approach that could be incorporated into regular mathematics
support offered through the learning support center, the problem I had identified might be
resolved.
I was then faced with the task of finding and/or developing a counseling approach
or approaches adaptable to the central mathematics learning task, compatible with the
educational setting, and, most importantly, perceptive of underlying causes. In this
chapter, I describe my search for such a counseling approach and demonstrate how my
research into existing theories in the fields of education and counseling psychology
provided the insight I needed to help the whole person doing mathematics.
16
CONTRIBUTIONS FROM THE FIELD OF
MATHEMATICS EDUCATION
First I asked if researchers and practitioners in the field of mathematics education
had also perceived the problem I had identified and, if so, what they had done about it. I
found that there is a large body of research on cognitive (cf Hiebert & Lefevre, 1986;
Piaget, 1969) and affective (cf McLeod, 1989, 1992, 1997; McLeod & Ortega, 1993)
factors of mathematics functioning and on the relationship between cognition and affect
(cf Boaler, 1997; Buxton, 1991; Skemp, 1987). Pragmatic approaches to improving
students' mathematics functioning problems range from those that focus primarily on
cognitive problems (changing mathematics pedagogy or curricula), through those that
focus primarily or affective problems (chiefly alleviating emotional symptoms such as
anxiety), to those that focus simultaneously on both cognitive and affective problems
(some dealing with affect and cognition separately, cf Nolting, 1990), others dealing
with them as interconnected factors (cf Carter & Yackel, 1989; Tobias, 1993).
Researchers and practitioners of mathematics education concur that a student's
mathematics functioning involves both cognitive and affective factors, although there is
little clarity on how these factors interact (cf McLeod, 1992; Schoenfeld, 1992). As a
minimum, they suggest m order to understand how a student is functioning
mathematically, a mathematics learning specialist needs to know what the student
understands of the prerequisite mathematics, how well she can apply that background
understanding in learning new mathematics concepts and procedures, and any affective
orientations she has developed that might affect that learning process.
17
Cognitive Factors
To know what the student understands of the prerequisite mathematics, a college
usually attempts to gauge her current level of competence using high school records and
course-taking history, a college-devised placement test, Scholastic Aptitude Test (SAT)
or American College Test (ACT) mathematics score, an interview, or some combination
of these. If course placement is mandated by this process, the student and the
mathematics learning specialist have some assurance that the level of difficulty of the
current course is within range of her capabilities. Other aspects of the student's cognitive
processing known to have affected her mathematics learning and present achievement
such as her preferred mathematics learning style," concept developmental levels, and
long- and short-term memory are generally not assessed, so Uttle is knovra except what a
learning specialist observes in tutoring. A student's awareness of her own learning
processes and her strategic study skills, when developed in relation to current
coursework, have also been found to be significant cognitive factors that are often linked
with achievement (Hadwin & Wiime, 1996).
The cognitive effects of the mathematics teaching approaches the student has
experienced may be even more significant. Students who have experienced
predominantly procedural rather than conceptual'" teaching approaches are likely to see
mathematics learning as memorization of procedures rather than understanding of
concepts and their connections, making security in the mathematics they know tenuous
and new learning more difficult (Boaler, 1997; Skemp, 1987). Students who have
experienced a teacher transmission and textbook exercise approach rather than a student-
18
centered, problem-solving approach are not likely to have developed effective strategies
for approaching new mathematics learning (cf. Schoenfeld, 1985, 1992).
While it seemed that in my approach I would need to be mindful of all these
cognitive fectors as potentially significant in a student's success, I was concerned about
the challenge of identifying aspects of mathematics affect that might be just as significant
and understanding how these factors interacted.
Mathematics Functioning: Affective Factors are Crucial
In academic support, I had found that research on affect — beliefs, attitudes, and
feelings — and its effects on students' mathematics learning and achievement is even
more difficult than research on cognition to translate into understanding an individual's
beliefs and feelings about her mathematics learning. It also seems more difficult to apply
this understanding to developing a plan to help her succeed in her course. Mathematics
and testing anxiety, locus of control, issues of learned helplessness, attribution, as well as
achievement motivation are all affective factors that have been demonstrated to be factors
in mathematics achievement (Dweck, 1975, 1986; Hembree, 1990; X. Ma, 1999).
Mathematics cognitive psychologists like Skemp (1987) who look at students from the
perspective of mathematics cognition have identified negative affective orientations and
outcomes linked with teaching and learning approaches. Others like Buxton (1991) who
have looked at students from the perspective of thefr affective difficulties with
mathematics have identified problems in their cognition and cognitive learning
approaches. To make this even more complex, demographic characteristics (gender,
socioeconomic status, age, first language, and race or ethnicity) may interact to magnify
19
or minimize individual effects of a student's past experience on her achievement
(Secada, 1992).
Existing Approaches that Attend to both
Affect and Cognition
Mathematics support personnel and researchers have struggled to understand
interactions among students' cognition and affect on their mathematics resilience and
achievement. Much work has been done in the attempt to develop ways of helping. From
this research, four major approaches have emerged. Each is a pragmatic attempt to help
adults overcome their underachievement, aversion, and fear of mathematics. The
approaches yield important information for my work, although I found that their
usefulness is limited by the fact that they are either not directly applicable to the setting 1
am investigating or they do not provide an adequate framework for holistic understanding
and counseling.
The First Approach: Freestanding Anxiety Reduction
Workshops or Short Courses
The most typical approach is a freestanding" course or workshop where the
participants do mathematics as they tell their mathematics stories. Through this they
become conscious of their own affect, habitual reactions, and beliefs about mathematics
and the effects on their mathematics identities (cf Buxton, 1991; Kogehnan & Warren,
1978; Tobias, 1993).
To this list Carter and Yackel (1989) added another: adults' enculturation in and
orientation to mathematics learning. They used Skemp's (1987) categories, distinguishing
between an "instrumental"^ mathematics orientation (characterized by a "just teach me
how to do it — I don't want to understand it" procedural approach) and a "relational""'
20
mathematics orientation (characterized by an "I want to understand why it is so and how
this relates to what I already know" conceptual approach). They found that an
instrumental approach was generally linked to heightened anxiety and passive behaviors,
while students taking a relational approach used active problem-solving strategies and
make positive attempts to construct mathematical understandings. Carter and Yackel used
journal writing, cognitive constructivist problem-solving approaches, and cognitive
behavior therapy techniques such as cognitive restructuring to help participants move
from an instrumental (procedural) to relational (conceptual) orientation to mathematics.
They found that students who made this change also experienced a significant reduction
in mathematics anxiety."'
One important limitation of Tobias's or Carter and Yackel' s approach for my
work, however, is the fact that it is freestanding and thus not linked with a college course.
Ahhough participants tend to become less anxious and gain confidence, there may be
little positive effect on their achievement in a college courses taken concurrently (E.
Yackel, personal communication, January 21, 2000). Notwithstanding, these researchers
do contribute some significant elements to the design of my approach: Practitioners like
Tobias or Carter and Yackel stress group work and focus on identifying (and challenging)
counterproductive thoughts and behaviors at the conscious level. Their successful use of
cognitive counseling techniques such as hypothesis testing of faulty beliefs"" and
cognitive restructuring" prompted me to investigate cognitive counseling further for its
possible contributions to my approach. Most importantly this process integrates focus on
research-supported conceptual mathematics approaches with linked affective outcomes
rather than treating cognition and affect separately. I determined to investigate how I
21
might incorporate Carter and Yackel's successful use of constructivist problem-solving
mathematics pedagogy to change counterproductive mathematics orientation and affect
into my approach in a college learning assistance context.
Second Approach: Study Skill and Anxiety Reduction Co-Courses
Linked to a College Mathematics Course
The second approach noted in the literature consists of a second course or lab
linked to a college mathematics course. Addressing counterproductive beliefs and habits,
these co-courses focus on developing skills for mastering the mathematics content of the
college course and are typically effective m improving students' achievement (cf
Stratton, 1996; Nolting, 1990). Even further improvement in achievement resulted for
students diagnosed with high external locus of control" when a brief course of individual
cognitive counseling aimed at internalizing locus of control and reducing helplessness in
the mathematics learning situation was provided (Nolting, 1990). This success
encouraged me in my pursuit of an individual counseling approach, but my experience
told me that it is not only students with high external locus of control who could benefit
from individual counseling; students like Janet (see chapter 1) have other forms of
emotional impediments to achieving their mathematics potential.
1 noted the consistent findings of Stratton, Nolting, and other researchers (Hadwin
& Winne, 1996) that students benefit more from study skills and negative affect reduction
courses or workshops that are linked to a particular academic course they are taking
simultaneously than from freestanding offerings that are not specific to a particular
course. This finding was a key incentive for me in pursuing an approach that could be
tailored to a particular course the student was taking and delivered simultaneously.
22
A Third Approach: Mathematics Instructors Addressing
Affect in the Classroom
The third approach involves the mathematics instructor herself incorporating
mathematics joumaling and/or history takmg, open discussion of feelings, and conceptual
understanding and problem-solving development into the course curriculum (cf
Rosamond, cited in Tobias, 1993, pp. 232-236). This type of self-contained situation
where instruction and support to overcome affective and cognitive challenges are
combined in the classroom is unusual. Its feasibility depends on the availability of
instructors who understand not only the importance of affect in mathematics learning but
also how to incorporate such understanding into classroom instruction of adults while
also covering the material mandated by the college mathematics department. I speculated
that if I were supporting students of such an instructor in the learning support center, I
might still find individuals for whom the whole class treatment of affect was not
sufficient. Importantly however, helping these students individually access and address
the core of their difficulties would almost certainly be facilitated by the significance
placed on affective issues by the instructor. This understanding made me conscious of the
importance of attending in my design to the effect of the current instructor and classroom
approaches on a student I was helping.
Approach Four: Individual Counseling Approaches
The fourth approach focuses on work with individuals who experience
psychological disturbances triggered by mathematics learning or directly impacting their
mathematics learning. There has been a long tradition of the use of behavior and cognitive
behavior counseling with individuals and groups adversely affected by mathematics anxiety.
23
using such techniques as desensitization, guided imagery, and relaxation training (Nolting,
1990; Richardson & Suinn, 1972).
Integrating Attention to Affect with Attention to Cognition: Summarizing Mathematics
Practitioners' Contributions — Implications for my Approach
There are particular mathematics education researchers and practitioners who
have studied concepts important to me as I developed this approach to college students'
mathematics mental health. Some were pivotal. Especially important is Carter and
Yackel's (1989) and Tobias's (1993) finding that participants must engage and succeed in
conceptual mathematics in order to improve their view of themselves as mathematics
learners and their mathematics mental heakh.
Factors related to mathematics fimctioning are expressed differently and lead to
different outcomes in different learning contexts. The helplessness that Nolting (1990)
noted in students with external locus of control as well as the passivity that Carter and
Yackel (1989) observed in instrumental (procedural) mathematics learners have been
linked with performance motivation"' in achievement situations (Dweck, 1986).
However, I noted that these results must be sensitively interpreted, since helplessness has
also been linked with learning motivation in high-achieving girls subjected to over-
procedural teaching. Understanding these factors within the current mathematics
classroom context and Ustening to the student helps avoid thoughtless direct application
of large group experimental findings to the individual (cf Boaler, 1 997).
The need for sensitivity again emphasizes the need for a whole-person approach
that may be conceptualized in terms of a students' mathematics mental health (see
chapter 1, endnote iv). Whether the designers of these approaches whom I have cited
state it explicitly as a goal or not, their workshops and co-courses that included in-class
24
journaling and discussions of emotion and in some cases narrowly specific individual
cognitive or behavioral counseling, helped participants to varying extents to become
aware of their mathematics mental health challenges. To the extent that participants were
supported in addressing these challenges, their mathematics mental health often
improved. Some even became willmg (and able) to learn the mathematics they needed
when they needed it (Tobias, 1993). In none of the approaches examined here, however,
was there the opportunity for an individual to explore her unique mathematics mental
health challenges with a suitably qualified professional while she was engaged in a
college course.
My focus on mathematics mental health as a way of conceptualizing students'
overall mathematics ftinctioning had become clear from examination of these approaches,
and affirmed for me the need in the field for an individual approach to helping a student
while she was taking a course. A logical next step was to identify or develop an
individual counseling approach that could provide a fi-amework for simultaneously
providing the mathematics cognitive support and acknowledging and addressing
students' affective problems.
CONTRIBUTIONS DRAWN FROM COGNITIVE THERAPY (CT)
When I explored the wide range of counseling psychologies that might be
applicable, the approach that first drew my attention was cognitive therapy (CT), which
was developed by Aaron T. Beck in the 1970s. I knew of CT's links with cognitive
psychology and had already noted the use of a number of its techniques in alleviating
mathematical affective problems. I found that techniques of cognitive behavior therapy
(CBT) which developed from Bandura's (1986) use of social-cognitive theory to merge
25
behavior therapy and cognitive therapy, had also been used effectively in educational
settings. I resolved to explore how CT theory (and CBT, where applicable) and
techniques might be adapted for my use as a basis for mathematics counseling.
Exploring Cognitive Therapy (CT) Theory as a
Framework for Mathematics Counseling
I wanted to help students become aware of how their past experiences and current
beliefs about themselves might be affecting their mathematics functioning; I also sought
ways to help students modify the underlymg orientation and overt behaviors that were
preventing their success. I wondered if cognitive therapy (CT) and cognitive behavioral
therapy (CBT) approaches with their focus on helping clients change their
counterproductive ways of thinking and behaving might offer what I was looking for.
CT conceives of awareness as a continuum rather than a dichotomy separating
conscious from unconscious experience. Beck (1976), the founder of CT, proposed that
"Man has the key to understanding and solving his psychological disturbance within the
scope of his own awareness" (p.3). Beck argued that his CT approach would change the
person's view of herself from "a helpless creature of [her] own biochemical reactions, or
of blmd impulses, or of automatic reflexes [as he contended that Freudian theorists
claimed]" to a person "capable of unlearning or correcting" the "erroneous, self-defeating
notions" she had previously learned such as, in this context, her supposedly genetic
inability to do mathematics (p.4). CT focuses more on how the patient distorts reality than
on why. In therapy, "the therapist helps a patient to unravel his distortions m thinking and
to learn alternative, more realistic ways to formulate his experiences" (p. 3).
26
Cognitive Therapy (CT) and Mathematics Depression
Students' emotional difficulties with mathematics often seemed to me to be
different from traditionally recognized mathematics anxieties or phobias. When I
examined CT's conceptualization of depression, I realized that much of what I had
observed could be seen as a type of situational mathematics depression. I have seen in
students' expressed negative views of their mathematics selves, mathematics worlds, and
mathematics futures, a more local or situational counterpart of negative views of one's
self, one's world, and one's future that, according to Beck (1977), characterize a
depressed person's orientation to life. I had also noticed that (as Beck, 1977, and
Seligman,1975, did in clients with generahzed depression) this mathematics depression
was almost invariably linked with helpless beliefs and behaviors m the mathematics
context. The promise of being able to differentiate depression from anxieties in the
mathematics learning setting added an important piece to my approach.
Other Cognitive Therapy (CT) Contributions
Dweck (1986), Beck (1977), and others have emphasized the importance of and
techniques for identifying and verbalizing erroneous and negative automatic thoughts in
order to test their veracity and defuse their power. Nolting (1990), Buxton (1991), and
others suggest the importance of students becoming consciously aware of their own
affect. Buxton (1991), Tobias (1993), Carter and Yackel (1989) and Stratton ( 1 996)
observe the therapeutic value of recognizing one's already existing mathematics aptitude
and finding oneself capable of doing mathematics. Cognitive and cognitive behavior
therapy (CT and CBT) and counseling techniques have been used effectively and
extensively in educational settings. As noted above, Tobias (1993), Carter and Yackel
27
(1989), and others use CBT techniques such as cognitive restructuring, hypothesis testing
of faulty beliefs, assigning aflbctive homework, and desensitization in their mathematics
anxiety reduction workshops. Nolting (1990) also demonstrates the efficacy of a limited
CBT cognitive restructuring approach (see endnote ix) to reduce the external locus of
control of certain beginning algebra students.
I determined that each of these CBT techniques might become part of my toolbox
to help students. CT in theory (though not always in practice) takes a constructivist,
problem-centered approach in that the client is seen to be the author of her own cure and
the counselor becomes a coach as they collaboratively identify key problems that the
client works to solve. This is the stance I chose as a mathematics learning specialist, to
take with my students. I saw an important advantage of CT/CBT's brief therapy mode in
college mathematics counseling. A course of CT/CBT therapy ranges from as few as
three to as many as thirty sessions, but is typically conducted in ten to twenty sessions, a
promising match for a college semester timeframe.
Limitations of Cognitive Therapy (CT) as a Framework for
Addressing Mathematics Mental Health Issues
CT still left unaddressed, however, how a student's present patterns of
mathematics fianctioning may have been influenced by her past experiences, which I had
identified as crucial for understanding and helping mathematics students. CT does not
consider the present role of the unconscious in sabotaging conscious motivations. I have
found that students are not dealing only with erroneous automatic thoughts that can be
identified and reasoned with; they often seem influenced by unconscious motivations out
of their awareness that stem from their past experiences and that are in conflict with their
conscious desires. As a mathematics tutor relating with the student, I also find myself
28
reacting and behaving in ways that puzzle me. In CT I did not find a way of
understanding these aspects of the student or myself or our interaction.
CT and CBT theorists contend that understanding the origins of a psychological
problem is not essential for producing behavior change (Wilson, 1995). The CT approach
thus helps identify and deal with symptoms but does not provide a way to unearth the
root of the problem. Perhaps, though, I reasoned, more than behavior change might be
needed for a student to succeed in mathematics. When I am confronted with a student's
puzzling behaviors she may be unaware of and contradictory automatic thoughts that she
does not even understand, it may be difficult to find ways to refute them even with good
present evidence or research or logic. With some, resolving the puzzle may require an
understanding of its begiimings and its developmental history.
CT/CBT counselors do not see a need to investigate unconscious motivations and
internalized relationships, nor do they examine present relationships to find clues to the
person's difficulties. It is precisely these motivations and relationships that I
hypothesized were key contributors to understanding a student's mathematics mental
health challenges. Although CT/CBT provided invaluable elements, I concluded that CT
could not supply the overarching framework for a holistic appraisal of a student's
mathematics mental health.
CONTRIBUTIONS FROM RELATIONAL CONFLICT
PSYCHOANALYTIC THEORY
In recognizing the need to address root causes of mathematics affective problems, I
returned to theorists of mathematics affect such as McLeod (1992) and looked more closely
this time at their endorsement of classical Freudian-type analysis and counseling approaches
albeit for cases of extreme mathematics emotionality (see McLeod, 1992, citing Tahta,
29
1993). In cases of severe disturbance some mathematics educators and therapists have
looked at or advocate looking at the role of individual students' unconscious in their
mathematics learning difficulties (cf Buxton, 1991; McLeod, 1992, 1997). As 1 noted in
chapter 1, Weyl-Kailey (1985) uses Freudian psychoanalytic techniques in a clinical setting
to probe and remediate puzzling mathematical behaviors as she uncovers and treats related
psychological disturbances."" Weyl-Kailey and others (see endnote xii) found that attention
to students' unconscious motivations gives insights that other approaches do not. These
researchers did not, however, use such approaches to understand and help "normal,"
struggling college students in the educational setting succeed in their current course, and it is
these "normal" students I planned to help.
Because my interest was in the mathematics mental health of ordinary students, not
just those with extreme difficulties, 1 had earlier rejected the utility of psychoanalytic theory.
I found no critical tradition in mathematics education that understood mathematics affective
and cognitive problems as symptoms of underlying causes rooted in each student's learning
history and expressed in her current patterns of behavior and relationships. But I now saw
the promise of psychoanalysis in its attention to the unconscious and the present effects of
the past on everyone. Indeed McLeod (1997) noted with interest Buxton's (1991) suggestion
that some struggles of such ordinary students with mathematics might well be understood in
terms of Freud's concept of the superego. I resolved to explore Freud's theory and the
theories that evolved from it.
The work of Stephen A. Mitchell (1988) emerged as highly relevant to my
research because it used a form of relational conflict psychotherapy derived from
Freudian psychoanalysis to help ordinary adults who had goals but were so embedded in
30
relational patterns with themselves and their significant others (both mtemal and
external) that those goals were not being fulfilled. Rather than seeing people through a
classical Freudian lens as largely driven by mstinctual pleasure-seeking and aggression
drives that continually engender internal conflict along a largely predetermined
developmental path, Mitchell's (2000) relational conflict theory recognizes that people
are hardwired for human relationships and that their drives, motivations, and conflicts are
focused around developing and maintaining those relationships with others and with
themselves. In 1988 Mitchell integrated the three major relational strands of
psychotherapy that emerged from Freud's classical psychoanalysis: self psychology,
object relations, and interpersonal psychology. Each of these strands emphasized one
dimension of what Mitchell termed as a person's relationality or her current behavior that
are the outcome of the development of her self her external and internalized objects, and
her interpersonal attachments (Mitchell, 2000). When I considered these dimensions in
the context of a student's mathematics learning experience, I interpreted them as follows:
1. Mathematics self or selves;
2. Internalized mathematics presences or objects; ™^ and
3. Interpersonal mathematics relational or attachment patterns.
Understanding a student's mathematics relational dimensions, how they are
positioned in relation to each other, and how they interact with one another to express her
relationality might provide the insight into the origin and development of her puzzling
behaviors and conflicts that I was seeking.
31
Relational Conflict Theory as a Framework
Relational psychotherapies rest on the premise that repetitive relationship patterns
derive from the human tendency to preserve the continuity, connections, and familiarity
of a personal interactional world."'" They recognize that the task of understanding the
person and helping her disembed from counterproductive interactional patterns may be
more complex and indirect than cognitive therapy concedes. Like cognitive therapists
and unlike classical Freudian psychoanalysts, relational theorists regard the person as
able to consciously choose to change her patterns of thinking and behavior (Mitchell &
Black, 1995).'"
Relational Theory, Development, and the Past
These msights from relational conflict theory promised to explain much of what
had puzzled me in the learning assistance center. Relational theory acknowledges that
human beings may proceed as if straightforwardly pursuing conscious goals but asserts
that, at the subconscious level, they seek to maintain an established sense of self and
patterns of relationship. In the learning center, I often found students who consciously
avowed a determination to succeed while they simuhaneously behaved in ways that
jeopardized that success. The self is not a static entity, however; it simultaneously affects
and is affected by internal and external realities. As Mitchell notes, the dialectic between
self-defmition and maintaining connection with others is complex and intricate. He
theorizes that humans "develop in relational matrices and psychopathology is a product
of disturbances in both past and present relationships and their interactions" (Mitchell,
1988, p. 35). Similarly, students' mathematics difficulties may be the product of their
mathematics learning experiences and relationships interacting with current situations.
32
Relational theory does not consider people developmentally arrested by early failures (as
object relations theorists believe), but rather that they have constricted relational patterns
that have developed in distorted ways in response to initial and subsequent environmental
and personal failures. This seems an apt depiction of both the beginning and the
outcomes of many students' mathematics learning histories.
These earliest experiences affect subsequent development. Understanding the past
is crucial... [because] the past provides clues to deciphering how and why the
present is being approached and shaped the way it is. ... [T]he residues of the past
do not close out the present; they provide blueprints for negotiating the present.
(Mitchell, 1988, p. 149, 150)
My puzzling students' normal mathematical development may have been
constricted by these negative experiences, and, as a result, subsequent relationships with
teachers, mathematics, and self became distorted. Their mathematics development had
also been affected by the effects of their own good and bad choices. The ways they relate
now to mathematics, to the instructor, and to me, the tutor, provide clues to their past and
to how to alter their present course.
Relational theory was offering me a way to understand the development of a
student's mathematics identity or what I came to call her mathematics self. This theory
offered me a way to understand how certain experiences and people might have been
internalized and might affect students' current perceptions of teachers. It also offered me
a way to understand how loss or change in mathematics and teacher relationships might
have affected their current relationships to the subject and to teacher.
Relational Theory and the Student-Tutor/Counselor Relationship
Mitchell's theory also challenged me with the prospect that a tutor would have to
take a stance toward the student quite different from the traditional stance. The tutor must
33
be prepared to see herself as an integral part of a current relationship with the student and
be willing and able to use her own feelings and reactions along with the student's
reactions to her as clues to understanding the student's past. These clues could be used to
work out with her what to do differently now so as not to reproduce counterproductive
relational patterns likely to hinder student success.
Following Freudian psychoanalysis, relational therapists observe and analyze this
relationship between the counselor and the client to collect key data germane to the
client's relational patterns. In this framework, a mathematics counselor would also
observe and analyze this relationship between herself and the student to provide key data
on the student's relational patterns. Relational therapy is not the same as mathematics
relational counseling, however. In relational therapy, the interpretation of a client's
transference of her past relationships into the relationship with the counselor and the
counselor's countertransference in reaction in her relationship with the client are central
to the psychoanalytic process. By contrast, although the mathematics counselor's
conscious awareness and examination of this transference-countertransference dynamic
will be key to her relational understanding, there is not likely to be time for lengthy
discussion of this dynamic, nor would the student's need for immediate mathematics help
or the educational setting make lengthy discussion appropriate. The admittance of
transference-countertransference as key to diagnosis in mathematics counseling will,
however, radically change the orientation to the student and her need for mathematics
support. Relational mathematics support is not only about the student but it is also about
how the mathematics tutor or counselor experiences the relationship with the student. The
34
ways the tutor feels free or constrained in the tutoring relationship become important
elements in understanding the student.
Limitations of Relational Conflict Psychoanalytic Theory
for this Setting
In embracing relational conflict psychoanalytic theory as the basis of a new
approach to improving students' mathematics mental health, I had to consider appropriate
boundaries. It is important to caution myself and the field that adapting relational
psychotherapies to an educational settmg without proper training is problematic. Even given
what I now saw to be the appropriate relational emphasis, the sphere of relational history
exploration needed to be kept limited to mathematics learning settings. Should the tutor
become aware of connections with more generalized mental health problems during that
exploration, referral to an appropriate mental health professional would be indicated.
Exploration of the present tutor-student relationship would also have to be bounded by the
educational setting.
Further, any history exploration would need to be conducted while they were
working on the mathematics. The traditional psychoanalytic leisure to explore at length the
person's relational past as well as the present therapist-client relationship would not be
possible or appropriate. Nevertheless bounded strategic engagement of the student in the
task of exploring and connecting present mathematical behaviors and relationships with past
experiences for the purpose of freeing her to change these behaviors and relationships, does
seem appropriate and is what this relational approach requires.
35
ADAPTING RELATIONAL CONFLICT THEORY TO HELP
STUDENTS DO MATHEMATICS
In order to explore the commonalities I saw between my own puzzling math
students and the adults for whom treatment with Mitchell's relational conflict therapy
was applicable, I needed to understand what a bounded and strategic exploration of a
student's mathematics learning history should entail from a relational perspective. In
particular, I had to investigate what the findings of the three major relational theories that
Mitchell integrated into his theory about relationality (self psychology, object relations
(internalized presences), and attachment theory) could tell me about how a student's
mathematics relationality might have developed and be expressed in the present. I also
needed to know about impediments to healthy development along the way, about what a
student's presenting symptoms tell about that development and current unconscious
relational conflicts that may impede her mathematical progress. I also needed to know
ways to improve her mathematics mental health.
For his conflict relational theory, Mitchell (1988, 2000) drew on (among others)
key theorists, Kohut (1977) for the self dimension, Fairbaim (1952) for the object
relations (internalized presences) dimension, and Bowlby (1973) for the interpersonal
attachment dimension, to explain how each of these relational dimensions differ from and
complement each other in understanding and helping chents. So these are the principle
theorists I chose as the basis for my approach.
In the foUowmg sections, I show how each of the three dimensions of a student's
relationality around mathematics learning, explained by the Kohut's theory of self,
Fairbairn's theory of internalized presences, and Bowlby's theory of interpersonal
attachments, yields a distinctive picture of one aspect of her mathematics identity and
36
how she likely developed in relation to the mathematical parenting she received. I show
how these distinctive pictures complement each other. When taken together, they yield a
useful picture of her relationality and the mathematics relational conflicts that now
challenge her, as I illustrate by applying the theories to Janet (see chapter 1 ) following the
discussion of each dimension.
The First Dimension: The Self and Mathematics Mental Health
Self psychology (Kohut, 1977; Mitchell, 1988) looks at adults' relational
difficulties to discover how their self development might have proceeded and what their
current self needs are. This perspective provides me a way of understanding the
mathematics self of an adult student, that is, the core of her mathematics identity. The
other dimensions then elaborate on interactions with that self The mathematics self may
be seen as part of a person's academic self, in turn situated in the person's nuclear self.
According to Kohut (1977), to develop a healthy self the child must experience
mirroring: unqualified recognition, delight, and admiration from a parent or primary
caretaker"™'. She also needs the opportunity and indeed the invitation to idealize and
incorporate into her self a parent image,""" first as part of herself (selfobject) and eventually
as ideals and values for the self (cf , the superego; Kohut, 1977, p. 185; St. Clair, 1990,
p. 157).
If we consider early elementary teaching to be analogous to early parent mg, the
development of a heahhy mathematical self requires the teacher to initially mirror the
child's developing mathematical identity, to recognize it, and to dehght in it, much as
Piaget"™'" (1973) and many cognitive constructivist theorists urge (Windschitl, 2002).
Simultaneously the teacher provides herself as the mathematical teacher image for the
37
student to idealize and to incorporate as part of herself. If early classroom conditions
facilitate this learning process the student's mathematics self development will likely
proceed in a healthy manner.
The elementary teacher's roles in nurturing and facilitating the growth of the
student's self, in particular her academic self, corresponds in a very real sense to the roles
of each parent; the mother provides the mirroring and the father provides the parent
image to be incorporated (see endnotes xvi and xvii). She must reflect back (mirror) to
the student her mathematics ability, she must allow the student to idealize and internalize
her mathematics values, and she must provide developmentally appropriate experiences
(both triumphs and disappointments). The teacher mediates between the formal subject
matter required by the mathematics curriculum and the informal mathematics the child
has already developed."'" As the child learns, interactions and connections are made
among her normal cognitive development, iimate curiosity and exploration, and the
environment (Ginsburg & Opper, 1979; Piaget, 1967; Vygotsky, 1986).
For growth to proceed, she must then experience tolerable reality. The self s
development needs the teacher to occasionally delay or fail to respond immediately to the
student's demands, thus forcing the self to develop abilities to meet her own demands.
The student needs to realize that she is not, after all, all-powerful or all-knowing"" (even
in her teacher's or parent's eyes) nor is her idealized teacher or parent perfectly able to
meet all her needs. The idealized teacher can no longer be the epitome of rectitude,
wisdom, or love she initially experienced. She becomes frustrated with the teacher's
imperfect mirroring and experiences tolerable disappointments with the idealized teacher,
along with broadening experience that supports her own ability to learn and grow. These
38
conditions contribute to the development of a self that integrates a realistic assessment of
the limits to her own prowess and value with a realistic assessment of the capacity and
limitations of the idealized teacher or parent.
The internalized teacher's mathematical values and ideals are integrated as the
student's own. These internalized values and ideals then provide structure and boundaries
as the child's own competence develops. When this process proceeds appropriately the
internal self-structure is consolidated and provides what Kohut (1977) calls "a storehouse
of self confidence and basic self-esteem that sustains a person throughout life" (p. 188,
footnote 8). This is the hallmark of a person who exhibits what Kohut refers to as healthy
narcissism. However, the need for mirroring and permission to idealize continues into
adulthood. This is a key understanding for a college mathematics counselor to consider.
If the teacher or parent responds to every demand or fails to respond at all, it
hinders healthy growth of the nuclear self because the student's own competence does
not develop in a healthy manner. If a teacher's failure to respond appropriately takes the
form of overindulgence (e.g., providing too easy tasks and unwarranted praise, having
high expectations with little pressure for the student to meet them) the student's
grandiosity is not appropriately challenged by reality and she develops what Winnicott
might call a false mathematics self (cf St. Clair, 1990). Her self-esteem remains low
because her competence does not develop appropriately but a defense is likely in the form
of unrealistic over confidence. She "knows" she can achieve if she wants/tries to. On the
other hand if the teacher's response is in the form of chronic neglect'™' (e.g., expecting
little when a student falters or seems slow to grasp concepts and subsequently ignoring
her need for challenge, tracking into low level tracks) she fails to see herself mirrored in
39
the teacher and her mathematics self fails to develop. In the extreme this may result in
what almost feels like the absence of a mathematics self (cf Cara in Knowles, 2001). Her
competence and therefore her self-esteem remam low as is true for the overindulged
student, but the neglected student's defense is likely to be different, in the form of
unrealistic under confidence. She is sure that she cannot succeed.
This study of self development allows me to see that a student whose mathematics
self is vulnerable because it is underdeveloped or undermined has likely developed
defenses (typically under or overconfidence and accompanying avoidance behaviors) to
protect this self from fiirther damage. Although her conscious goal is success in her
mathematics course, she likely acts in ways that jeopardize that goal. Her self-esteem is
compromised or low and she may have little underlying belief that she can succeed. Her
unconscious goals are in conflict with her conscious ones and she remains embedded in
her familiar patterns of relationship with self (cf Mitchell, 1988).
The Second Dimension: Internalized Presences — Objects Relations and
Mathematics Mental Health
Object relations theory principally focuses on the person's interior relational
world. This world is conceptualized as the person's self in relationship with internalized
and altered others (objects of the persons' feeling and drives), with split-off parts of self,
and with external others (objects). Whereas the focus of self psychology is on the
development of structures of the self, the focus of object relations is more on how early
interpersonal relationships are internalized and on how the irmer images of the self and
the other (object) are formed and shape perceptions and ongoing relationships with real
and internalized others (Fairbaim, 1952; St. Clair, 1990). From this perspective, a
student's internal reality is peopled by objects and selfobjects that affect her mathematics
40
self and the way she perceives external reality, in this context, the current mathematics
instructor and course.
If parenting is experienced as threatening enough to the self, bad internalized
presences are formed, creating internal conflict that distorts the person's perceptions of
present reality. Fairbaim (1952) contends that "internalized bad objects are present in the
minds of us all at deeper levels" (p. 65) and the degree to which they negatively affect us
in the present depends in part on how bad we experienced the original external other
(object) to he.'^
In the elementary classroom, a student cannot get away from the teacher and, in
fact, needs her. If the teacher humiliates the student or those around her, abuses her
verbally and or even physically, """ or otherwise creates a classroom environment that the
child experiences as unsafe, the child may cope with what feels like an intolerably unsafe
situation by holding the teacher to be good (right) and internalizing the bad part of the
teacher in order to feel safe, at least externally. She may then handle her now intolerably
unsafe internal situation by the defense of repressing the bad internalized object (the
teacher) or by a defense that Fairbaim (1952) calls the "defense of guilt" or "the moral
defense" (p.66). That defense is accompUshed thusly: The student or child is in a
situation where she feels surrounded by bad objects. Because this is intolerably
frightening, she converts this into a new situation where her objects (parents, caregivers,
teachers) are good and she herself is bad. A student or child who has suffered abuse or
neglect typically refiises to characterize the parent as bad, but is quick to admit that she
herself is bad.
41
It is not only students who have been abused who see themselves as bad and feel
shame and guilt; neglected students also feel shame for their deficiencies. The shame of
both abused and neglected students seems related to a sense of nakedness or sin, as if
their internalized mathematics object world is dominated by mathematics in the form of a
judgmental superego*""^ or by a bad mathematics teacher, threatening to unveil the
deficiencies of the vulnerable trying-to-hide mathematics seir"^ and the result is a
fearful, beleaguered mathematics self (cf Buxton, 1991).
A bad teacher presence (or object) assaults or conflicts with the student's
developing mathematics self and sabotages future relationships with teachers, even good
ones. What is pertinent for understanding the adult is not so much what actually occurred
between the teacher and child'""' but how the child experienced the mathematics teacher
and mathematics, how she internalized them, and how she as an adult now experiences
them. The student's initial transference relationship with the mathematics counselor and
the instructor is likely to reveal much about such presences. If her internalized good*"^" or
bad presences (especially internalized bad mathematics teacher-objects) are not brought
to consciousness and released, they may continue to control the present-day learning
relationships in a negative way.
Whether the student's efforts to deal with internalized bad teacher presences have
involved repression of bad teacher presences, moral conversion into herself being bad, or
another defense, when she enters the current classroom these unconscious forces are
activated and internal conflict develops between resignation to her mathematical badness
and her motivations to succeed in the class. Internalized presences may be so prominent
that they take precedence over current reality; the student may relate to the present
42
teacher as if she were in the original classroom. Conflicts arise when this mismatch
between her internal and external reality negatively affects her progress in the course. If
these conflicts are not resolved satisfactorily her desire to succeed or even survive in the
course may be thwarted.
The Third Dimension: Interpersonal Relational Attachments and
Mathematics Mental Health
The exploration of object relations gave me insights into how a student's internal
relational world might be configured and might now be affecting her. Attachment theory
promised to give me insight into the development, significance, and challenges of her
external interpersonal relationship dimension of relationality. In particular, attachment
theory examines the ways the person forms ongoing relationships with significant
persons in her life and work (Bowlby, 1965, 1982). Often her tendency towards
dependent, detached, ambivalent, or self-reliant relationships will provide clues to the
security of her early relationships and her subsequent experiences of loss or change in
those relationships. The extent to which a college student seeks the help she needs when
she needs it from her instructor, learning assistance personnel, or other suitably
knowledgeable person has been found to be an important factor in her success (cf
Downing, 2002, Zimmerman & Martinez-Pons, 1990). The student's established
attachment relational patterns may determine whether she is likely to make contact at all
with those who could help her, and if she does, how she proceeds to do so.
Attachment theorist John Bowlby (1973, 1982) and his colleagues found strong
evidence of a child's instinctive need for secure attachment to a particular parent figure.
The attachment-caregiving bond developed between child and mother figure'""'" is seen
as crucial to child's survival and forms the basis for any future attachment relationships
43
the child develops. The type of attachment achieved by the child varies according to the
type of caregiving the mother figure provides the child. Most important factors in
mother's caregiving are her responsiveness to the child's signals (e.g., crymg) and the
extent to which she initiates social interactions with her baby (Bowlby, 1982, pp.3 12-3 18,
referring to studies by Schafifer & Emerson, 1 964, and a study by Ainsworth, Blehar,
Walters, & Wall, 1978). Secure attachment is achieved when the caregiving by the
mother figure is characterized by being sufficiently available and responsive. The mother
figure becomes the secure base from which the child can move out and explore her
world, but return to for comfort and reassurance in times of distress.
Researchers have found that a child's insecure attachments can be explained by
the caregiver's behaviors towards the child. The caregiving that detached insecure
children receive is consistently detached, with the mother figure rarely responding to the
child's expressed needs and rarely herself initiating positive interaction with the child.
Children whose insecure attachments are ambivalent, alternating between demanding
contact with their mother figiire and resisting, receive inconsistent or conflicted
caregiving that the child finds unpredictable in its quantity or quality or both (Ainsworth,
Blehar, Walters, & Wall, 1978). Another insecure pattern, disorganized anachmsrA, is
characterized by fear of the caregiver or of her leaving or loss (Jacobsen & Hofinarm,
1997). '""
The peculiar mark of a securely attached child is her exploratory, adventurous
behavior, as long as she is assured of the availability of her attachment figure if needed.
By contrast, the insecurely attached child is preoccupied by frequently thwarted attempts
to avoid further separations from her attachment figure; she stays close and is afraid to
44
explore lest she be abandoned or punished, or she tries to meet her own needs, distancing
herself from her detached attachment figure. The secure person's behaviors lead to
learning; those of the insecure person's tend to inhibit it. Students' academic competence
through adolescence is also likely to be positively related to the security of their
attachments (Jacobsen & Hofmann, 1997). These outcomes are not unexpected. Many
educational researchers have demonstrated that the student's learning is dependent on her
investigating and interacting with her environment (cf Dewey, 1903; Piaget, 1973; and
others).
The subsequent ability of a person who has developed insecure attachments to
form relationships with others will be negatively affected and may be permanently
marred. By analogy, early experiences in a mathematics classroom where the teacher
does not understand or respond to the child's need for cognitive and emotional support,
challenge, and latitude for exploration may lead to a sense of insecurity and difficulty
with trusting the next teacher and subsequent mathematics material. Her beliefs and
behaviors may resemble anxious learned helplessness on the one hand or mistrusting
independence on the other.
People whose primary secure attachment relationships have not been unduly
disrupted usually develop into adults who form secure attachments. They are what
Bowlby (1973) calls truly self-reliant, "able to rely trustingly on others when occasion
demands and to know on whom it is appropriate to rely" (p. 359).'°°' Because these
people are confident that an attachment figure will be available to them when they need it
(a secure base), they are much less predisposed to intense or chronic fear than a person
who does not have that confidence (Bowlby, 1973; Sable, 1992; Weiss, 1991). They are
45
more resilient and able to negotiate difficult circumstances more successfully than those
whose early attachment bonds were insecure (see also Werner and Smith, 1982). In
contrast, adults whose attachment bonds were insecure or whose secure attachment bonds
were traumatized are likely to establish insecure attachments and have difficulty in
withstanding life's or the mathematics classroom's difficulties in a healthy manner.
Attachment bonds they form as adults are likely to be anxious, ambivalent, detached,
disorganized, or a combination of these.'"™
The teacher-student relationship, especially in the early years (generally through
third grade), is a type of attachment/caregiving relationship more than a relationship of
community'™'" although the teacher is not a substitute parent for her students. Even in the
early grades, there are important distinctions between parent figure roles and the teacher
roles. In particular, the teacher's relationship with the child should be characterized by
appropriate responsiveness and caregiving without the intense emotional involvement of
parental attachment (Katz, 2000).'™"" As the student gets older, the focus of the teacher's
"detached concern" care becomes a narrower one with more emphasis on providing an
academic secure base and less on emotional involvement (cf endnote xxxiii). A tutor or
learning counselor role is perhaps an intermediate one, with more emotional involvement
and partiality than is generally appropriate for a teacher. In a small college, for older
adult students, as well as for adolescent/young adult students, the power differential in
the 20- to 30-student classroom between the instructor and students and its similarities to
classrooms of the past can activate established teacher relational patterns that are more
akin to adult attachment than community relationships.
46
An unsafe or unsupportive classroom environment can certainly cause or
contribute to the development of insecure attachments to teacher or mathematics or both
(Dodd, 1992; Fiore, 1999; Jackson & Leffmgwell, 1999; Knowles, 1996; Mau, 1995;
Tobias, 1993). Students' subsequent avoidance of mathematics has been linked with
ambiguous and unsupportive classroom envirormients (Patrick, Turner, Meyer, &
Midgley, 2003).
It is not only student-teacher attachments that are affected by the way the teacher
manages the learning envu-onment. Student-mathematics attachments are also affected.
U.S. elementary teachers are likely to lack a secure base™"^ in the arithmetic they teach
(L. Ma, 1999), and those with insecure attachments are less able to provide secure
attachments to those in their care (Ainsworth, 1989; Bowlby, 1980). By extension we
may assume that in these mathematics classrooms, students' attachments to the
mathematics itself are vulnerable. Classrooms where the instructor provides either too
much or too little conceptual mathematics structure may inhibit students from making
healthy attachments to the mathematics. Likewise, teacher-as-authority mathematics
classrooms may also hinder healthy student attachments to the mathematics. Instead
students may develop an anxious attachment to mathematics that undermines their
confidence in feedback they get from working with the mathematics, and may keep them
unhealthily dependent on the teacher for decisions about whether they are proceeding
correctly. Confirming this, Skemp (1987) considers unhealthy dependence on the teacher
to be one of the chief drawbacks of an overly procedural approach to teaching
mathematics.
47
When the teacher's attachment to the mathematics is insecure, she is likely to
cling anxiously to procedures, not daring to explore or question, fearful that her
procedural grasp of the mathematics may be lost. She is less able to entertain students'
queries (much less, encourage their exploration) and is likely to respond with censure to
correct or logical approaches that differ from her grasp of the mathematics (cf Corwin,
1989; L. Ma, 1999). But if the procedural teacher has a secure mathematics base, the
prognosis for students' secure attachment to the mathematics is better even if it is
hampered by lack of encouragement to explore the mathematics for herself and construct
her own understandings with the teacher as guide. It is not only procedural transmission
pedagogical approaches that may jeopardize students' attachments to mathematics.
Students in laissez-fair classrooms are likely to lack a mathematical secure base™" and
even those in constructivist problem-solving classrooms may feel anxious and abandoned
xmless they are oriented to expect uncertainty as part of the problem-solving process and
appreciate the real availability of a mathematical secure base.™"'
Students' well-developed secure attachments to teachers and to mathematics can
be disrupted by a negative experience with a teacher or encountering a type of
mathematics or teaching style that result in a poor grade or failure. How well a person of
any age negotiates loss and avoids distortion of psychological development depends on
three factors.™"" The third factor: the continuity and quality of her relationship with
other primary attachment figure/s after loss or separation (Bowlby, 1980) is of particular
importance to a mathematics counselor working with students. It seems to me that a
counselor would fmd it easier to help students who had at some time experienced secure
48
attachment to mathematics to reattach to it than those who had never felt securely
attached to mathematics.
Change can also disrupt mathematics and teacher attachment relationships and
without support to negotiate the change students may remain stuck in a natural resistance
that could jeopardize their future success. Even when changes can be seen by outsiders to
be for the good, people are likely to resist or even reject™"'" those that cause disruptions
to their attachments to relationships and circumstances. This may help to explain why a
student repeating a course taught by a different instructor may resist approaches that are
different (and often preferable), even though the student initially failed with the
approaches she clings to. When students find themselves in a classroom whose approach
is different from the ones they are used to they are likely to experience what Marris calls
a "conservative impulse" to resist changes that call into question their familiar ways of
doing mathematics (cf Bookman & Friedman, 1998). I realized that helping these
students recognize and work through their resistance might free them to benefit from the
new course situation, but that would only be possible if I or the instructor or both
provided a secure base and the students could attach to it in the new situation. In order to
successfiiUy resolve the effects of loss or change experienced as loss, a person must work
through a grief process™"" to "retrieve the meaning of the experience and restore a sense
of the lost attachment that still gives meaning to the present" (Marris, 1974, p. 147, 149).
If a student has developed attachment patterns to mathematics teachers or tutors
that are characterized by investing either too little or too much reUance in the teacher or
tutor, their success or at least growth in mathematics learning may be compromised. They
face a likely conflict between maintaining their familiar but coimterproductive attachment
49
patterns and their willingness to risk trusting a relative stranger enough (e.g., the learning
counselor who is a mathematics "teacher") in order to attain a healthier balance between
their responsibilities and getting the appropriate help they need.
Application of Relational Conflict Theories to the Case of Janet
When I look again at Janet (see chapter 1) through the lens of relational conflict theory, it
seems likely that her first grade teacher's failure to mirror her already existing
mathematics ability and the teacher's developmentally inappropriate prohibition of the
use of concrete models to build understanding and provide transitions to internalized
knowledge had impeded the development of her mathematics self-esteem which is the
basis for a sound sense of mathematics self. This teacher had pushed underground her use
of fingers as a transitional object"' so that she had never developed beyond needing them
(at least emotionally) and still used them for security in an insecure world, despite the
risk of embarrassment, or worse, shame. This seems to have resulted in her seeing herself
as bad (at mathematics) because the teacher had to be good (or at least correct) in her
judgment of Janet and in her actions. It had thus distorted her sense of her mathematics
self.
Despite this inauspicious start it was now apparent that Janet's mathematics
competence had developed though it remained undermined and her self-esteem remained
low. She expressed her low self-esteem in an underconfident, resigned (perhaps
depressed) determination to proceed, with little hope of feeling secure in her grasp of the
material. It seemed that Janet had failed to develop initial secure attachments to
mathematics or to mathematics teachers and now her relationships with those from whom
50
she sought help seemed wary; she hid from them her shameful and illegitimate
techniques, expecting ridicule. Traditional understandings of Janet's affective problems
couched in terms of mathematics or testing anxiety and counterproductive beliefs related
to helplessness and her other unhelpfiil approaches, may now be seen as clues to her
underlying relational issues. Thus these affective problems could now be seen as
symptoms rather than causes of her difficulties. What was sound and healthy about her
affective orientation to mathematics learning could likewise be seen as symptomatic of
aspects of sound mathematics self-esteem.
Janet's mathematics cognitive knowledge, conceptions, and approaches can be
seen in context of and as outcomes of her mathematics relational history. Her current
patterns of mathematics learning and production can be understood as symptomatic of her
underlying sense of mathematics self.
Mathematics Parenting of Janet from the Three Perspectives
Each of the three relational perspectives gave me insight into parenting and
analogously into teaching as parenting. On reflection, two considerations stood out.
First, although it might appear that each says basically the same things about the
essentials and processes of early teacher-parenting, in fact that is not the case. Each
perspective does give different insights. Taken as a group of theories they are, as Mitchell
(1988, 2000) has shown, complementary with intersecting areas of interest.
Understanding the different related conflicts an adult might be experiencing, depending
on the dimension, promised to yield much in effectively diagnosing a student's
mathematical challenges. Second, I realized that Winnicott's (1965) concept of good-
51
enough mothering or parenting is a unifying concept that apphes in each perspective and
could be especially useful in my work with college students.
Different perspectives on teacher-parenting. Looking again at Janet, self
psychology's perspective would prompt me to examine her confidence level in relation to
her mathematics achievement to gauge the state of her mathematics self-esteem. When I
found that she expressed unrealistic underconfidence given her achievements I would
speculate that her early (and subsequent) teacher-parents failed to adequately nurture her
developing mathematics self As her counselor I would explore this speculation with her
and look for ways to re-parent her mathematics self now. I would find and help her
recognize and receive as her own her existing competencies and understandings (through
mirroring). I would expect and push the development of fiirther competencies and
understandings by initially allowing her to idealize and rely on me but progressively
challenging, frustrating, and disappointing her so that she would become more and more
reliant on her own competent self
An object relations perspective would lead me to clues to Janet's internal
mathematics relational life. I would now look for evidence of internalized teacher
presences, her use of repression as a defense, her moral conversion of herself as bad to
keep her bad teacher good, or other unconscious defenses in the face of her experiences
of traxuna in relation to teacher-parents (or parents as teacher/tutors). The teacher-
parenting central to this perspective is what the child experienced as traumatic. The
discrepancy between how she now relates to her current teacher and/or tutor, and how
they are in reality, is a clue to the influence of internalized realities. Applicable
counseling interventions would involve the counselor's providing herself as an especially
52
"good" teacher-parent and helping the student to become consciously aware of the
"goodness" of the current classroom teacher so that she can safely let go of detrimental
internalized teacher presences and incorporate instead the "good" teacher and the "good"
counselor.
From the perspective of attachment theory, I would notice Janet's occasional
wariness, and her intermittent dependence on me. She had little apparent relationship
with her classroom teacher, and lacked confidence in how she did mathematics even in
the face of good results. Taken together, these seem likely indicators of insecure
attachments to mathematics teachers and to mathematics. These attachment patterns
point to failure of past teacher-parents to provide a teacher secure base and a secure base
in mathematics.
Early teachers may have provided a secure enough base only to have that
disrupted by later teachers. In the case of Janet it seems that she had certainly not had an
early mathematics teacher who offered her the cognitive and emotional support,
challenge, and latitude for exploration that she needed to develop secure attachments to
teachers or mathematics. As a consequence, she had developed ambivalent patterns of
relationship. As a counselor I would provide myself as a consistent, safe secure base,
nevertheless challenging Janet, and pushing her to move away and explore and make
mistakes so she could experience returning to the base to find it secure and accepting.
While all three dimensions of a student's relationality should be the objects of a
counselor's curiosity, it is likely that any particular student's mathematics mental health
problems might be based more firmly in one of the areas than in the others at the time of
the brief counseling. Mitchell (2000) shows that as longer-term relational conflict therapy
53
proceeds and difficulties in one dimension are resolved, difficulties that emanate from
other dimensions will likely emerge to be dealt with.
Good-enough teacher-parenting. A good-enough mother, like a good-enough
teacher, provides sufficiently for the child to get a good start in Hfe by adapting
adequately to the child (or student) and her needs (St. Clair, 1990). This is an
empowering acknowledgement of the inevitable imperfections in parenting or teaching
that are nevertheless tolerable (or even necessary, within appropriate limits), for the
healthy development of the student's self Even if a student had experienced mathematics
classrooms as bad, had low mathematics self-esteem, and viewed her prospects as bleak
in the current class, I believed it was likely that we could find instances of good-enough
teaching and understanding so that some of the bad could be appropriately reinterpreted
and re-experienced as good-enough, providing bases for hope and progress. I use good-
enough to refer to the present, not only to current teaching and tutoring/counseling
conditions for the student but also to her process, progress, and outcomes. If the student
and I can let go of a perfect-or-nothing requirement and instead embrace good-enough for
ourselves, each other, and the teacher, we could perhaps make good-enough progress and
the student could achieve good-enough success.
Janet 's Relationality Summary
If I had offered Janet mathematics relational counseling, it would have involved
the kind of mathematics tutoring designed to help her recognize, draw on, and develop
her mathematical understandings and strengths while simultaneously attending to,
processing and dealing with her affective and cognitive symptoms of difficulty. A newly
developing self-esteem would likely have led to changes in her ways of seeing her
mathematics self, improvement in the way her internal mathematics world was
54
configured and in repaired attachments to the tutor, the teacher and mathematics. This
new freedom from formerly constricting relational patterns could lead to progress in
alleviating her negative symptoms, maximizing her mathematics potential, and achieving
good-enough success.
CONCLUSION: RELATIONAL CONFLICT THEORY AS A BASIS FOR
MATHEMATICS COUNSELING
Relational conflict theory had given me a way to explore how a student's self-
esteem and her beliefs, habits, ways of relating, and behaviors may be related to each
other. How the three dimensions of her relationality interact (her self, her internalized
presences, and her interpersonal attachment patterns), and the relational patterns she
employs to express that interaction give me the understanding I sought. Relational
theories point to some ways to identify and resolve her central conflict (Luborsky, 1 976;
Luborsky & Luborsky, 1995) and free her from the counterproductive relational patterns
limiting her progress. My adaptation of Mitchell's (1988, 2000) relational conflict theory
had given me a new way of looking at the student and at our relationship. I determined
that it was an approach that could include the insights and best practices of traditional
mathematics tutoring within a broader and deeper relational coimseling framework (see
Appendix A for a chart summary of the proposed mathematics relational counseling
approach).
In the next chapter I show how a relational conflict counseling approach could be
used appropriately and integrated with best practice mathematics tutoring in the setting of
the learning assistance center.
55
' In even numbered chapters, 1 use "she," "her," and "hers" for the third person generic singular.
" Davidson (1983) found strong linlcs to hemispheric preference and clearly defined analytic (her
Mathematics Learning Style I — left brain) and global (her Mathematics Learning Style II — right brain)
learning styles in terms of students' mathematical behaviors and approaches. Although these learning styles
have not been found to be directly related to mathematical achievement, Krutetskii (1976) found that
students who had a strong learning style preference and a relative inability with their other mode, found it
difficult, if not impossible, to begin a problem using the other mode's approach. Thus being forced to
approach problems using another's preferred approach greatly disadvantages these students. Learning
flexibility, however, can strengthen performance.
Students with an analytic/Mathematics Learning Style I use predominantly verbal-logical methods to
solve problems, use deductive approaches, and prefer to follow step by step procedures. Krutetski labels
them analytic. Students vnth a global/Mathematics Learning Style II use predominantly visual-pictorial,
inductive reasoning methods to solve problems and may know the answer to problems without being able
to explain how they arrived at it. Krutetskii labels them geometric. I hesitate to use Krutetskii's term
geometric because although he and others have generally found some relationship between success in
geometry and this visual processing right-brain preference learning style, analytic learners also achieve
success in geometry courses. Success in geometry is therefore not a clear indicator of a global learning
style.
'" To explain the difference between procedural and conceptual teaching, 1 offer the following example.
Teaching the fectoring of a trinomial X" + 7X + 12 by finding fectors of 12 that sum to 7 and putting those
into (X + )(X + ) gives a procedure that may be memorized but probably neither linked with prior learning
nor generalized to a more complex fectoring problem such as 3X^ + 16X - 12. Thus a new procedure must
be learned for this one, such as breaking apart 16X in a way that the coefficients multiply to equal -36 (i.e.,
3 x-12); then fector by grouping. Conceptual mathematics, as the term infers, is taught and learned as
concept-based processes that put less load on rote memorization and are more easily generalizable to new
more complex though related problems. For example the fectoring of the trinomial X' + 7X + 12 using a
conceptual approach might be linked with the earlier process of multiplying binomials X + 3 and X + 4
(and the geometric relationship of multiplying the length and width to get the area of a rectangle), and still
earlier distributive explorations of operations on number using two digit by two digit multiplication (also
area of a rectangle). The relationship between multiplication and division would be explored and
equivalence of division with fectoring made clear. Finding then that (X +3)(X + 4) = X"+ 7X + 12, and
relating this with the idea that 23 x 24 = 400 + 60 +80 + 12, that is, 20^ + 3x20 + 4x20 + 3x4, leads
students to explore the relationship between the 3 and 4 in the fectors and the 7 and 12 in the product and to
fiirther explorations and discoveries that are applicable to other problems.
" That is, not specifically related to a mathematics course the student is currently taking.
" Equivalent to what Hiebert (1986) and others refer to as "procedural" when they discuss mathematical
understanding.
" Equivalent to what Hiebert (1986) and others refer to as balanced "conceptual" understanding with the
requisite procedural knowledge.
"' Measured on Richardson and Suiim's (1 972) Mathematics Anxiety Rating Scale (MARS).
"" Hypothesis testing of a feulty belief might involve having a student viiio believes that people who do
well in math just see it immediately and do not need to work, interview some high achieving math students
who do have to work hard to understand and achieve.
" A cognitive restructuring exercise might involve having a student who sees herself a "bad" at math and
points out as evidence the errors on her quizzes and tests and any overall poor grades, develop the practice
of noticing instead not only the questions she did correctly but also her sound thinking even in the
56
questions she got wrong. If she combines this new practice with seeing the Hnic between insuificient or
inefficient preparation and her poor results, if she changes her preparation, and if she begins to see a change
in her results, her overall approach should change and her perception of herself doing math should also
improve. She has experienced cognitive restructuring.
" A person whose locus of control in a mathematics learning setting is external is likely to attribute her
achievement outcomes to factors that she feels she cannot change or control such as luck, the teacher, the
tutor, the weather, her health at the time, her lack of intelligence in mathematics (that she believes is a fixed
trait), etc. On the other hand a person whose locus of control in a mathematics learning setting is internal is
likely to attribute her achievement outcomes to factors that she feels can change or control such as her own
effort, her intelligence in mathematics that she believes can improve, getting the support she needs,
strategic planning for tests, etc.
" A student whose motivation for achievement is primarily performance is focused on passing or getting a
particular grade rather than on understanding the material. In contrast a student with learning achievement
motivation is primarily focused on understanding the material.
^ Others have looked into the unconscious and psychoanalytic symbols found in mathematics and
discovered there the roots and explanations for panic, aversion, and defenses against mathematics (of.,
Nimier, 1993; Tahta, 1993).
"" Object is used hwe in contrast with subject. In other words the object is the "other" in contrast with the
subject, which is the "self" According to object relations theory, early significant others become
internalized in various healthy and unhealthy ways as internalized presences that influence how the person
relates to others subsequently. I prefer to use the terms "other" or "presence" ratha than "object" because
"object" now has somewhat negative connotations implying a sense of persons as things.
'^ [PJsychopathology, in its infinite variations, reflects our unconscious commitment to stasis, to
embeddedness in and deep loyalty to the familiar... we experience our lives as directional and linear,
but like Penelope. . .we unconsciously counterbalance our efforts, complicate our intended goals;
seek out and construct the very restraints and obstacles we struggle against. (Mitchell, 1988, p.273)
"" Freud's view is that a person's choices are largely determined by unconscious instinctual drives and
forces outside of ha conscious control. Recognizing that the person, in contrast, is responsible for her
choices and actions, implies that helping her become conscious of her hidden motives should provide both
more insight into puzzling behaviors and also the possibility of modifying hidden motives in light of
conscious goals. The consistent relationship between academic/mathematics achievement and locus of
control (see Nolting, 1990, McLeod, 1992) is pertinent here. Students who fail to see their own
responsibilities in achieving success in a course, holding others or external fectors responsible instead,
consistently achieve less well than those who own that responsibility (internal locus of control).
"^ Usually seen as the mother although the role rather than the gender is the central factor. Winnicott
(1965) conceives of the function of mother as providing experiences to make possible a sense of
authenticity and reality; that is, to provide "good-enough" mothering that leads to "maturity and the
capacity to be alone ... [and] a belief in a benign environment"(p.32). Kohut (1977), in expanding
Winnicott's findings, also sees that the child's nuclear or core self arises as the result of the interplay
between her innate potentials and the responsiveness of the adult selves which the child internalizes as parts
of herself
'"^ Usually seen as the fether although the role rather than the gender is the central factor.
"™ Piaget (1973) contends that children's intellect develops primarily through self-directed activity, both
physical and mental. He asserts that all learning is "of a constructivist nature ... affirms a continuous
surpassing of successive stages. . . leads to placing all educational stress on the spontaneous aspects of the
child's activity... The basic principle of active methods will have to draw its inspiration from the history of
57
science and may be expressed as follows: to understand is to discover, or reconstruct by rediscovery, and
may be complied with if in the future individuals are to be formed who are capable of production and
creativity and not simply repetition" (p. 1 0).
"^ When a student tells a learning counselor that she has never been "good" at mathematics, even in first or
second grade, we must question her early experience of teacher-parenting. Research findings assure us that
barring a severe specific learning disability, developmental delay, emotional disturbance, or physical or
emotional abuse or deprivation, the average intelligent child is mathematically capable when she enters
school (Caufield, 2000; Hawkins, 1974; Kamii & DeClark, 1985; Kunzig, 1997). She has all she needs to
explore and learn developmentally appropriate number and operation concepts and their symbolic
representations, along with applications in solving problems based in her real world. If she does not
remember experiencing success it suggests many possibilities but most likely is that her early teachers did
not mirror her developing ability to do mathematics in a her ovm way or provide appropriate challenge and
frustration to promote her competence.
"' That is, her grandiose (to her, all-knowing and all-powerful) self is challenged and modified by reality.
™ Kohut (1977) maintains that it need not be specific traumatic events, but rather the chronic absence of
the parent's empathic responses to the child's need to be mirrored and to idealize that may lead to
pathology of self in the adult (p. 187).
'"" Fairbaim (1952) further asserts that
Whether any given individual becomes delinquent, psychoneurotic, psychotic, or simply 'normal'
would appear to depend, in the main, on the operation of three fectors: (1) the extent to which bad
objects have been installed in the unconscious and the degree of badness by which they are
characterized, (2) the extent to which the ego is identified with the internalized bad objects, and
(3) the nature and strength of the defenses which protect the ego from these objects, (p.65)
"^ For example, rapping knuckles, pinching, hair pulling. Although these activities are illegal in the U.S.,
they continue to be practiced, particularly in poorer communities where parents may feel less empowered
to challenge school practices.
'°^'' Or what Fairbaim (1952) calls the "internal saboteur."
""" This is a plausible explanation for what I have found to be the puzzling phenomenon of adult students
apologizing to me when they find or 1 point out an error in their work (especially an error in arithmetic)
saying, "I'm sorry," almost as if they have committed a sin and deserved punishment. Evidence of their
badness has been revealed and the effect is invariably shame. Are they ashamed because their early
teachers shamed them when they made such mistakes? Or are they ashamed that as adults they have
revealed incompetence at something a young child should be able to do?
"^ It is important here to distinguish between what actually happened, that is, what the teacher did in the
classroom, and how the adult student now remembers cognitively, affectively, and overall relationally
experiencing it. The forma- is impossible to verify and is not as relevant as the latter which is what is
affecting her now.
""^ Fairbaim (1952) referred to the internalization of good objects only in terms of the super-ego and the
development of principles and values much as Kohut (1977) saw the healthy modification of the parent
image. Bad objects on the other hand were internalized and interacted with the ego (operating part of the
self) causing conflict and splitting, that is, trouble when they were repressed or otherwise dealt with
internally.
'°™" Although this person is most often the child's biological mother, others, including the father or other
relative or unrelated person may be the mother figure for the child (Bowlby, 1982).
58
""^ There is some evidence that children of mothers who themselves suffer from unresolved attachment
trauma or loss are likely to develop this disorganized attachment (Main & Hesse, 1990). It seems that
many of the attachments formerly identified as ambivalent may be more accurately identified as
disorganized.
^ Bowlby distinguishes the concept of "self-reliant" from that of "independent," pointing out the cultural
stereotype of an independent person as one who relies only on self and repudiating or not needing the help
of others (Bowlby, 1973). Bowlby's concept of self-reliance is closely linked with Werner and Smith's
(1982) concept of resilience and Lillian Rubin's (1996) concept of transcendence. Werner and Smith found
that a key to a child's resilience under difficult circumstances was her significant relationship with an
accepting, approving, and challenging adult. Likewise, Rubin found that adults she studied who had
transcended abusive childhoods had all had such a relationship with an adult as a child, that had enabled
them to survive emotionally and become self-reliant adults themselves.
'°™ Disordered adult attachment behavior patterns linked with early insecure or interrupted attachment
relationships include
1 . Anxious attachment, characterized by over-dependence or clinging and severe separation anxiety,
thought to be linked to threats of abandonment by the childhood mother figure or to her forcing the
child to take on the caregiving role,
2. Insistent self-reliance, characterized by an apparent lack of any need for relationship or assistance,
thought to be connected with early rejection or prohibitions on expressing emotions or needs as a child,
3. Insistent or anxious caregiving, typified by exclusive formation of one-sided relationships in which
she is always the caregiver, thought to have developed from the experience of the mother figure's
expecting the child to mother her.
4. Detachment, characterized by emotional detachment and an inability to form stable bonds, stemming
from separations from the mother figure that were severe or prolonged (Bowlby, 1980; Sable, 1992).
'""^ Not all important relationships are attachment relationships. Attachment relationships, even for adults,
are characterized by "proximity seeking [to the attachment figure], secure base effect, and separation
protest" (Weiss, 1991, p.66). They contrast with community relationships "that link individuals to networks
of fellow workers, friends, or kin" (p.68) that also likely characterize peer relationships in a college
classroom.
'°™" Katz (2000) describes seven role dimensions where there are important distinctions between parenting
and teaching. They are: scope of fimction, intensity of affect, attachment of adult to child, rationality,
spontaneity, partiality, and scope of responsibility (p. 1 1 ). Of particular interest here are the dimensions of
attachment and partiality. Katz proposes that whweas parenting should be characterized by optimal
attachment with the child (essentially, secure attachment, appropriate caregiving), teaching should be
characterized by optimal detachment, or "detached concern," to use Maslach and Pines' (1977) term,
characterized by appropriate responsiveness and caregiving without the intense emotional involvement of
parental attachment (whether and how this optimal detachment is to be achieved may be related to the
teacher's ovm attachment history and to resolution of fransference and countertransference issues) With
regard to the partiality dimension, the parent's role is to be partial, biased towards her child; the teacher's
role is to be impartial, unbiased in relation to any one child but biased in her relationship with the class as a
whole.
These teacher and mother roles work best for the child if they are age-appropriate and complementary.
The teacher "is seen as wiser [academically] and sfronger [in relation to classroom management] and
therefore able to be protective at times when the self seems inadequate" (Weiss, 1991, p.68).
"""'' What Ma (1999) calls "a profound understanding of fijndamental arithmetic."
'°°" The effects of a laissez-feire classroom may be detrimental. Poorly planned discovery learning
situations where student are expected only to explore without knowledgeable and strategic teacher guidance
59
and support are unlikely to result in much mathematical learning (cf. G. Hein, personal communication,
September 1994). They are very likely to result in knowledge base gaps and insecure attachment to
mathematics.
""" A cognitive constructivist problem-solving situation is likely to increase emotionality and jeopardize a
student's relational attachments to teachers and to mathematics if not managed explicitly by the teacher
(McLeod, 1992; Szetela, 1997; Windschitl, 2002). In such a situation where the student is expected to
struggle over time with problems (with the teacher as guide or coach) she is likely to experience a range of
emotions that includes frustration and anxiety. If the teacher helps her to expect this emotionality as a
normal part of real problem-solving and to interpret and use it as a positive force in her process, her
attachmoits to mathematics and mathematics teachers should strengthen, especially if the instructor
provides herself as a reliable secure mathematics base whom the student can consult.
'°°™" 1. The honesty and openness with which the pason is prepared for or informed of the loss or
separation, is included in the mourning, and is allowed to mourn and to express her mourning over
time;
2 The quality of attachment to the mother (or attachment) figure before the loss or separation; and
3. The continuity and quality of her relationship with other primary attachment figure/s after the loss or
separation (Bowlby, 1980).
"°™'' Marris (1974) has observed what he calls the "conservative impulse" universally at work in people's
responses to loss, separation, and change. He proposes that this conservative impulse is based on the fact
that people develop meaning and purpose in the context of cumulative and long developed attachments in
relationships and circumstances. He notes that the cognitive process of assimilation of new understandings
into a person's existing cognitive schema, observed by Piaget, is similarly conservative. Changes that cause
disruptions to these attachments and that do not allow a person's engagement in the struggle to develop
new purpose and forge new attachments or assimilate the changes into former attachments, are likely to be
met with resistance and rejection, even when the changes can be seen by outsiders to be for the good
(Ginsburg & Oppw, 1979; Marris, 1974; Piaget, 1967).
""^ Reactions to separation or loss of attachment figure, or change impacting attachment bonds, have been
found to follow a common bereavement process, beginning with
1 . protest, involving confusion and searching for the lost object, sadness, yearning, anxiety, and
anger towards the lost attachment figure or agent of change, then
2. despair, depression, and disorganization, to
3. detachment from the attachment figure as defense, and finally to
4. acceptance of the loss (with ongoing sadness) if it is permanent, or to repaired attachment
(typically accompanied by anger, distrust, and anxiety) if the attachment figure returns and resumes
caregiving (Bowlby, 1980).
"' A child's fransitional object was typically a physical object such as a soft blanket used to smooth the
sometimes painfiil ttansition from complete dependence on her caretaker to her own autonomy (Winnicott,
1989). I speculate that the emotional role of fingers, counters, and other raanipulatives or physical
mathematical models, may be to fimction as transitional mathematical objects. These objects often smooth
the transition from externally verifiable to internally known mathematical understandings and they may be
comparable to a young child's fransitional object (or "blankie").
60
CHAPTER III
A NEW APPROACH: BRIEF RELATIONAL MATHEMATICS COUNSELING
In chapter 1 I describe the problem of mathematics support center professionals
not having what is needed to adequately help many typical college students to succeed. In
chapter 2, 1 discuss the scholarship that led me to a hypothesis that relational counseling
in conjunction with best-practice mathematics tutoring might address this problem. In this
chapter, I show how I generated a counseling approach by adaptmg the theories I had
studied to the realities of my practice as a learning center tutor. I describe how I used
relational conflict psychoanalytic theory as a basis for understanding best-practice
traditional mathematics support insights, as an approach to the student-tutor/counselor
relationship, and as a remedy to difficulties standing in the way of student success in a
learning support center context. I explain how the most important theoretical
underpinning of my approach — relational counseling — can be applied in a mathematics
academic support context to give a new way of looking at a student and at the tutor-
student relationship.
Drawing on my understanding of mathematics affective research findings and
cognitive therapy I describe the development of tools designed to facilitate my
understanding of students' affective and cognitive mathematics difficulties. I use the key
terms from relational counseling that I redefmed in chapter 2 in the context of
mathematics learning to show how relational counseling approaches may be used to
elucidate how symptoms are related, their underlying causes, and possible treatments.
Finally, I summarize ways mathematics tutoring and relational counseling can be
integrated in practice by describing roles of key participants.
61
THE CHALLENGES OF ADAPTING RELATIONAL MATHEMATICS
COUNSELING TO THE LEARNING CENTER CONTEXT
The Therapy Approach and the Problem of Time Constraints
How could a relational counseling approach be offered appropriately and effectively in
the college setting? Practical consideration led me first to consider time and institutional
constraints: For my purposes a major limitation of the relational conflict approach is the
necessary long-term nature of the therapy. A typical college semester is usually 15 or 16
weeks long. Realistically, potential contact time with a student is likely to be
considerably less unless the student begins the semester conscious of his' need for
assistance. Typically students recognize a need for support after the first quiz or exam
which may be several weeks into the semester. I wondered if the short tkne available
would be sufficient for a tutor to gain the in-depth understanding of the student that a
relational approach promised. I was also concerned about the appropriateness of a
therapeutic approach in an educational setting.
Given the educational setting, counseling, with its problem-centered approach and
counselor teaching/talking emphasis, seemed on the surface more appropriate than
therapy, which has a person-centered approach and relatively long-term mvestigative
emphasis based on close listening (Corsini, 1 995) . I considered the dual focus of
relational mathematics counseling: mathematics tutoring and counseling. On the one
hand mathematics tutoring is more problem-centered like counseling, since the focal
problem is the student's understanding and ways of doing mathematics. Here the tutor is
an expert in mathematics and takes on a coaching role as the student constructs new
understandings from his already existing knowledge. On the other hand, relational
conflict theory has generally been seen to involve client-centered therapy rather than
62
counseling. An approach to mathematics relationaUty then should be like therapy. It is
person-centered, with the student considered more expert than the counselor in his own
experiences, his personality, and his relationships. Here in contrast, the counselor's
expertise is in investigating, listening, and interpreting how these explain the student's
central mathematics relational conflict that needs resolution. While I call my developing
approach relational mathematics counseling, it could perhaps be more accurately
described as an integration of relational therapy into mathematics counseling.
I was aware of brief therapies, but most were problem- rather than person-
centered, like cognitive therapy, and I wanted an adaptation of relational therapy that was
both problem- a«J person-centered. This adaptation exists in Stadter's (1996) brief object
relations approach. Although some relational (psychodynamic) psychoanalysts resist
shorter courses of therapy for all but narrowly specified problems (cf Sifiieos, 1987),
brief therapy models such as Stadter's (1996) do apply relational (object relations)
counseling to time-limited settings. Brief therapy incorporates cognitive counseling
techniques and differentiates between the ongoing relational focus and the more
immediate symptomatic focus. A brief relational mathematics therapy approach needs to
incorporate the three relational dimensions, integrate pertinent CT/CBT approaches, and
allow the immediate focus to be on the learning of mathematics. Such an overarching
mathematics counseling framework could yield a nuanced understanding of students'
mathematics mental health that could lead to treatment in the limited time available in
college settings. It contains all the elements of an explanatory framework that can be used
to understand and support mathematics cognition and affect in the context of students'
63
mathematics relationships. Such an approach could appropriately be offered through the
academic support center.
The Use and Misuse of Assessment Instruments
As with best-practice traditional tutoring, the initial task in relational mathematics
counseling is to understand the student, his understanding, and his approach to
mathematics well enough to formulate an effective course of action. Understanding must
be followed quickly by effective and flexible implementation of the course of action
making constant adjustments in response to new insights and feedback from the student,
the results of assessments, and effectiveness of approaches. A relational counseling
approach differs from traditional practice, however, in how it changes the support
professional's ways of looking at himself, at the student, and at their relationship during
the tutoring process, as well as how it expands the scope of inquiry when investigating
and intervening in the student's mathematics learning.
A traditional approach to assessing or diagnosing a student's mathematics
functioning is to use formal and informal paper-and-pencil assessments. These are
generally used to identify the student's level on pertinent factors such as his mathematics
affect and his aptitude, achievement, and/or developmental level on the mathematics to
be attempted in the course.
Cognitive assessments I had used included in-house mathematics placement
instruments. Scholastic Aptitude Test (SAT) quantitative scores, and in-class tests and
quizzes. To assess affective orientation and identify possible affective symptoms of
mathematics difficulty I had previously used a number of diagnostic past-experience
questions, mathematics affect, and orientation surveys that explored students' beliefs.
64
attitudes and feelings. In researching for this study, I became familiar with other
instruments. My first inclination on lighting upon relational conflict theory as my
framework was to abandon these instruments and surveys, principally because of my
frustration with not knowing how to prioritize, understand, and use the data they
gathered. I quickly realized however, that given the short time available in a semester
worked against the relatively time-consuming relational therapy approach to data-
gathering so efficient data-gathering instruments would be necessary. Importantly, I
realized that the relational conflict framework was my key to prioritizing, understanding,
and using the data gathered by these instruments: Far from abandoning them, it seemed
that my new approach required their use.
I looked for assessments to help students become conscious of their present
condition with respect to their mathematics learning, both affective and cognitive, and
become aware of what that revealed about their established relational patterns. I was
aware that a new approach might deeply challenge not only the traditional uses of
assessments but also students' conceptions of what assessments could and could not say
about them.
Stephen J. Gould (1981) writes:
Few tragedies can be more extensive than the stunting of life, few injustices
deeper than the denial of an opportunity to strive or even to hope, by a limit
imposed from without, but falsely identified as lying within, (pp. 28-29)
Here Gould refers to the historical use of psychometric "biological labeling" to define
and limit the intelligence or abilities of groups or individuals in the U.S. I expected that
in introductory college mathematics-related classes in the U.S., there would be students
65
who had been subjected formally" or informally'" to such a denial of opportunity based in
a limit imposed by inappropriate interpretation of testing results in mathematics. I had
seen the effects of this denial to be affective, cognitive, and also relational. It negatively
impacted a student's overall mathematics fimctioning, that is, his mathematics mental
health. Accordingly I looked for assessment tools that could help the student become
consciously aware of his mathematics limits, of his beliefs about those limits, and of his
attitudes, emotions, and relationships related to his Umits so that we could explore and
detoxify the source of deceptive limits and constructively deal with real ones. I
determined that in any use I made of assessments I would keep central the possibility —
indeed, the expectation — of changes over time in the assessments for each student.
WHAT THE RELATIONAL MATHEMATICS COUNSELOR NEEDS
TO KNOW ABOUT THE STUDENT
Three avenues of inquiry emerged as important when I considered what
information I needed early in the process to begin to understand a student's mathematics
relational patterns and provide a way of discussing those with him. First, I wanted to
capture the student's sense of where he had come from mathematically, where he was
now, and what he thought were his key issues. Second, I needed to know how the student
was actually dealing with this mathematics class, the course instructor, and the content,
both in the classroom and out. Third, I hoped to fmd ways to see myself in relationship
with the student, and him in relationship with me to inform my interpretation of the &st
two.
Determining How the Student Sees Himself
In order to explore the student's sense of his mathematical progression, his
current placement, and what he considers his key mathematical issues, a two-pronged
66
approach seemed feasible: (a) First, during counseling sessions, I would use direct and
indirect questioning to analyze his mathematical orientation, approaches, and
background, and (b) second, outside of the counseling session (e.g., in class, for
homework), I would use strategic self-report surveying of factors I considered pertinent
to a student's mathematics relationality, such as his beliefs, attitudes, and feelings around
mathematics that I thought might be difficult to systematically gauge during counseling
sessions. I could use his survey responses in counseling as a vehicle to focus on issues
that might not otherwise arise.
During the Counseling Session
Mathematics background and experiences. In mathematics relational counseling,
taking personal history that focuses on the person's experiences with significant
mathematical others is likely the first essential to establishing a suitable relational focus
and to a proper understanding of how to deal with his particular mathematics learning
needs (for relational therapists' use of history taking, cf Luborsky & Luborsky, 1995;
Stadter, 1996). In mathematics anxiety reduction clinics (cf Tobias, 1991), in some
academic support settings, and in research studies (cf Mau, 1995), it has been standard
practice to invite adult students to tell or write their mathematics learning histories or
autobiographies to explore their present negative affect in the mathematics learning
situation, but is rare in the context of a college mathematics course, either in class or in
tutoring because of time pressures to focus on course content. Since my new approach
required it, however, I developed a Mathematics History Interview Protocol (see
Appendix A) based on findings from a qualitative research study I conducted into college
students' mathematics identity development and from my subsequent mathematics
67
academic support work with college students (Knowles, 1 998). Important areas of
inquiry include not only relational experiences with teachers, parents, peers, and others,
but also which completed high school and college mathematics and areas of self-
perceived mathematics competence and incompetence. I expected that because of time
pressure and the urgency of the current mathematics course focus, this history will
probably need to be gathered over several sessions, and history gathering would need to
be integrated into the ongoing mathematics tutoring process so that students can see its
connection and relevance to their current mathematics objectives.
Metaphor. "Metaphors are concrete images that require us to fmd the threads of
continuity and congruence between the metaphor and the primary subject" (Deshler,
1990). The primary foci for students in mathematics counseling should be mathematics
and themselves as mathematics learners. In my previous practice, I had asked students
write metaphors for their experience of mathematics but I had not known how to explore
beyond the obvious "threads of continuity and congruence" with students' mathematics
learning such as personal affective orientation to mathematics or beliefs about what
mathematics is. Now I realized that metaphor might also give students access to their
underlying relationships within their mathematics learning in an open-ended, indirect,
imaginative way. The relational perspective gave me a way to explore a metaphor with a
student, noticing clues to his sense of mathematics self, his internalized mathematics
presences, and his mathematics or mathematics teacher attachments. 1 could see how a
student's metaphor might provide a unifier or common thread to piece together other data
to understand the student's central relational conflict patterns. In counseling, I needed to
express my assumptions about his intended meaning in order to have the student clarify
68
or amend my perception. Such joint exploration seemed likely to unearth underlying and
possibly unconscious relational connections. During and at the end of the course of
counseling, students could reconsider their initial metaphors to see whether and how they
had changed and what, if any, changes might signify with regard to outcomes of the
course of counseling.
Mathematics negativity. In the mathematics learning situation, students with
negative beliefs about their mathematics world, their mathematics selves, and their
mathematics futures'^ tend to exhibit symptoms more like those of situational depression
than the more commonly assumed anxiety. This mathematics "depression" can be
debilitating in the learnmg situation, and students thus afflicted seem quite likely to give
up quickly, withdraw, or fail. The severity of the negative outlook may change from
week to week and, with that, the student's energy to struggle with the coursework, in an
inverse relationship between energy and severity of negative outlook. Dweck (1982),
Beck (1977), and others have found that having a person articulate her negative self-
statements may be the first step recognizing their irrationality and changing them. Having
clients respond to questions about their world, themselves, and their future each time they
met vnth a counselor has been found to help them and the counselor tackle negative self-
statements in an ongomg and timely manner (Beck, 1976, 1977). Therapists using this
method were also able to gauge the severity of the cUent's negativity/depression and
sometimes to prevent him from harming himself (Al-Musawi, 2001; Simon, 2002;
Sprinkle et al., 2002). Analogous to this self harming m the mathematics learning
situation is a student's sabotaging his chances of success by avoiding work or even the
mathematics class when his negativity and hopelessness become overwhelming.
69
In order to help my students become conscious of their thinking so they could
consciously deal with it rather than withdraw, for this study I developed a set of line
scales, each of which allows a range from positive to negative responses about the
student's current mathematics course, self, and future that week {JMK Mathematics
Affect Scales, see Appendix A). At each session the student will fill in the scales, and we
could compare his responses to previous ones and discuss changes in relation to external
circumstances, his progress with the course, and thoughts about himself We might look
for connections with his relational challenges and use this as feedback to help clarify the
focal relational conflict he is working to resolve. We might discuss changes in routines
and in his thoughts about hunself that he might try to implement over the following week
in response to the current evidence.
Outside the Counseling Session
The traditional means for finding out how a student sees himself as a mathematics
learner has been the self-report affect survey. Surveys requiring responses on a five- or
seven-point Lickert scale can be administered quickly in class or as a homework
assignment. I wondered if I might collect such data on affect that could help provide a
fiiller picture of the student that could be missed if I relied solely on conversation m the
counseling session.
Researchers have found two major areas of affect that interact directly with
mathematics cognition (albeit in complex and not always explicable ways) (cf McLeod,
1992). They are mathematics feelings (specifically, anxiety) and mathematics
beliefs/attitudes (and attributions based on these beliefs/attitudes). I determined that I
needed to find ways of observing or measuring students' levels of anxiety and
70
helplessness in new learning and testing situations (possibly indicating a damaged or
underdeveloped mathematics self) as well as curiosity and mastery orientation (possibly
indicating a healthy mathematics self) (Carter & Yackel, 1989; Skemp, 1987). 1 thought
a survey of each of these two areas — feelings and beliefs — that investigated key factors
linked with mathematics understanding and achievement might provide important points
of discussion and clarification in counseling. In addition, if used as a pre- and posttest, it
seemed possible that such surveys might reveal movement or change over the course of
counseling. I weighed the limitations of such surveys (e.g., closed questioning,
insensitivity to precision or depth or range of actual student feelings or beliefs) against
their benefits (e.g., quick assessment [using small constellations of items] of research-
confirmed key factors, and links with a student's underlying and overt relational patterns)
to assess what and whether surveys of affective issues could be helpful in the counseling
process.
Finding Out How the Student Does Mathematics Now
Mathematics diagnostics. Because I conjectured that students' mathematics
relational challenges (especially their sense of mathematics self) might be closely linked
with poor attachment to mathematics, I looked for diagnostics that could be administered
in class or during a counseling session that could discern between perception and reality
and thai were linked closely enough with current course content to be useful guides to
appropriate relational conversation.
Whatever the emphasis of an introductory college mathematics course,
arithmetical prowess in number (small and large) and operation sense and the student's
understanding of the algebraic variable seem to be pivotal areas to be explored. I
71
surmised that strategic use of an arithmetic and/or an algebra diagnostic could help both
student and counselor better understand the affective and cognitive impact of the
student's mathematics learning history. Once we had that information the student and
tutor could jointly plan strategic mathematics interventions for this course (see Appendix
B for the assessments I devised or adopted : Arithmetic for Statistics Assessment,
Knowles, 2000; the Algebra Test, Sokolowski's,1997, adaptation of Brown, Hart and
Kuchemann's,1985, Chelsea Diagnostic Algebra Test)
If the course had a specific applied emphasis (e.g., statistics) I wondered if a
specific diagnostic of that application could also be helpful. For an example see
Appendix B for Garfield's Statistical Reasoning Assessment used in my pilot study.
Mathematics course achievement. Mathematical tasks required in the course are
naturally central in coimseling. Students react to the grades they receive on course
assessments — exams, homework assignments, projects — differently, I surmise, because
of differences in their background experiences and relational challenges, and they also
react differently to these grades. A pivotal challenge in counseling is to analyze a
student's products with him in a way that helps him interpret his grades constructively.
The counselor must try to understand his reactions and to help modify them if necessary
in order that the student will approach the next assessment with a sense of responsibility
and with a developing sense of his mathematics self In this testing situation, the student
feels most acutely that his mathematics self is being judged. He may evidence conflicting
motivations and behaviors (e.g., wanting to succeed but also wanting to protect a
vulnerable sense of mathematics self by not trying, so as to avoid judgment of ability).
72
These heightened conflicts are likely to become clearest during exam analysis dicussions,
so these discussions create special opportunities for relational counseling.
Mathematics practices and behaviors. How the student actually does mathematics
may differ from how he perceives himself doing it. He may do it diflsrently in different
settings, and the counselor's observations and exploration of discrepancies should make
the student aware of approaches that he may need to modify. The settings where the
student does mathematics mclude the classroom, his home or dorm, and the learning
support center. Typically the mathematics counselor can observe the student directly only
in the learning support center, although he may be able to arrange classroom visits and/or
receive instructor observation reports (with student permission). It would seem however
that counselor observation of the student doing mathematics in different settings,
particularly in the classroom could be crucial for a clear understanding of the student's
mathematics relational issues.
The Student-Counselor Relationship
As I envisaged relational mathematics counseling, I realized that my relationship
with the student and his with me could be vital to understanding his core challenges, but
only if I purposely made our relationship a central object of inquiry and even, at times, a
topic of discussion. I noted the pivotal place relational therapy gives to the client's
transference of past analogous relationships to his relationship with the counselor and the
counselor's countertransference responses to the client, acknowledging that much of the
client's relationality is discernable through understanding and interpretation of this
mterchange. When I considered how I might integrate this observation and analysis of
our relationship into what the student understands to be essentially a mathematics tutor-
73
tutee relationship, I realized that if 1 self-disclosed when I became aware of my own
countertransferential impulses and asked about the student's sense of what was going on
and who they thought I should be and what they though I should be doing when I became
aware of being other than who I was, we might establish a place for exploring what it
might signify about their mathematics relationality. This approach seemed appropriate in
the learning support setting, but I was aware making countertransference and transference
issues explicit and be explored would likely differ markedly from student to student.
However, I could now admit my own countertransference and my experience of the
student's transference as data regardless of whether explicit discussion with the student
felt appropriate. As I considered the unportance of transference to a relational
mathematics counseling approach, it also became clear to me that I needed to arrange
supervision meetings in order to review and assess my transference-countertransference
interpretations with a person knowledgeable in counseling psychology.
Understanding the Student's Mathematics Mental Health Conditions
My interpretation of a relational view of mathematics mental health holds that a
student's relational patterns are adaptive. That is, he has developed ways of relating to
mathematics, instructors, and required mathematics courses that serve his sense of
mathematics self. His adaptations to mathematical circumstances may be conducive to
growth and positive development; they may be detrimental and skew or stunt his
development; or they may be somewhere in between. A student's state of mathematics
mental health may range from sound to poor, depending on the sense of mathematics self
he is attempting to maintain and the extent of conflict between contradictory goals he is
experiencing.
74
I had noted certain conditions or sets of indicators (or a syndrome) that could be
used to describe a student's state of mathematics mental health. These conditions could
be manifested as cognitive, as affective, or both. I believe that these may be best
understood in the context of a student's mathematics relationality. Understanding these
conditions or sets of indicators seemed key to helping a student focus quickly on his core
relational challenges.
Mathematics Cognitive Conditions and Relational Counseling
Research and experience have informed me that the cognitive conditions most
likely to negatively impact college students' achievement are: (a) a procedural approach
to mathematics learning, (b) the lack of a "profound understanding of fundamental
arithmetic" (L. Ma, 1 999) primarily number and operation sense, (c) weak connections
between arithmetic and algebra; (d) underdeveloped understanding of the algebraic
variable, (e) poor or counterproductive problem-solving strategies and monitoring and
control skills, (f) poor course management skills or (g) any combination of these. As a
mathematics counselor I would have to not only assess a student's cognitive standmg,
considering these categories, but also consider their impact on the development of his
mathematics self and his relational patterns. I would then have to prioritize tutoring
attention his cognitive conditions in relation to the demands of the course and his
limitations.
Cognitive Conditions Related to Personal Cognitive and
Environmental Attributes
Students with strong particular learning style^ inclinations may display learning
strengths or weaknesses depending on the particular learning environment.^' If there has
75
been a long-term mismatch between a student's learning style and mathematical learning
environments, unless he has been able to be flexible, he may have experienced less
success than his potential would indicate, along with an associated loss of confidence in
his ability.
Although there is powerful evidence that average children can learn mathematics,
many, and especially (but not exclusively) those from disempowered groups, are in
classrooms where their ability is judged inaccurately. They are often judged to be lower-
ability than they truly are and, perhaps worse, the ability they are considered to have is
judged as fixed (Downs, Matthew, & McKinney, 1994; Sadker & Sadker, 1994; Secada,
1992). Most U.S. students have experienced formal or informal tracking into ability
groups since the early elementary grades. Likewise students with diagnosed leammg
disabilities, ahhough cognitively capable, are likely to have been subjected to even lower
teacher expectations. Piaget (1973) goes further than Krutetskii (1976) in rejecting the
notion that some people have a math mind and many do not, but most U.S. college
students have entrenched beliefs about their own math ability that have restricted the
development of their ability and led to learning gaps. They may have been put in lower
tracks and given less coursework in high school, and they may have taken fewer courses
thus jeopardizing their achievement in college (cf Sells, 1976; Schoenfeld, 1992).
There are complex relationships among students' race, language, ethnicity, SES,
and gender, and their mathematics achievement (Secada, 1992). There is no credible
evidence that any of these factors or combination of factors affect potential to succeed.
There is, however, consistent evidence that schools' differential fmancial resources,
school cultures, and teacher race and ethnicity, attitudes, and expectations negatively
76
affect persistence in mathematics course-taking, achievement, and especially the
academic confidence of students from disempowered groups. Students from a
disempowered minority group who have been schooled in a majority setting where
teachers who are predominantly of the dominant culture is likely to experience minimal
respect for his own cultural norms or for the non-English language he speaks. Should this
be true, he has likely experienced minimal mirroring from the teacher and insufficient
support for his budding mathematics self
The development of a student's mathematics self, is affected by myriad personal
and environmental factors and their interactions. Students with underdeveloped or
damaged mathematics selves tend to blame their difficuhies on their own (imagined)
intrinsic inability or some other defect because they have been treated as if they are
inadequate. It has been relatively rare that a teacher is aware of and takes responsibility
for his part in his student's difficulties in learning mathematics. A relational approach to
students who have suffered such assaults on their mathematics selves should involve
carefial attention to what they can do mathematically, building on their abilities and
understandings using methods compatible with their learning styles, and refiitmg their
"no math ability" theories with evidence of their own work and thinking. In other words,
they need teacher-mirroring and support of their vuhierable and undeveloped
mathematics selves.
Mathematics Pedagogy and Cognitive Conditions
The mathematics self seems to be the most central dimension in the development
of healthy, flexible mathematics relational patterns. The principal means for this healthy
development is good mathematics teaching"" in an enviroimient where the student's
77
mathematics self is accepted, coached, and challenged. When that has not occurred or has
occurred intermittently, cognitive symptoms emerge, such as rigid reliance on memorized
steps or difficulty in adapting to slightly different wording or appearance that are
observable in the adult student's arithmetic, algebra, and problem-solving work in class,
on exams, and in the counseling session. How these cognitive symptoms interact with
students' affective symptoms and what they tell about the student's overall state of
mathematics mental health is investigated in this study. We can expect arithmetical
weaknesses and imcertainties to have deeper, more longitudinal and negative implications
to the mathematics self (identity) than algebraic weaknesses (if arithmetic is intact).
Number and operation sense weaknesses may be especially toxic, depending on their
severity and pervasiveness. As an example, Janet's lack of automatic access to her
multiplication and addition facts (see chapters 1 and 2) slowed her progress in
precalculus and seriously undermined her confidence. However, algebraic weaknesses
will invariably also strongly impact present functioning negatively. How cognitive
symptoms specifically affect an individual's present ability to learn new mathematical
content will be a function of a combination of the course difficulty, the way it is taught
relative to the student's needs, the relational and mathematics climate of the classroom,
the extra support available, and the way the student's mathematics relational patterns
interact with these factors. Vuhierable students may include not only those with cognitive
preparation deficits but also some whose cognitive preparation is adequate but who are
nevertheless not confident for other reasons.
Researchers such as Skemp (1987) and Buxton (1991) have shown links between
affective and cognitive symptoms that have their source in poor mathematics pedagogy.
78
In particular, predominantly procedural teaching with the teacher as the sole authority on
the mathematics leaves the student vulnerable to helplessness and anxiety because he has
recourse only to memory or the teacher's logic rather than to the connections he could
make himself if he has learned and understood it conceptually.
Cognitive Conditions and Relationality
Attachment to mathematics. Few elementary teachers have what Liping Ma
(1999) calls the "profound understanding of fundamental arithmetic" required to
understand the problems, and few are able to translate their understanding into practical
activities for their students. Thus they have to teach their students procedures rather than
concepts. These students tend to develop a narrow procedural knowledge of arithmetic
that links poorly with algebra because of the need to generalize beyond procedure to a
more abstract statement of relationship. Students' knowledge of and beliefs about
mathematics and about themselves doing mathematics may be distorted. If they have not
developed a secure attachment to mathematics that can enable them to be flexible and
venture into new learning this distortion may be extreme.
EMOTIONAL CONDITIONS AND RELATIONAL COUNSELING
Anxiety
Much of the negative affect that students experience while doing mathematics has
been lumped under the label "mathematics anxiety." Educational research supports a
relationship between mathematics anxiety and poor performance although that
relationship is not unequivocal nor is the effect always significant when it occurs
(Hembree, 1990; McLeod, 1992). According to the Yerkes-Dodson (1908) principle
(performance related to arousal roughly by an inverted U), students who experience
79
moderate levels of arousal (whether they interpret that as positive or negative) will do
better on a test than those who experience either too little or too much arousal. What
exactly mathematics anxiety is and what its causes are have been the matter of much
debate and many studies and factor analyses (Ma, 1999; McLeod, 1992). Part of the
difficulty is that its etiology, triggers, and expression differ from person to person. A
relational counseling approach, I believed, would provide the mandate and opportunity to
enable students to reveal and explore these individual differences. But because of
semester-long limitations, I wanted an instrument that would differentiate some factors in
mathematics anxiety and provide a starting point for discussion with individuals in the
counseling situation.
Analysis"" of the literature of attempts to define and measure mathematics and
testing anxiety have found a number of dimensions that affect students' performance in
sometimes singly, sometimes in combination, and always in relation to other dimensions
all in varying degrees. The pertinent dimensions are often agreed to be: (a) the
mathematical situations that engender anxiety (e.g., every day life vs. classroom; within
the classroom: testing vs. class work versus homework); (b) the type of mathematics
involved (e.g., arithmetic vs. algebra); (c) the cognitive precursors to anxiety (e.g., poor
exam preparation); (d) whether the mathematics activity is solitary, with peers, or public;
(e) to what extent the student suffers from strong chronic anxiety or experiences anxiety
easily (trait anxiety); (f) the type and intensity of anxiety engendered by the situation
(state anxiety, cognitive worry); and (g) the immediate and long-term physical, affective,
and cognitive effects of the anxiety.
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I determined that if I understood a student's mathematics affective history and its
effects on the different dimensions of his mathematics relationships and, flirther, if I
observed and experienced his resultant relational patterns, I might be able to
contextualize his anxiety. I searched among the many formal and informal instruments
for one that surveys affective response to mathematics cognitive and situational factors.
This seemed particularly urgent because of my perception of the centrality of
mathematics cognition in the development of the mathematics self. I chose Ferguson's
(1986) Phoebus (which I renamed as My Mathematics Feelings survey see Appendix B
and see endnote ix) to be used in conjunction with the student's and my observation and
discussion of his testing behavior. Other pertinent factors would emerge during
counseling and their relational etiology also could be explored.
I would first consider normal anxiety that is engendered by a dangerous situation,
before looking for a psychological cause originating from a disturbance of mathematics
self, internalized presences, or interpersonal attachments (Bowlby, 1973; Fairbairn, 1992;
Freud, 1 926; Kohut, 1 977). In this context such causes as inappropriate placement in the
class (indicative of prerequisite knowledge gaps), insufficient strategic preparation for an
exam, or poor problem-solving, monitoring and control skills would genuinely endanger
the student's chance of doing an exam successfully. These examples constitute
appropriate causes of normal anxiety.
Once such normal anxiety has been ruled out, I would consider the relational roots
of a student's anxiety.
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The Mathematics Self and Anxiety
Anxiety related to assaults on the development of self is what the founder of self
psychology, Heinz Kohut (1977) describes as disintegration anxiety, "an ill-defined but
intense and pervasive anxiety accompan[ying a sense that the] self is disintegrating
(severe fragmentation, serious loss of initiative, profound loss of self-esteem, sense of
utter meaninglessness)" (p. 103). I have seen this when a student with a deep sense of
his own inability to do mathematics becomes inarticulate and paralyzed when called on in
class or experiences panic, mental disorganization, helplessness, even physical pain when
taking a test. Could he be experiencing a form of the disintegration anxiety Kohut spoke
of? Is this part of himself so malformed or underdeveloped that when his mathematics
self is being scrutinized by a public question or a test, especially in mathematics class, he
feels his self disintegrating to the extent that it might even threaten the rest of his
developing academic self (cf Lenore in Fiore, 1999; Tobias, 1993)?
I envisaged that counseling help for a student suffering so could take a two-
pronged approach. The counselor could help the student to connect with mathematics, to
recognize and own his developing understanding, and to expand his tolerance of the
anxiety engendered by not knowing or understanding it all immediately; At the same
time, the counselor, student, and instructor might explore alternate alternative
arrangements in class work or testing designed to alleviate anxiety. For example, the
instructor could signal that the student will be the next person to be asked an identified
question so he has time to prepare an answer, or exam questions could be given one at a
time.
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Internalized Presences and Anxiety
When a student has developed and repressed bad internalized presences in
response to unsafe and abusive mathematics learning situations, or has established
mathematics as a punitive internal saboteur or superego, these internalized presences may
cause him to worry that his mathematical products are bad or wrong even when they are
not. He may have internalized his frightening third grade teacher who made him stand at
the board for long hours humiliated and unable to do the required problem and this
teacher's influence may be manifested during the college exam, insisting that he still
cannot do it and recreating the mind-numbing anxiety he experienced back then (cf Terry
in Fiore, 1999). During the exam, he may have to contend with the anxiety engendered by
the prospect of his exam grade pronouncing judgment on his worth as a person (cf
Buxton, 1991).
Interpersonal attachment and separation anxiety. Involuntary separation from a
person's attachment figure often causes distress and creates disturbance in that
relationship when the attachment figure returns, no matter how short the separation or
how well the separation was managed. If the person subsequently comes to believe there
is risk of fiarther separation he is likely to become acutely anxious (Bowlby, 1973). A
student may experience such acute anxiety if he has done well in mathematics and
enjoyed positive relationships with teachers but has been separated from these good
experiences and subsequently had a bad experience. He may have done badly in a course,
clashed with or been ignored by a teacher. Separation anxiety is a natural response in
children and adults'^ whose access to their attachment base is denied or threatened or
whose attachment figure is unresponsive. Maladaptive responses to separation, loss, or
83
change can be an apparent lack of response (i.e., detachment) or an mtense response (i.e.,
extreme anxiety or phobia) (Bowlby, 1973; Freud, 1926).
In a study of instructor-caused onsets of students' mathematics anxiety, Jackson
and Leffmgwell (1999) found that responses that could be classified as separation anxiety
arose from the perceived inaccessibility or lack of responsiveness of the mathematics
caregiver, the instructor." Experiencing inaccessibility or lack of responsiveness from
previous teachers can negatively affect students' responses to their current teacher's
offers of help as a secure base. Without understanding and intervention this separation
anxiety may persist.
Students who have once experienced success in mathematics but have
subsequently suffer a loss of competence because of poor teaching, course placement, or
other external events may experience separation anxiety in relation to the mathematics
itself They may be newly uncertain of its accessibility and reliability. Without
counseling mterventions to reconnect them to their once-secure base in mathematics and
their sound ability to negotiate the current course, this separation anxiety may cause them
to fail or do poorly in mathematics courses they are capable of mastering.
This exploration of the relational origins of mathematics anxieties led me to see
that once the student and I had determined through the My Mathematics Feelings survey
and conversation that his mathematics anxieties existed and were troublesome, we could
go further and distinguish their origin in different relational dimensions and devise
targeted interventions that could look quite different depending on the dimension of
origin.
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Learned Helplessness and Depression
Anxiety is not the only emotional response to mathematics stress. In my
experience, students who suffer from mathematics negativity (see above) expressed as
learned helplessness or even depression with or without anxiety are just as prevalent.
Learned helplessness has been linked with both situational and clinical depression
(Seligman, 1975). Dweck and Reppucci (1973) found that a student may come to believe
he is helpless under one set of circumstances but not under others. This supports
Seligman's (1975) notion of situational learned helplessness or depression. It may not be
so much the mathematics itself but the way it has been taught that renders students so
vulnerable to learned helplessness in its face (Boaler, 1 997; Carter & Yackel, 1 989;
Dweck & Reppucci, 1973; Piaget, 1973; Skemp, 1987) Mastery-oriented, positive
students may exhibit helplessness and depression- like symptoms in certain mathematical
contexts. Learned helpless and depressed people believe that the situation they are in is
beyond their control; there is little or nothing they can do to change the outcome.
It is not unusual to fmd one or two students in any class of 30 who view their
mathematical past, present, and iuture with despair. A mathematically depressed student
sees himself as mathematically deficient; he considers the present mathematical demands
excessive; and he views his fiiture as impossible. He may want to drop the course he is in
now and he will seek any alternative to the looming mathematics course to follow.
A depressed person's negative orientation and behavior influence other people
whose responses in turn influence the individual (Bandura, 1 977). For example,
emotional withdrawal may elicit rejection or criticism that in turn aggravates the patient's
negative self-cognition and thus his depression. A mathematically depressed student may
85
avoid classes, homework, or the learning support center. This avoidance behavior inay be
interpreted as laziness or irresponsibility and result in censure rather than sympathy.
Alternately, a mathematically depressed student may become excessively dependent on
the mathematics counselor or the instructor and seem unable to proceed on his own.
Student Beliefs, Helplessness/Depression, and Mathematics Pedagogy
Students develop beliefs about mathematics and their ability to understand it that
are closely linked with the beliefs and practices of their teachers and the effects on their
mathematics orientation and self concept. In the U.S., the most detrimental belief about
mathematics and mathematics learning that has the most far-reaching negative
consequences for students is: "Learning mathematics requires special ability, which most
students do not have" (Mathematical Sciences Education Board, 1 989, p. 1 0)."' The belief
that ability is a trait rather than a malleable quality has been linked to learned
helplessness in mathematics learning situations (Dweck & Wortman, 1982). It amounts to
a type of mathematics gene theory that is applied in both a positive and negative manner.
A student who identifies with a family member who is "good at mathematics" is likely to
believe he also has the potential to be "good at mathematics," but students who identify
with a family member or members who "could not do mathematics either" are more
common and are likely to find this belief debilitating. It has been found that a student's
beliefs about his achievement lead his to attribute outcomes to one of two central causes:
his ability or his effort. Thus a student who believes his ability is low and unchangeable
is likely to attribute a poor score on an exam to his (poor) ability. If a student attributes
both his failure to lack of effort and also success to his (soimd) ability, he is ascribing to
beliefs that generally underlie a healthy mastery approach. On the other hand these
86
attributions may instead be an all-powerftil, all-knowing (grandiose) mask for an
underlying fear that one might not be able to do it — and that one has no intention of
trying because of the risk of being found out (see below).
Student Beliefs, Achievement Motivation, and Helplessness/Depression
Achieving a high grade or some other recognition, also termed performance
achievement motivation, often becomes more important and more possible than learning
with understanding in the compulsory and competitive U.S. school systems. Piaget
( 1 967) sees learning achievement motivation to be related to two important factors: (a)
the "moderate novelty" of the new task, and (b) reasonable proximity and accessibility of
learning, given levels of prior understanding. An mdividual's curiosity is aroused by the
"moderate novelty" of an object in relation to his prior experience; this curiosity
motivates his to investigate, learn, and achieve understandmg. The students in this study
brought many different motivations to their tasks of succeeding in a mathematics course.
These stem from their prior experiences, are related to their present ambitions, and affect
how they would do in the course.
Both learning and performance achievement motivational patterns have been
found to be affected by students' sense of worth (Dweck, 1986). A sense of contingent
self worth and a belief that their intelligence is fixed typically lead students to make
performance as their primary goal in learning; they work only to be seen and judged to be
successful. They will not approach a task with confidence (mastery orientation) unless
they perceive their ability to be high for that task. If they perceive their ability to be low
they are likely to become discouraged and even helpless. If they have a choice of learning
tasks, some tend to choose tasks that are below their ability in order to ensure good
87
performance or they will choose tasks well beyond them that no one would expect them
to complete successfully.
Students who believe their intelligence is malleable show more adaptive
motivational patterns; they typically make learning their primary goal. These students
typically approach tasks with a mastery orientation regardless of whether they perceive
their ability to be low or high in relation to the task. They choose learning tasks because
the tasks are personally challenging rather than first considering whether they are able to
do well at them (Dweck, 1986). These students are likely to be discouraged and anxious
and become helpless in fast-paced, text-based, procedural classroom where they find
learning and understanding the mathematics difficult or impossible (Boaler, 1997). The
optimal conditions for learning achievement motivation to lead to understanding and not
be frustrated include these principles:
1 . Students need to be encouraged to make it their personal goal to solve the
problem; the tasks themselves need to be "appropriately problematic" (Hiebert,
et al., p. 51);
2. The culture of the classroom must be a secure base that supports and allows
time for struggle, reflection, and communication;
3. Students need to see ways to use the tools they already possess to begin the task.
(Hiebert et al., 1997)
Apparently similar classroom behaviors may stem from quite difl^erent
motivational orientations linked not only to the student's sense of mathematics self but
also to the mathematical tasks and learning environment.
88
Student Beliefs and Helplessness/Depression: Developing
a Survey Instrument
Students' beliefs about their mathematics selves, world, and future have been
researched extensively and the links between these beliefs and their mathematics course
persistence, behaviors, and achievement have been thoughtfully studied. As noted above I
had developed the JMK Mathematics Affect Scales (see Appendix A) to quickly gauge
students' immediate operating beliefs on a session-by-session basis. I wondered if in
addition I could develop or find and adapt an instrument that would survey underlying
factors researchers had linked to mathematics negativity or helplessness.
Whereas a mathematics anxiety instrument is intended to assess students' short-
term emotional responses, a belief survey looks more at stable long-term underlying
beliefs and attitudes. These may help to explain the student's short-term emotional
responses as well as established mathematical behaviors. I looked for a self-report survey
instrument around beliefs about self and mathematics that included statements about the
following:
1 . Mathematics as procedural or conceptual;
2. Mathematics self as learned helpless through mastery oriented in mathematics
learning situations;
3. Links between mathematics beliefs and mathematics self beliefs;
4. Achievement motivation: performance through learning motivation
5. Personal characteristics and societal myths: Fennema and Sherman (1976),
Fennema (1977), Kogelman and Warren (1978), Tobias (1993) and others
have shown links between these and mathematics anxiety and debilitation of
mathematics achievement.
89
I did not include locus of control as a factor, although 1 knew that a student's
locus of control (whether he sees himself or some external entity such as luck or the
teacher, as the controller of his outcomes) has been found to be an important factor in his
mathematics achievement (cf Nolting, 1990). I preferred to gauge this and also a
student's locus of responsibility (whether he sees himself or others as responsible for
what happens to him) directly from cues in the counseling setting.
With some modifications, Ema Yackel's (1984) Mathematics Beliefs Systems
Survey with some modifications fit my criteria. Its chief attraction for my purposes was
the careful investigation it provides of procedural versus conceptual beliefs about
mathematics, based on Skemp's (1987) analysis (see Appendix B). I reasoned that
analyzing clusters of items with the student could help him become conscious of beliefs,
attitudes, and conditions whose relational origins we could explore and that he would be
free to modify. I then considered what that exploration of relational origins of negative
and counterproductive beliefs might reveal. I found the relational dimensions of a) the
student's mathematics self and b) his interpersonal attachments to be particularly
vuhierable to development of different types of depression. Because of the difference
between the origins and hence potential remedies for these depressions I needed to clarify
how to distinguish them when their initial presentation was likely to be similar.
The Mathematics Self and Depression
While CT describes the manifestations of mathematics depression, relational
theory traces the origins of the depression and points to relational remedies. If a student's
mathematics self has been under-stimulated because of chronic teacher neglect, his
mathematics self will likely be underdeveloped. When such a student is faced with a
90
mathematics challenge, he is likely to experience a vague but pervasive sense of
depression and excessively low mathematics self-esteem. His depression will feel like
emptiness, a sense of not really being alive mathematically. He may believe his
mathematical self does not exist apart from the mathematics tutor. He might excessively
merge with the tutor. If so it will be the tutors' role to mirror his emerging mathematics
competence back to him and to provide timely tolerable frustrations. Then the student can
begin to discover his own prowess and learn to do mathematics on his own, with
appropriate support. Alternatively, if he believes, even unconsciously, that he is incapable
of imderstanding the mathematics he may try to memorize all procedures and will likely
defensively blame external factors when this is unsuccessful. Such a student may avoid
seeking help from teachers or tutors citing lack of interest or effort as the reason for his
lack of success.
Attachment Theory and Depression
Not all mathematics depressions are rooted in underdevelopment of self By
contrast, a student may be in the depression stage of a grief process. A student who is
used to doing adequately but then experiences doing badly may be thus affected. Another
student who has experienced a teacher's dislike, rejection, or humiliation after a history
of positive teacher experiences may also feel depressed in a subsequent mathematics
classroom unless he is helped to work through his depression.
This line of thinking about relational origins of depression had led me to seeing
that and how assessment results and counseling interventions for mathematics
depressions originating primarily in the self dimension (empty depression) might differ
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from those that origuiated primarily in the interpersonal attachment dimension (grief
depression).
Grandiosity
Self Psychology and Grandiosity
Although U.S. students perform much less well than students from other
developed countries, a persistent fmdmg in international mathematics studies is that, on
average, they think more highly of their mathematics ability than do the students who
outperform them (National Center for Educational Statistics, 1995, 1999). U.S. males are
more likely than females to think more highly of their prowess than their achievements
would suggest to be appropriate (Sax, 1994; Signer & Beasley, 1997). Struggling
students with a grandiose (all-powerful, all-knowing) view of their mathematics
functioning are rarely seen in the learning support center because they cannot consciously
face a need to get the help they need.
Grandiosity may be linked with an underlying poor mathematics self-esteem
because of early teachers' failure to provide the student with the tolerable reality that the
student is not all-powerful or all-knowing, even in his teacher's eyes. This in turn leads to
inadequate internal mathematics structure and values needed to curb his grandiosity via
idealizing and incorporating his mathematics teacher image. He is likely to deal with a
mathematics class or a specific mathematics problem by expressing his belief in his
ability to do it while he fails to put in the effort needed to succeed. He seems to be
unwilling to risk putting in the effort and risking that he may not be able to do
mathematics. That risk is too great for his vubierable and underdeveloped mathematics
self, so he may preserve his unrealistic sense of his ability by doing poorly or failing the
92
course and attributing this to his lack of effort. The challenges a mathematics counselor
might face in trying to help such a student seem considerable. The greatest challenge is
persuading him to get help and the counselor has to be very careful initially to accept the
student's grandiose view of himself while fmdiag ways to diagnose and remedy his
mathematics gaps and deficits.
Mathematics Mental Health Conditions: A Summary and Caveats
When I considered the ways a student might present himself to a mathematics
counselor, it was clear that the conditions I discuss above are far from exhaustive. Each
describes a dimension or continuum of cognition or affect common to every student.
Where a particular student's results are located in one dimension or combination of
dimensions will allow the tutor to determine the state of his mathematics mental health
from sound through poor. Whatever his state, growth is always possible. I expect that not
only students who consider themselves poor at mathematics could benefit from engaging
in this process of relational mathematics counseling. It was also clear, though, that a
relational counseling approach ensures that even if a student comes to counseling with a
condition different from those discussed here, the tutor will be able to understand him
well-enough to help him understand himself and improve his mathematics mental health.
I became mcreasingly aware that engaging in this process with a student likely involves
not only his change and growth but also mine. The role of the instruments I developed or
adapted must be adjunct rather than definitive; the role of the tutor and the relational
counseling approach should be preeminent in the growth and achievement of both the
tutor and student.
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IMPLEMENTING A BRIEF RELATIONAL COUNSELING APPROACH
Extreme scores on a student's Feelings and Beliefs in-class survey assessments
may alert him, me, and/or his instructor to the possibility of his benefiting by counseling;
he may have entered the course expecting to need assistance and comes early to seek the
regular help he believes he needs; or he might be prompted to come by a quiz or exam
grade below his expectations. Ideally he would begin a course of mathematics counseling
early in the semester, enough before the first big exam that at least half of the first session
could be devoted to gaining some understanding of his mathematical background and
experiences, and his current sense of himself as a mathematics learner. Student
expectations about how mathematics counseling might be similar and different from
traditional mathematics tutoring might need to be discussed; students are unlikely to be
consciously aware of the possible relationships between their mathematics relational
patterns (mcluding their sense of mathematics self) and their approach and achievement
potential in the current class. Reahstically, students are likely to exert considerable
pressure to focus on the mathematics content of the course from the beginning so the
process of gathering background information and the process of orientating them into a
mathematics counseling approach will need to be ongoing through the course of
counseling.
In that first session, I would ask the student to create the metaphor whose threads
and themes we could explore over the course of counseling. My curiosity about how he
came to where he is now would also form a thread running through sessions as we pull
apart the mathematics challenges he is facing. The counselor must be alert for his own
reactions, and for behaviors in the student that could be elucidated by the student's
94
Feeling and Beliefs responses, his metaphor, his mathematics background and
experience, and his present mathematics performance. Class assessment results, the
student's responses to them, and the mathematics patterns they reveal are likely to be
focal in counseling. The JMK Affect Scales filled out at each session would provide
regular opportunity to explore links among behaviors, beliefs, and exam results.
I realize that the student in mathematics counseling is part of a complex system of
important players. Each, including himself, is faced with multiple roles. In my study I
focus on the student and the counselor, but others, especially the instructor, play active
roles the student's and the counselor's mathematics relational worlds.
Roles Played in Mathematics Counseling
In this brief relational mathematical counseling approach, it is not only important
for the counselor to understand and integrate a great deal of information about the
student, but he also has to consider roles of all parties: the tutor/counselor, the student,
and other significant players (e.g., the instructor) within the college context. The
mathematics counselor or the mathematics counselor and the student together become
aware of the student's mathematics dimensional relationships as a vehicle for both to
know the student holistically and identify what and how he needs and wishes to change.
Approaches, assessments, and therapeutic contributions from each of three dimensions of
the relationship have been identified.
At this point I need now to discuss new and necessary orientations and
preparation of a mathematics relational counselor. By definition, the counselor
undertakes to view the student wdth unconditional positive regard. He imequivocally
believes in the student's existing mathematical intelligence and the potential for that to
95
grow. The counselor must also understand the counseling and mathematics learning
processes to be a collaborative effort. The counselor brings expertise in mathematics, in
mathematics pedagogy, and in relational counseling approaches and techniques;"" the
student brings his own reality, his mathematics understandings and potential and his
willingness to explore, consider, and apply insights that emerge in the counseling
process.
Relational Counseling Role: The Therapeutic Relationship
in Mathematics Counseling
The following roles emerged for me as ideal yet potentially attainable:
The Mathematics Counselor as Listener and Witness to the Student 's History
The counselor listens knowingly (mathematically and developmentally) with
curiosity rather than with judgment. He elicits the student's experience of his own
mathematics history. To test the efficacy of interviewing for understanding students'
mathematics identity (self) development, I developed a semi-structured interview outline
and piloted it with basic algebra students at a small liberal arts college in the Northeast
(see Appendix A). The interviews I conducted with these students about their
mathematics identity development corroborate Buxton's"'" (1991) fmdings and is the
protocol I developed that I use here (Knowles, 1 998).
Transference and Countertransference in the Mathematics
Counseling Situation
The counselor must be alert to how the student responds to him as a significant
mathematics figure from the past (transference); the counselor also watches for ways he
unconsciously responds to the student's transference or as a significant figure from his
own past (countertransference). This awareness and mterpretation of cUent transference
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and counselor countertransference in the counseling relationship are central to relational
psychotherapy. Close observation of the counselor-client relationship yields crucial data
for identifying relational patterns that are either conscious or unconscious, and that can be
either beneficial or counterproductive to the student's sound mental health. In the
mathematics counseling setting this requires the mathematics counselor to become
conscious of how he experiences the way the student relates to him and seems to expect
him to be as a teacher (transference). He must also become aware of his own reactions to
and hopes for the student, understanding direct responses to the student and knowing
reactions that are based on those from his own teaching or other relational experiences,
triggered by the relationship with the student (countertransference).
Insight, Central Conflict Identification, and Interpretation
in Mathematics Counseling
The counselor observes and hears patterns and unconscious contradictions among
aspects of the student's relationships that may help to explain the student's puzzling
mathematics-related behaviors and may yield clues to identifying his central relational
conflict (insight). He then discusses and clarifies these with the student (interpretation) so
the student may gain insight into his problems;
Mathematics Counseling Role: The Tutoring Relationship
in Mathematics Counseling
The counselor models healthy mathematical behaviors and interprets them in
relation to his own underlying healthy beliefs about the mathematics, himself, and the
mathematics learning situation. He cannot presume that the student will make these
connections between behaviors and beliefs without sometimes extensive mutual
interpretation.
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Mathematics Tutoring as Central to Mathematics Counseling
In standard relational counseling, the focus is both the client and his relational
problems; in mathematics relational counseling the focus is the student and his
difficulties learning mathematics. The counselor must balance therapy's client-
centeredness and counseling's problem-centeredness (Corsini & Wedding, 1995) by
adopting the dual focus of relational brief therapy.
Contributions of Conceptual Mathematics Tutoring and Mathematics Course
Management Counseling
The counselor is an experienced mathematics learning specialist who is aware of
the toxic effects of an exclusively procedural approach to mathematics and the
importance of strategic course management in a time limited college course setting.
Although these issues may not be focal in the eyes of the student, the counselor must be
alert to any need to incorporate them into successful cotmseling, Understanding the
motivations of the instructor (and the department) is also key since ambivalence about
what is valued as mathematics outcomes and how the instructor assesses these outcomes
may result in confusion for students between getting good grades and reall\- working for
comprehensive understanding'^' (Hiebert, 1999; Lee & Wheeler, 1987; Mokros. 2000).
The learning counselor typically has little if any influence on the curriculum or the
assessment so his role is to help the student adapt to the course in a way that is as healthy
as possible for him.
"Understanding is an ongoing activity not an achievement" (Kieran. 1994. p. 589)
but its hnks with mathematical self-esteem places the onus on the mathematics counselor
to discern compromises between achievement (of grades) and understanding: In addition
98
the student needs effective ways to adapt to the present mathematics classroom at the
same time that he repairs his mathematics self-esteem and succeeds in the course.
Constructivist Approaches: The Student as Author of His own Growth and
Healing in Mathematics Counseling Situations
The relational mathematics counselor believes the student has what he needs
relationally, intellectually, and especially mathematically to make the changes he needs
in order to achieve good-enough results. The approach to the counseling and to the
mathematics is thus a developmental constructivist approach. This, however, does not
preclude strategic direct teaching in the time limited setting.
Roles in Relation to Other Key Players
The Student and the Instructor
The student's relationship with his instructor is likely to be revealing not only of
his present mathematics learning approaches but also of his historical patterns of relating
with mathematics relationships. As the counselor becomes aware of the student's
perceptions of the relationship with the current teacher and as they are both able to
directly observe the relationship, the counselor may use discussions of the congruence
between the two to explore these patterns. How the student perceives himself in relation
to his classmates and relates (or not) with them is also likely to be of interest although not
as pivotal as his relationship with the instructor.
The Counselor and the Instructor
Effective tutoring involves not only supporting students in learning the content
covered in the syllabus but also in helping them understand the instructor's teaching
approach, assessment schemes, and priorities. This implies the tutor's knowing or being
able to understand the instructor's approach. A relational approach implies in addition
99
that the tutor/counselor know or be able to gauge how the instructor's pedagogy,
classroom management style, and relational patterns might impact the student. The
counselor must discuss the instructor's approach with the student (and possibly the
instructor), especially if it seems to be detrimental to the student. Ensuring that this
happens this is likely to be extremely important to the efficacy of counseling. The
counselor has to be conscious of his relationship with the class instructor and may have to
use this awareness in mathematics counseling to help the student fmd ways to negotiate a
constructive relationship with the instructor and class.
Supervision of the Counselor by a Person Knowledgeable in Counseling
Because a major source of insight for the counselor is the transference and the
countertransference in the counseling situation, he should be under supervision. This
means that at least once or twice during the semester he should present himself and his
student as cases to a person knowledgeable in counseling in order to confirm or challenge
his insights and approaches and to gain insight and inspiration in cases that he continues
to fmd puzzling.
CONCLUSION
I have situated brief relational mathematics counseling in the college learning
center context and pointed to the details of what it might look like. I have designed a
summary chart that illustrates its components and how I see they relate to each other (see
Table 3). What follows in this dissertation is a description of what happened in the pilot
study as I applied the theory explored in chapter 2 in ways that I envisioned in this
chapter.
100
In the next chapter I will describe the research methods I used to describe my
pilot implementation of brief relational mathematics counseling with students in a
statistics for psychology class at a small university college in the Northeast.
101
' Because this is an odd numbered chapter 1 use "he," "his," and "him" as the generic third person singular
pronouns.
" Through intelHgence testing, "aptitude" tests such as the SATs (Scholastic Aptitude Tests), or
standardized achievements tests with percentile rankings interpreted as ability measures.
"" Through school mathematics grades and "ability" grouping, and teacher/school and parental/societal
expectation.
'" As noted in chapter 2, people suffering from depression have been found by cognitive therapists to view
the world, themselves, and the future through a negative cognitive schema (Beck, 1977; Beck, Rush, &
Shaw, 1979). Martin Seligman (1975) has shovm that a person suffering from situational depression has
almost identical symptoms to those suffering from situational learned helplessness.
" How he processes and assimilates new learnings, accommodates his cognitive schema to these new
learnings, stores them in long-term memory, and retrieves them for application and in appropriating further
learning, constitutes the student's cognitive learning style (Davidson, 1983; Piaget, 1985; Schoenfeld,
1992; Skemp, 1987). Skemp (1987) uses Piaget's term "assimilation" as the process whereby the learner
assimilates the new learning into existing conceptual schema and at the same time "accommodates" the
existing conceptual schema to meet the demands of the new situation, resulting in a struggle to arrive at an
expanded schema and new greater understanding.
Davidson (1983) found strong links to hemispheric preference and clearly defined analytic (her
Mathematics Learning Style 1) and global (her Mathematics Learning Style II) learning styles in terms of
students' mathematical behaviors and approaches. Although these learning styles have not been found to be
directly related to mathematical achievement, Krutetskii (1976) found that students who had a strong
learning style preference and a relative inability with their other mode, found it difficult, if not impossible,
to begin a problem using the other mode's approach. Thus being forced to approach problems using
another's preferred approach greatly disadvantages these students. Learning flexibility, however, can
strengthen performance.
" A strong leaning towards one learning style/processing channel (e.g., visual versus auditory versus
kinesthetic) may present as a learning disability in an environment where another is favored but may be
celebrated as ability in an environment where the preferred style/channel is favored. Thus there are
individuals for whom having a learning disability may be more relative to the learning environment than
intrinsic to her.
™ In the high schools, teachers may know the mathematics content, but the pedagogy is often teacher-
centered and procedural, as procedural mathematics routines are transmitted to the students. Many
members of the mathematics education community have come to believe that exclusive exposure to a
transmission model of pedagogy is generally antithetical to the development of students' mathematical
power. Unfortunately it is what most U.S. students coming to college have experienced (Boaler, 1998;
International Association for the Evaluation of Educational Achievement, 2001; Skemp, 1987).
™' Analysis of attempts to measure mathematics anxiety reveals problems in the research in understanding
what exactly is being measured (McLeod, 1992). Use and development of Mathematics Anxiety Rating
Scale (MARS) (Richardson & Suinn, 1972) which has been normed and is perhaps the most used in the
field, is illustrative of the problem. MARS doesn't distinguish among different types of anxiety, for
example, cognitive worry versus affective emotionality that some theorists differentiate (cf Ho, et al.,
2000) or between state versus trait anxiety (cf Nolting, 1 990). MARS defines anxiety by a single affective
response — fright — and asks students to distinguish among five levels of fright in relation to mathematics-
related activities and situations. Factor analysis of MARS items yielded two relatively homogeneous
factors (15 items each): mathematics testing anxiety and numerical anxiety (Rounds & Hendel, 1980). An
additional factor, abstraction anxiety, important for college students but not addressed in MARS has been
identified by Ferguson and a resulting three scale (the first two of which use MARS items identified by
Rounds and Hendel) test — Phobus — developed (Ferguson, 1986). The last two scales of Phobus (numerical
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and abstraction anxieties) difierentiate between two types of mathematics, each, however, in different
settings: number/arithmetic (outside the classroom in every day settings) versus mathematics involving
algebraic variables and other literal symbols (in classroom and college settings). The first scale inquires
about mathematics testing-related situations before during and after the test. Items in Phobus can be further
classified according to whether the activity is likely to be solitary or public or either. No items inquire
about effects of degree of test preparation on anxiety levels nor do these scales ask about effects of anxiety
on cognition during testing. A deficits model of testing anxiety proposes that a student who is poorly
prepared and has poor test-taking skills will have high anxiety in testing situations (e.g., Tobias, 1985), and
an interference model of testing anxiety proposes that in testing situations, anxiety interferes with students'
recall and thinking.
" Freud (1926) believed separation anxiety to be a natural response to separation and loss only in children.
In adults he viewed it as pathological.
" Only 1 1% of the 157 above-average college students seeking certification in elementary education,
surveyed, reported only positive experiences in their own mathematics education. Of the others, when the
onset of their anxiety was in the 3"^ or 4* grade (as for 16% of the sample), among behaviors of instructors
cited, instructors were perceived to not respond to students' needs for clarification and tutoring or showed
anger or disgust when students asked for help (p. 584). Many of the students whose negative experiences
began in high school (26% of the sample) reported the same ignoring, rejecting, or ridiculing of students'
needs, as did many of the 27% of students whose problems began freshman year of college (p.584).
" This is a version of Kogelman and Warren's (1978) Myth 1 1 : Some have math minds and some don't; or
Schoenfeld's (1992): Ordinary students cannot expect to understand mathematics; or the National Research
Institute's first Myth: Success in mathematics depends more on innate ability than on hard work (National
Research Council, 1991, p. 10).
™In addition to access to a supervisory person knowledgeable in counseling psychologies, for a
professional mathematics tutor to engage in the brief relational mathematics counseling described here,
some preparation (i.e., coursework or at the very least, directed reading) in counseling psychologies,
including CT/CBT and relational conflict therapy would seem to be a minimal requirement.
'"' To understand (and ameliorate) adults' mathematics "panic," Buxton (1991) looked at individual in-
depth interviews, group study of mathematics problems, and discussions of affect. His participants' stories
invariably linked their mathematics panic and failure to achieve to parents, teachers, and their theories
about themselves. What they believed mathematics to be and how they experienced mathematics teaching
in their lives interacted significantly vnth their mathematical self-perceptions.
"Understanding can be characterized by the kinds of relationships or connections that have been
constructed between ideas, facts, procedures and so on... there are two cognitive processes that are key in
students' efforts to understand mathematics — reflection and communication" (Hiebert et al., 1997, p. 15),
both of which require the opportunity and time to do so.
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CHAPTER IV
METHODOLOGY TO STUDY BRIEF MATHEMATICS RELATIONAL
COUNSELING MODEL
LEARNING ASSISTANCE CENTER AND MATHEMATICS SUPPORT:
Finding an Appropriate Research Setting to Pilot Brief Mathematics Relational
Counseling
My responsibility as the mathematics specialist for the Learning Assistance
Center at Brookwood' State University is to offer support to students taking mathematics
and mathematics-related courses. Along with the mathematics peer tutors whom I help to
train, I offer mathematics tutoring in individual appointments, in open drop-in tutoring at
the Learning Assistance Center," and in study groups for specific courses. These
offerings are advertised to students via memos to instructors and by initial visits to the
classrooms by the peer tutor or me. Some students who need support fmd their way to
the Center in a timely fashion and often enough for the support to help; some come at the
last minute (e.g., just before an exam when it is often difficult to resolve their problems);
others do not come at all.
We do not see all of the struggling students and generally cannot directly observe
how the students we do see are handling their instructors' teaching and testing
approaches. For my study of whether and how relational counseling insights could
contribute to mathematics support, however, it was necessary to observe the classroom
environments and student behaviors and interactions there. I decided that my research
should focus on one mathematics course so that I could attend that class and offer
individual mathematics counseling to its members. This approach was modeled on a
tutoring practice already used in writing-intensive classes at Brookwood, where peer
tutors are class-support tutors'" who attend the assigned class and offer learning
104
assistance both within and outside the classroom. Although this approach had not yet
been used in mathematics classes, it was a familiar practice at Brookwood in other
subjects, and the advantages for my research seemed obvious.
To pilot my counseling approach I decided to focus on students in a class that was
considered to be at risk for high negative mathematics emotionality, withdrawal, and
failure. The PSYC/STAT 104'^ (Statistics in Psychology) class that I researched was a
one-semester introductory statistics course that fulfills the university's quantitative
reasoning core requirement. It is also a major requirement for nursing, psychology, and
biological science students. This course is taught for fifteen weeks in the fall and spring
semesters and for ten weeks in the summer" at Brookwood State University. Ann Porter"',
the PSYC/STAT 104 instructor, a tenure-track faculty member, agreed to host this
research in her classroom.
The specifics of the course, the students, the instructor, and the mathematics
counselor were particular to us. However, I was certain that my observations, diagnoses,
and the application of brief relational mathematics counseling approaches to Brookwood
students' challenges would provide insight into some broadly applicable ways that
mathematics students can be supported and shed light on changes needed in traditional
college mathematics support. I expected a fi-amework could emerge to help mathematics
support professionals to understand and deal with students' mathematics problems in a
way that would also promote their mathematics mental health while they are engaged in a
semester course. The emergent framework is grounded in relational conflict brief
counseling theories and cognitive constructivism.
105
The Course
The introductory level statistics courses, PSYC/STAT 104 and BUS/STAT 130,
are among the most failed and dropped first year college level classes at Brookwood.""
Although there is no stated mathematics prerequisite for PSYC/STAT104, there has been
ongoing pressure from academic counseling and academic support personnel on the
Enrollment Management Committee to make successftil completion of high school
algebra at least a strong recommendation.
PSYC/STAT 104 is offered through the psychology department. Some students
take it to fiilfill the quantitative reasoning requirement for a liberal arts degree. Nursing
and psychology majors are required to take it. Nursing faculty see it as a gatekeeper
course for the degree: If a registered nurse (RN) is not able to pass it with at least a C, it
is thought that she"" might not be a suitable candidate for a bachelor's degree.
Enrollment in the summer course is always lighter than in the fall/spring semester courses
and the course only takes 10 weeks to cover 15 weeks of material. During the summer,
students typically work full-time and take PSYC/STAT 104 and at most one other course.
The class offered in the summer of 2000 was typical, with RNs, psychology majors, and
others, all hopmg to do well enough to be able to proceed towards their larger goals.
THE STUDY SITE AND PERSONNEL
The University
Brookwood State University (not its real name) is a small commuter university
college with approximately 1,500 degree and continuing education students. The summer
enrollment is approximately 550. It is located in the small New England city of
Brookwood. The greater Brookwood area population is almost 200,000 and is
106
predominantly white with 5.6% non- white or mixed race residents concentrated in the
city proper. It is ethnically quite diverse. Thirty-one and one half percent of the
population has French or French Canadian ancestry and many maintain their ancestral
language and culture. Three percent identify as Latino and 14.4% of the population
speaks a language other than English, two-thirds of which are Indo-European languages.
There are more than 50 different languages spoken in the local schools. Six point six
percent of the population is foreign bom, half of these having entered the U.S. since 1990
(U.S. Census, Census 2000). Among these are considerable numbers of reiugees from the
Balkans, Africa, and the Middle East. The university's college credit and intensive
college-preparation summer English Speakers of Other Languages (ESOL) courses
attract between 30 and 50 high school students and adults per year, approximately 35% of
whom go on to degree programs in the university.
The average age of undergraduate students attending Brookwood is between 26
and 27 years and the student population approximately reflects the racial and ethnic
diversity of the greater Brookwood area (Brookwood University records, March 2003).
The university offers two-year associate's degree, bachelor's degree, and some master's
degree programs. Many students enroll in credit courses as non-matriculated, continuing
education students.
Until recently, many of the classes (including PSYC/STAT 104) were held
downtown at the Riverside Center, while the Learning Assistance Center and the
Computer Lab were located at the Greenville Campus on the edge of the city about five
miles from Riverside.
107
The Researcher and the Course Instructor
At the time of the study, I had been in the field of mathematics education for
abnost thirty years, the previous 1 2 at college level. My professional focus had been on
the teaching and learning of developmental and first-year college-level mathematics,
although I had tutored students and trained tutors across the undergraduate mathematics
spectrum. I had developed curriculum and placement testing and had taught mathematics
courses at a community college and small four-year liberal arts colleges in New England.
I had worked in academic support for these courses with general student populations and
special populations that included learning, sensory, and physically challenged students.
I worked at Brookwood State University as the mathematics learning specialist
and assistant director of the Learning Assistance Center. Prior to the study, I had only
briefly met Dr. Ann Porter (pseudonym), the course instructor, at all-college ftinctions
since she worked almost exclusively at the Riverside campus. We communicated by
memo and through peer tutors about study group and tutoring offerings for students in her
courses. I had not met any of the students in the class except Pierre, whom I knew by
name and face through the Learning Assistance Center's work with English speakers of
other languages.
I was aware of the negative reputation of the course among students who
perceived themselves to be shaky in mathematics because I had tutored one third of the
members of the PSYC/STAT 104 course'" at Greenville campus in the spring of 2000. 1
was aware of another third who were struggling and I perceived the dread of students in
the Learning Assistance Center who knew they would have to take it in the future.
108
Researcher as Mathematics Tutor and Counselor
My role in this study would extend beyond that of researcher doing naturalistic
observation to active intervention as a tutor and a counselor, so it was important for me to
engage in continual self-reflection before and during the time of contact with the class
and during the period of post-analysis of the data. In particular, 1 needed to reflect on
myself as a mathematics tutor and also as an emerging mathematics counselor as I put
relational and cognitive counseling theories into use.
Who I am as a Mathematics Tutor
I had always performed well as a student in the predominantly transmission'^
teaching, textbook-focused, and procedural"' mathematics classrooms of my elementary
and high school education. At university, I became more conscious of the larger concepts
underlying mathematics but it was not until I began tutoring students with learning
disabilities that I became uncomfortable with the prevailing pedagogy and its implicit
assumptions about students' learning processes; I realized how capable my tutees were
but also saw how incomprehensible they found much of the mathematics presented to
them in class. This began my struggle to understand their ways of thinking, to understand
the mathematics more deeply myself, and to find ways to help them understand and
achieve in a class that someone else is teaching, over whose curriculum or pedagogy I
had no control.
As a tutor I tend to help too much, by teaching and telling, more than to coach the
student to find his own way to understand the material. I tend to suffer irom "agenda
anxiety" on behalf of the students — knowing what they wUl be expected to cover but
109
worried that they might not recognize the urgency. I tend to try to push them too fast. I
find it hard to let them make the mistakes they need in order to grow.
I feel tension as a tutor of courses that other people teach, and this increases when
the curriculum or the instructor's pedagogy seems to increase the students' difficulty in
understanding the concepts and connecting related concepts. I feel even more tension
when the student experiences the classroom as abusive or unsafe. At times I allow this
tension to enter the tutoring session by siding with the student against the curriculum, the
system, or the student's past preparation or teachers. I generally do not join students in
criticizing the instructor but try to help them find ways to handle these conflicts in a way
that is constructive to them. I sometimes find myself defending teachers whom students
are attacking.
Who I am as an Emerging Mathematics Counselor
I am a white, university-educated, Australian, female, extroverted mathematics
teacher and tutor brought up in the suburb of a large city in a middle class home by
parents who were both tertiary educated professionals. In addition to working in
mathematics education, I also worked with several groups of Australian Aborigines doing
field linguistics and a trial literacy project. It was in the context of that work that I met
and married a rural, working-class. New England American who graduated fi-om a
technical high school program in the 1960s and works in the building trades.
My faith is grounded in the principle that all humans are made in the image of
God and are thus inherently creative and have the potential (indeed the obligation) to
learn and grow and understand. It has been an important basis for my interest in and
continually emergent acceptance of people whose backgrounds and characteristics are
110
different from mine. I am aware of my need to continue to work through my class-,
ethno-, religion-, gender- and extrovert-centric orientations and grow in appreciation and
acceptance of difference. As I explore and understand the challenges and opportunities
that my tutees and my characteristics and backgrounds have placed on our development
and our ability to understand and accept each other, I continue to find that some aspects
of who I am provide potential bridges and others create potential barriers; I know I have
blind spots that make understanding difficult.
I am female, non- American and from what is often seen by Americans as an
insignificant former British colony."" I live with a person from a low SES background
who experienced low expectations and less education because of this background and still
struggles with a sense of powerlessness. My efforts to accept and maximize my potential
within these identities give me some empathy with students from disempowered
groups — women, racial and ethnic minorities, and people of low SES — whose
mathematics selves, internalized presences, and attachments have been negatively
affected because of who they are."'"
My struggles with arithmetic details (I cannot keep my checkbook straight), visual
memory, visual-spatial reasoning, and directionality (I cannot tell left from right nor
connect the implications of up versus down without verbalizing) enable me to empathize
with students with learning disabilities or a strong mathematics learning style preference
(and concomitant wealaiess in the other) who believe their learning challenges prevent
their achievement in mathematics. I can also model struggle and success for these
students.
Ill
My being white, middle-class, university-educated, and successfiil in mathematics
may be initial barriers for students who feel disempowered because of who they are but I
have found that self-disclosure of my struggles can help break through. My family
background of trying to help a mother struggling with addictions makes me vulnerable to
co-dependently take on a student's responsibility to make any changes she needs to, or to
excuse his failure to take that responsibility himself I find the fme distinctions between
support and indulgence difficult. On the other hand, I find it difficult to (and would rather
not) work v\dth students who appear to be overestimating their abilities or knowledge or
who seem to be rigidly adhering to approaches that are counterproductive. Hence, I
recognize the particular importance in this study of attention to the student's transference
and my countertransference in the mathematics counseling situation.
The Instructor
Dr. Ann Porter is a young, energetic, white woman (in her late twenties at the
time of the study). She has a Ph.D. in experimental psychology and was actively engaged
in research with a geriatric population at the time of the study. Ann also served as faculty
advisor to the Student Government at the university. She had taught this course before.
Ann began at the university two years before the study. She stated that she taught with a
more "laid back" teaching style than she had experienced as a student (Interview 3,
archived). As she described her professors' transmission teaching methods for her
undergraduate introductory statistics and her later graduate statistics classes, she told me
that she believed students should instead be able to grapple with the mathematical
procedures during class with the opportunity to receive guidance rather than merely
watch the procedures being done on the board as had been her own experience (Interview
112
3). She had been comfortable with algebra in high school, had minored in statistics in her
doctoral program, and she was finding that teaching it to undergraduates was deepening
her enjoyment of the field.
Ann told the class she liked the assigned text, but disliked the required computer
program MINITAB — a late 1 980s version — because it was "somewhat archaic" (course
syllabus; Class 1, May 31, 2000). She told them of a more modem statistics software
program she used to analyze her own data and promised to bring it in to show the class.
She shared her own struggles with anxiety in a statistics class she had taken. She invited
the class to call her by her first name if they preferred. All did.
When I approached Ann before the course began she was hesitant about my
doing research in her class because the class time needed to cover the material was
reduced by several hours in the summer. "^^ She had committed herself to a very busy
summer and she was also concerned that my research project would add to her
workload. She was worried that my using a counseling approach with students for
their "psychological" problems might have unforeseen repercussions on students'
behaviors in the classroom and in relation to her, and make teaching the course more
difficult. She did not see students' affective issues to be within her purview and did
not want students to expect that of her. To allay her fears, I designed whole class
research explanation and surveying to take minimal time and we agreed to schedule it
just before breaks or after exams. Ann discovered her fears that my research would
increase her workload were unfounded; indeed, the reverse was true. She found that
in most cases students' negative affect and cognitive struggles actually became more
contained because of the support I was offering.
113
My Roles in the Study
I attended each class primarily as a researcher. In that capacity I took a small
amount of class time to explain my project and administer pre- and post-surveys of
mathematics affect to the class. Otherwise, I observed and recorded interactions in
the classroom — especially instructor-student interactions during lectures and student-
student and instructor-student interactions during problem- working sessions.'™
Increasingly I took the role of class-support tutor. In that capacity I led a weekly
study group for the class and during class I assisted students sitting near me by
working the problems in parallel with them, as Ann circled the room helping others.
Ann occasionally consulted me on mathematical questions when she was uncertain. I
also offered individual mathematics counseling to volunteers from the class and
because students were meeting with me, they were generally less demanding of Aim's
time outside of class.
THE CLASS AND INDIVIDUAL PARTICIPANTS
The PSYC/STAT 104 class of the summer of 2000 was typically small. There
were 13 students (7 women and 6 men) at the first class meeting on May 31, 2000. 1 have
given each a pseudonym to preserve anonymity. All were white and spoke English as
their first language except Pierre, a French-speaking black African; at least one (but
possibly three™) was first in the family to attend college, and most were long-time local
residents. They ranged in age from 19 to the mid- forties, and about half traditional-age
students. The class average age was around 28 years, somewhat higher than the
Brookwood average. All but three were fiill-time students. Because it was a summer
class, nearly half were from other colleges, a greater proportion than is usual in other
114
Table 4.1
Profile Summary of Students taking PSYC/STAT 104. Summer 2000 (N = 13)
Student
Where
Enrolled?
Student-Related
Data
High School Math
Courses
College Math
Courses
Work in
Summer
PARTICIPANTS
Eow^-R''
Autumn
4 sessions
SI/
W; Age ~20;Full-
Time; MjrPsyc
Mn'rBusiness to
Mj':Business; Mn:
Psyc
Algebra I: A", Geom: Finite Math,
A; AdvAlgebra II: C"; 2000: A"
Prob&Stat(I/2): A
Discrete Math( 1/2): A
Retail~30hrs
Ew-H? Brad BSl/ M"; Age~40s; Pt-time
4 sessions Repeat PSYC/STAT
104
Algebra I, II, Geom:
Bs??
PSYC/STAT,
1998: F/AF
Nursing-FT
Ew-H Jamie
5 sessions
SU W; Age: 20; Full-
Time; Mj: Psyc
Repeat PSYQSTAT
104
Algebra I: B/B", PSYC/STAT,
Geom: C"/D^ Algebra 1998: D"
II: B7C*, Precalculus: Finite Math,
C? 1999: W
Retail~30hrs
BSU
W; Age: 22; Part-time;
Algebra I; C?
PSYC/STAT,
Assistant
Eow-H Karen
Mj:Psyc, Mn:Educ
Geometry (struggle);
1998: F
TeachCT -
5 sessions
Repeat PSVaSTAT
Algebra II: C?
Elem. Special
Ed-FT
SU
W; Age:19;FuIl-Time;
Algebra I; Geometry;
Basic Math,
Camp
Ew-H Kelly
Mj: LibArts;
Algebra II
2000: D/C?
Counselor-
3 sessions
MjInt: Soc. Wk
FT
Eow-H Lee
BSU
W; Age:19;FulI-Time
Geometry: B; Algebra
Finite Math,
Dental Office
6 sessions
Mj -Psyc
II: A; Precalc/Calc: A
1999: A
Assist~30hrs
Eow-H? Mitch
BSU
M; Age:23;Full-Time;
Algebra I: F,A;
PSYC/STAT,
Retail: FT
4 sessions
Mj: EuropeHistory
Repeat PSYC/STAT
104
Geometry: F,C
1998: F
Finite Math,
1999: C
Eow-R Mulder
OU'
M;Age:21;FuII-
Algebra I; Geom;
None
Retail~30hrs
5 sessions
Time; Mj: Biology
Algebra II:C
Ew-H Pierre
BSU
M; Age~30s;Full-
Algebra through
Calculus I: D
Residential
8 sessions
Time; Mj Biology;
ESL/French
Calculus
Support: FT
Eow-R Robin
BSU
W;Age~mid40s; Part-
College Algebra A
None
Nursing: FT
3 sessions
Time; Mj: Nursing
CLASS ONLY
W;Age~40s; Full-
Calculus I,
(not individual
BSU
Time; Mj:Biology,
1999: A
participants)
Mn:Educ
Catherine
EUen
BSU?
W;Age~30s;
OC
M;Age~20-30s; Full-
Finite Math
Floyd
Time; Mj:Psyc
Note. ''Participant signed up for mathematics counseling: Eow = Every other week; Ew = Every
week. '' Participant's initial motivation for signing up was: -R = to help me with my research; -H
= to get tutoring help; -H? = apparently to get help. '^Institution where student was enrolled: OC =
Other College; BSU = Brookwood State University; SU = State U.; OU = Other U. '' Gender: W
= woman; M = man; ^ Mj: Major, Mn: Minor; MjInt =intended Major; 'FT = full-time work.
115
semesters. Three of these were from the affiliated State University. All of the participants
except Mitch had taken at least Algebra I, Geometry, and Algebra II in high school. Four
students were repeating PSYC/STAT 1 04; three had taken the course within the past 2
years and failed it (Karen, Brad, and Mitch) and one (Jamie) had earned too low a grade
to be counted for her Psychology major.
Ann briefly introduced me as a researcher at the first class meeting and in the
second I gave each class member information about my research project (see Appendix
D) and then administered class mathematics affect pretests (see Appendix C) to all who
agreed to be involved in the whole class study. All 12 students present completed the
pretests, thus constituting their consent to have me use them and classroom observations
of them as data. How they could signify consent was explained in writing in the research
explanation and I have archived pretests as consent agreements.'™" Ellen, who was not
present, had dropped the course. At this class meeting all students were invited to
volunteer to be individual research participants'"'" by signing up for one-hour
mathematics counseling sessions with me. Nine students volunteered by fdling out and
signing a volunteer agreement card. Four opted for counseling every week and five for
every other week. One other (Lee) initially checked "no" for one-on-one counseling but
e-mailed me the day before the first exam to ask to participate. Each of these signed an
Informed Consent Form (see Appendix D) during the first counselmg session."'" Of the
initial group often counseling participants, two failed to complete the course — one left
before the second exam and another, citing family responsibilities, in the ninth week of
the course. A summary of individuals' mathematics-related characteristics and history is
presented in Table 4. 1 .
116
The benefit of participating in my study was the individual statistics tutoring and
mathematics counseling, so monetary compensation for participants was not necessary.
DATA COLLECTION
The Research Schedule
Table 4.2
PSYC/STAT 104, Summer 2000 Class and Research Schedule
Course Week
1
2
3
4
5
6
7
8
9
10
Post
l"* Class
Class
Class
Class
Class
Class
Class
Class
Class
Class
"Comp
Mondays
2
4
6
8
10
12
14
16
18
(extra)
6:00 p.m.
Pre-
tests
EXAM
1
June 12
■PlB-
SRA
no
class
meeting
EXAM
3
July 17
MINITAB
Post-
Tests
EXAM
2""' Class
Class
Class
Class
Class
Class
Class
Class
Class
Class
Class
Wednesdays
1
3
5
7
9
11
13
15
17
19
6:00 p.m.
MINITAB
Cotnput
Lab
EXAM
2
June28
EXAM
4
July 26
EXAM
5
Aug. 2
Study Group
Study
Study
Study
Study
Study
Study
Study
Study
Study
Wednesdays
4:30 p.m.
Gp 1
Gp2
Gp3
Gp4
Gp5
Gp6
Gp7
Gp8
Gp9
4att.
3att
latt
4/5att
latt
2att
latt
5att
6+att
Drop-In
Ke,L
B,L
Ka,
L, J
Mu,
MAT120
Individual
Ka6/12
J6/20
Ka6/26
J7/3
Ka7/I0
Mu7/17
Ka7/24
P8/2
J8/6
Session
TCe6/8
A6/I2
B6/20
Mi6/26
L7/5
B7/10
Ka7/17
Mu7/25
L8/2
P8/7
Mi6/14
R6/14
Ke6/16
Mu6/21
P6/27
R7/5
J7/U
A7/I7
R7/25
P8/3
L8/7
L6/21
A6/28
Mu7/6
Mi7/12
L7/19
B7/25
Ke6/21
Mu6/29
R7/12
P7/26
P6/22
P7/13
B7/13
P7/14
J7/26
Mi7/26
A7/26
My Outside
Interview
Ann
Interview
Ann
Present
Interview
Ann
activities
Potter
Porter
cases
to Dr.
Porter
in relation to
May
July
P.
Aug
course
31
10
July 20
Study
Gp
before
EXAM
w/Ann
4.6 pm.
w/Jill
5.*pm
3
Note. "After Exam #1, 1 administered the Statistical Reasoning Assessment (SRA); ''Students could take
and optional comprehensive final after the course ended to replace a lower grade; '^A = Autumn; B = Brad;
J = Jamie; Ka = Karen; Ke = Kelly; L = Lee; Mi = Mitch; Mu = Mulder; P = Pierre; R = Robin.;
'The underlining in the table indicates the first individual mathematics counseling meeting for that
participant.
117
Once participants had volunteered for individual mathematics counseling, we
negotiated meeting times, and by the end of the fourth week of class I had met with each
of the participants at least once for mathematics counseling (see Table 4.2 for complete
schedule of the research). The number of counseling sessions ranged from three to eight,
with an average of five per participant.
I did not make my choice of individuals for the focal cases until after the course
was completed so that during the sessions I would be equally focused on all 1 0
participants. I audio-recorded counseling sessions and had ones I identified as key
transcribed. My roles in sessions varied with the participant and the timing of the session
(e.g., the proximity of an exam).
Mathematics, Affect, and Relational Data Collection and Use
Instruments for Assessment and Treatment
Because I was piloting the brief relational counseling approach, I knew I must
identify students' relational patterns and both affective and cognitive symptoms to be
dealt with in the brief time available. I devised, adopted, and adapted a number of survey,
emotional response, and mathematics cognition instruments, some of which I
administered to the whole class and others to individuals in counselmg sessions. In
chapter 3, 1 discuss my development and choices of individual instruments and indicate
my proposed use in counseling (see Appendix B for the individual instruments). Also in
chapter 3, 1 discuss my choices of class survey and mathematics instruments and indicate
my proposed use of the instruments in counseling (see Appendix C for the class
instruments). The individual case studies in chapter 6 reveal whether and how I actually
used them in counseling. Chapter 8 includes an evaluation of the instruments' use in the
118
counseling process and there I make recommendations regarding their further adaptation
and appropriate use.
Mathematics Data Collection and Use
I collected data about each student's mathematics skills using a statistics
reasoning test (the Statistics Reasoning Assessment or SRA), administered to the class at
both at the beginning and the end of the course, an arithmetic diagnostic (the Arithmetic
for Statistics Assessment), an algebra diagnostic (the Algebra Test), all class PSYC/STAT
1 04 tests,™ and participant-observation notes written during and immediately aiter
classes and individual and group meetings (see Appendix C). With all but Autumn, the
greater proportion of each session focused on the course's mathematics content. We used
student class and assessment products to identify issues with strategic preparation and
course management strategies. For example, exam analysis focused on accuracy of
students' perceptions of what would be examined and how, their preparation, type of
errors, and troubleshooting behaviors to enhance approaches to the next exam (see
Appendix E, Table E4).
Mathematics Affect Data Collection and Use
Each student's conscious affect around mathematics learning was appraised using
in-class pre- and post-feelings and beliefs surveys and discussion of his responses. The
following is an example of how I used this survey data with a participant in a counseling
session: When I pointed out Autumn's low score on the Learned Helpless/Mastery
Oriented subscale of the Beliefs survey scale relative to the scale and to the class, she
seemed a little surprised at first. Perhaps this was because she had offered to meet with
me for my research and did not perceive herself to be in need of mathematics tutoring or
119
counseling. When I explained the concepts, however, she agreed that she had acted in a
helpless way and at the same time revealed her disappointment with herself. It seemed
that her performance learning motivation and her learned helplessness had conspired
together to prompt her to a decision she later regretted.
JK: . . . You answered in a way that seemed as if under certain mathematical
situations, you would have a tendency to give up —
Autumn: Oh, YEAH.
JK: Or to not go ahead.
Autumn: Yeah.t (laughs)
JK: Okay. All right. That's— that's—
Autunm: Definitely!
Autumn: ... I remember now that when I was in 8th grade ... I was in the higher
level math class ... But I was only getting 70s, and I wasn't happy with
that so I wanted to go back, so I could get better grades. . . I went easier.
JK: Easier class?
Autumn: Because I'm a perfectionist, and that wasn't good enough.
JK: . . . rather than going and seeing how you could get your grade higher?
Autumn: Yeaht I just gave up and went down.
Autumn: But I didn't challenge myself, so — (Session 2)
Autumn had earlier expressed disappointment with her later mathematics achievement.
Autumn: Um, mathematical achievement. I'm somewhat discouraged because I
didn't really challenge myself enough in high school ... I kind of took
the easy way out.
JK: Ahh! So you feel that you could have achieved a higher level?
Autumn: Yeah. I definitely could have. (Session 1)
Autumn revealed in flirther discussion that she really was not "definite" that she
could achieve at a higher level; she had "challenged herself and tried a harder class
under difficult circumstances but had not gotten her required A. I surmised this was
probably because she did not go for help, but she seemed to have decided it was because
of an underlying inability to do harder mathematics — she took no further risks; al^er that
she chose only classes she knew she could get an A in. Autumn had given in. In the
conflict between safely preserving her high grades and achieving to what she hoped was
120
(but feared was not) her potential safety had won but Autumn was not happy. The learned
helplessness discussion was fruitilil in two ways. First, it showed me that a student's own
survey responses, while, in themselves, providing limited information, could form a
stepping off point for both the participant and me to explore more deeply. Second,
participants, at least in this case, will likely not reveal this type of information about
themselves and their motives through direct questioning; use of their survey responses
and my explanation of what these responses generally indicated about them seemed to be
the prompt for such revelations. To support this conjecture, although I had asked Autumn
about her mathematics course-taking experiences in Session 1 , she did not reveal her
performance achievement-motivated (see chapter 3), course-switching behavior m 8*
grade until Session 2 when I introduced her survey responses for discussion.
Relational Data Collection and Use
The principal means I used to collect data that linked affect and motives with
relationality of which the participant was less consciously aware were through the
metaphor and affect scales and through transference and countertransference.
Metaphor and Affect Scales Data Collection and Use
I gathered through individual metaphor surveys administered at the first
counseling session and as part of the One-on-One Evaluation at the end of the course, an
individual mathematics affect scales instrument administered at each counseling session,
an individual mathematics learning history interview protocol, and classroom and
individual meeting participant observation. Jamie was so unobtrusive in class that the
instructor, looking back at the end of the course, wondered if she had started three or four
classes later than the rest of the class (Interview 3). And she quietly slipped out of the
121
room whenever other participants were making their appointments. It was in discussing
Jamie's metaphor that the role of her mathematics history and her personaHty in her
present puzzling behaviors became clearer.
JK: Yeah. And then what happens during the storm? How do you, like,
handle the storm?
Jamie: Um — stay inside, [both laugh]
JK: So how does that relate to the math?
Jamie: Um. Well, you have to prepare for tests. I don't know how staying inside
does. (Session 1)
Jamie went on in discussion to talk about an elementary teacher who had yelled and
explained her reaction:
Jamie: Yeah. You want to sit down and shut up so you don't bother her.
JK: So maybe . . .you know, your presence in a classroom is very cormected to
that?
Jamie: Yeah.T (Session 3)
Bringmg this history, her metaphor, and my observations of her current classroom
behavior together enabled Jamie and me to realize that she was "staying inside" in this
class almost as if she were still in her 5th grade class not able to do anything but survive
the storm, but this behavior was jeopardizing her chances of success in the class. That
conflict became our counseling focus.
Transference and Countertransference Data Collection and Use
I noticed transference and countertransference in individual participants'
relationships with me as counselor and tutor. In some cases, we discussed it, providing
data about participants' and my own subconscious mathematics-related relationship
orientation. Here is an example of me slipping into a countertransference:
JK: Maybe then your resistance is: you say, "This is conceptual. I don't have
to do that." Maybe if you could say, "Ah this is not conceptual."
Rename it: "This is just mathematical."
Mulder: Pain in the butt!
122
JK: Am I a pain in the butt? [startled]
Mulder: No, that section of the test
JK: Well, you are doing a nice job of resisting, which is good . . . ( Session 5)
I was almost certainly included in Mulder's "pain in the butt" classification. Here,
as in previous sessions, I was giving advice, trying to fix his problem for him like a
mother of a child rather than trusting him or allowing him to find his own way, and
Mulder was actually resisting my countertransference with his rebellious teenaged-son-
to-mother transference "Pain in the butt!" as much as he was resisting the cognitive
challenges posed by the multiple-choice questions. In previous sessions I had scolded
him and pushed him to overcome his resistance to mastering the conceptual multiple-
choice part of the exams. In this session I continued:
JK: Come on! Keep going! You've got a bunch of these to do. You are
really resisting very well! . . . And what it does to me is like I'm thinking
this guy is so smart he could do so well and the mother in me comes out
and it's like "If I could only persuade him."
Mulder: Yeah, I don't think you can do this one. [ignoring me]
Mulder: Hey, I'm done, I'm done.
JK: Oh, but look — there are these.
Mulder: Oh, YEAAAH! Right on!! [very sarcastically]
JK: There are not too many!
Mulder: You make me really not want to come back here. (Session 5)
When I recognized Mulder's "teenaged son" transference and admitted to my
countertransferential indulgent but thwarted mothering approach I was able to recognize
the inappropriateness and ineffectiveness of this cycle we were in and soon after, I
removed myself from the cycle so Mulder could focus on his mathematical challenges
instead of on the power struggle with me (see chapter 6 for flirther elaboration). This
excerpt is an example of the transference and countertransference data collected and
shows how I used my understanding of the transference and countertransference in the
counseling process.
123
After each counseling session I examined and filed dated products from the
session and completed a Mathematics Counseling Session reflection (see Appendix B). 1
noted transferential, countertransferential, and relational dimension incidents. At the end
of each day I audio-taped flirther reflections on the class, individual counseling sessions,
study group, or other interactions that occurred that day.
Efforts to Obtain Triangulation of Data
Because much of the data I was gathering was subjectively experienced, and
because understanding the interrelationships among data was essential for effectively
helping participants' progress, I determined to work with a supervising counselor. There
were several participants with whom I was struggling, and my own blind spots were
almost certainly preventing me from seeing difficulties with others. After I had met
several times with each participant, I met with a psychological counselor. Dr. P.,
presenting each participant and my experience of her for clinical supervision, for an
expert perspective on subjectively experienced data and my responses to it, and for
support and suggestions for ongoing counseling interventions.
This meeting served the purpose of supporting, challenging, and focusing my
emerging counseling efforts with participants. It also served as a key triangulation tool
for the case study data, that is, it ensured that each participant's and my relational data
was experienced by another knowledgeable person who actively participated in the
relationship. The relational dyads between each participant became triangular — among
each participant, me and Dr. P. Dr. P.'s later responses to my analyses of courses of
counseling when the pilot study was completed further supported this triangulation
purpose.
124
Triangulation was also provided by the instructor's perspective on the progress of
the class and individual students. By the design of the study, in order to ensure that
students' course outcomes not be compromised, the instructor was blind to survey and
counseling data students gave me. I interviewed her before, during, and at the end of the
course to learn her perspective on her teaching, on the students in her class, and on the
effects of my presence in the class. Correlating her and my experiences of the classroom
provided valuable insights into students' processes and changes in the classroom and
assisted the progress of counseling. All mdividual meetings, the supervision session, and
interviews were recorded on audiocassette. One class and the lecture portion of another
were video-recorded. All material is archived.
Data Collection Summary
By the end of the summer course I had collected approximately 75 hours of
audiotaped data from 48 counseling sessions and nine study group meetings, and an
additional 25 hours from interviews, the supervision session, and my after-class
reflections. I had 56 class exams (all the exams taken by each class member)
approximately 20 completed pre- and post- feeling and belief surveys, 36 mathematics
assessments and approximately 50 in-counseling affect/relational assessments from the
ten participants. In addition, I had ahnost 100 pages of divided page course and
observations notes, and approximately 40 class seating, lecture interaction, and problem-
working session interaction charts. I also had copies of Ann's worksheets and the
worksheets I devised for use in counseling.
125
ANALYSIS OF DATA
Mathematics Educational Analysis
Increasingly, mathematics education research recognizes the value of carefully
conducted qualitative studies of teaching and learning processes and outcomes (McLeod,
1997). In this study, a case study analysis based on the considerable amounts and levels
of qualitative and quantitative""' data that I gathered and analyzed, best served to
illustrate the model of brief relational mathematics counseling, developing a full picture
that allows both researcher and reader to generate hypotheses that may be tested by
ftirther cases and more experimental approaches.
Despite the quantity of quantitative data I compiled, the conclusions I draw are as
much those of a therapist as of a social scientist. Fundamentally, my research is an
exploration of students' subjective experiences of mathematics, and of my subjective
experience as a tutor and counselor helping her. In the words of Pierre Dominice, "The
scientific model we have tried to respect in the educational sciences does not allow us to
explore the vividness of subjectivity" (Dominice, 1990, p. 199). The scientific
experimental method is not usually possible with human participants because all
variables except the being investigated cannot be held constant. Therefore, in education, a
quasi-experimental method is frequently used. Studies using this method attempt to hold
constant as many variables as possible while causing the one or two variables in focus to
change. In such studies, a multitude of variables, complex interrelationships among
variables, the uniqueness of each participant, are all seen to be difficulties or variables to
be reduced or at least evened out as much as possible to produce the uniformity necessary
to show the effects of one variable on another.
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In this study, however, not only do I choose not to ignore the complexities of
interactions among variables and the uniqueness of each participant, but I embrace them.
Complex human beings struggle with the influences of their conscious and subconscious
existence on current mathematics practices and outcomes; simplicity would be a
reduction of their educational reality. Case study analysis using both qualitative and
quantitative analytic techniques is therefore the optimum choice. Nevertheless, fmdings
from some elements of this pilot study may lead to the need for future quasi-experimental
studies to establish their effectiveness.
Dynamic Psychological Counseling Analysis
Psychotherapy research offers this justification for the case study method:
[T]he primary means of clinical inquiry, teaching, and learning has been and still
remains the case study method grounded in the tradition of naturalistic
observation. Statements about psychotherapy that are derived from group data
typically have little direct relevance for clinical problems that are presented to the
psychotherapist.™' (Jones, 1995, p.99)
Advances in quantitative methodology in single-case research are leading to greater rigor
and greater generalizability of the fmdmgs from such single cases (Jones, 1995).
Additionally, psychoanalytic research tradition development in standardizing
interpretation and treatment of clients' core relational challenges while taking care not to
minimize the uniqueness and complexity of each person seem to me to be directly
applicable to the mathematics counseling setting (Kemberg, 1995; Luborsky, 1976;
Luborsky & Luborsky, 1995).
Counselor-participant match is an important factor not only in counseling efficacy
but also in psychoanalytic research analysis of counselor and participant insight,
interaction, and change (Kantrowitz, 2002; Kemberg, 1995). Counselor-participant match
127
can be assessed in terms of particular conflicts that arise and more importantly, in terms
of characterological similarities and differences that may hinder or support participant
progress. In this study, the same mathematics counselor (I) met with each often
participants involved in the same focal endeavor — the PS YC/STAT 1 04 course. These
constants thus reduce to manageable proportions charting individuals' progress and
comparing their issues and changes through counseling. In psychotherapy, supervision is
considered crucial for helping counselors to identify blind-spots in their
coimtertransference (Kantrowitz, 2002). Because of the number and variety of
participants in this study, as discussed above, I turned to supervision by a person
knowledgeable in counseling psychology to help me become aware of patterns of
relationship with participants that were helpful for some but coimterproductive for others.
The patterns that emerged helped me identify characteristics and mathematical
relational patterns of students who ehcited similar or different countertransference
reactions in me. For example, the motherly reaction that Mulder elicited in me was
different from the one that Jamie elicited. I responded to Jamie with a nurturing,
controlling mothering reaction after I had overcome her "mathematics teachers are
dangerous; stay away from me" transference. On the other hand I responded to Mulder
with an indulgent but thwarted mother countertransference. A key part of my method in
relation to these focal participants was to analyze the countertransference they evoked in
me. It became clear that a focus of study was the student-counselor dyad rather than the
student or the counselor separately.
128
Integrated Cognitive and Relational Analysis
Used in this Study
Analysis of each participant's data and my relationship with him was ongoing and
evolved through the summer. I mapped the mathematical and emotional paths the class,
the individual students, the instructor and I walked, using data gathered principally to be
analyzed and used with students during the study to inform the direction of their
mathematics counseling. Data were also used in post-analysis of the study and in post-
analysis of the effects on participants during mathematics counselmg.
During the course, I studied the audiotapes, observation notes, and student
products continually so as to develop strategic cognitive, affective, and relational
interventions.
Relational Episode Analysis
A focal unit of study was the mathematics relational episode (cf Luborsky &
Luborsky, 1995, and see Appendix E). Each episode was analyzed and triangulated with
other data to determine what it revealed about the participant's central mathematics
relational conflict. In acknowledging that it is her unresolved mathematics relational
conflict that is preventing her desired achievement, it is important to see that this means
that the student is struggling with a conflict of which he is only partly conscious. She is
likely then to say and do things that appear contradictory, but it may be these very
contradictions that reveal most about his central conflict (see Appendix E).To identify
this central mathematics relational conflict, relational episodes were juxtaposed that
revealed insights into each of the participant's three personal dimensions identified by
Mitchell (1988): the mathematics self, internalized presences, and interpersonal
129
attachments. In Appendix E, I provide a discussion of analysis categories and the
procedures and theory used to develop them.
Conversation Analysis
In order to communicate what transpired in a mathematics counseling sessions I
needed to find ways of coding session transcripts to not only indicate transcription
technicalities such as impossible or uncertain transcription, but also to indicate a sense of
timmg, emphasis, and degree of agreement, and to allow for explanation of concurrent
activity. I found some of the conversation conventions developed by Deborah Tannen
(1984, p. xix) and those used by Anne Dyson (1989, Figure 1.1., p. 4) to be usefiil. I
developed some of my own for functions they did not address, modified some where their
Table 4.3.
Conventions used in Presentation of Transcripts
t marks enthusiastic agreement with other speaker
■I- marks hesitant or minimal agreement with other speaker
= marks somewhat agreement with other speaker
(+) marks positive affect in tone of speaker
(-) marks negative affect in tone of speaker
I marks a glottal stop or abrupt cutting off of sound.
NO that is, capitalized word or phrase, indicates increased volume.
{ } includes parallel or immediately contiguous speech of the other person of the
counselor-student dyad. If it is a person other than the counselor-student dyad
speaking, that person will be named.
* * indicates intentional waiting or pause time.
indicates omitted material.
/ / with no text included indicates that transcription was not possible
/ / with text included indicates uncertam transcription.
( ) includes notes referring to contextual and nonverbal information, for example
(laughs), (surprised), or (unconvinced).
[ ] includes explanatory information inserted into the quotation later by me.
[I use conventional punctuation marks (periods, question marks, exclamation points) to
indicate ends of utterances or sentences, usually [marked by conventionally agreed
intonation changes and] slight pauses on the audiotape. Commas [indicate] pauses within
sentence units. Dashes (— ) indicate interrupted utterances (Dyson, 1984, Figure 1.1., p.4).
130
distinctions were too fine for my purposes, and changed some for ease of word
processing (see Table 4.3.)-
Mathematics Behavior and Product Analysis
I developed different coding categories for student verbalizations and behaviors
during the major different in-class experiences. From analysis of the class lecture session
data I developed the following coding categories for student questions, answers, and
comments: (a) timing,'™'" (b) accuracy/relevance, (c) topic,'"™ (d) level of certainty
(affective and cognitive), (e) frequency, and (f) development. From analysis of student
behaviors in problem-working sessions, I developed the following coding categories: (a)
topic/task, (b) seating, (c) tools, '"^ (d) interaction with instructor, and (e) interaction with
researcher. From class exam data I developed coding categories for individuals and for
the class: (a) pre-exam input (class treatment, student reaction and counseling
preparation), (b) student's out of class preparation, (c) errors,""™ (d) trouble- shooting
efforts, (e) instructor grading, and (f) post-exam counseling (see Appendix E for chart
organizers of these coding categorizer schemes). What the analyses of class lectures,
class problem-working, and class exams revealed about the student's personal
mathematics relational patterns and central conflict I considered as the course
Counseling Use of Analysis.
As the study continued I devised ways to integrate data of different types so that I
could use them to counsel participants and clarify their challenges. They were also used
as interventions (e.g.. Survey Profile Summary Sheet, see Appendix B and chapter 6).
With each participant I used insights and suggestions from the supervision discussion of
131
their data into following counseling sessions. The integration of data m supervision
discussion increased my efficacy as a counselor.
Post-analysis of all data, including participants' final evaluations and exams
focused on relational episodes and their cognitive and affective links to relational
patterns. The timing and fit of the ongoing analysis and the researcher's understanding of
each participant's central relational conflict and related counseling interventions were
determined. It was then that the three focal cases for deeper post-analysis and
presentation were chosen in order to illustrate the brief relational counseling approach.
BRINGING IT ALL TOGETHER INTO A CASE STUDY ANALYSIS.
The PSYCH/STAT 104 Class as the Individual Case Context
In chapter 5 I narrate the story of the class as a whole. That narration provides the
basis for analysis of individuals' interpersonal relational patterns in the classroom
context. Since the focus of this study is on the individual counseling and the student-
counselor dyad, the particular value of examining the classroom context lies in the
context it gives for the focal student case studies I present in chapter 6. In addition, when
I conducted a comparative analysis of all participants' mathematics cognitive preparation
and relationality, I expected a student classification to emerge not unlike Tobias' tier
scheme. Tobias (1990; personal communication, March 16, 2001, May 20, 2003)
formulated a tier analysis of science and mathematics undergraduates as they appear to
academic support personnel. Given that Tobias's tier classification is accepted in the field
of developmental mathematics education, it is, in a sense, the null hypothesizes
classification scheme. As such I decided to use it for comparison purposes in describing
the classification scheme that emerged from this study. In addition, and perhaps more
132
importantly, I considered that Tobias's tiers describe students she sees to be increasingly
more vulnerable and in increasing need of academic support in order to succeed. It
seemed advisable for me to take this into consideration in choosing my focal cases: When
I choose from the ten participants, I chose students from vulnerable tiers. I had also to
consider however that my study might identify other criteria that should influence my
choice of focal cases. Tobias describes students in her tiers are as follows:
The First Tier
Students of the first tier are those who enter college well-prepared and confident,
that is, with mathematical power (NCTM, 1989, 2000). They have developed conceptual
understanding, are procedurally competent and are ready for new mathematical learning.
Academic resource centers or mathematics centers frequently recruit mathematics peer
tutors from this group.
The Second Tier
It is mostly the students in the tiers below who come to the attention of academic
support personnel. Tobias identifies students in the second tier as capable students who
have become convinced they "can't do mathematics." She observes that many of these
students have learning styles different from the learning styles favored in the traditional
mathematics classroom. They may be more verbal; they more often favor right-brain and
visual thinking; and they are usually divergent thinkers and global (in contrast with
analytical, c£ Witkin, Goodenough, & Karp, 1967; Davidson, 1983). It is not so much the
mathematics subject matter but the pedagogy that has been the stumbling block for them.
Depending on when and how these students experienced, "I can't do mathematics," they
are more or less mathematically prepared. Almost invariably they believe they do not
133
have mathematical minds. Because most of these students are college bound, however,
they may have struggled through three or even four years of traditional high school
mathematics, often through precalculus.
The Utilitarians ' Tier
Students in the next tier, whom Tobias has designated utilitarians, have in her
words "learned to play a mathematics game." According to her, they are procedural
learners who are competent but not interested in understanding the mathematical
concepts. They may have succeeded in traditional mathematics classes that emphasized
procedural competence but may be unprepared for and resistant to the greatly accelerated
pace and greater conceptual demands of some college mathematics courses. They may
become angry if they fail or do poorly and they may be resistant to suggestions involving
changing their ways of approaching mathematics.
The Underprepared Tier
In high school, many of these students were either not expected to attend, or did
not intend to attend college, or if they did they did not expect to have to do mathematics
in college, so they did little or no algebra. Others attempted some algebra in high school
but were never engaged or did it a number of years ago. Still others "succeeded" in
poorly taught or lower track classes. Whatever the reason, the underprepared have
serious gaps in their knowledge base and often a poor mathematical self-concept.
The Unlikelies ' Tier
These are students Tobias designates as those "we can never reach." They
include students who are hostile and "won't give us trust" (Tobias, S., personal
communication, March 16, 2001). But with the "unlikelies" Tobias hesitates to cite lack
134
of mathematical ability as a cause of their difficulties and poor prognosis. Academic
support personnel typically err on the side of faith in the ability of each student to
transcend her difficulties, given the right combination of circumstances, change of heart,
and support. However, most academic support personnel can point to students who would
not or could not budge. In my experience, students least likely to succeed were those who
are unable to confront their own difficulties honestly.
Choosing the Focal Participants
I chose three students, Karen, Jamie, and Mulder, for deeper case study analysis,
using a number of criteria. My most important consideration was how their mathematics
counseling illustrated different dynamics between the student and me involved in fmding
a central relational conflict and how we used this insight to improve the student's
mathematics mental health and success in the course. I also considered Tobias's tier
analysis, however. With respect to Tobias's tier analysis, Jamie would probably be
classified as second tier and Karen and Mulder had characteristics of the underprepared
and unlikeUes, and, even in some senses, utilitarian tiers. Their stories are presented and
analyzed in chapter 6.
The focal participants were in many ways typical of students in need of support in
their college mathematics course. Jamie and Mulder were traditional college aged, full-
time students and had at least one parent who had a bachelor's degree; Karen was a little
older, a part-time student, and the first in her family to pursue a bachelor's degree. Karen
had previously failed the class and said she had always been poor at mathematics; Jamie
had previously earned a D^ in the class and reported an uneven mathematics history,
doing well or badly at different times. Mulder had not previously taken a college
135
mathematics course, and reported a history of not trying in high school mathematics
classes and just getting by with Cs.
These students reported family theories about their mathematics ability — Karen
reported that hers was a reading and writing type family, Jamie said her mother's theory
was that the women in her family were not good at mathematics, and Mulder speculated
that he was probably capable of doing mathematics because his uncle and father were
"smart." Not only were the focal participants similar and different in their histories,
famihes and attributions, they also appeared immediately typically needy but for different
reasons and in different ways from the perspective of the learning support center. In
chapter 6 when I present the counselor-student dyad cases with Karen, Mulder, and Jamie
I will discuss further these similarities and differences and their significance to my case
selection.
I present these students in the context of the class in the next chapter and zoom in
on their courses of counseling in chapter 6 in order to illustrate the development and
appUcation of brief relational counseling to identifying and treating central mathematics
relational conflicts.
136
' I have given all institutions and locations mentioned in this study fictional names to preserve
confidentiality.
" The Learning Assistance Center has copies of all the mathematics course texts, student study guides, and
student and instructor solution manuals. Instructors are requested to file their syllabi and class handouts
with the Learning Assistance Center so that the peer tutors and I can keep pace with the courses as they
progress through the semester.
"' This class-support tutor has been variously labeled class-link tutor and class tutor. Typically this person
would be a peer tutor (usually an undergraduate who has successfiilly completed the course), but it is not
unheard of for a professional tutor to fulfill this role (M. Pobywajlo, personal communication, January 24,
2000; Petress, 1999).
" All course numbers have been changed to ensure confidentiality of the institution in which the research
was conducted. The first digit used here is designed to indicate the level. For example, the number 104,
with 1 as the first digit is a first year college level course. The course is described in the course catalog as
follows:
PSYC/STAT 104 (freshman level)
Design, statistical analysis, and decision making in psychological research. Substantive problems
as illustrations of typical applications and underlying logic. No credit for students who have
completed BUS/STAT 130 or BIO/ST AT 105 (fulfills quantitative reasoning general education
{core} requirement). Special fee. 4 cr. (From the on-line Brookwood State University Course
Catalog)
" The class was scheduled for Monday and Wednesday evenings 6:00 p.m. to 8:20 p.m. on the second floor
of the Riverside campus building and ran from Wednesday, May 31 through Wednesday, August 2, 2000.
" The names of all persons in this study have been changed to preserve their anonymity.
™ The average attrition rate (drop, withdraw, fail) from 1995 through summer 2000 for PSYC/STAT 104
was 26.6% over all. This breaks down to an average 3 1 .4% attrition rate for Fall/Spring semester courses
and a much lower 14.75% attrition rate for the summer courses (archived grade reports, Brookwood State
University).
"" In keeping with my former practice, as this is an even numbered chapter I use "she," "her," and "hers" as
generic third person singular pronouns.
'" This course was taught by an adjunct psychology professor.
" The teacher tells about and the students are expected to passively absorb the new mataial. See also
chapter 2.
" See chapter 2, endnotes xvi and xvii.
"" Although Australians are considered racially and ethnically similar and are generally well-liked by
Americans, there is an assumption that America and things American are bigger and certainly better than
things Australian, and that Australia and therefore Australians are cute but inconsequential in anything that
matters and are expected to agree and admire. I thus struggle with belonging in the U.S., with maintaining
an Australian identity, and with feeling "less than" because of who I am. On the other hand, because 1 am
not a vA\\te American, I am not implicated in the oppression of disempowered groups here (though 1 am in
my country of origin). Now after 23 years here, I am more an Australian American than an Australian but
continue to have a coimection with people who for whatever reason do not feel that they belong
comfortably because of who they are.
137
'"" I realize that merely belonging to a disempowered group or being married to someone from such a group
does not necessarily mean that I understand the challenges others from the same group face, nor how to
encourage them to achieve their potential nevertheless. Indeed, for example, if one is at a low level of
identity development, one is likely to buy in to the majority's negative assessment or low expectations of
one's group and/or be trying to distance oneself from one's group and be trying to be like the majority
(Ivey, Ivey, & Simek-Morgan, 1993; McNamara & Rickard, 1989).
Americans have particular frouble with SES — few admit to having a low SES, that is, to belonging to
the working class — and appreciation of values and cultures of the working class are rarely espoused
(Frankenstein, 1990). The deficits are well-known: students from low SES backgrounds with parents who
have not gone to college are less likely to go to college themselves or to succeed in college if they do. The
Federal TRIO grant program provides extra support for such students in post-secondary education. My
husband's experience of discrimination because of his SES background continues and we struggle with
appreciating each other's different class sfrengths and weaknesses. Again identity developmental level (in
this case class identity) is an issue, as is also an understanding of what might be involved (for my daughters
and for my students) in learning about and negotiating the culture of power — the predominant culture in
society and in academia (Delpit, 1988).
"^ The summer 10-week session allowed for 4 hours and 40 minutes per week for 9 weeks and one class of
2 hours and 20 minutes in the first week. Ann did not hold class on the Monday of the week of July 4. The
total class time available was then 42 hours. In confrast, during a regular semester she would have between
45 and 48 hours of class time to cover the same material.
From Class 2 on, I used a music-scale like form and class layout form to record professor-student
interaction for some portions of the lecture or lecture-guided problem portions of the class. During the class
I noted the time at regular intervals during the class. I used these forms in subsequent classes and I
developed an informal 2x2 charting procedure for diagramming interactions among students with each
other and with Arm during the problem-working portions of class. After each class I tape-recorded my
reflections on the class, professor, students, and on myself and my plans for the next class (I have archived
these notes and recordings). See Appendix C for copies of the forms.
™ Parents' college experience came up incidentally in counseling with some participants, but because I had
incomplete data, I surveyed participants in November 2000 by e-mail. Of the six who responded, Karen,
replied that her parents had not attended college. Lee's mother had an associate's degree from a technical
school but neither parent had attended a four-year college. I believe that of the others who did not respond,
Robin's parents (and possibly Kelly's) had not attended college.
""^ I sought approval for conducting the research from both the Lesley University Committee on the
Use of Human Subjects in Research, and from the Olfice of Sponsored Research's Institutional
Review Board for the Protection of Human Research Subjects for the state university system to which
Brookwood State University belongs, and was granted that approval. I have archived the official
approval documents I received.
^"' In this and following chapters I use the term "participant" to refer to students in the PSTC/STAT 104
class who participated in individual mathematics counseling with me.
"^ I have archived all original completed forms.
^ Aim followed department policy in not returning exams to students. Instead she briefly went over exams
in class with students and had them returned to her. However she agreed to allow me custody of exams to
use with participants in counseling sessions and gave me all students' exams at the completion of the
course. I have archived these materials.
™ I determined that the primary use of new and adapted instruments would be descriptive; early use for
individual affective, cognitive, and relational pattern recognition could be invaluable in helping the student
138
and me become aware and prioritize interventions. Already normed instruments might be usefial to develop
realistic goals in the context of a course. Post testing using the instruments should give students indications
of change in the factors surveyed, but the most concrete indicators of effectiveness of the mathematics
counseling, for the students at least, would be improvements in exams or quizzes. Causal factors for change
may be difficult to determine in such a study so hypothesized relationships among factors will need fijrther
study.
'""'In the past, research in psychotherapy into outcomes that involved pretreatment and posttreatment
experimental designs resulted in findings that do nor account for the real complexity and non-linear
experience personal processes. Research into process that involved time-sampling strategies and averaging
of readily quantified units such as grammatical categories of speech produced findings that seemed
disconnected fi"om the actual clinical experience and the theory behind the treatment. In any attempt at
quantitative research, the problem of quantifying the "relationship between therapists and patients" arises
but the fact is that this relationship "regularly appears in reviews as an important moderator of treatment
effects" (Russell & Orlinsky, 1996) (p. 713). More recently, "researchers have turned to systematically
conducted naturalistic studies to assess treatment effectiveness and clinical significance" (p.710). There is
an important trend for researchers to "sift through the complexities of interactional and relational meaning"
(p.71 1) and outcomes are being seen more as parts of a process rather than different phenomena.
'°^ Timing is judged in terms of the extent to which the student's verbalization is linked in a timely
manner with the instructor's utterance. For example, on a number of occasions Robin answered Ann's
question with the correct answer to a previous question; her timing was off
""^ Subcategories of topic developed were: (a) current content (mathematics; application; personal), (b)
course strategy, and (c) grading.
"" Subcategories of tools developed were: (a) text, (b) items provided by instructor, and (c) student
provided aids such as calculator, notes, . . .
■"^ Subcategories of errors developed were: (a) defining the problem: concepts, (b) planning the solution:
procedures, (c) carrying out the solution: algebra, (d) carrying out the solution: arithmetic, (e) conclusion:
Checking and reporting
139
CHAPTER V
AN ACCOUNT OF SUMMER 2000 PSYC/STAT 104 CLASS
In this chapter I will briefly reintroduce the students, introduce the physical
setting of the classroom, and then discuss features of the class and teacher that were
salient to the mathematics mental health of the students. Those include the curriculum
and the text, the instructor's pedagogy, her view of statistics and mathematics, the
emotional and mathematical climate established in the class, and how the students
interacted with the mstructor and with each other. 1 will show how these features played
out in the first few classes of the term and several typical or importantly different classes.
From that picture, I will discuss each participant's experience of the class in relation to
mathematics counseling interventions, highlighting the interactions among students'
relational patterns and the classroom dynamics.
Students
The class consisted of 8 traditional aged students (18 through 25 years of age), all
but one fiiU-time. The remaining 4 (5 if I include Ellen) non-traditional students who
ranged in age from early thirties through mid- forties were part-time bachelor's degree
students. Seven (or possibly eight) of the students were enrolled at Brookwood
University; three were enrolled at State University; and the other two were enrolled at
private colleges. All for whom I had data (I do not have that data on Catherine, Ellen,
Floyd or Mitch) were working during the summer, five at vocational positions they
maintained all year round, and four at temporary summer positions (see also chapter 4,
particularly Table 4.1).
140
Students' had differing degrees of familiarity with the college mathematics
courses. All had completed at least a year in college. Mulder and Robin were the only
students in the class who had not taken a mathematics course in college. Of those who
had taken college mathematics courses, only Autumn, Catherine, and Lee had been
successfiil; the rest had either failed or earned Ds.
Three students were repeating PS YC/STAT 1 04 because they had previously
failed it in the summer of 1998. 1 found out after the study was completed that Jamie was
repeating it because of a D^ on her first attempt in freshman year, not an acceptable grade
for a course in her psychology major. Eight of the students were required to take
PS YC/STAT 104 for their degree programs: Robin and Brad for nursing; Pierre and
Catherine for biology; and Floyd, Ellen, Jamie, and Karen for psychology. For the other
six, the motivation for taking the course was less clear. Two began the class with a
psychology major in mind, thus requiring PS YC/STAT 104, but one changed her mind
during the summer. The other began to waver on a psychology major, making it unclear
whether PS YC/STAT 1 04 would be necessary for her. Another was taking it for elective
credit to transfer. Mitch was taking the course to redress the messy situation of having to
repair his GPA because he had failed it before, even though he said he believed that he
should not have taken it in the first place. Kelly only needed to pass any college level
mathematics course, something she had thus far failed to do.
Only two students knew each other before the class began: Lee and Mitch. Mitch
was the only student in the class who knew Ann the instructor, outside of the classroom
setting; he was a member of Student Government for which Ann was faculty advisor.
141
Physical Settings
The class usually met in a room on the second floor of the renovated former mill
building that was the Riverside campus. The only classes not conducted in this room
were Class 5, the MINITAB computer orientation class run by Aim and Pat, the computer
lab assistant, which was held in the computer lab at the Greenville campus. Classes 4 and
9 when Exams #1 and #2 were given were held in a classroom across the hall which had
individual seats and attached desk-tops. Class 10 was not held as a class so that
MINITAB project partners could meet during that week.
Otherwise all classes were held in the same room. The space was almost entirely
filled with six 2.5 by 5 foot tables arranged to make one 5 foot by 15 foot table, with 14
or 15 chairs arranged around it (see Figure 5.1).
I found that, although there was considerable variation in students' choice of
seating, there were patterns that seemed to be connected to relational alhances, to
technical constraints (e.g., Pierre's audio-taping), to the timing of a student's arrival, and
I I chair
2.5'x 5' table
I><or left
akboai
d
-
right
back
Figure 5.1. Room and fiimiture configuration for PSYC/STAT104 class, second floor.
Riverside Center, Brookwood State University, summer 2000.
142
to Other less obvious relational factors. Ann had previously taught only in classrooms
with individual seats and attached desks facing the front and the chalkboard. She reported
that her students consistently sat in the same seats. In this setting she was surprised by
what she perceived to be almost random seating choices by students.
My own choice of seating was largely driven by my desire to observe the class
and individual members most strategically. I know that my seating choices affected
students and their experience of the class and also undoubtedly affected what I saw of the
class (particularly during problem-working sessions). In Appendix F I detail and discuss
seating choices — both the students' and mine.
PSYC/STAT 104: COURSE ELEMENTS AND EVENTS
Class Presentation Organization
Classes began at 6:00 p.m. The first part of a typical class consisted of Ann's
presenting theory or as she said, "the concepts," with the overhead projector and the
chalkboard. Aim always stood at the front of the room during the lecture portions of the
class, moving from her notes on the table to the board or the overhead projector and back.
The lecture took as long as the whole class period (i.e., from 2 to 2 hours and 20 minutes)
if there was lecture guided problem- working interspersed but more typically went until
break at around 7:00 or 7:15 p.m. (i.e., V* hour to 1 'A hours). Following the lecture. Aim
usually handed out worksheet/s requiring the application of the theory just presented.
She moved around the classroom checking over students' shoulders to see if they were on
track. If a student seemed to be struggling, Ann would sit with him. She usually carried
the worked solution so she could tell or show the students where and how their solutions
143
differed. When she had these worked solutions she would give me a copy so that I could
help the students in the same way."
Ann used arithmetical accuracy as a quick indicator of whether a student was
proceeding correctly. When I had my graphing calculator with me I would use it to enter
and analyze the data. Ann was unpressed with this as a quick way to fmd the
arithmetically correct answers when she hadn't previously worked them out.
When more than one topic was being covered during a class period, Ann typically
lectured on one of the topics and had students do a worksheet that was sometimes lecture
guided and sometimes done with her roving help. She then proceeded to lecture on the
next topic, go to another worksheet and so on (Class 2, for example). During the lecture
portion of the class Ann did not usually work problems on the board. Instead, she had the
students use her worksheets, their texts and her over-the-shoulder help to work them out,
sight unseen, during the problem-working portion of the class. In an interview, she told
me that she this was a preferred method because it forced the students to find out how to
do each problem themselves (Interview 3).
The Curriculum and Textbook
The text Ann used was Understanding Statistics in the Behavioral Sciences (5*
edition), written by Robert R. Pagano (1998). It is an introductory non-calculus based
statistics text using a typical sequence. The book treats descriptive statistics in the first 6
chapters, followed by inferential statistics in the subsequent 12 chapters. Probability,
random sampling and hypothesis testing concepts introduce the inferential section,
followed by a "cookbook" of parametric and non-parametric tests. Ann's curriculum
covered all but chapter 17 of the text, although some chapters were only partially
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covered. Ann lectured from notes that kept quite closely to the text; at times she dictated
directly from it.
Students were expected to read ahead in preparation for the lecture and to practice
procedures and solutions after the class.'" First a narrative introduction explains the
theory, next step-by-step procedures are provided, and worked examples are given, and
finally problem sets are assigned in each chapter, in that order. Material for the worked
problems and problem sets is situated in realistic behavioral science settings.
The first stated goal of the course is to familiarize students with the tasks and
tools of descriptive and inferential statistics so that when they take a subsequent research
methods course, they can assess others' use of statistics and begin to learn to design their
own studies. It is not expected that they do these things in PSYC/STAT 104; the
problems posed in the text and in problem-working sessions have all been worked
through to isolation of variables. There are no open-ended questions or non-routine
problems. The text contains no projects to give students experience with the process of
conceptualizing a hypothesis through data-gathering; the assumption is that these will
come later in the research methods courses. Nevertheless, the department had designed
MTNITAB computer projects'^ where students analyze given data sets and learn to
interpret results. Aim also teaches the Research Methods m Psychology (PSYC 220)
course and she told me that her expectations of how much was retained from
PSYC/STAT 104 were fairly low. If students have developed a basic idea of the
rudiments of descriptive and inferential statistics and their differences, she is prepared to
re-teach other pertinent PSYC/STAT 104 material during PSYC 220 (personal
communication, September 12, 2000).
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In order to "increase understanding and critical thinking about the statistics that
the media presents" (PSYC/STAT 104 Syllabus, see Appendix G) Ann raised some
common misconceptions around statistical ideas and discussed these briefly with the
class. She took time in Class 1 to introduce such a problem using a misleading
advertisement. I took this to indicate that she considered discerning misleading statistical
information as an important theme for the class. In Class 6, Ann distributed an article"
that claimed, that an increase in excise tax on beer would "lead to" a reduction in the
gonorrhea rate amongst teenagers, based on a correlational fmding. She pointed out to the
class the misattribution of a causal relationship, where a possible link was all that could
be claimed. Lee was the participant who showed the most curiosity about these issues and
was very eager to spend more class time than was given to explore them
Pedagogy and Student Responses
Ann's approach to mathematics teaching cannot be easily categorized. She did not
demonstrate how to do procedures; instead she employed a student-centered exploratory,
problem-working approach to mastering them, expecting that students had the capacity to
do it, with herself as coach. This approach would be considered pedagogically sound
from a cognitive constructivist point of view. Because of class time limitations and the
applied statistical focus of the curriculum, a compromise had to be made between
presenting conceptual links among and within procedures to the whole class and giving
students the opportunity to struggle with procedures so that they could master them. Ann
chose the latter alternative but helped students with conceptual questions and difficulties
on an individual basis during problem-working sessions.
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Lee's experience illustrated the difficulty an under-confident but conceptually
oriented student may have in a course like PSYC/STAT 104 even when the instructor and
the mathematics counselor are affirming of a conceptually curious orientation. Ann
admired Lee's inquisitive approach and her penetrating questions about the statistical
concepts but at times Lee was not able to articulate her question clearly or there was not
enough class time to pursue it. Lee's initially sound understanding that correlation cannot
be assumed to imply causation as well as her sense of Ann's ability to provide a secure
mathematics base were each undermined by her perception of Aim's and the text's
position. ^'
Ann's non-directive worksheets provided students with in-class experience of
working through problems on their own^" with her guidance (see Appendix G). This
process often challenged and even frustrated students. At the same time, each student did
experience successful completion of at least one problem of each type. I made note to
discuss in counseling both the appropriateness of their heightened emotions under such
circumstances and also the pedagogical benefits of this approach. Jamie claimed to be a
visual learner and said she found the worksheets very helpful, especially the ones with
the columns, because she felt they complemented her learning style. Mulder also
preferred to use visual learning approaches and found the worksheets helpful but he used
them unconventionally and studied by visualizing his successfully worked examples on
them.
Students with sound mathematical foundations (e.g., Lee) responded well to the
challenge of this approach and at times went beyond mere procedure on their own.""
Students whose mathematical foundations were poor (e.g., Karen and Kelly) found the
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exploratory, problem-working approach difficult and became anxious. Used to having
procedures demonstrated, Karen'" felt abandoned and helpless when she was expected to
negotiate such procedures on her own. Both Karen and Kelly complained that Ann had
not been "thorough" in covering the material before the first test. It may have been the
absence of familiar solution demonstrations they complained about. I was able to support
some students as they worked through ihistrations with Aim's exploratory approach (e.g.,
Karen). With this help, they found that they eventually benefited from having to struggle
to master the procedures on their own.
Students reacted differently to what appeared to them to be a laissez-faire
approach to linking the statistical concepts with their underlying mathematical basis and
to understanding the formulae to number to concept links. All students in the class except
Lee, Robin, Catherine, and perhaps Pierre were used to following a procedural approach
to mathematics. Because Ann allowed students to use formula sheets in exams with some
verbal identifiers and charts, there was a reduced load on memorization of formulae but
an increased call for understanding differences and similarities among formulae. Because
the concepts were not uniformly coimected to procedures during class, some students
found learning new formulae and procedures to be onerous and memory-dependent,
because they seemed new and different rather than being rooted in previously mastered
material.
Even though the more procedural learners were used to this experience, the fact
that they depended on their memory of dimly understood, individually mastered
procedures kept them vulnerable. Generally they were without the mathematical tools for
monitoring and checking and this kept them anxious and dependent on factors they often
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felt were beyond their control. Students like Karen and Brian tended to approach each
inferential test as if it required an entirely new procedure — another observation I used to
inform my mathematics counseling.
Conceptually oriented students found Ann's indirect approach to the conceptual
linking difficult in some ways, especially if their confidence in their own ability to
discover these conceptual links was shaky. Lee was the most vocal of the participants
about her difficulty with this approach but she struggled to make connections herself —
she attended study groups and met with me to explore and seek answers to her questions.
Lee spent little time doing homework on her own (20 minutes a week, see Appendix H,
Table HI) and expressed high anxiety. This may have been related to her difficulty in
acquiring a secure conceptual base more or less by herself
Pierre used an opposite tactic to try to gain a conceptual understanding of the
material. He spent many hours (17 per week at least, see Appendix H, Table HI) studying
the text and other materials he got from Ann and meeting frequently with Aim and me.
This broad-based, over-inclusive approach was done at the expense of mastering the
procedures to be tested and, therefore, at the expense of earning a good grade (at least
through Exam #3).
A challenge for me in counseling was to support students' strategic pursuit of the
conceptual links that were not provided in class and to help them embrace rather than
resist the real benefits Ann's approach afforded them in mastering the material.
Mathematical and Statistical Challenges
Aim was confident in her grasp of the statistical concepts, but she was less
confident of her grasp of the links between the statistical concepts and the mathematics
149
used to explore them. The mathematical challenge of this course Ues principally in being
able to understand, decode, and link data, and information about data, with appropriate
symbols or formulae, and in being able to adapt and apply mathematical understandings
to an unfamiliar problem situation. For example, the order of operations agreement
requires that to compute DX^ one must square all the Xs first before one adds them (i.e.,
work exponents before multiplication or division, which is, in turn, worked before
addition or subtraction), whereas for (SX)^ one must add the Xs first and then square the
result because of the parentheses that require attention to operations inside before doing
anything else (essentially allowing one to cut in line). In algebra an equivalent situation
might look like X^ + Y^ + Z^ where X = 2, Y = -3 and Z = 1 compared with (X + Y+ Zf
when X = 2, Y = -3 and Z = 1. If order of operations is not made explicit, students often
make errors that they would not if they were simply doing algebra. Because the text does
not make explicit the equivalencies despite the unfamiliar look, I realized that I should
include that discussion in counseling sessions.
Statistics and the Use of Already Derived Formulae
This course required very little" manipulation of algebraic variables as is typical
in a non-calculus based introductory statistics courses; there was a heavy emphasis on the
use of already derived formulae. A conceptual approach to instruction might involve
explormg the forms of these formulae in relation to their derivations and uses. Formulae
such as the one used to find the percentile rank of a score (see Class 2) comprise all the
steps of a multi-step process in one formula; this could be too complex for algebraically
challenged students because of the intricate interactions among letter symbols and
operations. I believed students might understand the finalized complex formulae if they
150
explored and mastered the process using estimation, proportional reasoning and
dimensional analysis. In Study Group 1, that is how we approached it (see below). We
extended beyond using a formula for a percentile rank to fmd its corresponding score
using the text's step by step approach, and it seemed that the work in Study Group 1 did
complement the text and class work and forged conceptual connections for some of the
students.
Early in the course Aim showed a preference for using an empirical (rather than
computational) process and formula for finding the standard deviation. She said she
wanted to help students develop a sense of how and why the formula was derived and is
used. She had students work the procedure in Class 3, but time constraints and most
students' procedural orientation led to a predominantly procedural focus for most
students. In Classes 6 and 7 the concept of deviations and squared deviations from the
mean reappeared (now in the context of two rather than one variable in correlation and
regression analysis). Now Arm had students use the computational formula rather than
the empirical one, and did not link the idea to students' prior work on deviations. This
was perhaps because it was now being applied to two variables and between the
variables, in two dimensions rather than one. Although the standard deviation concept
was the same, the uses and interactions may have been more complex than Ann felt the
students needed or had the time to explore. There were other mathematical themes that
Ann did not point out to students such as the fact that the fiinction of all the z and t
statistic formulae is the same."' I resolved to address these strategically in counseling.
For example to demonstrate the equivalencies of the t and z formulae, I decided to use
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comparative diagrams (for all) and algebra (only with students who had a level 4
understanding of the variable on the Algebra Test).
Multiple Uses of Letter Symbols
The multiple uses of letter symbols seemed to be the cause of much confusion
even for relatively algebraically confident students (as noted below in my discussion of
Class 2, Exam #1, and Class 13). These different uses are not usually discussed m
application classes like this one, yet they are particularly salient in introductory statistics
courses because of the heavy emphasis on the use of already derived complex formulae.
Philipp (1992) notes that current teaching practice in algebra does not address these
different uses of letter symbols explicitly. In introductory statistics courses instructors do
typically discuss the symbol classifications of random variable (a true variable),
parameter (constant for a particular population) and statistic (constant for a particular
sample). What Arm did was identify names and meanings of important letter symbols as
the text did. She required accurate memorization of these on the tests, giving up to ten
percent of test grades to symbol identification and meaning. However, she did not discuss
classes of symbol, nor draw attention to the multiple uses in one formula, or how the
symbols differed in their uses, and how they were related to the mathematical content and
each other."" Because Ann did not provide secure base support in this for students to
explore and develop these cormections, I took it to be part of my complementary teacher-
parent role to do so in mathematics counseling sessions.
Group Learning
From the first class, Ann provided opportunities for students to work together
both mformally and formally. The ways students did or did not take advantage of these
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Opportunities or form alliances independent of Ann provided important data about their
mathematics relational patterns that informed counseling. Although there was no effort to
organize students to work in groups in class, a paired getting-to-know-you interview in
Class 1 and pairing up to work on and present the MINITAB computer modules at the
second to last class presented opportunities for students to form study alliances. Aim
encouraged students to use the class contact sheet with e-mail addresses and phone
numbers to contact one another. The only pair of computer project partners to work
together on other aspects of the course was the Lee-Mitch pair who had known each other
before the class began.
Whether students worked together during the problem- working portion of each
class seemed to depend on where and beside whom they were sitting and on their
established interpersonal relational patterns. Lee (a social learner'^") initiated and
maintained contact with Mitch; Robin with Brad (both older and nurses) worked together.
They formed pairs that fairly consistently sat together and worked together on the
problems. Lee and Mitch were also MINITAB computer project partners. Mulder (who
was also a social learner) would work with whoever sat beside him unless it was a loner
who would not engage. Autumn, Karen, Catherine, (and Mitch if he weren't with Lee)
Jamie, and Pierre were all loners, rarely working with others, especially other students.
Autumn, Karen, and Catherine (and Mitch) seemed to be loners by choice (voluntary
loners), but Jamie and Pierre worked alone more because of constraints they seemed to
feel precluded choice (involuntary loners). Jamie and Pierre appeared to want to be more
involved with others.
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During problem-working sessions, in particular, how these distinctions played out
in class was affected by student seating choices and apparently affected the relative value
students received. For example, when Mulder, a social learner who found the lectures
difficult to process and relied on the problem-working session, sat between loners
Autumn and Pierre, he worked on his own (Class 3). That he did poorly when that
material was examined in the first exam may have been related. These distinctions also
seemed to affect the amount of support students received from Ann during problem-
working sessions. For example, because Jamie rarely used body language that would
invite Ann's intervention, such as moving to allow Aim to see her work as she went by,
Ann checked her work and offered her assistance less than she did the other students in
the class (cf video-recording of Class 16, archived). Because I observed how students
related (or not) with Ann (and me) in the classroom and I discussed with the student in
counseling, what that revealed about their teacher attachment patterns, some participants
were able to recognize and modify such behaviors they now recognized as
counterproductive.
My analysis of student seating choices indicates that, contrary to Aim's perception
of randomness, most students were quite consistent in their seating choices and that my
choices did not appear to influence theirs. My seating choice did affect the level of
interactivity of my immediate neighbors during problem- working sessions, however,
especially voluntary loners like Karen who would not work with her peers but would
work with me. Seating choices of those who were not loners did seem to be related to and
affected the level of collaboration during the problem-working sessions (see Appendix
F).
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Classroom Emotional Climate
The PSYCH/STAT 104 class had a generally positive emotional climate. The
course was taught in a manner that had the potential to develop, maintam, and repair
attachment relationships between teachers and students and between students and
mathematics. Ann provided the elements of such an environment, but that did not mean
that each student was aware of it nor received it as a benefit.
In Ann's course, Jamie and Karen, for example, each of whom came to the class
with a history of mathematics classroom experiences that had negatively affected them,
did not initially perceive Arm's classroom as safe for them and could not benefit from her
positive offerings. In counseling sessions, I saw an aspect of my role as helping them
investigate whether this classroom climate might be different and even positive for them.
Dimensions of a positive emotional climate emerged as (I) the creation and
maintenance of a positive interpersonal relational climate and (2) the creation and
maintenance of a positive classroom mathematics climate.
Creation and Maintenance of a Positive Interpersonal
Relational Climate
There were three crucial elements to the positive interpersonal relational climate
that Ann created in the class: herself as a secure teacher base, the classroom as a secure
base, and fairness in testing.
The teacher as secure base. Ann provided herself as, a good-enough, emotionally
secure base for her students so that they can find acceptance and reassurance when they
are uncertain, as well as the courage to move out to explore without fear of censure for
going away or for making errors. Arm set the scene in the first class by self-disclosing;
she described her own struggles with statistics learning and also how she managed to
155
overcome her uncertainties."'^ Ann did not hesitate to consult with me in class if she were
uncertain on the mathematical material, modeling an open exploratory approach that did
not require students or even teachers to have perfect understanding. 1, too, openly
expressed my puzzlements.'"
Another feature of Ann's approach was that she did not call on individuals for
responses to questions during the lecture discussion. I drew Jamie's attention to this
during counseling and she was then able to acknowledge to herself that, in this class at
least, she was safe. She came to realize that she did not have to worry that the instructor
might call on her.'™' This recognition freed her to relax and even to ask a question of Arm
in class.
Although Ann made herself available to meet with students and to help them with
the course material (because she believed mastery itself would allay anxiety) she did not
believe it appropriate for her to get involved directly with students' emotional problems
with mathematics or the class. She neither invited nor required student disclosures.
The classroom as a secure base. Aim modeled and monitored interpersonal
classroom behaviors to ensure that all students were safe. The way Aim deak with
incorrect or half-correct responses during the lecture sessions set the tone. She considered
the response, found what was reasonable in it, responded, and moved on respectfully.
Whether a student perceived this positively depended on his""" already established
interactional patterns. Karen gave an incorrect response to a question during the first class
but despite Ann's respectful response, subsequently responded only to questions
requiring a non-mathematical response.
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There were no incidences of student to student disrespect during the course. Lee
did object to the fact that (in line with department policy and for statistics education
purposes) Ann distributed a histogram of exam scores after each exam. Because it was
relatively easy to identify each person's grade given the small class size, Lee felt that this
was not respectful to students who did not wish to reveal their grades.
Fairness in testing. Aim seemed to make it a priority to be explicit and fair but
Karen and Kelly, for example, did not see that. This became a focus in counseling
because while they were extemaUzing their difficulties and scape-goating the instructor
they were not taking the control they needed to negotiate the course.
Before each test Aim was careful to give a study handout with a Ust of the
symbols that would be tested and specific homework problems from the text. She also
handed out solutions to even-numbered problems from the teacher's edition of the text
(for an example, see Appendix G). More importantly she made sure to teach everything
that she tested; in particular she made certain that each student completed each type of
problem correctly in class. Ann allowed unlimited time as well as the use of a formula
sheet on tests. She provided helpful organizers, including the Ust of six steps of
hypothesis testing, so that students could incorporate this into their formula sheet (see
Appendix G).
If a scheduled test time was inconvenient, students could take exams early, though
not after the scheduled time. Ann's optional comprehensive final could also replace one
missed exam and could be used to replace a poor exam grade during the course.'"'" The
relative proportions of the grade allotted (Ninety percent of the grade was earned from
exams and 10% from computer analysis projects.) seemed to accurately parallel the effort
157
and emphasis required in the course. The heavy weighting of exams may have
contributed to the class' collective mathematics testing anxiety remaining considerable: It
changed ixom 2.9 (on a scale of 1 : not at all frightened, to 5: very frightened), to 3.0 on
that scale. Only two individuals' testing anxiety levels fell substantially during the
course while three individuals' anxiety rose substantially and the others' remained
substantially the same (see Appendix C for the surveys and Appendix H, Table H3 for
student changes). One whose anxiety abated somewhat still expressed elevated anxiety
(3.6); in fact of the 9 students remaining in the class, 7 expressed anxiety levels of 3 or
above.
The Creation and Maintenance of a Positive Mathematics Climate
Aim provided herself and the classroom as a secure relational base, but even that
was not enough to create a good-enough mathematics classroom climate. Her attitudes
towards her students' ability and potential to learn mathematics and the way she taught
mathematics and supported students were also essential. In particular, her belief in every
student's potential to master the statistics (given adequate support) and her promoting the
authority of the mathematics over her own authority were key. This was evident in Ann's
willingness to acknowledge her own uncertainties about the mathematics and refer to
others (me in this case) who could not only help her understanding but also was there to
support her students mathematically.
In Ann's assessments of students' likelihood to do well in this class, her central
consideration was whether their mathematics background was adequate and whether they
would apply themselves sufficiently to succeed. She made no trait judgments that might
have locked students mto doing poorly because she expected it. She did not believe that
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some people could do mathematics and others not. Ann's expectations seemed to be
influenced by students' classroom behaviors"'" and by a constellation of age, gender, and
particular major. For example, she (and I, initially) expected Robin, an older (in her 40s)
nursing student who often appeared flustered and confused in class, to have trouble and
perhaps do poorly. On the other hand Brad whose classroom behavior was confident and
apparently relaxed, Ann expected to do well despite his being an older nursing major
(Interview 2). In each case Ann's expectations were challenged by the student's
achievements — Robin did well while Brad struggled to get C~s. However, I never
observed that Aim's expectations affected how she related to or graded a student.
This apphed course was taught by psychology rather than mathematics faculty,
and because of that an important complementary role emerged for the mathematics
counselor. When uncertain about the mathematical bases for the statistics, as noted
above. Aim was very open about drawing fi-om my mathematical expertise in class. Her
pedagogical approach, especially her use of problem- working sessions, reinforced the
statistics/mathematics as authority rather than the instructor. Mathematics counseling was
pivotal in complementing and supporting Ann's mathematics teaching because of varied
student comfort with and responses to it at least initially.
An important part of students' developing a sense of mathematical safety was the
support offered outside the classroom, especially for those whose low confidence made it
difficult for them to study and practice on their own. Ann repeatedly offered extra time
and help to students. Because I was so available and she was at State University in
another capacity several days a week, most students saw me more than her outside class
but they were well aware of her openness to helping them. She stayed after class to help
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anyone who came and helped out in several study groups before tests. All six students
who filled in the Class-Link Evaluation at the end of the class responded positively to my
contributions as a class-link tutor but only Lee, felt that Ann had relied too much on me
to give support to students (see Appendix C for the form; student responses are archived).
CHRONOLOGY OF PSYC/STAT 104, SUMMER 2000
The class chronology underscores the significance of understanding students'
mathematics relationality within the whole class system. To describe class process,
interactions, and student outcomes, I will describe in detail the first three weeks of the
course through the first test and I will discuss how this was the first of several cycles of
class, study group meetings, individual counseling sessions, that culminated in an exam. I
will then sketch key events that occurred during the remainder of the course (see
Appendix I for a complete calendar of events for the class).
The First Cycle through Exam #1
Class 1
The first class consisted of introductions along with an overview of the syllabus,
course schedule, and assessment procedures and an interactive lecture on the first chapter
of the text. All of the 13 students Aim expected were there except for Mitch who would
be at the next class.
Aim began to establish the relational climate that accepted struggle and
acknowledged the importance of collaboration and mutual support by self-disclosing her
own statistics anxiety (see endnote xiv), by asking the class to pair off, interview each
other, and mtroduce his interviewee to the class,'™ and by organizing an exchange of e-
mail addresses and phone numbers.
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After break, Ann used the overhead projector to show an advertisement for paper
towels that used misleading graphics and numbers to compare with its rival. Lee was
quick to respond accurately to Ann's questions about it. During the lecture, Ann directed
her questions to the class as a whole, not to individuals; if there was no response within
two or three seconds, she answered them herself Ann asked and then explained what
statistics was, using the defmition given in the text: "A way of organizing, summarizing,
and understanding data." Data is "information collected and generally understood at a
numerical level." All of the students wrote the defmitions in their notes. Next the class
discussed the scientific method and Mulder responded by referring to his research project
on caterpillar aggression.
The classroom interactions proceeded in the following pattern: Ann presented a
concept, she asked a question about it of the whole class, a student or group of students
responded (or Aim when there was no quick student response). Aim responded to student
responses, and then cycle began agam. Ellen, Robin, Mulder, and Brad responded during
this discussion. Robin seemed to have some concepts confused but Mulder, Brad, and
Ellen appeared to have a good grasp of the big ideas. Karen responded to a question
incorrectly. Ann dealt with this by respectfully considering Karen's answer, correcting it,
and moving on. Neither Catherine nor Autumn offered any responses but they appeared
to be actively and knowledgeably engaged in observing the interactions and they were
taking notes. Pierre also did not offer any reactions but he was working at his notes and
attending to the interchange. Jamie alone did not seem to be involved. She kept her eyes
lowered, not making any eye contact. She did take notes but at times I wondered if she
161
were asleep. I did not observe any interactions between students during this class other
than the paired mterviews.
Class 2
The plan for the class was to cover chapters 2 and 3 focused on basic
measurement concepts and frequency distributions including fmding percentile ranks and
percentile points.
Before beginning the lecture, Ann introduced me as researcher and academic
supporter (class-link) for the class. I invited students to a weekly study group before each
Wednesdays' class in room 207, where the class met. Mitch had joined the class but Ellen
was absent.
In the first half, the class worked on a teacher-directed, lecture-guided data sorting
exercises, classifying and sorting different types of data according to measurement
scale™ and fmding the median, mode, and mean of a set of ratio data (time in seconds for
20 rats to run a maze). Next came sortmg data into a grouped frequency distribution. Aim
provided worksheets for these exercises and she used the overhead projector to gather
class responses. The problem-working interactions were aknost exclusively between
mdividual students and Ann, rarely among students.
Just before break, according to prior agreement with Ann, I administered the
surveys I had prepared — the Mathematics Beliefs Survey, the Mathematics Feelings
Survey, and a short mathematics background survey (all class surveys are in Appendix C)
and invited volunteers for the individual mathematics counseling sessions. Nine students
of the twelve present volunteered to be participants.
162
The focus of the second part of the class was to learn to use the grouped
frequency distribution to find a score given a percentile rank. For example, we had to find
the number of seconds it took a rat to complete the maze given that it was at the 40"^
percentile rank in relation to the other rats' scores. Aim commented, "Students say this is
the hardest math in the course," but assured us that "the math gets easier; the concepts get
harder." Ann lecture-guided us through the steps delineated in the text as each individual
worked on the problem and reported his findings.
No one in the class, with the possible exception of repeating students, had seen
this procedure for fmding a score in grouped data, given the percentile rank. During the
procedure I felt lost; I did not have a sense of the end from the beginning, the rationale
for each step, nor any visual connection with the data — a very uncomfortable experience
for a conceptual learner like me. I made a note to explore the logical and visual
connections with students in study group and in counseling.
Perhaps more importantly for students was the fact that in the formula there were
six unfamiliar letter symbols, five of them with subscripts. I suspected that students might
find this procedure difficult on the test the following week.
By the end of class time the direct process had not yet been tried — fmding the
percentile rank for a given score. We were assigned this as homework — ^to find the
percentile rank of the rat that took 82 seconds to do the maze. A formula was given in
the text but there was no step by step procedure.
Study Group 1
The group formed before class at Riverside Center; Brad, Jamie, and Lee were
there, and Pierre came in a little after we began. The group gathered at the end of the
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table near the chalkboard. The first exam was scheduled for the Monday following, on
chapters 1 through 5 of the text. Students' pressing concern was solving the problem that
had been assigned at the end of Class 2.
I wanted the study group to be a setting where the students did the work while I
coached. I expected that the students might want help with strategic planning, knowing
what and what not to concentrate on for the test. Although I had tutored students taking
PS YC/STAT 1 04 in the semester before this, the instructor was different so like the
students, I was uncertain how we would be tested. Our natural questions were: Would
Aim examine students on what had been covered in class or on any concepts included in
the first five chapters? Would the problems be straightforward or tricky?
Even before the exam, we had some reassuring evidence that Aim's test would be
fair. She had handed out a study guide for the exam that included instructions on what
could and what could not be included on a student's formula sheet and a list of 13
symbols, including E, a, ji, P5o,and z, whose definitions were to be examined. She had
included an example of an acceptable defmition. I had observed Ann careflilly checking
her lecture notes, apparently to ensure that she had covered everything. She assigned only
certain homework problems from the text and handed out solutions to any even numbered
ones whose solutions were not in the back of the text. With this evidence I speculated
with the students at study group that Ann would test only on what had been covered.
There had been no opportunity for students to explore the derivation of the
formulae for percentile point and rank using proportional geometric reasoning, so I
thought the study group could try that. I put the grouped data on the board. Brad took the
chalk; I coached pointing to and drawing in the geometric proportions on the board; he
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had taken this class before and said that all he wanted was to know how. He did not want
a deep explanation.
As students worked, Lee found an anomaly in the formula to find percentile rank
of a score, namely /, in which the / that is the subscript of/ in the numerator locates the
interval in which the X score in focus is found (different for different values of X),
whereas the / in the denominator stands for the size of each class interval which is
constant for the distribution. The other students in the study group were struggling to sort
out the other symbols so were not engaged in Lee's and my discussion of this point. She
decided to use different symbols to keep them straight: an upper case / for the constant
size of the class interval and the lower case / for the subscript that indicated whatever
interval we were interested in. I was impressed with Lee's interest in and good analysis of
this use of letter symbols and showed my enthusiasm for her approach.
Jamie was actively following the discussion of the process but was not saying
anything, so I asked for her answer at one point. Although her face turned red, she
answered correctly. I hesitated to direct many more questions to her because of what
seemed to me to be her obvious discomfort, though I did some further questioning.
After completing the homework problem we tried one going in the opposite direction
and finally stopped as other students came into the room for class.
Class 3
Ann handed out written instructions on how to construct a formula sheet for the
test next class and a reminder of the symbols we needed to be able to define by giving
both a description and a statistical meaning. On the board, she went over the homework
165
question on finding the percentile rank of the time of the rat that ran the maze. Karen was
the only student apart from Catherine and those in the study group who had done it.
Ann went on to chapter 4 on measures of central tendency and variability.
Following a lecture discussion of mean, mode, and median. Aim handed out a worksheet
to teach the process of finding standard deviation from the mean using the rat maze time
scores in a column labeled X, and two blank columns labeled to facilitate correct
interpretation and use of the standard deviation formula.™' There was a brief discussion
of the different formulae for standard deviations of samples versus populations and Ann
called the Greek symbol for population standard deviation (o), "omega." Mulder
corrected her telling her it was called "sigma." Ann accepted the correction but
erroneously appUed it to s, the symbol for sample standard deviation, and continued to
call cj, "omega." I felt awkward as she continued and, perhaps somewhat
inappropriately, discussed the problem with Lee who was sitting beside me. She was
becoming confiised especially when Ann began calling s sigma.
This was a very real dilemma. I knew that letter symbol classification caused
students difficulty and now the instructor was confusing their names and was also mixing
up the Greek versus Roman letter symbol categorization. A relatively consistent
convention in statistical symboUzation is to assign Greek letters as symbols for
population parameters, corresponding to the first sound of the item labeled, for example
|x (Greek lowercase "mu") for population mean, and a (Greek lower case "sigma") for
population standard deviation. Roman letters are used to symbolize sample statistics, for
example, X is used for sample mean and 5 for sample standard deviation. In my
experience, explaining this convention to students struggling with many unfamiliar
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symbols had helped them considerably. 1 felt compelled to clear up Arm's confusion with
her so the students would not be confused but on the other hand, 1 was very aware of the
delicacy of trying to balance my multiple roles of participant as a student, class- link tutor,
researcher, and colleague m relation to Ann's instructor role, and I was not sure how Ann
would react to my introducing this concern.
When I did address it with Ann privately after class her reaction reaffirmed my
prior assessment of her healthy self-reliance (cf Bowlby, 1982) and of an appropriate
way for a class-link tutor to approach such an issue with a self-reliant instructor. Rather
than my telling her of the error, I pointed it out by using the text as the authority.
Although a Uttle embarrassed, Aim reacted positively to my addressing it in private and
to now knowing the correct designation for a and s. The way 1 handled the incident
seemed to contribute to Aim's confidence in my expertise because she encouraged me to
contribute it in the fiiture. At the next opportunity Ann addressed the confusion directly
with the class. I was then able to emphasize the Greek-Roman letter distinction for
population versus sample symbols with individuals without involving them in an
apparent conflict between believing what the instructor taught or what I told them. Kelly
was the only student in the class who remained confused labeling a "omega" in Exam #1.
Ann had filled in the bottom total row of the X - mean column on the worksheet
with the mathematical statement, S(X - mean) = 0, that is, the sum of the deviations of
scores from the mean is always zero no matter what scores are analyzed. This indicated
to me that Ann wanted the students to attend to that concept particularly but she didn't
stress it in class. Its mathematical inevitability, its universal application throughout
statistics, and its usefulness for checking ones' mathematical processes seemed so
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important, however, that I took mental note to point some of this out to students in
individual sessions.
At this point in my classroom observations I was using the dialogue and the class
plan as recording devices (see Appendix C) for interactions between instructor and
students during lecture discussions. I was not yet systematically recording interactions
among students during problem-working sessions. Ann circled the room helping
individuals and there seemed to be little interaction among students. Later I reaUzed that
students, who in later classes worked together, were not sitting together in this class,
except for Robin and Brad. Mulder who later would work with anyone who was willing
was between Autumn, who never worked with a fellow student in problem-working
sessions, and Pierre who rarely did. This setup definitely did not lend itself to
conversation.
After break Ann covered chapter 5 (T/ze Normal Curve and Standard Scores) in
lecture discussion using the rat maze mean times and standard deviation computed before
break to compute a transformed z score on the board. She also demonstrated use of the
standard normal tables to find the probability of obtaining a particular time score or less.
At the end of the class students who had formed pairs for the MINITAB computer
module project were given their projects. Lee had already arranged to partner with Mitch.
Otherwise, women sitting beside or near each other in the previous class seemed to have
already paired off when Aim made the announcement. (Kelly worked with Autumn and
Catherine with Karen.) Men who were sitting beside each other in class 3 paired off then:
Mulder partnered with Pierre and Brian with Floyd. Robin was left out — she seemed
flustered by the situation and initially said to me that she would just do the project on her
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own. When I told her that I understood that it was a requirement to partner with someone,
she agreed to partner with Jamie who did not yet have one.
Individual Sessions
The first test was to be held at the next class. I had individual mathematics
counseling meetings with Kelly on the Thursday and Karen on the Monday just before
the test. Both were very anxious, each seemed to have some fundamental arithmetic
confusion with decimals and percents, and both found the large amount of material to be
covered in the test overwhehning.
I had also been involved with Pierre who had to take his first test early because he
would be away when it was scheduled. He was going to take it in the library (located
across the hall from the Learning Assistance Center at the Greenville campus) on the
Thursday (June 8). He took the day off work so that he could study. I was expecting that
he would come to the Learning Assistance Center. At one point in the mid-afternoon
when there was a lull in Learning Assistance Center activities, I went looking for him and
found him in an otherwise empty conference room. He said that he had looked into the
Learning Assistance Center earlier adding, "You were busy." I encouraged him to come
in next time and begin working so I could help when I was free. There were several
similar situations in the following two weeks until we finally managed to meet for his
first mathematics counseling session towards the end of the fourth week of class.
On Monday, the day of the first test, Karen and Kelly came to drop-in
mathematics tutoring at the Learning Assistance Center. Karen came only briefly before
her mathematics counseling appointment at 4:00 p.m. Kelly seemed so overwhelmed that
she was seriously considering dropping the class. I felt as if she was trying to get me to
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tell her if she should. I told her that she had to make that decision but that perhaps taking
the test might help her decide. She decided to meet with Ann to discuss her options and
perhaps get her to make the decision for her. Kelly's mother had already called Ann and
me to discuss Kelly and we had independently encouraged her to let Kelly work it out.
Exam #1
The first test was held in a room across the hall from where the class usually met.
The exam room had individual seats with attached small right-handed desks all facing the
front of the room. I expected a rather high level of anxiety before this first test. Karen had
expressed some of this, perhaps seeing Ann as the cause, but students always face
unknowns that cause anxiety in all courses even when the instructor makes an effort to
prepare them, as it seemed Ann had.
On the surveys I had administered in Class 2, the class average level of
Mathematics Testing Anxiety of 2.9 on a scale of 1 (not at all scared) through 5 (very
much) with a range of 1 .5 through 4. 1 indicated a higher level of anxiety than for either
Number (2.2) or Abstraction (2.8) and high™'" if compared with Suinn's (1972) norms on
the Mathematics Anxiety Rating Scale (MARS) from which the items were drawn.
Jamie's reported testing anxiety level was the highest in the class at 4. 1 and Karen and
Kelly's were almost as high at 3.6 and 3.5 respectively (see chapter 6 and Appendices H,
K, and L). In counseling, we explored relationships among participants' testing anxiety
scores and factors such as their past experiences in mathematics exams, their preparation
for the exam, and their perceived ability to achieve on the current exams.
Everyone was in the classroom by 6:00 p.m. except Kelly. Kelly was considering
dropping the course but Arm had just persuaded her to try the first exam so she came in a
170
few minutes late. Each student had to hand his formula sheet'"'^ to Ann and pick up part
I, the part that Ann called conceptual consisting of multiple-choice questions and symbol
identification questions. Once a student completed part I, he took it to Arm who was
sitting at the table in the front of the room, and picked up the computational part of the
exam with his formula sheet that Aim had checked to make sure it met her criteria.
Students took between 20 and 30 minutes to complete part I of the test. Part II
consisted of 14 questions requiring various descriptive statistical analyses of small sets of
data, all from chapters 2 through 5 of the text. All required procedures had been covered
in class. Students had had the opportunity to struggle through all the procedures using
worksheets in class except for the z-score questions. Those had been covered in Class 3
on the board by lecture discussion only.
Ann had agreed that I could ask students to complete Joan Garfield's (1998)
Statistical Reasoning Assessment (SRA)™' after they completed their tests. I intended to
use this as a pre and posttest to gauge changes in students' statistical conceptions. As
each student finished her test she took the SRA. Autunm had arranged her first individual
session for immediately following the test so when she finished the SRA (around 7:20
p.m.) we went across the hall to our usual classroom and began. Autumn told me that she
thought she had done well on the test, that she had done quite a bit of statistics before,
and that she didn't think she needed any help on the current course. As the other students
completed their SRAs they came in and gave them to me. Catherine pronounced it "very
hard" and seemed anxious about it.
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Results of Exam #1
The students' test results are shown in Figure 5.2. My first impressions gathered
in class, in the study group, and from individual contacts led to my being surprised by
Jamie's high score because of her expressed anxiety, by Robin's high score because of
her apparent confusion in class, and by Lee's relatively low score because of her
insightfol conceptual approach at study group. (On the other hand she had e-mailed me
the night before the test expressing considerable testing anxiety and asking for my help.)
Mulder's low score also surprised me, particularly his score of 22 out of 40 on the
conceptual part (55%), the lowest in the class, because he had been outgoing and
articulate in class and seemed to have background in the use of statistics in research. I
found Brad's relatively low score was both surprising and not. On the one hand his
confident demeanor and class participation bode well for a high score but on the other
hand, his professed procedural approach in study group with the "only wanting to know
how" and his admission that he was repeating the class did not bode well.
Error Analysis of Class Performance on Part II (computational)
I expected students to have the most difficulty on the question asking, "What
score is at the 50* percentile point?" not only because of the complex nature of the
formula, the multiple uses of different types of letter symbols and the largely procedural
way this was approached in the text and in class, but perhaps even more because it was an
inverse mathematical procedure requiring students to begin in the right hand cumulative
percent column that they had to create themselves and proceed left to identify the score
corresponding to the given percentile. My expectation proved to be well founded.
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Only two students (Jamie and Catherine) did this problem correctly. The other
nine on whom 1 have data (I do not have Mulder's Part II of Test #1) lost at least half of
the points given for this question, and more students made errors on this than on any
other question on the test. Three of these made the inverse error — ^that is, they began with
0)
(D
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Individual Scores on PSYC/STAT 104
Test#1
120 -1
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I I 1
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6
7 £
\ 9 1
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□ Concept
28
28
28 2
2 32
34
30 2
6 38 3
8 38
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■ Symbols
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4 30
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4 41 4
3 50
52
STUDENT
1 2
9 10 11 12
Test Totals: | 42 | 59 1 62 1 63 1 68 I 72 i 76 | 78 | 86 |89 |95 1 100
Figure 5. 2. Individual's scores on Test #1, with each student's score broken down into
his Conceptual multiple-choice Part I) score, out of 40; his Symbol score (on Part I) out
of 8; and his Computational score (Part II) out of 52, total possible 100.
Note: the X-axis numbers refer to individual students as follows: 1: Floyd; 2: Kelly; 3:
Karen; 4: Mulder; 5: Pierre; 6: Brad; 7: Lee; 8: Mitch; 9: Autumn; 10: Robin; 11: Jamie;
and 12: Catherine.
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a score of 50 instead of a percentile rank of 50, but all three had not made a cumulative
percent column from which they should have begun.
Multiple uses of letter symbols seemed to be the cause of much conftision in this
question even for algebraically confident students. Students made the most errors
identifying the correct number indicated by the letter symbol, especially when it was a
multi-level symbol such as cum /l that not only requires careful interpretation, but also
requires a muhi-step approach to computing.'™" Six of the students made errors
in correctly identifying or computing the cum /l. Of the five who identified cum /l
correctly, three had attended the study group.
Jamie and Lee were the only students in the class to fmd and correct their own
initially wrong substitutions as they were taking the test.
Grading and Instructor Response to Test #1
Aim returned the tests to students at the next class with a summary of class results
m the form of a grouped histogram (see Appendix G) with the mean (74.2%) and
standard deviation (16.7%) of the scores. Students looked over their tests and then had to
return them to Ann. Since I was to be working with individuals, I arranged to get their
tests from Aim if they wished so we could do error analysis during individual sessions.
Aim seemed to use arithmetical accuracy as the main indicator of correct
procedure. No more credit was given for correct process with an inaccurate result than for
incorrect process with an inaccurate result. For example, in question 10, Brad lost 3 of the
4 points because he had replaced his initially correct cumulative frequencies and percents
with mcorrect ones and used those with the correct process to find the score given the
percentile rank. Floyd, Mitch, and Pierre also lost 3 points on that question when they had
174
made the much more serious inverse error, treating the given percentile rank as if it were
a score. With individual counseling participants who focused on grade as the gauge of
their mathematics ability, I feU an important strategy would be to affirm logical,
conceptually sound, albeit arithmetically inaccurate work as a more accurate way to
gauge their understanding and ability than the points they earned (or lost).
Relationality and Implications for Counseling
in the Next Cycle
Before Exam #1 I had met only with Kelly and Karen for individual mathematics
counseling. After Exam #1,1 met with the others who had signed up at various times and
by Exam #2 on June 26, 1 had met with them all at least once (see chapter 4, Table 4.2).
Kelly and Floyd had stopped attending class by Exam #2. All of the individual
counseling participants who took it except Jamie either maintained or improved their
scores on Exam #2 (see Table 5.1). Lee and Mulder's improvements were dramatic (up
two letter grades); Autumn, Karen, and Mitch improved by one letter grade; the others
made more modest gains. Jamie's decline was as dramatic as was Lee and Mulder's
improvement — by two letter grades.
Student MINITAB partners had been working with each other since the computer
lab class which was held in the computer center at the Greenville campus, but there was
no evidence in class of these partnerships leading to study alliances.
The first test marked an important point in the trajectory of the class not only for
the students but also for me. The test was a key piece in confirming my initial judgment
that Ann was providing a positive classroom climate where vulnerable students could
progress with will and strategic support. It also signaled who was vulnerable but in a
crude way, the grade signifying quite different things for different people. For example.
175
Jamie found her 95% quite unexpected, whereas Catherine's 100% was not a surprise at
all; Kelly reacted to her 59% flustered, casting about for someone to tell her whether to
continue in the class or not whereas Karen and Mulder with similar grades refused to take
them as a verdict on their course outcomes. Whether students perceived this positive
classroom climate or not depended on their past mathematics experiences, the status of
their mathematics preparation, and the relational patterns in which they were imbedded.
That was a challenge for counseling: to help students experience the current positive
relational reality rather than a negative reality from their past.
In this first cycle through the first test I found that Ann's provision of a positive
relational climate gave me the space to negotiate a comfortable position for myself as part
of the class community, and her healthy self-reliance made it possible for her to access
my support comfortably. As a result, in class and in counseling there had developed an
easy sense of our working together for the benefit of students. In counseling it seemed
that I would not have to do damage control for current relational assaults but rather
develop my role as a mathematics complementer of inevitably underemphasized or
missing material fi-om class while the student and I explored his particular relational
challenges.
The Post Exam #2 through Exam #3 Cycle
Exam #3 was based in chapters 10 through 14, moving into inferential statistics
with hypothesis testmg using sampling distributions. Ann confessed to not being really
clear on sampling distributions and how to explain them to the class. She invited me in
class to offer my fiirther explanation. I was somewhat nonplussed, not being sure what
she found confusing or what she felt students did not understand. Her explanation seemed
176
clear to me and I said so but I knew that unless students actually created their own
sampling distribution from a finite population the connections would likely not be clear
to them — telling is no substitute for experiencing the mathematics. There was no
opportunity for that now, however, except perhaps with Lee in a study group that only
she attended.
Class 13
The material to be covered in this class on Wednesday, July 12 was chapter 13 on
the Student's t test for single samples and chapter 14, on the Student's t test for correlated
and independent groups. The next class would be Exam #3 on all inferential statistics and
hypothesis testing through this class.
The first part of the class was a short lecture discussion. Ann explained the use of
the single sample t test for sampling distributions, comparing and contrasting it with the
single sample z test covered in the previous class. She briefly mentioned using the sample
mean to fmd a confidence interval to estimate the mean of sample means of a sampling
distribution of the population. She discussed power and Type II errors, and then wrote on
the board a t test confidence interval problem and handed each student a problem sheet
requiring hypothesis testing using a single sample t test.
Because of time pressure, Ann did not go beyond helping students work through
the procedure for finding a confidence interval estimate for a population mean. Students
then alternated between working alone and with the person beside them (except Jamie,
Autumn, and Pierre who only worked alone) on the single sample t test problem without
too much apparent difficulty.
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After break beginning at 7:35 p.m., Ann gave a ten minute overview of two-
sample t test hypothesis testing comparing the means of two samples. Students were
directed to work on a worksheet problem, using the text for formulae and as a procedural
guide. As the students began working on the independent samples t-test, individually
(Autumn, Jamie, and Pierre) or in pairs (Brad and Robin, Catherine alone and Mulder
checking with her, Lee and Mitch, and Karen and me), there was an audible reaction to
the formulae on page 331 (Pagano, 1998). Karen growled, Mulder sputtered in disgust
and they both proclaimed, "Yuck!" Robin frowned harder than usual and sighed. Mulder
demanded, "So where's the short version of this?"
There were a number of potential trouble-spots in the independent samples t
statistic formula, especially the complex subscript for the estimated standard error.
Although Ann had not explicitly taught the idea of subscript-as-label, students had
generally succeeded to this point, apparently by ignoring the subscripts that were
monomials (e.g., 3 inXsor a/2 in tan)- But they found this new binomial subscript (i.e.,
two terms as in A} - X2) with terms that themselves had subscripts, very confiising or
rather distracting. Now that they could no longer ignore the subscript, instead of
understanding that the subscript's function is labeling only, some students tried using it as
part of a formula, in this case to compute estimated standard error of the difference
between means {S _ _). Autumn, who always worked alone and rarely asked even
X, - X:
Ann a question during problem-working, had done this with the subscript. In an
unprecedented move, she got up from her seat and came around the table to me because
she knew that what she had done with this formula was wrong but she was not sure why.
She had given the subscript a numerical value of 0 (since she knew that the null
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hypothesized population mean difference was zero, that is, i^i - 1^2 = 0), had written the
population a instead of the sample S and had finished with ct - 0 as her interpretation of
S_ _ . Instead she should have seen this as the symbol only and computed its value
X, - X:
usmg a series of formulae given in the text. This type of error and confusion by even
confident and high-achieving students led me to predict that symbol and formulae issues
would cause more difficulty for students on the next exam than anything else (with the
possible exception of correctly deciding which statistical test was applicable).
As each student completed the independent samples t test problem Ann checked
it, and moved him on to the correlated samples / test. This did not cause as much
consternation as the independent samples / test probably because the formulae are less
complex and include only monomial subscripts. With each of these unfamiliar statistical
tests, students were able to use the familiar six-step hypothesis testing procedure protocol
that Ann had provided them in Class 1 1 — this provided welcome consistency.
The first students to fmish left at about 8:30 p.m., ten minutes after the class
officially ended, and others were still there after 9:00 p.m. This was the only class that
ran over time; as the Exam #3 was to be given at the next class students did not seem to
react negatively.
Individual Sessions
Between Class 1 3 and Exam #3 I had individual meetings with Pierre and Karen
twice and with Brad and Mulder once. Pierre and I had had two long sessions and I
realized that he was trying to master all the material in the text, not just what Ann was
requiring. While this seemed admirable it was leading to his not mastering in sufficient
detail any of the material, especially the material Ann was requiring, so his grades were
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poor (D* and C~ ) and I expected this exam to be the most challenging thus far. I
suggested he focus more on what was being covered in the course so he began to work on
the course specific materials I prepared. Mulder met with me the early on the morning of
the exam. He said he hadn't done much, if any, study and said he was feeling stressed
because ha had to work for the rest of the day and would have no fiirther chance to study.
Karen had met with me the week before the exam and we had worked on the
Mann- Whitney hypothesis test. On the day of the Exam #3 we met again after she had
already been at Drop-In mathematics at the Learning Assistance Center at the Greenville
campus for three hours. Jamie said she would come to Drop-In too but didn't. However
Ann was offering a special pre-exam study group/drop-in session in our classroom at
Riverside from 4:00 p.m. to 6:00 p.m. and Jamie was there when Karen and I arrived at
about 5:00 p.m. from Greenville. Lee had been there since 4:30 p.m. and was workmg
with Ann. Lee had called me earlier in the day panicked because she had to work all day
after being ill all weekend. She had hoped to meet with me at 4:00 p.m. but I was already
scheduled to meet with Karen then. Autuitm and Catherine were also there but each was
working on her own. Mitch arrived about 5:20 p.m. and seemed unprepared. He had
been hosting a visitor and said he had not been focused on his work.
Exam #3
My anxiety level on behalf of the students was higher for this exam than for any
of the others. I was especially anxious about how to counsel and tutor those who sought
help. I was also anxious about those I believed should have sought help and did not. My
anxiety stemmed not only from the fact that the inferential statistical and sampling
distribution material being examined was new to most students and was considerably
180
more complex than the previous descriptive statistics but also from my undoubtedly
inappropriate feeling that I could not quite trust the students to take responsibility for
themselves. I was also concerned with the number of statistical tests and procedures the
students had to master and wondered with them if they would be expected to identify the
applicable one from the problem statements on the exam. I e-mailed Ann to this effect
and she replied that she was not sure; maybe she would identify some and have students
decide on others.
To help students prepare for this exam I used my analysis of content of the
previous exams to develop strategic practice materials. Because the previous two exams
focused on problems like the ones worked in class, I erased the test labels on each of the
problems we had worked in class and copied them for individual counseling. I added a
normal z test problem from the text as we had not done one of those in class. I also
modified the decision flowchart the text used for choosing the appropriate statistical test
(Pagano, 1998, Figure 19.1, p. 473) to create a simplified flowchart including only tests
being examined on this exam, leaving the test name boxes blank for the students to fill m
(see Appendix J). Ann agreed that this modified flow chart, filled in by the student,
would be acceptable as a formula sheet to use in the exam.
The student who took most advantage of these materials was Karen, during her 4
hours at the Learning Assistance Center on the day of the test. On her second test she had
made significant errors because of missing work columns on her formula sheet and
because she had not sufficiently practiced all the problems to be tested. She remedied
both of these problems for this exam and was rewarded with an almost perfect score on
the computational section. She was one of only two students whose grades improved
from Exam #2 to Exam #3, before the extra credit was factored in (see Table 5.1).
Aim's strategy of having the students work through problems in class without first
showing them what to do forced each to negotiate the procedures required. The lack of
systematic group discussion of links among various aspects of the process and of known
trouble spots, except with individuals, left some vulnerable to memory lapses or
confusion in the exam unless we addressed these in a counseling session. For example,
explicit discussion of the direct relationship between the null hypothesis statement and
the relevant symbols and parts of the / statistic formulae may have prevented the error
some students made.'^^" In the exam Mitch, Mulder, and Robin (almost one third of the
class) used their non-zero mean of sample differences D for yi^ instead of zero even
though they each correctly stated in their null hypothesis statement that there was no
difference or change in the population scores before and after. They each then had to cast
about for improbable Ds because they had used theirs for jXa
Four of the ten students who took Exam #3 made errors in choice of degrees of
freedom in the independent samples question on the exam and one in the correlated
samples question. Others made errors negotiating the t table. This procedure was
introduced to students in Class 13, and its application is complex; it is somewhat different
for each of the three t tests taught in that class and different in significant ways from the
famiUar procedures for using the normal z table. Guided questioning in the form of an
assignment sheet might have helped students become more conscious of these
differences. In subsequent counseling sessions I noted the importance of walking
participants through the use of unfamiliar tables. Karen had resolved her initial
182
difficulties with the tables during her afternoon of preparation and she had
simultaneously designed her formula sheet to prompt correct usage.
Pierre was the only student who misinterpreted the subscript of the standard error
as part of a formula instead of as a label. Instead of calculating S (=1 .77) using
X, -X;
formulae on his sheet, he used X/ - X? not as a label, but as a factor, multiplying it by 5
to get S( X,- X2), that is, 9.73(43 - 39) or 38.92. The large size of his standard error
should have given him pause. Because there was no opportunity in class for discussion of
the expected relative sizes of the statistics, in relation to the mathematical processes
involved, in mdividual sessions I realized that it was important for me to model and
encourage students in this type of questioning and checking.
Grading and Instructor Response to Test #3
A new feature of Ann's grading emerged with the focus on hypothesis testing. In
her scheme, a certain number of points, typically 3 or 4, were allocated for the correct
decision at the end of the process (i.e., whether to reject or fail to reject the null
hypothesis) and for its meaning in terms of the problem at hand. On one question in
Exam #3, four students made errors in their calculation of the statistic and found its
magnitude to be less than the magnitude of the critical value. They therefore logically
decided to fail to reject the null hypothesis.™"" Ann penalized them the foil amount
because they made the incorrect decision, even though it was the one demanded by their
results. I was concerned in individual counseling to affirm students' sound mathematical
decision- making in a situation like this, and try to allay the negative impact of the lost
points on their self concept. At the same time as we discussed the validity of the
183
instructor's emphasis on the need for the correct decision'°"''and, therefore, the
importance in subsequent exams of checking the accuracy of one's computations.
Table 5.1
Grades Throughout the Course of all Individuals in PSYC/STAT 104, Summer 2000
Exam
Exam
Exam
Exam
Minitab
Minitab
Final
Optional
TOTAL
#1
#2
#3"
#4
Module
1
Presentation
Exam
(#5)
Comprehensive
Final
Percent of TOTAL
GRADE
20
20
20
20
2
8
10
Replace
lower exam
grade
100
Autumn 4^
86%
96%
(90+6)%
95%
100%
100%
100%
94.6%
Brad 4
72%
69%
(56+4)%
72%
100%
62%
Catherine
100%
92%
(91+6)%
100%
100%
100%
100%
97.8%
Ellen
Floyd
42%
Jamie 5
95%
74%
(84+6)%
76%
100%
100%
100%
71%
87%
Karen 5
62%
74%
(85+6)%
88%
100%
100%
96%
57%
83%
Kelly 3
59%
Lee 6
76%
97%
(83+6)%
81%
100%
100%
100%
77%
88.8%
Mitch 4
78%
87%
(62+6)%
82%
100%
100%
92%
82.2%
Mulder 5
63%
81%
(76+5)
91%
100%
92%
94%
81.96%
Pierre 8
68%
72%
(60+6)%
91%
100%
92%
96%
72%
79.56%
Robin 3
89%
87%
(77+6)%
88%
100%
100%
96%
89%
Notes: ' Names of counseling participants are bolded and the number beside their names is the number of
their counseling sessions. ''Because more than two thirds of the class experienced grade decline, some
severe, on Exam #3, and more showed a fundamental lack of understanding of the concept of statistical
power, Ann gave an in-class, open-book assignment worth up to 6 points to be added to the Exam #3 grade.
Karen and Jamie's scores on Exam #3 showed an improvement of one letter grade
over their scores on Exam #2. Everyone else except Catherine (whose score remained
about the same) dropped from one half to two letter grades. Ann was concerned not only
with the drop in scores but also with the evident lack of understanding of the concept of
statistical power. In Class 1 5 she assigned an open book extra credit assignment for 6
points on the topic of statistical power and the factors that influence it (see Table5.1).
184
From Exam #3 through the End of the Course
The ten students remaining in the class were all passing with grades ranging from
a D"(Brad) through A (Autumn and Catherine) after completing Exam #3. Karen was
showing steady improvement in grades and Jamie was recovering from her big dip in
Exam #2. Mulder's score on the multiple-choice conceptual section remained a
significant problem but he had done quite well on his computation despite his lack of
preparedness for the exam. Brad seemed quite crushed by his low score and 1 felt the urge
to "rescue" him from himself, convinced that he was sabotaging his own chances of
succeeding. Pierre had not followed my advice to focus on course material only and did
poorly again (a D~ before the extra credit).
The nine remaining counseling participants continued to meet with me
individually. Some also attended study group, Lee every week and others if the study
group was just before an exam. Their course grade progress is shown on Table 5.1. Their
progress as mathematics learners and course strategists and other changes in their
mathematics mental health are discussed in chapters 6 and 7.
END OF COURSE SUMMARIES AND DISCUSSION
Student-Instructor Interactions during Lecture Discussions
Analysis of the interactions between the instructor and individual students during
the lecture discussion portion of class revealed patterns relevant to the emotional climate
of the classroom and the individual's perception of it. In general Ann asked questions of
the whole class; she directed questions to individuals only in relation to a prior issue they
were discussing. At times several students responded together to Arm's whole class
questions. When students asked questions, some raised a hand to draw Ann's attention
185
(e.g., Lee in Class 16); others spoke into a silence or out of puzzlement with what Ann
had just said (e.g. Karen, Mulder, and Robin each asked Ann to repeat or clarify what she
had said in several instances).
There was almost no correlation between the number of students' responses or
questions and their grade in the class (see Table 5.2). There was, in fact, a small negative
correlation (r = -.244) between a student's average number of responses or questions and
final grade (for those who completed the course).
In addition, apparent accuracy and pertinence of student response was often
incongruent with grade. These phenomena make it very likely that any judgment of
student competency based only on class interactions could be quite misleading.
Table 5.2
Number of Individual Utterances During Lecture Portion of Classes and Final Grade
Class Number
1
2
3
(4)
(5)
6
7
8
(9) (10)
11
12
13
(14) (15)
16
(17) (18)
Av. &
Final
Grade
Autumn
0
4
3
3
0
1
0
0
0
2
0
0.82; A
Brad
4
2
2
12
2
2
0
0
3
1
4
2.9; D
Catherine
0
0
1
1
1
0
0
0
0
0
0
0.27; A
Ellen
3
Floyd
3
0
1
0
1
Jamie
0
0
0
0
0
0
0
0
0
0
0
0.0; B*
Karen
1
4
3
2
1
3
2
3
2
1
3
2.27; B
Kelly
1
3
2
1
0
2
1.33
Lee
2
4
4
5
2
8
5
3
1
3
3.36; A"
Mitch
3
1
2
1
1
0
2
1
1.38; B"
Mulder
6
3
3
3
0
3
7
5
1
1
2
3.09; B"
Pierre
0
4
0
1
1
2
1
1
1
1
1; B"
Robin
3
4
5
7
1
2
4
8
3
1
3
3.73; A"
186
Participation alone is clearly not enough. Factors related more closely to classroom
interaction were student learning style and preferred modality, personality, and previous
experiences in a mathematics learning environment.
Brad and Mulder interacted in a way that gave the impression of familiarity with
and grasp of the material, while Robin gave the opposite impression. Ann's initial
judgment of Brad's competence was dramatically modified by his poor grades and his
struggles in the class problem-working sessions. Her initial judgment of Robin's
incompetence persisted however, even despite her consistently good grades. Both Robin
and Mulder seemed to have difficulties with auditory processing of lecture material but
Robin's struggles were clearly discemable in her often puzzled demeanor, her checking
with Ann to see that she had understood correctly, and in the tentativeness of many of her
correct responses.™' In contrast, Mulder responded to questions only when he was certam
of the material; he dealt with his struggles to understand the lecture presented concepts by
focusing on parts of the lecture and ignoring others or by giving up and working instead
on the computation with fellow students during problem- working sessions if they were
willing. After Karen gave an initial response that showed conftision from then on she
restricted her responses to supplying data (e.g., her beer preference. Class 16; or sports
data).
Jamie was the only student who never asked or answered a question during the
lecture discussions (see Table 5.2). Her shyness was obvious from the beginning and she
typically kept her head down and eyes lowered. By the end of the course she was raising
her head and making eye contact but she still did not speak. The strongest students in the
class grade-wise, Catherine and Autumn, were among the quietest. Catherine (average
187
responses 0.27 per class, see Table 5.2 above) had an air of quiet confidence that
accurately reflected her easy mastery of the material. Autumn (average responses 0.82
per class) was more responsive during the first half of the course than later, perhaps in
part because she was more familiar with the material at the beginning and she was careful
only to respond when she was quite certain.
All students except Robin seemed to try to restrict their responses to answers they
felt they knew. For example, Mitch usually held his head stiffly on his hand and would
respond barely audibly, when he was sure of the answer (see Appendix E, Table E2 for
the criteria I used to analyze student's utterances during lecture discussions).
IMPLICATIONS FOR MATHEMATICS COUNSELING
I found that being in the class, doing the statistics, observing the students, and
taking the exams provided me with good data to use in order to plan and provide strategic
tutoring in the statistics/mathematics and the counseling of participants' relational issues.
Study Groups and Cooperative Learning
Because the study group was open,'™" attendance fluctuated from one to seven
students, apparently according to whether there was an exam immediately following (see
chapter 4, Table 4.2). Because attendance was not consistent, a group working approach
was difficult to establish. No one in attendance, except perhaps Lee, was oriented towards
working with peers to investigate a mathematics problem. They each related directly with
me and seemed to show little mterest in others' responses unless I directed them to
evaluate those responses. It was easier to have the students work together on a problem
when there was not an exam immediately following but even then the pressure to master
the procedures precluded open group explorations. The focus was on working assigned
188
problems with my guidance and coaching and with students taking turns in presenting
solutions to the group.
The study groups that met before exams were more like drop-in with me (and Aim
before Exam #3 and Exam #4) moving from individual to individual helping each with
his particular questions. An exception was the one before Exam #5. That was a round-
table discussion of problems Ann provided and I supplemented, during which I asked
individual students in turn (including Jamie) to respond with their solutions.
Mathematics Counseling and the Classroom — Relational Foci
I found that how a student interacted in class with the instructor, the mathematics,
and his peers provided me with important data. When considered with material that
emerged in individual counseling sessions, it helped me narrow and define that student's
relational focus — in other words, his core relational conflictual pattern. By itself,
classroom observation was certainly not adequate to identify students' entrenched
mathematics relational patterns as they affected their mathematics mental health and
prospects for success in PSYC/STAT 104. Observation seemed to even add to the
confusion at times. However when I used the relational dimensions to organize my
classroom observations, the data that were initially confusing often became important
clues to students' core relational conflict (see Appendix E, Table E3).
The Classroom and Issues of Mathematics Self
During lecture discussions, students' ways of interacting gave some clues to their
sense of mathematics self. Were they willing to reveal ignorance or only knowledge?
Were they interested m growing, in performing, or in merely surviving? Lee and Robin
were the most public, in different ways, in their attempts to grow and their willingness to
189
reveal ignorance for that purpose, indicating to me that they could have healthily
developing mathematics selves and learning motivation for achievement. Others, like
Autumn, Mitch, Karen, and Mulder, who only revealed their knowledge and were silent
when they were uncertain, signaled a more fragile mathematics self. Autumn and Mitch
acknowledged their performance motivation for achievement in their Mathematics Beliefs
survey (as I expected from their other behaviors), while Karen, and Mulder surprisingly
revealed more learning motivation. It seemed that their silence in class (except when
certain) may have been self-protective, with survival taking precedence over their
underlying desire to learn. Mulder, Floyd, and Brad all spoke and acted with confidence
in their own knowledge that their exam grades belied.
The student behavior 1 observed during problem-working sessions (particularly
with respect to arithmetical and algebraic comfort and level of confidence tackling new
material) gave clues to students' mathematical self development. In class discussions,
Karen just seemed to want to survive. There were fiirther clues in problem- working
sessions that her difficulties could be related to an underdeveloped mathematics self (e.g.,
her poor sense of decimals, poor operation sense, and low level understanding of the
algebraic variable).
The Classroom and Issues with Mathematics Internalized Presences
A discrepancy between how the student was experiencing the classroom and the
reality of the classroom was sometimes a clue to the effect of the student's internalized
mathematics presences from the past skewing the present experience. Jamie serves as an
effective example. Given the small class size, and the community-style seating
arrangements, and the positive classroom emotional climate created by Ann, Jamie's
190
almost complete lack of participation (in fact, her quite successful hiding) was a clue that
internalized past negative experiences might be skewing her perception of the present
class and making her feel unsafe in a safe environment. Likewise Karen's observable
defensive detachment and difficulty with the class in the beginning may have been
related to her difficulty with separating herself from past experiences, especially that of
previously taking the class. My awareness of the possible implications of these students'
behaviors prompted me to explore further in counseling (see chapter 6).
The Classroom and Mathematics Interpersonal Attachment Issues
Students' attachment patterns to teacher and mathematics also became apparent in
class. Kelly's behaviors indicated that she was anxious, disorganized, and dependently
clinging to Ann and me; she seemed to have an insecure attachment to mathematics and
to mathematics teachers. Although Lee seemed to have experienced a secure attachment
to mathematics at times and had a history of generally secure attachments to mathematics
teachers, she exhibited a lack of confidence in her ability to develop understanding on her
own. This and her difficulty with Ann's approach showed in her spending up to two and a
half hours a week with me in study groups and counseling while spending only about 20
minutes doing homework on her own. Karen was detached and defensive and kept Aim
(and me initially) at a distance — indicating the possibility of either a lack of secure
attachments to mathematics teachers in her history or a traumatic severance of such an
attachment with no subsequent reconciliation. Her confusion with decimals and her errors
with simple arithmetical procedures in problem-working sessions indicated a lack of
secure attachment to mathematics, almost certainly contributing to her expressed anxiety
during these sessions. In contrast, Jamie's anxious and disorganized attachment pattern
191
seemed to imply prior positive mathematics experiences with intervening negative ones
so that she was now uncertain and now had little sense of a secure base in the
mathematics or in mathematics teachers.
The ability to negotiate change was a particular issue for some students who were
repeating the class, particularly Karen and Mitch. I used my knowledge of the statistics
and my observations of Ann's teaching to build bridges from their past to their present
experience and facilitate their adaptation to this new course. However, the main challenge
was to help them acknowledge their conservative impulse reactions (cf Marris, 1 974),
recognize the differences between the current and their previous class, and take
responsibility for adjusting, rather than externalizing their discomfort by attributing
responsibility to Ann.
Whereas student's ways of relating with Ann and me seemed to fall into the
category of attachment relationships, their ways of interacting with peers seemed to fall
into the category of relationships of community (Weiss, 1991). These relationships of
community were evidenced in how students related to each other — as social,
independent, voluntary loner, or involuntary loner — and were apparent in problem-
working sessions and in study groups. In mathematics counseling, I explored further how
a student's pattern of relating with peers affected or was affected by his mathematics
mental health.
In the next chapter we will move from the classroom to the counseling setting
where I describe and analyze in detail the course of mathematics counseling with three
focal students.
192
' Excluding Ellen about whom I have no data.
" I remained seated during these problem-working sessions in order to observe, but I also assisted students
around me if they asked me.
'" Pagano (1998) does provide his own links between concepts and procedures in the text, verbally and with
diagrams, graphs, and illustrations (e.g., explaining the normal curve, pp. 81-86). He does not invite
exploration nor pose open questions for his readers to find links themselves.
" In the text, there is reference to statistical analysis computer software packages and optional companion
manuals for MfNITAB or Statistical Packages for the Social Sciences (SPSS) that may be used with the
text, but the only reference to them in the text is in chapter 1 (Pagano, 1998, p. 1 1 ) in a brief discussion on
the use of computers in statistics. Pagano, in his preface to the 5* edition, notes that he had removed the
cross references to computer software programs from the text at the request of teachers and students
(Pagano, 1998, p. xix).
The psychology department had developed a program of computer analysis projects, independent
of a text, using an old version of MINITAB to be completed by PSYSC/STAT 104 students. Because of the
accelerated timetable in the summer, instead of every student having to do each of 7 required MINITAB
computer assignment modules, Ann required Module 1 for everyone, worth 2% of the final grade. The class
then paired off to do one module per pair from modules 2 through 7 and these were presented at the second
to last class. This was worth 8% of the final grade.
" The text also addresses the issue of common misuses of and misconceptions about statistics in the seven
"What is the truth?" inserts scattered through the text where the author links statistical concepts to an
analysis of real-life mathematical or logical claims of advertisements, research reports, or news items in an
attempt to link the text with and perhaps challenge the student's reality. However, the answer is given and
there is no invitation for the student to examine his own beliefs and reactions to the material. Ann did not
use these in class but instead distributed a copy of a newspaper article that she invited the class to
critique — if there had been time. As it was she pointed out the errors in use of the statistics.
" Lee challenged two of the four possible conclusions from a high correlation coefficient that Anne dictated
fi-om the text (i.e., X caused Y and Y caused X) but eventually resolved this issue for herself by putting
caused in quotes so as not to be associated with what she knew to be an erroneous step of attributing
causation where "possible relationship" was the only valid conclusion. What Lee did not understand was
that Ann and the text were correct in giving possible real coimections between variables that would lead to
a high correlation, wiiereas Lee was rightly objecting to concluding that X caused Y or that Y caused X
because of a high correlation between them. Ann did not have time to resolve this to Lee's satisfaction.
Lee's subsequent response to a question about this showed that she had become conflised whereas on the
pre test she had shown a correct understanding (question 16 on the post SRA, see Appendix C). In
counseling 1 did not attoid to the real vulnerabilities of Lee's mathematics self-revealed by this situation, so
we did not address her conflict in counseling. This situation may also point to her vulnerability to authority
over reason.
™ The worksheets, with the exception of the one factor and two factor x' worksheets, consisted of a
question and, in some cases, an empty table v^dth column headings. However, the one factor and two factor
X' worksheets were different. Ann had posed the question at the top of the page and then provided a step
by step, fill in the blank, procedural format. Not surprisingly Lee objected strongly to it. Lee relied on
working out the procedure for herself, in class, to guide her in the tests; she found the fill-in-the-blanks
format confusing and distracting. She subsequently made an uncharacteristic and serious error on the two-
factor X" on the test which she attributed to the worksheet. Instead of apportioning the expected
fi-equencies proportionally among the four cells using the formula, /e= (Row Total)(Column Total)/(Total
Observed Frequency), Lee apportioned them equally. Jamie and Robin both made an even more serious
error adding the column and row totals to get a total expected frequency double the total observed
frequency — clearly not reasonable if they had thought it through. These errors seemed to be not only
related to the more directive worksheet that did not require the student to work his own way through the
193
procedure, but perhaps also to the absence of class or individual discussion of the mathematics or logic
inherent in the formula. Ann stated that in other classes she had taught, students preferred the directive
worksheets.
™' Lee's relatively sound arithmetical and algebraic background seemed to help her in this, but there was
not time in class for deeper conceptual explanations and connections. The lack of these made her anxious
so that when she came to study group and individual sessions our focus was on connections and
mathematical meanings (Level 4 on the Algebra Test, see Appendix H, Table HI and Appendix C).
" Karen had arithmetical difficulties (for example, uncertainties about values of decimal fractions and
placement of the decimal point) and her algebra background was shaky (Level 2 on the Algebra Test, see
chapter 6, Tables 6. 1 and 6.2 and Appendix C).
" The manipulation required is largely linear and usually direct, except, for example, when one has to find a
particular score given its percentile rank in a normal deviate distribution, which is an inverse procedure
requiring manipulation of linear terms. Ann didn't expect students to do this. In the least squares linear
regression analysis section, which she did require students to do, they are required to derive a linear
equation in two variables and use it to find particular points. This latter process requires only the
substitution of numbers for variables in the derived equation.
"" To transform the independent variable by translation and compressing or stretching in order to convert
the probability density fiinction of the data in question into a standardized form whose area (i.e.,
probability) values are accessible on statistical tables in the text (or statistical software package).
In statistical formulae the extensive use of subscripts as labels is complicated by the use of numbers,
single literal symbols, and even variable expressions as subscripts (see discussion of Class 13). In addition
in descriptive statistics the X(the mean of scores, ^Y, in a sample) in the first part of a course is a statistic,
that is, a constant for that sample; the score JT is the independent random variable in this distribution of
scores. However, in inferential statistics, the A' (sample mean) becomes the random variable because
sampling distributions are distributions of the sample means of all possible samples from the population of
a particular size (refer to the discussion on Class 13). Ann's sense of students' difficulty in understanding
sampling distributions and her own may have stemmed, to some extent, from the lack of discussion in the
text or elsewhere of this transition of the X from being a constant to being the variable (see From Exam #1
through Class 12 discussion).
'"'' I use social learner here to refer to a student's evidenced preference for collaborating with other students
in contrast to preferring to work alone (a solitary learner or loner). This categorization should not be
confused with Belenkey et al's connected versus separate knower which refers more to a student's
preference for personal connection with the material being learned. It also should not be confiised with
Skemp's (1987) categorization of relational versus instrumental mathematics learner which refers to a
preference for conceptual understanding in contrast with a preference for procedural (only) competence. In
addition to being a social learner Lee was also a connected and a conceptual (relational) learner, whereas
Mulder, who was also a social learner, was a separate and procedural learner by preference or at least by
socialization.
Ann then addressed what she expected to be some anxiety about the class by telling of her own
experience in learning statistics as a graduate student. In particular she referred to her metaphor of statistics
as a beautifully painted mural with all the elements separate, distinguishable, and in their correct places in
relation to each other. In an exam, under stress, however, it was as if the separate elements began to run
together to form a horrible brown indistinguishable mess; she couldn't tell one procedure from another.
She said she had recovered from this disaster, going on to master the subject at doctoral level. She went on
to explain to the class how her experience of teaching this course a number of times had increased her
confidence and her enjoyment of statistics and that she hoped the students would find taking the course an
194
"okay experience." She urged as a remedy to anxiety that students do their homework, study their notes,
and ask for help until they had "over-learned" the material.
" The only caution against this openness is the possibility of disturbing the trust of students in the received
knowledge phase (the first phase of epistemological development) who believe that the teacher or text — the
external authority — is the repository of all knowledge (Belenky, Clinchy, Goldberger, & Tarule, 1986;
Perry, 1968). Such a belief tends to develop too in students enculturated in transmission, teacher-as-
authority mathematics classrooms (see chapter 3). To help these students fmd their own ability to
understand the teacher as she expresses uncertainty can simultaneously model exploring the mathematics
and discovering it as a secure base.
"^ I did call on Jamie in study groups and though that made her uncomfortable, in the small group she was
able to respond. In a classroom setting when I am the instructor, my practice is to call on students for their
responses in order to ensure that students who do not voluntarily participate are involved. With shy students
who exhibit discomfort (and sometimes cognitive confiision) when called upon, I make prior arrangements,
letting them know ahead of time of the question that I will be asking them to respond to.
'™' Because this is an odd numbered chapter, 1 use "he," "his," and "him" as the generic third person
singular pronouns.
"^^ The feet that this extra exam was comprehensive, covering all the procedures and all the statistical
concepts from the course ensured that it was nol equivalent to a course exam. Instead it was more difficult
to do well on, especially without in-class review and specific preparation. Ann reported that no one in her
prior classes had taken advantage of this offer so she was surprised that more than a third of this class (4
students) chose to take it. My being available to provide preparation help was perhaps a factor. AH scored
below their course average (at least one letter grade below). Two of the students scored just well enough on
the exam for it to replace a lower course grade and ensure that they moved up into a higher final grade
category (Lee from a B* to an A' and Pierre from a C* to an B" ).
'"' My initial impressions of students were similarly affected. Meeting participants individually and
gathering multimodal data about them modified my first impressions however. See also final discussion of
student-instructor interactions.
'" Starting at the left front and going anti-clockwise, Ellen and Pierre, Lee and Robin, Jamie and Catherine,
Floyd and Brad, Autimm and Karen, and Mulder and I interviewed each other. As we were beginning,
Kelly rushed into class late, so she joined Mulder and me. Kelly and I interviewed Mulder and Kelly
introduced Mulder to the class, Mulder interviewed me and introduced me to the class. 1 interviewed Kelly
and introduced her to the class.
'™ The four types of measurement scale are: nominal, ordinal, interval, or rational scales.
'"" The blank columns were labeled X - mean, and (X - meanf respectively This sheet was designed for
students to compute the deviations of scores from the mean and then the squared deviations in order to
compute the sum of squares (of differences of scores from the mean) and from that the variance and finally
the standard deviation of scores from the mean. This procedure and
thus the formula from it, namely/ IXX - mean)' (for a sample; for a population the denominator is A^
V n-l
are labeled "empirical" because they reflect the actual process for finding how the scores vary from the
mean. Ann alluded to the alternative "computational" formula that does not, and told students "I like this
[empirical] way."
™" Overall, an average score of 3.2 (on the 98 items of MARS) is at the 95* percentile (Suinn, 1972),
indicating extreme mathematics anxiety (for further discussion of these scores, see chapter 6).
195
'^^ Each student prepared his own formula sheet to be used when doing part II of the exam, the
"computational" part. We could include formulae and descriptions of symbols but not their definitions, as
well as visual layouts for a procedure such as the labeled columns for finding the standard deviation, but no
worked examples.
"" 1 had obtained permission to use it from Dr. Garfield and from the Office of Sponsored Research's
Institutional Review Board for the Protection of Human Subjects. See Appendix C.
™" The cumulative frequency of the class group just below the class group from which you are trying to
finding the score for the corresponding given percentile rank. To identify this correctly required students to
first create a cumulative frequency column, interpret the subscript L to mean below (although L refers to
lower limit of the current interval wtien it is the subscript in Xl), and then find the cumulative frequency
immediately below the one for the interval in focus.
'™™ To compare two independent sample means, the null hypothesis is that there is zero or no difference
between the two population means (\x\ ^d H2 for independent samples), so that in the / test formula, (n, _ H2)
= 0. For the population mean of the differences for correlated samples the null hypothesis states that there is
zero mean of differences (hd)> or that there is no change, translates in the t test formula that hd= 0.
'°'™' For Question 3 in Exam #3 the correct decision was to reject the null hypothesis and conclude that
there is a relationship between amount of relaxarion and hot or cold baths. This decision is based on the
magnitude of the / statistic being greater than the magnitude of the critical value of the t with which it is
compared.
'""'' Ann's thinking might have been, in this case, that the importance of coming to the correct conclusion in
research justifies a severe penalty for the wrong one, in addition to the penalty already incurred for making
mechanical errors.
Robin asked "on task" questions and answered Ann's questions correctly approximately twice as often
as she questioned or answered incorrectly, tentatively, or off task. This was not substantially different from
Brad, for example, who made almost the opposite impression on Aim and me in class. Robin's almost
constant frovm of puzzlement and flustered air seemed to be related to her relative difficulty with auditory
processing of verbal material and her compensatory propensity to ask questions or check her understanding
whenever she was uncertain that she "got" it. Robin also seemed to be exhibiting the well-dociunented
tendency of women to be considerably more tentative about what they know than a man typically is (and
Brad certainly was).
'°™ In contrast are closed study groups to which students commit at the beginning and other students may
not join following the commitment period. These groups then have a consistent membership. Lack of
attendance may lead to a person being excluded.
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CHAPTER VI
UNCOVERING MATHEMATICAL RELATIONAL PATTERNS: THREE
PSYCH/STAT 104 CASE STUDIES
I have described the class in its context in the previous chapter; now is the time to
zoom in on the courses of brief relational mathematics counseling with the participants
from the PSYC/STAT 104, the focus of this study. What actually happened? As I looked
at the participant and at me and at us in a way that was different, that is, relationally, and
we explored the participant's relationality about mathematics as I supported her doing her
statistics coursework, what did that look like? Was it different in process or outcomes
from a traditional series of tutoring appointments? If so, how? In this chapter I present
three counseling cases in order to address these questions. Initially I wrote each case as a
profile of a student in the process of mathematics counseling within the context of the
class. But then I realized that although the student is the focus of attention in traditional
mathematics academic support, with this new relational approach I, as the counselor, also
came into focus. It struck me that it was, in reality, we — the student and I, and our
developing relationship — who were the object of this study. Before I present the cases
though, I will briefly review the counseling activity in the study and explain fiirther my
rationale for choosing Karen, Jamie, and Mulder from the ten.
Each mathematics counseling participant and I undertook the task of
understanding mathematics relational patterns (in particular central mathematics
relational conflicts) and pinpointing issues salient to a good-enough resolution of that
conflict while she' was taking the statistics for psychology course. The approach we used
was different from the typical treatments in its focus on joint understanding: That is,
students' class assessment results and survey responses became the object of discussion.
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modification, and deeper mutual understanding rather than pronouncements that locked
them in — in their minds and in mine. The relational counseling explored both conscious
and unconscious forces the student and I were experiencing, and the cognitive counseling
stressed continual conscious interventions using the insights we gained.
The ten mathematics counseling participants had between three and eight
individual sessions each, averaging close to five per person. I expected that only students
who were anxious or saw themselves as "bad at math" would volunteer to meet with me
for individual mathematics counseling. Instead almost the whole class signed up. The
group included students who were extremely anxious, some who were not particularly
anxious, and those who were somewhat ambivalent. Some wanted help with the
mathematics while others who did not think they needed mathematics help signed up to
help me with my research. Some might have accessed mathematics academic support if I
had not been in the class; others defmitely would not.
I found that the distinctions among the participants that were most indicative of
the soundness of their mathematics mental health were the level of mathematics
preparation (in terms of arithmetic [number and operation sense in particular], and in
terms of understanding of the algebraic variable), which seemed to directly affect their
mathematics self-esteem and interact with that to produce their particular condition of
mathematics self It was in talking about their mathematics learning histories and seeing
connections between those histories and their present patterns of mathematics
relationship, that participants' central relational conflicts around mathematics became
apparent. These conversations raised to the surface participants' and my awareness of
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these conflicts and supported some resolution. They provided key factors both for the
course and for their mathematics selves that could profit from brief therapy.
THREE CASES STUDIES
At the end of chapter 4 I alluded to my approach to choosing Karen, Jamie, and
Mulder as focal participants. Here I will explain more fiilly. Karen and Mulder were
mathematically underprepared students who acted quite differently but whose relational
patterns seemed to stem from a similar source. They, Karen more than Mulder, were
among the students most cognitively and relationally vulnerable to withdrawal, failure, or
inadequate grades — the students whom mathematics learning specialists most struggle to
imderstand and help in order to avoid disaster, often to no avail. Jamie, whose
mathematics background was more substantial than Karen and Mulder's, was, however
also surprisingly vuhierable to failure, even with relatively sound cognitive preparation.
She had serious relational challenges that jeopardized her chances of success. Karen
might, Jamie might not, and Mulder probably would not have accessed the traditional
mathematics academic support offered by the college. Each had mathematics learning
issues that emerged from different dimensions of their mathematics relationality. All
three had learning styles that had affected their mathematics relational patterns differently
and impacted how they were negotiating the present course. Though each is unique, taken
together, they represented atypical range of student issues that the Leaning Assistance
Center sees.
I faced quite different challenges dealing with Karen, Jamie and Mulder and
understanding myself in relation to them. I experienced Karen's holding me at arms
length as a challenge but I also found it finastrating and worrisome — I had to be content
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with her setting boundaries that I had to respect even when I beHeved they might be
counterproductive to her progress. Jamie's shyness and obvious discomfort when in focus
evoked my sympathy and protective impulses at the same time that I felt I needed to
tiptoe around her, anxious that I might harm her. Mulder was opinionated and stubborn
and he and I sparred — I found myself on the side of the opposition — which felt as if it
included Ann, the instructor, and perhaps his Mom. Each taught me about myself as a
tutor, a counselor, and a person; each learned about him or herself as mathematics
learners; and we all overcame mathematical and personal challenges to achieve success in
PSYC/STAT 104. Before I tell our stories I will quickly review the theoretical bases that
formed the framework for the relational counseling I employed.
Theoretical Bases and Case Presentations
The theoretical bases for brief mathematics relational counseling were discussed
in chapters 2 and 3. Essentially my approach involved embedding cognitive
constructivist, problem-solving, strategic tutoring in a brief relational conflict counseling
framework. This was a dynamic process that differed considerably from participant to
participant. What emerged from each participant's course of counseling, however, was a
common phenomenon that, while providing me with a pivotal key to understanding his or
her central relational conflict, also gave me a central organizer for presentation of these
three focal cases. That key was each participant's metaphor for mathematics or
themselves doing mathematics.
In presenting the cases then, after introducing the participant and me and our
relationship, I begin with the participant's metaphor and discuss the mathematics
relational implications of the metaphor that we discovered. This discussion leads into
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consideration of the participant's mathematics relationality and how we understood and
worked with it through the course of counseling. The participant's present ways of
relating with me, the instructor, and mathematics — his or her relational patterns —
illuminated each of the dimensions of relationality that Mitchell (1988, 2000) identified
and that I adapted to the college mathematics learning support context, namely, the
mathematics self, mathematics internalized presences — teacher/s (or parent) and
mathematics, and teacher and mathematics attachments. Disturbance in the development
of one or more of these dimensions led, for each, to present mathematics-related
emotional conditions, understanding which, in turn further clarified for us the
participant's relationality and central relational conflict. Understanding a participant's
central relational conflict, in the context of his or her mathematics relationality, helped
me develop a counseling focus. Finally, I follow discussion of this counselmg focus with
a summary of the course of counseling, session by session, to illustrate the processes,
demonstrate the changes we made, and present outcomes.
KAREN'S COURSE OF COUNSELING
Karen ''had to pass [PSYC/STAT 104] this time." I found this out by the vending
machines during break of the third class meeting. As we were choosing our snacks, I
commented on her being one of only two in the class apart from the study group to have
done an extra assigned homework problem. She told me then that she needed the class for
her psychology major but had failed it two summers previously. She sounded somewhat
desperate. Even then, before I had met with her one-on-one, after observing her only over
two and a half class meetings and despite her doing the homework problem, 1 had an
ominous feeling about how she would do. 1 had already observed her keeping all
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classroom personnel at arm's length, including the instructor and me. She seemed to be
positioning herself defensively. Karen's working alone during in-class problem-working
sessions seemed intentional and she had not attended the study group; instead she came
early to the classroom, sat at the back, and worked on her own while the study group
worked with me on the board (see chapter 5, Figure 5.1 and Appendix F)."
I was not sure how to interpret her signing up for counseling. It seemed
incongruous with her distancing stance but consistent with her expressed need to pass this
time although she did limit herself to signing up for once every other week not once a
week, which was the option I expected from someone who had already failed the class. I
wondered how it would be. I wondered if mathematics counseling would be any use. I
was worried that the task, that Karen's needs and her defensiveness, would overwhelm
both of us. I was anxious that Karen would especially resist my relational counseling
approaches but knew that these approaches had the potential to help her succeed this
time.
What I did in counseling was to go ahead anyway, tackling the statistics and
working side by side with her as we looked at the mathematics, I heard her voice and
together we challenged her negative sense of herself doing mathematics. At the same
time we evaluated the grounds for her defensive relational patterns. And I realized that
my initially overwhelming negative sense of her doing mathematics was also challenged.
Karen made better and better choices as she discovered a competency she had not
previously recognized and teacher support she had initially rebuffed. Her expertise and
confidence increased and her grades improved from a D' on the first exam to B^s and
'A's at the end with an overall 'B' for the course.
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As I worked with Karen I learned to attend to and manage my countertransference
reactions to Karen's initial defensive negativity. I experienced her transference as her
teacher who would "know" as she did the severe limits on the mathematics she could do.
She seemed very negative about her prospects for learning mathematics. "That's how I
am. I can't/won't be able to... I can plug in the numbers but I don't know why ..."
(Sessions 1, 2); and I felt firmly rebuffed as I imagined her former teachers did if they
tried to make a difference. In my countertransference I surmised that Karen's teachers
before me may have accepted as I had begun to do that she was unlikely to succeed; this
made me feel desperate and overwhelmed. But I (and she) challenged my
countertransference reaction and I chose to believe and act differently. By looking at
Karen from a relational perspective, I was able to help her find a real but underdeveloped
mathematics self and develop it further. By the end of the course, neither of us thought of
her any longer as someone who could not do mathematics. I was also able to challenge
her defensive detachment from Ann and me; Karen began to experience us as secure
bases on whom she could rely and from whom she could eventually venture out on her
own. Indeed, Karen still had mathematical challenges, true, but she could face them
knowing that she had found herself able to do well enough to succeed in this course.
Karen was a tall, blond, 22 year-old white, elementary school assistant teacher
who had dropped out of State University after a year and a half and was pursuing her
degree part-time at Brookwood State. She was the first in her family to pursue a
bachelors' degree although she reported that her parents had taken some post-secondary
technical courses. Karen wanted to become an elementary teacher and was majoring in
psychology but only because the university required prospective elementary education
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students to major in a non-education field (Class 1 ). In beginning this second attempt to
pass PSYC/STAT 104, Karen stated that she hoped for a B but expected a C/B (Pre-Test
Mathematics History Survey).
Karen's Metaphor: Mathematics as Cloudy
As the most representative of Karen's metaphors for mathematics: "black,"
"stormy," "cloudy," "bear," she chose "cloudy" "because there are some aspects of math
that are more clear to me, but mostly math is my worst subject and has always been hard
for me to understand" (College Learning Metaphor Survey). That she chose what seemed
to me to be the mildest image from her list surprised me. Karen's rather diffuse,
somewhat depressed, image of a cloud seemed to contrast with her almost aggressive
defensiveness, which made me expect her to select the image of defending herself against
a bear rather than seeing her way through a cloud. Still "cloudy" did seem congruent
with what I sensed as a resigned desperation, which to me felt as though she was
experiencing groping around in a cloud as fruitless.
As we proceeded with mathematics counseling I understood better what Karen
meant by her distinction between "more clear" and "cloudy" mathematics — it was partly
about the type of mathematics: "I'm better algebraically than I am geometrically. . .1 can't
do geometry at aU"(Session 1). But perhaps it was even more about Karen's sense of her
own limits: "[Mathematics is] my worst subject ...always hard for me to understand,"
She clarified this fiulher by responding "nothing" when I asked what she understood
about a new concept, explaining "see that's how I am"(Session 2). Her use of the word
"always" seemed to mdicate a long-term and global negativity about herself as a
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mathematics learner. When I asked her about it, Karen confirmed that 'always' meant
"Back — all through school" even in first grade.
Karen's Mathematics Relationality
Student-Mathematics Relationships: Karen 's Mathematics Self and Cloudy Mathematics
Since first grade Karen said she had found mathematics cloudy, "hard to
understand." In mathematics counseling when I asked her as an adult about her
mathematics metaphor her first statement to me was, "I hate math." I wondered what her
experience of mathematics had been through school for this to be the outcome.
JK: How have you been historically with math, you know, through the grades?
Karen: It depends on what kind of math it was. If it was like geometry or
something like that, I did horribly {okay} but Algebra and Algebra II, I
didn't do too bad on. I just don't like math {yeah} at all. I never ever, ever
have.
JK: Even in elementary school?
Karen: Nope I've always, I like reading and writing not math or science
At our first meeting Karen expressed an antipathy to mathematics requiring
interpretation of visual material (e.g., graphs and diagrams): "I'm better algebraically
than I am geometrically. I can't do geometry at all" and later "I hate those bell curve
things." Karen told me she had turned away when Arm had drawn a bell curve in the last
class (Class 3) because she disliked them so much. She believed, however, that
conceptual learning of algebra was beyond her. "If it's algebra, and it's just a matter of
plugging numbers into certain formulas, I can do pretty well with that... I can plug all
those things into that and I have no idea why, or what that means" (Session 1). I
mentioned to her that the study group had been working at understanding how and why
the percentile point and rank formulas worked and suggested that she might feel more in
control if she understood. Karen demurred, "Not necessarily; sometimes it's easier if I
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don't know why — I can just do it" (Session 1). I interpreted Karen to be saying that an
attempt to understand the procedure might undermine her tenuous grasp of how to do it. I
wanted to help her discover that she could understand, at least how this formula made
sense, but she did not want to risk it.
It was clear to me from these data that Karen's mathematics self-esteem was quite
low. She communicated that by describing her low confidence in her mathematics
capabilities ("That's how I am."), her low expectations ("I'll bomb the conceptual
portion."), and ahnost global negativity — possibly to protect her mathematics self from
further disappointment. She had little of what self psychologist, Kohut (1977) calls "a
storehouse of self confidence and basic [mathematics] self-esteem that sustains a person
throughout life [in the mathematics classroom]" (p. 188, footnote 8).
How realistic or accurate were her negative self judgments? Did she have enough
arithmetical and algebraic competence to build new learning on? Was she actually more
firmly attached to mathematics than she believed or felt? I gathered a more systematic
picture of Karen's arithmetic and algebra competence during posttesting and this
confirmed what I had found through observation of Karen's work in counseling and the
classroom during the course. Particularly with fractions and decimals, Karen's number
sense and operation sense were very weak (see Table 6.1). This made it difficult for her
to evaluate the appropriateness of the numerical results of her data analysis or to
troubleshoot her work in order to self-correct an error. In addition, Karen was operating
at a level 2 understanding of the algebraic variable, and here she was the lowest in the
class (see Table 6.2). That meant that she was able to coordinate operations with letter
symbols as objects but that she did not understand letter symbols as specific unknowns or
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generalized numbers (and in some cases as variables) and could not coordinate two
operations on them.'" Given this Karen was likely to find understanding and using letter
symbols in complex statistics formulae difficult. How she prepared the formula sheets to
use for exams could be crucial.
Student-Teacher Relationships and Cloudy Mathematics:
Karen 's Mathematics Struggles
I wondered how Karen's mathematics self development had proceeded for her for
her to have such crucial mathematics deficits and to feel so negative. What part had her
family and teachers played? Perhaps there was a family connection to her "always"
fmding mathematics "hard... to understand", I thought. It seemed that she had never
reflected on it before, but now she began to see it.
JK: What about your parents? Are they more like that [reading and writing,
not math people] too?
Karen: Yeeeah? [considering] Yeah=, yept (Session 2, see chapter 4, Figure 4.3
for coding conventions used)
Because of what I experienced as Karen's reticence in talking about anything
personal, I took the enthusiastic agreement I heard in "yept" to indicate that yes, she had
experienced her family culture as one where her not having an interest in nor doing well
in mathematics were accepted, perhaps even expected. I brought it up later and Karen
said, "I'd say we're more of the reading, writing type, the whole family" (Session 5), thus
confirming her sense that doing well in mathematics was not part of her family scene.
When she did do well on an exam (Exam #3) they were all surprised and delighted at her
success.
What about her teachers, then? What was their part in the development of Karen's
mathematics-as-cloudy self? I asked her:
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JK: Any teachers in math who, you know, who made you feel bad or better
about yourself?
Karen: No, not really. I mean 1 was never like worst in the class, you know. I was
always in the middle, middle to lower scale but I suppose I concentrated
less because I didn't like it as much so, you know? (Session 2)
Karen seemed to have managed to get by in class by being unremarkable. She was
not the worst so she did not attract negative teacher or peer attention, and she was
certainly not the best. But it seemed that she had not received positive attention either.
She had managed well enough to avoid attention, despite her perhaps defensive
"concentrate [ing] less." If how she was relating to Aim and me was any indication, she
had defensively kept her distance from them and teachers had let her be, accepting her
limitations as real and essentially neglecting her mathematics self development. This in
turn likely led to Karen's blaming herself, seeing herself as intrinsically bad (at
mathematics) and not seeing the teacher as responsible (cf Fairbairn, 1 972).
Confirming this, when I inquired whether there had ever been a negative incident
with a teacher she shifted the answer to herself by implying again that her present
mathematics situation was of her own making: "I was not interested in math at all. I don't
like it. That's why I don't do as well" (Session 5). Karen seemed to be using lack of
interest to avoid acknowledging what she really believed to be the reason: her underlying
lack of ability. Karen never spoke of a relationship with a mathematics teacher in either a
positive or negative sense. The only teacher Karen spoke of at all was her instructor from
the first time she took PSYC/STAT 104, and then it was to compare her teaching
approach with Ann's.
I considered the absence of direct information from Karen about her experience
with teachers, despite my probing, and realized that how she related to Ann and me in the
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present course might give me the clearest sense of the relationships she had with
mathematics teachers through the years. Karen was upset that Ann's teaching was unlike
that of her previous teacher for this course whom she described as "more thorough." She
believed that Ann would, nevertheless, expect her to know and use all the material in the
text even if it had not been covered in class. Later I realized that the discrepancy Karen
found most disturbing between the teachers was that her previous teacher had
demonstrated on the board how to do each type of problem (perhaps her idea of
"thorough") while Arm had each student tackle the problems herself in problem- working
sessions that were sometimes lecture-guided but more often accomplished with her
roving coaching help. Because of Karen's lack of confidence in her own ability — based
on her low mathematics self-esteem Ann's approach made her feel anxious and
insecure despite what I perceived to be Ann's adequate coaching support. Most
prominent for Karen seemed to be a sense of Ann's not being there for her in a way she
felt she needed. She seemed to feel abandoned. Past experience with mathematics
teachers appeared to have promoted her adoption of defensive detached patterns that
seemed to have been activated in this class.
At our July 10 interview (Interview 2), Ann had expressed disappointment with
her relationship with Karen: "I thought we would be closer." Karen sat as far at the back
of the classroom as possible and she did not connect with Ann outside of class time. On
several occasions (at least once in Ann's hearing) she expressed hostility about her
perception of what had been said about what to expect on the next test in contrast with
what Karen believed should have been said. This had to contribute to Ann's sense of
Karen's hostility and deliberate distancmg. I considered this aggressive detachment to be
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largely unconscious rather than deliberate on Karen's part. Keeping her distance seemed
to be Karen's established defensive way of negotiating a situation that exposed her
vulnerable, underdeveloped mathematics self
I began to believe that Karen had not ever developed a secure attachment to a
mathematics teacher. No mathematics teacher had offered herself as a secure base in a
way that she felt safe to connect with. She had learned to care defensively for herself and
expected little from the teacher. Such low expectations seemed to have made her angry
and anxious, even hopeless, because she knew she did not have what was necessary to do
it on her own and she needed support from the teacher. Aim's and my experience of her
defensively holding us at arm's length suggested that her demeanor may then have
become a factor inhibiting even good-intentioned teachers from reaching out to her.
Karen's defensive distancing may have been exacerbated in the college setting by the fact
the she was the first in her family to go to college. It was unfamiliar territory and she did
not have family experience and advice to help her negotiate it.
Emotional Conditions: Anxiety, Learned Helplessness, or Depression?
How did Karen respond emotionally to what seemed to be the underdevelopment
of her mathematics self? Was her reaction consistent with a diagnosis of
underdevelopment of mathematics self, expressed in underconfidence and defensive
detached relationality? Were her emotional responses interfering with her approach to
PSYC/STAT 104 to an extent that warranted emergency attention? The way I
experienced Karen at the first session felt confrising — I experienced her anxiety,
negativity (even hopelessness) and anger.
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Anxiety
Karen admitted to being very anxious before the first test. Her scores on the
Mathematics Feelings pretest survey confirmed that she consciously experienced
excessive anxiety in mathematics performance and testing situations. On her Survey
Profile Summary I had circled all three anxiety scales because they were all at almost the
top of the class range (see Appendix K, Figure Kl). The combination of her Abstraction
and Number anxieties, however, especially in conjunction with what I had observed of
her issues and approach did seem to be directly related to her underdeveloped
mathematics self, particularly her inadequate number and operation sense and low level
understanding of the algebraic variable (see Figures 6.1 and 6.2). Her testing anxiety
(second highest in the class) seemed also to be related to the inadequacy she felt when
she tried to recall how to do procedures she dimly understood. At least for the first exam
Karen's inadequate practice and unstrategic preparation contributed considerably to her
heightened anxiety. It seemed that the anxieties Karen experienced in mathematics
situations were normal reactions to threatening situations for which she felt inadequate. It
also seemed that her anxieties could be considerably alleviated by more strategic and
thorough preparation.
Depression and Helplessness
Karen expressed negativity about her mathematics self, mathematics, and this
class. I analyzed her responses to the Beliefs Survey that she completed during the second
class for underlying beliefs or constellations of beliefs that could better pinpoint her
negativity as well as others that could show healthy positive orientations. Karen's
average pre-score on the learned helplessness versus mastery orientation scale was
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worrisome: below the middle of the scale it was the third lowest m the class. On all three
belief scales, her responses fell below the class average. Nine of Karen's 14 Learned
Helplessness vs. Mastery Orientation responses were 2 or below, reflecting her belief that
learning mathematics involved having to be taught and then memorizing different
procedures for each new type of problem. This belief would make her helpless if she did
not memorize the right things. On the mastery oriented side, although she agreed that
some people could do mathematics while others could not, Karen believed that her
mathematics ability could improve, so it seemed that she was not locked into a frxed trait
belief about this ability. Karen also reported that when she could not immediately do a
problem she would not assume she could not do it and give up on it, and she usually tried
to understand the reasoning behind mathematics rules. Karen's negativity about her
mathematics self, world, and future did not preclude an underlying hope in the possibility
of change; she also had a view of herself not giving up when learning was difficult (see
Appendix K, Figure K2).
Karen's responses indicated that she was more motivated towards learning than
performance.'^ This surprised and encouraged me for Karen. Her focus was not just on
results; she wanted to understand the material. She did believe mathematics to be more
procedural than conceptual but her beliefs were not extreme (just below the midpoint)
and with her expressed learning motivation and strategic support to find she could make
the conceptual connections it seemed possible that her beliefs would improve (see
Appendix H, Table H3).
Karen's responses over time on the JMK Mathematics Affect Scales, however,
lent further weight to a diagnosis of entrenched negativity even depression. To monitor
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her negativity/positivity Karen filled in the scales at every counseling session except the
first. Karen's responses at the end of the second session were negative, all seven
responses falling at or below the mid-point. She was very much discouraged about her
problems with mathematics and she would withdraw from the current course if she could.
She expressed moderate to severe negativity about her mathematics self (scales 1, 2, 6,
and 7), about her current mathematics world — the class (scales 1 , 2, and 4), and about her
mathematics fiiture (scales 3 and 4) the three spheres Beck (1977) found to be significant
for people suffering from depression. As the course of counseling proceeded and Karen's
responses on the JMK Scales did not improve in proportion to her improving grades, my
awareness grew that it was mathematics situational depression (and related learned
helplessness) rather than anxiety that Karen was struggling with (see Figure K3, Table
K3, and Appendix B).
Identifying Karen 's Central Relational Conflict
As we began Session 1, 1 was already drawn into Karen's anger and anxiety. I
wanted simultaneously to rescue her from her plight and to defend Ann, the obvious
target of her anger. So that she might not be angry with me too, I tried to be on her side,
the fair, reasonable teacher she believed Ann wasn't. She kept her emotional distance
from me too though as if I were on the side of the opposition. I did not want to believe
her view that she was incapable of becoming more than a procedural mathematics
learner, although I worried that the time-limited situation might force me to help her
succeed only procedurally, thereby making her feel as if I agreed that she lacked the
conceptual ability. Her view of herself as a mathematics learner seemed to be globally,
diffusely negative, as if her mathematics self barely existed. Though she was trying to
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contain it using external means (blaming Ann, formula sheet, last minute tutoring),
Karen's sense seemed to be that this exam and this course were out of her control since
there was little inside her to draw on.
Karen's responses on the JMK Mathematics Affect Scales, taken with her low
indices on the Learned Helpless/Mastery Oriented Beliefs scale, her "cloudy" metaphor,
and her defensive detached stance in relation to peers, Ann, and me, pointed to a
diagnosis of moderate empty mathematics depression (cf Kohut, 1977, and see chapter 3,
pp. 91 ff.). This likely stemmed from Karen's deep sense of an underdeveloped
mathematics self rooted in her poor mathematics preparation and low self-esteem. Her
central relational conflict seemed to be between her strong desire and even need, to
succeed in this course and her fear that there were powerful forces outside her control,
including her own inadequacy and the instructor, which conspired to thwart that desire.
Her significantly underdeveloped mathematics self seemed to be the chief conspirator.
She seemed to be projecting her fear of her own inadequacy onto those around her.
Karen and Me: Dealing with the Clouds Now:
Relational Counseling for Karen
The Focus of Relational Counseling
I realized that, relationally, I had to provide myself as a guiding hand for Karen
to safely negotiate her way out of the clouds that she had felt trapped in. To help Karen
resolve her conflict I had to offer good-enough mathematics teacher-parenting to support
the emergence and development of a firmer mathematics self that could succeed in the
class. I planned to challenge her all-or-nothing thinking by mirroring her sound thinking
and achievements and at the same time I would provide myself as a mathematics parent
image that she could idealize and realistically incorporate into her increasingly competent
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mathematics self almost like her internal mathematics guide. 1 expected that this
development should go some way towards alleviating Karen's empty depression and
underconfidence.
I would have to work at overcoming Karen's emotional distancing enough that
she would accept my mirroring, though. To do this I had to resist agreeing with her about
her mathematics hopelessness. Although her transference of past teacher relationships led
me to believe that her low confidence was realistic, I had to resist that interpretation and
instead see it as unrealistic underconfidence; Karen was capable of doing mathematics. It
seemed crucial that Karen become free to avoid repeating her past experiences of doing
poorly in mathematics classes and failing PSYC/STAT 104. Importantly, this would
involve helping her recognize and take advantage of Aim as a secure mathematics teacher
base, rather than a neglectful but demanding teacher from the past.
Although her angry anxiety was a potential focus I decided that it was a symptom
rather than the root of her difficulties and could be ameliorated by helping Karen deal
more effectively with her sense that external forces controlled her course outcome. I
hoped that as her sense of her own competence grew, she would be increasingly able to
take more responsibility for strategic exam preparation, she could seek help from Ann or
me in a more timely manner, and she could make more effort to understand the
mathematics conceptually.
The Focus of Mathematics Tutoring
Mathematically I would provide myself to Karen as a mathematical co-explorer
with a flashlight and other tools that we could use to find our way through the cloudy
terrain. Given Karen's multiple mathematical concerns and her evident course
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management difficulties, I found that identifying a strategic mathematical focus initially
overwhelming but I soon focused on Karen's underdeveloped mathematics self I decided
to mirror back to her what I saw as her strengths in mathematics and her positive
approaches to the course. This was likely to help her begin to see her mathematics self
differently. I also set out to nurture and coach that developing mathematics self not only
helping her to develop further mathematics understandings and competencies but also to
recognize herself developing them. Then she might see herself finding her way through
the clouds into the clear light to day.
It seemed that if I worked beside Karen as she mastered new procedures
introduced in class and helped her link them to the concepts, and if she practiced she
would be able to understand enough and do new problems; she needed to also recognize
that she could. Karen's motivation for this deeper work could come from seeing her
growing ability to grasp these links herself Karen also needed to develop strategic
structures (guide ropes to hold onto in the clouds) to compensate for her underdeveloped
algebra, number, and operation sense.
Recognizing the useflilness of connecting the conceptual portion with the
computational part would give her increased control. The primary focus needed to
include her developing skill with letter symbols. Karen's mathematics self was affected
by her poor facility with decimals and percents as well as her underdeveloped number
and operation sense. Tackling this would, of course, depend on the time and emotional
energy Karen was willing to invest.
216
Karen's Course of Counseling: Session by Session
(see Appendix K, Table Kl for Karen's schedule)
Karen 's Sessionl
Karen came before her first appointment to drop-in and I observed her desperately
trying to practice problems she had not yet gone over. It was at drop-in that she angrily
denounced Ann for expecting the class to know all of the material in chapters 1 through 5
even though she had not covered it all in class. Karen had not gone to work that day
because she was not feeling well and it seemed that she had spent some time scanning the
chapters for material for her formula sheet (cheat sheet, she called it) and had become
increasingly upset as she found unfamiliar formulae and concepts.
Karen felt that Ann had not been clear about what would be on the exam so
studying the right material felt beyond her control. Karen did not interpret Ann's Exam
#1 study guide, her presentation of all the material in her notes, and having the class work
through specific problems from each chapter as likely cues that this was the material that
would be tested. All of the students were feeling some anxiety about this first exam, but
Karen seemed to be particularly misreading the situation. I wondered whether her failing
experience in the previous PSYC/STAT 104 class was so prominent at this point that it
was interfering with her ability to read the cues Ann was giving.
Her appointment with me was at 4:00 p.m. at Greenville campus and the exam
was scheduled for 6:00 p.m. at Riverside campus. I had her continue with the problem
she began in drop-in. I had reassured her that, based on the exam review guide Ann had
distributed, and my sense that Arm had been careflil to cover in class all that she would
examine, that her she could safely ignore the other material in the chapters and erase that
unfamiliar formulae from her sheet. She was already confident in the direct process of
217
finding the percentile rank of a given score procedure from chapter 3 of the text. This was
the one she had done correctly for homework: "Percentile rank, I've never gotten one of
those wrong." But she was not confident of this inverse percentile point procedure, the
one that we had done in class. After Karen did another of these to reassure herself, we
checked the exam study guide for the list of symbols. We reviewed her understanding of
the symbols to be tested for both name and meaning and she was quite confused." Karen
was not aware of the Greek versus "English" (Roman) letters distinction between
population parameters" and sample statistics,"" which I showed her. Although pleased
with this organizing idea for symbols and formulae, Karen was still somewhat
overwhelmed with the discussion of the concepts the symbols and formulae represented.
As we proceeded I began to understand that not only was algebra cloudy to her,
arithmetic was too. In finding 50% of 54"" Karen was content with 1 .08 (she had divided
54 by 50) as an answer; it became clear she did not understand percents, not even a
benchmark generally known.'" I wondered how pervasive were her arithmetic
uncertainties and what effect that might have on the current course. With less than half an
hour to go Karen announced "I have no clue on chapter 4 or 5." I accepted that global
statement on face value and anxiously joined in her desperate but seemingly impossible
race against time to cover that material before the exam.
Karen 's Session 2
"Horrible!" was Karen's response to her 62% on Exam #1. She was disgusted
with getting the range wrong when she knew it; it was such a "simple concept!" Karen
also reminded me that she hadn't been feeling well. The next exam was to be in two days
time so I felt some urgency to begin breaking down some of Karen's negativity towards
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Ann so she could take advantage of the structure and support she was offering. I also
planned to help Karen recognize what she could do to begin to break into her global
negativity about the rigid limits she placed on her mathematics self
Right away I asked her:
JK: How did you react to the exam itself? Better than you thought it was?
{Yes} just what she [Ann] covered {right} rather than the whole book?
K: Rightt
Karen agreed without hesitation that her fears before the first exam [that Ann
would examine material not covered in class] were baseless but later in the session she
brought up an assigned problem she had struggled with at home that she was pretty sure
Ann had not covered in class.
Karen: It's number 13 {number 13} right, and I don't remember doing that, using
this formula.
JK: Is this? Is this? No [hesitating]
Karen: So we don't even need to do that one then?
JK: Where is this? Is this on the list? Is this on our list to do? Do we have that
on our list? {Yeah} you can do it but [the problem m'os expecting student
to go beyond procedures taught in class]
JK: Where's your list? I saw you had it before. You seem to be neat, keep your
things in order.
Karen: Chapter 6 one through i *. No, it's nooot (-). See, I don't pay attention (-)
{laugh}
JK: She's fairly careful which ones she picks [to assign as homework
problems]. So I was thinking, "Why would she give us that?" So number
14, let's do number 14 (Session 2)
Karen had further good evidence that her fear that Aim might set her up with
impossible tasks were ungrounded and I was able to take advantage of the situation to
help Karen notice Aim's thoughtflil planning designed to avoid such student frustration.
Maybe Ann (and mathematics teachers) was more trustworthy than she thought. Maybe
Karen could begin to consider trusting her.
219
Exam #1 Analysis
When we analyzed Karen's exam together she had done better than she had
expected on conceptual section of the test on that section. She only missed one out of the
8 symbol questions" and contrary to her expectation "If anything, I'll bomb the
conceptual", was correct on 75% of the conceptual questions — it was the computational
section she failed.
Questions involving decimals gave her trouble,"' so I advised Karen to arrange an
extra meeting to do decimal exploration, "" since understanding and computing statistics
involves a lot of work with decimals, it was likely that her anxiety and negativity were
linked to her arithmetical uncertainties, and we were fmding that Ann stressed
arithmetical accuracy in her grading.
Karen had calculated all but one of the initial basic procedures accurately, and
also succeeded on the direct, percentile rank procedure that she had practiced thoroughly.
She was one of only four in the class to get this question entirely correct. Her response to
the inverse find-the-percentile-point procedure showed her understanding of the concept
but her anxious practice at drop-in and in the counseling session was not sufficient for her
to reproduce the required procedure and she earned no points. She saw that what she
failed was material she had not practiced at all from chapters 4 and 5.
This analysis revealed a mixed picture. Karen saw evidence that when she
practiced sufficiently she could succeed and her global negativity seemed unjustified.
Karen had not come to a help session early enough for Exam #1 but this session was two
days before Exam #2. Karen had realized she had not focused strategically for the first
exam so we planned to focus on problems like the ones done in class. Her defeatism
220
about her ability to do the mathematics and interference caused by repeating the class
seemed to have contributed to her difficulties on Exam #1 so in this session I began to
address these issues.
I asked Karen how her approach to the course work was different from what she
did for course she had failed. She seemed taken aback by my question. Her response was
"I think I just like, it just took me a while to get back into the [course] you know?"
indicating that she realized that she had begun preparation for Exam #1 later than was
wise. This seemed to contradict an earlier claim that she thought that she had been
prepared for Exam #1. Now it sounded as if she might be revising her sense of what
adequate preparation for an exam should entail for her. When I asked about her grades
the first time she took PS YC/STAT 1 04, 1 discovered that Karen had all her course
materials with her, including her test scores. The current test score 62% was considerably
higher than her 47% on the first test then, she had succeeded in getting one of the most
challenging questions correct, and she had overcome her confusion about the symbols, all
of which began to break through my internalized negativity about Karen's chances.
To prepare for Exam #2 we looked at the material that would be tested. When I
asked her what Pearson's r was, she responded, "Nothing (-). . .See! That's how I am. I
just plug in the numbers . . . That's why I have so much trouble." Her pronouncement
indicated a significant change from her earUer defensive response that it might be better
not to understand why. Now she conceded that not knowing was causing her difficulties.
Now I wanted to help Karen see that she could make her way through the clouds
and see clearly for herself I used a modified cognitive constructivist tutoring approach
and kept alert for relational opportunities to mirror her competencies. I provided myself
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as a model (a parent). The parallel modeling approach that we used looked much like
best-practice traditional tutoring but had the added effectiveness of intentional relational
attention.
As we individually set up and solved the problem side by side I talked through
it."'" We began by constructing a scatter plot of the data'"" and focused on identifying the
independent and dependent variables. I waited for Karen's decisions before revealing
mine. In the process, Karen found that her new understanding contrasted with her prior
confusion in class when Ann had briefly demonstrated the scatter plot construction
process, "Some of the points, either she [Ann] didn't do it right or I don't know where
she got them from." Although she was implying that not understanding it in class could
have been Ann's "fault" she also seemed to be conceding that it could also have been her
own issue.
Karen graphed the coordinate points without difficulty, but she was in trouble
once the scatter plots were drawn. As I questioned, coached, provided prompts, and
worked the problem beside her, Karen explored the relationships among the symbols and
their graphical representations and meanings. She gave no hint of her earlier anti-visual
position. She even reluctantly revived her hazy knowledge of coordinate graphing of a
straight line and explored that further, both graphically and algebraically. We did not
have enough time to calculate a standard error of estimate but Karen seemed to feel less
anxious about the upcoming exam. She had a much better idea of what to expect, she had
understood material she did not think herself capable of, she had two more days to
prepare, and she was a little more assured of Ann's care and good intentions. But just
before we left, Karen filled in JMK Mathematics Affect Scales (see responses labeled 2 in
222
Figure K3) and her responses were very negative (see discussion of Depression or
learned helplessness above).
Karen 's Session 3
Before Karen's Session 3 I had interviewed Ann and Karen arrived just as she
was leaving. Ann asked her how she was doing with her MINITAB computer module and
Karen had some questions so Ann offered to go with her to the computer lab to resolve
them. When they returned there was only half an hour left for our session. Ann resolved
Karen's concerns about materials for her presentation and left after she told us of the
research project she was launching the next day using an audiovisual presentation to help
elderly nursing home residents become more alert and care fill of their medications. This
encounter provided a natural opportunity for Karen to experience Ann's positive support,
an opportunity Karen would not have sought on her own.
Karen did better on the second exam but not as well as she hoped. When I
commented that her 76% was a lot better than her 62% on Exam #1 she demurred,
saying, "But they were so easy, the ones I missed." Her focus seemed to remain on the
negative. Unlike the first exam when her formula sheet had adequate column prompts for
formulae such as:
X
x-x
(X-X)^
to prompt the correct use of the sample standard deviation formula:
(x-xY
n-l
223
that she failed to take advantage of. this time her formula sheet did not have a necessary
column prompt so she failed to compute a statistic correctly."" Karen interpreted this error
as a procedural rather than conceptual failure on her part, but her failure to use her correct
formula as a prompt for the missing column indicated that she had not explicitly linked
the formula with each procedure she needed to follow — reasonable given her low level
understanding of the algebraic variable and the fact that this was the problem we had not
got to in Session 2 and that she had not therefore practiced. She had skipped another
question because she had not understood what the question was asking. Karen's strategic
preparation had improved but not sufficiently to compensate for her algebraic weaknesses
and because of this lack of preparation, her formula sheet was inadequate.
Karen was reporting on her exam, as it was not available (students returned exams
to Ann once they had looked at them — Karen had not arranged with Ann for her to give
the exam to me for our session.). Karen did not mention the complex questions I saw later
she had done and interpreted correctly on the exam. She had 78% of the computational
section correct — a significant improvement on her 52% on this section in Exam #1 and
on material that was mathematically and conceptually more complex.
At my suggestion we worked on an inference test problem worksheet Ann gave
out for students to try on their own in Class 1 1 (Mann- Whitney U test of separation of
two populations). The course focus had shifted from descriptive to inferential statistics
and Karen had grasped the strategic importance of following the same hypothesis testing
procedures for each test (using the step by step hypothesis testing procedure list Arm had
distributed, see Appendix G). Again I used parallel tutoring and diagrams to aid
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conceptualization and Karen struggled successfully creating a careful organizing scheme
to provide structure.
I noticed that Karen was fast and accurate at addition of whole numbers,
something I am not good at:
JK: ... they want you to add up the rank. [Karen circles the ranks that the c-
group got] there you go, there you go; there you go; right now add those
up
Karen: 16 ([immediately]
JK: Oh good on you! You did that boom! Wow quick! (Session 3)
And I found more and more opportunities to help her notice how she could move
out of cloudy misunderstanding into the clear light. For example, after Karen had begun
to add ranked scores instead of ranks we looked at them more closely and she saw it:
JK: so what is the difference? What is the difference between a score and a
rank?
Karen: Well this is just how, um, chronologically where each one falls and that's
the score that each one got {exactly} how many numbers, I mean how
many words they actually remembered
JK: Right, right and they have them ranked there because you put them in
order {right} but they are still each one are scores so the "rank" there is
almost an adjective {right} okay? {rightt}
**Time (1-2 seconds) as K adds up
Karen: I can't subtract worth a dime //
JK: But you're adding is like whoosh, right? (Session 3)
Again she added a string of numbers almost instantaneously, again drawing my
admiration despite her negative comment about her subtraction abilities, seemingly
designed to keep my attention on her deficits. She seemed unused to receiving positive
recognition for her mathematical work.
I continued to feel that Karen was maintaining emotional distance from me. In the
following interchange she seemed to be carefully considering each word so as to reveal a
225
minimum of information and she ended the exchange by abruptly turning back to the
statistics.
Karen: Usually I have this all done. I've been a little too harried
JK: oh^ oh! Other things going on in your life?
Karen: Well, no, Tve just been **, um, *** away.
JK: Away? (giggle) I don't know; do you find Fourth of July throws things off
a little bit {yeah = yeah=} oh, vacation! {yeptjand all of a sudden you
remember it's summer {mmm} and you think summer we're really not
meant to be doing this {right} is that what happens to you? {Yep} So
where'd you go?
Karen: Um I just went to visit *some friends {yeah} but um* Okay, so the ranked
score
Our interchanges about her former mathematics teachers and her family's
mathematics orientation were similarly limited. I was experiencing this as her not feeling
secure enough to trust me further and that seemed to have been a long established pattern
for her, at least with mathematics teachers.
At this session, because Karen seemed less stressed and more focused than in
Session 2 and she had done better on her second exam, I expected her responses to the
JMK Scales, to have become more positive (see Figure K3, responses labeled 3).
However there were only small positive changes on items 1 , 2, and 4, her sense of
hopelessness about her mathematics future had increased, and the other responses
remained the same. From Karen's point of view, it seemed her short-term fliture in this
class was not assured. She had made gains on Exam #2, but her control of the material
still felt uncertain. The last time she took this class, she had improved on the second
exam, too, but it had not been enough to pass. A D would not be sufficient; for her major
Karen needed, at minimum, a C. She was taking responsibility for doing more, but her
focus remained on mastering procedures, rather than concepts. She was no longer
blaming Ann, but now her self-criticism seemed, perhaps, too harsh.
226
Exam #3 was scheduled for the following Monday. Karen asked for an extra
appointment before the exam. This request indicated to me her growing sense of
responsibility and her ability to take some control by getting the help she needed when
she needed it. I wondered if it also signified that the strategic urgency she felt was
sufficient to override her defensive distancing from me at least with respect to the
mathematics.
Karen 's Session 4
Karen came to drop-in several hours before our scheduled appointment. I was
anxious for the students, especially Karen, on Exam #3 because it was the first one on
inferential statistics and hypothesis testing, and because of the number, variety, and
complexity of the inference tests to be examined (see chapter 5, discussion of Class 13, p.
175 and Exam #3, p. 179).
I offered Karen the materials I had prepared (see chapter 5, p. 179) and she
decided to use the flow-chart template to create her formula sheet for the exam.'"'
Karen's other organizer was her "Steps for Hypothesis Testing" list. During this drop-in
session, 1 was also working with other students, but I checked with Karen from time to
time to discuss and help her work through her struggles in deciding which test to use for a
particular situation. Deciding between the two-sample mdependent / test and the two-
treatment correlated t test was difficult for her so we discussed ways to decide which test
to use based on the situation described in the question.
At our scheduled meeting time. Karen had completed three of the six problems I
had given her, with only occasional help from me. I commented that she was much
calmer than she had been before the previous exams. She agreed to some extent but
227
qualified it. She seemed vulnerable to anything that went outside the structures she was
carefully building for herself.
You know if she changed a word or the order of the words just one little bit, I
wouldn't know what to do... that's what happened to me in the last test. She had
worded it differently so I sat there and looked at it [what I'd done] and I took it
out to her and she wouldn't say. . .1 was so confused. . .1 just left it because I had to
go on to the next one. (Session 4)
I asked her what statistical test the next question called for. She decided on an
independent t test and I asked about her reasons. She hesitated and asked for time to look
at the question, then said firmly, "It is an independent t test because she has given us the
two s- s." I affirmed her choice and her thinking. In retrospect my response surprised me.
This was quite different from my response to similar reasoning by Mulder. When he used
this reasoning I remonstrated, insisting that he link his decisions with the logic of the
setting by determining whether they originated from two different groups of subjects or
from one group of subjects tested twice.
Why did I not do that with Karen? I had had that discussion with her during drop-
in earlier but I didn't bring it up again now. I think I was responding to my sense that she
was carefiilly building up a fragile personal structure for negotiating the exam that I
hesitated to challenge too forcibly."^" I did not want to risk upsetting the procedural
control she seemed to be gaining over the material for the imminent exam by pushing her
to make these logical links.
For the remainder of the session Karen worked problems while she carefully
organized her formula sheet and I quizzed her on the definitions and the sample versus
population categorizations of each letter symbol. By the end of our session Karen had
completed at least one problem for each of the inference tests being examined.
228
I noticed out loud Karen's impressive grasp of the material and at first, she denied
it, attributing her success to external factors such as being in mathematics counseling not
in the exam room or "cheat[ing] in the book." I challenged that thinking pointing out that
it was not me helping her; she was doing it herself. And she challenged herself noting
that she hadn 't used the book only her formula sheet that she could use in the exam.
Karen: So I just know that this and that are the same thing!
JK: Right {alright (+)}. You are doing quite sophisticated mathematical
thinking!!
Karen: Yeah, but when it comes time for the test I'm not going to remember it (-).
Maybe if I look over it just before.
JK: You will. You are able to do this, you know. I'm not helping you at all
you're just doing it yourself
Karen: See this is going to be my problem. This was already done you know
what I mean? I cheated in the book. No, well I just looked at this [decision
chart formula sheet that she had been adding to and could use during the
exam] actually
JK: You really did and you'll have this on the test. Right? And you've become
aware of how you might be tempted to choose one [statistical test] rather
than another! Right? So I think you probably won't, right?
Karen: 'Cause I could probably get halfway through the problem and realize {I
would think so} that it wasn't right (Session 4)
Karen went on. She realized she had now intemaUzed the material and ways of
troubleshooting on the exam if she got into trouble. It was remarkable change from how
she had experienced her confiision during Exam #2. Karen was going into Exam #3 a
very different person from how she had gone into Exam #1 . She had prepared
strategically because she was comfortably aware of what would be tested; she could
compensate for her algebraic and arithmetic deficits; she had a carefiilly prepared and
strategic formula sheet, and now she had become aware of her own grasp of the material
and her ability to monitor and troubleshoot if things went wrong.
229
Discussion of Sessions 3 and 4 and Exam #3
My Relational Focus in Counseling
Karen's responses to the JMK Mathematics Affect Scales in Session 4 before
Exam #3 revealed that she felt considerably more able to make mathematical decisions
and she was more positive about herself mathematically than she was for the previous
exams (in Appendix K, Figure K3, compare responses labeled 4 with those labeled 2). On
the other hand, the material was much more complex, and Karen seemed to be relying
heavily on extrinsic clues and her formula sheet. Overall, her improved sense of
mathematical efficacy (items 5, 7) seemed to be breaking up her sense of hopelessness
(item 3), while the uncertainty of the imminent exam and her reliance on procedures and
extrinsic artifacts seemed to have kept her discouragement (items 2, 4, 6) from
dissipating.
Karen earned 85% on Exam #3 and she was very pleased. She lost only 2
points'"'" on the computational part, 12 points (6 questions) on the multiple-choice and 1
point on the symbol identification. As noted in chapter 5, Karen was one of only two
students in the class whose grades improved on this test. With the in-class extra credit
assignment where she earned 6 points, her overall exam grade was 91%.
Even though Karen was gaining control where she previously had felt helpless
and discouraged, even depressed, and her mathematics depression appeared to be lifting
somewhat, I was concerned that she had not made an extra appointment to address her
underlying number and operation sense weaknesses. I realized that her teacher relational
detachment issues would have made it very difficult to seek that appointment. It had
likely been difficult enough for her to make the extra appointment before Exam #3.
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Supervision
In presenting Karen at my own supervision session with Dr. P., I expressed a
history of my thinking about the changes she had made that I realized, as Dr. P. heard,
were considerable. "She's doing better and she is growing into a competency she didn't
know she had."
Dr P. suggested that Karen might have a new metaphor for herself doing
mathematics, and that it could be helpfiil for her to assess her own change. At each
session, I had been prompting Karen to look at what she was doing differently and seeing
differently. He suggested that reflection should continue at the next meeting. I wondered
how to help her reflect more deeply. Her underlying arithmetical and mathematics issues
and our limited opportunity to explore and discover her real ability to think conceptually,
dampened my enthusiasm about her current success. Was her success "good-enough?"
For this class, perhaps! But I worried about its strength for restoring a healthy
mathematics self Karen's opportunity for reflection would come when she did the final
evaluations and posttests.
Karen 's Session 5
Again Karen came early to drop-in, this time with a little grin and feeling "good"
about her grade on Exam # 3. She told me that her family were "waiting at the door" for
her, obviously very pleased. Then she began doing homework problems from the text.
In the individual session, Karen tackled a question where she was asked to fmd
the values missing from a one-way independent-groups ANOVA summary chart and
answer questions about it. She was stuck on the question, "How many groups are there in
the experiment?" (Pagano, 1998, p. 378). She said, "I don't know what to do." This was
231
the type of inverse reasoning question that Karen found difficuh. I suggested some ways
to think about how those numbers were derived and what they meant and I coached her to
think backwards to find the number of groups and, in the next question, the number of
people in each group (assuming equal numbers in each group).
Karen and I discussed strategy for preparing for the exams. In response to my
query about completing homework problems prior to coming to the Learning Assistance
Center, Karen replied, "Ah no, I did those tests [problems] that you gave me. They were
way more helpful than doing all this homework."
Karen had never directly attributed her difficulties in mathematics to lack of
ability. In the Beliefs Survey she had agreed that her ability to do mathematic could
improve but she also agreed that some people can do mathematics and other people can't.
In addition she labeled herself (and her family) as a reading and writing type [not a
mathematics type]. I took the opportunity in the following discussion to bring up the
topic of ability.
Karen: Right, I was not interested in math at all. I don't like it; that's why I don't
do as well
JK: Yeah that's got a lot to do with it probably, not much to do with abilityt
Karen: Probably not
JK: More to do with interest and CONFIDENCE
Karen: Right
JK: Because as you don't do as well, your confidence goes down {exactly}
You THINK you're not good at it (Session 5)
Karen's attributing her not doing "as well" to her lack of interest sounded
defensive. When I mentioned ability as a possibility and dismissed it she gave only
qualified assent "probably not." That was when I told her of Liping Ma's (1999) work
studying American elementary teachers' generally poor grasp of arithmetic. I speculated
with her that she most likely had teachers who themselves had not understood any more
232
than how to do the procedures. She agreed. I could tell that it would take more than her
good grade, my mirroring of her competencies, my logic, and research findings to
convince her of her sound ability to do mathematics, but all of these were making rather
large chinks in her armor.
Karen was pleased when I noted approvingly that she continued to follow the
hypothesis testing procedure meticulously. She commented that, in class, the other
students weren't doing it, but she was. Karen was beginning to recognize that more of her
mathematical behaviors were positive, in dramatic contrast with her former almost
exclusively negative evaluation of herself doing mathematics.
We worked on an ANOVA together. At one point I had a formula incorrect but
Karen had it correct so our answers were different. I questioned her but she held her
ground and then I realized that it was I who had it wrong.
Karen: I get more than that as the first one
JK: Why are you squaring that again?
Karen: I've just got the sum of x-one [X|]
JK: Oh you've got the formula copied wrong [inspecting Karen's work]
JK: Oh no, you don't! I'm doing it wrong. You're doing it correctly!
Karen: They're really big numbers though. (Session 5)
Karen's caring attempt to reassure me that my mistake was understandable:
"They're really big numbers though," marked a reversal. Karen had experienced tolerable
disappointment in me, the idealized teacher-parent, at the same time as she realized that
she had it right. Her competent self was emerging and could care for me the parent.
During class problem-working sessions, Karen continued to show no interest in working
with anyone other than "experts" in the mathematics, in this case Aim or me. While the
parent image was still prominent it was being modified by reality and incorporated into
her mathematics values structure.
233
And Karen had changed her mind about Ann. Now she recognized Ann's efforts
on her behalf and her defensive detachment had been replaced by a sense of secure
attachment, as illustrated in the following exchange.
JK Oh you'll plug all those into this. I can't imagine trying to do- 1 know it's-
I hope she gives us one with a lot smaller numbers [on the exam], that
would be better but no matter what
Karen: She usually does. {She does} Even when she gives us the practice
problems she never has the [large number of large numbers]. I mean, the
book is ridiculous sometimes like these aren't the biggest charts I've seen
like way back when we were doing just frequency distributions like a
whole page was writing; it was really long.
JK: Yeah, that's right it was wild. I think they do that because
Karen: She even made a comment too she said 'i'll never make them as long as
the book does." (Session 5)
It was in this session that I noted too a distinct change in Karen's emotional
distancing from me. When I commented on different national views on mathematics
ability and I mentioned Austraha's, Karen talked at length of her girlfriend's visit to
Australia.
As with Session 2 we had not covered all that would be on the exam. In fact, even
more would be covered in class tonight that would be on the Wednesday exam. But I was
confident that Karen had it well in hand and she was too. Although I did not know it at
the time, this was my last meeting one-on-one with Karen.
Karen 's Post Counseling Processes
Karen earned 88% on Exam #4. She was very satisfied. She lost no points on the
computational and symbol identification sections. All her points were lost on the
multiple-choice (6 out of 23 questions incorrect — a consistent result; see Appendix K,
Table K2).
234
After Session #5 we had scheduled an appointment for the following week that
Karen cancelled. I was concerned that we meet before Exam #5 because I knew the exam
would require students to decide on an inferential test using a specific decision flow
chart. I knew the questions on the exam would not contain the specific clues I thought
Karen might be relying on such as the '5- 's for the independent samples t test so I
suggested she come to drop-in on the day of the exam; she did not come.
In our final session her self-reliance had been remarkable. After that she felt she
could handle the rest herself, and she did. I struggled with my countertransferential
parental concern. It was hard for me to let her go and trust that she was in a good-enough
place, that she could do it on her own but I need not have been concerned. In fact, I
should have been pleased at Karen's growth. She earned a 96% on that exam (although
she did fail to correctly identify the independent samples / test!).
Optional Comprehensive Final
Karen decided to take the optional comprehensive final after class ended to
replace her lowest exam grade. I offered an appointment by e-mail, but she declined,
which made me quite anxious for her again. This was not my countertransference alone.
Students' grades on comprehensive mathematics exams, even with review, are typically
one-half to one whole grade lower than on their other tests. Karen earned a 57%, which
was lower than her lowest test grade so it did not alter her final grade, a B. She had badly
failed the conceptual multiple-choice part but on the computational part of this test Karen
earned a 75%. Although this 75% was considerably lower than she had been getting on
computational sections, given that the exam was comprehensive, and that she took it with
no class or tutorial review, it was reasonable for her. Even her overall poor result was
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relatively comparable with results of others who took the comprehensive final (see
chapter 5, Table 5.1) and unlike her each of them had had a fmal review session with me.
I wondered whether, however, without discussion of this overall low grade Karen might
allow it to dimmish in her mind the real gains she had made in her mathematical prowess
(Appendix K, Table K2).
Evaluations
Karen 's Evaluation of Her Changes
Karen said her initial "cloudy" metaphor for herself doing mathematics may have
changed "a little" but she was not specific. She predicted she would not "ever like math"
but that she was "more comfortable" with it. Karen attributed her own positive changes to
"1 on 1" and to the "amount of time I put in outside the class" (One-On-One Mathematics
Counseling Evaluation). She learned that she could "do a lot better than I thought" but
still found the "conceptual" aspects of statistics puzzlmg and would "pay particular
attention to the conceptual portions" of the next mathematics-related course she took.
My Evaluation of Karen 's Changes
Karen took the Feelings and Beliefs posttests in class, and the Algebra Test and
Arithmetic for Statistics assessment after she had taken the optional comprehensive final.
These two tests confirmed my sense of Karen's weak arithmetical and algebraic
understanding (see Tables 6. land 6.2, respectively) and they also confirmed her need for
compensatory structures and strategies to achieve the success she did.
By the end of the course Karen's overall defensive and detached pattern of
relationship in the classroom setting (or possibly the college setting) had eased. She
engaged in conversation with other students during the problem-working sessions but still
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she would not work with or check her work with anyone but Ann or me. Her initial
angry negativity towards Ann had reversed. She had begun to forge secure attachments
with trustworthy mathematics teachers — Arm and me.
Karen's sense of herself as a mathematics learner had become a little more
positive. By the end of her last individual session Karen's discouragement responses
(items 2 and 5) on the JMK Mathematics Affect Scales had lifted (see Figure K3,
responses labeled 5). She also indicated that she was less likely to withdraw from the
course. Until Session 4 (just before Exam # 3), Karen's responses on all scales were at,
spanned, or fell below the mid-points. Now at Session 5, in 3 out of the 7 scales, her
responses were above the midpoint (positive) and the others at least touched the
midpoint.
Karen's moderate mathematics depression had lifted somewhat in the context of
the mathematics counseling and the current course. However, her mathematics depression
appeared to have developed over many years of school mathematics in an envirormient
focused on procedural mathematics learning which she had little hope of understanding
and where her developmental needs were neglected. This resulted in an underdeveloped
mathematics self: she was underprepared mathematically and her mathematics self-
esteem was therefore low. Ann's course forced her to tackle procedures on her own, and
a formula sheet was allowed, so it was possible for Karen to gain control and succeed.
She developed a more positive sense of her mathematics self and moved from an
unhealthy detached independence to good-enough mathematical self-reliance. For a
lasting improvement and success in a more conceptual mathematics course, I believe
Karen would have to understand arithmetic better and develop her understanding of the
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algebraic variable. She still had finite mathematics ahead of her, and she planned to take
it in summer 2001 . She said she would return to the Learning Resource Center for
support "as long as Jillian is there" {Follow-up e-mail Survey). I worried that might not
be enough.
The changes in Karen's responses on the post-course Mathematics Feelings
survey (see Appendix K, Figure Kl) and Mathematics Beliefs survey (see Figure K2)
surveys seemed largely consistent with her changes and her success in the course,
although there were some apparent anomalies: Although her abstraction and number
anxieties had decreased substantially, her testing anxiety had increased (see Appendix H,
Table H3).
Evaluation of Counseling and My Changes
When I met Karen I was immediately drawn into her anxious, depressed,
negativity. With her I saw her deficits and limits and heard her anger at Ann and
despaired of her making it and of my being able to help her. But as I incorporated
relational counseling assessments and approaches into best practice modified cognitive
constructivist tutoring and course management counseling, I changed my mind about
Karen and about me. As I helped her see Arm and herself differently I began to see her
differently. My expectations of her rose, my role changed from motherly rescuing to
guiding hand and co-explorer and she rose to the occasion. We found ways for her to
compensate for her significant background deficits and my admiration of her grew. Going
beyond tutoring to incorporate relational approaches led to her not only doing the
mathematics but also to her recognizing herself doing the mathematics, and her
underdeveloped mathematics self developed. I (and Ann) had provided the opportunity
238
for her to forge secure attachments to mathematics teachers and she had availed herself of
that opportunity
Evaluation Summary
Karen's mathematical relationship patterns had begun to change. Her mathematics
self was becoming firmer; she found she could gain control over the mathematical
material to a greater extent than she had ever thought possible. I felt she was still quite
dependent on teacher/tutor input and external judgment of her mathematical correctness
rather than on her ability to judge the internal consistency and logic of the mathematics.
But once she worked out how to use structure and strategic effort to compensate for her
mathematical uncertainties, she did it on her own. Her final reflections indicated the
movement she had made towards an improved sense of her mathematical self and
mathematical self-reliance: She wrote, "I became more confident as the course went on
and I came [to drop-in and individual mathematics counseling sessions] more for security
in knowing I got the answers right" {One-On-One Mathematics Counseling Evaluation).
She apparently felt she had enough of a mathematical self to do it herself; she no longer
needed me except to check that she was on the right track.
Epilogue
Karen did enroll in Finite Math in the summer of 2001 and she did come to the
Learning Assistance Center to get help fi-om me. Following her pattern of summer 2000,
she came first just before her first exam, overwhelmed with the amount of work, resentful
that her transitional object — a formula and procedure sheet — was not allowed, and not
having practiced each type of problem. The instructor allowed her an extra few days but
she still did very badly. Karen regrouped and began to come regularly to Drop-In. She
239
did not like my going from person to person at Drop-In and not attending solely to her so
she began to work on her own in the cafe and would come down to the Learning
Assistance Center during Drop-In just to ask specific questions and then go away again. I
suggested we meet to deal with her arithmetic issues, which surfaced again but she never
made that appointment.
Karen's mathematical self-doubt remained a problem: Although she felt confident
with Venn diagram counting questions, on a take-home quiz she erased and changed her
answers when another student had different ones only to fmd out later that she had been
correct. "I always assume that I am the one that is wrong." Her belief that the teacher was
against her also returned though I did not feel included in that this time. As I suspected
her mathematics depression had deepened again since the end of PSYC/STAT 104 but
she persevered, and I continued to confront her globalizing self-negatives with proof of
their fallacy from her own work. Again, her grades improved. She made and kept two
individual appointments before the fmal, when I was more able to take a mathematics
counseling approach with insights from our earlier counseling sessions. Karen was
organized and knew what she needed to learn. She was allowed to use a restricted
teacher-developed formula sheet and went on to earn a B' on the cumulative final and a
C^ on the course. This was quite an achievement because it was a more mathematically
demanding course than PSYC/STAT 104.
For Karen this was a good-enough outcome. All the mathematics requirements for
her degree were completed. She will probably not take up the challenge of dealing with
her underlying operation sense, number sense and algebraic deficits, which are at the root
of her mathematics depression.
240
JAMIE'S COURSE OF COUNSELING
Jamie needed help with her statistics course. She decided so herself. I know this
because she signed up for mathematics counseling with me for once a week, not once
every other week, which was an option, and later, in her end-of-the-course evaluation,
she wrote that her initial motivation for signing up for counseling was "so that 1 could get
a better grade in the course," unlike other participants whose initial motivation was to
help me with my research""^ (cf Mulder, Robin, and Autumn). But if I had not crossed
lines with Jamie that are generally drawn in the helping professions, it is unlikely that we
would have worked together at all. As a helping professional I had learned that I should
wait for the person seeking help to approach me; it is usually considered unacceptable to
pursue the student in order to provide help, no matter how necessary that help seems to
be. Jamie, however, despite signing up for weekly counseling sessions and despite an e-
mail exchange between us about when, slipped quietly away after class night after night
until finally I decided to sit beside her in class in order to arrange the appointment she
had indicated she wanted.
Jamie was a tall, dark-haired, white, traditional-aged fiill-time student at State
University who had just completed her sophomore year. Her father was an engineer and
her mother was also college-educated. As a psychology major, Jamie needed
PSYC/STAT104'" but thought it might be easier to do it here at Brookwood State in the
summer; the small class size and focus on only one course, she thought, should more than
compensate for the course being faster than in a regular semester (ten weeks compared
with 1 5 weeks to cover the same material). Jamie had withdrawn fi-om Finite Math in the
fall of 1999, without penalty because of illness, although she was failing at the time
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(Session 1 ), so the last mathematics course she reported that she completed was pre-
calculus in high school in which she earned a "C?'""' {Pre-Test Mathematics History
Survey, see Appendix C). Ann, the instructor, thought she was "VERY quiet" and used
the word "fragile" to describe her (Interview 2). Jamie wrote that she hoped for a B in
PSYC/STAT 104 but expected a C {Pre-Test Mathematics History Survey). Her summer
job was in a department store in a mall.
What struck me most about Jamie at the first class was her demeanor — she was
sitting straight up with her eyes lowered. At times I wasn't sure if she was asleep but her
expression did not seem to change and she did not make eye contact or interact with
anyone, except during the paired introductions interview when she told her interviewer
that she was "not keen" on mathematics or doing this course.
I found out that Jamie was cognitively capable and well-enough prepared
mathematically to succeed, yet in two attempts at mathematics courses in college she had
not succeeded. Jamie's personal and mathematical style and challenges induced her to
accept my offer of help but dissuaded her from accessing it. And hers contrasted
markedly with my personal and mathematics style and challenges. Mine induced me to
cross accepted helper boundaries to give her the help she needed but caused me to
struggle with helping her fmd her voice when mine was so loud and hers so quiet. How
we understood and struggled with, negotiated, and made use of our differences together
forms the substance of this account of Jamie's and my growth as tutee and tutor over the
summer of 2000. As I used the relational counseling approach that I delineated m
chapters 2 and 3, 1 looked at her and at myself differently from how I would have in my
former practice. Both Jamie and I benefited — she "realized it was more about my feelings
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and confidence in my math ability, th[a]n any real problems with the math course work"
and she earned a B^ and I learned how attending to our relationship helped me understand
her and myself better and modify my approach with a student who was so different from
me.
Jamie 's Metaphor: Mathematics as Stormy
By the time Jamie and I met for the first time in the fourth week of the course, she
had received the results of the fu-st exam and to her dehght and surprise, had scored a
95%. Nevertheless, her metaphor for mathematics was a [violent] thunderstorm. She
explained her choice: "stormy because it is usually very tough for me to do and
imderstand math, even though I did good on the test I'm afraid the 'storm' will come
back again" (archived College Learning Metaphor, see also Appendix B).
For a storm, Jamie said, she would, "prepare for it; before it comes, like, get your
water or flashlights." When I asked how she would handle the storm when it came, Jamie
replied that she would "stay inside." She saw how her storm preparation related to
mathematics: "Well, you have to prepare for tests," but she wondered "how staying
inside does." We did not initially explore what the storm itself was to Jamie — I assumed
it was mathematics itself, in particular, mathematics tests. I did not pick up then on the
connections between one of her other metaphors "shark," her use of the word "afraid" in
her "stormy" metaphor, and the link to my countertransference experience in the first
study group: my experiencing being potentially dangerous to Jamie (see chapter 5, Study
Group 1). I also didn't attend to her wondering what "staying inside" out of the storm
might have to do with her doing mathematics.
243
Over the course of the first three meetings Jamie told me of her stormy
experiences with previous mathematics classes. Her most recent experience, she told me,
was withdrawing failing from a finite mathematics class at State University'"" and high
school had been mixed. The storms began in elementary school, however.
Student-teacher Relationships as Stormy: Jamie 's Internalized Teacher Presences and
her Mathematics Self
Jamie's early elementary experience of mathematics sounded calm: "first grade
and second grade and stuff, you know, I got 'A's in everything," and she remembered
she'd liked her fourth grade teacher. Her experience of 5"^ grade had been different: Her
5* grade teacher "yelled" though not at her, and not particularly about mathematics.
Jamie attributed the start of her doing poorly in mathematics and science to the
frightening classroom situation tills 5* grade teacher created, though her reading and
writing achievement remained unscathed. She remembered:
But in 5th grade, my teacher kind of yelled a lot, and stuff, and I didn't do good
[in mathematics] . . . Science, I think, too. . . I did good in writing and reading, that
kind of stuff. . . It was from then on. . .1 think she had a short temper, I guess.
(Session 1 , June 20)
In Session 3, as I was asking Jamie about her shy, non-interactive demeanor in
Aim's class, the effect of her 5' grade teacher came up again. Jamie explained further
"You want to sit down and shut up so you don't bother her [the 5* grade teacher]." I was
struck with how closely this described Jamie's current behavior that I observed in class. I
was also aware of how much Ann, the instructor's, approach differed from Jamie's
description of this 5'^ grade teacher.
244
In high school, to Jamie's surprise ("because I don't do good in math") she "did
good" (a B or B^) in Algebra I. The storm hit again, though, in precalculus that she took
with the same teacher she had for Algebra I. Her experience in precalculus was so "bad"
that by the end she said she didn't understand anything and she found the teacher to be
"stand-offish, like, 'You should know this.'" She remembered needing little help in
Algebra I. It had gone smoothly ("I didn't do bad and good and I wasn't up and down"),
but when she did need help in pre-calculus, she (and the other students, she said) found
the teacher to be unavailable. Jamie conceded though, "Well, part of that not getting help
is partly me." Jamie's unwillingness to seek help from the teacher (i.e., Jamie 's
unavailability), she believed, contributed to her problem of not getting the needed help. It
seemed to me, however, from what she said that she had been inhibited, not only by her
"stay inside" relational pattern, but also by her observations of other students' difficulty
in getting a response from the teacher. She perceived this to constitute a negative change
in the teacher from her Algebra I experience of her. Mathematics teachers had become
potentially dangerous to her. It was as if she had internalized bad mathematics teacher
presences through whom she saw Ann and me or any mathematics teacher. And all her
difficulties she attributed to her own inability to do mathematics.
Student-mathematics Relationship as Stormy: Attachment
and Jamie 's Mathematics Self
I reflected on the probable effects of Jamie's stormy history on her sense of
mathematics self Her attachment to mathematics and to mathematics teachers had been
secure through fourth grade. Then in fifth grade her expected secure teacher base was
withdrawn: She could no longer safely explore and ask for or expect the support she
needed. Thus began her sense of isolation, separation from a secure teacher base and.
245
from then on, from a secure base in mathematics — she could no longer be sure that she
understood it, sometimes she did well, other times she did not, but she could not ask why
because she was no longer sure of the availability of the teacher.
Jamie and Me — Dealing with Storms Now: Relational Counseling for Jamie
As I reflected on what "stormy" meant to Jamie, these understandings clarified for
me the effects in this class of her current expectation and fear of these storms continuing.
Our differences became more apparent but I also became more aware of what I needed to
be for her. "Stormy" seemed to have multiple meanings to her, all negative, but the
consistent theme was absence of calm — teacher "yelling" or "ups and downs" in
understanding or grades. I, on the other hand, enjoy storms, especially the thunder and
lightning, and calmness bores me. I would have liked to persuade Jamie that "stormy,"
like mathematics, might have positive aspects — challenge, excitement, darkness lit up by
the lightning. As — but I gradually realized that none of these (probably including 'A's)
would feel agreeable to Jamie. If I could offer myself as a smooth, level path with no
surprises around the comer, only more of the same, or perhaps a gradual ascent, nothing
that would startle her or trip her up, that would be perfect.
Jamie 's Mathematics Relationality
Interpersonal Relationships and Self: Family and Personality Interacting
At Jamie's second session, just after she had taken Exam #2 but before she
learned her grade, I asked about her family's reaction to her 95% on the first test. Jamie's
Dad was pleased and had expected it to continue; her grade was proof to him that she
could do well in mathematics. Jamie saw it differently; this was not proof but rather an
246
anomaly, not likely to be replicated. She knew she had not done as well on Exam #2 and
she was not surprised.
She had negotiated her panic on Exam #1 when she found and corrected an error
so I presumed that that success and her high grade would result in reduced anxiety for the
next exam. On the contrary, Jamie said she had higher anxiety on the second exam
because of her family's (in particular, her father's) higher expectations. I realized that I
had to navigate my own assumptions and expectations of Jamie. As I proposed a
conjecture and learned to listen to Jamie's responses, including her hesitations,
qualifications, and tone of voice, she changed my mind and revealed herself Although
she never contradicted my conjectures about her, Jamie's unconvinced "maybe"s
contrasted with interested and curious "possibly"s; her hesitant "yeah4'"s contrasted with
her somewhat "yeah="s and her firm, in-agreement "yeaht"s, laughs, and "I know"s.
Jamie usually qualified her own theories with "I guess"s and "maybe"s but I had to watch
and listen to clues to how deeply she held these theories (See chapter 4, Table 4.3 for
transcription coding conventions I use.).
JK: And the anxiety in the second was just to do with that confusion about the
~ [Jamie had just told me that she had known how to do the various
correlation and regression computations on the exam but had been
confused about what each one were called.]
Jamie: Yeah=, that and I think I might have been more [anxiety] for the second,
actually
JK: You were more anxious on the second one?
Jamie: Yeah, I think so.
JK: That's interesting, (surprised) Does that happen to you? Like for the first
one in your course you're not quite as anxious?
Jamie: Maybe>l' (unconvinced)
JK: Why do you think you were more anxious for the second one?
*Time* [here I waited for Jamie to answer — several seconds]
Jamie: Umm, Well, I know this time why I was.
JK: Okay, why?
Jamie: It was because of my 95! !!
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JK: Ahhh! That's interesting! That made you more nervous? Now why?
Jamie: Weil, I guess 'cause my parents {Alih!} were expecting it to be maybe a
similar grade.
JK: Oh, so there was this high grade and it was really possible not to get that?
{yeah t} put a lot of pressure on you? So do you think the actual level of
confusion [also] contributed to your conflision?
Jamie: Possibly=
JK: Shouldn't tell them [your parents] your grades... {(laugh) I knowt} keep
that to the end, but you were so excited it would be hard to keep that to the
end...
Jamie: Yes. (Session 2, July 3)
When I offered the suggestion that Jamie might be relieved to get a lower grade
on the second exam I didn't feel as if I was putting words in her mouth and her strong
"YeahT"s confirmed this.
JK: So it actually may be a relief to get a little bit of a lower grade?
Jamie: Yeah t
JK: And then you won't feel so much pressure on you for the next one.
Jamie: Yeah t (Session 2)
She seemed to have experienced the 95% grade as much as a storm as she might
have a really low grade, an "up" that she seemed to dread as much as a "down" — ^the
absence of calm. And I was surprised and curious. How could this be? It was hard for me
to entertain the possibility that an A might constitute a burden for someone. When 1
considered where I stood in Jamie's world, I had been more with her parents than with
her, not only in my own mind but also perhaps in hers. I heard her conflicting
motivations — to "do better on the course" but also to maintain calm, that is, not to do too
much better, not to raise hopes, not to elicit external pressure to maintain to her, an
impossible standard. As she explained herself in contrast with her parents, I became more
aware though, of how my expectations of her might differ not only from hers but also in
some ways from her parent's. Could I hold high expectations of her without exerting the
accompanying pressure that made her so anxious? Yes, I decided, because, unlike
Jamie's parents, I was positioned to be able to help Jamie explore to what extent these
expectations were realistic and to own them for herself if they were.
248
In the next session (3), I asked about her parent's reaction to the 74% she had
earned on Exam #2.
Jamie: Um, I don't know. I guess my dad was just kind of Hke, "Why did you get
a 74?" or something.
JK: Really? {Yeah} Especially when you got the 95, right? And what did you
say?
Jamie: ...I just kind of said, "To me it was more surprising that I got the 95 than
the 74." You know? (Session 3, July 1 1).
Jamie seemed calmer; as her parent's expectations had been reduced so also was
the pressure. Jamie and I could continue to explore and challenge her expectations with
evidence of her prowess and achievements.
I used the word "quiet" for her when we discussed her reaction to the 5* grade
teacher and her demeanor in class; it was held to be self-evident in our discussions. Jamie
agreed that she was quiet, like her Dad. During the course, she never used words like
"shy" about herself although in the Follow-up E-mail Survey she did. "I'm kind of shy,"
she said, "and don't really like to ask for help, even when I need it, (especially from
someone I don't know)." She reported her sister to be "the exact opposite," like her Mom.
Jamie saw herself as not so "bad" now, particularly in smaller groups of people she
knows. Giving presentations used to be hard but is doable now.
Jamie: Yeah, I've grown a lot since then, believe it or not ... Like, I used to be
worse.
JK: Really, a lot?
Jamie: Yeah.
JK: Oh dear. You say, 'worse,' as if this is a bad thing.
Jamie: Yeah.t
JK: Like, if people say, 'This is bad, you need to speak more', or?
Jamie: Well, like, whenever I had to do oral reports and stuff, it was very
traumatic.
JK: Oh, dear.
Jamie: Whereas now I'd be able to get by. (Session 1)
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In Session 3. 1 raised the question of wiiether Jamie felt I assumed that her quiet,
non-participatory style was all "bad" as she labeled it. I asked her what she saw as
advantages of her style and she immediately responded that she was "able to listen more.
'Cause some people don't listen; they're just talking all the time." I experienced this as
illustrative of how Jamie might experience me at times and I began to explain my own
efforts as an outgoing extrovert with shy, introverted family members to modify my
behavior and listen. I told Jamie (and reminded myself) how difficult I found it to listen
to quiet people, to wait long enough for them to form their thoughts and answer; I was
aware of how important in the recovery of her mathematics self it was for Jamie to fmd
and express her voice and I had to allow that to happen.
This led Jamie to discuss her mathematics ability in relation to family beliefs. She
reported that her mother often said she passed her own "not good" mathematics genes to
Jamie and her sister. "My dad, he's very good at math. My mom always said that
unfortunately, me and my sister got our math genes from her 'cause she's not good"
(Session 3). I questioned her mother's theory and reminded Jamie that we were gathering
evidence that reflated that claim.
Jamie used the words "good," "not bad," "bad," "not good" or "worse" to classify
how she and her family did mathematics, to describe her progress in dealing with her
shyness in school, and to describe her feelings. I wondered whether Jamie meant them as
polarized judgments and if so how much they might be locking her into particular
positions — if she (and the females in the family) was "not good" at mathematics or
"do[es]n't do good at math," if storms were all "bad," if only others like her Dad were
250
"good" at mathematics, even contradictory evidence such as her Algebra I experience or
her 95% could be discounted as anomalies.
Jamie 's Attachment to Mathematics
My first impression was that Jamie's mathematics cognition functioning level was
very different from and considerably higher than Karen's. Her 95% grade and the story
she told of how she achieved it spoke of a firm mathematical knowledge base, good
trouble-shooting skills, and an ability to perform under pressure. Because we didn't have
Jamie's Exam #1 with us,'"'" she had to recount her experience from memory.
Jamie: Well, there was one part. . . I started doing it, and then I was like, 'Wait.
That's not right!' So I went back and I changed it. Like, it was one of the
ones that had to do with some of the earlier problems too. . . So I went back
and I had to change everything, because I was getting all in my brain, like I
was . . .how to do the wrong . . . wrong equation. Like I was doing the right
one for a different one,'"'^ but not-- ... So I was getting them mixed up. ...
But then I realized it, and I went back and fixed them all. ... 'Cause I was
having problems, and I was, like, 'Why is this not coming out right?' ...
And then I figured it out.
Great! So did you feel good when you went back? And you were like,
"Yeah!"?
Yeah,t because I wasn't really sure at first; I was confused if it was right or
not. ... It didn't really look — you know... So — but then after I fixed it, I
was confident...
And so what about it made you feel it was not right?
Um, I think it was the answer I got. . . . Like, I think it was the 'z' score
[standard normal deviate score — a transformed score indicating how many
standard deviations a score is from the mean] or something. . . .And I got a
really high number that was. . .not even on the chart. ... so I figured it was
probably wrong, if it wasn't even on the chart.
JK: Right. Cause zs only go up to, like 3 something-
Jamie: Yeah, so then I went back and I was like, 'Oh no,' and I was all panicky,
and then ... I realized what I did. So it was okay. . . I think I was doing the
wrong thing for that [the sample standard deviation].... But then I noticed,
so I fixed it. (Session 1)
JK:
Jamie:
JK:
Jamie:
In this interchange, there was clear evidence of Jamie's robust number sense, her
understanding of the statistical concepts, her use of letter symbols, her self-monitoring.
251
and her problem- solving strategies under stress. When we did look at her first test
(Session 5 just before the optional comprehensive final), I saw what she had done (see
Appendix L, Figures LI and L2). Jamie had remembered accurately. Jamie's number
sense was illustrated by her realization that a z score that was too large was caused by a
standard deviation 5 that was too small since the 5 is in the denominator of the z formula:
Z = X - X . Jamie was clearly pleased with herself that she tracked down and
corrected her error, especially because the realization of her error pushed her from her
customary anxiety into a panic. As I listened to her I affirmed her masterflil handling of
the situation.
Unlike Karen's sense of "always" having struggled with mathematics, Jamie's
variable history, including positives such as getting As through fourth grade and doing
"good" in Algebra I, and her less categorical "usually" very tough pointed to the
probability that her mathematics self had developed soundly-enough — arithmetically and
algebraically — despite the storms.
This evidence of good-enough mathematics iunctioning was tempered, however,
by Jamie's high Abstraction Anxiety score'™' on the Feelings Survey pretest and her
repeated declaration that "I don't do good at math." I wondered whether there could be a
cognitive base for her high abstraction anxiety. In Session 2, 1 suggested Jamie complete
the Algebra Test (see Appendix C) to see if her concept of variables was indeed related to
her high abstraction anxiety. The results of the test showed that she was comfortably at
Level 4, the highest level identified by the compilers of the test, but not at Level 5 the
highest level postulated by Sokolowski (1997) whose adaptation of the test I used (p.97)
(see Table 6.2 and Appendix L, Figure L3). Jamie thus began the course with an
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understanding of the variable that I expected should be more than adequate for the
task.'™" I interpreted the results of the Algebra Test to her: "You're a powerhouse,
woman! This discounts my theory that your abstraction anxiety might be related to poor
understanding of the variable. You're very sound right through level 4! Amazing! Not
really amazing!. ..so it's just this issue, learning to ask for the help you need when you
need it..." Jamie had more than a good-enough concept of the variable to negotiate this
course successfully; she grinned. Perhaps her parent's expectations raised by Exam #1
were not so unfounded!
Jamie had some uncertainties about operations with the variable, but she was able
to problem solve and check herself as she had on the first exam. Even on Test # 2, where
she earned 74%, she had tried an inventive (though incorrect) strategy, on a problem
dealing with the probability of success of .7 to solve a binomial probability question
using the table that gave probabilities through only .5.
It seemed that it was her stormy experiences with algebra, not an actual inability
to do algebra that caused her abstraction anxiety to be so high. I hoped that these results
might help allay her imcertainty about her mathematical ability, confirm that her algebra
base was secure, alleviate much of her abstraction anxiety, and give her more confidence
that she could do well in PSYC/STAT 104. 1 had also seen strong indications (e.g.. Exam
#1, see Figures LI and L2) that Jamie's arithmetical understanding (including her
operation and small number sense) were sound. This was confirmed when she took the
Arithmetic for Statistics assessment with the posttests in class (see Table 6.1 below). I
saw the main cognitive focus of our meetings then to be continued efforts to reconnect
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her with her good-enough mathematics self, a process already begun with our Exam #1
discussion and her Algebra Test resuhs.
Emotional Conditions: Anxiety, Learned Helplessness, or Depression?
Although her Testing anxiety score on the Mathematics Feelings survey was high,
Jamie had successililly used mathematical trouble shooting in a crisis in Exam #1, even
though the crisis put her into a state of panic. This was not the type of testing anxiety that
interrupts or derails cognition. Rather, it seemed that it was a type of mathematics social
anxiety confounded with mathematics and mathematics teacher separation anxiety (see
chapter 3) that prevented Jamie from clarifying what she understood and from getting the
help she needed, even when it was readily accessible. In class, during problem-working
sessions, I had observed that Ann generally spent considerably less time with Jamie than
with other students, although she checked over her shoulder almost as often as with
others. During these sessions, Arm used a combination of roving checking over shoulders
(and offering help if she saw trouble) and responding to cues from students: a raised
hand, a head up as she went by, a verbal plea. Jamie gave such cues less frequently than
other students. Ann's sense that Jamie was "fragile," seemed to inhibit her from offering
Jamie more help (Interview 2).
Jamie's beliefs about mathematics were slightly more procedural than conceptual
on the continuum (a 2.7 on the 1 through 5 scale) and were more towards the
toxic/negative rather than healthy/positive (a 2.5 on a 1 through 5 scale; a 3 is middle of
the scale). On learned helplessness versus mastery orientation, Jamie had the most
learned helpless score of the class (a 2 on a scale of 1 (learned helpless) through 5
(mastery oriented) see Figure K4). However, her noticing and troubleshooting her error
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on Exam #1, despite her panic, indicated a more mastery oriented than learned helpless
approach in that situation. Again it seemed that it was not so much cognitive but a kind
of social learned helplessness that was impeding her ability to take the initiative to get
help she needed when she needed it.
I wondered if Jamie's reported learned helplessness was indicative of the often
linked situational depression but examination of Jamie's responses on the JMK
Mathematics Affect Scales seemed to rule that out (see Appendix L, Figure L5). After the
first session Jamie's responses had been largely positive. This was not surprising to me
since she had just found out about her 95% on her first exam, although her responses
during this second session while lower were still at or above the midpoint of the scale
(even though she knew she had done worse on the second test). The only responses that
were of some concern because of the level of negativity expressed (average of 46.5%
positivity; five responses at or below the 50% mark) were Jamie's responses at Session 3
by which time her expected low grade on Test #2 had been confirmed — a 74%. After that
her responses bounced back and remained largely positive. Taking this positive affect
with the strong indications that her anxiety was more central suggested that mathematics
depression was not a real concern for Jamie.
Identifying Jamie 's Central Relational Conflict
My experience of Jamie's transference was that she saw me as no less dangerous
than the teacher who had first sent her into hiding. If I had reacted to this transference as
Aim did by staying away in order not to hurt Jamie, I would not have pursued Jamie to
begin counseling. Her insights into her own shyness and introversion pointed to a
conclusion that her central mathematics relational difficulty was multi-faceted anxiety
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and should be the focus of the counseling. This anxiety not only increased in mathematics
testing situations, it kept people at arm's length and stopped her from getting the help she
needed and therefore frustrated her goal of "doing good" (but not too good) in
mathematics. My interest was in the domains, triggers, and origins of the anxiety. Jamie
spoke of her social anxiety and her success in overcoming it in public speaking. But this
social anxiety influenced her behavior in other domains, specifically the mathematics
classroom, and in relating to mathematics teachers. Her lack of interaction with anyone in
class, effectively hiding while we faced each other around one rectangular table, her
failure to make a follow-up appointment with me, her discomfort when I asked her a
question in Study Group 1, and my worry (influenced by Jamie) about asking her a
question in front of her peers, all spoke of her anxiety in relating to people, her wish to
avoid them, and the demeanor that dissuaded them from interacting with her (at least in a
mathematics setting).
In the one-on-one setting I found Jamie more open and willing to connect with me
than Karen had been, although at times, especially in the beginning, she exhibited
discomfort (In my notes written immediately after Jamie's first session I wrote "At times
Jamie seemed close to tears."). Jamie had done well in mathematics and related
positively to her teachers through 4* grade, secure in a mathematical learning base. The
mathematics separation anxiety that was connected to her 5* grade experience seemed to
have been exacerbated by the subsequent experience of finding then losing a secure base
in her Algebra I teacher, who she perceived as unavailable when she needed her later.
This precalculus class experience also seemed to have made her attachment to the
mathematics itself, particularly algebra, feel insecure. At this point Jamie did not know if
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or how she might do well or poorly and she did not perceive mathematics teachers (or
tutors) to be safe enough to ask for input and support.
In addition to her mathematics social anxiety and her separation anxieties she
seemed to have experienced a debilitating performance/fear-of-success anxiety on her
second exam related to a combination of shyness and her family dynamics — her father's
expectations and pressure versus her mother's acceptance of Jamie's lack of mathematics
ability.
The central conflict that was keeping her stuck seemed to be between her desire to
succeed in the course, her uncertainty about her ability to succeed, and her sense that
becoming conspicuous might endanger her in some way. She seemed trapped far from
home, separated from mathematics teachers and her mathematics self and she saw no
means to reconnect safely and inconspicuously.
Central Counseling Focus
I realized that if I stayed where Jamie's transference put me (leaving her alone so
as not to endanger her) she would not become aware of her issues in a way that would
allow her to change. In counseling therefore my focus was to reach past the protective
shield Jamie had built up for herself, to disempower the objects of her anxiety. I had to
provide myself as a secure mathematics teacher base, a smooth path with few surprises
around the comers, and I had to help her find a way to assess the level of safety of the
class and the instructor so that she could choose to access her as a safe support base.
Ann's non- intrusive, respectful approach made the present class a good-enough secure
base where this could happen for Jamie, if she were able to see it and was willing to take
advantage of it. I also had to help Jamie recormect with her own sound mathematics
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cognition so that she could proceed with her mathematics learning, secure in her
mathematics (arithmetic and algebraic) base.
Jamie's Course of Counseling: The Process of Brief Relational
Mathematics Counseling, Session by Session
(see Appendix L, Table LI for Jamie's schedule)
Jamie 's Sessionl
I found out a lot about Jamie's mathematics experiences and her current
orientation at our first meeting. Because she had done so well on her first test and the
next test was not imminent, there seemed to be the necessary leisure. Telling her story did
not seem easy for Jamie — at times her color deepened and she seemed uncomfortable,
sometimes close to tears — but that did not impede her or me from our exploration. My
curiosity about her story, the connections, and the apparent contradictions seemed to help
her become conscious of it in a piece for the first time — no one had asked about it before.
Themes of Jamie's shyness and social anxieties evident in class and in study group, her
variable success, the impact of teachers on her success and sense of mathematics self, and
her personal preference for calm rather than storms — the absence of "up and down," —
emerged.
When we worked on linear regression problems I noticed that Jamie's arithmetic
and use of algebra seemed adequate and she did not seem anxious as she worked. At the
end of the session, as we worked some of the assigned homework problems in parallel,
another student came for his appointment. Jamie finished the problem she was on and left
without arranging our next meeting.
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Jamie 's Session 2
Again, as for the first meeting, although there was ample opportunity, Jamie did
not approach me to make an appointment for a second meeting. I eventually approached
her and we arranged to meet during the July 3 cancelled class time. Our second meeting
took place during the sixth week of the course when she knew that she had not done as
well on the second test but didn't know her grade.
At this meeting we discussed a picture I thought seemed to be emerging. I showed
Jamie her Survey Profile Summary (see Appendix L, Figure L4) where her testing anxiety
and helplessness scores were the most extreme in the class and her abstraction anxiety
high, exceeded only by that of another student. Discussion of learned helplessness (and
its counterpart mastery orientation) and Jamie's extreme score led us away from the
cognitive domain (Jamie's mastery recovery on Exam #1 seemed to discount helplessness
in that domain. See p. 251 .) towards the relational domain.
I asked if Jamie would have eventually approached me for an appointment if I
hadn't approached her. She didn't know but from past experience it seemed unlikely,
even though she had not done as well on Exam #2. I suggested that her learned
helplessness might be more about this apparent inability to access help even when she
knew she needed it — a type of relational helplessness — and that this was perhaps also
illustrated by her behavior in class. When I asked Jamie why in class, for example, she
didn't ask the questions she had in her mind, she replied, "That's my fear that I'll be
wrong." When she was not convinced by my response: '"Asking a question, you don't
have to be right!" I realized that to safely even ask a question, for her, it would have to be
a "right" question, one that fit logically and unobtrusively into the context. Even during
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problem-working sessions when students were working in pairs or alone and Jamie
always alone, she would not ask Ann a question though Ann made herself available by
circling the room and checking on each student's progress. Neither did she answer any
questions during lecture-discussion sessions even when several students responded
together and she knew the answer. The unconscious subtext seemed to be that she would
draw attention to herself if the question or answer were "wrong." Why was I so
concerned that Jamie speak in class? Perhaps it was just her style and of no consequence?
I asked Jamie what she did if she did not understand what was going on or how to
proceed and she replied, "Nothing." That concerned me and I believed it should have
concerned her. We discussed a possible relationship between my asking her questions in
study group in order to get her to verbalize and clarify her thinking (even when I knew it
made her uncomfortable) and her exam results.
I told Jamie that I would no longer approach her to make appointments. To do so
would allow her to continue in her pattern of getting help only if she was required or if
people like me pursued her (even though her "don't hurt me" demeanor made that
unlikely). I wondered out loud if she might be able to practice getting the help she
needed, and I suggested two homework assignments for her: (a) to set herself an
assignment to ask or answer a question in class, (b) to approach me to make the
appointment for our next meeting if she wanted one. I confessed to Jamie that I found it
hard to let go of my practice of making appointments with her, and allow her to choose to
do it or not:
That's risky to me {giggle} 'cause . . .1 have the sense that you have so much
potential and I have a sense that here are some of the clues as to why you don't do
as well as you could and that's exciting to me {laugh}. I think, "We could really
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get her over this hump," you know, doing math, so Td really, I'm like, '1 want to
go get Jamie!" and that's my mother thing. (Session 2)
When Jamie then took the Algebra Test to explore possible connections between
Jamie's high Abstraction Anxiety score and her understanding of the algebraic variable,
her sound level 4 pleased and surprised her and moved my "sense" of her mathematical
potential to conviction of it. Jamie left after rehearsing her two relational assignments.
During the next class after Session 2, Jamie asked Aim some questions during the
problem- working session and she was pleased with herself At the end of the class, she
came round the table, beaming, calendar in hand, to arrange an appointment with me.
Jamie was making a move to throw off the hold of the bad internalized teacher presences
from the past, overcome her social anxiety, and alleviate her separation anxiety, both in
Arm's class and with me.
Jamie 's Session 3
Jamie wanted to go over her second test and Ann had given it to me so we could.
After we had briefly discussed Jamie's significant achievement in asking Ann questions
and making an appointment with me, that is much of what we spent the session on —
analyzing her work in relation to her preparation and affect. This was the test for which
she had been so anxious because of pressure she felt from her family to get the imlikely
95% she had on the first test. Although her anxiety was elevated, Jamie again did not
seem to have been cognitively derailed by it. Her difficulties, she realized lay mainly in
insufficient preparation — "like I knew how to do the things but I didn't know what they
were called" and she had only put formulas on her formula sheet, not what they referred
to. That she lacked understanding of what it was she was (accurately) computing affected
or was affected by her poor preparation for the letter symbol understanding section. We
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had both made the same error in the multiple choice section and I shared with Jamie a
significant computational error I had struggled with and corrected just in time. I was
attempting to challenge the lines Jamie had drawn between "good" and "bad" at
mathematics; I, like her, struggle and make errors.
We began talking about Jamie's now feeling able to ask Ann some questions in
class. She attributed some that to Arm: "That's partly because it was just her," which led
to a discussion of how Jamie's "staying inside" behavior in mathematics class seemed to
have begun with the distress of her 5* grade classroom experience and its effect on her
shy personality. I suggested that as a 1 0-year-old, she had to survive what to her was a
frightening situation, so she did what she could — "sit down and shut up so you don't
bother her [the teacher]." But I wondered out loud with her whether, as a young adult,
now Jamie might have more choices. Maybe she could now assess the safety of the
classroom situation and decide whether she could participate. Jamie agreed that the small
classroom and the positive, supportive emotional atmosphere Arm had created made the
PSYC/STAT 104 classroom was such a situation and she had chosen to participate.
Jamie had been filling out a JMK Mathematics Affect Scales at the end of each
session (see Appendix L, Figure L5). At the end of Session 3, despite her having made
such strides in personal interaction in relation to the class and getting help, her scores
were the lowest yet, three of her seven responses (Items 2, 5, and 6) falling below the
midpoint towards the negative end for the first time and another (Item 3) remaining there
from before. Overall her responses seemed to indicate the presence of mild mathematics
depression that had not been evident earlier.
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Supervision
On July 20, 1 met with Dr. P. for supervision. By then Jamie had taken Exam #3
and earned an 84%, improving a whole letter grade over her Exam #2 grade (see
Appendix L, Table L2). She and Karen were the only student participants whose grades
did not drop by at least a half letter grade. I expressed my struggle as an out-going
extrovert to be quiet and listen to Jamie, a shy introvert who preferred not to speak. Dr. P.
encouraged me to invite Jamie to reflect on the changes she had made. "Commend her,
give her a bouquet. Have her write a new metaphor." And he encouraged me to continue
to my struggle to listen more and talk less, allowing, more, encouraging Jamie to express
her voice (Dr. P., Supervision).
Jamie 's Session 4
Jamie again approached me for an appointment for a fourth session. And she did
compose a new metaphor:
JK: So your metaphor was a storm; what would it be now?
Jamie: I kind of see it like it would be different for this class ...not necessarily
math in general ...maybe partially sunny ... maybe bring an umbrella in
case it turns to rain but it's okay to go outside, maybe, more, you know,
because it's sunny I can go out in it, but I would still take my umbrella.
(Session 4)
This metaphor shows significant changes from Jamie's prior sense of
endangerment in the mathematics class. Jamie's behavioral changes in the classroom
situation — her little smiles, making eye contact, asking Ann questions in problem-
working sessions, and her continuing to make appointments with me — were all outward
indications of the changes she experienced. I observed myself doing better at waiting and
listening for Jamie and she was now receiving Ann's offer of respect and safety. She felt
safe to "go outside." She seemed to be resolving her conflict between fear of being
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noticed and humiliated, and wanting to succeed in the course. She was finding that by
going "outside" she had reconnected with mathematics teachers who were available and
had not caused her damage, and had helped her reconnect with her mathematics self that
she found to be good-enough for success in the course.
Jamie's additional 6 points on the extra credit in-class power assignment brought
her grade on Exam #3 to a 90%, an A". She reported that her father was very pleased.
She did not show the increased anxiety she had in response to her high score on the first
test, however. Jamie herself seemed encouraged and quietly determined, I surmise,
because she was feeling more firmly attached to and was drawing on her own good-
enough mathematics self Following Session 3, she had been more active and strategic in
her preparation and found that she could change the outcome, so her father's expectations
were no longer felt as external pressure to pull off another flukish feat, but rather were
now more in line with her own realistic expectations, given what she now knew of her
sound mathematical base and the importance and possibilities of strategic preparation.
We worked together on questions Jamie might encounter on the Exam #4
scheduled for that evening. When Jamie filled out the JMK Mathematic Affect Scales
there was no longer any of the mild mathematics depression that she seemed to be
experiencing at our last meeting (see Appendix L, Figure L5).
Study Groups and the Final Exam (Exam #5)
I did not meet again with Jamie one-to-one until after the final exam. She was at
the study group just before Exam #4 with Ann and me, but she kept to herself; Jamie
earned a 76%. Her symbol identification was perfect as for Exam #3 but her score on the
multiple-choice had not improved irom the previous exam and this time she also lost 12
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points on the computational section. Only one third of these errors were from lack of
preparation or conceptual understanding of an analysis. So this score though similar to
her score on Exam #2 meant something quite different about Jamie's grasp of the
material and the process.
Jamie came to the study group immediately before Exam #5. 1 asked individuals
in turn to name the appropriate statistical test for scenarios I compiled from the text, and
then we discussed the responses. Jamie responded incorrectly about a scenario requiring a
two-way ANOVA when it was her turn but through discussion she understood the
solution. Each student was involved in the others' questions. Jamie went on to earn a
100% on Exam #5. 1 felt less anxious about causing Jamie trauma by askmg her questions
at that final study group than I had at the first study group, and again the outcome was
good. It gave her the opportunity to express and evaluate her thinking whereas merely
thinking about it might have left misconceptions unchallenged.
She had earned a 1 00% on her MINITAB presentation with Robin where she was
poised and showed no signs of embarrassment. Thus, with her 100% on the fmal exam
Jamie was getting a B^. With her father's encouragement she decided to take the
comprehensive final to replace her lowest test score of 74%, hoping to bring her final
grade up to an A". She asked me if we could meet once again to review all her exams as
preparation.
Jamie 's Session 5
Session 5 was a marathon at a coffee shop on the Sunday evening before the
comprehensive final exam. We reviewed each of her exams. Because the grade earned
on each test did not necessarily reflect the level or quality of her mathematical thinking, I
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decided it was important for Jamie's growing mathematics prowess to identify where she
had thought well and to reduce the role of the grade as sole measure of her ability. It was
also important, however, for Jamie to realize that issues other than mathematical
understanding, such as clear communication and correct solutions, can be so important in
real life application, that instructors use severe point penalties to emphasize this on an
exam. Twice on Exam #4 Jamie had made the logical decision based on her (incorrect)
calculations of statistics, but had 4 points deducted on each because these were incorrect
decisions for the problem. Arm had also deducted points for Jamie's technically accurate
but poorly communicated defmitions of symbols. Jamie was able to see that her grade on
Exam #4 undervalued her actual mathematical thinking and ability; I nevertheless
emphasized that the grade J/J enforce the importance of her improving accuracy and
clarity for her chosen field of psychology.
We also discussed changes in Jamie's responses to the Mathematics Feelings and
Mathematics Beliefs surveys. She had made substantial changes on each of the anxiety
scales. By the end of the course her score on testing anxiety had gone down from high to
moderate (a 17% decrease,'""" see Jamie's post-scores on Figure L2, Appendix L). My
anxiety about Jamie and her mathematical learning and my inclination to control and
mother had also decreased as she took more control and internalized realistic expectations
of herself. Jamie's beliefs on the Learned Helpless versus Mastery Oriented scale had
changed very little but her social learned helplessness, at least in this setting, had abated
considerably. Jamie earned a 71% on the comprehensive final exam (80% on the
computational part), and could not replace any of her earUer test grades, so she ended the
class with a B^.
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Jamie 's and My Final Evaluations
On Jamie's One-On-One Mathematics Counseling Evaluation she described her
initial motivation for signing up to meet with me was "so that I could get a better grade in
the course." Since I only learned after the course from Jamie that she had been repeating
PSYC/STAT 104, 1 speculate that initially she had not disclosed this because of her "fear
that I'll be wrong" and thus conspicuous and censured by a mathematics teacher whom
she did not know (me) and who was not to be trusted to do anything but humiliate and
abandon her, as past teachers had done. Her end-of-course written comment about "a
better grade" may have been an indication of her now feeling safe to let me know,
perhaps also affirming her trust that I could know that she had not done well in the course
before without rejecting her.
Asked whether her motivation had changed during the course, Jamie indicated
that she recognized the focus was primarily relational, "Kind of, I realized it was more
about my feelings and confidence in my math ability, th[a]n any real problems with the
math course work" {One-on-One Counseling Evaluation, archived).
Jamie had learned how to ask for help in this course but it seemed this experience
was not enough for her to do it in a new class. Unless the class was structured like her
English composition class with required meetings with peers and instructor, or had a
resident class-link tutor who initiated the contact, I concluded Jamie would probably
continue to be an involuntary loner. Although it had for this class, for other mathematics
classes, her conflict between wanting to do well (and knowing that likely means getting
help from and working with strangers) and fear of being conspicuous had not been
resolved and the latter would probably predominate. She /za<i however, become aware
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through counseUng that she could judge the safety of the situation and the instructors and
helpers and not feel compelled to hide no matter what.
Jamie had done well in the PS YC/STAT 1 04 by reattaching to her sound-enough
mathematics self and to safe mathematics teachers/classroom. I had learned to wait,
listen, and affirm her strengths as well as challenge her to confront her fears. I (and Ann,
once I had helped Jamie see) had provided her with good-enough objects to replace the
bad 5"" grade teacher internalized object (presence). I had become a secure enough
base — a smooth level path, with a gradual incline — from which she could experience this
class, not as a storm any longer but now as a "partially sunny [day]" where she could go
"outside." It is not possible to say what grade Jamie would have earned in PSYC/STAT
104 without counseling, but she almost certainly would have remained hidden, the
instructor would have tiptoed around her, she would have remained isolated from her
peers except for the required contact over the computer module presentation, and her
questions and comments would have remained unspoken. Most importantly Jamie's sense
of her mathematics self would likely not have changed. If I had not examined her
transference and my countertransference reactions I might not have pursued counseling
with Jamie at all. If I had only gone by Jamie's responses on my traditional anxiety and
belief surveys and not delved with her for the underlying meanings they signaled, if I had
not explored her metaphors and tracked her progress with the JMK Scales, in other words
if I had taken a traditional approach instead of brief relational mathematics counseling
with Jamie, she is likely to have remained an under-confident involuntary loner achieving
variable results over which she felt little control — always afraid of the storms of
incomprehension, anxiety, and unwanted attention.
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Epilogue
Jamie decided not to try Finite Math again to flilfill her quantitative reasoning
general education requirement, but to take a logic course instead. She has not let me
know how it went. Jamie now knows she could assess the relational safety of the
instructor and the class to see if she might go "outside," ask questions, and ask for help,
and she has a budding understanding that she was in fact quite capable of doing
mathematics. If Jamie perceives a new situation as benign enough so that she does not
regress and go back into hiding with "you are dangerous to me; don't come near me"
transference, an attentive instructor or class- link tutor might feel less reluctant to
approach her to offer help and she is more likely to accept such offers. If the mathematics
counselor or tutor waits for Jamie to come to the Learning Assistance Center or make
contact with the instructor or even the class link tutor, it is likely they will wait in vain
and Jamie will not receive the help she needs.
MULDER'S COURSE OF COUNSELING
Mulder™"" exuded an outgoing social energy. During the first class lecture
discussion on the scientific method, he was actively involved™", telling classmates of his
research project on centipedes' attacking postures. During problem-working sessions he
always worked with any neighbor willing to engage. Ann thought he seemed "smart" and
"on the ball [with respect to] his research experience into caterpillar aggression" — but not
likely to put in the effort needed to succeed in the class and not very committed
(Interview 2, Class 1).
When he foimd out that I was available as a tutor, Mulder was enthusiastic. He
didn't think he would need much help with statistics (He had used some statistics for his
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biology research projects.), but he thought he might for his finite math class — that
seemed more challenging to him. So Mulder signed up for mathematics counseling once
a week. He struck me as a charming scallywag. In fact, I called him that once. He
seemed busy, mischievous, stubborn, and somewhat of an opportunist but he was
confident that he could handle PS YC/STAT 1 04 fine, so I believed we would focus
mostly on finite math.
However, Mulder and I soon found reason to suspect that his confidence was
perhaps overconfidence. He earned a 'D"' on his first exam and thus began a quest unlike
any either of us had been on before. I found I could deal with anxious and underconfident
students like Jamie, using relational approaches to get at the roots of her anxieties; I could
even overcome the despair that depressed and underconfident students like Karen threw
me into because both of these students knew that they needed help. But how was I to use
relational approaches to recognize that a student with a social, confident, and up-beat
demeanor might actually be overconfident and that he might then be drawing me into
believing he was less needy than he really was? Then once I recognized this, how could I
help when he seemed to have all the answers? And for Mulder this seemed to be new too.
It turned out that he had never really tested his "I know I can do math" theory by actually
trying to do it well and he had not worked with someone who was trying to support him
in that endeavor.
Sometimes Mulder developed what appeared to be indirect and to me illogical
schemes to improve his achievement; at other times he stubbornly resisted the
mathematics he found did not yield to these devices. He did improve his computational
grade and then his grade on symbol identification improved, but on exam after exam he
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failed to improve his conceptual section score. The narrative that follows chronicles how
we struggled and how relational counseling insights and approaches I used not only
helped Mulder resolve the conflicts that hindered his success but also helped me grow as
a mathematics counselor.
Mulder was a 20 year-old white man who was a biology major at a small
university in the Midwest. He was home for the summer, taking Finite Mathematics —
MATH 120— in addition to PSYC/STAT104 at Brookwood State. The fmite
mathematics course was required for his major but not statistics. He had the option of
transferring his statistics credits for elective credit if he did well enough.
Mulder was an only child. He was short, muscular, and fit — ^participating in both
soccer and track (Class 1 ). The last mathematics course he took was Algebra II as a
junior in high school; he reported that he earned Cs in mathematics classes then. He
mdicated that he hoped for a B and expected a B in the PSYC/STAT course (Pre-Test
Mathematics History Survey), both overestimates perhaps, given his history.
Mulder 's Metaphor: Fox Mulder Searching for Aliens
Mulder asked if he could think about choosing a metaphor "because I really — I
don't know that I could say for a while" so I suggested we come back to it later. When
we did come back to it I asked him if he would rather do a drawing of himself doing
mathematics, but a metaphor came to him, "For me math is like Mulder searchmg for
aliens. I am searching why I make math so difficult for myself" He referred to Fox
Mulder from The X-Files,™^ a popular science fiction television program. Mulder
explained further, "I have confidence in everything else I do. It's not that I don't have
confidence [in my ability to do mathematics], but it's just like — ^I know what I'm doing.
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but I can't explain it to other people." I wondered how this related to his "mak[ing] math
so difficult" for himself And what if anything did the metaphor tell us about how Mulder
saw mathematics?
Student-Mathematics Relationships: Mathematics as a Search for Aliens
When I asked Mulder about how mathematics had been for him, he responded
"It's never been my favorite." Later, "It's, like, it's the only thing that ever gives me any
problems." It was from freshman year of high school that mathematics seemed to have
become an issue for him. His theory was that it was his lack of effort rather than low
ability that accounted for his difficulties, yet he had not tested his theory by putting in
that effort even after he "realized" that was his problem. He avoided mathematics
altogether his senior year because he knew he would not do well in it: He wanted to "save
[his] grade point average".
I formed this attitude in high school, you know, high school, if I had really, really
tried in high school, I could have done really, really well [in mathematics]. It
wasn't 'till the end of my freshman year I realized and I still don't think I try as
much as I should, you know ... It's just a matter of applying it and taking
advantage of it ... study skills in high school weren't that great [I did] three
sports a year. I did real well my senior year because I took no math ...I was
enrolled in trigonometry and precalc but 1 dropped it because I wanted to save my
grade point average. (Session 3)
Mulder had done better in Geometry than in Algebra II even though he had
"thoroughly slept through it" and he put that down to the difference between the teachers
rather than differential ability or a preference for that type of mathematics.
I got 'C's in all these [mathematics] classes. This one [pointing to Geometry on
the list] I thoroughly slept through; I'm not lying... ironically enough I did better in
this class [Geometry] than in this one [Algebra II]. It was the teachers. (Session
3)
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He could attribute his low performance to not trying or to sleeping, so his belief in
his potential ability to do it was preserved especially since "my uncle is getting his
masters, great student and my dad's really smart so it's kind of like a thing I know I
have" (Session 3).
I found my initial reactions to Mulder and his prospects in the class were quite
different from my reactions to Karen, even though her grade on the first test was almost
the same (62% compared with Mulder's 63%). His metaphor was active, if somewhat
self-defeating, and he seemed willing to engage. Mulder was positive about his
mathematical potential. I was drawn into his confidence and considered then neither that
his knowledge base might be weak nor his underlying mathematics self-esteem low.
Because he had earned Cs through Algebra II with lots of sleeping, and not really trying,
the result may have been a relatively underdeveloped mathematics self I eventually
found considerable evidence to support this conjecture.
After the course it was confirmed that Mulder did have a minimal algebra
background for college though this was not obvious to me during the course. When he
took the Algebra Test after the conclusion of the course, he tested at a low level 2,
indicating that he, like Karen, had not yet developed an understanding of letter symbols at
least as specific unknowns or generalized numbers (and in some cases as true variables)
nor could he coordinate operations using them (see Table 6.2). That perhaps explains
why his formula sheet for the first test had been so inadequate for his needs: he, like
Karen, needed detailed formula sheets for exams that interpreted formulae into columns
and step by step procedures. Unlike Karen though, he seemed relatively arithmetically
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sound, with accuracy and confidence in his number and operation sense (see Session 1
and Table 6.1).'°°^
Initially I was taken in by Mulder's sound-enough arithmetic and confident take-
charge approach. I did not become conscious of his real deficits with respect to the
algebraic variable and related concepts until later in the course. I now see that Mulder's
low understanding of the algebraic variable, his more procedural than conceptual beliefs
about mathematics (2.9 on a scale of 1 through 5 on Beliefs Survey, see Appendix M,
Figure Ml), and his poor high school preparation seemed to have combined to make the
statistics almost alien to him, especially the conceptual aspects that required him to
understand and communicate in earthly rather than alien terms. Unconsciously at least,
these factors were almost certainly calling into question for him his own ability. Maybe it
was not just about effort. Maybe he really could not do it.
Student-Teacher-Self Relationships:
Mulder complained about Ann's lecturing style (She "jumps around a lot.") Later
in that first session he said of Aim's lecturing: "It's not that she goes through it so fast;
it's just I have a hard time following her" (Session 1). His references to past teachers
were in a similar vein. He attributed doing better in Geometry that Algebra to his
teacher's different approaches.
Mulder: It was the teachers.
JK: You seem to react to teachers
Mulder: Yeah, I do
JK: And it seems to affect how you do in class?
Mulder: Yeah. (Session 3)
Present struggles with teachers seemed to be closely Imked to Mulder's struggles
to understand the course content and gave me clues to the nature of his past struggles. He
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strongly preferred his finite teacher's direct approach, "[S]he tells us how to do it and she
tells us why and how to use it" (Session 3). This teacher made explicit the links among
concepts, procedures, and applications and she demonstrated the procedures. By contrast
Ann had students work out how to do the problems for themselves during problem-
working sessions, after she presented concepts involved. It seemed hard for Mulder to see
how the concepts discussed in the lecture related to the problems worked later, even that
they were related. And the struggle seemed to be exacerbated by Mulder's auditory
processing difficulties and his compensatory visual memorization strategies.
Learning/processing style and Mulder-teacher relationships. A pattern of
Mulder's difficulties in understanding and expressing new knowledge through his
auditory and verbal channels emerged. This was evidenced in his relative difficulty with
finding a metaphor, following Ann's lectures, and in his linking his "Mulder" metaphor
to difficulties he made for himself in mathematics, especially in explaining what he
understood. Initially my realization of this difficulty was masked by Mulder's outgoing
social learning style and I speculated that other mathematics teachers may have been
similarly misled. I began to wonder whether Mulder was making it hard for himself or if
it was a processing difficulty that he blamed on himself Perhaps it was a combination of
factors. Perhaps what he labeled as his laziness was, in part, avoidance of these primarily
verbal study tasks he found difficult.
The seemingly illogical visual memorization strategies I observed him using
perhaps served to compensate for his auditory struggles, and his perception was that he
did better on assessments that required visual recognition of material. For example, as
Mulder tried to understand why he did not do as well as he expected on certain exams he
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cited testing anxiety as a factor for all exams except "practicals," (those requiring
identification of visually discriminated materials: "Like, a bone practical [where he had
to identify and describe the function of bones of humans or other animals]" He always
got As on those [Session 3].) Did Mulder's have a global, visual-pictorial, mathematics
learning style II (Davidson, 1983; Krutetskii, 1976)? Or was his approach the result of
continued use of strategies he had developed to compensate for auditory/verbal
processing difficulties? Or some combination?™"' It seemed that he was not easily
classifiable but I began to wonder if his atypical approaches to mathematics learning
might not only have negatively affected his level of mathematics understanding, but also
how he was perceived by his teachers.
I needed to explore with him what effect these approaches had on his mathematics
self development. I needed to know how teachers had reacted to him and what effect that
had on him. I myself reacted with amazement and sometimes horror to what seemed to
me to be a lack of observable logic in some of his tactics (see Sessions 4 and 5). Mulder's
approach seemed consistent with his metaphor at least; he did indeed seem to be using
alien methods to search for his aliens.
But these methods looked enough like attempts to avoid hard work that I
speculated that his mathematics teachers had not only perceived him as capable (because
of his confident upbeat demeanor) but lazy, but also labeled him thus. Indeed this was
how Ann saw him: smart but not likely to put in the effort. Mulder's constant concern
that he might be perceived to be lazy ("I hate doing this ... It's just — it makes me feel
lazy" when admitting to putting work off 'a lot' when he filled in the JMK Affect Scales,
Session 1) and his repeated description of himself as lazy about doing mathematics
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supported this conjecture that he was used to being labeled thus and had taken it on
himself. Perhaps the "capable but lazy" label had become a shield for his possibly
incapable self and, if so, it may also have functioned as a trap, hindering him from doing
what he needed to do to develop his capabilities and deterring teachers, whose help he
needed, from helping him.
Emotional Conditions: Anxiety, Learned Helplessness, Depression,
or Grandiosity?
Anxiety. Mulder certainly didn't strike me as anxious. But after his poor showing
on Exam #1, he brought up testing anxiety as one of the factors he believed was operating
against him, especially on tests like mathematics tests that were not visual memory
oriented "practicals." I had not highlighted any of his average anxiety scores on the
Feelings Survey for discussion with him (see Appendix M, Figure Ml) because each fell
in the middle of the class range and was not extreme compared with the class. However,
his testing anxiety averaged at a little above moderate (3.1 on the 1 through 5 scale) and
could be considered high for a physical science-oriented student and even for a social
science student if compared with means Suinn (1972) reported on the Mathematics
Anxiety Rating Scale (MARS). '""'"
Was Mulder's anxiety a normal reaction to a challenge he was not adequately
prepared for or something more than that? It seemed feasible that it was linked with his
history of not having done well on mathematics exams and an underlying belief that he
may not be able to do it. That combined with lack of strategic preparation for this exam
to compensate for his mathematics deficits (e.g., well constructed formula sheet, strategic
practice of target problems) would give good reason for considerable but normal anxiety.
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Depression. And Mulder gave me the impression of being anything but negative
or depressed. This observation seemed to be confirmed when he completed the JMK
Affect Scales during his first session. Apart from his extreme negative response (a lot) to
putting work off" all other responses were at or above the mid point (see Appendix M,
Figure M2, responses marked 1). His average positivity on the scales was approximately
55% or 64% if the "putting off work" item were removed (see Appendix M, Table M3).
Given Mulder's poor showing on Exam #1, rather than indicating mathematics
depression, his responses perhaps showed the opposite, mathematics optimism.™"^ .
Learned helplessness. Although depression was not an issue for Mulder, learned
helplessness did seem to be. When I showed him his low learned helpless score average,
Mulder responded, "I think it's math. Any other thing I'd be up here [pointing to the
Mastery Oriented end of the scale]" (see Appendix M, Figure Ml). Perhaps this was a
chink in his up beat armor that I initially did not explore. Although Mulder saw
mathematics as somewhat more procedural than conceptual and his approaches seemed
procedural, his achievement motivation became more for learning than for performance
over the summer (Items 4, 7, 9, and 10, Part I, Beliefs Survey and Appendix H, Table H3).
Perhaps like Karen, he wanted to understand the material but used procedural approaches
both by habit and also because he did not feel capable of achieving that understanding.
Grandiosity. Mulder's emotional response to his mathematics challenges did not
seem to be marked by anxiety or depression that could be considered unrealistic. The
evidence seemed to be pointing to grandiosity. It seemed that Mulder might have
developed an overconfident demeanor combined with relative lack of effort and indirect
approaches in order to protect an underdeveloped mathematics self that was
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compromised by his learning style challenges: "I know I can do really, really well ...
[but] I don't really try/I thoroughly sleep [through class]."
Identifying Mulder 's Central Relational Conflict
Mulder now faced a dilemma. He wanted a B in the two summer mathematics
classes he was taking to make it worth transferring the grades. His high school tactics
would not work but if he actually tried he risked being found out. On the other hand he
did not want to be considered lazy. When I asked him about how much work he did for a
finite exam, at first he denied doing any work. This seemed to express his grandiose
stance (I can do well; I don't need to work at it.). But then he conceded that he had
practiced but only some of the problems (perhaps his "I don't want you to think I'm lazy"
stance)
Mulder: And I don't — ^I've never really sat down and done practice problems
before a test.
JK: And you didn't do practice problems?
Mulder: No.
JK: Was it stuff you were already familiar with?
Mulder: No.
JK: No? But you got it Irom the class?
Mulder: Yeah. I knew how to do it. I did — I did some of the homework; I don't
do all of the homework but I do some of it.
JK: Just pick a few things?
Mulder: That I need — that I need to work on. (Session 1)
Confounding his difficukies were the very real challenges that his auditory
processing difficulties, his compensatory visual strategies, and his poor mathematics
preparation posed, especially as Mulder did not seem to be aware of them or their
potential for sabotaging his success.
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Mulder and Me Dealing Jointly with Aliens:
Relational Counseling for Mulder
The Focus of Relational Counseling
I realized that relational counseling should focus on helping Mulder and me
become aware of the conflict between his competing goals — to do well in the course but
also to protect his underdeveloped self. Perhaps the very defenses he was using to protect
his mathematics self were what were "mak[ing] mathematics so difficult for [himjself " I
would have to recognize that Mulder's grandiosity might be masking an underdeveloped
and vulnerable mathematics self. He was so convincing and I found myself believing his
grandiose view and not attending to his real challenges. I was likely falling into a pattern
of former teachers — believing him, being disappointed, getting frustrated, scolding and
pushing hum, and even giving up — and not offering him what he really needed. Not only
would Mulder need to become aware and change, but I would also have to change my
approach in order for him to feel safe enough to drop his counterproductive defenses.
And he might need me to change before he could. I realized that we were unlikely to
resolve his conflict unless I could work out how I should change.
The Focus of Mathematics Counseling
Because my preferred learning style is strongly auditory, I had to be aware of the
risks of devaluing Mulder's mathematical learning approach, simply because it was
different from mine. Instead I needed to accept and try to understand how his visual-
memorization, his procedural mastery, and his social style both facilitated and impeded
his mathematics learning. How could I help him use his strengths and preferences to help
rather than stand in the way of his grasp of the mathematics? It became clear to me that
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the strategic mathematics tutoring focus should be on Mulder's finding a way of seeing
this alien mathematics in a more accessible, logical, earthly form.'™"
Mulder's Course of Counseling Session by Session
(see Appendix M, Table Ml for Mulder's schedule)
Mulder 's Session 1
Mulder was doing well in his finite math course, but he had done poorly on the
first PSYC/STAT exam (63%). He had failed the multiple-choice conceptual section,
with the lowest score in the class (see chapter 5, Figure 5.2). At times he had failed to
follow directions ^°°"' and at others he did not know the information adequately so he
guessed rather than trying to work them out from the context. His computational score
was less extreme but still only 65% correct. He had lost only one point on the symbol
identification part, but he had only named the symbols and not defined them as was
required. Ann said she had been lenient in her grading on this section because it was the
first exam.
When we examined how Mulder had prepared for the first test where he had done
so poorly he identified the fact that his formula sheet was not adequate and he had no
direct information (e.g., a quiz) to guide him to work out how Ann tested so he had not
prepared sufficiently or strategically enough. These factors seemed to give good reason
for the testing anxiety he said he had suffered.
Although he had not done well on the computational part of the exam,'™'"" Mulder
seemed to have solid number sense and no problems with decimals so I was not alarmed
as I was for Karen about his prospects. The questions involving number or decimal sense
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(e.g., deciding on real limits for the weight of 0.35 grams of cheese) were answered
correctly.
As we discussed how to prepare for Exam #2, Mulder seemed to be astutely
assessing the mathematical tasks required for him in the computational section. "They're
not really word problems, you know. The information's there and the equation's there
and she shows us how to set everything up, and I understand all that." Although perhaps
globally positive rather than realistic, he did seem to have pinpointed a crucial problem
with his first exam, that is, he had not set up his formula sheet adequately. If he had, "It's
easy to write down the equation, say what the ground rules are, and then plug the
numbers in." He saw the mathematical tasks as procedural and felt that he could manage
that. Mulder seemed confident that he could remedy the situation in Exam #2 by
improving his formula sheet and studying the procedures now he knew how Aim tested.
Mulder seemed to consider the computational and conceptual sections of the
exams as separate, requiring different types of preparation and despite his low score on
the conceptual he commented, "I did all right on the conceptual part" and for the next test
it was the computational part he was going to focus on. Later in the session Mulder did
concede though, "Obviously, I need to spend more time on the conceptual."
Although Mulder's decision on how to improve his computational preparation did
not include understanding and linking the concepts, I did not pursue it, thinking maybe
his plan would work. I was concerned about his conceptual understanding of symbols (on
Exam #2, Ann would require that) but when I made a suggestion, he was defensive and
claimed he had already done what I suggested. I also suggested ways of tackling his
multiple-choice challenges but felt some resistance to my reconmiendations.
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When we turned to working on the statistical procedures to be tested on Exam #2
using both notes and diagrams, Mulder had some questions about the statistics but
seemed to have control of straight line equations needed for regression.
It was in this first session that I became aware that Mulder might have verbal and
auditory processing problems. He talked about his struggles with following Ann's
lecturing approach, he had more trouble than any other participant in composing his
metaphor, and he had failed the conceptual portion on Exam #1.1 decided that at our next
session I should try to help Mulder work out ways to compensate for his processing
difficulties so he could make the conceptual connections with the procedures that he
seemed to think he was capable of mastering. At this point in our relationship I thought
that this would simply involve beginning with his procedural competence and working in
parallel as I modeled finding conceptual connections.
Mulder's Session 2 and Session 3: The Conflict Emerges
Mulder 's Session 2
Although the focus of this session was on exam preparation for a fmite
mathematics exam in an hour,™"'" Mulder also tried to make sense of what was
happening in PSYC/STAT 104 in terms of how he had previously done in mathematics
courses. He had taken Exam #2 the night before and he knew that he had done badly
again on the conceptual part of Exam #2. He had lost 19 points out of 50, including 5 of
the 6 points for symbol identification — a D; Aim had shown him his score when he
handed in his computational section. However, he was fairly certain that he had the
computation 100% correct. Mulder had conflicting theories about whether historically he
was better at the "math part" or the "conceptual." Now he was irritated by the fact that he
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had done ' A^' work on what he considered the mathematics but was being denied
recognition for that because of this "other stuff."
I shared my ideas about what might be happening and proposed that he had a
theory about what was and what was not mathematics and that, according to that theory,
the conceptual part of PSYC/STAT was not mathematics. In addition, I told him that I
saw him as a strong-minded person who acted on his theories, and in this case he was
rejecting the conceptual aspects and concentrating on what he saw as real mathematics —
the computation. Mulder agreed that I was accurate but explained, "I've always thought
math was the harder part for me so that's what I've been concentrating on in the lecture.
Everything seems to be centered around that formula, so I concentrate on that formula, on
how to do that formula rather than taking it all in." With his poor result on the
computational part of Exam #1, concentrating on that formula was indeed an important
element of his recovery strategy. After all, it had led to success on the computation
section of Exam #2. 1 was concerned about his rejection of "taking it all in" because it
seemed tantamount to his deciding to dismiss the conceptual aspects of the course and not
make the conceptual link to the computational. Did Mulder think that he was not able to
do both or that he should not have to do both? It seemed that unconsciously he felt he was
not able; consciously he insisted he should not have to.
We were at this point still living in the initial transference-countertransference
relationship. I had assumed Mulder's transference of past teachers so I had higher
expectations of him that were reasonable and 1 was becoming frustrated when he would
not or could not deliver on his confident plans. My countertransference reaction was to
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push him, to accuse him of avoiding tasks (e.g., the conceptual), and to nag him with
direct advice, expecting that he could get his act together.
Mulder 's Session 3
At this point Mulder was frustrated, "I don't know what to do! ... I don't know
what to expect on the multiple-choice. ...We've had two exams now and I can't work out
what it'll be." I also felt at a bit of a loss. I recalled from Session 1 that Mulder was
ambivalent about doing the homework problems from the text because he was confident
about the computational part of the exam, so I asked if he had done the homework this
time. I also asked about the first text problems from each chapter set that were
conceptual questions like: "What is the range of values that a correlation coefficient may
take?" and "From each scatter plot in. the accompanying figure (parts a-f, on page 124)
determine whether the relationship is ...positive of negative ...perfect or imperfect"
(Pagano, p. 123). Mulder replied somewhat indignantly, "Those are the ones I did ... and
I wrote them down [the answers]." But when we looked at his conceptual Exam #2
errors, though, it seemed that he had not linked the concepts from this homework to their
numerical meaning. For example, he had responded with 0.75 as the correlation
coefficient that indicated the greatest strength on one question where the correct response
was -0.80 because its magnitude is greater. Mulder insisted that he had read his class
notes and the book a few times to prepare for the conceptual section. I suggested that
only reading was probably too passive to be helpful. Further I suggested that he might
even be he was even stopping himself from really learning it in some ways, resisting it
because he did not beheve he should have to learn it. His sheepish reaction seemed to
confirm my supposition.
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As we continued to discuss what might be hindering Mulder's success I brought
up his low learned helpless score on the Beliefs survey. When he said it was only with
math that he was like that, I suggested that he could change it even in mathematics: He
seemed to be doing well in his finite class, and could do so even in this statistics course.
Mulder seemed skeptical until he remembered that he had recently experienced not
giving up on a math problem. On the computational part of Exam #2 he had initially
made an error that led to what seemed to hrni to be anomalous results. "I sat there a long
time [looking at it] and I realized my standard error was wrong. It was way too big." He
went back to find and correct the initial error in his calculations and then to fix all the
computations affected: a mastery oriented response he now recognized. But could he do
that with the conceptual section on the test?
I recommended that he use study guide multiple-choice questions to prepare
better for the conceptual section. That we focused on the contents of the previous test
rather than on material for the upcoming test was not strategic but tackling these
multiple-choice questions highlighted Mulder's misconceptions about material that would
continue to be needed"™" and his ineffective study methods, especially on symbol
defmitions and their links to the calculations that would be on the next test. Mulder
agreed that this new tack of working on multiple-choice questions should help. I gave
him copies of sets of multiple-choice practice questions for each chapter to be covered on
the next exam for homework. He emphasized as he left, however, that at our next session
just before Exam #3 we should review all the symbols because he had done so badly on
the symbol section of Exam #2.
Mulder 's Session 4: The Central Relational Conflict Becomes Clearer
286
Session 4 took place on the day of Exam #3. Mulder was tired, grumpy, and
oppositional. I asked about what he had done to prepare for the exam. I had prepared
practice materials using the problems we had done in class but I had removed any
reference to the type of statistical test required to solve them so students could practice
also identifying the test. I had also prepared an empty flow chart template for the
statistical tests that would be on the exam for students to fill in as their formula sheet if
they desired. Mulder was taken aback that he might be required to identify which
statistical test was appropriate because he had been certain Ann had said that she would
tell us what statistical test to use with each problem and we would not be expected to
identify what test was necessary until chapter 19 and Exam #5. I told him that my e-mail
exchange with Aim on the subject left the question open. He was not happy, grumbling
about curve balls.
Mulder grabbed one of the problem sheets, declaring that by looking at a question
he could identify the statistical test required. Rather than analyzing the problem
statement, he tried to remember by the look of the problem and the order it had been
presented in class, but he remembered incorrectly. Another strategy was to identify the
type by whether the data were presented in columns or as an already computed statistic.
Again, he was incorrect. He made little attempt to read the questions and understand the
situation or experimental design. I remonstrated and insisted that as a "bright man" he
could and should think about the questions.
Mulder ducked my comments and moved to a discussion of symbols. I set up a
divided page to sort the population symbols from sample symbols and we began to
discuss how to make decisions on tests. The single sample tests went smoothly but when
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we returned to the two sample tests Mulder again tried to use his memory of the class
when students worked them. He seemed to enjoy my frustration with his approach. I
realized Mulder had not understood a central concept — ^that the words "independent" and
"correlated" described the groups or samples not the data numbers. However, he insisted
that to do a problem correctly on the exam he did not need to understand such
distinctions, saying, "I would have figured it out because I would have looked at my
equations and I would have figured out what went where." He then predicted that Ann
would give the alternative type (SS| and SS2) on the exam because one type of already
computed statistic (s,^ and Sj^) had already been given on a class problem. This
speculation seemed illogical and risky to me. I was quite alarmed by how Mulder was
orienting himself to the exam and he seemed to be enjoying my alarm.
Next we engaged in a much-needed discussion of symbols; I drew a reluctant
Mulder into making links with symbols he already knew. We discussed how to identify
sample mean symbols'^ and looked at what might be a logical value for the population
mean of difference scores, [Id ; because of the null concept of no difference in the
problems the class was mastering, |a,D should be zero in the null hypothesis statement and
therefore in the formula. Mulder ultimately got that wrong on the test (see discussion in
Session 5). I coached Mulder as he applied this logical classification and linking process
to the definitions he had prepared, modifying them in ways that made more sense to him
or that were necessary to be accurate. I encouraged him to link this process with the use
of the statistics they represented in the computation but he seemed bent on keeping the
sections separate in his mind.
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We briefly discussed his plans to complete his test preparation during the day, and
he left to do his chores and go to work-
in this session, I pushed Mulder to make logical decisions and connections and he
quite vigorously resisted, using visual memory and pattern finding of generally
extraneous details as benchmarks for decision making rather than exploring the logic of
the material.
The only thing Mulder had done differently to prepare for the multiple-choice
section had been what we did in Session 3; he had not used the practice multiple-choice
questions I had given him to do at home. He had written out defmitions for the symbols
but his resistance to changing his approach or to doing more than memorize patterns and
procedures remained entrenched. The resistance may even have grown and Mulder
seemed to use considerable amounts of energy for this resistance. The more I reacted the
more he resisted. I urgently needed to understand this resistance and help Mulder fmd a
way to put his energy and intelligence into preparing for and taking his exams.
I was getting frustrated! I experienced sessions with Mulder as enjoyable. Even
when he was tired and grumpy, I found him quick and funny. I tried laughing at his
outrageous strategies, appealing to his intelligence, scolding, and cajoling him but
seemingly to no avail. I was acting out of my countertransferential role of mother of a
rebellious teenaged boy. I tried but could not manage to get him engaged in anything
other than sparring with me. He was certainly not interested in addressing his issues
seriously while I was trying so hard to change him.
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Supervision
By the time I presented this case for supervision the results of Exam #3 were
known. Mulder had earned a 76% but with the in-class open-book extra credit assignment
he earned 5 of a possible 6 points his final grade for the Exam was 81%. He had
significantly improved his symbol identification score but had lost 18 points on the
multiple-choice section, 59% correct (on Exam #2 he had lost 14 points). In addition
Mulder had lost 6 points on the computational section, most because he had failed to set
his |Xd= 0, despite our discussion of this in Session 4 the morning of the exam.
I presented Mulder to Dr. P. and realized that what I wanted to do was to talk with
Mulder about how he stands in his own way and to propose to him that he had the choice,
that he might be able to make the choice to stop doing it. Dr. P. suggested instead a
strategy established by Alfred Adler (Mosak, 1995) and called paradoxical intention by
Victor Frankl (1963) that might help Mulder make that choice. The theory suggests that,
"The symptomatic patient unwittingly reinforces symptoms by fighting them... to halt this
fight, the patient is instructed to intend and even increase that which he or she is fighting
against" (Mosak, 1995, p. 74). Dr. P suggested that in the next session I have Mulder
experiment with truly resisting on an exam. His directions were clear:
Ask him how he resists; suggest that as an expert in resisting that he let me know
what strategies he uses to do that, so that he paradoxically really exaggerates this
thing; it's his life but as long as he is into resisting he might as well do it really
well. (Dr. P., July 20, 2000)
In presenting other cases for supervision (cf Brad and Autumn) I had revealed my
tendency to tell rather than ask participants how they might make helpful changes using
the insights we uncovered. Dr. P. gently but firmly helped me recognize how
counterproductive that was. I needed to see how by my telling Mulder I was likely
290
exacerbating his resistance — he was not only resisting a conceptual understanding of the
mathematics, he was resisting me as a "teacher" and perhaps even a "mother."
Dr. P. also suggested that I have Mulder explore the implications of the metaphor
that he was standing in his own way. "What's he doing and then what's the part of him
that's standing in his way doing? What gesture? What sound? What stance? Is he
tripping himself over? Is he holding himself back?" (Dr. P., July 20, 2000)
Finally he suggested I compliment Mulder on his insight into how he gets in his
own way, "Insight saves you a lot of trouble, not having to say, 'I don't know what's the
problem here'" (Dr. P., July 20, 2000). I puzzled over this. What exactly did Dr. P. mean?
In some ways I felt that Mulder might be expending too much energy struggling for
insight (or was it an excuse?), so I had to concede achieving insight should certainly free
that energy for actually doing the coursework. And the importance of congratulating him,
mirroring his achievements in insight and cognition was becoming clearer to me.
Session 5: Honing Resistance Strategies
I had videotaped the class on the evening before Mulder's and my Session 5.
Mulder sat beside and around the comer from me and was very interested in my research
activities. He noticed that I was observing students and taking notes when no one else
was writing. He "acted up" during the problem- working session in class to the extent
that, at one point, I called him a "scallywag." He seemed pleased.
At the start of tutoring I told Mulder I had found out more about Fox Mulder and
he gave an appreciative laugh. I said that Fox Mulder seemed to me the kind of antihero
who does things opposite to how others think they should be done. I asked Mulder
whether that was the characteristic that appealed to him and he agreed that Fox Mulder
291
was like that, doing everything in a way that was "definitely indirect" but he denied that
was the element he identified with. He insisted what appealed to him was Fox Mulder's
constant effort in looking for the truth. It seemed to me that he was denying a reality that
was largely unconscious (opposition) while consciously espousing a desire that was not
yet a reality (fmding truth).
I suggested Mulder's own search for "why I make mathematics so difficult for
myself might entail exploring his underlying and seemingly growing resistance to
mastering the conceptual part of the course (that Ann tested using multiple-choice
questions) that seemed to be making it harder for him to succeed. Paradoxical intention
theory suggests that getting Mulder to consciously and vigorously resist as he answered
multiple-choice conceptual questions (i.e., enacting the very behavior he needed to stop)
should result in his overcoming his resistance.
I proposed to Mulder that he take a mini-test of conceptual multiple choice
questions from the textbook's study guide'^' that I would give him, but that he should
strongly resist while talking aloud about his resistance. He seemed intrigued but initially
challenged my instruction to really resist with, "What, not do it? I can do that!" Mulder
began working silently so I asked how he was doing. Rather defiantly he replied, "You
tell me!" but in a few more minutes he said, "I don't like this one" and to my query,
"Because I don't know if it's really, really easy or if I'll have to do some work to fmd it."
292
I began talking Mulder through the problem by tapping into his existing
knowledge of the process of finding critical values using tables in the appendices in the
back of the book. As we did this I probed a little, "Is that part of your resistance because
you think about 'is that easy or is that hard?'" Mulder conceded, "Probably."
He decided the answer to his original query (is it easy or will it require work?)
was, "Too much work for me!" so we talked about how much work he was doing
comparatively for the fmite mathematics class. When I asked if it was about the same as
for the statistics class he demurred, "I don't know; next to 'bout none." I pursued this
further since he had clearly been doing better in the fmite class from the begiraiing
Mulder insisted it wasn't because he was doing more work or because it was
coming more easily to him but rather that, "I just don't have to do conceptual questions!"
I wondered, "Maybe then, [for] your resistance you say, 'This is conceptual. I don't have
to do that.' Maybe if you could say, 'Ah this is not conceptual' Rename it: this is just
mathematical. . ." Mulder blurted out "Pain in the butt!" I was prescribing and telling so
his resistant reaction to me should not have surprised me yet I was startled and asked if
he was calling me a pain in the butt but no, he insisted, it was "that section of the test." I
responded, "Well you are doing a nice job of resisting, which is good!" I could not
consciously acknowledge to myself that it was almost certainly me he was calling a "pain
in the butt" and resisting.
Mulder went on with his multiple-choice mini-test. He grumbled as he went, at
one point exclaiming "Crap!" when he picked the wrong value for N. I guided him to
interpret the table using the given a value of .01 and when he did it correctly I
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commended him. He interpreted that as condescension on my part declaring indignantly,
"I can readV I came back with, "You can also think!"
I gave him the package of class problems with the test name whited out, intending
him to use those after he was done with the multiple-choice, but he immediately grabbed
them and began naming the test using the same type of pattern memory "logic" that he
had exhibited in Session 4. As he expected, I scolded him reminding him of all the
multiple-choice questions he was now avoiding and commenting again on how well he
was resisting.
At that point, I interpreted what I saw happening between us from my point of
view, "I'm thinking this guy is so smart he could do so well and the mother in me comes
out and it's like 'If I could only persuade him'" I spoke from my side of our power
struggle. Mulder continued with the next question, appearing to ignore me. In a few
minutes as he began to work on new questions, I commented.
Every now and then you stand up against your own resistance by speaking [the
resistance, but then you stop resisting and do the work]... First of all you say I
don't want to do this, I don't want to come back here, she's a pain in the neck
...but then [you see] what happens when you try them [the questions]. (Session 5)
After this interchange I felt a distinct change in Mulder's demeanor. It felt like a
turning point. He began to ask "why" questions. For example, "I have a question: Why do
we find the total of the degrees of freedom?" Mulder became actively engaged in the
discussion of my responses. When I admired his thinking he was spurred on, but when I,
drew his attention to this apparent change and put it in terms of his forgetting about his
resistance he became defensive and seemed to return to it. It would have been better if I
had asked him "Does this feel different to you?" rather than telling him of the change I
felt. By not doing so I was pulling him back into our power struggle.
294
I wondered out loud if his resisting was really letting the instructor win, a form of
"I can't work this through... I'm wimping out here." Mulder ignored me and went on
working, even suggesting that he do some from chapter 1 8 and triumphant when he got
those correct.
I went further with the Mulder-has-two-parts theory. I reminded him of his
reaction to losing points only on the conceptual/symbol definition section in Exam #2. I
suggested that he had seemed almost proud that he had proved the pomt that he couldn't
do the multiple-choice conceptual part," but he disagreed, "Not proud of that just pleased
I got all the calculations right . . .1 got a B!" He sounded indignant that I didn't seem to
appreciate that fact and admire his competence. I wanted to give him a little more
feedback.
There's a part of you that is really listening and engaging in it and there's another
part that's like "uugh" [pushing away with my hand]. It's almost like you've got
this little battle going on [Mulder chuckles]. You don't think that's happening?
(Session 5)
Mulder continued with his mini-test as I looked at what he had done on Exam #3.
I continued to alternate between exploration of Mulder's mathematics approaches and his
resistance. I interrupted him to ask him about an error on Exam #3. He said that forgotten
some details that Ann had told him so he made something up. I had also gone over this
fact and the logic of it with him in Session 4 the morning before the exam, but he was at
the time resisting my push to link symbol defmition to the computation. He clearly didn't
recall that we had discussed it. Mulder had however successfully used his visual
memorizing of in-class worksheets on parts of the exam He described the process, "That
one I just sat there forever and ever until I came up with it. ...I kept thinking of the
worksheet in my head until I came up with it." I wondered about what might have
295
happened if he had sat for ever thinking about the multiple choice questions. I suggested,
"It's like a cooperative part that does this [the computational part] and a resistant part
[that doesn't do the multiple choice]"
Mulder continued working, talking out loud as he did liberally sprinkling his
reactions with "crap" and "turd,' and at one point calling himself an idiot for not dividing
accurately. He seemed intrigued with the concept of his two conflicting parts doing battle
even though he said, "Let's not remind me!" when I noticed that he was doing lots of
good thinking so "the cooperative side of Mulder is working." At one point when he
called himself an idiot I was wondering out loud about his sense of his own competence,
which he thought was fairly robust, and I hesitated thinking I was interrupting his work.
But I was surprised when he said, "Keep going, I'm listening," so I did:
JK: In the test when you're doing this part is there anything that your I-want-
to-do-better; I-can-think-about-this; This-may-be-logical part of you can
say to the resistant part that says "turd" or "I'm screwed!" or "I don't like
this!" or "this is ridiculous!"
Mulder: You quoting me still? You're going to write your dissertation and say this
person said all this stuff about math.
JK That could be part of it but the main thing is what you're saying to
yourself (Session 5)
Mulder sidestepped my question by drawing me into a difficulty he was having
with x^ , but I felt that I was really being invited to witness his internal dialogue as he
found his error and mastered the procedure and the concept — ^that was his answer to my
question. Mulder was demonstrating his answer — that his smart, achieving self was in
charge now.
I became aware that in my counseling role, even my tutoring role, I should have
taken a neutral stance with respect to Mulder's warring parts. I was truly moving closer to
that but I was so used to takmg sides that I continued to betray my partiality for Mulder's
296
"good" side, risking as I did, propelling him towards his "bad" side and getting us back
into our power struggle.
After Mulder successfiilly negotiated more questions on x^, I moved in again as
tutor. I wanted to help him make more visual and logical connections:
JK: Have you seen a picture of a x^? [JK draws a graph]
Mulder: So it's going to be positively skewed. . .
She didn't show us this [resisting what he saw as additional material].
JK: [chuckles] That's true But it's not going to do her in if you get this wrong;
it's going to do you in!
Mulder: Yes.
JK: Yes and you want to be done in to prove your pomt.
Mulder: Yes that's my goal in this test: to fail miserably on the multiple-choice.
JK: To prove your point.
Mulder: So I have to take that fmal.
JK: Ah, there you go!
Mulder: I'm not going to take it if I don't have to. . . if I could do well on these next
two tests I wouldn't have to take the [optional] fmal.
JK: There you go! This is that logical sensible person. Don't speak to me!
Speak to that resistant part that has to get a bad grade on the multiple-
choice. (Session 5)
Mulder continued, alternating between resistant grumbling and engaged
cooperating. But the focus of his grumbling was changing. It was less about her [Ann]
and having to do the conceptual work, and more about the cognitive demands of the
conceptual work. He grumbled about a change in how to identify significance required in
a new test, about a question we both agreed was badly worded in the study guide, and
especially about the practice problems I had prepared for the exam. He tackled the rest of
the multiple-choice questions in an engaged, positive way. And I stayed out of the battle.
At the end of our session Mulder's fmite math instructor came in and Mulder told
her in discussion "I have a resistant side of me." To her dismissive "Don't we all," he
insisted, "I lost 18 points on the multiple-choice." This was the first time that Mulder had
verbalized the theory of his two battling sides and owned it.
297
Final Discussion: Mulder and Mathematics Counseling
Mulder completed the practice problems, spent time on his formula sheet, and
even came to study group before the Exam #4 the next evening. And he earned a 91%
overall and an 82.6% on the multiple choice section! He went on to earn an A on Exam
#5, an A" on the MINITAB presentation, and a B in the class. He was satisfied with that
and decided not to take the optional comprehensive fmal. He dropped in a couple of times
at the Learning Center, once to have me check over his MINITAB presentation paper (I
suggested corrections that he did not make because he did not have time.) and once to
help him prepare for his finite math fmal.
Session 5 was a pivotal session both for Mulder and for me. At the start of the
session, I believed it was primarily his stubborn resistance against the assessment that
prevented him from improving although I suspected that, in some ways, I was
exacerbating that resistance by my mother-of-a-teenager countertransference. Though I
planned to use Dr. P."s suggestion to try paradoxical intention, I found myself telling him
what I saw of his resistance and suggesting how to fix it. Ah, that was it, I realized. When
I tried to get him to do or know or believe something, he resisted me. His resistance to
Ann and the conceptual part of the test was confounded with his resistance to me, and
that in turn made me push harder. It was when Mulder and I assumed the same stance in
looking at his approaches that the change in his self-awareness began. When I implied
that he might be betraying or letting down his resistant self by the intelligent engagement,
things didn't go well — I was taking sides. When I couched it in terms of two legitimate
parts of himself that were engaged in battle, Mulder went with that and I was able to
withdraw myself from the battle. I no longer had to battle Mulder's resistant side trying to
298
persuade him to capitulate and cooperate. He could fight the battle himself, using his
intelligent engaged self, and I could let him go. It felt good (but a little scary) to pull
myself out of the fight and let Mulder battle himself and fight his own demons. He did
this successfiilly, and I congratulated him.
Evaluations
On the post-surveys, Mulder's learned helpless beliefs changed significantly
towards mastery orientation (see Appendix M, Figure Ml and Appendix H, Table H3).
He said his motivation for coming to mathematics counseling changed irom helping me
with my research to getting help with his strategies because the help with the statistics
was "great," but his metaphor had not changed much: the Truth (mathematics?) was still
"out there." Mulder had found that despite a relatively underdeveloped, vulnerable
mathematics self, he could do well in a mathematics course if he got out of his own way
and tried to think strategically and conceptually. That very success could contribute to the
development of that self There was still room for growth but now Mulder might draw on
this experience and risk trying to understand rather than using illogical alien approaches
or overconfidently avoid trying.
I had learned to attend to Mulder's transference of past relationships with teachers
into our relationship as he alternated among confessions of laziness, pronouncements of
his potential, and theories about what mathematics was that precluded the parts he was
struggling with. I learned to attend to my countertransference reactions: I was the
frustrated, cajoling mother of a young man who seemed to be his own worst enemy, and
he sparred with me and appropriately resisted my efforts to fix him. Mulder's own
metaphor was the key to resolving the conflict. In supervision with Dr. P., I began to
299
understand better what was going on as we examined Mulder's metaphor and I shared my
transference-countertransference insights. In our last counseling session Mulder resolved
his central conflict when I withdrew from my countertransference stance. I truly used a
relational counseling approach, the outcome for Mulder was good, and I learned how
powerfiil and counterproductive countertransference reactions can be even when one
theoretically knows about their reality. I learned that although it is difficult, examining
my countertransference reactions and choosing consciously to do things differently is
crucial.
REFLECTING ON THE COUNSELING CASES
Crossing previously drawn lines — that seemed to be a common thread through the
course of mathematics counseling with Karen, Jamie, and Mulder. Indeed a relational
approach required it. I crossed lines and so did they, and we crossed lines together.
Although mathematics was our primary activity, my persistent curiosity into how they
did well and why they struggled led us to new ground. Of the three Mulder was the most
willing on the former and the most resistant on the latter. Karen put strict limits on her
responses to what she perceived to be non-mathematical discussions but she crossed her
own previously drawn lines in mathematics effort. Jamie was willing to cross lines with
me after I crossed lines to draw her into counseling in the first place. But we stayed
within boundaries acceptable in the Learning Assistance Center context.
Each of these students made progress academically. Each achieved a grade as
high as or higher than they had hoped (see Appendix H, Table HI .). Mulder and Karen
who earned D" s on the first exam went on to earn Bs in the class. Both did so despite
significant deficits in their mathematics preparation (see Tables 6.1 and 6.2). Jamie foimd
300
Table 6.1
Focus Participants ' Levels of Understanding of the Variable on the Algebra Test
(Sokolowski, 1997; Brown et al., 1985, p. 17; see Appendbc C)
Participant
Number
Correct (of
53)
Level 1
Level 2
Level 3
Level 4
Level 5
Level of
Understanding
Jamie
42
6/6
5/7
8/8
6/9
1/3
Level 4
Karen
30
6/6
5/7
3/8
2/9
1/3
Level 2
Mulder
25
6/6
7/7
3/8
3/9
0/3
Level 2
that, contrary to her belief, she was adequately prepared mathematically (see Tables 6.1
and 6.2). She and Karen were repeating the class and they saw and did things differently
and did well this time.
And each of these students gained new insights into themselves as mathematics
learners. Jamie realized that her difficulties with mathematics were not to do with her
ability but rather with relational issues; Karen found that she could achieve well in
mathematics despite her considerable arithmetical and algebraic deficits and lingering
doubts; and Mulder overcame his resistance to aspects of mathematics he found difficult
because of his auditory processing difficulties and in defense of his vulnerable
mathematics self and found that he could do well.
I crossed lines and found a new way of looking at myself and them and us that
gave me new power to reflect, monitor, and change my approach and steer the
counseling. At the same time, this new way of looking gave me new ways of listenmg,
observing, and responding to them so that they could and did choose their way and
modify mine. The counselor-student dyad indeed was the key to the changes we all made.
301
Table 6.2
Focus Participants ' Understanding of Arithmetic on the Arithmetic for Statistics assessment
(Appendix C and chapter 8 discussion)
Participant
Class
work
Small
(<1000)
Large
Integer
Fractional
number
Place
Value/
Operation
Sense
Open Ended
Arithmetical
Statistical
Sense
Integer
Number
Sense
Decimal
thinking/
Number
sense
Sense
problem-
sense
solving
Jamie
adequate
100%
43%
67%
|a|<5:
80%
95%
88%
adequate
inadequate
-adequate
97%
adequate
|a|>5:
100%
adequate
adequate
adequate
adequate
Karen
marginal
45% of
17% of
33% of
|a|<5:
20% of
35% of
45% of
total; 100%
total; 30% of
total;
65%
total: 33%
total; 42%
total;
of attempts
inadequate
0
attempts
inadequate
75% of
attempts
adequate?
-adequate
|a|>5: 40%
inadequate
of attempts
inadequate
Ofattempts
inadequate
62.5% of
attempts
-adequate?
Mulder
adequate
100%
56%
66%
|a|<5: 90%
100%
76%
56%
adequate
marginal
-adequate
adequate
|a|>5: 69%
adequate
adequate
adequate
marginal
good X>85%
adequate: 70% < X < 85%
-adequate: 60% <X < 69%
marginal : 50% <X < 59%
inadequate: X < 50%
inadequate? X > 50% but < 50% attempted
adequate?: X > 70%. ofattempts
adequate?: 60% <X < 69% of attempts
In the next chapter I briefly profile the remaining nine students in the class and
discuss the developing theory that emerges from this pilot study. In particular I propose
criteria for a new way of categorizing students as mathematics learners that surfaced from
302
analysis of participant profiles. I then analyze and present what I see as the essentials of
the brief relational mathematics counseling approach that emerged.
303
' Because this is an even numbered chiapter 1 use "she," "her," and "hers" as the third person singular
generic pronouns.
" This pattern continued except for two occasions later in the course before exams when she did allow
herself to be drawn in to some of the study group's discussion.
■" I saw the impact of this limited understanding in the first exam when she used the deviation of only one
score from the mean instead of the required deviations of all values of the variable (the scores), to find the
standard deviation of all the scores fi-om the mean.
" Performance vs. learning achievement motivation questions were numbers 4, 7, 9, and 10 of Part I of the
Beliefs Swvey (see Appendix C)
■" For example, she mistakenly thought o (sigma, the standard deviation of the population.) represented the
mean of a population.
" For example, \x = population mean, and a = population standard deviation, both constant identifiers for a
particular population distribution.
™ For ex2mp\e, X sample mean, and 5 sample standard deviation, each constant identifiers for a particular
sample.
"" She had to find the median, P50 , of a set (distribution) of scores.
" She agreed with prompting that her answer didn't make sense but when I instructed her in the use of her
calculator to get the correct answer she remained baffled, "I don't really know why though. I just plugged
in what you told me." In this session there was no time to teach Karen the concept of percent.
" Despite her emotional state in Session I, Karen had managed to successfijlly correct, learn, and retain
these symbol designations during the session.
'" For example, for the real limits of the weight of a slice of cheese of 0.35 grams, Karen had answered
d. may be anywhere in the range of 0.34 -0.36, instead of the correct
b. may be anywhere in the range of 0.345 - 0.355.
Karen's answer to this and similar questions showed an understanding of the concept being tested but
anything beyond the first place (tenths) of decimals confijsed her. Karen was able to give 8.5 and 9.5 as the
real limits for 9 and other whole numbers but not for 0.9 or 2.9. She misnamed decimal places, calling
hundredths tenths and vice versa. She did not seem to have a firm sense of the relative size represented by
the places nor the places' relationships with each other.
"' In reality, when Karen took the Arithmetic for Statistics assessment after the course ended, she showed
that she c/zii understand relative sizes represented by the places, on the number line graphing questions
though not by using numerals alone. If Karen had made that extra appointment, I would have given her the
Arithmetic for Statistics assessment then and coached the exploration beginning with her number line
understEinding.
™' We each copied the data fi-om the class question onto our own sheets of graph paper. Next we copied the
formula as I read it out loud and constructed columns beside the X and the Y columns with column
headings corresponding to pertinent elements of the formula for /% namely X", Y", and XY.
'"" I suggested we use the questions for the assigned problem in the text (Pagano, 1998, chapter 6, Problem
14, p. 124) and the data from the in-class question because the questions were more delving and the data
were less complex.
304
'" Karen had not computed the standard error of estimate of Y accounted for by X correctly because she
had not constructed the squared deviations (Y -Y'f column. She had the (Y -Y') but not the (Y -Y')^
column on her formula sheet, so she used the (Y -Y'). This should have summed to zero prompting her to
check her formula (which correctly included the squaring) and create the (Y -Y")^ column. Instead of
addingKaren tried multiplying the (Y-Y')s to get 0.00015 Iwhich she then wrote as 1.51 tomakeitmore
reasonable. She knew this was not correct, however, as she wrote on her exam "still can't figure out where
I screwed up."
'^ I had checked with Ann about student use of the blank flow chart and she was agreeable.
""' Karen knew from class that being given 5" s or the SSs (sums of squared deviations of scores from the
mean) would indicate an independent samples t test in contrast with two sets of data that would indicate the
correlated groups t test. / knew, however, that in the real world of data gathering and analysis, students
could be given two sets of scores for eitlier situation, independent or correlated, and coi/W calculate SSs and
r s from that data for either.
™" Karen made an error in one inference test that did not result in her losing points. On the normal deviate
z test, Karen had compared the magnitude of the p value she obtained (0.0013) with the z score she had
obtained (-3.01) instead of with the critical alpha level of 0.05. She came to the correct conclusion though
so Arm did not deduct any points.
'"" Still later, a semester after the end of summer PSYC/STAT 104, however, Jamie revised her stated
motivation to helping me: "...I'm much more of helper, which is why I think I signed up to do this with
Jillian, cause 1 saw it as helping her with her project. If it had been just for my benefit I don't really know if
I would have approached her or not."
"" Jamie was, in fact, repeating this course, but I did not find that out until after the course was over. With
appropriate permission, I obtained the printouts, without names, of the grades of all students of the
PSYC/STAT 104 for the 5 years before the summer of 2000. It included the data from this class with some
that suggested that Jamie was repeating the class. When I sent a post-study e-mail survey to check that and
other data 1 was unsure of, Jamie replied to my assumption that she was not repeating the course, that in
fact she was repeating it because she got a D* the first time and that was not adequate for her psychology
major.
^ Jamie's "?" indicated her own uncertainty about her exact grade.
'"' She did not tell me of her D* in her first attempt at PSYC/STAT 104 at State University.
'™" See Chapter 5 for a discussion of Ann's policy regarding students' tests.
'""" Here Jamie indicates her belief that her error lay in using "the right one [equation] for a different one"
perhaps thinking of the different formula for a, the standard deviation for a population for which the
denominator is N rather than the n - 1 for the s, the standard deviation or a sample. In feet, the class had
learned no formula for which the denominator is EX - Ithe one she ad initially used.
"^ 3.7 on a 1 through 5 scale — close to the highest in the class (see Figure L4, Appendix L)
'°™ Sokolowski's three college student subjects who achieved a level 4 of the algebraic variable, had each
succeeded in at least one college level mathematics course, was at the time of her study an A/B
mathematics student, and succeeded in combinatoric-/probability-/statistics-related mathematics (p. 70, 98).
'°™" Since a score of 1 represents zero anxiety, the drop of 0.5 in Jamie's Mathematics Testing Anxiety
form 4. 1 represents a 0.5/3. 1 that is a 1 7% decrease.
305
'°"™ This participant chose this pseudonym for himself when the question arose during his mathematics
counseling Session 5 on July 25, 2000. Fox Mulder was also his metaphor for how he approached
mathematics.
""" Over the course he talked during the lecture portions of the class in every class but one, averaging three
interactions — answers or corrections — per class. This placed him as the third most involved in these lecture
discussions, after Robin and Lee (see chapter 5, Table 5.2).
"" A science fiction television series featuring FBI paranormal detective Fox Mulder (and his partner
Scully) in search of the aliens who he believed had abducted his sister.
"""^ I was able to check on this more formally when I gave Mulder my Arithmetic for Statistics assessment
(Appendix C) as a posttest on July 31, 2000. He asked if he could fill it in later and eventually sent it to me
in March 2001 . See Table 6.2 for Mulder's results, all of which were adequate except for his statistical
number sense and large integer number sense which were marginal. These last areas (tested on this
assessment) were not tapped during the course.
'°°"' His approach lacked a number of the identifying features of learning style II; he did not seem to grasp
the gestalt of a situation or use an inductive (rather than deductive) reasoning approach nor did he have
difficulty with details and step by step procedures. In these areas he seemed more analytically procedural
(like Davidson's mathematics learning style I) though he used visual memorization rather than verbal
tactics. On the other hand his finding solutions without being able to satisfactorily explain how and his
sense of appropriate sized solutions supported a learning style II conjecture. It was also not clear to what
extent he had adapted his approach to handle mathematical tasks that seemed beyond him.
■"^ Suinn (1972) found on his 98 item Mathematics Anxiety Rating Scale (MARS) from which all the
testing and number anxiety items of my Mathematics Feelings survey are drawn, that mean scores were as
low as 1 .47 for physical sciences majors (sd = 0.4), and 1 .7 for social sciences students (sd = 0.6) which
would seem to imply that Mulder's 3.1 shows high anxiety (more than 2 standard deviations above the
mean). But because Suinn's scale was found to confound testing and number anxiety factors (see Rounds
and Hendel, 1980) and students' number anxiety scores were on average 0.75 points lower than their
testing anxiety scores when separated here in the Mathematics Feelings survey, 1 would suggest a higher
average for testing anxiety and a lower average for number anxiety than Suinn's should be considered
moderate on my Feelings survey. Given this consideration Mulder's 3.1 testing anxiety score could still be
considered well above moderate even for a social sciences student.
'""""The fact that he was 75% satisfied with his mathematics achievement (Item 6, responses marked 1) and
75% confident about his mathematics future (Item 3, responses marked I), and 75% positive about the
course he was taking now (Item 4, responses marked 1 ) indicated a positivity that did not seem justified by
his history or his performance on Exam #1 (see Appendix M, Figure M2 and Table M3).
'°°" That is, to have him discover that developing a conceptual understanding of the procedures he had
mastered by Exam #2 would help ensure continued success in the computational part and mastery of the
conceptual part of the test.
""^ He pointed to question 3.
3. The 20 subjects constitute a
a. population
b. sample
c. parameter
d. variable
Mulder asked me who the 20 subjects were; what was that about? 1 found that he had not realized that the
first 5 questions were referring to an experiment described and bolded at the top of the page. He got three
of these five questions wrong (and lost 6 points).
306
'"""' Ann mislaid the computational section of Mulder's first exam so we were unable to analyze his errors
on that section.
'°°™" Mulder wanted to "memorize rules for doing a SIMPLEX problem." He felt he had the material under
his control and was really just checking that he had it correct. He knew the material procedurally but was
not able to explain to me nor did he want to know why he had to do what he was doing. He was using the
SIMPLEX method to maximize profit given a system of linear constraints (Rolf, 1998, chapter 4). He knew
that an equation had to be changed to two inequalities, in particular, inequalities in which the variable sum
was less than the constant, before slack variables could be added, but he did not know why, probably
because he did not understand the meaning and use of the slack variables. He was able to perform the
necessary procedures.
'°°™' For example, Mulder thought 5 stood for "sample" (rather than sample standard deviation).
"^ For example, we worked out what D meant knowing already that X was the mean of scores for a
sample. 1 pointed out the links: the bar conveys the idea of mean and the D represents the list of data being
analyzed (in this case the differences between pairs of before and after scores).
^' The chapters covered were: Chapter 15: Introduction to Analysis of Variance, chapter 16.' Multiple
Comparisons and chapter 1 8: Chi-Square and Other Nonparametric Tests in Understanding Statistics in the
Behavioral Sciences {Pagano, 1998).
307
CHAPTER VII
DEVELOPING THEORY: STUDENT CATEGORIES AND
WAYS OF COUNSELING
In this study I gathered mathematics cognitive and affective data from 12 of the
students of PSYC/STAT 104 and I counseled ten of them using cognitive constnictivist
tutoring and relational and cognitive counseling approaches described in chapters 2 and
3. The results of the study are described in chapters 5 and 6. As I analyzed these results I
noticed a number of interesting interlocking patterns that I discuss in this chapter. I will
demonstrate how this analysis supports a categorization scheme of mathematics learners
that emerged from this research. I will then present my analysis of this brief relational
counseling approach as I found it relates to students thus categorized.
When I analyzed the three in-depth cases and the briefer profiles of the other nine
students in the class I found that categories of mathematics self development emerged
from interactions between two dimensions — mathematics preparation and relational
experience. These interacting factors produced relatively well-defined categories that can
be compared and contrasted with Tobias's tiers described at the end of chapter 4. These
categories although similar to Tobias' tiers are distinct in important ways.
This result was of particular interest because I also found that different relational
mathematics counseling approaches and the relative balance among its components
(degree of cognitive constructivism, amount and kind of mathematics tutoring, amount of
course management counseling, and cognitive and relational counseling) were differently
applicable to specific categories of student.
308
Mathematics Preparation
Students in the class fell into three broad categories according to the adequacy of
their mathematical preparation for the class: well prepared, adequately prepared, and
underprepared (see Table 7.1).
Table 7.1
Criteria for Determining Level of Mathematical Preparedness of PSYC/STA T 104
Participants
Well Prepared
Adequately
Prepared
Underprepared
Course grades:
Exam #1
B+ through A
D through A
F through C
Final Course Grade
A' (B^) through A
B" through A~
AF through B
Algebra:
Algebra Test
Level 4 or 5
Level 4
Level 1 or 2 (or 3?)
and class, exam and
counseling session
work
Arithmetic:
Arithmetic for
Good (>85%)in all 8
Adequate or above
Ranges from
Statistics
categories
(>70%) in all but
adequate or above
Assessment and
one or two number
(>70%) on at most 6
class, exam.
or operation sense
categories to
counseling session
categories; variable
inadequate (<50%)
work
in other sections
or marginal (50% <
X < 59%) on three
or more categories
Evidence for how participants placed in these categories was gathered throughout
the course. Not all participants took the Algebra Test (Robin, Brad, and Kelly did not.) or
i\\Q Arithmetic for Statistics Assessment (Robin, Mitch, Brad, and Kelly did not.) but in
these cases there was sufficient evidence from their exams and work in class and
counseling by the end of the course to place them with reasonable confidence. The three
criteria that served best to categorize students in this sample were (a) understanding of
the algebraic variable (measured on the Algebra Test, see Appendix C), (b) understanding
309
of and facility with arithmetic (measured on the Arithmetic for Statistics Assessment, see
Appendix C), and (c) performance on the first exam of PSYC/STAT 104. When I
considered students' high school and college course-taking and grades as an additional
criterion for this sample, there was not enough consistency for this to be usefiil (although
with larger groups of students this might be found to be a factor). The students who
were well prepared rwAhemdiiic&Wy had a high level understanding of the algebraic
variable (see also Appendix H, Table HI), were arithmetically confident and competent,
had always done well in mathematics, and did well on the first exam in the course (see
Table 7. 1). Those who were adequately prepared had a high-enough level understanding
of the algebraic variable. However, while their arithmetic was generally sound they had
some deficit areas, they had each had variable success in previous mathematics courses,
and their performance on Exam # 1 ranged widely from D~ through A. Those who were
tinderprepared had a low level understanding of the algebraic variable, deficits in
arithmetic that ranged fi"om significant to mild, and they did poorly on the first exam of
PSYC/STAT 104.
It is possible that using this approach to classifying students with larger groups
might result in the imderprepared group' s being split into more categories. With this
small group, placing the student/s who were weak in both arithmetic and algebra in the
same category with student/s who were weak in algebra but sound in arithmetic makes
sense given other identifying criteria. Further evidence may suggest otherwise.
Mathematics Self: Mathematics Preparation and Self-Esteem
After I sorted students according to their mathematics preparation (see Table 7. 1),
further analysis revealed that students' level of mathematics self-esteem roughly matched
310
the preparation categories and that these taken together gave a measure of students'
mathematics self development. As noted in chapter 2, self psychologist, Kohut (1977)
proposes that healthy self development leads to internalized values and ideals that
provide structure and boundaries as the person's own competence develops. When this
process proceeds appropriately the internal self-structure is consolidated and it provides
what Kohut calls "a storehouse of self confidence and basic self-esteem that sustains a
person throughout life" (p. 188, footnote 8). From this study I found that it was a
student's mathematics competence (preparation) taken with his level of his self-esteem
that indicated that self development level: Category I (sound), II (undermined), or IE
(underdeveloped) (see Table 7.2). I found that his level oi confidence . realistic, under, or
overconfidence, was an initial cause of confusion in assessing a student's category of
Table 7.2
Emerging Categories of Mathematics Self Development
Mathematics Preparation
Level of self-esteem
Well Prepared
Adequately
Prepared
Underprepared
Sound self-esteem
Category I students
with sound
mathematics selves
Compromised self-
esteem
Category II students
with undermined
mathematics selves
Low self-esteem
Category HI
students with
underdeveloped
mathematics selves
mathematics self (see Ja/w/e, Karen, and Mulder in chapter 6). I found that a student's
level of self-esteem, however, was directly related to his mathematics preparation
311
(competence) level. The levels of self-esteem I found in students in this sample were:
sound, compromised, or low (see the student profiles below for discussion of how I
discerned these levels). The shaded cells in Table 7.2 indicate that I found, as I expected,
no student whose level of self-esteem was not directly related to his level of mathematics
preparation.
Mathematics Self Category and Relational Malleability
I found that students in the second and third categories of mathematics self could be
further sorted according to the extent of malleability (willingness to change beliefs and
behaviors) versus inflexibility (resistance to changing beliefs and behaviors) in their
mathematics relational patterns. This malleability versus inflexibility seemed to stem
from personal characteristics interacting with past mathematics experiences in the current
course environment. Students in Category II fell into these two subcategories according
to how they had handled their compromised self-esteem: they had developed mathematics
relational patterns that were either malleable or inflexible for the brief semester
timefi"ame. Students in Category 11 of mathematics self similarly fell into these two
subcategories according to how they handled their low self-esteem (see Table 7.3 for
criteria I used to gauge malleability). This classification became important from early in
the course because a student's willingness to engage in the struggle early in the course
and to change if he' was persuaded that he needed to was, not surprisingly, a pivotal
factor in his success. This was especially important for underprepared students with low
self-esteem (i.e., students with an underdeveloped mathematics self).
Interestingly, students I found to be inflexible seemed to fit Tobias' categorization
of students as "utilitarian" (see chapter 4) and Mercedes McGowen's categorization of
312
Table 7.3
Criteria for Determining Malleability of PSYC/STA T 104 Participants
Malleable Relational Patterns
Inflexible/unstable Relational
Patterns
Achievement motivation"
Stated learning achievement
Stated performance achievement
motivation /learning motivation
motivation
when he believes he is capable
Learned Helpless versus
Mastery orientated in beUefs
Learned helpless in beliefs and
Mastery Oriented
and/or behaviors
behaviors
beliefs''/behaviors
Procedural versus Conceptual
Conceptual mathematics beliefs
Procedural mathematics beliefs
Mathematics
and/or behaviors and/or change
and behaviors
beliefs%ehaviors
towards conceptual
Problem-solving/trouble
Engagement in problem-solving
Avoidance of problem-solving
shooting beliefs/behaviors
practices/behaviors
practices/behaviors
Changes over course
1 . Behaviors
Willingness to change/
Resistance to change over
resistance to change that
course
changes to willingness to
change during course
2. Beliefs:
a) Mathematics'
a) Substantial change in
a)Limited positive changes in
behaviors/beliefs over
beliefs/behaviors over course
course — some positive
(especially focal beliefs or
emotions), some negative
b) Fixed trait beliefs
b) little need for change or
b) limited change
about personality.
positive change for this
limitations, and/or
situation at least
mathematics
potentiaf
3. Attachment Patterns'*
Secure; avoidant to secure;
Remains detached or
dependent to secure
dependent or ranging between
both
Note: " Achievement motivation beliefs were gauged initially by averaging 1 through 5 responses on a
subscale of the Beliefs survey: Part 1, Questions 4. 7. 9, and 10 (see Appendix C). Achievement motivation
behaviors and fiuther explanation of beliefs were gauged through observation and conversation in
counseling.
'' These beliefs were gauged initially througli responses on the Beliefs subscale; behaviors and fiirther
explanation of beliefs were gauged through observation and conversation in counseling.
" Changes were gauged through posttesting oi Beliefs and Feelings surveys, by conversatioa and by
observation of responses (verbal and behavioral) to counseling interventions and in course achievement.
■^ Changes in attachment patterns gauged in counseUng through transference/countertransference etc.
313
students as "rigid" (personal communication, April 1 1, 2000). However, in addition,
some malleable Category III students (e.g., Karen and Mulder) initially presented as
inflexible (utilitarian/rigid) but their apparent inflexibility turned out to be defensive in a
way that was adjustable with appropriate relational counseling.
I am very aware that my own personality and behaviors might have been a factor
in the extent to which a participant exhibited malleable or inflexible behaviors in the
counseling situation. This may not be entirely intrinsic to him. My particular challenge as
a counselor may be to develop ways of helping students I perceive to be relatively
inflexible to bend. Effective ways to achieve that are certainly not by advising, lecturing,
or scolding. A relational understanding led me to see that inflexibility may be at least in
part in defense of compromised or low self-esteem. It was in students' responses to the
surveys and their interactions in the classroom, with the course material, and with me that
a malleable or inflexible profile emerged. Whether and how students changed over the
course also helped confirm such a malleable or inflexible profile (see Table 7.3 for details
of criteria and individual characterictics and Table H3 in Appendix H for student
changes). No one met all the criteria identified for a profile but each participant had a
predominance of characteristics of one type with relatively fewer of the other.
I will now present brief profiles of the students in the study showing how they led
me to develop the categories and sub-categories I have identified.
Category I Students with a Sound Mathematics Self: Mathematically
Well-Prepared with Sound Mathematics Self-Esteem
In this class there were only two students who fit the Category I mathematics self
profile and both had malleable relational patterns. It is possible that Category I students
could exhibit inflexible relational patterns but I believe that to be unlikely unless such
314
students are faced with mathematics challenges well beyond what they are prepared for.
Then the inflexible/malleable distinction might surface.
Sound Mathematics Self Students with Malleable Mathematics
Relationship Patterns
Two students in the class (Catherine and Robin) each had a constellation of
characteristics that identified them as mathematically well-prepared students with sound
self-esteem: Each earned a good grade on Exam #1, had experienced prior steady success
in mathematics and had no crucial knowledge base gaps in arithmetic or algebra, had
more conceptual than procedural beliefs, showed mastery orientation to mathematics
learning, and had low to moderate anxiety. They had learning (rather than performance)
motivation for taking the course, and each was realistically confident and exerted a
realistic amount of effort towards mastery in the current class. In other words, each had a
well-developed mathematics self, no toxic internalized mathematics presences, and
current patterns of mathematics relationship that were flexible and constructive.
Although Lee had the highest conceptual beliefs score in the class on the pre-
beliefs survey and had recently succeeded in a finite mathematics course in college, her
high anxiety scores, low confidence about her mathematics (related to struggle and
variable success in prior mathematics courses), relatively low Exam #1 score, and
underdeveloped arithmetic operation sense ruled her out of this group. Autumn's
performance motivation, procedural beliefs, learned helpless orientation, and history of
uneven mathematics course performance also ruled her out of this group despite her high
expectations and good grade on Exam #1 . Robin signed up for mathematics counseling to
help me with my research, but Catherine declined the offer.
315
Catherine. Catherine," a non-traditional biology major who had just completed
Calculus I with an A, was confident but quiet in class. Her high conceptual belief score,
low anxiety, and high course outcome expectations seemed congruent with her presence
in class and she was not considered to be at-risk in a statistics class. Initially, the
instructor and I both thought she would do well without help. I wondered about how she
would handle her own expressed need for conceptual understanding of the mathematical
procedures {Beliefs survey) because these links were not generally made in class, but she
did enough work on her own (5 hours per week) to make the conceptual links to the
mathematical procedures that she needed in order to master the material. She did not ask
for nor seem to need mathematics counseling.
Robin. Robin's mathematics successes were much more distant in time, and she
obviously struggled in class in both lecture discussions and during problem-working
sessions. She was not initially recognizable as a Category I student. Her membership in
one of the groups traditionally at risk for Brookwood (older, female, nursing'" students)
and her classroom presence initially raised questions for both the instructor and me about
her prognosis in the class. I had more early information about Robin than the instructor
because of my pre-course surveys, but it was in such contrast to how she presented
herself in class that I questioned its reasonableness. She seemed to need mathematics
counseling so I was not surprised that she chose to participate — I thought she would need
considerable emotional and cognitive help. It was in the counseling setting that I
observed Robin's competence and confidence. I discovered then that she volunteered for
the study to help with my research, and not because she believed she needed help.
316
Robin revealed her positive mathematical self-esteem and history through her
metaphor for herself — Belle (from Walt Disney's animated movie Beauty and the Beast),
an intellectually curious and competent feminine woman. She explained this in terms of
her family's identifying her with a mathematical grandmother. Her success in school
mathematics was tempered by her parochial school teachers' censure when she knew an
answer but could not explain how she arrived at it. Robin seemed to be a global learner'^
with some auditory processing difficulties. She had not taken a mathematics course for 25
years.
My counseling support consisted of helping Robin become conscious of her
positive mathematics self-concept by interpreting her Belle metaphor, inviting her to tell
her story, and affirming her achievements and her current approach to mathematics
course material. I chose parallel conceptual-to-procedural link tutoring to help her feel
more grounded in her competent, conceptually oriented mathematics self Given how
long it was since she had taken a mathematics course, Robin expended a realistic 10
hours per week on homework. She struggled successftilly to compensate for her learning
style challenges and make the necessary conceptual-procedural connections. Although
the instructor's perception of her as a struggling nursing student never changed, Robin's
confidence improved, she mastered the material to her satisfaction, and she earned an A~.
Even without the mathematics counseling, it is probable that Robin would have done well
but her mathematics base for further mathematical study became much more secure
because of the affirmation of her good mathematical abilities by a mathematics expert.
317
Category II Students with Undermined Mathematics Selves: Adequately
Mathematically Prepared with Compromised Mathematics Self-Esteem
I found that more than one third of the students were adequately prepared to
succeed in PSYC/STAT 104 but because of the interactions of past experiences with
personal characteristics, they had developed relational patterns that could compromise
their mathematics success in this class. They had relatively sound mathematics selves that
had been undermined. Within this group there seemed to be two subtypes that I
characterized as: a) students with malleable mathematics relational patterns and b)
students with inflexible mathematics relational patterns (see Table 7.3). Students from
these subtypes seem to have reacted differently to similar assaults on their developing
mathematics selves.
Undermined Mathematics Self Students with Malleable Mathematics
Relationship Patterns
The students who fell into this group were Lee, Pierre, and Jamie. They saw
themselves as successful students in all but mathematics (and perhaps the sciences).
They either underestimated or were ambivalent about their mathematical ability because
of mixed mathematics success in the past. This caused moderate to severe affective
problems in the Ann's mathematics class, particularly anxiety (for the women) and an
expectation that they might do worse than they hoped. Generally, they had sound
algebraic and arithmetical conceptual understanding but each had important gaps. Their
beliefs about mathematics ranged from slightly more procedural than conceptual to
conceptual, and they responded with positive mastery orientation to the challenge of
developing a conceptual understanding once they believed they could. In other words.
318
each had an underlying sound-enough but undermined mathematics self from which he
felt separated.
These 3 students signed up for mathematics counseling with an initial motivation
of getting help to negotiate the course. Although it became clear that each had a good-
enough mathematical knowledge base"^ to succeed in this course (despite variable Exam
#1 results, see chapter 5, Table 5.1), each had secondary problems that could have
jeopardized this success. The women had developed anxiety problems expressed in their
Feelings survey responses that were confirmed by observation and in discussion; the man
had developed over-inclusive study practices that were counterproductive. Whatever the
complexity, these students were willing to change their course approach in order to
understand the concepts and achieve good grades.
Lee. From the beginning, Lee was the most mathematically insightful of the
participants. She was interested in how different elements of statistical analysis related to
each other (see chapter 5, discussion of Study Group 1, pp. 175-176). She was the second
most verbally responsive student in the class, with an average of 3.36 questions or
answers per lecture discussion. Most of her questions were about exam strategy and
concepts. Lee initially had the most conceptual beliefs in the class and was significantly
more mastery oriented than learned helpless (on the Beliefs survey), but all three of her
anxiety scores were high; her testing anxiety and number anxiety were each the second
highest in the class and her abstraction anxiety the third highest on the Feelings survey.
She signed up for mathematics counseling the day before the first exam because of
anxiety, but we could not meet until after the exam. She had blossomed in a mathematics
environment where she was required to think and explore deeply. She was driven to the
319
point of anxiety in classes where the conceptual connections to the procedures were not
explored and where she felt that only mathematics procedures were being taught.
However, she was convinced that, because the mathematics was not immediately clear to
her and she had to work hard to understand, she was not good at mathematics.
Lee found the PSYC/STAT class difficult because of a lack of in-class guidance
linking concepts to procedures. She did well with the instructor's problem- working
approach because it forced her to explore and master the procedures herself It seemed
that she did not feel secure in her relatively sound mathematics self because of variable
past success in her past and her self-comparisons with peers who "just got it" without
having to work hard at it as she did. Lee's strong performance on the Algebra Test and
Arithmetic for Statistics assessment helped allay her concerns somewhat (see Appendix
H, Table HI), but she performed poorly (< 50%) on the operation sense section of the
Arithmetic assessment. This significant gap seemed to affect her mathematics self and
probably contributed to her anxiety.
In our sessions I focused on affirming Lee's conceptual problem-solving
orientation and providing a secure base for her to explore the concepts and the
cormections that she did not experience in class. Lee relied on these sessions perhaps too
much. She reported at the end of the course that she did only about 20 minutes homework
a week. That was likely a factor in the high testing anxiety that increasing over the
course.
Another issue in Lee's anxiety may have been linked to the fact that she valued
the conceptual understanding of the mathematics but may have undervalued the
importance of thoroughly mastering the procedures. Her grades fluctuated, apparently
320
linked to whether she and I practiced the mathematical procedures or not, but she finished
the course with an A after taking the optional comprehensive final to replace a lower test
grade.
Mathematics counseling was beneficial for Lee. It provided a secure conceptual
base so she could repair her undermined attachments to mathematics and supported her in
making the conceptual links to the procedures. In mathematics counseling I should have
given more attention to providing bridges of understanding between her and her
instructor (given their different priorities). I did continue to affirm her sound ability,
learning motivation, and mastery orientation to achievement tasks, and Lee became more
mathematically self-reliant.
Pierre. Pierre had been in the U.S. for only two years and his English was
difficult to understand. He had earned a D in the calculus course he had just completed so
he signed up for individual counseling once a week but we did not meet until the end of
the fourth week of class because of miscommunication. He reported no difficulties with
mathematics in his early schooling. His anxiety scores were low and his Belief savvey
results indicated a mastery-oriented approach to mathematics learning although his
beliefs were somewhat more procedural than conceptual (2.5 on the I to 5 scale).
We first met after Exam #1 where Pierre earned only a 68%. He put this down to
having to take the exam early because of a prior obligation but his C^ on Exam #2
seemed to point to something more. Pierre was in the B"/B* range on the conceptual
multiple-choice and symbol section but in the D"/F^ range in the computational section. It
did not seem that he had any fundamental problem with his arithmetic or algebra,
although his operation sense (like Lee's), was inadequate (Appendix H, Table HI). He
321
seemed to have an over-inclusive approach to his learning. In his reported 17 hours per
week of homework he surveyed and studied the greatest amount of material possible
including extra material he asked Ann for and Pierre met with Ann and with me often.
Because Pierre gathered and worked on so much, he was not mastering the
mathematical computational material focused on in class, and he at times confused the
extra material for material he was meant to use. In addition he approached the
mathematical computation in a very procedural way, separate fi"om its conceptual base.
For the third exam I suggested that he focus on the course material. When he did not and
earned a D , I forceflilly confronted him before the fourth exam with the likelihood that if
he did not change his approach he would get another D. He seemed a little shocked by
my forthrightness but this time he listened. On Exam #4 Pierre earned a 91%, losing only
one point on the computational section! When he came to tell me, he was very pleased
and a little surprised at how much difference this strategy change had made.
Pierre's English language difficulties contributed the most telling perspective on
his performance in the classroom. It was clear that he had to use much of his energy to
comprehend the material and to understand the organizational decisions. He did not
collaborate with other students during problem-working sessions. Pierre did contribute a
little in class (an average of once per lecture discussion) but his English continued to be a
challenge for him and his peers. Although it dominated his class presence, it was not the
main issue in his struggle to get a good grade; rather that issue was whether he was
willing to give up his over-inclusive strategy to take a strategic approach.
With much improved grades on Exam #5 and the MINITAB projects and a
reasonable score on the optional comprehensive final to replace his lowest test grade.
322
Pierre went on to earn a B" in the course, much better than the D he was earning through
the third exam. He retook Calculus I in the spring of 2001 and with this new approach
earned a B^ to replace his original D.
Jamie. Since Jamie is a focal student (see chapter 6) I will review her profile only
briefly, chiefly to explain why I believe she falls in this category. As with Lee and Pierre,
once we had ruled out arithmetic and algebra knowledge base issues as a central concern
and began to reconnect Jamie to her secure mathematics base, counseling could focus on
her central affective issues, which in Jamie's case was her severe anxiety as revealed in
her Feelings survey, metaphor, and presence in class. Her shy personality had interacted
with classroom teachers and family theories, and caused her to question her ability in
mathematics Work on repairing damaged mathematics and mathematics teacher
attachments, replacing her negative internalized teacher presences with positive ones, and
supporting healthier interactions with the mathematics classroom personnel resulted in
significant reduction in her anxiety, an improved sense of her mathematics self, and a B"^
in the course. However, her slightly more procedural than conceptual beliefs did not
change and her performance orientation remained (see chapter 6 for a detailed account of
Jamie's course of counseling).
Students with Undermined Mathematics Self and Inflexible Mathematics
Relationship Patterns
Autumn and Mitch fell into this group. Like Lee, Jamie and Pierre, they had
sound-enough mathematics preparation and compromised self-esteem emanating from an
undermined mathematics self but unlike Lee, Jamie, and Pierre they did not seem willing
to change their counterproductive ways of protecting their undermined mathematics
selves. Their primary achievement motivation was for performance (certain grades)
323
rather than learning. They had achieved quite well in mathematics at times in the past but
had also gotten disappointing results. They saw themselves as capable procedural
mathematics students, but feared and resisted both problem-solving and the conceptual
demands that were made on them. They did not want to risk exploring conceptual links.
This approach resulted in a learned helpless orientation in conceptually demanding or
problem-solving situations. Their underlying understanding of the algebraic variable was
good-enough to support some conceptual exploration and their facility with arithmetical
processes was adequate, although there was some question in my mind about operation
sense. They tended to avoided open-ended questions (cf Autumn's efforts on Arithmetic
for Statistics assessment, archived). Both had an overall negative attitude to themselves
doing mathematics that could be classified as mild to moderate mathematics depression.
They maintained detached distance from mathematics teachers and peers.
These students seemed to have the most difficulty of all students in the class with
any change of approach in how a class was taught and managed; their strong conservative
impulse (cf Marris, 1974) led to strong resistance against change. It seemed that painflil
or disappointing experiences with mathematics in the past had led to their building
defensive barriers around their relatively sound but fearful mathematics selves to guard
against scrutiny or further assault. They seemed inflexible and unwilling to give up their
defensiveness in order to risk growth in understanding and achievement.
Autumn. Autumn said she signed up for mathematics counseling to help me with
my research. Although she reported disappointment with herself for not pursuing and
succeeding in the algebra through calculus sequence, she was confident of success in
PSYC/STAT 104 and did not want to explore conceptual connections or try to develop
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her admittedly poor problem-solving abilities to become more mastery-oriented. She
wanted a good grade rather than a conceptual understanding of the material and she
maintained a performance motivation to learning statistics (see Table H3). She was a
voluntary loner in class and maintained a detached distance from both the instructor and
me.
From her middle and high school mathematics history it became clear that
Autumn's performance motivation had prompted her to take an easier class in order to
earn an A. Her detached independence prevented her from getting the help she needed
when she did try a harder class, particularly her advanced Algebra II class where she had
a poor background because of the easier Algebra I class she had taken to get her A. Her
low grade in advanced Algebra II had in turn contributed to her disappointment with
herself, her compromised mathematics self-esteem, and mild to moderate mathematics
depression that was evidenced in her Metaphor and responses on the JMK Affect Scales.
Autumn's depression was not allayed by her consistently high grades in the course.
If Autumn had participated in counseling designed to help her understand these
connections and also supported her in exploring conceptual links and problem- solving,
the current course experience might have developed her self-reliance and sense of
mathematics self and perhaps even broken up her mathematics negativity. As it was, in
counseling Autumn was willing to report her mathematics history, discuss her survey
responses, and take the Algebra Test (a sound level 4) and Arithmetic for Statistics
assessment (see Appendix H, Table HI), but she resisted doing exam analysis or
exploring statistical procedures and concepts. Over the course, she remained relatively
inflexible. Her procedural beliefs and learned helpless orientation changed little and her
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abstraction anxiety score increased from 2.9 to 3.3 on the 1 to 5 scale although she
reported that her confidence in her mathematics ability had improved (see Appendix H,
Table H3).
Mitch. Mitch signed up for mathematics counseling because he needed to erase an
F from his GPA. That goal was admittedly limited but his self-reported rigidity and
resistance to change jeopardized his achieving even such a limited goal. He did not want
to explore his affective problems with mathematics although he alluded to them. If he had
been willing to explore his metaphor of Inspector Javert"' as mathematics relentlessly
chasing him through the years, he might have felt less beleaguered. Since he was not
willing, what we did in the mathematics counseling was to work on the statistical
problems at hand as I affirmed Mitch's sound mathematics self (e.g., his level 4
understanding of the algebraic variable on the Algebra Test) and tried to help him
reconnect with it. I helped him notice that not changing his approach from his failed
attempt at the course was negatively impacting his attempt to do better this time. Through
the third exam he used a formula sheet of the type his former teacher had allowed despite
my pointing out this instructor's more generous criteria that allowed the inclusion of
more information. His extreme negativity on the JMK Mathematics Affect Scales at the
first session did abate somewhat but only two responses were on the positive end of the
scale by his last session (6: mathematics achievement, and 7: making mathematical
decisions). He made good-enough adjustments, earned a B to replace the F, and he is
finally safe from Inspector Javert's pursuit; he never has to take another mathematics
course at least as an undergraduate.
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Category III Students with Underdeveloped Mathematics Selves: Mathematically
Underprepared with Low Mathematics Self-Esteem
The same number of students in the class had underdeveloped mathematics selves
as had undermined mathematics selves. Those with underdeveloped selves fell into
similar sub-types as those with undermined mathematics selves, that is, malleable and
flexible/disorganized.
Underdeveloped Mathematics Self Students with Malleable
Relationship Patterns
Karen, Mulder, Brad, and possibly Floyd were students in this study with
underdeveloped mathematics selves who evidenced malleable mathematics relationship
patterns. They had a history of struggling and/or not trying, poor mathematics
achievement, and little (if any) feeling that they had ever understood. Like the adequately
prepared students, they experienced relatively more success in other subjects. They were
interested in understanding mathematics but felt capable of learning it only procedurally,
if at all. They were more learning- than performance-motivated and were open to
developing conceptual understanding once they believed they could, but all (particularly
the men) seemed to fear risking the effort to understand, in case they found that they were
incapable.
Karen, Mulder, Brad and Floyd each had mathematics knowledge gaps evidenced
in a low understanding of the algebraic variable and possibly also in arithmetical number
and operation sense deficits. In their attempts to deal with the discomfort engendered by
being in a setting where they felt lost and incompetent, these students had developed
compensatory procedures and approaches that included avoidance, busy work,
memorization techniques, under or overconfidence, external blame, and hostility. Each
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had an underdeveloped and shaky mathematics self that produced negativity and empty
depression on the one hand, or unrealistic bravado and resistance on the other.
These underprepared, malleable students differed from the inflexible, adequately
prepared students because they retained their learning motivation and an openness to
learning conceptually despite their having experienced mathematical neglect."" In
contrast, the inflexible adequately prepared students who also presented with
mathematics depression, had experienced some mathematical success and had developed
a good-enough knowledge base. Nevertheless they exhibited independent detachment and
personal rigidity, performance motivation and resistance to problem-solving and
conceptual learning.
Karen. Since Karen is a focal student I briefly review her profile in terms of her
mathematics preparedness and self characteristics. Karen's negativity about herself, the
class, and mathematics, along with her hostile detachment relational pattern with teachers
and peers and her knowledge base gaps were evident early. A picture of her moderate
empty mathematics depression emerged as relating to an underdeveloped mathematics
self (see chapter 2, Self Psychology). She was learning- rather than performance-
motivated but took a procedural approach to mathematics because she did not believe she
co?//£/ understand conceptually (although she wanted to). At the start of the course Karen
consciously attributed bad outcomes to external sources, and in the counseling setting I
had to overcome my countertransference reaction of feeling hopeless and depressed on
her behalf I challenged her external control beliefs with evidence, provided mirroring of
her tentative self, helped her see the instructor as on her side rather than against her, and
offered enough structured guidance and course management that she was willing to
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consider changing her external control to internal and to take responsibility for what she
realistically could do. Her mathematics depression abated somewhat, she became more
self-reliant, her grades improved, and her beliefs about mathematics became significantly
more conceptual. Her overall confidence in her own ability to understand conceptually
grew only slightly, however. Since the course was taught with manageable limits that she
could handle with strategy and effort, what we did was good-enough (see chapter 6 for a
detailed account of Karen's course of counseling).
Mulder. Because Mulder is a focal student I highlight his characteristics briefly.
Mulder had not really experienced success in mathematics, at least in high school. He
"knew," based on his theory of family genes that he could succeed but he had not really
tried. When he did try after Exam #1 in the class, he found that he could handle the
mathematics computations but he struggled with the conceptual multiple-choice
questions. Rather than mirroring his emerging prowess and supporting its application to
the difficult multiple-choice, I was somewhat dismissive of that success. I pushed him on
the multiple-choice and he resisted.
It was not until we tried a counseling intervention suggested to me at my
supervision session and I withdrew my counterproductive countertransference stance that
he was able to overcome and succeed (see chapter 6 for a detailed account of Mulder's
course of counseling).
Brad. Although Brad's bravado was more extreme and more unrealistic than
Mulder's, it seemed to have stemmed from a similar source — his underdeveloped
mathematics self It brought forth a similar but more extreme countertransference
reaction in me. I bristled at his we're-the-adults-here way of relating to Ann and me in
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class. Unlike Mulder, Brad had tried PSYC/STAT 104 once and failed it, a fact that he
seemed to almost inadvertently let slip in study group. He had, in contrast, written on his
survey that he expected an A in this class and had earned a B in his last mathematics
class. Algebra. He wrote on his metaphor survey that "anyone can do well" if he allows
enough time and energy, yet he seemed ambivalent about doing that himself He was
surrounded by women at work, and had a woman as his superior. His motivation for
doing this class was to get a degree that would allow him to change to a more male-
favored position. His conflict seemed to be around a fearful sense of not being capable of
doing the mathematics, combined with a desperate need to be able to do it. He was taking
a risk enrolling again, and my scolding and pushing him rather than supporting him in
this effort was not helpfiil to him. Unlike Mulder he did not stand up to me but oscillated
between avoidance and non-strategic effort in a way that did not achieve any more than
marginal results.
Floyd. The data I gathered on Floyd (from class surveys, the Statistics Reasoning
Assessment and his Exam #1) revealed a similar bravado and resistance to getting the
help he needed that Brad and to some extent Mulder exhibited. Like the other men in this
group (it seemed) his grade hopes and expectations (both As) were unrealistically high,
especially in light of his 42% failing grade on Exam #1 (see Appendix H, Table HI). He
exuded confidence in class and declined the offer of mathematics counseling. However,
like the other malleable underprepared students, Floyd's achievement motivation was
more learning- than performance-oriented and his sound understandings on the Statistics
Reasoning Assessment were the fourth highest in the class (10 of the 20) (see Appendix
H, Table H2).
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Analysis of Floyd's Exam #1 efforts"" revealed what appeared to be minimal if
any prior study or practice, a somewhat surprising ignorance of basic statistical concepts
such as median and mode (the only student in the class to show such ignorance), and
probably a poorly constructed formula sheet. He did not make errors that indicated
arithmetical gaps or misconceptions but there was too little data to assess that accurately
or to assess his understanding of the algebraic variable. He overcame his resistance to
getting help too late. He asked me for an appointment (just before Exam #2) but he did
not come and then stopped attending the class.
Underdeveloped Mathematics Self Students with Inflexible or Disorganized
Mathematics Relational Patterns
Kelly. Kelly had a history of poor mathematics achievement. She had deficits in
number sense, operation sense, and understanding the algebraic variable. She had
performance motivation and procedural beliefs, high levels of anxiety on all scales, and a
learned helpless orientation to mathematics learning. Kelly had belief and anxiety scores
similar to Karen's (a malleable student with an underdeveloped mathematics self) except
that Karen was significantly more learning-motivated (3.5 compared with Kelly's 2.5). In
addition, Kelly's externalized surprising-to-her "sudden storm" metaphor for
mathematics, her and her mother's blaming her mathematics difficulties on something
she felt was out of her control (a learning disability), and her relational pattern of
dependence on both the instructor and me filled out a picture of her periodic sense of
mathematical self disintegration. I allowed myself to be drawn into this vortex and was
not able to help Kelly avoid another failing experience. My suggestions for counseling
that may help such a student avert failure are discussed below.
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Summary of Mathematics Self Categories
From this analysis of characteristics, behaviors, and responses to counseling the
three student categories emerged according to how students' mathematics selves had
developed and what that implied about their present approaches to mathematics learning.
These categories are summarized in Figure 7.1.
Categories of Students According to Mathematics Self Development
Category I Students with Sound Mathematics Selves: Mathematically Well-
Prepared with Sound Mathematics Self-Esteem. Defined by soundness of arithmetical
and algebraic knowledge base and absence of any experience of assault or questioning of
mathematics ability or achievement, resuhing in a sound current mathematics self
Type A: Sound Mathematics Self and Productive Relationship Patterns:
e.g., Catherine and Robin
Category II Students with Undermined Mathematics Selves: Mathematically
Adequately Prepared but with Compromised Self-Esteem. Defined by sound-enough
arithmetical and algebraic knowledge base and a variable experience of achievement with
or without outside assauh on student's mathematics self concept, resulting in a relatively
sound but undermined and vulnerable mathematics self
Type A: Undermined Mathematics Self and Malleable Relating Patterns:
e.g., Jamie, Lee, and Pierre
Type B: Undermined Mathematics Self and Inflexible Relating Patterns:
e.g.. Autumn and Mitch
Types A and B are differentiated by their affect and relational patterns developed around
vulnerable and ambivalent mathematics selves.
Category III Students with Underdeveloped Mathematics Selves: Mathematically
Underprepared with Low Self-Esteem. Defined by serious algebraic and/or arithmetic
deficits or underdevelopment and a history of poor achievement resulting in an
underdeveloped mathematics self
Type A : Underdeveloped Mathematics Self and Malleable Relating
Patterns: e.g., Karen, Mulder, Brad, and possibly Floyd
Type B: Underdeveloped Mathematics Self and Inflexible or Unstable
Relating Patterns: e.g., Kelly
Figure 7.1. Mathematics self development categories of PSYC/STAT 104 participants.
After I analyzed the student participants fi"om PSYC/STAT 104 into these
categories, I was able to sort the counseling approaches I found to be appropriate and
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helpful against these categories. I found that approaches indicated were closely related to
the categories and also to students' affected dimension of mathematics relationality.
Integrating Relational Mathematics Counseling
with Mathematics Tutoring: An Analysis
Dimensions of participants' mathematics relationality were interdependent, but
some students had more pronounced difficulties in one dimension than the others. The
categories of participants' mathematics ftinctioning identified in this chapter (see Figure
7. 1) seemed to be related to the problematic dimension (particularly the self dimension)
and the depth and type of the relational difficulty.
As I have demonstrated, mathematics knowledge base deficits interacted
predictably with students' mathematics self development and it was these two factors and
their interaction that pinpoint a student's profile type. Past negative teacher-student
experiences formed internalized presences that interfered in the present and had caused
damage to mathematics selves. These negative teacher-student experiences had also
caused damage to mathematics and mathematics teacher attachments (cf Jamie).
Difficulties with establishing or maintaining secure attachments to mathematics
and/or mathematics teachers also strongly affected their present relational patterns and
mathematics functioning (cf Jamie and Karen). Mathematical and counseling
instruments and techniques for diagnosing and treating difficulties in one dimension at
times resulted in improvements in another; in other cases they proved inappropriate and
even counterproductive in dealing with another dimension.
In the following three sections I present my analysis of mathematics relational
counseling for each of the three relational dimensions that Mitchell (1998) identified,
which form the basis for my approach. In this analysis I show how a student's category of
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mathematics self identified above affected the counseling needed and its possible
outcomes.
Mathematics Counseling and the First Dimension: Self
Some students with pronounced defects in their mathematics selves presented as
either unrealistically negative (underconfident) or positive (overconfident; in either case,
unrealistic) about their mathematics self (Category III, Type A students, cf Karen,
Mulder). Others clung to the counselor with little or no sense of having an independent
mathematical existence (the Category III, Type B student, Kelly). Students whose
mathematics self was relatively sound but had been undermined so that they were no
longer confident in it were likely to present with inappropriately severe anxiety (Category
n. Type A women, cf Jamie, Lee) or with a rigid resistance to change or risk (Category
n, Type B students, cf Autumn, Mitch) depending on their attachment patterns. I was
able to use the following means to explore disordered self relational patterns:
1 . Investigation of the mathematics knowledge base: using diagnostic assessments of
arithmetic and algebra, class exams, and/or learning modality and style checklists,
and
2. Investigation of self relational patterns by:
(a) Investigation of pronounced ongoing negativity/depression on the JMK
Mathematics Affect Scales, Learned Helpless-Mastery Oriented Scale,
generalized negativity, and underestimate of the mathematics self
(underconfident). Clues lie m Metaphor, in self-statements in counseling, and
in my countertransference feeling of depression or despair for the student's
prognosis (cf Karen);
334
(b) Investigation of pronounced discrepancies between a student's elevated
perception of his mathematics self when compared with a realistic assessment
of mathematics self (overconfident). Clues lie in Metaphor, in self statements
in counseling, and in my countertransference of first believing and then
wanting to dispute inflated and unrealistic self assessment and to deflate it (cf
Brad, Mulder);
(c) Investigation of anxieties that seemed disproportionate with measured
levels of mathematics competency on the algebra and arithmetic assessments
and/or exams. Clues lay in Feelings survey scores [and possibly Metaphor]
relative to sound mathematics diagnostic scores [and possibly class exam
scores] (cf Jamie);
(d) Investigation of inappropriate dependence on counselor combined with
lack of focus, willingness, or belief in ability to engage cognitively in the
mathematics. Clues lie in Metaphor, in self statements in counseling, and in
my countertransference feeling of being sucked into a bottomless pit (cf
Kelly);
(e) Investigation of a marked discrepancy between personas in different
settings, for example, in class compared with the counseling setting (cf
Robin).
Students with moderately to severely underdeveloped mathematics selves were
underprepared mathematically and their self-esteem was consequently low. They had
inadequate scores on the algebra diagnostic (and some also on the arithmetic diagnostic)
and low scores on the first exam in the course. Where I treated these problems
335
effectively, I mirrored sound mathematics thinking and course strategy practices so
students' sense of their own competence would become both realistic and hopeful. I
provided myself as a good-enough mathematics parent image for students to idealize and
model themselves on but I subsequently provided manageable finstrations and
disappointments so they could withdraw dependence and grow into their own
competence. I found this easier to accomplish with Karen, who presented with symptoms
described in 2. (a) above.
I did not deal as well with the students described in 2. (b) Brad, or 2. (d) Kelly
because I did not discern soon enough that their root problem also lay with their
underdeveloped and vulnerable mathematics selves. With Mulder and Brad, for example,
instead of mirroring areas of real competence I tended to act out my countertransference
reaction to deflate their overly positive opinions of themselves. My inappropriate
approach tended to increase their overt grandiosity and their resistance to or avoidance of
the task but with Mulder, supervision advice and my becoming aware in time of the part I
played in his resistance, a positive outcome was achieved. With Kelly, I was drawn into
her vortex and tried frantically to give her all she thought she needed instead of mirroring
her evidenced competencies and providing bounds she could not establish for herself
Category II students who had suffered some short-term and/or long-term blows to
their mathematics selves did have adequate underlying mathematics selves (and
knowledge bases) but they had been undermined. They needed not so much to develop
their mathematics selves through mirroring and permission to idealize, but needed rather
the offer of a secure base and help with repairing damaged attachments (i.e., techniques
of the interpersonal attachment dimension, see below).
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Mathematics Counseling and the Second Dimension: Internalized Object
(Internalized Presences from the Past).
All students' behaviors and expectations are influenced by their prior mathematics
learning experiences. But students suffering from the undue negative influence of their
internalized mathematical presences (internalized objects) typically behaved towards the
instructor or peers or tutor in ways that were incongruent with present realities. In this
study Jamie was most affected this way, but it does not appear that a problem in this
dimension is restricted to or indicative of a particular student category. I found that
certain learning style differences and learning modality preferences could be confounded
with a problem with this dimension, however, so assessment needs to be careful. The
most important diagnostic data came from:
1. Investigation of the mathematics knowledge base: using diagnostic assessments of
arithmetic and algebra, class exams, and/or learning modality and style checklists,
and
2 . Investigation of internalized mathematics relational patterns by :
Investigation of obser\'ed behaviors in class or study group or counseling that
seem incongruent with the way class members related to each other, the
teacher or the counselor/tutor. Clues lie in Metaphor and History Profile; in
my sense of student's transference that was very different from reality [e.g.,
dangerous to Jamie]; in my countertransference feelings that I should act
differently from what I believed would be appropriate [e.g., stay away and not
ask questions so as not to cause damage].
I found that discussion of metaphor and mathematics history quickly uncovered
bad internalized teacher presences that interacted with personality and caused present
337
mathematics teachers to seem dangerous, for example. Counseling involved support in
close analysis of the instructor and the mathematics counselor to see if they could
displace the bad object (presence), and devising relational assignments (cf Jamie's
assignment to ask Ann a question and make an appointment with me) based on a new
more realistic evaluation. If a student's life were constrained by extremely bad
internalized mathematics presences, however, such a straightforward process would
likely not be possible. In that case, the mathematics counselor should not proceed except
as a team member with a mental health counselor.
Mathematics Counseling and the Third Dimension:
Interpersonal A ttachments
Students in the study who had developed insecure attachment patterns with
teachers presented as avoidant, overly dependent, ambivalent, or fearfiil of the teacher or
counselor. Certain personality styles seemed to be conflated with this dimension,
however, so diagnosis has to be carefiil. Some students suffered from an insecure
attachment to the mathematics, they presented most often as procedural in their
mathematics (cf Autumn), with associated uncertainty about their ability to do
mathematics, and separation anxiety in exams. The most important diagnostic data for
insecure attachment to teacher or mathematics came from the following:
1 . Investigation of the mathematics knowledge base using diagnostic assessments of
arithmetic and algebra, class exams, and
2 . Investigation of interpersonal mathematics relational patterns by :
(a) Investigation of observed m>oidant or dependent attachment behaviors in
class or study group or counseling. Clues lie in Metaphor and History Profile;
in my sense that a student's transference was keeping me at a distance
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personally (cf. Karen), or that she was excessively needy for my presence; or
that she was ambivalent; in my countertransference that I should try to gain
her approval or that I wanted to escape, or that I was confused and continually
moving between the two reactions;
(b) Investigation of a student's apparently mrwarranted insecurity in her
ability to do the mathematics at hand. Clues lie in responses to the Beliefs
Survey and Procedural/Conceptual and Learned Helplessness sub-scales;
history of intermittent success and relative failure in mathematics; more
anxiety on tests than seems appropriate given preparation and mastery of the
material. This lack of a secure base in the mathematics seemed to be the result
of a history of procedural transmission teaching and never having truly
understood the mathematics or a history of having been suddenly separated
from a secure mathematics base.
When students evidenced an insecure teacher attachment pattern, my counseling
role was to provide a secure teacher base where they felt mathematically accepted and
safe so that they could begin to explore on their own and risk taking paths that might be
wrong, so as to eventually become self reliant. I also needed to help the participant
reevaluate the present instructor and her approaches and begin to receive rather than
reject her good offerings (cf Karen""). With students showing an insecure mathematics
attachment pattern, my counseling role was to help them rediscover the existing sound
basis in mathematics from which they had been separated (cf Category II students,
Jamie, Mitch). These students' mathematics separation anxiety dissipated as their security
in the mathematics grew. When I helped insecure procedural learners link their
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procedures with the underlying concepts they began to estabUsh their own secure
mathematics base.
Summary of Brief Relational Counseling Analysis
I have analyzed here the application of brief relational mathematics counseling
according to the dimension of student relationality: self internalized presences, and
interpersonal attachments. I have also shown the interactions among the categories of
students according to mathematics self development, which I identified earlier in the
chapter, and the counseling approaches that are applicable. It is not possible to establish
a causal link between student outcomes and the counseling because of the many variables
at play. However, in the process of the counseling, the participants and I did identify
relational conflicts by attending to patterns of relational episodes and we attempted to
resolve them. As could be expected counseling Category II students required less
emphasis on the mathematics itself than for Category III students because their greater
level of mathematics preparation enabled them to proceed without as much mathematical
support once they were reassured of their competence. Category III students needed more
mathematical support throughout and each category of participants benefited from
relational counseling to help them resolve the relational conflicts that had arisen over
their learning histories. In most cases, counseling worked well enough that focal
participants and others felt their originally questionable course prognosis changed and
they succeeded.
In chapter 8 I will reflect on my relational approach and its components and
suggest directions for further research.
340
' Because this is an odd numbered chapter, I use "he," "him," and "liis" for the generic third person.
" Catherine reported tliat she Hked matliematics and was confident in her ability to do well. She was more
conceptually than procedurally oriented towards mathematics learning (3.5 on a scale of I procedural
through 5 conceptual, on tlie pet-Mathematics Beliefs Survey and 3.8 the highest in tlie class on the post-
survey). Her presence in the class did not cliange over the course. She seemed comfortable; she was quietly
(contributing only an average of .27 responses or questions per session) confident during the lectines,
appearing to be processing and understanding tlie material; she worked on her own during problem-
working sessions but was willing to interact witli a neighbor if Uic neiglibor initiated it (e.g., Mulder, Class
7). Since Catlierine, unlike Lee, did not seek conceptual links during lecture discussions or problem
working sessions nor participate in study groups or individual counseling, she may have been using some
of her 5 hours per week of homework time doing that. Her mathematics testing anxiety was initially the
lowest in the class (1.5 on a scale of 1 [none] through 5 [extreme]) but uicreased considerably to 2.1,
nevertlieless remaining relatively low.
■^ RN to BSN RN = Registered Nurse; BSN= Bachelor of Science in Nursmg.
" She had the characteristics of Davidson's (1983) Mathematics Learning Style II learner or Krutetskii's
visual-pictorial (see chapter 2).
*" Evidenced in their Level 4 understanding of the algebraic variable on \hc Algebra Test hxA somewhat
variable Arithmetic for Statistics responses.
" From Alain Boublil and Claude-Michel Schdnberg's musical adaptation of Victor Hugo's novel Les
Miserables.
™ Karen did have a detached pattern of relating but it was defensive rather tlian independent and her
mathematics depression was more prominent. All Category II, Type A students had issues of control but for
Category II, Type B students it was much more prominent.
™ For only four of fourteen questions did Floyd apjjeared to understand what the question was asking for,
know the formula, use the formula correctly and achieve a realistic answer; for two questions he used
appropriate formulae but did not seem to understand the meaning of the question so substituted incorrect
(though not unrelated) elements and achieved unreahstic answers; on six questions he did not seem to
understand the meaning of the question; and on the other two he showed only partial understanding of the
question and of the procedure required achieving a somewhat realistic answer in one of them.
" I accepted her anger and her depression and offered mathematics counseling as a secure base from which
she could do the course. Towards the end of the course she had developed more self-reliance and chose
when she did and didn't need to come for help. She also clianged in her stance towards the instructor
learning to appreciate and rely on her. See chapter 6 for fiirther discussion.
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CHAPTER VIII
EVALUATION OF THE PILOT STUDY AND RECOMMENDATIONS
FOR FURTHER RESEARCH
When Jamie and I first met, mathematics was a storm that she was afraid would
come back so she stayed inside. By the end of the counseling, it had changed to a partly
sunny day and she could go out with her umbrella. Jamie's relationship with mathematics
changed as she went to class, met with me in counseling, and struggled to understand and
resolve the central conflict that had been sabotaging her conscious desire and her ability
to do well in mathematics. Through this study I have developed a new way of providing
mathematics support over a college semester, one that incorporates relational and
cognitive counseling approaches. Here I turn from looking at the particulars of each
participant's experience of counseling to looking at what those particulars might tell
about the counseling approach itself as it emerged in this pilot application.
People can change. I found that out in this study. By crossing the lines drawn in
traditional mathematics support in order to incorporate a relational counseling approach, I
first changed how I looked at students and at myself, and they in turn changed how they
looked at themselves, at the instructor and me, and at mathematics. We found that we
could disembed ourselves from our entrenched theories about ourselves and each other
and change our counterproductive patterns of relationship in mathematics learning when
we recognized, explored, and challenged those patterns. To do so we each had to cross
traditional lines to widen appropriate and useful objects of attention in academic support
settings. We had to consciously attend to our relationships.
To explain fiorther, I first evaluate this process of counseling and its elements, and
evaluate the student categorizmg system that emerged. Next I assess the limits of the
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approach as it developed. Finally I discuss the limitations of the study and suggest future
directions for research and development of this approach.
Understanding the Student: Who is She' and How Do I Know?
How the Counselor Role Changed What I Knew
As an educator I used tests and surveys to try to classify each student's
mathematics cognition and affect; I took what she told me about herself at surface value.
As a counselor I gradually learned to listen not just to her "I" statements but to her
behaviors, her metaphors, her "she should" and "they are" and "it is" statements, her
transference (the role participants seemed to be casting me into), and to my
countertransference (my feeling constrained to act out or react against the role imposed
on me by the participant's transference). I looked for links between her history and her
current mathematics performance. Jointly we looked at her exams and reviewed her
grade, her thinking, her feelings and beliefs, her effort, and her contradictions. The chief
difficulties 1 faced in understanding the student were (a) the now-dynamic nature of our
relationship (The student changed, I changed, and our interaction changed.) and (b) the
reality and power of the unconscious: The student spoke honestly about her realities, yet
there was often good evidence that seemingly contradicted what she said; sometimes she
reacted in surprising ways that seemed incongruent with the present reality. These
difficulties provided the richest sources for understanding her (and me) as we worked
together.
When I heard or found or sensed contradictions, at first I was angry and mentally
accused the student of falsehood or cowardice (i.e., Brad's assertion that he earned a B in
his last mathematics class when he had actually failed his last class). At times I felt hurt
(i.e., Jamie only hinting and only at the end of the class that she was repeating
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PSYC/STAT 104). I chose sides and went with one assertion, dismissing or refuting the
other side." But when I drew on relational counseling insights through supervision and
further readings, I recognized these contradictions and the conflict that they created to be
equally genuine realities for the student. It was that very conflict that needed to be
brought to consciousness and resolved. Formerly, in my educator-only role, I was not
aware of wrestling with such nagging contradictions so I had not brought them to the
student's attention. They remained as the "elephant in the living room" — ^known about by
both of us at some level but unacknowledged. This lack of awareness or resolution of a
student's conflicts may have precluded the possibility of unproved mathematics mental
health and success in mathematics courses.
When I encountered what seemed to be willful refusal to allow me to see areas of
vulnerability or to help change behaviors or approaches (particularly with Autumn and
Mitch), I felt frustrated. / could see what their problem was — why would they not discuss
it with me, explore it, and resolve it? Through supervision some of my blind spots were
identified. I had difficulty allowing a student to choose her own path, especially when
that path seemed counterproductive to me. Instead I tried to get students to see their
difficulties as I saw them and to change. In the cases of Autumn and Mitch, my behavior
probably contributed to their becoming more entrenched in what I saw as their
counterproductive approaches.
Working through a relational conflict perspective allowed me to understand that
students had developed their current patterns of relationships in their attempts to protect
and defend their vuhierable mathematics selves. When I brought together my educator
and counselor roles I began to understand how a student's mathematics history might
have influenced her current ways of functioning within her overall relationality. This
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integration yielded the three broad categories of mathematics student (each with at least
one subcategory) that I described in chapter 7 (see Figure 7. 1 ).
Developing Categories to Understand Students.
Why Categorize Students?
In my endeavor to effectively support the whole person doing mathematics, I
needed to understand the range of variations in students' responses to their mathematics
learning histories. The significance of these variations and similarities helped me notice
details as part of a whole rather than being distracted by them. These variations also
helped me understand that quite different- looking symptoms could stem from similar
sources and might call for a similar counseling approaches (i.e., Karen's under-
confidence and empty depression and Mulder and Brad's over-confidence and
grandiosity both were expressions of underdeveloped mathematics selves that stemmed
from mathematics underpreparation and low self-esteem).
Emergent Categories
I developed categorical descriptions that were determined by interactions between
a student's history and adequacy of mathematics preparation, and her mathematics self-
esteem. My analysis of the case data gathered in this study led me to suggest that the
condition of a student's underlying mathematics self may be classified into one of three
categories: (a) Category I: A sound flinctioning mathematics self; (b) Category II: A
relatively sound but undermined and vulnerable mathematics self; and (c) Category III:
An underdeveloped mathematics self. Mathematics preparation and related self-esteem
were the principal discriminator of these three categories (see chapter 7, Tables 7.1 and
7.2).
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With my course participants, categories were further refined according to how a
student had handled her compromised self-esteem, that is, by developing either malleable
or inflexible (or unstable) mathematics relational patterns (see chapter 7, Table 7.3).
These categories are not exhaustive, however. Conceivably if these distinctions are
applied to other groups of mathematics students, other subcategories could be identified
within one or all of the three broad categories. I suspect that these three broad categories
are sufficiently explanatory to encompass all students. It is possible however, that
Category III might be helpfully divided into two categories according to whether the
student had a low level of the algebraic variable with adequate arithmetic skills or
inadequate levels of both. Further support for my categorization scheme is shown in the
fact that students from my preliminary research on practice are relatively easily
categorized with this scheme (e.g., Mary as Category II, Type A and Jane and Cara in
Category III, Type A, see Knowles, 1998, 2001)
Comparison with Other Schemes
The only similar attempt to classify mathematics students is the tier sort Sheila
Tobias proposes (personal communication, March 16, 2001; see chapter 4, pp. 131-134).
Tobias' &st, second, and third ("utilitarian") tiers more or less correspond to my
Category I, Category II: Type A, and Category II: Type B, respectively. Her
"underprepared" fourth tier and "unlikelies" fifth tier do not comfortably parallel any
categories I found but my Category III: Types A and B have "underprepared" and
"unlikely" characteristics. Tobias has researched her second tier in science classes
(personal communication, April 5, 2003; Tobias, 1990 and her other tier categories come
from experience and observation, again primarily based on science students although she
applies them to mathematics students. The major contrast in our schemes is that mine is
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based on underlying cognitive and relational differences in mathematics self development
and learning history, while Tobias' focuses more on current cognitive preferences and
behaviors. I find that mine provides more direction for counseling support and
intervention.
Evaluating Cognitive Categorizing and Counseling Instruments
The principal means I used in this study to gauge students' mathematics cognitive
functioning levels were: course assessments, the Algebra Test, the Arithmetic for
Statistics assessment, and the Statistics Reasoning Assessment (SRA) (see Appendix C).
All but the last provided valuable data both for categorizing students in order to develop a
mathematics tutoring focus as well as for relational mathematics counseling. Taken
together the first three data sources helped me sort students into the three categories of
mathematics preparation: well prepared, adequately prepared, and underprepared (see
also chapter 7, Table 7.2). A most important finding of this analysis for this group of
students is that course the first exam grades did not provide in themselves an accurate
indication of membership in a category or subtype of a category except perhaps for the
students in Category I and to some extent Category III. The Algebra Test was a better
indicator for the students who took it. It distmguished between students most
appropriately described by Categories II and III. Students' arithmetic levels (as gauged
by the Arithmetic for Statistics assessment and/or arithmetic samples gathered from
exams, counseling and in class) discriminated well among all three categories.
The Algebra Test (Brown, Hart, & Kuchemann, 1985; Sokolowski, 1997)
I found the Algebra Test (see Appendix C) useful in mathematics counseling with
students who scored at high concept levels of the variable but who had also developed
negative or ambivalent beliefs about their own mathematical ability — Category II
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students.'" A high level provided some proof that their negative views were not accurate,
and this was more objective than my reassurances or even the evidence of their
coursework. It was worthwhile to use some mathematics counseling time to take this test
because their difficulties did not seem to be fundamentally mathematical.
Once I established, for instance, that Jamie's level of the algebraic variable was
high and not an issue for counseling, I determined to use this good result to refute her
negative beliefs about her mathematics ability. The other three participants to whom I
administered the Algebra Test during the course were all at Level 4 (see Appendix H,
Table HI) and in each case this good result was used in counseling to allay concerns
about each one's mathematical ability. Because Category II and III students' first exam
results discriminated their category relatively poorly, the Algebra Test seemed to provide
a more accurate way to clarify early her level of cognitive preparation especially when
taken with Exam #1 grades and arithmetic preparation (see chapter 7, Table 7.2 and
discussion).
Arithmetic for Statistics (AFS) Assessment
Each of the five students at level 4 or above on the Algebra Test took the
Arithmetic for Statistics (AFS) assessment (see Appendix C) and performed adequately
on it on at least seven of the eight categories tested.'^ The two students who were at level
2 on the Algebra Test (see chapter 6, Table 6.1). Karen and Mulder performed quite
differently on the Arithmetic for Statistics assessment. Mulder performed adequately on
all categories except large integer number sense and statistical sense. Karen however did
not perform adequately on any category and her performance on operation sense, place
value/decimal sense for numbers of magnitude greater than 5, and open ended
arithmetical thinking/problem-solving was inadequate (see chapter 6, Table 6.2).
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I developed the AFS assessment as the course was proceeding in order to more
clearly isolate participants' arithmetical conceptual and procedural difficulties related to
the mathematical requirements of the course (see Appendix C). Despite its limitations^ it
revealed more precise data about participants' arithmetical issues than I could observe
anecdotally in class or counseling. With modification, I believe it should be administered
early in the counseling process, so that arithmetical issues may be addressed more
systematically with the arithmetically weaker students. Adjusted to satisfy issues raised in
endnote v, it should be a useful tool to be added to the Algebra Test and used at the
beginning of the counseling process. This would help students with specific weaknesses
or problem areas that impact their confidence and progress in mathematics.
Statistics Reasoning Assessment (Garfield, 1998)
The SRA was not useful in category placement or diagnosis for the strategic
mathematics counseling of students taking PSYC/STAT 104 (see Appendix H, Table
H2). Changes in scores fi-om pretest to posttest did not parallel other changes students
made over the course. This was not surprising because the course's design, direction, and
implementation were not focused on confronting and changing individual students'
misconceptions about statistics or probability, which the SRA was designed to measure.
The primary focus of my counseling was to support students in their coursework so much
of what is assessed by SRA did not match.
Evaluating Affective and Relational Categorizing and
Counseling Instruments
College Learning Metaphor Survey
Metaphor writing and analysis quickly provided rich, deep material that was
directly relevant for both the participant and for me; it was key in establishing the central
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conflict and relational focus (and in some cases the mathematics focus) for a participant's
counseling. Given the brevity of available time the quick collection of data that revealed
underlying issues was important. All but one participant found no difficulty in creating a
metaphor and nearly all were open to jomtly interpreting and exploring the meaning of
personal metaphors.
The chief limitation in using metaphors lay in my tendency to assume that I
understood when I should have remained open and probed more. It was easy to be
diverted by other data and in some cases I initially failed to use those data in conjunction
with the metaphor in order to see a clear common focus. In order to disembed the student
from her own metaphor, both the student and I probed its meaning; explored its links to
current practices, reflections, and automatic thoughts; explored ways to change; and
finally, the student created a new metaphor to reflect on changes made.
The shared analysis of the meaning of students' metaphors and what I learned of
their deeper meaning to the student often provided a unifier or common thread and even
provided vital missing clues to the relational conflict, the mathematical focus or both. I
discussed these insights with some participants, and we explored the implications
together. However, in these cases I initially understood only part of the meaning; as
counseling progressed more data emerged from the metaphor in the context of the
student's approach to the counseling, to the course, and to the mathematics.^'
With some participants I found the initial link between the metaphor and other
presenting data was less accessible to me. Thus I found a conscious formulation of the
central conflict and dynamic foci more difficult. For example, along with persistent
negativity on the JMK Mathematics Affect Scales that contrasted with their course
performance, the metaphors of Category III: Type A woman and the two Category II:
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Type B students were an important piece in the diagnosis of mild to moderate
mathematics depression, despite behaviors that initially indicated otherwise. The
Category III: Type A men's metaphors at first seemed active and positive but in light of
these students' initial poor performance, somewhat grandiose. However, further analysis
revealed that these men's metaphors indicated a sense of being outside of the
mathematics; their metaphorical characters used elusive and discoimected clues to try to
understand the alien or mysterious mathematics, and I saw that the metaphors truly
provided an accurate representation of how the students viewed and approached
mathematics.^"
The Category III: Type B student's metaphor held rich though indirect material
and early indicated a lack of realism on her part about how she might need to change in
order to succeed in the course.
Only one student refused to engage in exploring the meaning of a metaphor that
seemed to me directly linked with his problems with the course (a Category II: Type B
student). Even so, I was able to use the insights I gained to provide interventions such as
giving him the Algebra Test to reassure him that he had the ability to opt out of his
metaphor by passing PSYC/STAT 104.
JMK Mathematics Affect Scales
Both content and structure of this instrument made it extremely useful in the
counseling situation (see Appendix B). The scalar design allowed for open-ended
responses and its repeated use proved invaluable. The range from positive to negative
allowed students to see their changes over time. Our shared discussion linked these
changes to changes in their life circumstances, personal decisions, automatic thoughts,
and unconscious patterns. Scale topics focused on students' immediate sense of their
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mathematics self, world, and future (cf. Beck, 1977). The selected topics proved to be
important but I found that they were difficult to address verbally at each session. My
asking direct questions might have been perceived as accusatory or confrontational, and
asking them at each session might have seemed to be nagging. The use of these scales
avoided that conflictual situation.
People who are negative about themselves, their world, and their futures often rate
themselves more negatively compared with their peers than may be warranted. To
measure this I would add a new item to the JMK scales to investigate this perception:
Compared with others in this class, I do mathematics better than/as well as/worse than
most of them (see Appendix B for the original and revised versions).
There seemed to be a relationship between some students' metaphors and their
responses to the JMK Scales. When a student's metaphor was negative, stable, and either
passive (e.g., "cloudy" or "overcast") or indicating persecution (cf Inspector Javert) there
seemed to be an underlyuig mathematics depression as measured on the JMK Scales, yet
when a student's metaphor was negative but unpredictable (e.g., storm), mathematics
depression did not seem to be generally present — anxiety seemed to be more of an issue.
Beliefs Survey
I found each scale: Procedural vs. Conceptual; Toxic vs. Healthy; and Learned
Helpless vs. Mastery Oriented, taken with other data, to be especially relevant for
different participants. The first scale differentiated Category I students from the others
and discriminated somewhat between Types A and B in Categories II and III. In most
cases when a scale was highlighted with a participant in counseling, she became more
aware of its relationship to her approach to the mathematics. She was usually able to
clarify how it was manifested in mathematics testing and learning situations. From this
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she often became more able to change in positive directions. If a post-course meeting to
discuss post-scores, changes, and current beliefs had been possible with each participant,
developing and discussing a long-term plan for each participant's mathematics future
might have been feasible
A cluster of questions surveying learning versus performance motivation emerged
in the post analysis as a discriminator between Types A and B in both Category II and III
students, with Type A students being more learning-motivated than Type B students. A
revised short Beliefs survey that highlights this factor is presented in Appendix C {My
Mathematics Orientation).
Feelings Survey
Each of five students (except Lee) who reported very high mathematics testing
anxiety"" signed up for mathematics counseling during the pretesting session at the
second class of the course when I offered coimseling to all. Lee initially refused
counseling but contacted me just before the first exam requesting support, citing her
mathematics anxiety. These students also had the highest abstraction anxiety scores in
the class (from 3.2 through 4.2 on a 1 through 5 scale). Jamie was the only one of these
five for whom we eventually established the primary relational focus to be anxiety and
the only one whose anxieties on this instrument all decreased substantially. I found this
instrument to be useful in conjunction with other instruments in establishing a diagnosis
although it did not seem precise when used alone (see Appendix C for the My
Mathematics Feelings survey).
Mathematics testing anxiety of the class increased slightly overall but the class'
average responses to individual items are of even more interest. "Signing up for a math
course" or "Walking into a math class" now evoked considerably more anxiety than at
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the beginning of the course (from 2.6 to 3.1 and from 2.1 to 2.7 respectively). In contrast,
"Waiting to have a test returned" and "Receiving your final math grade in the mail" now
created considerably less anxiety than at the beginning of the course (3 to 2.6 and 3.1 to
2.4 respectively). When taken with other evaluative data, these responses seem to
indicate that in the context of this class students' anxiety levels had decreased as their
control and achievement had increased but that this improvement did not generalize to
future or other mathematics classes. In fact, the prospect or memory of other courses now
evoked more anxiety.
I paid little attention to the number anxiety results during the study since all
number anxiety mean responses were at or below the mid point (3) of the scale and thus
seemed to indicate low to moderate anxiety, especially when taken m contrast to the
reported testing and abstraction anxieties that went as high as 4. 1 and 4.2 respectively on
the pretests. I realize now that the two participants whom I early recognized to have poor
number and operation sense (Kelly and Karen) had the highest number anxiety scores in
the class at 3 and 2.9 respectively. Lee, who was considerably more competent
arithmetically than they, had a relatively high score of 2.8. On the other hand, on the
Arithmetic for Statistics assessment, although Lee was generally adequate, she
nevertheless had a marginal operation sense that likely contributed to heightened number
anxiety. I would now flag scores in the middle of the number anxiety scale for immediate
investigation of a student's number and operation sense.
For some individuals the changes in their feelings on survey responses confirmed
the direction and efficacy of the mathematics counseling. For example, Jamie's testing
and abstraction anxiety showed an overall significant decrease, with some aspects
increasing while others decreased (see Appendix H, Table H3). In chapter 6, 1 discussed
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the course of counseling with Jamie in detail. Her dynamic focus was specifically social
anxiety'" that was intensified in the mathematics learning environment, and it was this
and her related practices that we worked together to change. All aspects of Jamie's
testing anxiety that had a self-focused social public component ("Walking into a math
class," "Raising your hand in a math class to ask a question,"" and "Waiting to have a
math test returned.") decreased over time. The aspects of Jamie's abstraction anxiety that
had a self-focused social public aspect also decreased. By contrast, Jamie's anxiety about
taking a final math exam in class increased. This seemed to have a mathematics self-
competence, performance focus for her rather than a social self-focus.
The Role as Relational Counselor Transform My Tutoring Work
Counseling Use of Transference and Countertransference
The new need to attend to transference and countertransference immediately gave
me conscious access to a fund of analyzable and usable data that I had previously largely
ignored in mathematics tutoring practice. In typical tutoring situations, transference
usually remains unplicit as both student and tutor often continue to act out old patterns of
interaction without the conscious reflection that my new approach encourages.
Relationship patterns based on the student's internalized teacher presences of the past
may pull or push the tutor mto assuming the teacher role they demand. She may on the
other hand react against assuming roles she believes are toxic for the student's
mathematics mental health. Because these relationship patterns are not brought into the
open the student may resent the refiisal of the tutor to take on the expected role and the
student's expectations are not realigned. But my new approach to the participants in this
study allowed for this material to be brought into the open and dealt with consciously so
that we could each adjust to more productive ways of relating.
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In the academic setting, my use of insights gained from transference and
countertransference was necessarily quite different from a psychoanalyst's use. Since the
focus was on the mathematics and not on the resolution of personal psychological
problems, interpretation and specific working through of transference was not
appropriate. What was appropriate was noticing it and checking with the participant
about shared insights. Most important was looking at patterns of interactions over time
including the transference and countertransference so that a central relational conflict
could be identified.
I found it challenging to attend to the student's transference and to my
countertransference. In the past, I had found myself on occasion acting in ways that
surprised and concerned me — for example, believing whiners and joining them, almost
doing a student's work for her, agreeing to work with a student much more than
appropriate, scolding, or panicking with them. It had not at all been my practice in the
mathematics tutoring situation to consider what these behaviors might be telling me about
the students' history, personality, approach, and practices, nor to consider my behaviors.
During this study I needed to develop this reflection as a new practice. In the relational
counseling situation, even in brief counseling, it is usual for counselor and cUent to
discuss the transference and countertransference. Where the focus was mathematics
learning, would that be appropriate or necessary? In the brief counseling situation in a
college setting, the challenge was to estabhsh for myself parameters for if, when, and
how to use the transference and countertransference material in counseling with the
student. I found that the following practices were appropriate in the mathematics
counseling setting and allowed for effective use of the data from both transference and
countertransference:
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Incorporating data from instruments and from observation to consider students '
conscious and unconscious expectations about their relationship with the present
mathematics teacher and tutor. With every participant, I listened for, observed, and asked
how she experienced the present class. I asked students questions about their past and the
present to help them discover the ways they might be appropriately or inappropriately
bringing their past into the present.'" If it became clear that a student's experience of the
class was discrepant from the present reality I drew her attention to it and invited her to
consider how she might adjust to this new awareness.
Developing reflection and self-awareness regarding countertransferential
reactions to the tutees. I filled out the mathematics counseling session summary sheet
after each session to help me reflect immediately on the session. I listened to my tape
recordings of counseling sessions and study groups and studied transcripts in order to
observe myself in relation to participants. Supervision was central in some cases to
recognizing my countertransferential reactions. Not surprisingly, my session notes were
often ahead of my counseling practice.
Using self-revelation of countertransference. I found that when I did self-reveal in
the counseling situation, both the participant and I became clearer about the relational
patterns that might be keeping us both stuck. We were then more able to change our
behaviors and to extricate ourselves from counterproductive patterns."" I found that I
needed to present my experience of countertransference in a manner compatible with
student's learning style or risk her not understanding and optimally benefitmg."'"
Indirect use of transference and countertransference observations in situations
where the student rejects or avoids a counseling approach (Category II: Type B students)
or has a detached avoidant relating pattern (a Category III: Type A student in this study).
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I could only talk about transference and countertransference indirectly with some
students by noticing behaviors and perceptions and asking them to verify whether they
were seeing the present relationships as different from the past, inviting them to notice
the present relational reality, suggesting they evaluate the appropriateness of their beliefs
and practices in relation to the present reality, and affirming their helpful choices and
changes to appropriately deal with present reality.
Supervision by a person knowledgeable in counseling. I needed a knowing
dispassionate ear to share my actions and judgments, particularly my subjectively
experienced transference and countertransference. Preparmg my cases for supervision
forced me to reflect on each participant in a more global way than I had till then.
Supervision itself provided me with affirming and challenging feedback on my progress
thus far with each participant. It forced me to pay close attention to my own reactions and
my tendency to impose my agenda on participants rather than facilitating their own
choices and movement.'^^ It furnished me with possible new approaches for stuck
situations (cf. paradoxical intention for Mulder). An even earlier supervision meeting
may have helped me decide to do things differently from unwittingly acting out my
countertransference.'"
Counseling as Good-enough Tutor-Parenting
Winnicott's concept of good-enough freed me in a number of ways to be more
available to my tutees and to help them be strategic in their choices. I am not neutral with
respect to procedural (only) versus conceptual (including procedural) mathematics
pedagogy, for example. In my experience, conceptual learning helps make students
secure in their mathematics base. The tutoring role differs from the teaching role in that
control over the curriculum lies with the teacher not with the tutor; the tutor must support
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the student in mastering the curriculum whether the tutor "approves" of the curriculum or
not. In the context of the 10- or 15-week mathematics course where students had the
opportunity to struggle on problems in class with coaching support from the instructor,
the conceptual aspects were only linked with the mathematical procedures when
individuals asked the instructor during problem-working sessions. Opportunities in
counseling to help tutees attain a more conceptual understanding of the material were
limited by time and content, especially with students who were akeady deeply embedded
in a procedural approach. I found that to support a student in doing well on a
PS YC/STAT 1 04 exam, there were times when procedural advice superceded conceptual.
I was able to see my mathematics tutoring as good-enough ui providing for my students
although it was less than (my) perfection. In line with good-enough parenting I also had
to me learn to better tolerate students' mathematical goals when they differed from mine
in contrast to my former approach of trying to badger or cajole them to take on my goals
for them. On the other hand, I had to be carefiil not to allow this good-enough concept to
lull me into lowermg my expectations for what they could achieve.
Challenges and Limitations of this Approach: Integrating Counselor and
Tutoring Roles into Mathematics Counseling
I found that to be a good-enough mathematics counselor is very difficult. My
"successes" from my long enculturation and experience in traditional mathematics
teaching had only relatively recently been called into question by the nagging failures
that drove me into my doctoral program. A cognitive constructivist, conceptual, problem-
solving approach to teaching and learning mathematics was the solution, I was
convinced, but I found it difficult to be that teacher, to facilitate that learning. I had
always been the one who worked out what the problem was and structured the solution
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and told the student, who ran with it, or puzzled over it, or denied it, or ignored it. I grew
to believe that the essence of constructivism was in the student seeing the problem and,
with the teacher as guide, finding a solution for herself, but how to be a guide? I now
know that telling spoiled it by making it mine and not hers (even if I was "right"). Now I
have discovered that counseling is the same. I had learned through lay counseling
ministry training and experience that a constructivist approach was essential for healing
and growth. That was confirmed in my doctoral psychological counseling coursework.
Now in this study I had to integrate my emerging but tentative constructivist teaching role
with a constructivist counseling role to be a good-enough mathematics coimselor.
The Challenge of Learning to be a Relational Counselor
Mulder taught me about coimseling perhaps more than any other participant
because he would not accept my telling; he resisted it and stood up to me and I learned to
step aside and let him fight his own battle. Not that my input was not helpful — indeed it
was! On his own, it is almost certain that Mulder would not have made the changes he
did but in the end they were his own changes. If I had not stepped aside he may not have
made the final crucial change. With other participants my propensity for prescribing my
solutions for them was not as clear to me although my experience of transference and
countertransference gave me clues. Dr. P. saw it and helped me to begin to see it in
supervision. With some participants, though, it was only as I analyzed the transcripts, my
session notes, other data, my own initial analysis of the student's needs, and Dr. P.'s
persistent supervision-style queries of that analysis that I finally heard myself telling and
scolding and prescribing. And I finally realized how I could have done it differently, in a
constructivist manner, because along with the telling I did some of that (work in a
constructivist manner) and Dr. P. also pointed that out to me. All along I had the insights
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and approaches of relational counseling to use; when I did use them students did well and
found their own feet. In the end what I did with each participant (except for Kelly, Brad,
and perhaps Autumn) was good-enough for them to gain insight into their restrictive
mathematics relational patterns. This equipped them to make the changes necessary to
succeed in PSYC/STAT 104.
The Challenge of Learning to be a Relational Tutor
I learned relational mathematics tutoring from the participants, especially Karen,
Mulder, and Lee. I found that when I used <:onstructivist, relational counseling
approaches such as mirroring sound thinking (even in the midst of errors or low grades)
to build up tentative and vulnerable mathematics selves, participants began to move into a
competence they did not know they had and then to develop that competence. When
mstead I was drawn into participants' focus on the negatives (the errors or the low grade)
and tried to fix it by telling the answer and teaching them more, I cut them off from that
tentative mathematics self so that it could not grow.
Likewise when I heard their mathematical questions and responded to their
pressing felt needs by telling, things did not go well; when I responded by eliciting from
them what they already knew and we went from there (e.g., parallel problem-solving),
they grew. In the end what I did with each participant (except for Kelly and Brad) was
good-enough for them to gain access to their growing mathematics competence, develop
insight into counterproductive mathematical beliefs and practices, and make the changes
necessary to succeed in PSYC/STAT 104.
Relational counseling is based on the idea that the counselor and client are both
adults, and the client chooses her path while the counselor supports her. The tutor-tutee
relationship is usually an expert-novice relationship with regard to the mathematics
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content and (theoretically at least) a novice-expert one with regard to the student's own
affective experience of the mathematics. I had been learning how to negotiate the
mathematics content in a constructivist, reciprocal way, but not the student's affective
experience of the mathematics. In my prior tutoring practice the tutor-tutee relationship
with regard to the student's own experience of the mathematics was more often a parent
to child one. From this study, I found that that is the challenge for me in the practice of
mathematics relational counseling — ^to learn how to be constructivist, non-directive, and
supportive, while also learning from the student. This was needed not only when we dealt
with the mathematics content, but also when we explored and gained insight into the
affective areas of her mathematics cognition and her underlying relational patterns.
Just as a crucial assumption of this approach is the reality of student choice and
responsibility for choices, this assumption applies equally to the counselor. The benefit
for the student is in helping her become conscious of her choices and the extent of her
power to choose differently. The danger in this approach is to appear to hold a person
responsible for things she has little power to change. Thus Jamie could choose to sign up
for individual mathematics counseling, but the shyness and prior negative experiences
that dominated her interactional patterns, led to her choosing to hide and disappear rather
than relate and approach. Because she was not consciously aware that she was making
that choice, she seemed to remain powerless to choose differently. My choice to approach
her was perhaps going against one of the maxims of counseling (Wait for the person to
seek your help; that will mean she is ready and willing to receive it.) but because my
choice was good-enough in this case, Jamie became aware of her choices and her power
to chose differently. At other times my choices to be parent rather than peer with the
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student were not good-enough (cf. Brad and Autumn). My awareness of my own power
to choose my roles and the importance of my choices grew as the study proceeded.
BRIEF RELATIONAL MATHEMATICS COUNSELING: A SUMMARY
The traditional model for providing mathematics academic support typically
excludes from consideration many aspects of the student's relationship to mathematical
learning and compartmentalizes what is considered into content knowledge, and some
aspects of affect. Tobias' concept of mathematics mental health provides a different
perspective for viewing the struggling mathematics student. A brief relational counseling
approach prioritizes students' mathematic mental health problems and provides a means
of dealing with them in a hoUstic and productive way, without ignoring or minimizing
important elements. My categorization scheme uses both the relational and cognitive
diagnosis as a way of understanding and dealing with complexity.
It is important to highlight here crucial ways that this new approach differed from
more traditional approaches to mathematics support. But first I must point out that this
study alone, while it puts forward considerable evidence of student change, does not
provide quantifiable comparisons between the effects of this approach and the effects of
traditional mathematics support. For comparisons, the sample was small, there was no
matched sample to receive control treatment (traditional mathematics support) and
changes reported and the processes that led to these changes were in many ways not
quantifiable nor easily verifiable. Nevertheless the differences I observed were striking.
Needy students do not necessarily access traditional mathematics academic
support. Jamie would likely not have opted for academic support unless meeting with a
tutor were a course requirement or there was a class-link tutor she could get to know first.
Jamie's dilemma presented me a disturbing possibility that I had previously only vaguely
363
considered — there are likely unknown numbers of students who might benefit from this
course intervention but would never come to a traditional Learning Assistance Center;
Class-linking provides unique observation opportunities for more thorough and
effective counseling. Being able to observe student behaviors and practices in the
classroom'"' provides the tutor with data for timely and focused counseling interventions.
If Jamie had come to a traditional Learning Assistance Center, the tutor would not have
had the benefit of observing her in class, nor have been aware of how important that
observation was. In my former role I would likely have worked with Jamie on her
mathematics and found it to be relatively sound. We would not have foimd the real root
of her mathematics learning issues nor explored ways she could see herself differently as
a successful mathematics learner. Other students in the study who needed assistance also
indicated that they would not have accessed my help if I, as the tutor, had not been in the
classroom.'""
In contrast with current learning support experience, early and thorough
diagnostic assessment of both mathematics cognition and mathematics relational issues
is possible. Such timely diagnosis is key to growth in mathematics skill and improved
mathematics mental health for students who are willing to explore both. Even for students
who do not wish to explore their mathematics relational problems, their assessments can
be used to design mathematics-only counseling interventions that assist them to make
academic and indirectly, relational progress;
Without a relational counseling approach focused on the student 's transference
and the counselor 's countertransference, the counselor might not be able to identify and
deal with students ' core mathematics mental health issues. If I had not identified Jtunie's
transference towards Ann and me as frightening, dangerous teachers, I might have acted
364
out the same kind of countertransference that Ann did (staying away in order not fulfill
Jamie's fears). Instead I broke through and came close and was not dangerous;
For counseling to be efficacious, the importance of the classroom emotional
climate established by the mathematics instructor cannot he underestimated. The positive
emotional climate established by the instructor in this study created an environment for
most students where damaged attachments to mathematics teachers could be repaired,
where underdeveloped or vulnerable mathematics selves could grow, and where no
further damage was done. By contrast, counseling students taking a course where they
experience the instructor and the classroom as indifferent or abusive would likely have to
take a different direction and would invariably have reduced efficacy in achievement and
emotional healing for the student.
A counseling use of the mathematics addresses the various mathematics mental
health problems caused by the ways the mathematics content has been and is being
taught. Because a procedural approach to mathematics is closely related to conceptual
linking difficulties (i.e., linking procedures with their conceptual base) and a tendency
towards learned helplessness in the mathematics course environment, individualized
mathematics-focused counselmg approaches (e.g., mathematics course management and
conceptual problem-solving counseling) may be called for. One use of exam analysis
counseling is to help negative students break a negative focus by affirming or mirroring
sound mathematical thinking, thus building up their underdeveloped selves. Conceptual
linking counseling offers students a secure mathematics base they may not have
previously experienced. Such intentional uses of mathematics tutoring as counseling
contrast with traditional uses and demonstrate promise for improving the student's
mathematics mental health.
365
LIMITATIONS OF STUDY
It is important to evaluate the conclusions of the study in terms of the sample, the
measurement instruments and their uses, and the research methodology.
Sampling limitations. The number of participants in the study was small and they
were taken from a small urban New England commuter university. The nature of the
study necessitated a small sample but the fact that students were from widely varying
backgrounds enhances its value. The small size of the sample restricted the use of
quantitative results of the instruments to descriptive support for qualitative results within
the sample, aiding the ongoing counseling process, and understanding of individual
outcomes. The findings from the instruments may not be generalizable to students
enrolled in other mathematics or statistics courses nor attending other types of college or
university although uses of some instruments (e.g., the Metaphor and the JMK Affect
Scales) seem applicable for counseling purposes in any setting.
Quantitative instrument limitations and uses. The only quantitative instrument
reliably calibrated on large samples was the Algebra Test (see Appendix C). Apart from
the first 20 questions of the Feelings Survey that were taken from the 98 item MARS
which was normed thirty years ago (Suiim, 1972), all instruments with quantitative
outcomes, except for the class exams and other class evaluations, were created or adapted
and the results evaluated by the researcher. Individual results were compared with those
of the (small) class group and individual changes are described in comparison with other
researcher-observed changes and class achievement changes.
Researcher bias. The participants' words and actions were filtered through
researcher bias. Relational data were collected via counseling session interaction where
the researcher and the participant were working for change, and the interactions did
366
change both. Although ongoing analysis of the interactions by the researcher (with
participant feedback) and clinical supervision (and later evaluation of researcher analysis
by the clinical supervisor) were designed to monitor, interpret, and neutralize this bias,
interpretations of the data by others might yield different conclusions about the relational
outcomes of the study.
Possible omission of important student factors. My understandings of key aspects
of mathematics functioning (cognition and affect)'"'", were applied, integrated, and
adapted in this study. I adapted Mitchell's relational conflict theory and his concept of
three dimensions a person's relationality, and revised Tobias' five tier categorization of
college mathematics students. In addition to Dweck, Seligman and Beck's work on
learned helplessness and depression, the researcher's own findings were also used to
ground the study. None of these, separately or together, has been used in a holistic study
of college students' mathematics mental health or of interventions to improve it while the
student was taking a college mathematic course. Because of this it is certainly possible
that important aspects of students' mathematics mental health were not addressed. All
findings in this study should be interpreted in this light.
RECOMMENDATIONS FOR FURTHER RESEARCH
Based on fmdings in this study and the limitations, I make the following
recommendations for future research:
Counselor characteristics subtype B students and counselor-student match. My
use of brief relational mathematics counseling helped students from all three categories
identified. Students who benefited less were from subtype B of both Category II (Autumn
and Mitch) and Category III (Kelly) — students whose ways of dealing with the
vulnerability and under-confidence of their mathematics selves were relatively inflexible
367
or unstable. Counselor characteristics may have been a factor in this relative lack of
success. Additional research needs to be done when offering this counseling to such
students. It will be vital to investigate counselor characteristics, counselor-student match
and interventions that may help them succeed.
Quasi-experimental studies. My adaptation of Mitchell's (1988) relational conflict
theory to mathematics support in this study yielded an understanding of the three
dimensions of participants' relationality and their central relational conflicts that enabled
us to resolve that conflict well enough in the brief time available for them to be
successful in the course (with the exceptions noted above). It is not possible to say what
their outcomes would have been if they had not participated in the counseling
intervention. Additional research using this approach with other college mathematics
students, comparing their process and outcomes with those of matched samples of
students who receive tutoring support only, and with matched samples of students who
receive no support would further our understanding and test its generalizability,
particularly the finding of increased achievement and improved mathematics mental
health.
Gender differences. I found interesting gender differences in the way similar core
problem were expressed. Men and women, especially those within Category II: Type A
and Category III: Type A groups, whose core problems were the same, differed markedly
in their emotional conditions, practices and ways of relating. With Category II: Type A
students, the women expressed their insecure attachment to mathematics with anxiety; the
man expressed his with a frantic and counterproductive attempt to learn it all. With
Category III: Type A students, the woman expressed her underdeveloped mathematics
self with underconfidence, empty depression and hostile detachment; the men expressed
368
theirs with overconfidence, unrealistic bravado (grandiosity) and resistance. Further
research on gender differences within and across types may be called for to confirm (or
disprove) that the presentation of similar core problems consistently differs predictably
according to gender.
Counselor-student match and gender. I had considerable difficulty in overcoming
my countertransference reactions to grandiose men of Category III: Type A. I wanted to
deflate their inflated sense of prowess. My reactions seemed to come at least in part from
our gender difference. One of these men, who spoke of his difficulties working with
women, made only minimal progress in counseling. I wonder if a male counselor may
have been more successful in supporting and developing his underdeveloped mathematics
self. Research into the effects of counselor-student match by gender could shed light on
this.
Identifying mathematics situational depression. This study suggests that some
students may suffer from mathematics situational depression. The College Learning
Metaphor (pre-and post) and repeated use (administered at every session) of the JMK
Mathematics Affect Scales analyzed together seemed to aid diagnosis and help to monitor
this condition. Further, it can alert the counselor to a need for a specific and timely
intervention. Ongoing research using both the College Learning Metaphor Survey and the
JMK Mathematics Affect Scales conducted with large numbers of students would
investigate a possible relationship. Such a finding would investigate the simultaneous use
of both instruments to rule out mathematics depression and aid accurate diagnosis
Mathematics relational counseling and other classroom conditions. This study
was conducted in the context of a classroom where the instructor created a positive
relational climate, where the mathematical demands were somewhat more procedural
369
than conceptual, and where the conceptual content was taught by lecture discussion and
the mathematics procedures developed in problem-working sessions. Relational
counseling in other contexts is likely to look different and have different outcomes for
different categories of student. Classes may differ in how mathematics is taught: they
may stress non-routine problem-solving; they may be designed to challenge student
misconceptions; they may involve mathematics procedures only being demonstrated by
the teacher on the board. Classes may differ in relational climate: the teacher may be
disdainful of what she perceives to be students' low ability and poor understanding; she
may be judgmental of certain student approaches; she may ignore or insuk student who
struggle. Further studies of the use of relational counseling to support students in
different settings accordmg to how mathematics is taught and according to relational
climate would contribute to our understanding of the efficacy and limits of its use.
CONCLUSION
I learned during the summer of 2000 to open my learning specialist eyes wider
and to see through the lens of relational conflict theory. This at once complicated and
clarified my task. New complexities arose in having to look now not only at the student's
mathematics tasks but also at her whole approach to the mathematics course, her
mathematics self, her intemaUzed presences, and her patterns of mathematics
interpersonal interactions.
In this study I determined it was indeed possible for a mathematics learning
specialist with some exposure to the field of psychological counseling to hoUstically help
traditional and non-traditional aged college students taking an introductory level college
mathematics course. Most came to understand their mathematics learning issues and
found their own coherent explanatory frame for how the aspects of mathematics
370
cognition were personally configured within their relational history. Students became
conscious of their areas of embeddedness as well as how they contributed to their own
immobility, they made changes, and they improved their mathematics mental health.
They attained "good-enough" success in the current course and some even seemed to
develop the heart to tackle future mathematical challenges more effectively.
I found that the concept of relationality with its three dimensions did provide an
adequate frame for me to understand and focus on each participant's particular relational
conflicts and I found that the approaches of relational and cognitive therapy were useful
as elements of an approach designed to address those difficulties. The approach
highlighted my role in the counselor-student dyad and I found that to the extent that I
reflected on how I reacted and interacted with the student, the insights I gained led me to
change in ways that promoted student growth. In sum, I found that the relational conflict
perspective has given me a new, more nuanced, and authentic way of seeing students and
helping them to see themselves and their interactions, in the academic support and course
classroom setting.
371
' As this is an even numbered ciiapter, I use "shie," "her," and "hers" for third person generic pronouns.
" For example, I agreed with the Karen who said she probably could not do it and at first ignored the Karen
who expressed and showed surprising competence; I agreed with the Mulder who said he could do it and
dismissed the Mulder who expressed frustration and struggle.
'" Each of these students (Autumn, Jamie, Lee, and Mitch) was ambivalent about her own mathematical
ability. None saw him/herself in the category of "some people can do math" (question 9, Part 11,
Mathematics Beliefs survey) although all but Jamie initially believed her "ability in mathematics" could
improve (question &, Part II). The Algebra Test is not a test of ability; it shows a student approximately
where she is on a developmental path. I was wary of feeding into any fixed trait beliefs about mathematical
ability even the "I'm one of those people who can do mathematics" belief that saw others as not being able
to. In feet, I took every opportunity to dispel such fixed trait beliefs. I deemed a developmental view that
saw growth of self and others as always possible and expected improvement in relation to intelligent effort
as much healthier. In the context of a 10 week college mathematics course, however, being at a higher
algebra concept developmental level certainly gave a student an advantage over a student at a lower level.
" Two of these five did not exhibit an adequate operaticm sense (Pierre's was inadequate and Lee's was
marginal), one did not exhibit adequate open ended arithmetical thinking /problem-solving (Autumn was
inadequate), and one did not exhibit an adequate large (>I000) integer number sense (Jamie' was
marginal).
" I found the AFS assessment to be too long, with a number of questions not relevant to this statistics course
or discerning enough. The graph related questions not varied or discerning enough, there were not enough
operation sense questions, and relevant categories such as order of operations were not addressed. In
addition, within each category, the questions were not designed developmentally to reveal levels of
understanding.
" For example, Jamie's "fear of the storm coming back" metaphor was not principally about the
mathematics itself but about a dangerous classroom environment with dangerous teachers. It took us some
time to link this with Jamie 's own behaviors in the storm — staying inside in order to keep safe in this
dangerous situafion. Mulder himself linked his "Fox Mulder searching for aliens" metaphor with making
mathematics hard for himself, so that became our initial focus — the ways Mulder did mathematics that
made it hard for him. We at first missed the link for Mulder between the object of the search — aliens — and
mathematics. An important piece to Mulder's difficulties was that he was indeed seeing mathematics as
alien, so he was using alien search techniques to master it ratha- that exploring and mastering it logically
and conceptually.
"" Mulder's metaphor was Fox Muldw seardiing for aliens and Brad's was Sherlock Hoknes trying to crack
a mystery.
"" Each had an average score of 3.5 or above on a scale of I through 5.
^ This social anxiety was related to and complicated by Jamie's fear of too much success that invited
attention, expectation, and pressure from ha- father for fiiture performance.
" Jamie answered 4 on her pretest on this but on her posttest she answered: 3 (not in front of class,
individual work), 4 (our size class) and 5 (math lecture size class like at State University)
" I did this somewhat unevenly. For example, I indulged Lee because we were pals (staying wdth her
transference and my countertransference); I did not become conscious enough of how her positive feelings
towards me contrasted with her negative feelings toward the instructor so I did not help her to evaluate that
against the reality and we did not ask ourselves how our reactions might have a detrimental effect on how
she approached PSYC/STAT 104. We should have, for example, questioned the discrepancy between her
effort in the class and on homework versus how much time she was spending with me.
372
^ For example, with Mulder, I told him I felt like his scolding mother — that helped me in my process of
moving out of that inappropriate role; I could ha\e done it more clearly with Karen, e.g., "When I listen to
what you say about yourself doing mathematics I feel depressed, but when I look at the mathematics you
are doing and the ways you are changing I admire you and am hopefiil that you can learn it and get a good
grade in the class." I believe that this would have had a positive effect on the development of her
underdeveloped mathematics self
™ For example, with Robin a visual learner with auditory processing difficulties, when I told her of my
dual reactions: that in class she seemed to be acting as an intellectually incompetent female and I felt
irritated and alarmed at the same time, whereas one-on-one in the mathematics counseling situation I found
her to be intellectually competent and 1 admired and respected her ability to manage the content, she did not
understand because I did not use visual pictures that she had introduced in her metaphor of the ditzy village
girls versus Belle.
™ During supervision my strong fiiistration with Autumn became apparent and my judgment of her self-
containment, her emotional distance, "1 want to shake her" (Jillian, July 20, 2000). I found myself reporting
a very similar reaction to Brad, but more for his refijsal (from my perspective) to face the reality of his
situation, and his bravado in the face of the realities. With both of these participants 1 had been very
directive, partly the cause, Dr P gently suggested, for their digging in and my fi-ustration. His reminding me
that it was their motivations for change or stasis that needed to be revealed and respected and not my
motivations for them imposed. He encouraged me to find what in me had been triggered by their behaviors
and attitudes.
'" In my session notes, I found that my insights and determinations were at times ahead of my actions.
Earlier supervision might have alerted me to those discrepancies and led to a different approach with Brad
or a timely intervention with Kelly.
What being a class-link contributes to the efficacy of the brief relational mathematics counseling model
is: Counselor presence in the central context, leading to
1. Participation by the students in the counseling,
2. Inside perceptions of the instructor and the Course that could be used in counseling, and
3. In-class perceptions of student practices that could also be used in counseling
""^ Seven of eight participants who finished the course and who responded to my post-study survey
(December 2000) acknowledged that they would not have accessed mathematics learning assistance at the
Learning Assistance Center if I had not been in the class with them. Of these seven, three, Karen, Mitch,
and Pierre, said they will now access the Learning Support Center in the future when they are taking a
mathematics course, one of these though, Karen, only "...as long as Jillian is thae because you really
helped me in my last math class." Another, Lee, indicated a very conditional, "if I do [take another math
course], and I find it difficult, 1 will use the Learning Assistance Center facilities if 1 think they will best
help me." The others. Autumn, Jamie, and Mulder, said they probably would not access the Learning
Support Center in the future when they are taking a mathematics course, Jamie because she is "kind of shy"
and a "helper" who doesn't "really like to ask for help," and Mulder because "Unless I am really struggling
I do not go for help. I like to figure things out on my own." Autumn gave no explanation.
'^" Suggested by Schoenfeld (1992), Dweck (1975; 1982; 1986), Tobias (1993), Skemp (1987), Boaler
(1997) and others.
373
374
Appendix A: Table Al
Proposed Brief Relational Mathe matics Counseling
Relational
Student's Mathematics History
Mathematics Affect Now
Mathematics Cognition Now
Relational Diagnosis
Proposed Treatments
Dimension
How does this develop?
What can go wrong?
Expected Affective symptoms
Expected Cognitive symptoms
Expected Central Mathematics Relational
Conflicts
Mathematics
1 . Mirroring
1 . Neglect . . . chronic lack
I . empty depression; learned
1 . sound mathematics preparation
Self: Conflict between conscious
For Self. Counselor
Self
2. Idealization of teacher
of mirroring =>
helplessness
2. adequate mathematics
ambition/desire to succeed in course and
mirrors studait's
image
underdevelopment of self:
2. grandiosity
preparation
underlying belief in inability to succeed in
mathematics self.
3. (a) Internalization and
low self-esteem, under-
3. underpreparation
course (low/undermined self-esteem)
provides self for
transformation of teacher
con fidence
idealization.
image into values and
2. Failure to provide
provides manageable
ideals... superego, (b)
realistic curbing of
frustrations to push
manageable frustration =>
grandiosity =>
student to
development of student's
underdevelopment of self:
development &
competence.
low self-esteem.
realization of
confidence and basic
ambivalent/over-
competence
mathematics self-esteem:
confidence
healthy narcissism
Mathematics
1. Installation of bad
1 .Experience of
1. guilt, shame => depression
1 . intern al ized presences
Internalized Presences: Conflict between
For Bad Internalized
Internalized
internalized teacher
endangerment by bad-
2. fear of judgment => anxiety
supportive or at least not
conscious desire to and perhaps belief in
Presences:
Presences
presence in the unconscious
enough teacher => moral
detrimental to mathematics self
self for success in course and
Counselor provides
(Note: degree of badness)
conversion to self as bad
and internalized mathematics
Internalized presences insisting that one is
self(and points to
2. Identification of the ego
internalized presence or
values (superego)
bad/cannot succeed
instructor) as good
with the bad internalized
repression of bad
2. internalized presences
replacements for bad
presence (Note: extent of
internalized teacher object
undermining mathematics self
presences and
identification) =>
2. Experience of
3. punishing mathematics
refutes claims of bad
development of defenses to
mathematics as punitive
superego ("internal saboteur")
internalized
protect the ego from these
internal saboteur:
making mathematics self feel
presences
bad internalized presences
superego => sense of
moral failure
guilt/ shame
Mathematics
Mathematics teachers:
Mathematics teachers:
1 . grief/loss =>
1 . sound mathematics attachment
Attachments: Conflict between conscious
For Compromised
Attachments
1 . Teacher provides good-
Teacher unavailable
depression
2. traumatized mathematics
desire to succeed in course and detached
Attachments:
enough caregiving:
and/or unresponsive =>
2. separation anxiety from teacher
attachment
attachment pattern that prevent studait
Counselor provides
responsive & available =>
student develops insecure
and/or mathematics
3. failure of mathematics
from getting the help he/she needs or
self(and points to
teacher as secure base
attachment to teacher:
attachment
dependent relational pattern that prevents
instructor if
2. Student develops secure
anxious, ambivalent, or
student from taking responsibility with
applicable) and
attachment => able to
disorganized attachment
support or ambivalent unstable
promotes
explore and return to secure
Mathematics:
attachment pattern
mathematics as
base when needed
Teacher does not know
secure teacher and
Mathematics:
and/or teach mathematics
secure mathematics
1 . Teacher has good-enough
well enough=> student
bases on which
grasp of ftindamental
develops anxious.
student can rely
arithmetic/ transitions to
ambivalent, or
algebra/algebra
disorganized attachment
2. Teacher promotes
to arithmetic and/or
mathematics rather than self
algebra
as authority for correctness
3. Teacher believes in
student's prowess and
provides developmentally
appropriate mathematical
tasks; student has necessary
tools => student develops
secure attachmait to
mathematics
1
375
APPENDIX B
Individual Mathematics Counseling
Assessment and Treatment Tools
1 . Mathematics Counseling Session Reflection
2. Student Mathematics History Interview Protocol
3. College Learning Metaphor Survey
4. Negativity/Positivity Survey
a. JMK Mathematics Affect Scales
b. JMK Mathematics Affect Scales, revised
5. Survey Profile Summary Sheet
6. One-On-One Mathematics Counseling Evaluation
376
1 . Mathematics Counseling
Session #
Name Course/Semester_
Counselor Date
Notes
Transference/Countertransference
Self
Object
Space-in-between
Summary:
Thoughts for the next Session:
©Jillian M. Knowles, UNHM, Summer 2000
377
MATHEMATICS HISTORY INTERVIEW PROTOCOL
NAME AGE
DATE MAJOR
1 . Tell me how you usually feel when you are doing mathematics.
In the class you are in now, how does it feel to be:
• in class?
• doing homework?
• doing an exam?
2. Describe your best experience doing mathematics?
Why;
3. Describe your worst experience doing mathematics?
Why*^
4. A. Have you always felt this way about doing mathematics? [If not, when
and why and how did how you feel doing mathematics change?]
5. Is doing mathematics the same as or different from doing other
subjects for you? How?
6. Do you do mathematics outside of classes — when do you do it in your
daily life?
378
7. Tell me how well you do in mathematics courses, in daily life.
8. What is mathematics anyway? How would you describe it to a friend?
9. If doing mathematics is different for you from doing other activities,
why do you think that is so?
10. How important do you think math/stats will be for you in your future?
How does that make you feel?
11. Are any parts of math comfortable for you to do? Tell me a little about
it...
12. What are your least favorite types of mathematics to do? Tell me a little
about it...
379
13/14/15 In Elementary/Middle/HIgh school, what was mathematics like
for you?
What type of mathematics did you do?
What tools did you use?
Do you remember the teacher?
Was there anything about you/your family that you felt made a
difference in how the teacher treated you/her expectations of you?
How did you get through school math when it got hard? [when you felt
unable to do it well]
Did you receive any extra help? How was that for you?
How do you think math should have been taught/the learning
environment should have been for you to have done better in it?
How do you think YOU could have done things differently in
mathematics for you to have done better in it?
380
16. List the mathematics courses you took in high school, the year you took
each, and the grade you earned in each:
Mathematics Course Year (e.g., 1995)
1th
9'" Grade
10"^ Grade
11 '''Grade
12"^ Grade
Grade earned
1 7. Did your parents work with you with math at home? How was
that? Theu- attitudes to math? to you doing math? Any brothers? Or sisters?
© Compiled by Jillian Knowles, Summer 1997; revised Summer 2000
381
3. College Learning Metaphor Survey:
The College Learning Metaphor was administered at the beginning of the first
session to all participants. When possible, it was also administered in the final session to
assess any changes (but see also One-on-One Counseling Evaluation below).
COLLEGE LEARNING METAPHOR SURVEY
Name Date
1. Make a list of metaphors that show how you FEEL about
MATHEMATICSA^OURSELF DOING MATHEMATICS. For
example, if it were a color what color would it be? If it were weather or
an animal or a fictional character or . . . what would it be?
Now choose one of the metaphors from 1. that most closely describes
your relationship with MATHEMATICS and write more about why this
metaphor describes your relationship with MATHEMATICS.
3. As you reread your metaphors, what do they tell you about your attitudes
as you do MATHEMATICS? your expectations of yourself doing
MATHEMATICS? your predictions about your success in
MATHEMATICS?
Adapted from; Gibson, H. (1994). "Math is like a used car": Metaphors reveal attitudes towards
mathematics. In D. Buerk (Ed.), Empowering students by promoting active learning in
mathematics (pp. 7-12). Reston, West Virginia: National Council of Teachers of Mathematics.
© Jillian Knowles, revised December 1999.
382
4 a.
JMK Mathematics Affect Scales
Name Date
On this questionnaire is a group of scales. Please read each scale carefully. Then
indicate the part of each scale which best describes the way you have been feeling while
doing mathematics during the PAST WEEK. INCLUDING TODAY. If an interval on
the scale better describes your range of feelings rather than point, indicate that range with
a line. If the words on the scale do not accurately describe your feelings, supply your
own.
1 . When I think about doing mathematics,
I tend to put work ofif:
never a lot
sometimes
2. If I think about how I experience my problems with mathematics,
I tend to feel discouraged:
never very much
sometimes
3. When I think about my mathematics future,
I feel: I feel:
confident hopeless/nothing can
improve
4. When I think about the mathematics course I am taking now,
I: I:
would withdraw if I
like it could
5. When I think about how I do mathematics,
383
I:
feel pride in
how I do it
I:
feel ashamed/ _
all the time
6. When I think of my mathematical achievements,
I:
feel satisfied
I feel
discouraged
I:
feel like a complete
failure/
7. While I am doing mathematics,
I can:
make
mathematical
decisions on my
own
I get
confused
I can:
not make
mathematical
decisions on my own
©Jillian Knowles, Lesley College, MA, 2000. Not to be used without permission.
384
4. b. I revised the JMK Mathematics Affect Scales following the study by adding an
eighth scale to gauge responder's sense of himself in relation to the rest of the class. See
chapter 6 for discussion.
8. When I compare myself with others in my mathematics class,
lam:
better at
mathematics
than most of them.
I am about the
same level as most of them
©Jillian Knowles, UNHM, Fall 2001 Not to be used without permission.
I am:
much worse
at mathematics
than most of them.
385
5. As a way of integrating students data and using it for ongoing insight and intervention
in counseling, I placed an individual's scores with the class extreme scores for each scale
and discussed the concepts and implication with participants during counseling sessions.
See chapter 5 and 6 for discussion.
Name
Not at all
U
1
Not at all
U
Survey Profile Summary Sheet
Class Pre/Post
MATHEMATICS FEELINGS
Math Testing Anxiety
Number Anxiety
Date
very much
— ►
very much
— ►
1
lU
Not at all
U
1
Abstraction Anxiety
very much
— ►
III
MATHEMATICS BELIEFS SURVEY
Procedural Math Conceptual Math
4 ►
Toxic /Negative
Learned Helpless
*
1
OVERALL SUMMARY
Negative
Healthy/Positive
4 >
Mastery Orientated
►
5
Positive
■« ►•
©Jillian Knowles, Lesley College, Summer 2000
386
5. One-On-One Mathematics Counseling Evaluation. Participants were asked to respond
to whether and in what ways they had changed in their approach to mathematics learning
during the course and to write about factors to which they attributed any changes. As part
of this they were invited to write a different metaphor if a new one was applicable. See
chapters 6 and 8 for a discussion of responses.
I administered this to individual counseling participants during class posttesting, July 31.
2000 _______^ ^^____
One-On-One Mathematics Counseling Evaluation
Name (optional) Date
Please answer the following questions as honestly as possible from your point if view.
Please be open with any criticisms, questions or suggestions you have. Use the back if
necessary.
1 (a) What was your initial motivation for signing up to meet with me for one-on-one
mathematics counseling?
(b) Did that motivation change? If so, how and why?
2. Did the way you see yourself as a mathematics learner change in any way as you were
doing PS YC/STAT 1 04this summer? If so, in what ways did you change? Did
your math metaphor change? To what? What, do you think, were the main factors in
that change? (e.g., the way the class was taught?, the professor? the testing style?,
meeting with me?, the math content? a personal change? ...a combination?)
3. Do you think your meetings with me affected how you were approaching
PS YC/STAT 104?
If so, in what ways?
4. Do you think your meetings with me affected your success in PSYC/STAT 1 04? If
so, in
what ways?
387
5. With regard to Question 4, how do you.define "success in PSYC/STAT 1 04" ?
6. How. if at all, do you think your overall experience in PSYC/STAT 104this summer
will affect how you will approach your next mathematics-related course or
challenge?
©Jillian Knowles, Lesley College, Cambridge, MA, Summer 2000
388
Appendix C
Class Assessment and Observation Tools
1 . Beliefs Surveys:
a. Modified Mathematics Beliefs Survey, June 5, 2000
b. Modified Mathematics Beliefs Survey, Revised Version, August 2002
c. My Orientation to Mathematics Survey, Short Revised Version, May
2003
2. Mathematics History, Feelings and Evaluations Surveys:
a. Pretest Mathematics Background Survey and My Mathematics Feelings
survey
b. Pretest Mathematics Background Survey Revised Version, August 2002
c. Posttest Course Reflection and Evaluation Survey that preceded the
posttest My Mathematics Feelings survey
d. Class-Link Evaluation.
3. Arithmetic Assessment:
a. Arithmetic for Statistics (AFS) Assessment
b. Arithmetic for Statistics (AFS) Assessment Profile form
4. Statistical Reasoning Assessment (SRA)
5. The Algebra Test and sample scoring sheet
6. Observation Tools:
a. Music Staff Class Interaction Observation Chart
b. Class Layout Observation Form
c. Problem Working Session Interaction Chart (Class 13)
d. Class Summary analysis sheet
389
1 . Belief Surveys:
When Ema Yackel, with counseling psychologist Ann Knudsen (and later
Carolyn Carter) developed and ran a mathematics anxiety reduction course, they aimed at
challenging and changing students' procedural, helpless, and mythical beliefs about
mathematics and themselves and reducing anxiety levels while the students learned
conceptual mathematics usmg a problem-solving, constructivist approach (E. Yackel,
personal communication, January 2 1 , 2000; Carter & Yackel, 1 989).
Yackel created a two-part mathematics beliefs survey as a before and after
instrument for the course. The fu-st part assesses beliefs about mathematics along a
continuum from beliefs about mathematics as conceptual (Skemp's (1987) "relational"
mathematics) through mathematics as procedural (Skemp's "instrumental" mathematics).
In the second part Yackel had included questions that she felt from her experience as a
mathematics educator to be important for a healthy approach to mathematics, questions
she "found interesting" (personal communication, January 21, 2000). Because Carter and
Yackel used Kogelman and Warren's (1978) anxiety reduction approach in their
workshops, I reviewed Kogebnan and Warren's Ust of myths and used in my survey ones
related to the topics I surveyed:
1 Men are better at math than women.
2 Math requires logic, not intuition.
3 You must always know how you got the answer.
4 Math is not creative.
5 There is always a best way to do a math problem.
6 It's always important to get the answer exactly right.
390
7 It's bad to count on your fingers.
8 Mathematicians do problems quickly, in their heads.
9 Math requires a good memory.
10 Math is done by working intensely until the problem is solved.
1 1 Some people have a math mind and some don't.
12 There is a magic key to doing math.
These beliefs can be grouped into three broad categories: Some of these myths
relate to an erroneous or procedural view of mathematics and self (e.g., 2, 3, 5, 6,
8, and 9); some relate to learning style bias and constricted /jerfagogy (e.g., 3, 4,
7); others are embedded in American cultural tradition (1, 11, 12). Yackel's
survey used versions of myths 1, 4, 5, 6, 9, 11 isolated by Kogehnan and Warren.
I added question 19 (Part II) that Kogelman and Warren isolated (cf their #7) and
Yackel had not included.
I also added some perceived usefulness questions (Part II questions 22
and 23) to touch on Sherman and Fennema's (and others') usefulness factor found
to be related to mathematics learning motivation and achievement although
Yackel had already included two usefukiess items. Yackel's survey touched on
male domain and mathematics-related affect factors identified on Fennema-
Sherman Attitude scales and I added a parent/teacher item (Part II #21) that I
believed may be linked to learned helplessness. Yackel and I did not include any
success items (Fermema & Sherman, 1976; Mulhem & Rae, 1998).
391
In order to elucidate student's beliefs around their control of the situation 1 modified
the second part, adding questions related to learned helplessness (Licht &Dweck, 1984)
such as:
7. 1 think my ability to do mathematics can improve. SD D U A SA
(SD means "strongly disagree," SA means "strongly agree,"), that asked whether the
respondent has a fixed trait mathematics theory about herself or not.
A number of questions relating to learned helplessness were already in the first part
since, for example, a belief in mathematics as procedural sees the mathematics as outside
one's control, leading to a helpless response if one does not "recognize" the problem or if
one "forgets" the procedure. I thus created a Learned Helplessness through Mastery
Orientation (LM) Scale within the larger scales. There were fourteen questions that
pertained to student beUefs and behaviors on this continuum. (Since these questions are
embedded in the larger survey I labeled the Learned Helpless/Mastery Oriented questions
as LM and signed them LM- to indicate a Learned Helpless and LM + to indicate
Mastery Orientated belief or behavior respectively. The label LM -and the LM+ signs did
not appear in the student administered version of the Modified Mathematics Belief
Scale.)
During post analysis as I looked for factors that discriminated among the categories
of students I identified (see chapter 7), I found that questions m this Beliefs Survey ( Part
I, Items 4, 7, 9, and 10) that related to achievement motivation contributed to that
identification. Since these questions are embedded m the larger survey I labeled the
performance/learning achievement motivation questions as P/L and signed them P/L - to
indicate a performance achievement motivation and P/L + to indicate learning
392
respectively. The labels P/L -and the P/L + did not appear in the student administered
version of the Modified Mathematics Belief Scale.
Thus my Modified Mathematics Beliefs Systems Survey yielded three measures of
belief and attitude:
1 . mathematics as procedural through conceptual,
2. mathematics learning approaches and attitudes as toxic through healthy, and
3. learned helpless through mastery orientation, and
to provide a starting point for discussion, challenge, and reeducation m the mathematics
counseling setting, and a fourth: performance through learning achievement motivation,
to aid post analysis .
I used this information with each participant by discussing their positions on their
individualized Surveys Profile Summary (see Appendix B), by investigating individual
item responses, and by explaining the concepts involved and their ramifications to
mathematics learning.
I gave the Mathematics Beliefs Systems Survey as a posttest to ascertain if any
changes had been made over the summer. I had opportunity to discuss these changes with
only one participant, Jamie. See chapters 3, 6, and 7 for further discussion.
393
1. a. Modified Mathematics Beliefs Systems Survey administered as a pre- and posttest to
the class on June 5. 2000 and on July 3 1 , 2000 respectively.
Modified Mathematical Belief Systems Survey
Date
Name/Number
All individual responses to this survey will be kept strictly confidential. Your responses
will be used to study relationships among student beliefs about mathematics, past
teaching methods used, effects of mathematics learning assistance and certain other
variables such as mathematics background.
For each item, circle the response that indicates how you feel about the item as indicated
below. PLEASE add your own comments or questions at any point in the Survey.
Strongly
Disagree
Disagree
Undecided
Agree
SD
D
U
A
Parti
1. Doing mathematics consists mainly of using rules.
2. Learning mathematics mainly involves memorizing procedures SD D U
and formulas.
3. Mathematics involves relating many different ideas.
4. Getting the right answer is the most important part of P/L-
mathematics.
Strongly
/\gree
SA
SD D
U
A
SA
SD D
u
A
SA
SD D
u
A
SA
SD D
u
A
SA
5. In mathematics it is impossible to do a problem unless
you've first been taught how to do one like it.
LM-
6. One reason mathematics is so much work is that you need to
learn a different method for each new class of problem. LM-
7. Getting good grades in mathematics is more of a motivation
than is the satisfaction of learning the mathematics content. P/L-
8. When I learn something new in mathematics I often continue
exploring and developing it on my own. LM+
9. 1 usually try to understand the reasoning behind all the rules
I use in mathematics. P/L+ LM+
SD D U A SA
SD D U A SA
SD D U A SA
SD D U A SA
SD D U A SA
lO.Beingable to successfully use a rule or formula in mathematics SD D
is more important to me than understanding why and how it
works. P/L-
U A SA
11 . A common difficulty with taking quizzes and exams in
mathematics is that if you forget relevant formulas and rules
you are lost. LM-
SD D U A SA
SD
D
U
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
394
12. It is difficult to talk about mathematical ideas because all you SD D U A SA
can really do is explain how to do specific problems.
13. Solving mathematics problems frequently involves SD D U A SA
exploration.
14. Most mathematics problems are best solved by deciding SD D U A SA
on the type of problem and then using a previously learned
solution method for that type. LM-
15. 1 forget most of the mathematics Ilearn in a course soon SD D U A SA
after the course is over. LM-
1 6. Mathematics consists of many unrelated subjects.
1 7. Mathematics is a rigid uncreative subject.
18. In mathematics there is always a rule to follow.
19. 1 get frustrated if I don't understand what I am studying
in mathematics.
20. The most important part of mathematics is computation. SD D U A SA
Part II
1 . I usually enjoy mathematics.
2. Mathematics is boring.
3. When I work on a difficult mathematics problem and I can't
see how to do it in the first few minutes, I assume I won't be
able to do it and I give up. LM-
4. When I read newspaper and magazine articles I skip over SD D U A SA
numbers, graphs, and numerical material.
5. I only take mathematics courses because they are required.LM- SD D U
6. I think mathematics is fiin and is a challenge to learn. LM+ SD D U
7. I think my ability to do mathematics can improve. LM+ SD D U
8. Mathematics/statistics, in my experience, has no connection SD D U
to the real world.
9. Mathematics is a subject that some people can do and others SD D U A SA
can't. LM-
10. My overall feeling towards math is positive.
11. Mathematics is used on a daily basis in many jobs.
1 2. Mathematics is easy for me.
SD
D U
A
SA
SD
D U
A
SA
SD
D U
A
SA
A
SA
A
SA
A
SA
A
SA
SD
D U
A
SA
SD
D U
A
SA
SD
D U
A
SA
SD
D
U
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
395
13.1 like to work on hard mathematics problems. LM+
14. Most mathematics courses go too fast for me.
1 5. Mathematics is a subject men do better in than women.
1 6. 1 would like to learn more about mathematics/statistics.
17.1 was better at Geometry than at Algebra.
18. 1 have to understand something visually before I can "get' it
auditorily/verbally
19. 1 think having to use fingers or other calculating SD D U A SA
manipulatives is childish and shows you are not very good
at mathematics.
20. 1 have avoided/delayed taking a mathematics class because of SD D U A SA
my worry about my ability to succeed in it. LM-
21. 1 have had a math teacher/guidance counselor/parent who SD D U A SA
has made me feel I did/do not have the ability to take higher
level math classes.
22. I'll need mathematics/statistics in my future schooling.
23. I'll need mathematics/statistics in my future work.
Other Comments and Questions:
©Adapted, with permission, by Jillian Knowles, Lesley College, Cambridge MA, Summer 2000, for the
purposes of her Doctoral Dissertation Research, from Ema Yackel's 1984 Survey created for a Purdue
University Continuing Education Reducing Mathematics Anxiety course
SD D U
A SA
SD D U
A SA
396
1 . b. Modified Mathematics Beliefs Survey, Revised Version, August 2002, recommended
for use after analysis of dissertation data.
Modified Mathematical Belief Systems Survey
Name/Number Date
Strongly
isagree
Undecided
Agree
Agree
D
U
A
SA
All individual responses to this survey will be kept strictly confidential. Your responses
will be used to study relationships among student beliefs about mathematics, past
teaching methods used, effects of mathematics learning assistance and certain other
variables such as mathematics background.
For each item, circle the response that indicates how you feel about the item as indicated
below. PLEASE add your own comments or questions at any point in the Survey.
Strongly
Disagree
SD
Parti
1 . Doing mathematics consists mainly of using rules. SD D U A SA
2. Learning mathematics mainly involves memorizing procedures SD D U A SA
and formulas.
3. Mathematics involves relating many different ideas. SD D U A SA
4. Getting the right answer is the most important part of SD D U A SA
mathematics. P/L-
5. In mathematics it is impossible to do a problem unless SD D U A SA
you've first been taught how to do one like it. LM-
6. One reason mathematics is so much work is that you need to SD D U A SA
learn a ditferent method for each new class of problem. LM-
7. Getting good grades in mathematics is more of a motivation SD D U A SA
than is the satisfaction of learning the mathematics content. P/L-
8. When I learn something new in mathematics I often continue SD D U A SA
exploring and developing it on my own. LM+
9. 1 usually try to understand the reasoning behind all the rules SD D U A SA
I use in mathematics. P/L+ LM+
lO.Being able to successfiilly use a rule or formula in mathematics SD D U A SA
is more important to me than understanding why and how it
works. P/L-
1 1 . A common difficulty with taking quizzes and exams in SD D U A SA
mathematics is that if you forget relevant formulas and rules
you are lost. LM-
12. It is difficult to talk about mathematical ideas because all you SD D U A SA
SD
D
U
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
397
can really do is explain how to do specific problems.
13. Solving mathematics problems frequently involves SD D U A SA
exploration.
1 4. Most mathematics problems are best solved by deciding SD D U A S A
on the type of problem and then using a previously learned
solution method for that type. LM-
15. 1 forget most of the mathematics I learn in a course soon SD D U A SA
after the course is over. LM-
16. Mathematics consists of many unrelated subjects.
17. Mathematics is a rigid uncreative subject.
18. In mathematics there is always a rule to follow.
19. 1 get frustrated if I don't understand what I am studying
in mathematics. Item broken into two parts and moved to Part II
20. The most important part of mathematics is computation. SD D U A SA
Part II
1 . I usually enjoy mathematics. SD D U A S A
2. Mathematics is boring. SD D U A SA
3. When I work on a difficult mathematics problem and I can't SD D U A SA
see how to do it in the first few minutes, I assume I won't be
able to do it and I give up. LM-
4. When I read newspaper and magazine articles I skip over SD D U A SA
numbers, graphs, and numerical material.
5. I only take mathematics courses because they are required.LM- SD D U A SA
6. I thinlc mathematics is fun and is a challenge to learn. LM+ SD D U A SA
7. 1 think my ability to do mathematics can improve. LM+ SD D U A SA
8. Mathematics/statistics, in my experience, has no connection SD D U A SA
to the real world.
9. Mathematics is a subject that some people can-de understand SD D U A SA
and others can't. LM-
1 0. My overall feeling towards math is positive. SD D U A SA
(new) I rate my ability in mathematics as:
poor; below average; average; above average; excellent (circle one)
1 1 . Mathematics is used on a daily basis in many jobs. SD D U A SA
398
12. Mathematics is easy for me. SD D U A SA
(new) Pt 1, 19 (a) I am able to learn mathematical procedures SD D U A SA
(no score on scale)
(new) Pt I, 19 (b) I do not expect to be able to understand what SD D U A SA
I am doing in mathematics or why
13. 1 like-teworii en-hard on mathematics problems until LM+ SD D U A SA
I master them.
14. Most mathematics coxirses go too fast for me.
15. Mathematics is a subject men do better in than women.
16. 1 would like to learn more about mathematics/statistics.
17. 1 was better at Geometry than at Algebra.
18. I have to understand something visually before I can "get' it
auditorily/verbally
19. 1 think having to use fmgers or other calculating SD D U A SA
manipulatives is childish and shows you are not very good
at mathematics.
20. 1 have avoided/delayed taking a mathematics class because of SD D U A SA
my worry about my ability to succeed in it. LM-
21.1 have had a math teacher/guidance counselor/parent who SD D U A SA
has made me feel I did/do not have the ability to take higher
level math classes.
22. I'll need mathematics/statistics in my future schooling.
23. I'll need mathematics/statistics in my fiiture work.
Other Comments and Questions:
SD
D
U
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
SD
D
u
A
SA
SD
D U
A SA
SD
D U
A SA
©Adapted, with permission, by Jiliian Knowles, Lesley College, Cambridge MA, Summer 2000, for the
purposes of her Doctoral Dissertation Research, from Ema YackePs 1984 Survey created for a Purdue
University Continuing Education Reducing Mathematics Anxiety course
399
I.e. My Orientation to Mathematics Survey, Short Revised Version, May 2003
This is a shorter Learned Helpless-Mastery Oriented focused Revised Version of
Modified Beliefs Survey. This shortened form includes items from the Beliefs Survey that
investigate students' beliefs about (a) the nature of mathematics (conceptual versus
procedural), (b) ability and effort beliefs and attributions items (that include some U.S.
cultural beliefs), (c) student mathematics practices items, and (d) achievement motivation
items, but no usefulness beliefs, or mathematics attractiveness attitudes items. I have
added a category of items (e) to investigate student social practices related to social
learned helplessness in accessing support (see chapter 6 and 7).
MY ORIENTATION TO MATHEMATICS LEARNING
Name Course Date
SD = Strongly Disagree; D = Disagree; N = Neutral; Agree; SA = Strongly Agree
1 . In mathematics it is impossible to do a problem unless SD D N A SA
you've first been taught how to do one like it.
2. One reason mathematics is so much work is that you need to SD D N A SA
learn a different method for each new class of problem.
3. When Ilearn something new in mathematics I often continue SD D N A SA
exploring and developing it on my owti.
4. 1 usually try to understand the reasoning behind all the rules SD D N A SA
I use in mathematics.
5. A common difficulty with taking quizzes and exams in SD D N A SA
mathematics is that if you forget relevant formulas and rules
you are lost.
6. Most mathematics problems are best solved by deciding SD D N A SA
on the type of problem and then using a previously learned
solution method for that type.
7. 1 forget most of the mathematics I learn in a course soon SD D N A SA
after the course is over.
8. When I work on a difficult mathematics problem and I can't SD D N A SA
see how to do it in the first few minutes, I assume I won't be
able to do it and I give up.
9. I only take mathematics courses because they are required.
1 0. I think my ability to do mathematics can improve.
1 1 . I rate my ability to do mathematics as: (circle one)poor| below |average|above lexcellent
{average | |average|
12. Mathematics is a subject that some people can do and Others SD D N A SA
can't.
SD D N
A SA
SD D N
A SA
SD
D
N
A
SA
SD
D
N
A
SA
400
13. I can do mathematical procedures.
14. 1 don't expect to be able to understand what 1 am doing in
mathematics or why.
15. In the past, working hard has not changed how I did in SD D N A SA
mathematics.
16. 1 work on hard mathematics problems until 1 master them. SD D N A SA
17. If I get a good grade in mathematics it is only because I work SD D N A SA
hard, not because I am smart.
18. I delay taking mathematics classes because of my worry SD D N A SA
about my ability to succeed in them.
19. If I got a bad grade in mathematics it is only because I didn't SD D N A SA
work hard.
20. Teachers should not pick out particular students to answer SD D N A SA
questions in class.
21. I would never volunteer to answer a question a teacher asked SD D N A SA
in class even if I knew the answer.
22. If I didn't understand what the professor was saying about a math problem I would
a) ask her in class SD D N A SA
b) go to her office hours to ask her SD D
c) ask a student sitting near me SD D
d) go to the Learning Center to ask a tutor SD D
e) do nothing and hope it would be covered in the next class SD D
23. Getting the right answer is the most important part of SD D
mathematics.
24. Getting good grades in mathematics is more of a motivation SD D N A SA
than is the satisfaction of learning the mathematics content.
25. Being able to successfully use a rule or formula in mathematics SD D U A SA
is more important to me than understanding why and how it
works.
N
A
SA
N
A
SA
N
A
SA
N
A
SA
N
A
SA
1.
SD
D
N
A SA
5
4
3
2 1
2.
SD
D
N
A SA
5
4
3
2 1
3.
SD
D
N
A SA
1
2
3
4 5
4.
SD
D
N
A SA
1
2
3
4 5
5.
SD
D
N
A SA
5
4
3
2 I
6.
SD
D
N
A SA
5
4
3
2 1
7.
SD
D
N
A SA
5
4
3
2 1
8.
SD
D
N
A SA
5
4
3
2 I
9.
SD
D
N
A SA
5
4
3
2 1
10.
SD
D
N
A SA
1
2
3
4 5
11.
P
b/a
av
a/av e
12.
SD
D
N
A SA
5
4
3
2 1
13.
SD
D
N
A SA
14.
SD
D
N
A SA
5
4
3
2 1
15.
SD
D
N
A SA
5
4
3
2 1
16.
SD
D
N
A SA
1
2
3
4 5
17.
SD
D
N
A SA
5
4
3
2 1
18.
SD
D
N
A SA
5
4
3
2 1
19.
SD
D
N
A SA
1
2
3
4 5
20.
SD
D
N
A SA
21.
SD
D
N
A SA
22 e)
SD
D
N
A SA
5
4
3
2 1
23.
SD
D
N
A SA
5
4
3
2 1
24.
SD
D
N
A SA
5
4
3
2 1
25.
SD
D
N
A SA
5
4
3
2 1
401
Score My Mathematics Orientation
©Jillian Knowles, Lesley University.
Fall 2002.
Not to be used without permission
TOTAL:
Learned Helpless
21
Mastery Orientated
42
63
84
105
402
Subscales From MY ORIENTATION TO MATHEMATICS LEARNING
Scale 1 : Mathematics as:
Procec ural
Total from Questions I, 2, 5, 6, 7:
Cone ;ptual
1>
10
15
20
25
Scale 2: My mathematics practices as
Learned Helpless
t
Total from Questions 3, 4, 8, 9, 16, 21:
Score 21. SD D N A SA
5 4 3 2 1
P lastery Orientated
12
18
24
30
Scale 3: My beliefs about my mathematics self as
Detrirnental
13.
SD D N A SA
12 3 4 5
6 12
Total from Questions 10, 1 1, 12, 13, 14, 18:
Score II. p b/a av a/av e
2 2 4 4 3
Constructive
18
24
30
Scale 4: Attributions as:
Unhealthy
Total from Questions 15, 17, 19:
Hea thy
12
15
Scale 5: Social/ Accessing Support as:
Independent
14
Total from Questions 20, 2 1, 22 a, 22b, 22c, 22d, 22e:
Self-reliant
21
2S
Score
20.
21.
22a.
22b.
22c.
22d.
22e.
SD
D
N
A
SA
5
4
3
2
1
SD
D
N
A
SA
5
4
3
2
1
SD
D
N
A
SA
1
2
3
4
5
SD
D
N
A
SA
1
2
3
4
5
SD
D
N
A
SA
1
2
3
4
5
SD
D
N
A
SA
1
2
3
4
5
SD
D
N
A
SA
5
4
3
2
1
Scale 5: Achievement Motivation as:
Performance
Learning
♦■
16
2b
Total from Questions 4, 23, 24, 25:
403
2. Mathematics Feelings Surveys.
The Mathematics Anxiety Rating Scale (MARS) has been normed and is perhaps
the most used in the field (Richardson & Suinn, 1 972). It is long however (98 items),
only yields one measure, but seems to address anxiety in a number of different settings
that it would be helpful to differentiate. Rounds and Hendel did a factor analysis of 94 of
the items of MARS and identified 30 items that they found measured two relatively
homogeneous factors (15 items each) they called "mathematics testing anxiety" and
"numerical anxiety respectively" (Rounds & Hendel, 1 980). Ron Ferguson created a
three-factor instrument from this using the twenty items that loaded most heavily on these
factors (10 each) and adding ten items to measure a factor he labeled "abstraction
anxiety" to make an instrument more applicable to a college setting. Factor analysis
showed that his items did measure a factor different from the two that Rounds and
Hendel identified (Ferguson, 1986). I have slightly changed some of Ferguson's items
and adopted his instrument, calling it Measuring Mathematics Feelings rather than
Ferguson's suggestive "Phobus" (a moon of Mars and the root of the word phobia).
Fergsuson has placed his items in the public domain and I have purchased MARS (adult
form) from Dr. Suinn so that I could use the 10 mathematics testing anxiety and the 10
numerical anxiety MARS items that Ferguson used from Rounds and Hendel' s factor
analysis. Ferguson's instrument is not normed but, as my primary use of it is in the
counseling situation, its ability to quickly assess three pertinent factors of a student's
anxiety, two of which relate to the type of mathematics, thus providing a point of
discussion, made it more usefiil for this study than the full MARS. The principle reason
for assessment in this study was not to compare an individual or group with equivalent
404
people in the wider population, but to compare an individual with herself as she made
changes.
I used this information with each counseling participant by discussing their
positions on their individualized Survey Profile Summary (see Appendix B), by
discussing individual item responses, and by explaining in more detail the concepts
involved.
I gave the Measuring Mathematics Feelings as a post test to ascertain if any
changes had been made over the summer. I had opportunity to discuss these changes with
only one participant, Jamie. Discussing her changes on the instrument highlighted
another aspect of anxiety that was particularly pertinent to her — ^the interaction of social
anxiety with the mathematics learning or performance situation. It was this element of
Jamie's mathematics anxiety that had been reduced over the summer.
I have therefore coded each item of Measuring Mathematics Feelings as:
1 . Position in relation to others:
a. P for primarily public,
b. S for solitary,
c. S/P for solitary with a public component,
d. P/S for public with a solitary component, depending on the relational
setting implied or explicitly referred to in the question, and
2. Setting of activity
a. CI to indicate primarily classroom setting for the activity and
b. Cl/H for an activity that occurs both at home and in the classroom.
405
I have done this to aid analysis with the student responder and for post analysis.
My coding may change in discussion with a student who feels the question situation as
more or less public or more or less solitary. For iurther discussion see Jamie and Me
chapter 6.
Note: Dr. Richard Suinn has given me permission to include here as samples (to
be used by readers only with his permission) ten of the twenty items that I took from his
Mathematics Anxiety Rating Scale (MARS) that form parts I and II of the My
Mathematics Feelings survey. I have deleted the other ten items but retained the above
categorization of them.
2. a. Measuring Mathematics Feelings Pretest Survey, administered in class, June 5,
2000. This survey also includes the Pretest Mathematics Background Survey questions.
Statistics in Psychology PSYC 402, Summer 2000
Please fill in whatever of the foUowmg you feel comfortable sharing. All the data will be
kept confidential. Participation or non-participation in this study will not affect your
grade in this class in any way.
406
Name/Number Date
Major Is this class required for your major? If yes, why do
you think it is required?
Last math class taken before this one Year Grade
What statistics have you studied before?.
What, in your opinion is the relationship between mathematics and statistics?_
What grade do you hope for in this class? What grade do you expect?_
Measuring Mathematics Feelings
Each question below describes a mathematics-related activity or situation. Please
indicate on the scale of 1 through 5 how much you are scared by that mathematics-related
activity or situation nowadays.
N
I. S. Signing up for a math course.*
2.P/ClWalking into a math class.*
3. P/CLRaising your hand in a math class to ask a question.* 1
4. S/P/Cl.Taking an examination (fmal) in a math class.*
5.S.**
6.S.**
7.S/P/C1 Waiting to have a math test returned.*
8.S.**
9.S. Receiving your final math grade in the mail.*
lO.S/P/Cl**
* Sample items from the Mathematics Anxiety Rating Scale. The Mathematics Anxiety Rating Scale
(MARS) is copyrighted by Richard M. Suinn, Ph.D. Any use of the MARS items requires the permission of
Dr. Suinn: suinn(Silam ar.colostate.edu. I retained these items because class and/or individual response
changes on them over the course were notable (see chapter 8 for fiirther discussion)
** The items from MARS used are omitted here as per agreement with Dr. Richard Suinn.
PART II
Not at all Very much
1 .P. Determining the amount of change you should get back from a purchase involving
several items.* 12 3 4 5
tall
1 2
3
4
Very much
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
407
2.P. Listening to a salesperson show you how you would save money by buying his
higher priced product because it reduces long-term expenses.*
12 3 4 5
3.P.** 12 3 4 5
4.S. Reading your W-2 form (or other statement showing your annual earnings and
taxes).*
12 3 4 5
5.P. 12 3 4 5
6.P. Hearing iriends make bets on a game as they quote the odds.*
12 3 4 5
7. P/S.** 12 3 4 5
8. S. ** 12 3 4 5
9. S. ** 12 3 4 5
lO.S. ** 12 3 4 5
* Sample items from the Mathematics Anxiety Rating Scale. The Mathematics Anxiety Rating
Scale (MARS) is copyrighted by Richard M. Suinn, Ph.D. Any use of the MARS items requires
the permission of Dr. Suiiin: suinn(g),lamar.colostate.edu. I chose to retain these items because
they elicited the highest anxiety responses or changed most over the course.
** The items from MARS used are omitted here as per agreement with Dr. Richard Suiim.
408
PART III
Not at all Very much
l.S/Cl. Having to work a math problem that has x's and v's instead of 2s and 3s.
12 3 4 5
2.P/C1. Being told that everyone is familiar with the Pythagorean Theorem.
12 3 4 5
3.S/P/C1. Realizing that my psychology professor has just written some algebraic
formulas on the chalkboard. 12 3 4 5
4. S/Cl. Being asked to solve the equation x^ - 5x + 6 = 0
12 3 4 5
5. P/Cl. Being asked to discuss the proof of a theorem about triangles.
12 3 4 5
6. S/Cl. Trying to read a sentence full of symbols such as: SSxx = Sx^ - (£x)^
N
12 3 4 5
7. P. Listening to a friend explain something she just learned in calculus.
12 3 4 5
8. S, CVH. Opening up a math book and not seeing any numbers, only letters, on an
entire page.
12 3 4 5
9.S. Reading a description from a college catalog of the topics to be covered in a math
course.
12 3 4 5
lO.P, Cl/H. Having someone lend me a calculator to work a problem and not knowing
which button to push to get the answer. 12 3 4 5
The 98 item Mathematics Anxiety Rating Scale (MARS) was developed by Richardson and Suinn in 1972
(Richardson & Suinn, 1972). Ron Ferguson created Phobus(a moon of Mars) by first choosing 20 items
from MARS, ten found by Rounds and Hendel to be related to Mathematics Test Anxiety (Part I) and the
other ten to be related to Number Anxiety (Part 11) (Rounds & Hendel, 1980). Ferguson then added ten
more items to measure what he calls Abstraction Anxiety (Part 111) (Ferguson, 1986, 1998). 1 have slightly
changed items 2 and 3 of Part II and 6 and 7 of Part III.
©JiUian Knowles, Lesley College, Summer 2000.
409
2. b. Following the study I found that there were a number of background details I had
failed to ascertain from participants. To remedy this I sent an e-mail survey to which
most participants responded. The following is my revision of the Pretest Mathematics
Background Survey that I would recommend to avoid these difficulties I encountered.
Revision additions are bolded. Revision deletions are shown as strike throughs.
Statistics in Psychology PSYC 402, Summer 2000
Please fill in whatever of the following you feel comfortable sharing. All the data will be
kept confidential. Participation or non-participation in this study will not affect your
grade in this class in any way.
Name/Number Date
Major Is this class required for your major? If yes, why do
you think it is required?
Last high school math class taken Year (e.g., 1997) Grade (e.g.. A)
Last college math class taken Year(e.g., 1997) Grade (e.g.. A)
Have you ever repeated a mathematics course? If so what course and when?
Are you repeating PSYC/STAT 104?
Have you taken the Brookwood State University mathematics placement test?
If yes, what mathematics course was recommended?
Did you take that course? If yes, when and what grade (e.g., B)
What statistics have you studied before?.
What, in your opinion is the relationship between mathematics and statistics?
What grade do you hope for in this class? What grade do you expect?_
Describe your worst experience in a mathematics class?
How old were you?
Describe your best experience in a mathematics class
How old were you?
410
2 . c. The Posttest Course Reflection and Evaluation survey questions preceded
Measuring Mathematics Feelings Posttest Survey, administered in class, July 31, 2000.
The Measuring Mathematics Feelings part was identical with that on the Pretest Survey
so it is not included here. _____^
Statistics in Psychology PSYC 402, Summer 2000
Please fill in whatever of the following you feel comfortable sharing. All the data will be
kept confidential. Participation or non-participation in this study will not affect your
grade in this class in any way.
Name/Number Date
Please describe how taking this summer course, PSYC 402, was for you?
What did you learn about yourself as a mathematics learner doing this course?_
How much time did you spend on studying/homework per week on the course?
What about the statistics covered in this class is still puzzling to you?_
Will you try to fmd out more about it? How?_
What is the most meaningfiil concept/idea you learned about statistics in this class?
Why?_
What grade did you hope for in this class? What grade are you getting?
How satisfied are you with this grade? With what you learned? With your own approach?
411
2. d. Class-Link Evaluation. This is an evaluation form designed by Learning Assistance
Center personnel for students to evaluate the class-link tutor and the instructor's use of
her. 1 administered this form to the class during posttesting, July 3 1 , 2000. See chapters 8
for discussion.
futor's Name:
Qass Link Evaluation
Course:
Instructor's Name:_
Yean
[ ]FaU
[ ] Spring
1. How often did you see a tutor for this course?
2. Describe how the tutor worked with you. For example, did s/he demonstrate? Ask questions? Read aloud?
. 3. In what specific ways was the tutor helpful?
4. Were there areas where you feel s/he could have been more helpful? If so, how?
5. Did the tutor ever confuse you? If so, how?
Please answer the following questions by circling the most descriptive response.
The scale ranges from strongly disagree (1) to strongly sgiiee (5).
Qass links are an asset to a class.
12 3 4 5
I would prefer to deal with the instructor rather than the
class link.
12 3 4 5
I sometimes feel the instructor used the class link in order
to avoid student contact.
12 3 4 5
I do not like having another student involved in my work.
12 3 4 5
The class link was knowledgeable about the course
content.
12 3 4 5
I was usually able to get in contact with my class link when
a need arose.
12 3 4 5
The class link was reliable in keeping appointments.
12 3 4 5
The class link should have been better informed about the
requirements and materials of the course.
12 3 4 5
\
The class link was easy to talk with.
12 3 4 5
My work iinproved through my association with the class
link.
12 3 4 5
The instructor and the class link communicated
sufficiently.
12 3 4 5
The class link made me feel comfortable in the learning
process.
12 3 4 5
I sometimes felt that the class link was too critical.
12 3 4 5
It is generally helpful to have a class link with whom to
discuss ideas.
12 3 4 5
I feel that the instructor relied too much on the class link.
12 3 4 5
ThxrkyQufcfrtcJimgthetmBto<xmfktethkeidi0ticn Please
add cany other aomTBnts you inish to make aba-it dx tutaifs) and/or
the Learning Center on the lack (fthisjbrm
413
3. I developed the Arithmetic for Statistics instrument during the course as a diagnostic
for participants in response to the type of arithmetical reasoning errors I saw and lack of
the type of arithmetical reasoning that if used might have led students to correct their
errors. I used operation sense and number sense questions suggested by Marolda and
Davidson (Marolda & Davidson, 1994), items used by Liping Ma in her assessment of
elementary teachers' profound understanding of fundamental arithmetic (Ma, 1999a), a
proportional reasoning question, number line scale questions investigating small
(decimal) and large numbers, some normal curve area under the curve and horizontal
scale questions, a coordinate graph question and a pie graph question. Some questions
were open-ended; others closed; some asked for a written explanation. 1 was able to use it
with some of the participants during the course but others completed it during the post-
testing session after the MINITAB project presentations in the second to last class
meeting (July 3 1 , 2000) and others mailed theirs to me. Robin and Brad did not complete
theirs. For fiuther discussion see chapters 6, 7, and 8.
414
This diagnostic was administered during individual counseling sessions with some
students and given to all the students who had not already taken it during the July 3 1 ,
2000 post- test session after the MINITAB project presentations.
Arithmetic for Statistics
Assessment
Name Date
1. When you multiply 61.2 and 3.5 the product is 21.4; 264.2; 2,142 or 214.2?
2. When you divide 12 by 0.12 you get a number smaller than 12? A number smaller than
1? A number larger than 12?
3. When you multiply, you always get a number bigger than the one you started with?
Yes/No Explain.
4. When you divide, you always get a number smaller than the one you started with?
Yes/No Explain.
5. If you earn 10% interest per year on your investment of
$ 1 million, how much would you earn?
$1 billion, how much would you earn?
$1 thousand, how much would you earn?
$1 hundred, how much would you earn?
$10.00, how much would you earn?
Now work out your earnings if the interest rate is 8%
6. Does Va lie between 7/12 and 2/3? Explain?
Explain.
8. Given that 1 is the largest probability you can get, what could you say about a
probability of 0.099? 0.99? 0.119?
9. Is .099 closer to 1 or to 0? Explain.
10. a. Which is a better sale, 2/5 off or 40% off or .04 off? Why?
415
b. In a group of 48 students, 1 out of 8 is of African origin, 2 out of 8 is Latino, and 4
out of 8 is of European origin, and the rest are of Asian origin. How many students are
there from each racial category in the whole group?
1 1 . On this line place the point 9.9.
01 2 3456 789 10
12. On this line, place the points 9 and 0.9 and 0.09 and 0.009
01 2 3456 789 10
13. On this line place the point 0.99
01 2 3456 789 10
14. On this line place the point 4.19
01 2 3456 789 10
15. On this line place the point 3.99
01 2 3456 789 10
416
16. On this line place the point 6.49
0 1
7 8 9 10
17. What fraction of the area under the curve is colored yellow? What percent? What
amount, given that the ,^ — ' [ ~\ total area under the curve is 1 unit"^
What fraction of the area under the curve is colored ^|? What percent? What
amount, given that the total area under the curve is 1 unit?
18. Z = -1.645. Where should it be on this Standard Normal Graph?
-2-1012
19. Fill in the missing number labels for the points on the line;
0.02
417
20. Fill in the missing number labels for the points on the line
2.15
21. a. For the following normal distribution of continuous data, fill in the missing number
labels for the points on the line.
^ = 25 A
CT = 2.5 /
^
\
X
-2-1012 Z
b. How is this standard normal distribution graph related to the one above with ^ = 25
and a = 2.5?
22. Fill in the missing number labels for the points on the line:
0.1
0.6
23. Place the points 1 .85and -1 .85 on this number line.
-2-10123
24. Fill in the missing number labels for the points on the line:
108
108.45
418
25. From the function graph below find the value of Y for which the X value is 5.
Think of a situation in which one variable is related to another in the way shown on the
graph below. Fill in the table with data of all the points shown on the ftinctiop line.
X
X
10
26. On each of the following three number lines think of three different numbers
appropriate for that scale and plot them.
■1000
-500
500
1000
-10
10
-.1 -.05 0 .05 .1
Create a scale on this number line to plot these numbers and then plot them: 25, 150
Create a scale on this number line to plot these numbers and then plot them: 0.04, 0.45,
3.05
Create a scale on this number line to plot these numbers and then plot them: 1 800, 85
27. Block out 0.35 of this pie graph. How much is left?
28. The pie graph below represents the population of 1,500 students at a small liberal arts
college. 35% are freshmen; 25% are sophomores; 25% are juniors and the rest are
seniors. How many are in each class? Show them on the pie graph. 60% of each of the
freshman and sophomore classes are women. 44% of the junior class and 40% of the
senior class are men. Create a chart to show the make-up of the college by gender. Show
it on the other pie or other graph.
© Jillian Knowles, Lesley
University, Summer 2000
420
3. b.
_'s Arithmetic for Statistics Understanding Profile
Course:
Date:
Numb
Statistic
Small
Large
Proper
Place Value/
Operation
Open Ended
Overal
er
al sense
(<1000)
Integer
Integer
Number
Fractional
Sense
Decimal/ Percent
Sense
Sense
Arithmetical
thinking/
1 level
Correc
Number
sense
problem
t
sense
solving
17a(3)
5c.
5a.
6.
a <5
1.
3b
17b(3)
5d.
5b.
7.
8(3)
2.
4b
18
5e.
51(2).
10a.
9.
3
6b
19b(3)
5f(3).
26a(3)
17a(3)
10a
4.
7b
21a(5)
26b(3)
26f(2)
17b(3)
12a
7.
8(3)
21b
26d(2)
12b
9b
25(6)
12c
12d
13
14
15
18
19a(4)
20(4)
22(4)
23(2)
26c(3)
26e(3)
25(1)
26a(3)
26b(3)
26c(3)
26d(2)
26e(3)
26f(2)
27(2)
28(6)
Ta >5
1.
11
16
19b(3)
21a(5)
24(4)
Totals
22 or 16
11
9
9 or 3
la|<5:33 lal>5:15
5
33or
27w/o28
%
Not
Attempted
Comments
©Jillian Knowles, Lesley University, Cambridge, MA, June, 2001
421
4. Statistical Reasoning Assessment. I was aware at the beginning of the study of a
number of factors that that led me to conjecture that the chief aim of PSYC/STAT 104
would not be to change students' misconceptions about probability and statistics or to
develop their statistical reasoning. Instead, I supposed the aim would be to use a
traditional lecture and test approach to have students become familiar with standard
means of sorting and describing data (descriptive statistics) and with recognizing when
and knowing how to use standard parametric and nonparametric statistical analysis to test
hypotheses about populations (inferential statistics), that is, to introduce potential social
scientists to procedures they would later use to do their own research (see chapter 5).
This is not to say that these two aims are necessarily incompatible but it has been
demonstrated that even with deliberate and concerted effort and active student
involvement with data the former aim is very difficult to accomplish, and without such
effort extremely unlikely (Garfield, 1992; Shaughnessy, 1992). In mathematics
counseling, however, I hoped to have opportunities to address mathematical and
statistical misconceptions, so I felt that a pre- and post- statistical reasoning assessment
might reveal changes related to that. Joan Garfield's 20-item multiple choice Statistical
Reasoning Assessment is well constructed and investigates such faulty heuristics as
representativeness (e.g., items 9, 1 1, 14), the gambler's fallacy (e.g., item 10), base-rate
fallacy (e.g., item 12), and correlation as causality (e.g., item 16).
I gave this assessment as a pre-test at the beginning and a posttest at the end of the
course. The Statistical Reasoning Assessment was used with permission its author Joan
Garfield (1998) for purposes of research. See also chapter 8 for discussion of usefulness
of this instrument in this study.
Tlic following pages consist of multiple-choice questions about probability and statistics. Read
tlie question carefully before selecting an answer.
1. A small object was weighed on the same scale separately by nine students in a science class.
Tiie weights (in grains) recorded by each student are shown below.
6.2 6,0 6.0 15.3 6.1 6.3 6.2 6.15 6.2
Tlie students want to determine as accurately as they can the actual wcigiit of this object. Of
the following methods, which would you rcconiniend they use?
. a. Use the most conunon number, which is 6.2.
b. Use the 6, 15 since it is the most accurate weighing.
c. Add up tlic 9 numbers and divide by 9.
d. Throw out tJie 15.3, add up the other 8 numbers and divide by 8.
2. Tlie following message is printed on a bottle of prescription medication:
WARNING: For applications to skin areas
there is a 15% chance of developing a rash. If a
rash develops, consult your physician.
Wliich of tlie following is tlie best interpretation of this warning?
a. Don't use tJic medication on your skin — tliere's a good chance of developing a rash.
b. For application to the skin, apply only 15% of the recommended dose.
c. If a rash develops, it will probably involve only 15% of the skin.
d. About 15 of 100 people who use tiiis medication develop a rash.
e. There is hardly a chance of getting a rash using this medication.
3. Tlic Springfield Meteorological Center wanted to dctcmiine the accuracy of tlieir weatlicr
forecasts. They searched their records for those days when tlie forecaster had reported a 70%
chance of rain. They compared tliese forecasts to records of whether or not it actually rained
on those particular days.
Tlie forecast of 70% chance of rain can be considered very accurate if it rained on:
a. 95% - 100% of Uiose days.
b. 85% - 94% of those days.
c. 75% - 84% of tliosc days.
d. 65% - 74% of tliose days.
c. 55% - 64% of those days.
4. A teacher wants to change tlie seating arrangement in her class in tlie hope tliat it will increase
tlie number of comments her students make. She first decides to see how many comments
students make with the current seating arrangement. A record of tlie number of comments
made by her 8 students during one class period is shown below.
Student Initials
A.A.
R.F.
A.G.
J.G. C.K.
N.K.
J.L.
A.W.
Number of
comments
0
5
2
22 3
2
1
2
She wants to summarize tliis data by computing tlic typical number of comments made that
day. Of the following methods, which would you recommend she use?
a. Use tlie most common number, which is 2.
b. Add up the 8 numbers and divide by 8.
____ c. Tlirow out die 22, add up the other 7 numbers and divide by 7.
d. Tlirow out the 0, add up tlie other 7 numbers and divide by 7.
5. A new medication is being tested to determine its efTectiveness in tlie treatment of eczema, an
inflammatory condition of the skin. Tliirty patients with eczema were selected to participate in the
study. Tlie patients were randomly divided into two groups. Twenty patients in an experimental group
received the medication, while ten patients in a control group received no medication. The results after
two months are shown below.
Experimental group (Medication)
Improved 8
No Improvement 12
Control group (No Medication)
Improved 2
No Improvement 8
Based on the data, I think the medication was:
1 . somewhat effective
2. basicallv ineffective
If you cliose option 1. select the one explanation
below tliat best describes your reasoning.
a. .40% of the people (8/20) in the
experimental group improved.
b. 8 people improved in tlie experimental
group while only 2 improved in the
control group.
__ c. In the experimental group, the number of
people who improved is only 4 less tlian
tlie number who didn't improve (12-8),
while in the control group tlie difference is
6 (8-2).
_ d. 40% of the patients in tlie experimental
group improved (8/20), while only 20%
improved in the control group (2/10).
If you chose optios 2. select the one explanation
below that best describes your reasoning.
a. In the control group, 2 people improved
even witliout the medication.
b. In the experimental group, more people
didn't get better than did (12 vs 8).
c. Tlie difference between tlie numbers who
improved and didn't improve is about tlic
same in each group (4 vs 6).
d. In tlie experimental group, only 40% of the
patients improved (8/20).
6. Listed below are several possible reasons one might question the results of tlie experiment
described above. Place a check by every reason you agree with.
a. It's not legitimate to compare the two groups because there are different numbers of
patients in each group.
b. The sample of 30 is too small to permit drawing conclusions.
c. Tlie patients should not have been randomly put into groups, because the most severe
cases may have just by chance ended up in one of the groups.
d. I'm not given enough information about how doctors decided whether or not patients
improved. Doctors may have been biased in their judgments.
e. I don't agree with any of these statements.
7. A marketing research company was asked to detemiine how much money teenagers (ages 13 -
19) spend on recorded music (cassette tapes, CDs and records). The company randomly
selected 80 malls located around the country. A field researcher stood in a central location in
tlie mall and asked passers-by who appeared to be the appropriate age to fill out a
questionnaire. A total of 2,050 questionnaires were completed by teenagers. On tlie basis of
tliis survey, the research company reported that tlie average teenager in tliis country spends
$155 each year on recorded music.
Listed below are several statements concerning tliis survey. Place a check by every statement
tliat you agree with.
. a. Tlie average is based on teenagers' estimates of what they spend and tlierefore could
be quite different from what teenagers actually spend.
b. They should have done the survey at more than 80 malls if tliey wanted an average
based on teenagers throughout the country.
c. The sample of 2,050 teenagers is too small to permit drawing conclusions about the
entire country.
d. They should have asked teenagers coming out of music stores.
e. The average could be a poor estimate of the spending of all teenagers given that
teenagers were not randomly chosen to fill out tlie questionnaire.
f. The average could be a poor estimate of the spending of all teenagers given tliat only
teenagers in malls were sampled.
g. Calculating an average in this case is inappropriate since tliere is a lot of variation in
how much teenagers spend.
li. I don't agree witli any of tliese statements.
8. Two containers, labeled A and B, are filled with red and blue marbles in the following quantities;
Container Red Blue
A 6 4
B 60 40
Each container is shaken vigorously. After choosing one of the containers, you will reach in
and, witliout looking, draw out a marble. If the marble is blue, you win $50. Which container
gives you the best chance of drawing a blue marble?
a. Container A (willi 6 red and 4 blue)
b. Container B (witli 60 red and 40 blue)
c. Equal chances from each container
9. Which of the following sequences is most likely to result from flipping a fair coin 5 times?
a. H H H T T
b. T H H T H
C. T H T T T
d. H T H T H
e. All four sequences are equally likely
10. Select one or more explanations for tlie answer you gave for tlie item above.
a. Since tlie coin is fair, you ought to get roughly equal numbers of heads and tails.
b. Since coin flipping is random, die coin ought to alternate frequently between
landing heads and tails.
c. Any of the sequences could occur.
d. If you repeatedly flipped a coin five times, each of these sequences would occur
about as often as any other sequence.
____ e. If you get a couple of heads in a row, the probability of a tails on the next flip
increases.
f Every sequence of five flips lias exactly the same probability of occurring.
11. Listed below are tlic same sequences of Hs and Ts tliat were listed in Item 8. Which of tlie
sequences is least hkely to result from flipping a fair coin 5 times?
a. H H H T T
b. T H H T H
C. T H T T T
d. H T H T H
e. All four sequences are equally unlikely
12. The Caldwells want to buy a new car, and tliey have narrowed their choices to a Buick or a
Oldsniobile. Tliey first consulted an issue of Consumer Reports, which compared rates of
repairs for various cars. Records of repairs done on 400 cars of each type showed somewhat
fewer mechanical problems with the Buick than witli the Oldsmobile.
The Caldwells then talked to three friends, two Oldsmobile owners, and one former Buick
owner. Both Oldsmobile ovmers reported having a few mechanical problems, but nothing
major. Tlie Buick owner, however, exploded when asked how he liked his car:
First, tlie fiiel injection went out — $250 bucks. Next, I started having
trouble with tlie rear end and had to replace it. I finally decided to sell it after
tlie transmission went. I'd never buy another Buick.
Tlie Caldwells want to buy tlie car tliat is less likely to require major repair work. Given
what they currently know, which car would you recommend that they buy?
a. I would recommend that they buy the Oldsmobile, primarily because of all the
trouble their friend had witli his Buick. Since tliey haven't heard similar horror
stories about the Oldsmobile, they should go with it.
b. I would recommend that they buy the Buick in spite of their friend's bad
experience. That is just one case, wliile tlie information reported in Consumer
Reports is based on many cases. And according to that data, tlie Buick is
somewliat less likely to require repairs.
c. I would tell them that it didn't matter which car they bought. Even though one of
the models might be more likely than the other to require repairs, they could still,
just by chance, get stuck with a particular car that would need a lot of repairs.
Tliey may as well toss a coin to decide.
13. Five faces of a fair die arc painted black, and one face is painted white. Tiic die is rolled six
times. Wliich of tlie following results is more likely?
a. Black side up on five of tJie rolls; white side up on the other roll
b. Black side up on all six rolls
c. a and b are equally likely
14. Half of all newborns are girls and half are boys. Hospital A records an average of 50 birtlis a
day. Hospital B records an average of 10 births a day. On a particular day, wliich hospital is
more likely to record 80% or more female births?
a. Hospital A (with 50 births a day)
b. Hospital B (with 10 births a day)
c. The two hospitals are equally likely to record such aii event.
■^
30 40 50 60 70 80 90 100
Test Scores: No- Sleep Group
30 40 50 60 70 80 90 100
Test Scores: Sleep Group
Examine tlie two graphs carefully. Then choose from the 6 possible conclusions listed below
tlie one you most agree with.
a. The no-sleep group did better because none of these students scored below 40 and
the highest score was achieved by a student in this group.
b. Tlie no-sleep group did better because its average appears to be a little higlier than
the average of tlie sleep group.
c. Tliere is no difference between the two groups because there is considerable
overlap in the scores of tlie two groups.
d. There is no difference between the two groups because the difference between their
averages is small compared to the amount of variation in the scores.
e. The sleep group did better because more students in this group scored 80 or above.
f. The sleep group did better because its average appears to be a httle higher than the
average of tlie no-sleep group.
\'
15. Forty college students participated in a study of tlic effect of sleep on test scores. Twenty of i |
tlie students volunteered to stay up all night studying the night before the test (no-sleep ■ '
group). Tlie otlier 20 students (the control group) went to bed by 1 1 :00 p.m. on tlie evening
before tlie test. Tlie test scores for each group are shown in the graphs below. Each dot on ; ' :
tiie graph represents a particular student's score. For example, the two dots above the 80 in [ ,
tlie bottom graph indicate Uiat two students in the sleep group scored 80 on the test.
16. For one month, 500 elementary students kept a daily record of the hours they spent watching
television. The average number of hours per week spent watching television was 28. The
researchers conducting tlie study also obtained report cards for each of the students. They
found that the students who did well in school spent less time watching television than those
students who did poorly.
Listed below are several possible statements concerning the results of this research. Place a
check by every statement that you agree with.
a. The sample of 500 is too smallto permit drawing conclusions.
b. If a student decreased the amount of time spent watching television, his or her
performance in school would improve.
c. Even though students who did well watched less television, this doesn't necessarily
mean that watching television hurts school performance.
d. One month is not a long enough period of time to estimate how many hours the
students really spend watching television.
e. The research demonstrates that watching television causes poorer performance in
school.
f I don't agree witli any of these statements.
17. The school committee of a small town wanted to determine the average number of children per
household in their town. They divided tlie total number of children in the town by 50, the total
number of households. Which of the following statements must be tnie if the average children
per household is 2.2?
a. Half the households in the town have more than 2 children.
b. More households in the town have 3 children than have 2 children.
c. There are a total of 1 10 children in the town.
d. There are 2.2 children in the town for every adult.
e. The most common number of children in a household is 2.
f None of the above.
18. When two dice are simulataneously tliiown it is possible tliat one of the following two results
occurs:
Result J: A 5 and a 6 are obtained.
Result 2: A 5 is obtained twice. ''
Select tlie response tliat you agree with tlie most:
a. Tlie chances of obtaining each of tlicse results is equal
b. Tliere is more chance of obtaining result 1 .
c. Tliere is more chance of obtaining result 2. :
d. it is imposible to give an answer. (Please explain why)
19. When three dice are simultaneously thrown, which of the following results is MOST LIKELY ;!
to be obtained? Ji
a.;toM//V:"A5, a3anda6" ^1
b. Result 2: "A 5 three times" ;
c. Result 3: A 5 twice and a 3"
d. All tliree results are equally likely
20. When three dice are simultaneously thrown, which of these three results is LEAST LIKELY to
be obtained?
a. Result I: "A 5, a. 3 and a 6"
b. Result 2: "A 5 three times"
c. Result 3: A 5 twice and a 3"
d. All three results are equally unlikely
432
5. The Algebra Test, adapted from the Chelsea Diagnostic Algebra Test. This test is one
often designed as a diagnostic instrument to be used "both for ascertaining a child's
[aged 12 through 15+ years] level of understanding and to identify the incidence of
errors" by the mathematics research team of the British Social Science Research Council
Program 'Concepts in Secondary Mathematics and Science' (CSMS)(Brown, Hart, &
Kuchemann, 1 985). The research was carried out "broadly within a Piagetian
framework." In particular, the algebra test specifies four levels of understanding of the
algebraic from level 1 at which a letter can be evaluated by recalling an arithmetical
relationship and letter objects to be collected are all of one type, through level 4 at which
the letter is understood at least as specific unknowns or generalized numbers (and in
some cases as variables) and two operations can be coordinated. Sokolowski designed a
fifth level at which the letter is understood as having "a range of numbers (a dynamic
view) that is, as a true variable and coordinated operations can be reordered and
reconfigured" (Sokolowski, 1997, pp. 97-98). Sokolowski's level 5 items have not been
subjected to the rigorous clustering and leveling analysis applied by CSMS to the level 1
through 4 questions, however. Sokolowski also made minor language and setting
changes in the test to make it comprehensible to students in the New England area of the
U.S.
In this study it was expected that some participants' difficulties with the
mathematics could be linked directly to weak mathematical backgrounds, gaps, and
primitive understanding of the algebraic variable. Others' difficulties were expected to be
in spite q/" sound mathematical and algebraic concepts. I beheved that the Algebra Test
would be a valuable tool for helping pinpoint a symptomatic (mathematical) focus for the
433
former, and an explanation (removable by education) other than intrinsic inability for
their troubles. For the latter it could be used as evidence to reflite their negative opinions
of their mathematical functioning that were contributing to their helplessness and poor
achievement. I used the Algebra Test with permission. See chapter 5 and 6 for further
discussion. The Algebra Test was used with permission for the purposes of this research.
See chapter 6, 7, and 8 for further discussion.
^^"^ - T>«che
Algebra Test
Practice Item I
1 . What number does a + 4 stand for if a = 2
-
ifa = 5_
Practice Item 2
2. Fillin the blanks:
Work down the page
X-* 3x X-* x^3
X -* 7x
X —*■
2-> 6 5—8
2 — ►
3 ->
5 -> 4 ->
n —*■
x + S
1 . Fill in the blanks: .t — * .r + 2 x —* 4x
6 ->■ 3 -> _
2. Write the smallest and the largest of these: smallest largest
n + 1, « + 4, n-3, n, - «-7
3 . Which is larger, 2n or /i + 2?
Explain.
4. 4 udded to n can be written as « +4. n multiplied by 4 can be written as 4/i .
Add 4 to each of these: Multiply each of these by 4:
8 n + 5 3/1 8 /i + 5 3«
5. Ua^b =43 If n -246 = 762 If<?+/ =8
a+b+2 = 71-247 = e+f+g=_
6. What can you say about a if a + 5 = 8
What can you say about b if h + 2 is equal to 2h
1 . What are the areas of these shapes?
A =
A =
10
A =
c 2
A =
8 . The perimeter of this shape is equal
to 6 + 3+ 4 + 2, which equals 15.
What is the
perimeter of this shape? P = .
9. This square has sides of length ^.
So, for its perimeter, we can write P = 4^.
What can we write for the perimeter of
each of these shapes?
P =
\s 5
P =
Piut of this ligun: is nol
dnwn. Tliere arc n sides
altogelher, all of length 2.
P =
10. Small apples cost 8 cents each and small pears cost 6 cents each.
If a stands for the number of apples bought
and p stands for the number of pears bought,
what does 8a + 6/7 stand for?
What is the total number of fhiits bought? .
1 1 . What can you say about u if « = v + 3
and V = 1
What can you say about m if m = 3/? + 1
and rt = 4 _
12. If John has y compact discs and Peter has P compact discs, what could
you write for the number of compact discs they have altogether?
13. a + ia can be written more simply as 4a.
Write these more simply, where possible:
2a + 5a =
3a-{b + a)= .
a + 4 +a-4 = .
3a -b + a =
la
^5b =
{a + b) + a =
2a
^5b +a =
(a-
-b) + b =
ia + b) + {a-b) = .
14. What can you say about r if r = ^ + r
andr + i+r=30?
15. In a shape like this you can determine the number of diagonals from one vertex
by taking away 3 from the number of sides.
So, a shape with 5 sides has 2 diagonals;
a shape with 57 sides has diagonals;
a shape with A: sides has diagonals.
16. What can you say about c if c + J = 10
and c is less than d
17. Mary's basic wage is $200 per week.
She is also paid another $7 for each hour of overtime that she works.
If h stands for the number of hoiu-s of overtime that she works, and
if W stands for her total weekly wages (in $),
write an equation connecting W and h..
What would Mary's total weekly wages be if she
worked 4 hours of overtime?
18. When are the following true - always, never, or sometimes?
Underline the correct answer:
A + B + C = C + A + B Always Never Sometimes, when .
L + M + N = L + P + N Always Never Sometjmesi when .
19. a = h + 3. What happens to a if * is increased by 2? ,
/ = 3^ + I What happens to / if g is increased by 2? .
20. Bagels cost b cents each and muffins cost rn cents each.
If 1 buy .4 bagels and 3 muffins,
what does 4b + 3/n stand for?
21. If this equation (x + 1)3 + jc = 349 is true when a: = 6,
then
what value of .r will make this equation , {5x + 1)^ + 5x = 349 , true?
x =
22. Fine point black pens cost $3 each and medium point red pens cost $2 each.
I went to Staples in Salem, New Hampshire, and bought some of each type of pen,
spending a total of $25.
If & is the number of black pens
and if r is the number of red pens bought,
what can you write about b and r?
23. You can feed any number into this machine:
Can you find another machine that
+ 10
—^
X 5
has the same overaill effect?
X — w~ +
Note: The Chelsea Diagnostic Algebra Test (Brown et al., 1985) was used for this research with the written
permission of its publishers.
Name_
Date
Algebra
Levels of Understanding
Course/Semester
Last Math Course
Level 1
Level 2
Level 3
Level 4
i-^v'el ^
5(a)
7(c)
4(c)
3
I'^CL
6(a)
9(b)
5(c)
4(e)
as
7(b)
9(c)
9(d)
7(d)
8
11(a)
13(b)
13(e)
9(a)
11(b)
13(h)
17(a)
13(a)
13(d)
14
18(b)
15(a)
15(b)
20
16
21
22
4/6
5/7
5/8
6/9
Totals
1:
3:
Math Course Taking History
High School
Freshman
Sophomore
Junior^
Senior
College
Freshman _
Sophomore
Junior
Senior
442
6. Observation Tools:
a. Music Staff Class Interaction Analysis Chart
Start Time: End time:
l:S"
i;Q
_S^
S:S
Start Time: End time:
I:S
1:Q
S:Q
.iS.
Start Time: End
time:
I:S
I:Q
S:Q
S:S
TS indicates Instructor's Statement; I:Q indicates Instructor's Questions; S:S indicates Student's
Statement; S:Q indicates Student's Questions
6. b. Class Layout Observation Form
Front Right
443
Front Left
444
6. c. Problem Working Session Interaction Chart: With seating for Class 13
Time
pa
A
R
B
Jillian
Ka
C
M
J
L
M
Note. A = Autumn; B = Brad; J = Jamie; Ka = Karen; Ke = Kelly; L = Lee; Mi = Mitch; Mu = Mulder; P =
Pierre; R = Robin.
^Seated with Pierre front right going counterclockwise to Mitch seated front left.
445
6. d.
Class #
Summary
Course/Semester
Professor
Date
Location
Classroom configuration and individual's locations:
Summary of class
Interactions
Teaching/Learning
Participant/observer issues
Thoughts for next class:
©Jillian M. Knowles, Lesley University, Summer 2000
446
Appendix D
Research Information and Informed Consent Forms
Learning Assistance for a College Undergraduate Mathematics Class
Doctoral Dissertation Research
Jillian Knowles Summer, 2000
My aim in this research is to investigate the role of a number of different types of
learning assistance interventions in helping students who are taking a required
undergraduate mathematics course to not only pass the course, but also to improve their
grasp of and approach to mathematics. In order to do this, in all aspects of the research, I
will be investigating each participating student's own ideas and feelings on his or her
issues around mathematics learning at college.
I, as the participant researcher, will be:
• attending and observing all the Summer 2000 PSYC/STAT 104 classes. This will
include my giving the class two pre and post surveys on beliefs and feelings around
mathematics. [Complete confidentiality is assured.]
• I will be organizing and observing (including audio-taping these sessions) a weekly
study group that will meet before class at 4:30pm on Wednesdays m University
Center, Room 254 [Complete confidentiality is assured.]
• I will be offering Drop-In and by-appointment mathematics tutoring in the Learning
Assistance Center, Greenville campus, Room203. [Complete confidentiality is
assured], and, finally,
• I will be offering one-on-one mathematics counseling (audio-taped, transcribed and
analyzed) to volunteers who want to work on their emotional and mathematical
background issues in order to improve their approach to and achievement in
mathematics in life and in college. There is a possibility of follow-up of individuals
per mutual agreement with me. [Complete confidentiality is assured.]
Please Note:
1 . The personal identity of each participant in this study will be kept confidential. Each
participant will be assigned an assumed name (You can choose!). All analysis and
reporting will use these assumed names and the setting will be disguised.
447
2. Your participation or non-participation in this study will in no way affect your
grade in this course. If you do not wish to participate you will indicate that by not
filling in the class surveys and by not signing the permission sheet at the study-group
and Drop-In. Alternately, on the class surveys, you may be willing to complete them
using a number rather than your name, remembering the same number for the post
tests. In that way, there will be complete data for the class but your individual
responses will not be directly linked to you.
3. You are encouraged to take advantage of any or all of the above learnmg assistance
offerings. Being involved in one does NOT mean you cannot take advantage of
others.
Analysis will involve some quantitative and much qualitative work. Quantitative analysis
of the pre and post surveys using, amongst other tests. Student's t test difference of
means for dependent groups wall be clarified using qualitative data. Qualitative analysis
will involve developing grounded theory. This means that I will have to be continually
noting and setting aside my own assumptions about what are your key issues around your
mathematics learning and listening to and hearing you. I vWll work at producing draft
theories for you to look at and critique, until a grounded theory is developed. This study
will then be reported in my doctoral dissertation for Lesley College, Cambridge MA.
Jillian Knowles,
Local Identification and Contact Information
448
A Call for VOLUNTEERS
I'm looking for people who want to learn how to do their mathematics more effectively.
I need several volunteers who will agree to meet with me regularly (for 1 hour per week
or once every other week for 1 hour per session) for the duration of the Statistics in
Psychology PSYC/STAT 104, Summer 2000 course, to engage in one-on-one
mathematics counseling — working on both your mathematics and also your emotional
issues around mathematics.
If you have issues around mathematics learning that you feel may make it harder to
succeed in this course, maybe this could help. I have worked with college students,
teaching, tutoring, and helping them with their mathematics for many years. In my
doctoral studies I have been looking for better, more effective ways to do this. In this
dissertation research project, I wish to explore these new ways with students who want to
improve how they do mathematics. It will be completely confidential and should lead to
improved ways of doing mathematics.
If you would like to work with me, please respond "Yes" on the attached index card
which I will collect with your surveys. If you want more time to think about it, come to
see me at the Learning Assistance Center, Room 203, Greenville campus, call me at the
Learning Assistance Center at or at home or e-mail me at ^or at .
I DO need to know by Wednesday, June 7, because the course time is so short, so you
could let me know at the study group or m class on June 7.
Jillian Knowles
Local Contact Information
Doctoral student
Lesley College,
Cambridge, MA
449
DISSERTATION RESEARCH INFORMED CONSENT FORM
I, , a) affirm that I have read and Jillian Knowles has
explained the objectives of her research, the procedures to be followed and the potential risks and
benefits. yes/no
b) understand that my participation or
nonparticipation in this research project will not affect my grade in Dr Paglia's Statistics in
Psychology PSYC 402 Summer 2000 class yes/no
c) understand that I am free NOT to respond to
any part of the research yes/no
d) understand that I can withdraw from the
research at any time yes/no
e) affirm that I have volunteered to be involved
in this research of my own free will, without coercion by Jillian Knowles or any other person
yes/no
f) agree that the information I give may be
discussed only with Jillian Knowles' dissertation committee members at Lesley College,
Cambridge, MA, using my name/under an assumed name, and used to write her dissertation for
her doctoral degree. Otherwise all materials and information about me she gathers will be
kept completely confidential — in particular, they will NOT be shared with any persons or
institutions within the University of New Hampshire at Manchester
yes/no
g) assert that if Jillian Knowles chooses at some
time to include any information I give in a published article/book, she may do so with/without my
written/verbal consent yes/no
h) Jillian Knowles will not publish materials
about me without having allowed me to review the relevant part of article/book first yes/no
i) Jillian Knowles will keep audio-tapes and
transcripts of this data in a secure place and will only allow direct access to it by her dissertation
committee. Access by others will only be allowed with my verbal/written permission yes/no
Signed by me this day of ,
20
Name
Address
Phone e-mail
450
Individual Mathematics Counseling
Sign-Up Card
A personalized copy of this 4inch by 6inch response card was given to each student in the
PSYC/STAT 104 class during the second class of the course. All students responded and
returned their cards at that time.
Student Name 6/5/00
I would like to meet with Jillian Knowles for
I I 1 hour per week
1 hour every other week
[Please check one]
beginning this week (if possible) until the end of the summer
2000, PSYC/STAT 104 Statistics course, to do one-on-one
mathematics counseling. [Please
Yes/No circle
Signed by (optional) one]
451
Appendix E
Coding and Analysis
In this study looked at students' sense of mathematics self, their mathematics
internalized presences, and their mathematics attachments to better understand their state of
mathematics functioning or mathematics mental health that would lead to strategic
approaches to helping them negotiate their college mathematics course, I analyzed our
interactions, their behaviors and utterances in class, study group, and in counseling sessions,
their responses to the instruments and their mathematical products in terms of these three
dimensions. I wished to determine if the three dimensions provided a reasonable framework
for understanding their mathematics functioning but also if there were important elements
that could not be understood this way. I wanted to see if students' affective and cognitive
symptoms of dysfunction could be better understood via this framework.
The central task for the relational mathematics counselor in this study was
continually culling relevant data from the voluminous observations and then processing the
data in order to help the student grow in his mathematical functioning and relationships.
That processing as Arlow (1995) and others in the psychoanalytic tradition point out has "an
aesthetic [aspect] that depends on empathy, intuition, and introspection" (p. 144) and a
cognitive aspect that "depends on rationally assembled, methodologically disciplined
conclusions from the data of observation" (p. 44). Since in psychoanalysis as in mathematics
counselmg, life and class events change the context and meaning of observations, the many
variables are impossible to control; hence the need to limit the dimensions of the issues
under investigation in any empirical investigation (Arlow, 1995). The stance that
psychoanalysis and empirical investigation are antithetical is giving way to more and more
nuanced standardized methodologies such as using guided central relationship measures to
guide the therapy more systematically and allow for more empirical evaluation of
452
techniques, their underlying rationales, and outcomes (Luborsky & Luborsky, 1995). For my
purposes here the use of a modified guided central mathematics relationship measure to
guide the counseling and, via analysis, to trace its path retrospectively, seems appropriate.
My task in tracing the path of mathematics counseling and analyzing its efficacy is
in some senses easier than the task of the psychoanalyst. Since the central symptomatic
focus for each tutee is mastering the mathematics course, his mathematical behaviors in the
classroom (see Table E2 and Table E3) and in the counseling sessions and his mathematical
products for the course: homework, projects and especially exams(see Table E4), provide
central data for charting his progress. I was also able to follow targeted mathematics
affective symptoms and their changes through pre and post feeling and belief/attitude
surveys. It was the relational changes that I hypothesize underlie his mathematical cognitive
and affective changes that I need a guided central relationship measure to gauge (see Table
El). In this also I have an advantage over the psychoanalyst, who only sees the client in the
counseling setting, since I see the student not only in the counseling setting but also in the
central forum of his present mathematics life — the classroom^ — so what he reports in the
counseling session of his experiences in class I and the instructor also observe (see Table
El).
On the other hand the major disadvantage in trying to formulate a student's central
relational pattern or conflict lies in the fact that the central focus in the mathematics
counseling is on the student doing mathematics rather than on his relational, albeit
mathematics relational, conflicts. This means that relatively little time is spent in a
mathematics counseling session in talking about his past and present mathematics
relationships. Therefore there is substantially less direct student-initiated relational data from
the sessions, especially relational data with respect to the counselor. This may be related to
the predominance, from the student's perspective, of the tutor role over the coimselor role in
453
this setting and his concomitant expectations of and desires for what mathematics
counseling would entail — that is, mathematics tutoring.
The basic organizational unit I used to do identify a student's central mathematics
relational pattern was the relational episode. This involved first locating and identifying
narratives' (called relationship episodes) and then reviewing the relationship episodes and
extracting the central relationship theme from them (Luborsky & Luborsky, 1 995). Three
components that Luborsky (1976) finds prominent in these relational episodes are: what the
patient wanted from other people; how the other people reacted; and how the patient reacted
to their reaction. Other researchers include disguised allusions, acts of self, expectations of
others, consequent acts of others towards self and consequent acts of self towards self (cf
Gill and Hof&nann, 1982; Schacht et al., 1984)
My adaptations for mathematics achievement settmg are: Central relationship pattern
with respect to self:
1 . what student wants/expects from self;
2. student's achievements;
3. how other people reacted to student's achievements;
4. how the student has reacted to others' reactions to his achievements; and
5. mathematicsaspart of self
Central Relationship Pattern with respect to internalized presences:
1 . what student wanted/expected from other people;
2. how other people reacted to student;
3. how the student reacted to their reactions; and
4. mathematics as internalized other
454
Central (interpersonal) Attachment Pattern:
1 . what student wants/expects from other people;
2. how other people react to student;
3. how the student reacts to their reactions; and
4. attachment to mathematics
A crucial concern in understanding a student's central relationship pattern from a
relational conflict perspective is the understanding that the student is dealing with patterns of
conflict, parts of which he is conscious and parts of which he is unconscious. This means
that his verbal statements and behaviors will likely include ones that appear to and some that
do contradict his basic wishes. In order to identify a student's central relationship pattern
given this difficulty I adopted the following four principles developed by Luborsky and
Luborsky(1995):
1 . The central conflictual relationship theme may have an opposite conflicting less
conscious theme,
2. A wish frequently expressed may have a less frequent (but perhaps more intense)
version of that wish in reduced awareness,
3. Instances of denial are likely to point to content that is in reduced awareness,
4. If a student refers to a history of difficulties with awareness this might infer present
similar difficulties that he does not acknowledge, (p. 345)
455
Table El
Analysis scheme for Counseling Session Data: Student's Mathematical Relationality
Mathematics Self
Mathematics
Mathematics Interpersonal
Internalized Presences
Environment
Relational
Mathematics Identity
Object relations
Attachments
Assessment
Central Relationship Patterns
Cenfral Relationship
Central Relationship
Categories
1. With self
Patterns
Patterns
2. With mathematics
1 . With internalized
1 . With others now
others
2. With mathematics now
2. With mathematics
Central what a student wants/expects
relationship from self; a student's
measure achievements; how other people
categories reacted to student's
achievements; how the student
reacted to others' reactions to his
achievements; and mathematics
as part of self
Metaphor Survey"
what student
wanted/expected from other
people; how other people
reacted to student; how the
student reacted to their
reactions; and
mathematics as internalized
other
Metaphor Survey
what a student's
wants/expects from other
people;how other people
react to student; how the
student reacts to their
reactions; and attachment to
mathematics
Metaphor Survey
Mathematics Testing or mathematics anxiety
Affect as extinction anxiety: History (re
teacher 's mirroring and
invitation to idealize), Feelings
Survey, Metaphor Survey; Test
Taking behaviors. Mathematics
depression, learned helplessness
as empty depression;
mathematics grandiosity
History (mirroring and
invitation to idealize). Beliefs
Survey, JMK Affect Scales; class
and counseling behaviors
Testing or mathematics
anxiety as social anxiety,
adjustment disorder, PSTD,
phobia, ...
Feelings Survey, Metaphor
Survey, History (re critical
incidents); classroom
behaviors versus classroom
" reality. " Mathematics
depression related to a
severe mathematics super
ego/ internal saboteur
Testing or mathematics
anxiety as separation (from
teacher or mathematics)
anxiety
History (re separation, loss,
change)
Feelings Survey Metaphor
Survey; teacher and
counselor related behaviors.
Mathematics depression
related to separation or loss
Mathematics Elementary Mathematics:
Cognition PSYC/STA T 1 04 Exams,
Arithmetic for Statistics
Assessment; High School
Mathematics: HS courses/
grades. The Algebra Test; Intro
Statistics: PSYC/STAT 104
Exams
Elementary Mathematics:
PSYC/STA T 104 Exams,
Arithmetic for Statistics
Assessment; High School
Mathematics; HS courses/
grades. The Algebra Test;
Intro Statistics: PSYC/STAT
104 Exams
Elementary Mathematics:
PSYC/STA T 104 Exams,
Arithmetic for Statistics
Assessment; High School
Mathematics: HS courses/
grades. The Algebra Test;
Intro Statistics: PSYC/STAT
104 Exams
Counselor's
counfertransference
" The items in italics are instruments, protocols, mathematics products, demographic and behavioral data
that were used in conjunction with audiotaped counseling session data to develop a profile of a student's
mathematics fimctioning and his central mathematics relational pattern.
456
Table E2
Analysis of Lecture Session Student Exchanges with Instructor
Student Questions
Student Answers
Student Comments
Timing^
Timing
Timing
Relevance
Accuracy
Relevance
Topic:
Topic:
Topic:
1. current content:
1. current content:
1. current content:
mathematics;
mathematics;
mathematics;
application;
application;
application;
personal
personal
personal
2. course strategy
2. course strategy
2. course strategy
3. grading
3. grading
3. grading
Level of certainty:
Level of certainty:
Level of certainty:
1 . affective,
I . affective,
1 . affective.
2. cognitive
2. cognitive
2. cognitive
Frequency
Frequency
Frequency
Development
Development
Development
Implications re student's
Implications re student's
Implications re student's
1. mathematics self
1 . mathematics self
1 . mathematics self
2. internalized
2. internalized
2. internalized
presences
presences
presences
3. attachments: to
3. attachments: to
3. attachments: to
teacher; to
teacher; to
teacher; to
mathematics
mathematics
mathematics
Implications re student's
Implications re student's
Implications re student's
auditory processing
auditory processing
auditory processing
Central relational conflict or theme
''Timing is judged in terms of the extent to which the student's verbalization is linked in a
timely manner with the instructor's utterance. For example, on a number of occasions
Robin answered Ann's question with the correct answer to a. previous question; her
timing was off.
Table E3
457
Analysis of Student's Problem Working Session Behaviors
p
Topic/Task:
Seated beside:
Tools:
Interaction
Interaction
A
1. Left
1 . text
with
with
R
2. provided by
Instructor
researcher
A
2. Right
instructor
M
3. student
E
aids:
T
E
R
S
calculator,
notes,...
Cognitive
Peer relational
Mathematics
Student-
Student-
preparation:
behaviors:
learning style
teacher
researcher
I . Background
I. social
Behaviors:
relational
relational
2. Homework
learner '
I. Analytic
behaviors:
behaviors:
2. voluntary
(Mathematics
I. Secure
1. Secure
loner"
Learning
attachment ■*
attachment''
3. involuntary
Style I)"
2. Insecure
2. Insecure
loner"
2. Global
(Mathematics
Learning
Style II)''
3. HarmonicI'
4. Harmonic ir
avoidant ''
3. Insecure
dependent "*
4. Insecure
disorganized'^
avoidant '^
3. Insecure
dependent ''
4. Insecure
disorganized''
Changes
R
E
Implications
Implications
Implications
Implications
Implications
L
Re:
Re:
Re:
Re:
Re:
T
1. Self
1. Self
1. Self
1. Self
1. Self
O
2. Internalized
2. Internalized
2. Internalized
2. Internalized
2. Internalized
N
A
presences
presences
presences
presences
presences
L
I
T
Y
3. Attachments
3. Attachments
3. Attachments
3. Attachments
3. Attachments
Central relational conflict
or theme
" I designated as social learners in this group Lee, Mulder, and Robin because they always chose to work with
people beside them if they were willing; I designated as involuntary loners Pierre and Jamie because they
seemed to be working alone not by choice but because of personal issues; I designated as voluntary loners
Autumn, Catherine, and Karen because they showed no interest in working with others (except Ann or me).
Catherine was willing to help someone if he asked (e.g., Mulder) but never asked to check with anyone. ""See
Davidson, 1983; Witkin et al, 1967 and chapter 2 discussion. '^ Harmonic I balance of Mathematics Learning
style I & 11 more I; Harmonic II balance of Mathematics Learning style I & II more II. See Krutetskii, 1976,
and chapter 2. '' See Bowlby, 1973 and chapter 2.
458
Table E4
Protocol for Analysis of Exam Question Solutions
The
question
Pre-Exam:
Class
treatment,
student
reaction and
Counseling
preparation
Student's
out of class
preparation
Errors
Trouble-
shooting
efforts:
Instructor
Grading
Post-exam
Counseling
Defining
the
problem:
concepts
1 . understanding
the question
2. misconceptions
3. confusions with
4. other
1. affective
2. cognitive
Planning
the
solution:
procedures
1 . formula sheet
2. data-symbol
linking
3. choice of
formula
4. strategy/
layout
1. affective
2. cognitive
Carrying
out the
solution:
algebra
1 . multiple uses of
letter symbols
2. algebra
1. affective
2. cognitive
Carrying
out the
solution:
arithmetic
1. arithmetic
2. order of
operations
1 . affective
2. cognitive
Conclusion
Checking
and
reporting
1 . reasonableness of
solution
2. units
3. interpretation of
solution
1 . affective
2. cognitive
Individual
Patterns
Class
Patterns
' These narratives are extracted not only fi-om direct student reports but also from discussion of classroom
interactions, metaphor and survey responses and mathematics focused interactions with the counselor.
459
Appendix F
Researcher and Student Seating
Ann in her final interview (Interview 3) noted that she had not taught a class
before in which there seemed to be so much change in seating arrangement. She
wondered if the different physical arrangement of the classroom from the usual rows of
individual chairs was a factor. In previous classes she had taught students had mostly
maintained the seating positions they had taken in the first class, changing only to sit in
seats adjacent to the original. We both also wondered about the effect of my choices of
seating on the choices by the students.
Researcher seating. I had struggled with my seating choices throughout the
course (see Figure FI). In my role of researcher, I wanted to be as much an observer and
as little a participant as possible. As the class progressed, and I realized that I was
No. of Which
Position Times Classes
□
■
(Class 14)
(Class 8)
(Class 11)
(Class 17)
(Classes 6,7,& 12)
(Classes 1,2,&3)
(Classes 13, 15,16,18,&.
j/o
tjor
left
DDHna
front
chalkboa
D
D
SX-
nnannn
right
back
Figure Fl. Jillian's seating positions for the PSYC/STATS 104 course, second floor.
Riverside Center, Brookwood State University, Summer 2000
choosing to sit only on the right side, albeit in various positions, I decided that I needed a
perspective of the class and students from the left. Eventually, towards the end of the
460
course, I decided that a better perspective of the whole class might come from the end of
the table beside Karen (see Figures 1 and 2).
Student seating. Karen (12 times) and Catherine (10 times) (left back comer, see
Figure 12) were the most consistent in their choice of seating of the class, although
Autumn (9 times) mostly right middle and Jamie (9 times) left middle were almost as
predictable. Brad (right back comer), Robin (right side mostly next to Brad in the right
back comer), Mitch (left middle to left front), and Lee (mostly left side beside Mitch)
each had her or his discemable partem (7 to 8 times each). Pierre usually sat close to the
front evenly on left and right sides — ^presumably to maximize the use of his tape-recorder
for each lecture. Mulder showed the most inconsistency, perhaps because he was usually
a few minutes late to class (he had to transport his mother) so he had to find an empty
space — he was more often on the right or at the back (See Figure F2 for a most
representative seating arrangement).
B Mitch [^ Karen Door/
B Lee Q Brad
■ Jamie V^ Robin
E^ or lH Catherine front chalk boaid
^m Autumn
left
[— 1 or
Pierre
DBD
nnannn
n
n
right
tack
461
Ellen only came to the first class and sat on the left at the front. Kelly dropped
the class the day of the second test (the 9"" class). She sat towards the front either on the
left or the right as she tended to be late and there was usually a seat or two unoccupied
towards the front. Floyd came to only four classes in this room and sat at the back each
time. This data contrasts somewhat with Arm's perception of constant change in student
seating. Most students were relatively consistent in their seating choices or patterns.
462
Appendix G
PSYC/STAT 104 Instructor Syllabus and Selected Handouts
PSYC/<T'AT- Statistics in Psychology
j 0%
Summer 2000
M/W 6:00 -8:20 May 31" - August 2"''
Professor: Aia v^ 'Pov-i'-e^i^ Ph.D.
Office Information:
Phone: e-mail:
Summer Office Hours: by appointment
Due to several advisory responsibilities at and
my summer schedule is extremely inconsistent. PLEASE
DO NOT take this to mean I am inaccessible, just that my schedule
fluctuates from week-to-week. Please, feel free to contact me anytime
to schedule an appointment.
Required Text Book:
Pagano, R.R. (1998). Understanding Statistics in the Behavioral Sciences - Fifth
Edition. Pacific Grove, CA. Brooks/Cole Publishing Company.
Course Overview:
Psyd/^hxf'iO^. - Introduction to Statistics in Psychology will provide a
comprehensive overview of the basic statistical concepts utilized in psychological
research. Many, if not all, of these concepts are utilized in other disciplines as
well. In order to comprehend statistics, it will be necessary to initially learn the
material at a conceptual level. Calculations and computer modules will be
required to advance your understanding of the statistical concepts.
Computers are an essential part of the psychology program at and are
extensively used in the field psychology. These computer assignments are
intended to illustrate the ease that computer statistical analysis provides with
large data sets. Although there will t»e some initial frustration, as you become
familiar with the computer program itself, you will witness the convenience that
computers provide to statisticians. Familiarity with Mini-tab, a computer program
available for statistical analysis, is a university-wide requirement for this course.
Mini-tab is available on the mainframe computer. This version of Mini-tab is
somewhat archaic, but will provide the necessary exposure to statistical analysis
on the computer.
Course Goals:
1. To provide a basic overview of the statistical concepts utilized in empirical
research.
2. To facilitate a comfortable relationship with statistical concepts.
3. To increase the conceptual understanding of the various statistical
analyses utilized in research.
4. The increase understanding and critical thing about the statistics that the
media presents.
5. To increase familiarity with statistical calculations and computer analysis.
Course Requirements:
1. You are expected to attend class on a regular basis. You are expected to
read the text and compute statistical calculations in preparation for class 7
lectures and the tests.
2. Tests: There will be a total of 5 tests. Each of the first 4 tests will
be worth 20% of your final grade. The 5"' test is a conceptual
comprehensive exam and is worth 10% of your final grade. You will have
the full class time to complete the exams. All 5 of the combined exam
grades will determine 90% of your final grade.
3. Computer Assignments: You will be required to complete one
computer module independently for 2% of your final grade. Additionally^
you will be required to complete an additional computer module with a
few of your classmates. In addition to completing the module as a group,
you will be required to present this information to the class. The group
presentation and paper are worth 8% of your final grade. Handwritten
papers will not be accepted! Computer Assignments must be typed!!! If
papers are turned in after the due date, you will loose one letter grade for
each date that the paper is late.
The group presentation and paper should include:
• All members of the group contributing to the oral presentation.
• A review of the method that the module illustrates.
• A visual display of the entire statistical analysis.
• An overview of the "Interpretation" portion of the assignment.
• A brief written commentary (specific form will be distributed to the
class) of the efforts of the group (ex. meetings, attendance at
meetings, designation of tasks, etc.) completed separately by each of
the group memt)ers. If any member does not contribute to the group
assignment, it will be reflected on that individual's grade for this
assignment
4. Homework: "Questions & Problems" are located at the end of each
chapter in the text. You are not required to turn in the homework to me,
but be sure to do these assignments, as they are essential to your
understanding of the course material. These "Questions and Problems"
provide an excellent review for the tests.
UNH Grading Scale:
Final grades will be based on the following scale:
B+ = 87% - 89%
C+ = 77% - 79%
D+ = 67% - 69%
A = 93% ■
-100%
A-
= 90% •
■ 92070
B = 83% ■
- 86%
B-
= 80% •
- 82%
C = 73% ■
•76%
C-
= 70% ■
- 72%
D = 63%
- 66%
D-
= 60%
- 620/0
F= 0%-
S9%
Course Policies:
A calculator with a square root key is required for this course.
Rescheduling/ Missed Exams:
With good reason & advanced notification, you may take an exam earlier
than the scheduled date. If you miss class on the date of the exam, you
will be required to take a comprehensive exam (conceptual & calculations)
at the end of the summer session in place of the missed exam - no
exceptions. If you have not missed an exam, you may take the
comprehensive exam to replace your lowest exam grade. If you choose
this option and the comprehensive exam grade is lower than your lowest
exam grade, the grade will not be averaged with your final grade.
Absence on an exam date mav be subject to the approval of the Dean of
the College
Policy on Cheating / Plagiarism:
DO NOT CHEAT OR PLAGIARIZE!!!
Any student caught cheating or plagiarizing will be penalized in
accordance to the policies stated in the 1999-2000 UNH Student Rights,
Rules, and Responsibilities. (NO EXCEPTIONS!)
Students with Disahiiities:
If you have a disability that requires soecial accommodations, you must
obtain written documentation from Sv.'fe:' c.
Course Schedule (All Dates Are Subject to Change!)
••••• Computer Orientation - June 14"'*****
Assignment
Scheduled Date
Chapters 1-5
Exam #1 - Monday - June 12*
Computer Orientation
Tentatively Scheduled
Wednesday- June 14*
Chapters 6-9
Exam #2 - Wednesday - June 28*
Work independently
or in groups on
computer projects!!
Monday - July 3'"
Chapters 10 - 14
Exam #3 -Monday -July 17*
Chapters 15, 16, & 18
Exam #4 - Wednesday - July 26*
Minitab Projects &
chapter 19 review
Presentations - Monday - July 31"*
Chapter 19
Conceptual Comprehensive Exam #5 -
Wednesday - August 2""
50 55 60 65 70 75 80 85 90 95 100
Column 1
= 77
^ \2.0S
2e^\^^ £^^i^^1
Ch. 15
1^,6,8. 9. 10, 11, 12, 15, 16 & 17(formula used in class)
Ch.16
1, 2, V^l(!dvX^^<^9SiQ<S't<iiS!^^e,<^^)
Ch.18
1-3,5.6,8-10,12,13,15,22
Symbols:
a
X^obt
Fobt
X^crit
Fcrit
fo
Sw
Practice Problem #1
To determine the effect of Ginko-Biloba on short-term memory, an
experimenter gave a list of 50 words to two groups. One group has
received Ginko-Biioba, the other received no Ginko-Biloba. Each group is
allowed to study the list for 5 minutes and then asked to recall as many
words as possible. The numbers below represent the number of words
recalled. Use the Mann-Whitney U to evaluate the results (a=.05 i tan).
Control Group =
5
9
17
3
Experimental Group =
1
8
28
20
18
Rank
Score
_
Group
Rank
Procedure for Testing the Null Hypothesis p. . /
1. State the Null Hypothesis (symtjcis ^^/or words) yj ^.Jt;]^ ^ ^
2. State the Alternative Hypothesis (sy misols &/or words) J^r » vv ^^-^
directional (1 tailed)} non-directionaiX2 tailed) "^ . /^qj^
3. Choose an alpha level / decision rule
4. Determine the most appropriate statistical analysis
5. Compute calculations
6. Make a decision (reject or fail to reject the null hypothesis)
7. Draw conclusions in tiie context of the problem
For the following, determine null/alternative, alpha, & the most appropriate statistical
analysis:
1 An ecologist suspects that kingbirds found in Switzerland have more feathers than the
rest of the kingbirds in the riatfefi An exhaustive worldwide study was conducted last
year to assess the number of feathers on all of the kingbirds in the-aatioa, ^^ <=^'lA
2. An investigator conducts an experiment to determine the importance of frequency of
psychotherapy on depression for men and women. Men and woman suffering depression
are randomly assigned to one of three frequencies of treatment conditions (3X per week,
IX per week, IX per month). The depression scores are assessed after 6 months.
3. Prior to the superbowl, a survey was conducted to determine whether there was a
relationship between gender and team preferences (Tampa Bay Buccaneers or New
England Patriots).
4 A health educator wants to evaluate the effect of a dental film on the frequency with
which children brushed their teeth. Eight children were randomly selected for the
experiment. First, a baseline of the number of times children brush their teeth in a month
was established. Next the children are shown the dental film. Again, the numbers of teeth
brushings are recorded for a month.
5. A student at Midwest college is interested in whether women or men take more time in
the shower. 8 women & 8 men are randomly selected to determine weekly shower time.
6. A traffic safety officer noticed that he was giving more speeding tickets to older people,
so he conducted an experiment to determine whether there is a relation between people's
ages & driving speeds.
7. A professor of women's studies is interested in determining if stress affects the
menstrual cycle. Ten women are randomly sampled & divided into two groups. One of
the groups is subjected to high stress for 2 months, while the other group lives in a stress-
free environment for 2 months. The professor measures the menstrual cycle for all of the
women.
8. A researcher believes that women in her tovvTi are taller today than in previous years.
The researcher compares her data to that of a local consensus collected 20 years ago.
9. An investigator conducts an experiment to determine the importance of frequency of
psychotherapy on depression. Subjects suffering depression are randomly assigned to one
of three frequencies of treatment conditions (3X per week, IX per week, IX per month).
The depression scores are assessed after 6 months.
10. During the past 5 years there has been a consistent inflationary trend in milk prices.
You have yearly average in milk prices for the past year. You are an elementary school
administrator and need to predict the cost of milk in 2005.
471
APPENDIX H
Descriptive and Comparative Data for PSYC/STAT 104 Class of Summer 2000
Table HI
Students ' Expectations & Hopes in Relation to Effort, Grades and Scores, Summer 2000
Pre:
Grade
Hoped
For?
Mitch 4
(II B)
Pre:
Grade
Expected
in
course
(June 5)
Grade
in
Exam
#1
Post:
Post: Grade
Grade So
Hoped Far?
For? (July
31)
Time on Prior High
H'wlc School
per Highest
week Math
Algebraic Final
Variable Grade in
Level" PSYC/STAT
104
Autumn 4"
(II Bf
A
A
B^
(E=Rr
A-
A or
A^
2hrs/wk
Algll.Disc/Stats
(Finite Math in
college)
4/5 [50]
A
(E=R)''
Brad 4
(III A)
A
A
c
(E>R)
p/c-]
Algebra ?
(Fin
PSYC/STATS
in college)
WP
(E>R)
Catherine
(I)
A
A
A
(E=R)
A
A
5hrs/wk
?(Calclin
college)
5 [50]
A
(E=R)
Ellen
Floyd
(111 A)
A
B
F
(E>R)
[F]
?
AF
(E>R)
Jamie 5
(II A)
B
C
A
(E<R)
C
B
5hrs/wk
Precalculus
(D* in Psyc
Stats in college)
4 [41]
B*
(E<R)
Karen 5
(III B)
B
C
(E>/=
R)
C"
B
6-7hrs/wk
Algebra 11
(Fin
PSYC/STATS
in college)
2 [26]
B
(E<R)
Kelly 3
(III B)
B"
B"
F/D"
(E>R)
[F/D-]
Algebra 11
AF
(E>R)
Lee 6
(II A)
A
A
C
(E>R)
B
B
20min/wk
Precalc/calc
(Finite math,
college)
4 [45]
A"
(E=R)
(E>/=
R)
3-5hrs/wk Alg I, Geom 4 [43]
(repeat),
(Fin
PSYC/STATS
in college)
B
(E>/=R)
Mulder 5
(III A)
B
B
D
(E>R)
B
?
3hrs/wk Algebra 11
2 [25]
B
(E=R)
Pierre 8
(II A)
A
B
(E>R)
B
B
17hrs/wk College prep
4 [44]
B^
(E=R)
Robin 3
(I)
A
A
B"
(E=R)
A
B
lOhrs/wk "College"
Alg. (inHS)
A"
(E=R)
Notes: ^Levels of understanding of the algebraic variable on the Algebra Test from 0 the least,
through 5 the most sophisticated (see Appendix C). The number in the [ ] is the number of items
correct out of 53. ''Names of individual counseling participants are bolded and the number beside
their names is the number of their counseling sessions. 'Category Type number (see chapter 7).
''E=grade expectation, R=grade reality; = less than one grade discrepancy; < or > more than one
grade discrepancy
472
Table H2
Student Tier (Tobias) and Category (Knowles) in Relation to Class Rank after Exam #1
and Pre- and Post-Statistical Reasoning Assessment (SRA) scores.
PRE-Statistics Reasoning
Assessment [SRA]
(6/12/2000)
POST-Statistics Reasoning
Assessment [SRA]
(7/31/2000)
Student in
order of
score (Is)
on pre
SRA
Tobias' Tier
level/Knowles
Type
Exam
#1
class
Rank
Number of
Is (correct
reasoning)
Number of Os
(misconceptions)
Number
of Is
(correct
reasoning)
Number of Os
(misconceptions)
Catherine
1 ^ Tier/Category 1
V
13
4
13
(+0)
4
(+0)
Robin
1" Tier/ Category 1
3rd
12
6
Kelly
Unlikely/ Category
III, type B
11'"
11
6
Floyd
Unlikely/ Category
III, type A/B?
12'"
10
6
Mulder
/ Category
III, type A
g't>
9
3
7
(-2)
6
(+3)
Autumn
Utilitarian/
Category II, type B
4«i>
9
4
9
(+0)
4
(+0)
Brad
/ Category III,
type A
7'^
9
4
Mitch
Utilitarian/
Category II, type B
5,h
9
7
9
(+0)
4
(-3)
Pierre
ES0L2°'' Tier/
Category II, type A
gth
7
8
7
(+0)
9
(+1)
Jamie
2°" Tier/ Category
II, type A
'^nd
7
10
7
(+0)
9
(-1)
Lee
2°" Tier/ Category
II, type A
6-^
6
9
7
(+1)
8
(-1)
Karen
/ Category III,
type A
10'"
3
11
4
(+1)
6
(-5)
Class
Average:
8.75 (n=
=12)
6.5
(n=
12)
7.875
(n=8)
6.25
(n=8)
Notes: ESOL: Pierre was an English Speaker of Other Languages
473
Table H3
Students ' Pre and Post Positions on Feelings and Beliefs Surveys with Net Number of
Changes
MATHEMATICS FEELINGS
MATHEMATICS BELIEFS
Testing
Anxiety
Number
Anxiety
Abstraction
Aaxiety
Procedural
to
Conceptual
Toxic to
Healthy
Learned
Helpless to
Mastery
Oriented
Performance
to Learning
Achievement
Motivation
N
E
T
F
I
N
A
L
CATEGORY I
Type A
Catherine
okT**ok
oktok
oktok
oktok
oktok
oktok
oktok
«; voK
Robin
oktnok
ok=ok
oktok
oktok
oktok
oktok
oktok
+2
CATEGORY
II Type A
Jamie
noki*nok
oki*ok
noki*ok
nokinok
noktnok
noktnok
nokt~ok
+ 1/ 20K
+4/
Lee
noktnok
nok^ok
noktnok
oktok
okiok
ok=ok
ok=ok
+5
+1/ 50K
-1/
Pierre
oki*ok
ok=ok
oktok
nokt~ok
okt**ok
oktok
nokJ'nok
+1/ 60K
+3/
+2
TypeB
Autumn
oktnok
ok=ok
oktnok
nok=nok
oktok
noktnok
noktok
-21 20K
0/
Mitch
nokinok
okJ'ok
oki**ok
noktnok
nokt*ok
nokinok
noktnok
+1
.1/ 30K
+4/
+5
CATEGORY
III
Type A
Brad
nok
ok
nok
ok
ok
ok
ok
Frank
nok
ok
ok
ok
ok
ok
ok
Karen
nokt*nok
nokinok
nok>l'*ok
nokt*ok
oklok
nok=nok
okiok
+21
+ 1' 40K
0
Mulder
nokinok
okiok
ok^lok
noktok
okt*ok
nokt*ok
oktok
+2/ 60K
+6/
+7
TypeB
Kelly
nok
nok
nok
nok
ok
nok
nok
Notes: On scale of 1 through 5: nok: 3.5 to 5 (anxiety) or I to 2.5 (beliefs); nok: 3 to 3.4 (anxiety) or 2.6
to 3 (beliefs); ok: 2.6 to 3 (anxiety) or 3-3.4 (beliefs); ok: 1 to 2.5 (anxiety) or 3.5-5(beliefs) ; t/i:
increase/decrease; t*/i*:significant increase/decrease (p < .05);t**/ i**:significant increase/decrease (p <
.01) using Student t test of difference between means, dependent samples.
474
Appendix I
Summer 2000, PSYC/STAT 104 Class Calendar of Events
WEEKl
Interview 1 with Ann Porter May 3 1 , 2000
Class 1 Wednesday, May 3 1 , 2000, Introductions, the syllabus and schedule, and
chapter 1 : Statistics and the Scientific Method
WEEK 2
Class 2 Monday, June 5, 2000, chapter 2: Basic Mathematical and Measurement
Concepts and chapter 3: Frequency Distributions. I administered pretest feelings
and beliefs surveys and invited volunteers to participate in individual mathematics
counseling.
STUDY GROUP 1. Wednesday, June 7, 2000 4:30 p.m. 5:45 p.m.
Riverside Center
Brad, Lee, Jamie, Pierre (later)
Class 3. Wednesday, June 7, 2000, chapter 4: Measures of Central Tendency and
chapter 5: The Normal Curve and Standard Scores
Individual Sessions. Kelly June 8, 2000
WEEK 3
Drop-In. June 12, 2000 Kelly, Karen
Individual Session. Karen June 12, 2000
Class 4 Monday, June 12, 2000 Exam 1 on chapters 1 through 5. 1 administered
the Statistics Reasoning Assessment as a pretest.
Individual Session. Autumn June 12, 2000
Brad June 13, 2000 cancelled
MitchJune 14,2000
STUDY GROUP 2 Wednesday, June 14, 2000 4:30 p.m. 5:45 p.m.
Greenville campus in the Learning Assistance Center, Mitch, Lee,
Kelly
475
Class 5 Wednesday, June 14, 2000, Minitab Computer Orientation Computer Lab
Greenville campus
Minitab Module 1
Individual Session. Robin June 14, 2000
Kelly June 16, 2000 1 1 :30 a.m.
WEEK 4
Class 6 Monday, June 19, 2000, chapter 6: Correlation
Individual Session. Jamie June 20, 2000 5;30 p.m.
Brad June 20, 2000 6:30 p.m.
Mulder June 21, 2000 9:30 a.m.
Lee June 21, 2000 3:20 p.m.
STUDY GROUP 3 Wednesday, June 21, 2000
Lee
Class 7 Wednesday, June 21, 2000, chapter 7: Linear Regression
Individual Session. Kelly June 21 , 2000 8:20 p.m.
Pierre June 22, 2000 6:00 p.m.-9:00 p.m.
Floyd June 23, 2000 9:00 a.m. cancelled
WEEKS
Individual Session. Karen Jime 26, 2000 4:00 p.m.
Mitch June 26, 2000 5:00 p.m.
Class 8_Monday, June 26, 2000 chapter 8: Random Sampling and Probability and
chapter 9: Binomial Distribution
Individual Session. Pierre June 27, 2000 6:30 p.m.
STUDY GROUP 4 Wednesday June 28, 2000 4:30 p.m.
Mitch, Lee, Robin, Jamie, Karen (from back of the room). Brad
(watching). Autumn (with her own questions, Pierre (late), Carol (just
checking)
Class 9 Wednesday June 28, 2000, Exam 2
Individual Session. Autumn June 68, 2000 7:30 p.m.
Mulder June 29, 2000 8:00 a.m.
476
WEEK 6
Class 10 Monday July 3, 2000, no class meeting
Individual Session. Jamie July 3, 2000 7:00 p.m. - 8:30 p.m.
Lee July 5, 2000 3:30 p.m.
STUDY GROUP 5 Wednesday July 5, 2000
Lee
Class 11 Wednesday, July5, 2000, chapter 10: Introduction to Hypothesis testuig
Using the Sign Test; entirely lecture ...didn't get to Mann Whitney
Individual Session. Robin July5, 2000 7:30 p.m.
Mulder July6, 2000 12:00 noon
WEEK 7
Interview 2 Ann Porter July 1 0, 2000 3 :00 p.m.
Individual Session. Karen July 10, 2000 4:00 p.m.
Class 12 Monday July 10, 2000, chapter 1 1 : Mann- Whitney U Test and chapter
12: Sampling Distribution of the sample means, the Normal Deviate (z) Test
Individual Session. Brad July 10, 2000 8:20 p.m.
Jamie Julyl 1, 2000 10:00 a.m.
Drop-In Learning Center (with Jillian)
Lee Julyl2, 2000 1 :00 p.m.- 3:00 p.m.
Individual Session. Mitch Julyl2, 2000 3:30 p.m.
STUDY GROUP 6 Wednesday July 12, 2000
Mitch, Lee
Class 13 Wednesday Julyl 2, 2000, chapter 13: Student's t Test for Single
Samples, chapter 14: Student's t Test for Correlated and Independent Groups
Individual Session. Mulder Julyl2, 2000 8:20 p.m. cancelled
Pierre Julyl3, 2000 1 1 :00 a.m.
Brad Julyl3, 2000 6:00 p.m.
Pierre July 14, 2000
Robin July 15?, 2000
477
WEEKS
Individual Session Mulder 9:00 a.m. July 17, 2000
Drop-In: Learning Center (with Jillian)
Karen 1:00 p.m. -4:00 p.m.
Jamie didn't come
Individual Session Karen 4:00 -5 :00 p.m. July 1 7, 2000
Drop-In: Riverside (with Ann 4:00 p.m. - 6:00
p.m., with Ann and Jillian 5:00 p.m. - 6:00 p.m.)
Lee 4:30 p.m. - 6:00 p.m.
Jamie
Autumn (doing her own thing)
Catherine (doing her own work)
Mitch 5:20 p.m.
Karen 5:00 p.m. (doing her own thing)
Class 14 Monday July 17, 2000, Exam 3
Individual Session Autumn July 17, 2000 7:40 p.m.
Lee July 19, 2000 3:30 p.m.
STUDY GROUP 7 Wednesday July 19, 2000
Lee
Class 15 Wednesday, July 19, 2000, chapter 15: Introduction to the Analysis of
Variance chapter 16 Multiple Comparisons, did one-way, talked about setting up
two-way
Individual Session Rohm July 1 9, 2000 8:20 p.m. cancelled
WEEK 9
Individual Session Karen July24, 2000 4:00 p.m.
Class 16 Monday July24, 2000, chapter 18: Chi-Square and other Nonparametric
Tests, namely, one-way and two-way x^ and Wilcoxson Matched-Pairs Test
Individual Session Mulder July25, 2000 9:00 a.m.
Robin July25, 2000 ??
Pierre July26, 2000 8:00 a.m.
Jamie July26, 2000 1 0:00 a.m.
Mitch July26, 2000 3:3
478
STUDY GROUP 8 Wednesday July 26, 2000
Mitch, Lee, Autumn, Jamie, Mulder, Pierre [Robin, Brad,
Catherine came later]
Class 17 Wednesday July 26, 2000 Exam 4
Individual Session Autumn July26, 2000 7:40 p.m.
WEEK 10
Drop-In Mulder 1 :00 p.m.
Individual Session Karen 4:00 p.m. cancelled
Class 18 Monday July 31, 2000, Minitab Project Presentations, chapter 19:
Review of Inferential Statistics; I administered research posttests. Brad absent
Drop-In Karen cancelled
Individual Session Pierre, August 2, 2000 8:00 p.m.
Lee, August 2, 2000 3:30 p.m.
STUDY GROUP 9 Wednesday August 2, 2000
Lee, Mitch, Autumn, Pierre, Robin, Jamie, Catherine
Class 19 Wednesday, August 2, 2000 Exam 5, Brad absent
Individual Session Pierre August 3, 2000 8:00 a.m.
Interview3 Aim Porter, August 3, 2000, 1 :30 p.m.
Drop-In: Learning Center (with Jillian)
Mulder August 3, 2000 (for Finite Math)
WEEK 11
Individual Session Jamie August 6, 2000 7:30 p.m.
Pierre August 7, 2000 8:00 a.m.
Lee August 7, 2000 9:00 a.m.
479
Appendix J
Sample Mathematics Counselor Tutoring Handouts
r\ university is considering implementing one of the following three ^^'
gradmg systems: (1) All grades are pass-fail, (2) all grades are on the 4 0
system, and (3) 90% of the grades arc on the 4.0 system and 10% are
pass-fail. A survey is taken to determine whether there is a relationship
between undergraduate major and grading system preference. A random
sample of 200 students with engineering majors, 200 students with arts and
science majors, and 100 students with fine arts majors is selected Each
student is asked which of the three grading systems he or she prefers '
2 V A physician employed by a large corporation be-
lieves that due to an increase in sedeniarv life in
the past decade, middle-age men have become
fatter. In 1970. the corporation measured the per-
centage of fat in their employees. For the middle-
age men. the scores were normally distributed
with a mean of 22%. To test her hypothesis, the
physician measures the fat percentage in a ran-
dom sample of 12 middle-age men currently em-
ployed by the corporation. The fat percentaces
found were as follows: 24, 40, 29, 32. 33, 25, 15,
22. 18. 25, 16. 27. On the basis of these data, can
we conclude that middle-age men employed by
the corporation have become fatter?
^' WdT.r" "^ ' ""tritionis, who has been
asked to determine whether there is a diff.r^
(.. _ A professor has been teaching statistics for many
^^ years. His records show that the overall mean
for final exam scores is 82 with a standard devia-
tion of 10. The professor believes that this year's
class is superior to his previous ones. The mean
for final exam scores for this year's class of 65
students is 87. What do you conclude?
A
r^ neuroscientist suspects that low levels of the brain neurotransmitter
serotonin may be causally related to aggressive behavior As a to s7eD in
^ -vest.gat.ng this hunch, she decides tf do a correlaUve study TnvoW.^;
^ . mne rhesus monkeys. The monkeys are observed daily for 6 monZ 2
the number of aggressive acts recorded. Serotonin levels in the striatum ^a
bram re^on associated with aggressive behavior) are also measureHi e
A
n get 1 additional hour per week in wh.chl 1'*^"'' '^"'^*"8 '""hod
"nder the guidance of the processor Sint In'' '°"' *""^'^^''^^ P^b'ems
'"how the methods affecflTems ^ di fen' '"^
volunteers for the experiment ZZhLJT^ niathematical abilities
ab.l.ty info superior, average id poor 't'f 'T''"^ '° mathematical'
group are randomly assigned fo method I .hh"!' ^'^ ''""^"'^ f™'" «=ach
to method II. At the end of the coui^e a , 30 f h "'' '™'" ^^'^^ 8^°"P
exam. The following final exam scor« resuUed '"^ ''"^ ''*= ^^"^^ «-'
a° ti't,i:ie';"hr^'' 'r '^^'^'^ '-""'''-■
-7 "me schizopSi :;end"'"" ''^ ^'"°"'" °^
/' director of trainm^ T '" ^" '"«""t'on. As
aeree.ole h?r. .^ ' ' "^^^''J' institution, you
mean duration tZi T"' '"^'«"''on. The
mstaufon is 85 we ks S'"'"""' '''' ^' ^""^
«f 15 weeks The^cnr ^ "^""^^'^ "deviation
The results oI^L? ''' """"""^ distributed,
'ients.eaedhvrh """"'""' ^""^ '*>=' 'he pa-
a few years ago) show that the Ebiiam. '"7,""' ^^tires (collected
Swedish. 8% Irish. 5% German anStrH^,:"^^^^^^ ^"'"^S'^"- ^^^^
With percentages under 2% have nm h. f ^^°'^ •''^' nationalities
random sample of 750 intbtnts" tateVTnd fn'-^ ""^ '"' ^'^ "^'-^ "
the following table: ''''"• ^"'^ 'he results are shown in
. A sIcL-p icM.'urclici conilucis an experiment It)
- determine whethe-r sleep loss affects the ability to
maintam sustained attention. Fifteen individuals
are randomly divided into the lollowing three
groups of 5 subjeos each: group 1. which gets
t^he normal amount of sleep (7-8 hours); uroup
C4 2. which is sleep-deprived for 24 hours: and Iiroup
3. which IS sleep-deprived for 4S hours. Alfthrce
groups are tested on the same auditory vigilance
task. Subjects are presented with half-second
tones spaced at irregular intervals over a I -hour
duration. Occasionally, one of the tones is slightly
shorter than the rest. The subject s task "is to
detect the shorter tones. The following percent-
ages of correct detections were observed:
caUntS h'' ""'"''^''^ - •«« for mechani-
cal ap tude. He wants ,o determine how reliable
\ O month r °''!' **° admintstrattons spaced by 1
\ <J month. A study .s conduaed in which 10 students
are g'ven two administrattons of the test, with
Ifter the"fi ' rT^'*" "''"^ ^.ven 1 month
after the first. The data are given in the table.
A physical education professor believes that ex-
ercise can slow down the aginu process. For the
past 10 years, he has been conducting an exercise
j ^ class for 1 4 individuals wto are currently 50 years
old. Normally, as one ages, maximum oxygen
consumption decreases. The national norm for
maximum oxygen consumption in 50-vear-oid in-
dividuals is 30 milliliters per kilogram'per minute
with a standard deviation of 8.6. The mean of
the 1 4 individuals is 40 milliliters per kilogram per
minute. What do you condude '
T
mou°nrr'".'"*''"f '°'°"==^^^ g^°""«- the government is considering
I -? rr ? "^ '^"y-'l^ conservation campaign. However, before doing so
i Z °"/., " '^"- " ^'"^^ '° =°"'*"" ^" expenment to evaluate the
effectiveness of the campaign. For the experiment, the conservation cam
pa.gn IS conducted in a small but representative geographical area. Twelve
amihes are randomly selected from the area, and the amount of gasoHne
hey use is monitored for 1 month prior to the advenising campaign and
for 1 month following the campaign. The following data are coHeCed
licvos thai ihc amount of smoking by women
has increased in recent yeais. A complete census
\ 3- ° ''''^'■'" - ^'^^'■^ -"SO of womtn living m a neigh-
" ''"''"S ^"y -showed that the mean number of
cigarettes smol<ed daily by the women was 5.4
with a standard deviation of 15. To assess
her belief, the professor determined the daily
smoking rate of a random sample of :(H) women
currently living in that city. The data show that
t^he number of cigarettes smoked daily by the
200 women has a mean of 6.1 and a standard
deviation of 2.7.
A
Jr^ college professor wants to stetermine the best way to present an
/ important topic to his class. He has ihe following three choices: ( 1 ) he can
^5 • lecture, (2) he can lecture plus assign supplementary reading, or (3) he
can show a film and assign suppiejuentary reading. He decides to do an
expenment to evaluate the three options. He solicits 27 volunteers from
his class and randomly assigns 9 to each of three conditions. In condition
1. he lectures to the students. In comition 2. he lectures plus assigns supple-
mentary reading. In condition 3, tbe students see a film on the topic plus
receive the same supplementary reasEng as the students in condition 2. The
students are subsequently tested oa the material. The following scores
(percentage correct) were obtained:
/\ clinical psychologist is interested in the effect that anxietv ha. nn ,h.
the effect of anxiety depends on the difficulty of the new material An
/ experiment .s conducted in wh.ch there are three levels o7anxie,vh,fh
low anx.ou fjl5 " "' '"""''^- °"' «^ ^ P°°' °f volunteers, 15
'A group of researchers has devised a stress ques-
t,onna,re cons.stmg of 15 Ufe events T^ey are
.nterested m determining whether there is Ls^
iustrl ^T'"' °" '^" '"'^"^^ ^"'°"n' of ad-
fndl.H ^^^.^"'^"^^"s and 300 Italians. Each
•nd.v dual ,s mstructed to use the event of -mar
required n marnage. Marriage is arbitrarily
given a value of 50 points. If an event is iudoed
ve"?rr"" ^'^"""'="' "^^" --^i" '^e
event should receive more than 50 points How
adrtrnT''""^'^'''^"'^^""''---hmor:
adjustment is required. After each subject within
eac culture has assigned poin. ,o /hTlS liL"
events, the points for each event are averaged
The results are shown m the table that follows.'
APPENDIX K
Data about Karen
Table Kl
Karen's Individual PSYC/STAT 104. Summer 2000 Participation Profile
Week
486
10
Post
l"* Class
Class
Class
Class
Class
Class
Class
Class
Class
Class
Comp
Mondays
6:00 p.m.
2
Pre-
Tests
4
EXAM
#1
JuneI2
6
8
10
no
class
meeting
12
14
EXAM
#3
July 17
16
18
Minitab
Present,
Post-
Tests
(extra)
EXAM
2"" Class Class
Class
Class
Class
Class
Class
Class
Class
Class
Class
Wednesdays '
6:00 p.m.
3
5
Minitab
Partner
Catherine
K absent
7
9
EXAM
#2
June28
11
13
15
17
EXAM
#4
July26
19
EXAM
#5
Aug. 2
Study Group
Wednesdays
4:30 p.m.
Study
Gp4
Kpaitial
Drop-In
Jime
July
July
August
2
K
cancelled
12
17
24
K'/.hr
K3hrs
Ky4hr
Karen's
June
June
July
July
July
July
Individual
12
26
10
17
24
31
Sessions
K
cancelled
Meet with
Instructor
Extra
Stud
Gp.
before
EXAM
w/Jill 5-
6pm
IC partial
487
Table K2
Karen's Progress in Tests in Relation to Mathematics Counseling Interventions
Exam#l
Exam #2
Exam #3
Exam #4
MINITAB
Exam
Optional
[20% of Final
[20% of Final
[20% of Final
[20% of
projects
#5
1 10% of
Final
Exam
Grade]
Grade]
Grade]
Final
[10% of
(to replace
Grade]
Final
Grade]
Grade]
lower
grade)
General Stra
egies: average homework 6-7 hours per week, work by se
f (voluntary) or with experts (me/Ann) in class, 1
organize, access Individual Mathematics Counseling, and Drop-hi at the Learning Assistance Center |
Before
6/12
6/26 Individual
7/10 Individual
7/20 My
7/10
1 offered
Individual
Math
Math
Supervision
went
meeting but
Math
Counseling 2
Counseling 1/n
meeting =>
with /\nn
Karen
Counseling
days before:
Drop-In: 3hrs +
have her
to Comp
didn't want
just before
discussed Exam
Individual Math
assess her
Lab for
one.
exam, very
# 1 . strategy:
Counseling: Ihr
own
help on
anxious, angry
enror analysis,
course strategy
=> Karen gain
control, deal
with math
depression
just before: —
decision flow
chart + unlabeled
problems
=>formula sheet;
math depression
lifting
change, a
new
metaphor?
7/24
Individual
Math
Counseling
Modi
Test
MC:-12,S:
MC:-I4+2,S:
MC:-12,S:-1,
MC:-12,S:-
Modi:
96%
MC:
Results
-l,Calc:-25
-3,Calc:-ll;
Calc: -2
0,Calc:-0
100%;
12/40
Total: 62%
Total: 74%
Total: 85 + 6%
Total: 88%
Present
with
Catherine
Mod 3
100%
(30%)
Calc:45/60
(75%)
Total: 57%
Analysis
Unhelpful
Formula Sheet
Math
Karen now
Formula
issues?;
Computation
felt she had
Sheet; literal
ALARM: Q6—
"good enough"
it in hand
symbols (N,
decimals.
but compared
(except for
E) issues
operations
"apples' with
issues with
statistical
issues and
"cheese"; Now
MC)
concepts
literal symbols ;
has symbols in
issues
Language
hand; Still
Decimals
issues-MC and
Math
language/
concept MC
fiizzy
Computation
issues;
Preparation
and S issues;
Preparation
issues
Preparation
improving
much improved
Post
Individual
Individual Math
Individual
Cancelled
Will come
Strategies
Math
Counseling:
Counseling:
7/31
to Learning
Counseling:
focus: mirror
focus: provide
individual
Center for
focus: mirror
her developing
bearable
appointment.
finite
her embryo
mathematics
frustration;
didn't come
mathematics
mathematics
self; develop JK
promote growth
to drop-in
next
self; develop
and Annas
of realistic self-
before
summer "if
JK and Annas
secure bases
esteem, secure
Exam #5
Jillian is
secure bases
teacher
attachments
there"
KAREN'S Survey Profile Summary
488
II
III
Not at all
1
Not at all
1.1 1.2
MATHEMATICS FEELINGS
Math Testing Anxiety
3.6
Number Anxiety
2.9
2.8 3
Abstraction Anxiety
very much
1.1 1.5
T
1.1 4.5
'A T
o
4.5
very much
1,4
2.8t:
.0
A
A '
3.9 very much
3.5 T 4.2
-* Ht ►
1
a
2.8 3
T Pretest Class Extreme
Scores
▲ Posttest Extreme Class
Scores
T Karen's Pretest Scores
[placed above the scale]
a Karen's Posttest Scores
[placed below scale]
Figure Kl. Karen's pre and post scores on the pre Mathematics Feelings Survey in
relation to class extreme scores.
Mathematics Beliefs Scales
Procedural Math
I
1
Toxic /Negative
n 'I
©
1
Lean led Helpless
KAREN'S Survey Profile Summary
MATHEMATICS BELIEFS SURVEY
2.9
1.9) 2.25
— " — Jr
3.55 3.75
T A-
3.1
2.5 2.7
T A
3.6 4
3.9
-15 —
3.1
2.6
2.1
■rr-
a
2.7
3. .5
4
3.91
4.4
— A-
Process/Conceptual Math
5
Healthy/
Positive
5
Mastery
Drientated
T Pretest Class Extreme
Scores
A Posttest Extreme Class
Scores
T Karen's Pretest Scores
[placed above the scale]
a Karen's Posttest Scores
[placed below scalel
12 3 4 5
Figure K2. Karen's pre- and post-summary scores on the Mathematics Beliefs Survey in
relation to class range scores.
JMK Mathematics Affect Scales
1 . When I think about doing mathematics,
I tend to put work off;
never 3
a hi
sometimes
2. If I think about how I experience my problems with mathematics,
I tend to feel discouraged:
never 5 3 2 very much
<: > <: [>
1 1 4-,
sometimes
3. When I think about my mathematics future,
I feel:
confident
'<^
=>
=>
489
2 Session #2, June 26, 2000
3 Session #3, July 10,2000
4 Session #4, July 17,2000
5 Session #5, July 24, 2000
<IlI^ Karen's indication of her
ranges of response.
[ ] Karen's written comments
I feel:
hopeless/nothing can
improve
4. When I think about the mathematics course I am taking now,
I: 5
Ilikeil
<c=
o
I:
"would withdraw if
I could
5. When I think about how I do mathematics,
I:
feel pride in 2
how I do It
4
:0
[somewhere
in here]
I:
feel ashamed/ _
all the time
6. When I think of my mathematical achievements,
I: 2
feel satisfied 3
<;=
^
7 While I am doing mathematics.
I feel
discouraged
lean:
make
mathematical decisions -
on my own
k=^
I:
feel like a complete
failure/
I can."
not make mathematical
decisions on my own
I get
confijsed
Figure K3. Karen's responses on the JMK Mathematics Affect Scales^ (in Mathematics
Counseling Sessions 2 through 5)
490
Table K3
Karen's JMK Mathematics Affect Scales numerical responses.
JMK Mathematics Affect
Item
Item
Item
Item
Item
Item
Item
Average
Scale
1
2
3
4
5
6
7
JMK2:76/26
0.4
0.12
0.5
0.12
0.5
0.5
0.5
0.38
aflE1:62%
Karen
JMK3:7/10
0.5
0.25
0.37
0.25
0.5
0.5
0.5
0.41
aflE2:74%
JMK4;7/17
0.5
0.25
0.5
0.25
0.75
0.5
0.5
0.46
befE3;91%
JMK5;7/24
0.5
0.62
0.37
0.3
0.62
0.63
0.5
0.51
befE4:88%
0.48
0.31
0.43
0.23
0.59
0.53
0.53
0.44
' See Appendix B for a discussion of the development and rationale for the use of these scales and a copy of
the survey.
491
Table LI
Appendix L
Data about Jamie
Jamie's Individual PSYC/STAT 104. Summer 2000 Participation Profile
Week
i ? 3 4 5 6 7 8 9 io~
Post
1^ Class
Mondays
6:00 p.m.
Class
2
Class
4
EXAM
1
Junel2
Class
6
Class
8
Class
10
no
class
meeting
Class
12
Class
14
EXAM
3
July 17
Class
16
Class
18
Minitab
Post-
Tests
Comp
(extra)
EXAM
2"" Class Class
Class
Class
Class
Class
Class
Class
Class
Class
Class
Wednesda '
ys 6:00
p.m.
3
5
Mini tab
Partner
Robin
7
9
EXAM
2
June28
11
13
15
17
EXAM
4
July 26
19
EXAM
5
Aug. 2
study Group
Wednesdays
4:30 p.m.
Study
Gpl
Study
Gp4
Study
Gp8
Study
Gp9
Drop-In
No
Show
Individual
Session
June
20
Jillian
Initiate
Julys
Jillian
Initiate
July
11
Jamie
initiate
July
26
Jamie
initiate
Aug.
6
Jamie
initiate
Meet with
Instructor
Extra
S.Gp
before
EXAM
w/Ann
4-6 pm
w/JUl 5-
6p-m
492
Table L2
Jamie's Progress in Tests in Relation to Mathematics Counseling Interventions
Test#l
Test #2
Test #3
Test #4
MINITAB
projects
Tcst#5
Optional
Test
June 12
June 28
July 17
July 26
July 31
August 7
[20% of Final
[20% of Final
[20% of Final
[20% of Final
August2
Grade]
Grade]
Grade]
Grade]
[10% of
Final
Grade]
[10% of
Final
Grade]
(to replace
lower
grade)
General Strate
gies: average homework 5 hours per week, work by self (involuntary) or with Ann in class (leaves questions unasked, answers |
unspoken), listen, problem solve. Individual Mathematics Counseling — Jamie initiates last 3 sessions, attend 4 of 9 Snidy Groups (positive if |
speaks, negatii
le if not)
Before
6/7 Shidv
6/20 Individual
7/3 Individual
My Supervision
7/26
8/6
Group 1:1
Math Counseling:
Math
meetmg =>
Study
Individual
asked her a
analyze Exam# 1 ;
Counseling Did
commend her.
Group 9: 1
Math
question. J
metaphor: inside in
The Algebra
give her a
asked J a
Counseling
responded
storm.
Test
bouquet, have
question.
Meeting at
correctly
her write a new
Jamie
Starbucks —
6/28 Study Group 4:
7/ II Individual
metaphor.
responded
analyze all
J watched, listened
Math
Counseling
before t-tests
covered in class
7/l7ExtraSmdv
Group with
Ann: J watched
and listened
7/26 Individual
Math
Counseling:
new metaphor
Partly sunny day
1 can come out
in
7/26 Study
Group 8: J
watched,
listened
incorrectly
Tests. . . see
her own
competence
see changes
from pre to
post test
surveys e.g.,
reduced
anxiety
Test Results
MC:-2,S:-I,C:-2
MC:-4,S:-5,C:-17;
MC:-14,S:-0,C-
MC:-I2,S:-0,C-
Modi:
100%
MC: 23/40
2
12
100%;
(58%) Calc:
Total: 95%
Total: 74%
Total: 84+ 6%
Total: 76%
Present
with
Robin
Mod
100%
48/60 (80%)
Total: 71%:
Too low to be
used
Analysis
Former
More anxious than
Math
Math
knowledge plus
before Test #1
Computation
Computation:
good problem
solving even
with panic,
because of 95% on
Test tt 1 and family
pressure; Stady
"good enough";
Now has
symbols in
one analysis not
understood —
illogical use of
sound number
confusing;Formula
hand; language/
literal symbols
sense
Sheet issues?;
Language issues on
Math Computation,
and problem solving
didn't "work'; MC
"good enough";
symbols a problem
strategy MC
issues
cf , numbers,
one careless fix;
two logical
conclusions for
incorrect calc —
no credit; MC
still an issue
Post
6/20 Individual
7/3 Individual Math
Individual Math
Probably will
Strategies
Math
Counseling: focus —
Counseling:
not go to a
Counseling:
Exam Analysis
focus —
learning
focus —
Counseling
center for
commend
help in
problem solving
future; will
access course
related group
help
493
fa»
(1^-l.Y
M
q
•
' — - — w
o iiio
4
J N-±
datrtbuHon. (Spolnti)
n W TDU Aout Bib MM ■oora* kr
II
When Jamie realized lier z in
question 1 3 was unlikely, she went
back to question 1 1 , to the s. She
had found that she had divided the
sum of squared deviations by 1 00 -
1, that is, U(- 1, instead of by the
correct 10-1, that is, « - 1 . When
she corrected herself, her incorrect 5
= .63 changed to i' = 2. 1 1
Figure LI. Jamie's responses on Questions 1 1 and 12 on Exam #1. Note her self-
corrections in question 1 1 .
~v
■■za
ccnblnflAulQ
(Be**.)
-L-
s
-
-
.3_
-
i«a»i.Mi
^
iXf
" MWdDMIhaztcoraMirajRadXialy AaitlnAUMIQ.npoMi]
TKa. 7. score. Htib L((X/-+KajV indl^/iduoul
& 30+" (X. -fesV" ^corc ixOLj d^i^® oJao/t^ y
o-i*»ai-\5 TVms in^-vibocul is (XT, ejt(>^vYjlCcif ^
When Jamie substituted her
new 5 = 2. 11 for the
incorrect 5 = .63 in the z
formula, the incorrect z of
4.76 became a more
reasonable z = 1 .42.
Figure L2. Jamie's responses on Questions 13 and 14 on Exam #1. Note her self-
corrections in questionl3.
494
Jamie 's Responses on the Algebra Test
Jamie's level 4 score meant that she was able to treat letters appropriately as
specific unknowns in some cases, as generalized numbers in some cases, and as variables
7.(d)
A =
( e + 2) 5
Figure L3. Jamie's solution to question 7 (d) Find the area of the figure. Algebra Test
(Sokolowski; see Appendix D). Note Jamie's initial error that she scratched out and
replaced with the correct properly coordinated area solution.
in others. Jamie's use of letter symbols:
1. as specific unknowns in some cases (as in Q.14: ...if r = s + t and r + s + t = 30,
[what is] r?: r = 15 (Jamie's correct response in bold),
2. as generalized numbers in some cases (as in Q.I8: When [is this] true...? L + M +
N = L + P + N Always Never Sometimes, when: M =P (Jamie's correct
response in bold), and
3. as variables in others (as in 5. (c) If e + f = 8 then e + f + g =: 8 + g (Jamie's
correct response in bold). (Appendix C, Algebra Test)
In addition, she was able to resolve ambiguity by coordinating two operations. For
example, to determine the area of a rectangular figure she corrected her initial impulse to
incorrectly use only one operation, multiplication, to get lOe, to the coordination of
addition and multiplication, to obtain (e + 2)5 (see Figure L3).
495
JAMIE'S Survey Profile Summary
MATHEMATICS FEELINGS
Math Testing Anxiety Pre 4.1
Not at all
1.1
'*-r
1.5
T
4.1
3
Post 3.6
Abstraction Anxiety
Not at all
1.1 1.2
ffl
Pre 3.7
••^T-
3.5
—r
Post 3.0
4.2
very much
4.5
s. —
Not£
tall
P
1.4
re 2.1
T
Numbe
2.. 1
r Anxiety
veryn
11
T (S
A Post 1.5
•1
very much
►■
▼ Pretest Class Extreme Scores
A Posttest Extreme Class
Scores
T Jamie's Pretest Scores
[placed above the scale]
a Jamie's Posttest Scores
[placed below scale]
MATHEMATICS BELIEFS SURVEY
Process/Relational Math
1.9
I '■'!
T
3.55 3.75
1
Tox
1 :
ic /Negative
I
Post2.6
Pre2.5
T
2.5 2.7
T A
3.9
Healthy
J 4.4
5
/Positive
n
T ▲
' A '
Lear
1 :
Pr
ned Helpless t
2
I
b2
2.1
Post2.7
5 -!
3.5 3.9
Masteiy
5
Orientated
in
Post
A
2.1
T A
Figure L4. Jamie's mathematics pre- and post-feelings and beliefs Survey Profile
Summary in relation to class range pre-and post-scores
Session 1. We discussed briefly Jamie's anxiety average scores on the Mathematics
Feelings pretests that I had plotted with the class extreme scores on her Survey Profile
Summary (see Figure L4). Jamie's Number Anxiety was low (2. 1), close to the middle of ■
the class range, but her Abstraction Anxiety was high. She was not surprised by her high
Math Testing Anxiety score (the highest in the class at 4.1).
JIVIK Mathematics Affect Scales
496
1 . When I think about doing mathematics,
I tend to put work off:
never 2
1 3
S 4, 1
a lot
sometiities
2. If I think about how I experience my problems with mathematics,
1 tend to feel discouraged:
never 12 3 verv much
5 4
sometimes
3. When I think about my mathematics future,
Ifeel:
1
confident
1 Session 1, June 20, 2000
2 Session 2, July 3, 2000
3 Session 3, July 11,2000
4 Session 4, July 26, 2000
5 Session 5, August 6, 2000
Ifeel:
hopeless/nothing can
improve
4. When I think about the mathematics course I am taking now,
I: 1:
would withdraw if
I like it
4 2
5 3
5. When I think about how I do mathematics,
I:
feel pride in
how I do it
1 5 4
could
1:
feel ashamed/_
all the lime
6. When I think of my mathematical achievements.
feel satisfied 4
5
2
7. W
1 can:
make
Ifeel
discouraged
hile I am doing mathematics.
mathematical decisions
on my
own 4 1
2
I:
feel like a complete
fa lure/
I can."
not make mathematical
decisions on my own
I get
Confused
Figure L5. Jamie's responses on the JMK Mathematics Affect Scales, Sessions 1-5
497
Table L3. Numerical scores and averages of Jamie's JMK Scale responses (see Figure K5)
JMK Mathematics Affect
Item
Item
Item
Item
Item
Item
Item
Average
Scale
1
2
3
4
5
6
7
JMK1:6/20
0.87
0.75
0.37
0.9
0.82
1
0.75
0.78
aftE1:95
"Jamie"
JMK2:7/3
0.75
0.5
0.37
0.63
0.5
0.7
0.5
0.564286
anE2;74
JMK3:7/11
0.75
0.25
0.37
0.63
0.37
0.4
0.5
0467143
befE3:90
JMK4:7/26
0.75
0.5
0.5
0.75
0.7
0.75
0.75
0.671429
befE4:76
JMK5:8/6
0.87
0.75
0.63
0.75
0.75
0.75
0.75
0.75
befEC:71
0.80
0.55
0.45
0.732
0.63
0.72
0.65
0.646571
498
APPENDIX M
Data about MULDER
Table Ml.
Mulder's Individual PSYC/STAT 104, Summer 2000 Participation Profile
Week
10
Post
1^ Class
Class
Class
Class
Class
Class
Class
Class
Class
Class
Mondays
6:00 p.m.
2
Pre-
Tests
4
EXAM
#1
Junel
2
M
late
6
8
10
no
class
meeting
12
14
EXAM
#3
July
17
16
18
MINITAB i
Post-
Tests
2"" Class Class
Class
Class
Class
Class
Class
Class
Class
Class
Class
Wednesdays 1
6:00 p.m
3
5
Minitab
Partner
Pieiic
7
9
EXAM
#2
June28
11
13
15
17
EXAM
#4
July 26
19
EXAM
#5
Aug. 2
study Group
Wednesdays
4:30 p.m.
Study
Gp8
Drop-In
July 31
Minitab
paper,
Aug. 3
MATH
120
Finite
Individual
June
June
July
July
July
July
Session
21
29
6
12
M
17
25
Stats
Finite
math&
Stats
Stats
cancelled
Stats
Stats
only
only
only
only
Meet with
Instructor
Extra
Gp
before
EXAM
w/Ann
4-6 p m
w/Jill 5-
6 p.m
Table M2
Mulder 's Progress in Tests in Relation to Mathematics Counseling Interventions
499
Test#l
Test #2
Test #3
Test #4
MINITAB
Test#5
Test
6/12/00
6/28/00
7/17/00
7/26/00
projects
7/31/00
8/2/00
[10% of
(10
replace
[20% of Final
[20% of Final
[20% of Final
[20% of Final
[10% of
Final
lower
grade)
Grade]
Grade]
Grade]
Grade]
Final
Grade]
Grade]
General Strat
egies: average homework 3 hours per
week, work with others in class, visualize/memorize access
Individual M
athematics Couns
eling, and Drop-In at the Learning Assistance Center
Before
6/21 First
7/6 Individual
My
7/31
Individual
Math
Supervision
dropped
Math
Counseling:
meeting=>
in to have
Counseling;
Finite Math
trial MC
me read
analysis of
7/17 9:00am
resistance Test
his and
Exam# 1 -lack of
Individual
at 7/25
Pierre's
prep, lack of
Math
Individual
MINITAB
knowing what
Counseling
Math
project
to expect
focused on
Exam #3; focus
on symbol
links; choosing
and doing
hypothesis test
Counseling
write up
Test
MC:-18,S:-
MC:-14,S:-
MC:-I8,S:-
MC:-8+2,S:-
Mod:
94%
Results
l,Calc:-18
5,Calc:-0
0,Calc:-6
0,Calc:-3
100%;
Total: 63
Total: 81
Total: 76 + 5
Total: 91
Present
with
Pierre
Mod
92%
Analysis
Poor Formula
Has Math
Math
Mulder has
Sheet; MC
Computation
Computation
mastered the
issues; lack of
more in hand;
still OK; Now
last hurdle:
study
Verbal-MC. S
issues
has symbols in
hand; STE.L
big MC issues
MC!
Post
Individual
6/29 Individual
Individual
Drop-in
Probably
Strategies
Math
Math
Math
help with
Finite
Word
won t
goto
Counseling;
Counseling:
Counseling:
a
focus — overall
[half on finite
focus MC,
Problems
LcurnjQg
Center
approach;
math: simplex
resistance
when
Formula Sheet
method]
focus — verbal
connections
especially
symbols
taking
math
in
fiiture
Not at all
I*
MULDER'S Survey Profile Summary
MATHEMATICS FEELINGS
Math Testing Anxiety
3.1 very much
1.1 1.5
'
4.1 4.5 1
< A T
i!
' A '
1 2
Number Anxiety
Not at all 2.1
n
1 2
Abstraction ArLxiety
Not at all
very much
1.4
T 2.8
h«
^ T a
▲ 1.8
M
2.7
T Pretest Class
Extreme Scores
APosttest
Extreme Class
Scores
X Mulder's
Pretest Scores
[placed above
the scale]
1.1 1,2
T
3.5
4.2
^T A c
1.9
A
T *
veo'much ^ lyiuijer's
111
1 2 3
MATHEMATICS BELIEFS SURVEY
Procedural Math 2.9
3.051
Posttest Scores
[placed below
scale]
1.9
i 2.25 X
3.55 3.75
^ ■f
A
a T A
1
Toxic /Negative
3.2
II
2.5 2.7
y A
4
39B
Process/Relational Math
Healthy/Positive
1 2
^ — -Learned Helpless
2.6
(^
2 2.1
"S —
3.5
3.5
4
3.91
4.4
Mastery Orientated
500
Figure Ml. Mulder's responses to the pre- and post-Mathematics Feelings and
Mathematics Beliefs surveys in relation to class extreme scores.
JMK Mathematics Aifect Scales
1 . When I think about doing mathematics,
I tend to put work off:
never
a hi
1
501
I Sessional, June 21 2000
3 Session #3, July 6, 2000
5 Session #5, July 25, 2000
2. If I think about how I experience my problems with mathematics.
I tend to feel discouraged:
never 3
very much
3. When I think about my mathematics future,
I feel:
confident 1
3
5
I feel:
hopeless/nothing can
improve
When I think about the mathematics course I am taking now.
/ like it
5. When I think about how 1 do mathematics,
1:
feel pride in
how I do it 5 3 1
6. When I think of my mathematical achievements,
I: 1
feel satisfied 3
5
7. While 1 am doing mathematics,
lean:
make
mathematical decisions
on my own
5 3
I feel
discouraged
Iget
confused
would withdraw if
could
feel ashamed/ _
all the time
feel like a complete
fa lure/
lean;
not make mathematical
decisions on mv own
Figure M2. Mulder's responses on the JMK Mathematics Affect Scales for Sessions 1,3,
and 5.
502
Table M3.
Numerical Scores and Averages of Muldefs JMK Affect Scale Responses
JMK Mathematics
Item
Item
Item
Item
Item
Item
Item
Average
Affect Scale
1
2
3
4
5
6
7
JMK1:6/21
0.0
0.5
075
0.75
0.5
0.75
0.60
0.55
aftE1:63'
Mulder
JMK3:7/6
0.25
0.75
0.75
0.75
0.63
0.75
0.60
0.64
aftE2:81befE3:81
JMKS:7/25
0.5
0.5
0.75
0.75
0.75
0.75
0.75
0.68
befE4:91'=
Average:
0.34
0.58
0.75
0.75
0.62
0.75
0.68
0.64
Notes: ^ after Exam #1 where he earned 63%; ''after Exam #2 where he earned 81% and
before Exam #3 where he earned 81% (with extra credit); '^ after Exam #4 where he earned
91%.
503
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