ESSAYS ON
MATHEMATICAL EDUCATION
BY
G. ST. L. CARSON
WITH AN INTRODUCTION BY
DAVID EUGENE SMITH
LONDON AND BOSTON
GINN AND COMPANY, PUBLISHERS
1913
L& \64S-
COPYRIGHT, 1913, BY GINN AND COMPANY
ALL RIGHTS RESERVED
SI3.6
GINN AND COMPANY PRO-
PRIETORS ' BOSTON -U.S.A.
INTRODUCTION
It has always been hard for people to judge with any accuracy
the work of their own age, and it is hard for us to do so to-day.
In spite of our optimism and of our certainty that we are pro-
gressing, what we conceive to be an era of great educational
awakening may appear to the historian of the future as one in
which noble ideals were sacrificed to the democratizing of the
school, and the twentieth century may not rank with the sixteenth
when the toll is finally taken.
It is, therefore, with some hesitancy that we should assert
that we live in a period of remarkable achievement in all that
pertains to education. That the period is one of advance is in
harmony with the general principle of evolution, but that all that
we do is uniformly progressive is not at all in accord with general
experience. Certain it is that the present time is one of agita-
tion, of the shattering of idols, and of the setting up of strange
gods in their places. Nothing is sacred to the iconoclast, and he
is found in the school as he is found in the church, in govern-
ment, and in the social world.
Among the objects of attack in this generation is " the science
venerable" that has come down to us from Pythagoras and
Euclid, from Mohammed ben Musa and Bhaskara, and from
Cardan, Descartes, and Newton. And yet it does not seem to
be mathematics itself that is challenged so much as the way in
which it has been presented to the youth in our schools, and
to most of us the challenge seems justified. With all the excel-
lence of Euclid, his work is not for the child ; and with all the
value of formal algebra, the science needs some other introduction
than the arid one until recently accorded to it
345267
iv MATHEMATICAL EDUCATION
It is on this account that Mr. Carson's work in the English
schools and before bodies of English teachers has great value.
He is thoroughly trained as a mathematician, is a product of the
college where Newton studied and taught, is a lover of the
science in its purest form, and has had an unusual amount of
experience in the technical applications of the subject; but he
is a teacher by instinct and by profession, and is imbued with
the feeling that mathematics can be saved to the school only
through an improvement in our methods of teaching and in our
selection of material. He stands for the principle that mathe-
matics must be made to appeal to the learner as interesting and
valuable, and he has shown in his own classes that, after this
appeal has been successful, pupils need to be held back rather
than driven forward in this branch of learning.
It is because of this feeling on the part of Mr. Carson that
his essays on the teaching of mathematics have peculiar value
at this time. They will encourage teachers to continue their
advocacy of a worthy form of mathematics, at the same time
seeking better lines of approach and endeavouring to relate the
subject in a reasonable manner to the various other interests of
the pupil. The problem is much the same everywhere, but the
ties of a common language, a common spirit of freedom, and a
common ancestry make it practically identical in English-speak-
ing lands. On this account we, in the United States, feel that
Mr. Carson's message is quite as much to us as to his own
countrymen, and we shall appreciate it as we have appreciated
the noteworthy work that he has already achieved in the
teaching of mathematics in England.
DAVID EUGENE SMITH
TEACHERS COLLEGE
COLUMBIA UNIVERSITY
NEW YORK CITY
CONTENTS
PAGE
SOME PRINCIPLES OF MATHEMATICAL EDUCATION ... i
INTUITION 15
THE USEFUL AND THE REAL 33
SOME UNREALISED POSSIBILITIES OF MATHEMATICAL
EDUCATION 47
THE TEACHING OF ELEMENTARY ARITHMETIC .... 63
THE EDUCATIONAL VALUE OF GEOMETRY 83
THE PLACE OF DEDUCTION IN ELEMENTARY MECHANICS 113
A COMPARISON OF GEOMETRY WITH MECHANICS . . . 123
SOME PRINCIPLES OF
MATHEMATICAL EDUCATION
(Reprinted from The Mathematical Gazette, January, 1913)
SOME PRINCIPLES OF MATHEMATICAL
EDUCATION
Of all the problems which have perplexed teachers of
mathematics in this generation, probably none has been
more irritating and insistent than the choice of assumptions
which must be made in each branch of the science. In
geometry, in analysis, in mechanics, one and the same
difficulty arises. Are we to prove that any two sides of a
triangle are greater than the third ? That the limit of the
sum of a finite number of functions is equal to the sum
of their limits ? That the total momentum of two bodies
is uninfluenced by their mutual action ? And in every
such case, on what is the proof to depend ? A clear under-
standing of the answers to such questions, or, better still, a
clear understanding of principles by which answers may
be found, would go far to co-ordinate and simplify elemen-
tary teaching ; the object of this paper is to state such
principles and indicate their application.
AXIOM, POSTULATE, PROOF
It is first necessary to lay down definitions, as precise
as may be possible, of the terms " axiom," " postulate,"
" proof." It is not implied that these definitions should be
insisted on, or the terms used, in elementary teaching;
nothing could be more likely to lead to failure. But a full
comprehension of each is essential to every teacher of
mathematics, and is too often lacking in current usage.
3
4 ' MATHEMATICAL EDUCATION
' 'Ap- axiom, oiy" cpmmon notion " in Euclid's language,
is a statement which is true of all processes of thought,
whatever be the subject matter under discussion. Thus
the following are axioms : " If A is identical with B, and
C is different from B, then C is different from A" " If B
is a necessary consequence of A, and also C of B, then C
is a necessary consequence of A" But " Two and two
make four " and " The straight line is the shortest distance
between two points " are not axioms, although they may
be considered no less obvious. A statement is not an
axiom because it is obvious, but because it concerns uni-
versal forms of thought, and not a particular subject matter
such as arithmetic, geometry, and the like.
A postulate is a statement which is assumed concerning
a particular subject matter ; for example, " The whole is
greater than a part " (subject matter, finite aggregates) ;
"All right angles are equal" (subject matter, Euclidean
space). It is essential to observe that, whereas an axiom
is an axiom once for all, a postulate in one treatment of a
science may not be a postulate in another. In Euclid's
development of geometry, the statement that any two sides
of a triangle are together greater than the third side is not
a postulate, because it is deduced from other statements
(postulates) which are avowedly assumed ; but in many
current developments it is adopted at once, without refer-
ence to other statements, and is therefore a postulate in
such cases. To use an unconventional but expressive term,
postulates are " jumping-off places " for the logical explo-
ration of a subject. Their number and nature are immaterial ;
they may be readily acceptable, or difficult of credence.
Their one function is to supply a basis for reasoning,
PRINCIPLES OF EDUCATION 5
which is conducted in accordance with the axioms. Postu-
lates are thus doubly relative : they relate to one particular
subject matter (number, space, and so on) and to one par-
ticular method of viewing that subject matter.
A statement which is deduced, by use of the axioms,
from two or more postulates is said to be proved. There
is thus no such thing as absolute proof. Proofs are related
to the postulates on which they are based, and a demand
for a proof must inevitably be met by a counter demand for
a place to start from, that is, for some postulates. When
a statement is said to have been proved, what is meant is
that it has been shown to be a logical consequence of some
other statements which have been accepted ; if these
statements are found to be incorrect, the statement which
is said to be proved can no longer be accepted, though the
logical character of the proof is in no way impugned.
Thus the type of a proof is, " If A, then B" ; relentless
and final certainty surrounds " then " ; but A, which is
assumed in the "if," may nevertheless be utterly fantastic
as viewed in the light of experience.
THE THREE FUNCTIONS OF MATHEMATICS
The first application of mathematics to any domain of
knowledge can now be explained. Starting from postulates,
the truth of which is no concern of mathematics, sets of
deductions are evolved by use of the axioms ; agreement
of the results with experience strengthens the evidence in
favour of these postulates. If this evidence be deemed
sufficient, as, for example, in geometry and mechanics, then
deduction yields acceptable results which could not other-
wise have been predicted or ascertained.
6 MATHEMATICAL EDUCATION
It is here that the prevailing concept of the power of
mathematics ends ; but such a concept presents a view of
the subject so limited and distorted as to be almost gro-
tesque. The process just described may be regarded as an
upward development ; a downward research is also possible,
and no less valuable. It consists of a logical review of the set
of postulates which have been adopted ; in the result, either
it is shown that some must be rejected, or the evidence
in favour of all may be considerably enhanced. This review
consists of two processes, which will be described in turn.
It is first necessary to ascertain whether the set of pos-
tulates is consistent ; that is, whether some among them
may not be logically contradictory of others. For example,
Euclid defines parallel straight lines as coplanar lines which
do not intersect, and proves in his twenty-seventh proposi-
tion that such lines can be drawn ; for this purpose he uses
his fourth postulate, which makes no allusion to parallels.
If he had included among his postulates another, stating
that every pair of coplanar lines intersect if produced suffi-
ciently far, and had omitted his definition of parallel lines,
his postulates would not have been consistent ; for the
twenty-seventh proposition proves that if the fourth postu-
late be granted, then the existence of non-intersecting co-
planar lines must be admitted also. It is essential to realize
that the contradiction implied in the term "inconsistent"
is based on logic, not on experience ; assumptions which
are contrary to all experience are not thereby inconsistent.
There is nothing in logic to veto the assumption that, for
certain types of matter, weight and mass are inversely propor-
tional ; or that life may exist where there is no atmosphere,
as on the moon. Such assumptions are not inconsistent
PRINCIPLES OF EDUCATION 7
with the other postulates of mechanics or biology ; they
are merely contrary to all experience gained up to the
present time.
Here, then, is the second function of the mathematician
the investigation of the consistence of a set of postulates.
And the task is not superfluous. Physical measurements
are perforce inaccurate, and a set of inconsistent assump-
tions might well appear to be consistent with actual obser-
vations. More accurate measurements must, of course,
expose the discrepancy, but these may for ever remain
beyond our powers ; logic renders them superfluous by
demonstrating the consistence or otherwise of each set
considered.
The next investigation concerns the redundance of a set
of postulates. Such a set is said to be redundant if some
of its members are logical consequences of others. For
example, any ordinary adult will accept without difficulty
the properties of congruent figures, the angle properties of
parallel lines, and the properties of similar figures, as in
maps or plans, regarding them as " in the nature of things."
And electricians may, by experiment, convince themselves
first, that Coulomb's law of attraction is very approximately
true ; and secondly, that within the limits of observation
there is no electric force in the interior of a closed con-
ductor. In neither the one case nor the other need there
be the least suspicion that the statements are logically
connected, so that they must stand or fall together. Yet
so it is, and the fact is expressed by the statement that
the assumptions are redundant.
The investigation of redundance, and the demonstration
that sets of postulates are free therefrom, forms the third
8 MATHEMATICAL EDUCATION
function of the mathematician. Its value, in connection
with any subject matter to which it may be applied, may
not at once be evident It is based on the fact that all
experiments, necessary and inevitable though they be, are
nevertheless sources of uncertainty ; it reduces this uncer-
tainty to a minimum by removing the redundant assump-
tions into the category of propositions, and exposing the
science in question as based on a minimum of assumption.
And more ; it can offer several alternative sets of assump-
tions for choice, that one being taken which is most nearly
capable of verification. The labours of Faraday resulted
in the offer of such a choice to electricians ; either, they
were told, you can base electrostatics (inter alia) on Cou-
lomb's experiment, or on the absence of electric force in-
side a closed conductor ; it is logically immaterial which
course you adopt. The latter experiment, being far more
capable of accurate demonstration in the laboratory, is
chosen as the primary basis for faith in the deductions
of electrostatics a faith which is, of course, very much
strengthened as such deductions are found to accord with
our experience. But these considerations are for the physi-
cist ; the task of the mathematician is ended when he has
put forward, for choice by the physicist, alternative sets of
assumptions which are at once consistent with each other
and free from redundance. In this way does he free the
physicist, so far as may be, from the uncertainties of
assumption, and assure him that no further increase of
such freedom can be attained. 1
1 The antithesis between mathematician and physicist does not imply
that the functions are of necessity performed by different individuals ;
it is used merely to enforce the argument
PRINCIPLES OF EDUCATION 9
Such is the range of application of mathematics to other
sciences. When complete it reveals each science as a
firmly knit structure of logical reasoning, based on assump-
tions whose number and nature are clearly exposed; of
these assumptions it can be asserted that no one is incon-
sistent with the others, and that each is independent of
the others. There is thus no fear that contradictions may
in time emerge, and no false hope that one assumption
may in time be shown to be a logical consequence of the
others. Finality has been reached.
The acute critic may, of course, ask the mathematician
whether his own house is in order. What is the precise
statement of the axioms which are the basis of his science,
and can they be shown to depend on a set of consistent
mental postulates, free from redundance? Here it need
only be said that the labours of the last generation have
done much to answer these questions, and that their com-
plete solution is certainly possible, if not actually achieved ;
to go further would be beyond the limits of this paper.
THE DIDACTIC PROBLEM
The complete application of mathematics to any branch
of knowledge being thus exhibited, the didactic problem
can now be stated in explicit terms. In any given science
geometry, mechanics, and so on what is the right point
of entry to the structure, and in what order should its
exploration be made ? What results should be regarded as
postulates, and should their consistence and possible inter-
dependence one on another be investigated before upward
deduction is undertaken ? Should the minimum number be
chosen on the ground that the pupil should at once be
10 MATHEMATICAL EDUCATION
placed in possession of the ultimate point of view? Or
should some larger number be taken, and if so, on what
principles should they be chosen ?
Bearing in mind that the pupils concerned are not
presumed to be adults, it is easy to indicate principles
from which answers to such questions may be deduced.
One of the few really certain facts 'about the juvenile
mind is that it revels in exploration of the unknown,
but loathes analysis of the known. It is often said that
boys and girls are indifferent to, and cannot appreciate,
exact logic ; that it is unwise to force detailed reasoning
upon them. Few statements are farther from the truth.
Logic, provided that it leads to a comprehensible goal, is
not only appreciated, but demanded, by pupils whose in-
stincts are normal. But the goal must be comprehensible ;
it must not be a result as easily perceived as the assump-
tions on which the proof is based. Let any one with expe-
rience in examining consider the types of answer given to
two problems ; one, an " obvious " rider on congruence,
involving possibly the pitfall of the ambiguous case ; the
other, some simple but not obvious construction or rider
concerning areas or circles. In the former, paper after
paper exhibits fumbling uncertainty or bad logic ; in the
latter, there is usually success or silence, and more usually
success ; bad logic is hardly ever found. The phenomenon is
too universal to be comfortably accounted for by abuse of the
teachers ; the abuse must be transferred to the crass methods
which enforce the premature application of logic to analysis
of the known, rather than to exploration of the unknown.
The natural order of exploration should now be evident.
Let the leading results of the science under consideration
PRINCIPLES OF EDUCATION II
be divided into two groups : one, those which are accept-
able, or can be rendered acceptable by simple illustration,
to the pupils under consideration ; the other, those which
would never be suspected and whose verification by exper-
iment would at once produce an unreal and artificial atmos-
phere. Let the former group which in geometry would
include many of Euclid's propositions be adopted as
postulates, and let deductions be made from them with full
rigour. Wherever possible, let the results of such deduc-
tions be tested by experiment, so as to give the utmost
feeling of confidence in the whole structure. Later, when
speculation becomes more natural, let it be suggested that
gratuitous assumption is perhaps inadvisable, and let the
meanings of the consistence and redundance of the set of
postulates be explained. Finally, if it prove possible, let
the postulates be analysed, their consistence and independ-
ence be demonstrated, and the science exposed in its
ultimate form.
These second and third stages are even more essential
to a " liberal education " than the first, for they exhibit
scientific method and human knowledge in their true aspect.
It is not suggested that they can be dealt with in schools,
except perhaps tentatively in the last year of a long course.
But it is definitely asserted that the general ideas involved
should form part of the compulsory element of every Uni-
versity course, even though details be excluded, for they
are of the very essence of the spirit of mathematics. The
method of developing such ideas remains to be considered.
It may be presumed that the pupils concerned have some
knowledge of arithmetic, geometry, the calculus, and
mechanics, each subject having been developed from a
12 MATHEMATICAL EDUCATION
redundant set of postulates. In which, then, of these four
branches is it most natural to suggest the analysis of these
assumptions ?
Since analysis of the known may still be presumed to
have its dangers, the branch chosen must be that one in
which the investigation bears this aspect least prominently.
Now the main ideas of arithmetic, geometry, and the cal-
culus are so firmly held by boys and girls, that any attempt
to discuss them in detail produces revolt or boredom. Such
attempts account for much ; the writer can well remember
his feelings on first seeing a formal proof that the sum of
a definite number of continuous functions is itself a con-
tinuous function ; and at the same time he realized to the
full that the proposition might well be untrue if the num-
ber of functions were not finite. Ground such as this is
unfavourable for the development of this new analysis.
The same is by no means true of mechanics. Here the
postulates, acceptable though they be, have been elucidated
within the memory of the pupils, and they may reasonably
be asked to examine the facility with which these assump-
tions were made, and to consider whether the evidence can
in any way be strengthened. This being done, the ideas
of consistence and redundance can be developed, and some
idea of the structure of a science imparted. Even then it
may probably be wise to lay little stress on analysis of the
geometrical postulates ; if the ideas are realized in connec-
tion with mechanics, we may well leave the seed to mature
in minds to which it is congenial.
In the view of mathematics here taken, its various
branches are regarded as structures with many possible
entrances, and the discussion has been concerned with the
PRINCIPLES OF EDUCATION 13
choice of entrance and the route to be taken through the
edifice. We cannot hope that our pupils will ever know
more than the outline of each structure. Even we who are
the guides cannot know each detail of any one ; the laby-
rinth is too vast. But the best guide to a structure is he
who knows its main outlines most completely, and a teacher
who has clear ideas of mathematical principles can do much,
in leading his pupils through such avenues of the structure
as they can attain, to give them a view of the whole. Of
the import and beauty of this view more need not be said.
INTUITION
(An address delivered to the Mathematical Association, and reprinted
from The Mathematical Gazette, March, 1913)
INTUITION
If there be one duty more incumbent than any other
upon mathematicians, it is to have a clear and common
understanding of every term which they use. I do not
say a formal definition, though that is most advisable if
and when it can be obtained ; but a class of entities must
be known and recognised before it can be defined, and no
term should be used unless it at least gives rise to definite,
recognisable, and identical images in the minds of the
speaker and listener. It cannot fairly be said that mathe-
maticians are at fault in this respect, when dealing with
their own special subjects ; but I fear they cannot so easily
be acquitted when discussing the didactic side of their
work. Concrete and utilitarian, axiom and postulate, intu-
ition and assumption ; how many of us have definite
meanings for these terms, and can feel certain that they
represent the same meanings to others ? The term which
I have chosen as the title of this paper is one of the most
commonly used and, as it seems to me, most often misun-
derstood ; at the same time, the ideas and processes for
which it stands lie at the root of all elementary teaching.
I have therefore thought it worth while to discuss its
meaning and to show the bearing of the process on math-
ematical education.
There is, I think, little doubt that to most of those who
use the term " intuition," it connotes some peculiar quality
of material certainty. Take, for example, the equality of
17
1 8 MATHEMATICAL EDUCATION
all right angles, or the angle properties of parallel lines,
and ask one who understands these statements with what
degree of certainty he asserts their truth. It will be found
almost invariably that he regards them as far more certain
than statements such as "the sun will rise to-morrow
morning " or " all men are mortal " ; these, he admits,
might be upset by some perversion of the order which he
has regarded as customary, but the geometrical statements
appear to be of the essential nature of things, eternal and
invariable verities. So much, indeed, is this the case that
the very idea of practical tests is grotesque. Who has ever
experimented to ascertain whether, if two pieces of paper
are folded, and the folds doubled again on themselves, the
corners so formed are superposable ? If the individual
under examination be questioned as to the basis for this
faith, he can only reply that it is the nature of things, or
that he knows it intuitively ; of the degree of his faith
there is no doubt. It is to statements asserted in this
manner that the term " intuition " is commonly applied ;
other facts, such as the mortality of all men, which are
justified by the fact that all human experience points to
them, are not classified under this heading nor, as I have
said, are they accepted with the same faith.
These alleged certainties can of course be dissipated by
purely philosophical considerations concerning the relations
and differences between concepts and percepts ; but " an
ounce of practice is worth a ton of theory," and I propose
here to show, mainly from historical considerations, that
there is no ground for absolute faith in certain intuitions,
however tenaciously they may be held. Take first the idea,
still held by many, that a body in motion must be urged
INTUITION 19
on by some external agent if its velocity is to be main-
tained. Until the time of Galileo this belief was held uni-
versally, even men of eminence who had considered the
subject being convinced of its truth. Now this faith was of
just such a kind, and just as strongly held, as the faith in
geometrical statements which I have mentioned ; it was,
and still is by many, regarded as in the nature of things
that a body should stop moving unless it is propelled by
some external agency. And yet others, of whom Galileo
was the forerunner, see the nature of things in a light
wholly different. They regard it as utterly certain that a
body can of itself neither increase nor retard its own mo-
tion. Ask a clever boy who has learnt some mechanics,
or even a graduate who has not thought overmuch on the
foundations of the subject, which he regards as more un-
likely : that an isolated body should, contrary to Newton's
first law, set itself in motion, or that the secret of immor-
tality should be discovered. He will tell you that the
second might happen, though personally he does not be-
lieve that it ever will ; but that a body can never begin to
move unless it has some other body " to lever against."
We thus see two contradictory intuitions in existence, each
held with equal strength.
Coming to more recent history, let me remind you of
the development of the theory of parallels, and the rise
of non-Euclidean geometries. Until the last century it
may fairly be said that no one had ventured to doubt the
so-called truth of the parallel postulate, though many emi-
nent mathematicians had endeavoured to deduce it from
the other postulates of geometry. The genius of Bolyai
and Lobachewsky, however, put the matter in quite another
20 MATHEMATICAL EDUCATION
light. They showed that a completely different theory of
parallels was just as much in accord with the nature of
things as that hitherto held; and that, to beings with
more extended experience or finer perceptions than ours,
this different theory might appear to correspond with ob-
servation while the current belief failed to do so. In other
words, they showed that there are several ways of account-
ing for such space observations as we can make with our
restricted opportunities ; just as it was then well known
that there were two theories which fitted the observations
of astronomers, of which Newton's was the more simple
and self-consistent.
. It thus becomes clear that intuitions are no more than
working hypotheses or assumptions ; they are on the same
footing as the primary assumptions concerning gravitation,
electrostatics, or any other branch of knowledge based on
sensation. They differ from these in that they are formed
unconsciously, as a result of universal experience rather
than conscious experiment ; and they are so formed in
regard to those experiences space and motion which
are forced on all of us in virtue of our existence. It is not
implied that their possessor is even fully conscious of them ;
ask some comparatively untrained adult how to test rulers
for straightness, and he may be at a loss or give some in-
effective reply ; but suggest placing them back to back
and then reversing one, and he at once assents. He re-
gards this not as new information, but as something so
simple and obvious that it had not occurred to him. It is
to him the essential nature of things ; he has held this
view from so early an age, and it has remained so entirely
free from challenge, that he revolts at the suggestion that
INTUITION 21
things, viewed from another standpoint, may appear to
have a different nature.
The formation of such working hypotheses is the normal
method by which the mind investigates natural phenom-
ena. After observation of a certain set of events, a theory
is formed to fit them, the simplest being chosen if more
than one be found to fit the facts equally well. This theory
is developed, and its consequences compared with the
results of further observations ; so long as these are in
accord, and so long as no simpler theory is found to ac-
commodate the fact, the first theory holds the field. But,
should either of these events occur, it is abandoned ruth-
lessly in favour of some better description of the recorded
observations. There are famous historical cases of each
event; Newton's corpuscular theory of light yielded de-
ductions in actual disaccord with observation, and was
therefore abandoned. The ancient theory of astronomy,
wherein the stars were imagined to be fixed on a crystal
sphere on which the planets travelled in epicycles, was
abandoned in favour of the modern theory, not because it
could not be modified to accord with observation, but be-
cause of its greater complexity. In every such case the
question of absolute truth is irrelevant and beyond our
reach ; the problem is to find the simplest theory in accord
with all the facts, abandoning in the quest each theory as
a successor Is found which better fulfils these requirements.
Shortly, then, we may say that intuitions are merely a
particular class of assumptions or postulates, such as form
the basis of every science. They are distinguished from
other postulates first, in that they, with their subject
matter for example, space or motion are common
22 MATHEMATICAL EDUCATION
from an early age to every human being endowed with the
ordinary senses ; and secondly, in that no other assump-
tions fitting the sensations concerned ever occur to those
who make them. Their formation is forgotten, and they
are therefore regarded as eternal ; they hold the field un-
challenged, and are therefore regarded as inviolable.
Before passing to the consideration of the bearing of
intuition on the teaching of mathematics, it may be well
to illustrate what has been said by the consideration of a
few particular cases.
First, suppose that one sees a jar on a shelf, and puts
his hand up to find out whether it is empty. Is the act
based on an intuition from the appearance of the jar?
This is not the case ; if asked before the act, one would
not express any final certainty that the hand could enter
the jar ; it might have a lid or be a dummy. The in-
dividual can make more than one assumption which corre-
sponds to the sight sensation ; the first assumption made
that the hand can enter the jar is merely the most
likely as judged by experience.
Next suppose that a knock is heard in a room. The
natural exclamation " What is that ? " is based on intuition,
for it expresses the now universal conviction that such a
noise is an invariable accompaniment of some happening
which, given opportunity, will also appeal to the other
senses. Accumulation of human experience has led to the
belief that such is invariably the case ; but belief it is, and
not certainty. If the reply were, " It is nothing ; under no
circumstances could you have correlated any sight or feel-
ing with that sound," it would be received with complete
incredulity.
INTUITION 23
Consider again the statement that, given a sufficient
number of weights, no matter how small, one can with
them balance a single weight, however large. No one
would doubt this or treat it as anything but the most obvi-
ous of truisms, and yet it is a pure assumption, formed
unconsciously as the result of general experience ; it an-
swers in every respect to our definition of an intuition. It
may be thought by some that the statement can be proved
arithmetically, but in every such alleged proof the assump-
tion itself will be found somewhere concealed. We have,
in fact, no warrant for assuming that the phenomenon
called weight retains the same character, or even exists,
for portions of matter which are so small as to be beyond
our powers of subdivision.
Finally, consider the statement, "I knew intuitively that
you would come to-day." In what respect do those who
use it regard it as differing from, " I thought it was almost
certain that you would come to-day " ? It may fairly be
said that the former expresses less basis of knowledge but
more feeling of certainty than the latter ; it means, " I
don't in the least know how I knew it, but I did know
beyond all doubt that you would come." Such ideas, with
or without the use of the actual term " intuition," are com-
mon enough. They are here quoted to justify the statement,
made above, that the term connotes to many of those who
use it some peculiar degree of certainty. Such statements
are not intuitions ; they are mere superstitions, and those
who are subject to them fail to realise how often they are
unjustified by the event. Belief in the absolute truth of
the angle properties of parallels or of the Laws of Motion is
equally a superstition, though these are, until now, justified
24 MATHEMATICAL EDUCATION
by the event. The truth is that they can never receive this
absolute justification, for no material observation is beyond
the possibility of error, nor can it be certain that some
simpler theory will not be formed, accounting equally well
for the observations ; it is the belief in this impossible
finality which constitutes the superstition.
Turning now to the more educational aspect of the
subject, the first problem which confronts us is this :
children, when they commence mathematics, have formed
many intuitions concerning space and motion; are they
to be adopted and used as postulates without question, to
be tacitly ignored, or to be attacked ? Hitherto teaching
methods have tended to ignore or attack such intuitions ;
instances of their adoption are almost non-existent. This
statement may cause surprise, but I propose to justify it
by classifying methods which have been used under one
or other of the two first heads, and I shall urge that com-
plete adoption is the only method proper to a first course
in mathematics.
Consider first the treatment of formal geometry, either
that of Euclid or of almost any of his modern rivals ; in
every case intuition is ignored to a greater or less extent.
Euclid, of set purpose, pushes this policy to an extreme ;
but all his competitors have adopted it in some degree
at least. Deductions of certain statements still persist, al-
though they at once command acceptance when expressed
in non-technical form. For example, it is still shown in
elementary text-books that every chord of a circle perpen-
dicular to a diameter is bisected by that diameter. Draw a
circle on a wall, then draw the horizontal diameter, mark
a point on it, and ask any one you please whether he will
INTUITION 25
get to the circle more quickly by going straight up or
straight down from this point. Is there any doubt as to
the answer ? 1 And are not those who deduce the propo-
sition just quoted, from statements no more acceptable,
ignoring the intuition which is exposed in the immediate
answer to the question ? All that we do in using such
methods is to make a chary use of intuition in order to
reduce the detailed reasoning of Euclid's scheme ; our
attitude is that statements which are accepted intuitively
should nevertheless be deduced from others of the same
class, unless the proofs are too involved for the juvenile
mind. We oscillate to and fro between the Scylla of ac-
ceptance and the Charybdis of proof, according as the one
is more revolting to ourselves or the other to our pupils.
At this point I wish to suggest that a distinction should be
drawn between the terms " deduction " and "proof." There
is no doubt that proof implies access of material conviction,
while deduction implies a purely logical process in which
premisses and conclusion may be possible or impossible of
acceptance. A proof is thus a particular kind of deduction,
wherein the premisses are acceptable (intuitions, for exam-
ple), and the conclusion is not acceptable until the proof
carries conviction, in virtue of the premisses on which it
is based. For example, Euclid deduces the already accept-
able statement that any two sides of a triangle are together
greater than the third side from the premiss (inter alia}
that all right angles are equal to one another ; but he proves
that triangles on the same base and between the same
1 There is often apparent doubt; but it will usually be found that
this is due to an attempt to estimate the want of truth of the circle as
drawn.
26 MATHEMATICAL EDUCATION
parallels are equal in area, starting from acceptable prem-
isses concerning congruent figures and converging lines.
The distinction has didactic importance, because pupils
can appreciate and obtain proofs long before they can
understand the value of deductions ; and it has scientific
importance, because the functions of proof and deduction
are entirely different. Proofs are used in the erection of
the superstructure of a science, deductions in an analysis
of its foundations, undertaken in order to ascertain the
number and nature of independent assumptions involved
therein. If two intuitions or assumptions, A and B, have
been adopted, and if we find that B can be deduced from
A, and A from B, then only one assumption is involved,
and we have so much the more faith in the bases of the
science. Herein lies the value of deducing one accepted
statement from another ; the element of ddubt involved in
each acceptance is thereby reduced.
Next, to justify the statement that intuition has been
attacked. Both Euclid and his modern rivals knew well
enough that their schemes must be based on some set of
assumptions ; they differed only in the choice. Each agrees
that intuitive assumptions are undesirable, but the modern
school regards the extreme logic entailed by Euclid's
principle of the minimum of assumption as impossible
for young pupils. There is, however, a third school which
pursues a different course ; it professes to replace intuition
by experimental demonstration. Pupils are directed to draw
pairs of intersecting lines, measure the vertically opposite
angles, and state what they observe ; to perform similar
processes for isosceles triangles, parallel lines, and so on.
Instead of being asked, " Do you think that, if these lines
INTUITION 27
were really straight, and you cut out the shaded pieces,
the corners would fit ? " they are told to find out, by a
clumsy method, a belief which they had previously held,
though it had never, perhaps, entered definitely into their
consciousness. The question suggested is, in these homely
terms, just sufficient to bring the idea before them, and it
is at once recognised as according with the child's previous
notions ; he does not regard it as new, but merely as some-
thing of which he had not before thought so definitely.
It is this type of exercise in drawing and measurement
which I regard as an attack upon intuition. It replaces
this natural and inevitable process by hasty generalisation
from experiments of the crudest type. Some advocates of
these exercises defend them on the ground that they lead
to the formation of intuitions, and that the pupils were not
previously cognisant of the facts involved. But in the first
place, a conscious induction from deliberate experiments
is not an intuition ; it lacks each of the special elements
connoted by the term. And as to the alleged ignorance of
the elementary idea of space, it appears to me to be a mis-
taken impression, based on undoubted ignorance of mathe-
matical terminology. If you say to a child of twelve, "Are
these angles equal?" he has to stop to think first, what an
angle is, and next, when angles are equal ; by the time he
has done this his mind is incapable of grasping the pecul-
iar relations of the angles in question, and he is labelled
as ignorant of the answer. The real difficulty, and it is
not a small one, is to lead the child to express familiar
facts in precise mathematical terminology; to say "angles
equal " rather than " corners fit." Until this terminology
is thoroughly familiar, the effort of using it must absorb
28 MATHEMATICAL EDUCATION
a large part of the child's attention, leaving little available
for the matter in hand. This paper is not concerned with
the methods or practice of teaching, but I would strongly
urge all those who are concerned with young children to
guard against this danger, by constant transition to and
fro between common and technical phraseology, appealing
at once to the former at the least sign of doubt or hesita-
tion. 1 The learning of technical terms should not appear
as part of the definite work, or it will inevitably be regarded
as the major part ; it should come incidentally and by
gradual transition, as I have suggested.
The only alternative to this evasion or suppression of
intuition is to accept it from the commencement as the
natural basis for primary education. But to be of any avail,
the acceptance must be unquestioned and complete ; every
intuition which can be formed by the pupils must, without
suggestion of doubt, be adopted as a postulate, none being
deduced from others which are themselves no more easy
of acceptance. Such a course leads, it need hardly be said,
to considerable simplification in the early treatment of any
subject. For example, in geometry the angle properties of
parallel lines, properties of figures evident from symmetry,
and the theory of similar figures (excluding areas) appear
as postulates ; in the calculus it is not proved that the
differential coefficient of the sum of a finite number of
functions is equal to the sum of their differential coeffi-
cients ; the statement is illustrated by, say, consideration
1 It is no good to say, " Come now, what is an angle ? " Appeal first
to the tangible fact in the child's mind by saying, " Cannot you see that
those corners must fit?" and then remind him that "equal angles"
merely means the same thing.
INTUITION 29
of some expanding rods placed end to end, and at once
commands acceptance. Here the question of terminology
again arises ; I have often been struck, in teaching school-
boys and students, by their slowness to accept this and
similar results in the calculus ; the clue was given to me
by a boy who remarked that it was taking him all his time
to remember what a rate of increase was, and he could not
manage any more at the moment. Since that time I have
avoided many seeming difficulties with elementary and
advanced pupils by appeal from technical to familiar terms,
always of course rephrasing the result in the proper form
before leaving the matter in hand.
It will, I know, be thought by many that this adoption of
all natural intuitions involves an appalling lack of rigour.
But I would ask those who are of this opinion to do one
thing before passing judgment, and that is, to define and ex-
emplify with some care the meaning of the term "rigour."
When they have done this, I think they may be disposed
to agree with the answer to their accusation which I am
now going to put forward. It is that the scheme suggested
is perfectly rigorous, provided that every deduction made
from the postulates adopted is logically sound; on the other
hand, it is admitted that the mathematical training thus
imparted is not complete, because no attempt has been
made to analyse these intuitive postulates into their com-
ponent parts, showing how many must perforce be adopted
in the most complete system of deduction. In other words,
we may be rigorous in regard to logical reasoning, or in
regard to lessening the number of assumptions which form
the basis of a science. The view for which I contend
is, that in all stages of mathematical education, deductions
30 MATHEMATICAL EDUCATION
from the assumptions made should be rigorous ; but that
in the earlier stages every acceptable statement or intuition
should be taken as an assumption, the analysis of these,
to show on how small an amount of assumption the science
can be based, being deferred.
To avert misapprehension, let me say again that I pro-
pose that, when all intuitions are accepted as postulates,
this should be done without question or discussion other
than that necessary to give them some precision. To em-
bark on a discussion of their nature, or to appear to cast
doubt upon them, would be fatal, as fatal as has been the
apparently futile process of deducing one accepted state-
ment from another. The pupil is already in possession of
a body of accepted truth ; let us build on that and defer
its analysis, or anything that pertains thereto, until he is
sufficiently mature to appreciate the motive.
The first course of mathematics would, then, range
from arithmetic and analysis through geometry to mechan-
ics. In this last subject there is little scope for intuition.
Most of the mechanical intuitions formed by the race as a
whole have been mistaken, and it is just this fact which
gives some indication of the proper commencement for the
second course, in which the intuitive postulates are to be
analysed and reduced as far as possible. Let the student
learn something of the history of mechanics, realising that
ideas which he regards as impossible and absurd were
held, by men of great eminence, with faith just as strong
as that which he places in his geometrical postulates. Then
let it be suggested to him that this renders care in regard
to assumption of vital importance, and so commence an
analysis of the mechanical postulates, hitherto redundant,
INTUITION 31
obtaining deductions of one from another to show their
inter-connection. This completed, and the task is not a
large one, it is natural to suggest that the postulates of
geometry deserve some examination, and so, according to
the time available and the ability of the pupil, we may pass
backward through a review of the foundations of geometry
to an examination of the foundations of analysis and arith-
metic. It is not, of course, implied that every student of
mathematics can reach this goal ; few can ever get beyond
some consideration of the foundations of geometry, with a
clear understanding of the end to be attained in its general
application to all sciences. But I do wish to put forward,
with such emphasis as I can, this general scheme of math-
ematical education ; namely, an upward progress, based on
intuition, from arithmetic through geometry to mechanics,
followed by consideration in the reverse order of the founda-
tions of each branch, the upward progress constituting the
first course, and the downward review the second course.
It would, I believe, give an intelligible unity to the whole
subject, and would do something to restore that purely
intellectual appreciation which has so largely declined
during the past generation.
Mathematics is a useful tool, but it is also something
far greater, for it presents in unsullied outline that model
after which all scientific thought must be cast. I have
endeavoured to show how this outline may be developed,
starting from those intuitions which are common to us all,
and ending in an analysis demonstrating their true nature.
The concrete illustrations, so necessary and illuminating
in elementary teaching, are so many draperies, fashioned
to render this outline visible to those who cannot otherwise
32 MATHEMATICAL EDUCATION
appreciate it. Even the several branches analysis, geom-
etry, mechanics serve the same end; behind them all
is the one pure structure of mathematical thought. They
who most appreciate the structure will best fashion the
draperies, and so render it most clearly visible to those
whom they instruct.
THE USEFUL AND THE REAL
THE USEFUL AND THE REAL
Among the many changes in mathematical education
during the last twenty years, and among the many and
often conflicting ideals which have directed these changes,
one element at least appears throughout ; a desire to relate
the subject to reality, to exhibit it as a living body of thought
which can and does influence human life at a multitude of
points. The old scholastic ideal of development in the
most abstract way, the realities being allowed to take care
of themselves, is exploded for this as for most branches of
education ; it is recognised that the separated mediaeval
worlds of thought and action must be replaced by a single
world wherein each exerts profound influences on the other.
Our children must learn to thmk > _ajld to think about the
'-wdrld asTt now is and the manner of its evolution. Some
few there may be who can with profit to us all devote
themselves to one or other side of this world of thought
and action, but the mass of men must be fitted to play
their part between the two.
So far all are agreed, but community of pious opinion
has before now been known to result in discord, and dis-
cord none the less acute because due to diversity of policy
alone. Such has been the case with mathematical educa-
tion ; the community of ideals just described has not
resulted in community of action ; it is more nearly true
that each man is a law unto himself in his method of for-
warding them. Like most disorganised armies we have
35
36 MATHEMATICAL EDUCATION
our shibboleths, and among the most prominent are "real,"
"useful," "concrete." An examination of what these do
and should represent may not be without profit.
Starting from agreement that the world of thought is to
be related to the world of action or reality (not thereby
dependent upon or limited by that world), the natural
course is to attempt to form some concept of the particular
world of reality with which we are concerned. Suppose
that one desires to explain the principles of the calculus to
an assemblage of doctors ; tables of population, mortality,
and the like form one obvious world of reality from which
thought can be developed ; if to an assemblage of mer-
chants, statistics of trade and finance would form such a
world, and so on for other avocations. But suppose that
the assembly consisted of men engaged in no one pursuit ;
the difficulty would be greatly enhanced, for there would
be no obvious world of reality common and familiar to
them all. So also in his dealings with young children
must the teacher of mathematics determine fitting worlds
of reality and develop his instruction for them.
But what is reality ? What considerations determine
the entities which have this attribute ? For me, my hands,
my furniture, this town, England, Cromwell, Macbeth, the
binomial theorem, are all real ; but Cromwell's hands, the
furniture in a strange house, Hepscott (I take the name
at random from a gazetteer), Fanning Island, Ben Jonson,
Hedda Gabler, nitrates all lack reality. Each one of the
first is related to some definite recognisable sensation or
concept of my own, but each of the second is (for me) a
mere name which bears no relation to any such sensation or
concept ; I know nothing of Fanning Island, nor sufficient
THE USEFUL AND THE REAL 37
of Ben Jonson to distinguish him from other writers
of his time; I have not read Ibsen, and I know little
chemistry. The essence of reality is thus found in definite
recognisable percepts or concepts, and is therefore a func-
tion of the individual and the time ; what is real to me is
not necessarily real to another, and much that was real to
me in childhood is no longer so. It is for the teacher to
determine the realities of his pupils and exemplify mathe-
matical principles by as many as are suitable for the purpose.
He will also find it necessary to enlarge their spheres of
reality, but he must avoid confusion between a name and
a thing ; he must, for example, make sure that his pupils
know what a parallelogram is before they use the name.
It is at this point that various policies have arisen, des-
pite general agreement on ideals. One of these confuses
the many worlds of reality, different for each individual,
with some absolute world of reality supposed to be com-
mon to all. This absolute world is usually based on those
applications of mathematics which have some commercial
or scientific utility, such utility being considered to involve
reality for the pupil. The result of this confusion of the
useful with the real is seen in problems which deal with
such mysteries as resistance in pounds per ton weight, the
extension of helical springs, efficiency and load, ton-inches
of twisting moment. To all children (and many adults)
these phrases are as meaningless as the symbols of the
purely scholastic algebra of thirty years ago; they are
merely a cumbrous way of writing the x and y of that alge-
bra and imply as little to those for whom they are intended.
But their use may tend to impart an idea that realities are
being dealt with an idea thoroughly vicious in that it
38 MATHEMATICAL EDUCATION
replaces entities by words. We may name entities which
are direct sensations, and we may name entities which are
pure creations of the imagination ; but to imagine that a
name which is co-related to neither sensation nor imagina-
tion possesses any sort of reality is the grossest of errors.
Too many teachers are content to use words for which
they have no definite meanings, and to allow their pupils
to imagine that they have acquired something in learning
such words ; but we need not go out of our way to spread
this error, the more so as we are concerned with the one
subject which should suppress it most completely.
It is not, of course, suggested that any existing courses
of mathematics are limited to such applications alone. But
there is an obvious tendency to judge applications by such
standards, attributing more and more importance to those
which accord with them. An excellent instance of such
judgments is the condemnation of the traditional problems
dealing with tanks which are emptied and filled simultane-
ously by different pipes. It is argued that no adult ever
deals with a cistern in this way, and that the problems
should therefore be replaced by others having more reality.
The term "reality" begs the whole question, for it has no
absolute meaning for all people at all ages and must be
defined by those who use it. And it is here confused with
utility, a very different attribute. The essence of the con-
tention is that no application should be used in education
unless it is of actual use in some branch of science or walk
of life. This is a far cry from the pupil's world of reality ;
the formalist attempted to transport him to a world of ab-
stract thought wherein the entities are typified by letters
x and y, but our utilitarian proposes to limit the play of
THE USEFUL AND THE REAL 39
his imagination to matters used by adults, no matter how
far these may be removed from his cognisance or interest.
There is at bottom little difference between the two, but
the formalist is the more open in that he does not cloak
his meaning under a mass of words which are full of sound
and signify nothing.
A variant of this school may reply that they are being
unjustly accused ; that they are in entire accord with the
rejection of matters such as voltage and twisting moment
on the ground that they have no reality for the pupils con-
cerned, but that there are plenty of applications which are
real and also useful. This may be so, though examination
of modern text-books hardly supports the claim ; but in
any case we cannot on such lines develop mathematical
thought from any large portion of the pupil's world of
reality ; it is related to those parts only of that world which
coincide with the worlds of various adults, and these may
well be neither the most interesting nor the most familiar
portions of his own world.
A second policy, exemplified for the most part in con-
nection with geometry, interprets the child's world of reality
as the world of his senses, and more particularly the senses
of sight and touch, and so is allied with the concrete rather
than the useful. It endeavours to develop thought from
manipulations and measurements performed by the pupil
himself, and is thus limited to the perceptory or concrete
portion of his realities. In itself and so far as it goes this
is an entire and most valuable gain as compared with the
practice of thirty years ago ; the pupils feel that they are
dealing with matters within their own personal cognisance
instead of abstractions which are evidently familiar only
40 MATHEMATICAL EDUCATION
to men with whom, intellectually, they have and will have
little in common.
Unfortunately, however, there has been a strong tend-
ency to limit the work to this concrete domain, refusing
any part to that world of imagination which is, especially in
children, just as real and a great deal more vivid. Intro-
ductory courses of geometry consist of the construction
and measurement by the pupil of figures whose dimensions
are prescribed. They develop a detailed knowledge of per-
ceptory space but make no use of that much larger and
more important conceptual space wherein the creations of
his imagination move and have their being. Travels, ad-
ventures, romances, history, and the hundred and one
utterly useless but apparently practical things which interest
a boy are situated in this space, and here, as well as in the
smaller concrete space of the senses, should the world of
thought be exemplified, for these things also are realities
for him.
Three distinct policies have now been discussed : the
first rejects all applications and insists throughout on devel-
opment in abstract terms ; the second insists that illustra-
tions must be drawn from applications which are relevant
to some branch of science, industry, or commerce ; and the
third insists that development must originate in the imme-
diate evidence of the senses. Of course, no man or body
of men holds one of these views to the entire exclusion of
the other two, nor is the world of imagination entirely
ignored in current practice. But most text-books and writ-
ings on mathematical education are influenced mainly by
some one of them, and may be placed in a class which
holds that particular policy as paramount. There is most in
THE USEFUL AND THE REAL 41
common between those who hold the second or third view,
for they give a common allegiance to the use of reality and
differ only in the scope of the term. Many teachers are,
indeed, influenced by considerations of utility in algebra
and considerations of concrete reality in geometry, their
utterances on one subject often contradicting those on
the other.
Now among the many uncertainties and conflicts which
surround these (as all) questions of education, two state-
ments at least stand out as certain beyond dispute. The
first is that the operations and processes of mathematics
are in practice concerned at least as much with creations
of the imagination as with the evidences of the senses ; it
is enough to mention points, complex numbers, ether,
electric charges, to make this plain. The second is that
the purpose of mathematical education is to put the pupil
in a " mathematical way " ; to permeate his whole being
with the elementary principles of the science so that he
will apply them spontaneously in considering any matter
to which they may be relevant. The formalists held that
if principles were imparted in their utmost generality, each
individual could and would make such applications as he
might require, a statement not justified by experience and
not in accord with such knowledge of the mind as we
possess ; the moderns believe that principles can only be
seen by their exemplification throughout the world of reality
of the pupil. The formalists thus seek unity of treatment
for a class in generality of presentation, the moderns seek
it among the experiences and concepts of the various pupils.
Fortunately for education in general, this modern search
is certain to prove successful as regards children, because
42 MATHEMATICAL EDUCATION
their experiences and imaginations run in grooves more or
less alike ; they are interested in puzzles, hidden treasures,
travels, railways, ships, and the like, and problems concerned
with these entities are real to them no matter how absurd
they appear from the standpoint of practical life. Their
educational utility is not to be measured by their commercial
or scientific value, but by their degree of reality for the
pupils under instruction.
Putting the matter in more or less mathematical phrase-
ology, we may say that the mathematical instruction of a
beginner must be exemplified by a maximum number of
his realities in order that the principles may permeate his
whole being ; in dealing with a class we must therefore
find the greatest common measure of their realities and
work from that. If the class is composed of adults having
varying antecedents, this common measure may be small
compared with the realities of any one member ; but if the
pupils are children, it is large in comparison with their
individual realities, and the task of the teacher is corre-
spondingly simplified.
Leaving generalities which may have appeared somewhat
vague, we may now consider a few problems which are real
for children but not directly useful to them or any one else.
First take the type already mentioned, which deals with
the emptying and filling of a tank. There is no doubt that
this is sufficiently real for any child ; he can visualise the
whole process, and its value is increased because the entities
are imagined and not perceived through the senses. The
purpose served is the exemplification of the method of
adding or comparing several rates by reducing all to a
common unit, an idea sufficiently important in after life.
THE USEFUL AND THE REAL 43
Those who attack such an illustration must find others
which will serve the same purpose and satisfy their test of
utility, and in doing this they will in all probability pass
beyond the limits of reality. There is no doubt that the
emptying of cisterns, the coincidence of clock hands, and
other seeming trivialities do exemplify the handling of rates
in ways which are more real to young children than others
which have more actual utility, and they are therefore to
be welcomed rather than condemned. The mistake in their
treatment, and as gross a mistake as could well be made,
has been their grouping by subject matter instead of prin-
ciple. All questions which deal with one principle should be
grouped together and the subject matter varied continually.
Next consider the Progressions, which have of late been
attacked on the score that they are comparatively useless
in mathematics or anywhere else. This is true, and ad-
vocates of their retention have done their cause no good
by saying vaguely that they have their uses and then fail-
ing to give specific instances. They do, however, provide a
number of problems which have reality for children, and
they exemplify three most important matters : the concept
of a series, the value of which extends far beyond math-
ematics ; the insight which can be gained by a proper
grouping of various entities ; and the construction of a
formula or law to cover any number of discrete cases.
Consider again the well-known problem in the calculus
of a man who is on a common and wishes to reach a point
on a straight road, along which he can walk more quickly
than on open country, as soon as possible. If such a problem
ever has practical utility, it is not for one man in ten thou-
sand, and to regard it as in any way generally useful is
44 MATHEMATICAL EDUCATION
obviously grotesque ; but again it is real for those who study
it, and it exemplifies the comparison of different modes of
transition from one state to another, and the selection of
the most suitable.
Another illustration is provided by the use of statistical
graphs in the introduction of the calculus. Such graphs
are of service in exemplifying the meaning of a differential
coefficient and a definite integral by means which possess
reality for the students, and their whole function is described
in this statement. Now it may well happen that a set of
statistics which have no practical use of any kind, or are
even in actual disaccord with the results of some branch of
science or industry, may serve these purposes better than
others which have some direct use or are in accord with
experience. For example, excellent problems can be made
concerning the consumption of coal by locomotives, but
they would never occur in the practice of any engineer, nor
would the numbers which happen to give good graphs
occur in the working of any conceivable locomotive. But
this is in no way to the detriment of the problem for the
purposes of instruction. The inaccuracy of the information
contained in the figures is surely immaterial if students are
told that actual numbers can be found in any handbook for
engineers should they ever chance to need them, and no
other objection seems relevant to the purpose of the problem.
It exemplifies principles through illustrations which are real
for the particular students, 1 and thus fulfils its aim.
These examples exhibit the tests by which applications
should be judged. They must exemplify those leading ideas
1 They would not be real for a class of locomotive engineers, and the
example would not be used for such a class.
THE USEFUL AND THE REAL 45
which it is desired to impart, and they must do so through
media which are real to those under instruction. The reality
is found in the students, the utility in their acquisition of
principles.
The outcome of our discussion is, then, that illustrations
must above all be real ; they must be useful as well, if
that be possible, and particularly with reference to other
branches of study such as physics ; but reality is the crucial
test. And reality is a function of the individual and the
time, so that no absolute schedule of the more and the
less real can be devised ; but there is sufficient community
between children of the same age to handle them in
groups, while adults might, on the other hand, require
classification in regard to their realities before they could
receive efficient instruction in groups. Many problems
which interest and even excite children are to them hope-
lessly banal, and others must be used more in touch with
their particular spheres of reality. Finally, each principle
must be exemplified in as many ways as possible so that
unity may be perceived in principle rather than subject
matter.
We have travelled far from the useful applications of
mathematics in our quest for fitting illustrations ; we have
been led to consider reality as the proper criterion, and to
recognise that the term is essentially relative. But so also
is "useful " a relative term ; what is useful for one purpose
is useless for another, and it may well be said that many
applications of mathematics which are grotesquely useless
in any branch of science or commerce are of the utmost
use in education for their vivid illustration of ideas so
abstract as to be otherwise vague or invisible.
SOME UNREALISED POSSIBILITIES
OF MATHEMATICAL EDUCATION
(An address delivered to the Mathematical Association, and reprinted
from The Mathematical Gazette, March, 1912)
SOME UNREALISED POSSIBILITIES OF
MATHEMATICAL EDUCATION
The last half-century has seen a great and significant
change in the popular estimation of mathematics. Formerly
the subject was regarded as utterly unpractical and there-
fore useless in the narrow sense of this term, though it
was recognised as providing a training unique in its char-
acter, in logical thought and in accurate expression. Now
it is regarded, and correctly regarded, as having enormous
practical importance in science and engineering. Most, if
not all, of those discoveries and inventions which are so
profoundly modifying civic and national life have found
their origin, or development, or both, in the labours of
mathematicians, and this fact is widely known. The
mathematician is no longer regarded as a dreamer of
dreams ; he is classed with the doctor, the engineer, the
chemist, and all those whose specialised labours have had
immense import for the human race.
But simultaneously a change of no less magnitude has
taken place in the mathematical world. The type of in-
vestigation which bore such fruit in the hands of Faraday,
Clerk Maxwell, Kelvin, and many others no longer occupies
the attention of those who are in the forefront of mathe-
matical investigation. The theories of pure number, of
space, of functions, and such names as Dedekind, Cantor,
Grassmann, Klein, and, in our own country, Hobson,
Whitehead, and Russell, have little or no connotation for
49
50 MATHEMATICAL EDUCATION
the outer world. In so far as this outer world is cognisant
of their existence, these theories, and the men to whom
they are due, appear as chimerical and unpractical as would
the labours of Clerk Maxwell have appeared to the Lan-
cashire cotton spinner of 1850. And I fear that this view
is too often shared, consciously or unconsciously, by mathe-
maticians themselves, and especially by those who teach
the subject. Here, they say or think, is a type of thought
or investigation of great interest to those who can appre-
ciate it, but it is utterly and permanently out of touch with
the world at large. It can have no relevance or import for
the ordinary boys and girls who learn mathematics at school,
and can in no way assist them to become efficient citizens.
But is this really the case ? He would be a bold man
who would say with certainty that any branch of scientific
investigation must be regarded, once for all, as having no
bearing on the development of the individual or race. Is
not the better answer that the practical import of these
investigations has not yet been perceived ; that it behoves
all mathematicians, but especially those who are engaged
in teaching, and therefore have some knowledge of the
youthful mind, to do what they can to correlate this work
with the outer world, and to examine to what extent it can
now influence the manner or matter of teaching in our
schools ? The question will probably receive an affirmative
answer from each one of you, but you may perhaps add
that I am walking in the mists which hide from us the
development of future centuries ; that sufficient unto the
day is the vision thereof ; and that the ground to which I
invite you is a morass which may conceivably be made
firm by our great-grandchildren.
SOME UNREALISED POSSIBILITIES 51
Nevertheless, I am going to ask you to bear with me
while I endeavour to convince you that we can now com-
mence to bridge the morass. I admit that it is one.
Hesitating and imperfect our endeavours may be, but I
am honestly convinced that the time is ripe for a com-
mencement, and that the future of mathematics as a
universal subject in the curricula of schools depends, in
some part at least, on this commencement being made at
once. My ground for this conviction is best stated tersely.
I believe that the modern theories of pure mathematics ./ A*
are destined to illumine our understanding of the human
mind and of cities and nations, just as the pure mathe-
matics of fifty years ago has already illumined the previ-
ously dark and chaotic field of physical science ; that
modern mathematics is or will be to psychology, history,
sociology, and economics as has been the older mathe-
matics to electricity, heat, light, and other branches of
physical science. For example, it may well be that the
theory of sets of points or the theory of groups will find
fruitful application in economics. You will see that I am
suggesting that the range of applied mathematics may be
widened far beyond its present scope. It was asserted
recently at a meeting of head-masters that the reign of
pure mathematics was closed. Would it not be more
accurate to say that pure mathematics has of late extended
and co-ordinated its dominions to an amazing extent, and
that corresponding extensions of applied mathematics have
yet to be found ? If I am right, then our subject has an
irresistible claim. We may trust our lives to engineers
and scientists, just as we entrust our bodies to doctors and
surgeons ; but each member of a human society should,
52 MATHEMATICAL EDUCATION
so far as he may, be competent to analyse and estimate
for himself the workings of his own mind and the devel-
opment of the society of which he is a unit. In the more
detailed remarks which I am about to make, I will ask
you to bear in mind that their main inspiration and justi-
fication lies in what I have just said that they represent
an individual attempt to relate mathematical education to
human thought and social development.
Mathematics has been defined by Russell as the class
of propositions, " If A, then .#," and is applied to classes
of entities concerning which certain propositions A are
assumed ; the truth of these is no concern of the subject.
The entities form the universe of discourse. They can be
ordered in respect of each of the attributes which charac-
terise their class. This universe of discourse may be of
any number of dimensions from one upwards ; in arith-
metic it is one-dimensional, and in geometry it should be
three-dimensional, but is more often two-dimensional. I
may remark in passing that some attempt to estimate the
number of dimensions, that is, of quantities required for
exact specification, of the entities discussed in such subjects
as economics would often throw considerable light on these
subjects. The abstract idea of entities and their dimen-
sions is too often wanting. Hence arithmetic forms the
basis of mathematics, since it explores the properties of one-
dimensional fields. Any treatment of arithmetic which fails
to explore the whole domain of such fields is ipso facto
incomplete, and its victim is in possession of an imper-
fect instrument which cripples him alike in concrete and
abstract applications. My first plea is, therefore, for a mathe-
matical treatment of arithmetic from the earliest stages.
SOME UNREALISED POSSIBILITIES 53
There is much which might be said concerning integers
and fractions, and in particular scales of notation. My
omission of these subjects is only to be interpreted as an
admission that decimals and the theory of exact measure-
ment are of more immediate importance, and must occupy
such time as I can devote to arithmetic. To my thinking,
young children are hurried on to fractions far too soon.
There are many unexplored fields of concrete problems,
possessing real interest for young pupils, the study of
which would give a much firmer basis for future develop-
ments than is now obtained. And the proofs of such
simple rules as " casting out the nines " may provide easy
exercises in deduction, not without value.
To commence, then, with measurement. When, in
actual practice, one measures a length, there are three
distinct objects, any one of which may be in view. The
purpose may be either (i) to state a length greater than
that of the given object, but as little greater as may be,
or (2) to state a length less than that of the given object,
but as little less as may be, or (3) to state two lengths as
close together as may be, between which the given object
lies. I venture to suggest that training in measurement
can only become of any value (other than manipulative)
if it proceeds on these lines, phrases such as " nearly "
and " exactly " being abolished as inexact, and therefore
unscientific. " Nearly " is useless until we are told how
near or within what nearness, and " exactly " only means
" as nearly as I can see." By the use of a vernier the
theory of which should be included in every course of
arithmetic children should learn how nearly they can
see, and then say, for example, 13-4 cm. within 0-2 mm.
54 MATHEMATICAL EDUCATION
We should thus sweep away all the loose statements which
are, I honestly believe, responsible for much of that lack
of accurate thought which is the subject of present com-
plaint, and replace them by a training in the exact expres-
sion of practical measurements, the final form being of
the type " between 7-38 and 7-39 cm."
The ground is now prepared for the extension of the
idea of number, this being done, probably, in connection
with mensuration, that is, by questions such as " Find the
length of the side of a square whose area is 2 sq. in."
Few trials are necessary in order to ensure conviction of
the fact that the number of inches is not a fraction, and
systematic approximation from above and below is at-
tempted. By actual trial, using multiplication only, it is
found that the following pairs of numbers are respectively
smaller and greater than the number required: (i, 2),
(1-4, 1-5), (1-41, 1-42), and so on. So far nothing more
appears than can be realised by measurement, but it is at
once seen that (i) this process can be continued indefi-
nitely, given time and energy; and (2) that there is no
limit to the closeness of the approximation. The human
mind, by this systematic approach, has thus ridden rough-
shod over the imperfections of physical measurement
The latter leaves, and must always leave, an unexplored
gap which cannot be diminished, but the method of suc-
cessive approximation enables us to diminish the gap
below any limit, however small.
Now this process, if carefully developed, is not beyond
the comprehension of young pupils, and it may fairly be
said to contain the germ of any proper study of functions
and the calculus, whether this be undertaken on a graphical
SOME UNREALISED POSSIBILITIES 55
or analytical basis. In either case this method of inclu-
sion between converging pairs is essential to any exact
comprehension of the subject. And beyond this it develops
the theory of pure number so far as to give the pupils
however unconsciously an early example of a perfect
mental structure, fashioned by extension from concrete
experience, and it gives them the only true ideal for the
exact estimation of any set of phenomena.
Shortly, then, I suggest the continuous development
of the idea of a cut or Schnitt of the rational numbers,
commencing it at an early age in connection with a
scientific treatment of simple measurements, the purpose
being to give a true concept of number in its relation to
measurement.
I next make some reference to algebra, stating first
that I am not to be taken as implying that the subject
should be taught before geometry. On the contrary, I am
convinced from actual experience that geometry should
have been studied for two years at least before algebra is
commenced.
At the risk of appearing to raise needlessly large issues,
I must ask the question, What is an algebra for our
present purpose, and what educational purpose may be
served by its study ? To my mind there are two essential
steps in the development of an algebra : the first is the
development oa_symbolismjwhich is usually suggested by
certain combinations of entities, for example, a + b = b + a,
ab ba; and the second is the extension of this symbol-
ism to cases which bear no interpretation in terms of these
entities, and its subsequent application to other classes of
entities. By this I mean the interpretation of symbols such
56 MATHEMATICAL EDUCATION
as 3 5, .zi, 3 -f- V 7, in each of which the entities origin-
ally considered are found to form part of a larger class. I
propose to allude shortly to each of these steps.
As regards the first I have little to say, for the unrealised
possibilities with which I am concerned are here not con-
spicuous. But I do feel that the laws of algebra have re-
ceived far too little attention in current and past teaching,
in that their interpretation is so exclusively confined to the
domain of pure number. Any ordinary boy or girl of 1 5
is able to realise that a -f b b + a and a + (b -f c) =
a + b 4- c are true when a, b, c are vectors, and to make
simple deductions therefrom, as, for example, the proof of
the median properties of a triangle. Such work, even if
only a little time be devoted to it, gives a larger and truer
view of algebra as a language with more than one inter-
pretation. And it gives the idea of an algebra relevant to
any field of human thought, an idea far more stimulating
and fruitful for the ordinary man or woman than the nar-
rower view of one absolute algebra, which is too often the
only result of our teaching. But, when all is said and
done, this first part of the subject only presents itself as
the formation on methodical lines of a shorthand language ;
every step in the solution of equations, factorisation, or
what you will, can be expressed in words whether the
entities be numbers or vectors, and no new methods are
involved.
But now take the second step, the interpretation of alge-
/
braic symbols such as x q or V 7, which have at first no
meaning. The process involved is, or should be, purely
logical. We assume that such laws of combination as
x x x n = x* n and (x m ) n = x mn must also hold in cases
SOME UNREALISED POSSIBILITIES 57
which already bear interpretation, and then find that the
/
one interpretation x q = ~^ ' x p is consistent with each of these
laws. It is too often assumed without proof that, because
the one law x x x n = x m + n leads to this interpretation,
the other laws, such as (x m ) n = x mn , must also be true in
this case. I do not believe that complex exercises in the
manipulation of fractional and negative indices can be of
any profit, but I am convinced that a complete and logi-
cal interpretation of these indices, if only in particular
numerical cases, can and should form part of every course
in algebra. It is one of the best examples of constructive
logic to be found in elementary mathematics, and it gives
a sense of new methods for the discovery of hidden fields
of entities which is hardly to be found elsewhere.
Passing now to imaginary expressions, I would suggest
that the geometrical interpretation of these is not beyond
the capacity of pupils of seventeen or eighteen years of
age, and, further, that it provides a valuable link between
the symbolism thus far developed and geometry of two
dimensions. Not much knowledge of trigonometry is re-
quired in order to understand the expressions a 4 bi,
(a 4 bi) (c 4 di\ nor is it necessary to plunge into useless
elaborations. The pupils have ample scope for exercise in
written descriptions of the processes ; for example, in show-
ing that this interpretation satisfies the laws z^+z^z^+Zy
z l (z 2 4- #3) = z^z^ -f ^g. Work of this kind provides excel-
lent material for short essays, a side of the work which has
received scant recognition. The power of logical thought
is a poor thing if its possessor is incapable of clear expres-
sion of his ideas, and this type of writing is well calculated
to stimulate expression.
58 MATHEMATICAL EDUCATION
At this point the pupil may well review his experience
of algebra. One after another apparent impossibilities of
interpretation have been surmounted. Is there an end to
the process, or can we go on in this manner indefinitely ?
The answer is, of course, that the performance of any
algebraic operation on a quantity of the type a -f bi pro-
duces another quantity of the same type, and the process
is closed. I would suggest that there is no inherent diffi-
culty in the proof of this, granted a knowledge of elemen-
tary trigonometry, and that the view of algebra so gained
is of real value as showing that the exploration of the field
of entities under discussion has been completed. 1 If the
boys and girls of the future can reach this point, they may,
I admit, forget, and rightly forget, many of the details of
their education, but this idea of the exploration of a field
of entities, and the demonstration that this exploration is
complete, may remain with them. If this be so, I do not
think that you will question its value in dealing with the
problems which present themselves or should present
themselves to every citizen of a modern state.
Finally, I must make some reference to geometry. The
primary value of the subject is, in my opinion at least, that
it develops a power of dealing logically with manifolds of
two and three dimensions. When we prove that, if A and
B are fixed points, and the point P moves so that the angle
APB is constant, then P must lie on one of two arcs of
circles, we are selecting from all the points of the plane
1 The entities are typified by the points of a plane, denoted by sym-
bols such as 2, , vj, 7 + 4 *', and the exploration is not carried to
three dimensions, as might have been expected after the extension from
a line to a plane.
SOME UNREALISED POSSIBILITIES 59
those which enjoy a certain property, and are showing that
a certain other property is a necessary consequence of this
principle of selection. And we develop the consequences
in order to encourage the dormant faculty of selecting some
set of a class of entities (the points in the plane) and ex-
amining their properties, not by the imperfect method of
measurement, but with the relentless certainty of logical
reasoning. But I will not dilate further on this aspect of
geometry, as it can hardly be called an unrealised possi-
bility of education.
My first suggestion in regard to geometry is that some
simple idea of methods of transformation, such as projec-
tion and inversion, should form part of every course, at
any rate for pupils who continue the subject until they are
eighteen or nineteen years of age. Such transformations
contain the idea, not illustrated so completely elsewhere
in elementary mathematics, of a correlation between two
sets of entities, such that to each entity of one set corre-
sponds a definite entity of the other set; and from the
known properties of one set we derive properties of the
other set. It may, I know, be said that the study of graphs
involves this idea, but graphs deal with one-dimensional
sets only, and a general idea cannot be gained by one il-
lustration. The problems which concern both the ordinary
citizen and the workman in the trades must often involve
sets of entities of several dimensions, and if he has at-
tained to some idea of the correlation of such sets, and
the examination of a new set in the light of known proper-
ties of an older set, he must thereby have more likelihood
of forming some definite conclusions instead of floundering
in vague uncertainties.
60 MATHEMATICAL EDUCATION
My last suggestion, and perhaps the most startling at
first sight, is that older pupils should be given some idea
of the nature of non-Euclidean geometry. One of the
most vicious fallacies with which we are encumbered is
the idea that our postulates of space, and in particular
the parallel postulate, possess an absolute certainty which
is denied to every other statement that is the result of
experience. Most of us regard the parallel postulate as
more obvious and certain than, say, the statement that all
men must die some day, and we are utterly wrong in so
doing. An outline of the idea and history of non-Euclidean
geometries I would refer especially to Poincare's illus-
tration and the recent paper by Carslaw 1 is sufficient to
dispel the idea, and to exhibit our space postulates as mere
assumptions which fit our experience more simply and
nearly than any others which can be made. I am not
speaking at random ; I have aroused keen interest in a
form of classical specialists whose knowledge of geometry
was distinctly limited. To what end, you may ask. In
showing the true relation between thought and experi-
ence, the manner in which the mind deals with the sensa-
tions which reach it from the outer world. Far as we have
progressed, the saying " Man, know thyself " still has
force. No experience with which I am acquainted shows
so conclusively the relation of each of us to the universe
as the discovery that the supposed certainties of space
are pure assumptions ; as much so as Newton's laws of
gravitation and motion, or Darwin's theory of evolution.
1 See J.W. Young, Fundamental Concepts of Algebra and Geometry; also
the article by Professor Carslaw in Proceedings of the Edinburgh Mathe-
matical Society, Vol. XXVIII ; W. B. Frankland, Theories of Parallelism.
SOME UNREALISED POSSIBILITIES 61
You may, I fear, regard me as an unpractical visionary
who has put before you a host of ludicrously impossible
suggestions. But I would ask you to stop and consider
whether they really are impossible, and I would remind
you that I have suggested nothing that has not been at-
tempted, in outline at any rate, with ordinary pupils, and
with some measure of success. And I would ask you to
remember one thing more. The whole world is going
through a transformation, due in part to scientific and
mechanical invention and in part to the growth of sepa-
rate nations, each with its own methods and ideals, of which
no man can see the outcome. Our function, the function
of all teachers, is to produce men and women competent
to appreciate these changes and to take their part in guid-
ing them so far as may be possible. Mathematical thought
is one fundamental equipment for this purpose, but mathe-
matical teaching has not hitherto been devoted to it, be-
cause the need has but recently arisen. But now that it
has arisen and is appreciated, we must meet it or sink, and
sink deservedly. Neither the arid formalism of older days
nor I say it in no spirit of disrespect the workshop
reckoning introduced of late will save us. The only hope
lies in grasping that inner spirit of mathematics which
has in recent years simplified and co-ordinated the whole
structure of mathematical thought, and in relating this
spirit to the complex entities and laws of modern civil-
isation. Even though every suggestion that I have made
be fallacious and impossible, this one statement remains,
and the future lies with those who first achieve success in
directing mathematical education to this end.
THE TEACHING OF ELEMENTARY
ARITHMETIC
(An address delivered to the Southeastern Association of Teachers of
Mathematics, and reprinted from the Journal of the Association,
March, 1912)
THE TEACHING OF ELEMENTARY
ARITHMETIC
I propose to commence my discussion of this subject
by raising the question, a question joyous of sound to
many a boy and girl wearied with obvious futilities
Why should we teach arithmetic at all ? I raise it in no
whimsical or revolutionary spirit, but in order that we may,
if possible, agree upon the motives which determine the
appropriation of so many valuable hours in the life of
a child to this one subject. Our mission is not merely to
occupy our pupils' time, nor to make them efficient but
unintelligent beasts of burden ; it is to educate them to
take their places as efficient citizens of a free community.
It is in the interpretation of this mission that subjects
should be included in the curriculum, and in its furtherance
should guidance as to matter and method of teaching be
found.
But, some may say, what has this to do with teachers
themselves ? Are they not in the hands of those who draw
up schemes and syllabuses, and can they with profit do
more than carry out such instructions as they receive ?
The question is not unnatural, but it is based on grave
misconception of the duties and privileges of each and
every teacher. Schemes and syllabuses there must be, and
by them all must be bound within reasonable limits, or
anarchy will result ; but their interpretation is in the hands
of the teachers themselves. This interpretation can be
6s
66 MATHEMATICAL EDUCATION
performed either as a mechanical duty, or in free and will-
ing co-operation, and those who ask the question I have
suggested imply that they regard their duties as mechani-
cal rather than co-operative ; that they attend to the letter
rather than the spirit. The better method is, surely, to
endeavour to appreciate the motives of the schemes under
which we work, and to shape them to the best advantage
for our pupils. In so far as this is done, in so far will our
profession acquire its proper influence in the general con-
duct of education, and associations such as this may do
much to that end. Frankly, I am one of those who think
that the body I wish I could say corporate body of
teachers should have more voice in educational affairs
than is now the case, and the remedy is largely in our
own hands. It is by consideration of the why, as well as
the how, of teaching that we shall best utilise our unique
experience among the children themselves, and so gain
our true position.
Why, then, do we teach arithmetic? First, of course,
because a certain minimum knowledge is essential to the
conduct of life. We must all be able to use money, keep
our own simple accounts, and so on ; but to how much
does this amount ? At most to simple operations in sums
of money, lengths, and so on ; certainly not to the cum-
brous barbarisms which disfigure the pages of so many
text-books.
" Find the cost of 17 tons n cwt. 7 qr. 14 Ib. at 9
1 6s. 4^d. per ton," or " Find the compound interest on
,273 1 6s. 7d. for four years at 3| per cent, per annum
paid half-yearly." Who on earth wants to do these sums in
everyday life but a merchant's clerk, and what conceivable
TEACHING OF ELEMENTARY ARITHMETIC 67
mental value can they have ? If the need for such results
does arise, may we not like the clerk use a ready reckoner
as we use other time-saving devices made for us by the
specialised labour of others ? Every one should know
what compound interest is, and why more frequent pay-
ments increase the amount, but simple examples and few
of them suffice for this. If, then, utility of the narrow
personal kind is the only reason for teaching arithmetic,
let us ensure full proficiency in the operations of everyday
life, show the use of ready reckoners when needed, and
utilise the time so gained in teaching something more likely
to assist in the production of capable citizens. Provided
that children can perform simple calculations with fair
speed and accuracy, they should learn the proper use of
ready reckoners before they leave school, in accordance
with the modern tendency to use labour-saving devices and
so obtain greater efficiency. The individual is thus freed
for the performance of other functions, and so increases
his power of production. Those who deprecate this sug-
gestion might as well deprecate the use of sewing-machines,
and their introduction into girls' schools. In neither the
one case nor the other is there a loss of independence ;
on the contrary, there is a gain.
But can arithmetic fulfil no other function in our schools ?
I am probably preaching to the converted when I say that
there are two other aspects of the subject which not merely
recommend but enforce its study in schools of all types.
They are its application to the social life of cities and states
(for example, to the intelligent consideration of schemes
of insurance and pensions), and the concept of orderly
and precise methods of thought which it may convey, in
68 MATHEMATICAL EDUCATION
hardly less degree than the study of geometry. The two
are indeed linked together, for these methods of thought
find some of their best applications in the study of con-
crete problems which have some touch of reality for the
children concerned.
To sum up, then, we base our teaching of arithmetic on
three foundations : practical use, furtherance of the proper
understanding of social and political problems, and develop-
ment of power of independent thought ; and we accept
the use of labour-saving devices wherever possible, even
though we are now compelled to waste time which will be
better employed when our examinations are more rationally
conducted.
First, then, for practical utility. We have to ensure the
ready and accurate use of figures in concrete problems,
and their combination by addition, subtraction, multipli-
cation, and division ; and the idea of a fraction must be
gained for its utility alone it is a necessary part of the
equipment of every civilised being. At the base of all these
things lies our scale of notation. Now I do not propose to
enlarge upon the way in which this should be explained.
I have never, unfortunately, taught it to young children.
I would only commend the use of the abacus to those not
familiar with it, as having historical sanction and being
justified by modern experience. But I do wish to enlarge
upon the importance of a correct understanding of the
method of the scale, not only for its own sake, but for its
applications also. It is the first example of orderly classi-
fication reached by the child, and as such deserves full
elucidation, for if he once acquires the idea that things are
taught to him which he need not and cannot understand,
TEACHING OF ELEMENTARY ARITHMETIC 69
the impression will dog him and his teachers as an evil
spectre for many a weary year.
The difficulty, such as it is, lies in the fact that only one
scale of notation is presented as such, and the underlying
principles cannot be grasped from a study of this or any
one case. It is too little realised that our English systems
of money, weights, etc., are also scales of notation. The
notations
hundreds tens units
7 3 5
that is, 7x10x10+3x10+5 units ; and
pounds shillings pence
7 3 5
that is, 7 x 20 x 12 + 3 x 12 + 5 pence
are exactly similar in method, though not in detail, for
they each form groups of groups : tens of tens of units in
the one case, twenties of twelves of pence in the other ;
and this is done merely to save time. Shillings and pounds
are not necessary, but they are convenient as saving time
and labour in speaking, in writing, and in carrying money.
There is even a somewhat vague scale of notation in geog-
raphy hamlet, village, town, county, country, continent.
I believe that such general considerations can and should
be brought before children during their education, for they
enlarge the mind and lead to the formation of general
concepts from particular cases. They may invent examples
for themselves, finding, for example, what 235 would mean
for a race of beings who, having only eight fingers, counted
in eights.
70 MATHEMATICAL EDUCATION
Beyond this we have to deal with money, weights, and
measures, and simple sums concerned with them. Here
again there is little of practical value to be said ; methods of
teaching and working have been thrashed out ad nauseam.
The only plea one can make is for the utmost speed and
accuracy in simple mental calculations, such as the cost of
2\ Ib. of tea at is. /d. Frequent practice for short periods
is the only way to ensure this, and such practice has its
reward in an increase of general alertness and vigour.
I wish, however, to suggest a type of easy problem
which has hitherto been neglected in elementary teaching.
Take, first, an illustration. " Four boys A, B, C, D are to
sit on one bench, and the teacher knows that A and C, if
placed together, will talk. Show all the ways in which he
can seat them." The solution should be systematic; A and
C may be placed in six ways, thus :
A C
C A
A C
C A
A C
C A
and then B and D may be placed in two ways in each case,
giving twelve ways altogether. The question can then be
narrowed ; A and B may have to share a book, and so on.
Other problems are easily devised. " In how many ways
can four people sit round a table ? " " Three men are to
be chosen out of five to perform a piece of work ; A and B
refuse to work with C. How many teams can be chosen?"
Such questions not only give training in classification,
TEACHING OF ELEMENTARY ARITHMETIC 71
they develop the idea that some things can be done in
several ways, and that it may be worth while to reckon up
all the ways and choose the best.
We now consider fractions, remarking first that they
may and should be taught before long multiplication and
division. The whole theory can be taught in concrete ap-
plications without the use of large numbers, and is only
obscured by their introduction. Harder examples are ac-
cessible without further theory, once the fundamental proc-
esses are fully assimilated.
The first point is to develop the idea of a fraction, and
the last way to do this is to commence with the notation f .
This should come late much later than is usually the
case. The symbol means three-fifths of something, say of
a pound or a line on the blackboard, and it should be re-
garded as three units of a new size. There is much diffi-
culty in getting as far as this. A few concrete examples
taken orally may suffice, written work, if there be any,
being expressed with the denominator in words ; thus,
3-fifths. It is essential that some unit be stated, 3-fifths
of a pound or an apple ; the abstract 3-fifths is far too
general a concept at this stage.
Although there is not much difficulty in imparting the
idea of a fraction, it is vital that this, as any other mathe-
matical concept, should acquire living reality for the pupil,
and not remain an arid tract of schoolroom formalism.
The best safeguard against this danger is considerable
practice in estimating one magnitude as a fraction of an-
other two lines drawn on the board, the areas of two
pages, the sizes of two pieces of wood (to be tested by
weighing), and so on. A little practice in this, the pupils
72 MATHEMATICAL EDUCATION
being told which estimates were most nearly accurate, will
soon induce that sense of proportion which is of the es-
sence of fractions, and is so essential in practical life. And
here we come to a method, which is two thousand years
old, for the exact specification of one line or other mag-
nitude as a fraction of another.
Suppose that two lines AB, PQ are to be compared, and
let AB be the shorter. Lay off lengths equal to AB along
PQ as far as possible, and let the remainder, if any, be RQ.
Then lay off lengths RQ along AB and let the remainder,
if any, be CB. Then lay off lengths CB along RQ, with
remainder, if any, SQ, and so on. The remainders will
soon become indistinguishably small, so that AB and PQ
are expressed, with such accuracy as our instruments allow,
as exact multiples of the smallest remainder visible, whence
the fraction is at once obtained.
There is much more to be said of this process, but it
pertains to the education of older pupils. Any one who can
understand the process can, however, realise also that he
has in this method a logical process which will continue
until his instrument fails him ; in other words, it beats the
instrument every time and so illustrates the superiority of
mental process over empirical measurement.
Next comes the addition and subtraction of fractions,
still in concrete problems. For example : " A farmer wishes
to sow two-thirds of his land with barley and one-quarter
with wheat ; what fraction is left for other purposes ? "
The best way to surmount the very considerable difficulty of
this question for young children is to lay stress on the idea
of change of unit. We may commence by saying, " Can you
add 7 dollars to 4 francs and call it 1 1 ? No ! What do you
TEACHING OF ELEMENTARY ARITHMETIC 73
do ? Convert them both to pence : 7 x 50 pence +4 x 10
pence = 390 pence = 39 francs, or 7 dollars and 80 cents.
In the same way we must convert 2-thirds and i -quarter to
the same kind of thing before we can add them." Now draw
a line and demonstrate on it that 2-thirds = 8-twelfths and
i -quarter = 3 -twelfths ; we can then say 2-thirds + 1 -quarter
= 8-twelfths + 3-twelfths = 1 1 -twelfths. It is unnecessary
to enlarge upon this process ; its nature is evident, and
text-books contain many suitable examples in the chapters
on ratio and proportion and in other parts. The essentials
are constant verbal expression and continual illustration by
division of a line or area until real comprehension is attained.
There is no need to be particular about the lowest common
denominator ; we may well allow our pupils to say :
i -quarter + i -sixth = 6-twenty-fourths + 4-twenty-fourths
= lo-twenty-fourths
= 5 -twelfths,
for at this stage the aim is clearness and accuracy, not
brevity gained at their expense.
As soon as the pupils have become fluent in such state-
ments as
2-thirds -h 3-quarters = 8-twelfths + 9-twelfths
= 1 7-twelfths,
the fraction notation may be introduced on the ground that
it saves time. Some stress should be laid upon this point,
and it should be illustrated by analogies. Thus we save
time by having one word "school," instead of saying "a
place where children are taught " ; " chair " instead of
" a thing to sit upon." The fractional symbol thus assumes
74 MATHEMATICAL EDUCATION
its proper aspect as a short expression of an idea already
comprehended, and the child is receiving a valuable lesson
on the meaning and use of language. Even when the
notation is in use, frequent practice should be given in the
verification, by division of lines or areas, of such statements
as f = y|. Unless they are understood they will inevitably
be misapplied.
Next comes the multiplication of fractions. Here I wish
to make a strong protest against the usual premature use of
the term "multiply" in such statements as "f multiplied
by |." What does the child understand by multiplication ?
Surely nothing but repeated addition. If, then, we say to
him "|^ multiplied by 5," he can see that this means five
times two of a certain thing (thirds), and is therefore ten of
these things. But to tell him that " to multiply two fractions
you multiply their numerators and denominators " confuses
the term hopelessly. It has had one meaning, clearly com-
prehended, and now acquires a second which is apparently
a mere juggle with figures. All sense of logic and exact
use of language must depart with this step.
It is well to recognise that there is no obvious or easily
apparent justification for the use of the same name for
these two processes. Things receive the same name be-
cause they have something in common. We are all called
human beings because, amidst much diversity, we have
certain common attributes. Now the common element is
not at all manifest, at first sight, in such statements as the
following :
7X3 = 7 + 7 + 7 = 14 + 7 = 21,
3 x 2 = 3 x 2 = 6
8 5 8x5 40 '
TEACHING OF ELEMENTARY ARITHMETIC 75
D E
K
Unless we can see the common element we have no right
to name them alike, and until the child perceives a common
element it is absolutely pernicious to suggest a common
name, for in so doing we debase that most wonderful
creation of the human race language as a clear expression
of thought. So soon as we do this we may say farewell
to clear thinking or exact expression on the part of the
pupils. I honestly believe that this one step is responsible
for most, if not all, of the doubt and haze which hang like
a nightmare over many children in their dealings with
fractions.
A good introduction is to consider questions such as, "A
man left f of his land to his children, and | of this to his
eldest son. What fraction of
the land did the eldest son
get ? " Representing the land
by ABCD, we divide it into
thirds by PQ, RS, and then
into fifths by EF, GH, IJ,
KL. Then \ of \ of the land
D o
is seen to be POGD, and this
is seen to be -fa of the land. With many such questions
the general idea that f of J of a thing is the same as ^
of that thing is imparted, and the formal rule is seen to
hold in all cases. The drawings can be discarded, except
for revision, when this formal rule is grasped and not be-
fore. At no stage need they, or should they, be made
with great accuracy. Freehand sketches are better than
drawings with instruments, for they enforce the lesson that
the process is essentially one of reasoning and not one of
measurement.
H
76 MATHEMATICAL EDUCATION
Finally, we come to the division of one fraction by
another. Here again, and for precisely the same reason,
this term should be discarded until its application can be
comprehended. By concrete questions the problem is
raised, " Two-thirds of a thing is taken, and three-quarters
of the same thing. What fraction is the former of the
latter ? " and the idea is evolved from such discussions.
It is not pretended that this work is easy, or that it can
be learnt by rote ; but experience shows that it is within
the comprehension of ordinary boys and girls of twelve
years of age or even less, and it has far more value, as a
practical mental training, than purposeless juggling with
numbers. A child who can perform these processes feels
that he is using his own mind to answer definite questions
with logical certainty. The mere appreciation of this fact
raises him from drudgery and gives him an ideal of men-
tal independence which he may, perchance, in some part
retain in after years.
Before leaving the subject of fractions some further
reference should be made to the premature use of the terms
" multiplication " and " division " as applied to such num-
bers. First take the statement, " To multiply | by j, do
to | what is done to unity to obtain |." This definition
is hopelessly defective in that it omits to state exactly what
is done to unity. Is ^ subtracted from it, or is it increased
by 2 and the result divided by 4, or which other of the
innumerable ways of obtaining | from it is meant ?
The upholders of this definition will reply that this is
splitting hairs ; that every one knows it to mean that
unity is to be divided into four equal parts and three of
these parts taken. Certainly this is so, and the statement
TEACHING OF ELEMENTARY ARITHMETIC 77
as thus amended loses its gross ambiguity. But what
analogy is there between this process and the original view
of multiplication as repeated addition to justify the same
name for both processes ? This question must still be
answered before the definition can be accepted.
The justification is twofold. First, it can be shown that
the latter definition includes the former as a particular
case ; that multiplication by - according to the definition
for fractions amounts to the same thing as multiplication
by 7 according to the first definition. And secondly it can be
shown that the formal laws such as ba = ab, a (b +- c) =
ab -f- ac, which are so easily seen to be consequences of the
first definition, are consequences of the second also ; that
is, they are true when a, b, and c are fractions as well as
when they are integers. The coincidence being complete,
the use of the same term is justified, but it is evident that
considerations of this nature can hardly be included in a
first course of arithmetic. For pupils who are revising the
subject for the second or third time they may be interest-
ing and profitable, but before then the use of the term "of,"
as in | of \ y is preferable to the apparently ambiguous
" multiply by."
The considerations above may serve to show the spirit
which it is suggested should inform the treatment of the
theory of arithmetic ; we now pass to a discussion of some
elementary applications which may go towards fitting the
pupils to be capable citizens as well as efficient clerks.
Of all the various applications which appear in very ele-
mentary text-books of arithmetic, the theory of averages
suffers perhaps more than any other from the banality of
its treatment. And yet no other application possible in
78 MATHEMATICAL EDUCATION
such books is possessed of equal ease and interest ; nor
are there many, if indeed any, others which have so direct
a bearing on almost every question of national or civic im-
portance. There are few such questions into which num-
bers or statistics do not enter in some shape or form, and
their correct treatment by averaging is almost invariably
essential to a proper view of the facts.
The one, and usually the only, thing which is taught in
connection with averages is the rule for obtaining the
average of a set of numbers ; it is then applied without
intelligence to problems fit and unfit for the purpose.
Consider the two sets of numbers :
10, 7, 12, 8, 9, 9, 8, n, 14, 12;
and o, 23, 2, 45, 6, 15, o, 7, i, i.
Each has 10 for its average, but it can at once be seen
that there is no real significance in this statement. In the
first set the numbers are grouped closely round this average,
but in the second they bear no special relation to it ; it is
nothing more than a levelling up of things widely diverse
one from another, and has little or no other import.
Considering the first set a little more closely, we may
make a table showing how many of the numbers come
within different percentages of the average, thus :
Percentage of average 10 20 30 40
Number within 4 8 9 10
Percentage within 40 80 90 100
Such a table shows with what nearness the average repre-
sents the group of numbers, and enables us to compare the
relative values of different averages. For example, the
TEACHING OF ELEMENTARY ARITHMETIC 79
cricket averages of different players may be treated in this
manner, when much information is gained as to their
steadiness of play. Or the average age of each form and
of the whole school can be so compared ; the contrast be-
tween the irregularity of the distribution in many small
forms and its uniformity for the whole school will convey
its own lesson. The results can be exhibited graphically,
laying off horizontally differences of i, 2, 3, ... percent,
from the average, and vertically the percentage of the
whole number of observations which fall within each limit.
All such graphs should be drawn to one uniform scale, so
that a glance will indicate the relation of the average to
the set of numbers, and oral or written statements of what-
ever can be seen from the work should be insisted on in
every case.
A second application of averages concerns the " smooth-
ing out " of a series of statistics which, though liable to
large irregular variations, obey on the whole some definite
law of change. Suppose, for example, that the shade tem-
perature is observed each day at noon for a period of six
months ; the results will be very irregular but will show on
the whole a steady increase, and the object is to eliminate the
irregularities so far as may be possible and thus exhibit
the general law of increase. This is done by taking the
average for a number of consecutive days (say five) and
assigning it to the middle of the period, this being done
for every such period in the six months. The accidental
irregularities, due mainly to the direction of the wind and
the amount of cloud, are thus spread out and the increase
corresponding to the change of season becomes apparent.
In practice, periods of five days would be too short, as the
80 MATHEMATICAL EDUCATION
wind often holds in one direction for a longer time, but
they commence the smoothing process and longer intervals
may be considered afterwards. The comparison of results
for different periods is of interest as showing how the
effect of a long period of high or low temperature is
gradually eliminated.
Another method of obtaining the same final result, and
one which is simpler in itself, is to take the average tem-
perature over a series of years for each particular date and
so smooth out the irregularities in a different way. But
this method would be impossible in other cases, such as the
study of the mortality from consumption or the price of
corn in London, for these phenomena are not recurrent
like the seasons and we cannot, therefore, eliminate acci-
dental irregularities by reviewing several cycles of change.
Moreover, the temperature averages obtained from a period
of years assume that every season is the same apart from
irregularities, and so conceal any possible change in the
seasons from year to year ; but a comparison of these aver-
ages with those obtained by the method of consecutive
days will reveal such changes if they exist.
Other materials for the application of this method of
smoothing out will be found in any book of reference and
in most text-books of arithmetic. It can be applied to any
set of statistics, but consideration of weather records is of
special use, partly from their interest, but more from the
repetition from year to year which has just been discussed.
Yet another important application of averages is the
method for obtaining the mean value of a continuously
varying quantity, such as the height of the barometer or
the depth of a tidal river. In a certain town (there may
TEACHING OF ELEMENTARY ARITHMETIC 81
be many such) the maximum and minimum temperatures
for each day are recorded and the mean is taken to be
their half-sum, which is solemnly written down as the
mean temperature for the day. Now it is obvious that this
method will give results which are too low in summer and
too high in winter ; for in summer the temperature stays
in the neighbourhood of its maximum for the greater part
of the day, and in winter it lingers near its minimum. This
application will show how a proper estimate of the mean
may be obtained.
It is clear that the truth would be shown more nearly
by averaging readings taken every three hours, and that
still better results would be gained if the readings were still
more frequent. Such averages should be taken and the
results compared with each other and with the mean of
the maximum and minimum readings ; statistics are easily
obtained from the charts given in many newspapers 1 or
from an instrument dealer who has recording instruments.
With such charts we can, however, obtain an even better
estimate of the mean by finding the height of a rectangle
whose base is the horizontal width of the graph and whose
area is equal to the area under the graph, for it is obvious
that this height is the true average of the heights of all
points on the curve. The area of the curve can be esti-
mated in the usual way by counting squares, and the aver-
age height is then found by dividing by the base. This
method of estimating mean values is of much importance
in theory and practice, and examples of its use are not
lacking in interest.
1 The London Daily Telegraph, for example. In the United States,
the reports distributed freely by the Weather Bureau may be used.
82 MATHEMATICAL EDUCATION
It has seemed worth while to discuss averages in some
detail, even to the exclusion of other applications of arith-
metic, for the work conflicts little, if at all, with the syllabi
to which most schools are subject, and combines ease,
interest, and value in exceptional degree. But I would
mention also the understanding of insurance tables (not
the formulae from which they are constructed) and the mean-
ing of the value of money and its variations in time and
place as matters which should be considered by teachers of
arithmetic ; the fallacy of the thirty-shilling wage would find
no wide acceptance if education were all that it should be.
I have endeavoured to suggest some simple and perhaps
novel considerations concerned with the teaching of ele-
mentary arithmetic. It may perhaps be felt that they have
some slight interest and value, but that it is hopeless to
attempt the application of some at least, in view of prevail-
ing custom and requirements. This frame of mind, excel-
lent as a balancing factor, is nevertheless to be regarded
with much caution, for salvation from our present difficul-
ties can come only from the efforts and experiments of
teachers themselves. Educational matters are in a ferment.
Men are asking more and more insistently why this and
that are done, and they are right in their insistence. Un-
less fitting answers are ready, our work will stand con-
demned ; the degradation of our subject to the domain of
purely immediate utility will surely follow, as also the loss of
that higher mental training which is so essential to the for-
mation of an efficient citizen. A man who has no power of
intelligent numerical thought is to this extent a serf intel-
lectually, and it is hard to believe that teachers as a whole
will fail to point out the evil and insist upon its avoidance.
THE EDUCATIONAL VALUE OF
GEOMETRY
(Reprinted by permission of the Controller of His Majesty's Stationery
Office from the Special Reports of the Board of Education on The
Teaching of Mathematics , No. 15, 1912)
THE EDUCATIONAL VALUE OF GEOMETRY
" Every great study is not only an end in itself, but also a means of
creating and sustaining a lofty habit of mind ; and this purpose should
be kept always in view throughout the teaching and learning of mathe-
matics." BERTRAND RUSSELL
The title of this paper has been chosen to indicate that
the discussion will not be concerned with the value of
geometry as applied to other sciences or to practical ends,
nor even with its place and importance in schemes of
mathematical education. The purpose is to state the rea-
sons which appear to have led to the universal acceptance
of the subject as a necessary element in education, to as-
certain to what extent geometrical teaching in this country
can find justification in them, and to give some slight ac-
count of experiments in teaching made on this basis by
the writer and his colleagues at Tonbridge School. Lest
it should be thought, however, that this avoidance of the
practical importance of the subject and its relation to other
branches of knowledge imposes unreasonable limitations,
it may be well to state the reasons for it.
The danger of giving undue importance to considera-
tions of practical utility need hardly be enlarged upon, since
it is not proposed to consider geometry from this point of
view. I am more concerned to point out that if the advo-
cates of the subject rest any part of their case on such
considerations, they at once enter into competition with
a host of other interests, many of which have, on such
85
86 MATHEMATICAL EDUCATION
grounds, much higher claims. The parents of a boy who
is to adopt a business career will rightly prefer, if his edu-
cation is guided by his future requirements, that he should
spend his time on geography or economics, arguing that
surveying and bridge building can have no relevance to
his future interests ; while those who take a wider but
still utilitarian view will insist that subjects such as civics
and the chemistry of food have stronger claims to a place
in the education of every child.
Still more dangerous is the plea that every educated
man should have some idea of a subject of such wide util-
ity. Apart from the claims of many other branches of
knowledge, it has a further demerit in that the object
of teaching the subject is implied to be the acquisition of
encyclopaedic knowledge, rather than the development of
the mental faculties. The old conception of education as
the acquisition of information is dead, and it least becomes
mathematicians to do anything to revive it. The use of
justifications of this type, even though it be only in sec-
ondary positions, is likely to defeat the aims of those who
advance them and to do much harm to educational ideals.
A discussion of the value of geometry in relation to
other branches of science would be appropriate in a paper
dealing with the co-ordination and relative importance of
these branches. My object here is, however, to show that
the subject has for its own sake a claim to a place in the
education of every human being. Such a discussion could,
therefore, give only a secondary and relatively weak sup-
port to this claim, a support which only becomes valid
when the claims of these other subjects to a universal place
in education have been admitted. If the view here taken
EDUCATIONAL VALUE OF GEOMETRY 87
is unjustified, geometry must then make its own place in
such volume of scientific knowledge as may be found
necessary to a liberal education. This place would be an
important one, especially in view of the now almost uni-
versal teaching of natural science, but it is hoped that the
considerations to be stated in support of the stronger view
are such as will meet with agreement among mathemati-
cians and convince laymen of its truth.
This is, of course, no new claim. Plato inscribed over
the entrance to his Academy, " Let no one enter who is
ignorant of geometry," and almost every university now
imposes a similar condition. Such recognition of the sub-
ject by educationalists who are not mathematicians implies
an inherent value which must be expressible in non-tech-
nical terms, and it behoves all those who teach it to assure
themselves that they appreciate this value and that the
education in schools is such as to realise it as fully as may
be. In this country at the present time the duty is espe-
cially urgent. Educational systems and ideals are changing
with some rapidity, and almost every subject in school
curricula has been challenged to justify its place, geometry
being one of the few exceptions, if indeed it still be one.
It is almost certain that the motives for this forbearance
are of utilitarian type, combined, perhaps, with some vague
idea that the subject may train a boy to chop logic and
hold his own in argument. Thus the lay advocates of the
subject, on whom its continuance must depend, base their
support on reasons which are open to successful attack
from those who take a too material view of education, and
are almost beneath attack from those who have higher
ideals. Of this weakness they must become conscious ;
88 MATHEMATICAL EDUCATION
signs are not wanting that this is in process, and unless
mathematicians themselves take the initiative in defence
they may find the attack developed with some sudden-
ness. If the boy who specialises in science obtains exemp-
tion from the study of Greek, his fellow who specialises
in classics or history will almost certainly claim exemp-
tion from mathematics, as also will those who intend
to devote themselves to subjects such as law, medicine,
or music.
The question for mathematicians is, then, whether they
can convince others that the appropriation to the study of
geometry of a portion of the school time of every boy and
girl is really expedient. To do this it will almost certainly
be necessary, even though those who are to be convinced
have themselves had some portion of their school time so
appropriated, to explain in some detail what geometry
really is. The first element in the explanation must be
that the subject is based on agreement as to a certain num-
ber of cardinal facts, this agreement resting on foundations
of general experience common to every civilised human
being. 1 The equality of vertically opposite angles, the
angle properties of parallels, and those properties of a circle
which can be perceived from considerations of symmetry
are instances. It is essential to understand that these facts
should not depend, even for their elucidation, on numerical
experiments made in class-rooms or laboratories. Rough
descriptive illustrations there may be, but their only pur-
pose is to recall or intensify conceptions previously formed
1 The latter part of this sentence defines the subject as taught in
schools. It is, of course, an essentially vicious limitation from a more
general point of view.
EDUCATIONAL VALUE OF GEOMETRY 89
subconsciously, for it is this universal acceptance of postu-
lates without conscious experiment which differentiates
geometry from physics or chemistry. The necessary ex-
periments and inductions are made in infancy without the
aid of ruler, compasses, and protractor ; the dog who fol-
lows his master by a " curve of pursuit " fails to describe
a curve which is grasped by young children, although it is
the one actually taken by infants who can only just crawl.
They, in their apparently aimless wanderings, are in the
true geometrical laboratory, performing experiments and
making inductions.
Trite as this statement may appear to many, it is of some
importance here. The appreciation of common conviction
should, when possible, be the first aim of teacher and pupil,
as of all those who would make a concerted effort, and
here there is found a number of facts of which all pupils
can say, " Yes, I know that these things are true." 1
They feel that they are not using arbitrary rules, as in the
study of languages ; nor dealing with facts asserted by
others to be true, as in geography or history ; nor dealing
with experiments put before them by some one who knows
what will happen, as in experimental science : here they
rest on their own convictions. I need not enlarge upon
the value of this consideration, but I would point out that
its importance can be realised by the layman and should
influence him considerably. Amid the many arbitrary rules
and asserted facts which, perforce, find place in education,
the presence of schemes of deduction based on statements
1 The basis for and meaning of this assertion do not enter into the
question. The fact that it is made so universally is the point of im-
portance.
90 MATHEMATICAL EDUCATION
which find universal acceptance as descriptions of our space
impressions must make for good in the child's development.
The peculiar nature of the premisses on which geom-
etry is based having been explained, it might appear suffi-
cient to complete the description of the subject by stating
that it consists of a series of deductions from these prem-
isses and therefore supplies a useful training in the art of
deduction. But this statement would, if not amplified,
be so bald as to mislead. The full sequence of processes
involved is:
1. The separation of essential from irrelevant consider-
ations involved in the appreciation of points, lines, and
planes and their mutual relations.
2. The erection on this appreciation of continuous chains
of reasoning, one result leading to another in such a way
that each chain can be comprehended as an ordered whole
and its construction realised as fully as that of each sepa-
rate link.
3. A discussion of the interdependence of the various
premisses and their precise statement.
Since some such sequence is common to every human
construction, the educational value of a subject which pro-
vides a training in these processes is indisputable ; for this
purpose geometry stands alone in that its bases can be
appreciated, and the deductions performed, at an age earlier
than is possible in the study of any other subject having
the same purpose.
There is yet one more consideration, and that not the
least important, to be urged in favour of the subject. The
appreciation of literary and artistic beauty has of late re-
ceived increasing recognition as a necessary element in the
EDUCATIONAL VALUE OF GEOMETRY 91
training of the young. The study of the English language
now includes literature; drawing and music advance in
importance in schools of every type. But the element of
intellectual beauty has not yet received general recognition,
despite the supreme position ascribed to it by almost all
schools of thought from the Greeks downward, and the
loss is a great one. The contemplation of unassailable
mental structures such as are found in mathematics cannot
but raise ideals of perfection different in nature from those
found in the more emotional creations of literature and
art. It must induce an appreciation of intellectual unity
and beauty which will play for the mind that part which
the appreciation of schemes of shape and colour plays for
the artistic faculties ; or again, that part which the appre-
ciation of a body of religious doctrine plays for the ethical
aspirations. Fanciful though this may appear to many, I
believe that it may be an important factor in determining
the retention of geometry in schools. The conception of
a body of truth invulnerable on all sides, 1 a conception
which finds one of its best and most common expressions
in the quotation, " Four square to all the winds that blow,"
appeals to most men, and they will welcome an effort to
bring it to their children in one of its purest forms.
Such being the basis on which it is thought that the
universal teaching of geometry may be justified, it remains
to develop the leading principles of a scheme of education,
and to examine to what extent teaching practice in this
country accords with them.
1 Euclidean geometry is still invulnerable in that it is based on the
simplest known description of our space perceptions, and such de-
scriptions form the foundation of most " truths."
92 MATHEMATICAL EDUCATION
The division of the processes of geometry into three
classes implies a corresponding division of the period of
education into three epochs. In the first, the imagination
is stimulated and developed and some general power of
reasoning should be acquired without any formal presenta-
tion ; in the second, ordered systems of reasoning are
developed from facts which are now within the scope of
the imagination ; and in the third, the true basis for the
assertion of these facts is discussed and their interconnec-
tion investigated. It is at least doubtful whether this third
stage would be a subject proper to a school course if it
were within the pupils' grasp ; and it is almost certain that
its nature and the problem involved are as much as can
be brought home to them. It must therefore be assumed
that a school course should end with the second of these
epochs, and the first two only need be considered in detail.
The increase in power of imagination, which is the main
object in the first epoch, can only be effected by extension
from those impressions in which this power is already
developed to some extent. A child may know what is
meant by a point and a line, and be able to recognise
them at sight, and yet not be able to think of points and
lines, just as an adult may recognise the whole of a tune
when he hears it and yet not be able to reproduce a single
note. It is therefore inadvisable to commence by drawing
points, lines, and angles, and performing constructions and
measurements, because this opens up a new field of un-
familiar ideas having no connection with any of that knowl-
edge which has been so far assimilated as to be a possible
subject for imagination. Houses, roads, mountains, islands,
and the like can all be imagined by a child of ten years of
EDUCATIONAL VALUE OF GEOMETRY 93
age, and his geometrical imagination is developed by stat-
ing problems in such terms, but the construction of tri-
angles having given sides leads nowhere at this age and is
a mere gymnastic.
It is not hard to devise problems on these lines which
stimulate the imagination, excite the spirit of research, and
provide exercise in the simpler forms of geometrical reason-
ing. They may be divided into groups, in each of which
one of the following methods is introduced :
1 . Construction of triangles and polygons when lengths
only are given.
2. Simple constructions for heights of buildings, ships'
courses, and the like, depending on compass bearings and
angles of elevation.
3. Construction of triangles and polygons when lengths
and angles are given.
4. Extension of all the above to problems in more than
one plane.
5. Determination of a point as the intersection of two
loci, or its limitation to lie inside or outside two or more
loci. These loci need not be lines or circles and are con-
structed by actual plotting.
It may be worth while to give a few specimen problems
to show what is intended, and it should be stated that these
and subsequent suggestions are the outcome of experience
and have been tested in Tonbridge School. Each of these
problems is well within the scope of a boy of eleven or
twelve years of age, after he has received a reasonable
amount of teaching in the shape of questions involving the
same principles ; the concrete terms should, of course, be
varied constantly.
94 MATHEMATICAL EDUCATION
a (Type i). A straight road runs east and west ; a house
is 95 yards north of the road and a well is 40 yards north-
east of the house. Draw an accurate plan and locate upon
it the position of a cow-shed which is to be built 65 yards
from the well, within 70 yards of the road, and as far as
possible from the house.
b (Types 2, 3, 4). The height of a tree or chimney
may be found from measurements in one or in two planes,
the measurements being made by the pupils themselves.
c (Types 2, 3, 4). A hill rises due north at a gradient
of i in 10. Find the direction of a road which rises to the
eastward on the hill at a gradient of i in 30.
d (Type 5). A lightship is 2 miles from a straight
shore. A submarine, which is cruising in the neighbour-
hood, explodes and sinks, the only survivor being the man
in the conning turret ; he can only say that the wreck is
equidistant from the lightship and shore, and the watch on
the lightship says it was 3 miles away when the explosion
happened. Show on an accurate plan where the wreck
must lie.
e (Type 5). Two towns, A, JB, are 5 miles apart, and
a man who lives with his parents at a place 6 miles from
A and 7 miles from B, bicycles to A, then to B, and then
home every day. After a time he wishes to reduce his
daily ride to 14 miles. Show on an accurate plan all the
places where he can live, and indicate those places which
are nearest to and farthest from his parents' home.
The first three types require no further explanation, but
since the fourth and fifth involve departures from current
practice, discussion of these may be of some value. The
early introduction of solid geometry (examples b, c) may
EDUCATIONAL VALUE OF GEOMETRY 95
cause some surprise, especially in problems of such appar-
ent difficulty. But I am convinced, from the experience of
myself and my colleagues at Tonbridge, that boys of twelve
or thirteen grasp these ideas with much more ease and rapid-
ity than those who have deferred the work for two or three
years, and that their whole outlook is improved in conse-
quence. The first problems should concern things which
can be seen ; for example, the height of one corner of the
class-room can be determined by measurements taken from
the ends of a desk not in line with it, or the diagonal of a
block of wood may be found by construction of two right-
angled triangles. A problem such as c above is simply
illustrated by using one corner of a book half open to
represent the hill-side. These aids to imagination are
soon found to be unnecessary, fairly complex construc-
tions being undertaken without their assistance, and there
is no lack of material from which examples may be
constructed.
Since the method described in the fifth type and exam-
ples d and e is, except for the introduction of other loci,
used in the preceding sections, its separate mention may
appear redundant, but it is easily seen to involve principles
which can hardly receive too much attention. The statement
of the laws of selection which define two or more manifolds
in such form as to exhibit the common elements (if any)
of them all is one of the most familiar forms of mental
activity, and experience has shown that children can under-
stand and perform the process in cases such as those above.*
In so doing they apply it, not to new and unfamiliar ab-
stractions, but to classes of objects so familiar as to be
possible subjects for mental operations.
96 MATHEMATICAL EDUCATION
Such applications form the second of four stages into
which education in logical processes may be divided. In
the first they are applied to definite objects, as in the
methodical selection from a number of tiles of all those
which have given shape, size, and thickness ; in the second,
to mental images of classes of objects, as when " a house "
is not thought of as any particular house ; in the third, to
abstractions, such as point, line, colour, sense, instinct;
while in the last the processes themselves are considered in
their utmost purity. For our purpose the first stage is dealt
with in the kindergarten and in courses of practical meas-
urement, and the third in formal geometry ; the second
does not, in my opinion, receive sufficient recognition in
the teaching of geometry at the present time, and the
exercises above are intended to remedy this deficiency.
The course may also be regarded as an introduction to
the ideas of a manifold and a function. Their importance
has of late been recognised and need not occupy us here,
but their difficulty, especially in regard to manifolds, is
hardly realised. If a child is directed to mark on paper
a number of points such that the sum of their distances
from two given points has a given value, he will do so
without any idea that all such points form a continuous
curve, and if he is told this he will not grasp the fact. But
allow him to go on marking points and he will ultimately
attain to a conception of the whole and their assemblage
along a continuous curve. This cannot be taught ; the
teacher must wait for the child to reach it, and the value
of the work is lost if mechanical means of describing the
curve continuously are adopted. Such a process has its
own value, but in this connection it obscures the idea of
EDUCATIONAL VALUE OF GEOMETRY 97
the curve as a manifold of points selected from the mani-
fold which forms the plane.
The early formation of such habits of thought, even
though subconscious and unsystematised, has a technical
importance which deserves mention despite our general
limitation to educational considerations. Recent research
has exhibited the whole structure of mathematics as founded
on the processes and conceptions which underlie them, and
has gone far, with their aid, to perfect a co-ordination be-
tween the different branches of the subject. These funda-
mental ideas are now seen to be essential to the under-
standing of even elementary mathematics, and their absence
is responsible for most of the difficulties which occur in the
introduction to more advanced work. The difficulty of the
child who in commencing algebra says, " Yes, but what is
xl" of the boy who cannot solve a simple rider, and of
the student who finds the calculus puzzling, are now all
traced to this common origin. The moral for the teacher
is obvious. Until the young child can in some way attain
to such ideas in relation to matters within his own experi-
ence he has not grasped the groundwork of mathematical
thought and so cannot erect the structure on proper foun-
dations, and the present problem for teachers is to bring
them to him in as many forms as possible. For this pur-
pose some understanding of the philosophy of arithmetic,
geometry, and algebra is essential, and I believe that the
light which this philosophy throws on problems of ele-
mentary education constitutes by no means its least value.
The transition from the preliminary stage just described
to a first course of formal geometry is marked by the in-
troduction of connected reasoning leading from one result
98 MATHEMATICAL EDUCATION
to another, and a gradual cessation of the aid to imagina-
tion involved in the use of houses, roads, and the like.
The assumption of postulates as a basis for this reason-
ing should not be stated formally ; unless the preliminary
course has failed in its object the pupils will accept them
without difficulty as they are required. But it is essential
that the choice of these postulates should not be left to
chance or dictated by convenience of presentation ; it
must be guided by the considerations which determine
the adoption of the subject as a mental training.
If these considerations be such as are indicated above,
there is no difficulty in specifying the nature and extent
of the assumptions which should be adopted as facts in
a first course of geometry. It must be possible to induce
the child to accept them without the aid of numerical
measurement of any kind, and every statement for which
this is possible should be regarded as a postulate. This
definition includes the following : ( i ) the equality of verti-
cally opposite angles ; (2) the angle properties of parallel
lines ; (3) properties of figures which are evident from
symmetry ; l (4) properties of figures which can be dem-
onstrated by superposition, a method which should be
once for all discarded as a proof.
The meaning and intention of this definition is best
illustrated by reference to recent developments in the
teaching of geometry. There has been a tendency in
preliminary courses to associate such ideas with numerical
measurement, if not, indeed, to profess demonstration by
1 Demonstrations by folding are not proofs as the term is here used.
They lack the essential element of deduction from two or more state-
ments already admitted.
EDUCATIONAL VALUE OF GEOMETRY 99
its aid. For example, children may be instructed to draw
two straight lines, measure the vertically opposite angles,
and state what they observe. It is hard to see what good
can be derived from such exercises, and they may do much
harm. If it be true that the education of the child should
follow the development of the race, they are condemned at
once, for it is inconceivable that these ideas were suggested
by such processes, or that there was a geometrical Faraday
who announced them to the ancient world. They can only
be regarded as intuitions from rough experience ; a person
in whom they are wanting is ignorant of space, and a
knowledge of (Euclidean) space is neither more nor less
than their full comprehension a comprehension resting
on foundations wider by far than any schoolroom experi-
ments. An attempt to aid its formation by numerical
exercises may not unfitly be compared to an attempt to
teach music by explaining the mechanism of a piano, and
the relation between notation and keyboard, before the
pupil has heard a single tune. It is a crippling of subjec-
tive growth at its most sensitive stage by the crudest form
of materialism. A mind which has ranged over all its ex-
perience and has made these intuitions has gained a sense
of power and accepted truth which cannot be induced to
an equal extent by any substitute for the process.
The statement that all possible intuitions must be taken
as postulates is hardly less important than the definition
(for our purpose) of a postulate as any statement of com-
mon acceptance. The object of teaching geometry has
been stated as twofold : the sense of logical proof is to be
developed, and the conception of a chain of proofs is to
be formed. There are many ways of doing this ; they vary
100 MATHEMATICAL EDUCATION
with the selection of the theorems used and the order of
their sequence, but each should be subject to the condition
that every theorem should prove some new fact which
could not have been perceived by direct intuition, sym-
metry, or superposition. If this condition be violated the
pupil's sense of a proof is confused ; he appears to arrive
with much painful labour at a result which he and every
one else knew before to be true, and the impression of
inevitable but unforeseen truth is not imparted. Take,
for example, the theorems concerning bisected chords of
a circle : if a boy cannot perceive the truth of these with,
possibly, a little stimulus, his knowledge of space is so
meagre that he ought not to have commenced formal
geometry. If, on the other hand, he can see their truth,
there can be few processes more destructive than the
elaboration of deductions from other intuitions with the
intention of imparting a sense of proof.
The details of the propositions and the order of their
sequence is hardly germane to this paper, but the neces-
sity for presenting them in sequences each of which can
be grasped as a whole deserves further mention. If the
Euclidean tradition be ignored, such sequences can still
be found. For example, experience has shown that the
angle properties of polygons, arcs of circles, and tangents
to circles can be arranged in one sequence, so that the
unity of the group can be appreciated by pupils to whom
it is the first example of formal geometry. Another sub-
ject is found in the theory of the regular polyhedra ; the
limitation of their number, the investigation of their shapes
and dimensions, and their construction, can be grasped by
boys of ordinary ability at an early age. It will probably
EDUCATIONAL VALUE'. QFGEOMETRV' i id*
be agreed that a considerable element of solid geometry
is essential ; the tetrahedron, cone, and sphere are other
themes which have been found possible and valuable.
It is, of course, true that Euclid presented geometry in
a series of such sequences, and despite the logical defi-
ciencies in his scheme his genius is probably still unrivalled.
But his theme was the reduction of the number of postu-
lates to a minimum and the introduction of each as late as
possible. As soon as it is granted that all possible intui-
tions should, in a first course, be accepted as postulates,
his sequence fails to have value for this purpose, and has,
indeed, demerits. It is therefore a necessary consequence
of the theory of geometrical education here developed
that a deliberate effort should be made to replace this
scheme by one more suitable. The need is not met by
allowing propositions to be treated as intuitions when
possible and retaining his sequence, for this destroys the
whole meaning of that sequence ; his motive enforces the
introduction of solid geometry as late as possible, while
ours demands it as early as possible, and numberless other
contradictions arise.
This conclusion induces doubts whether even this resid-
uum of Euclid's propositions, taken in any order, forms
the best material for geometrical training of an educative
value. Some of it undoubtedly deserves its place, both for
its own value and its necessity in other parts of mathe-
matics ; but stripped of unessentials this is small in content
and less than can be acquired in an ordinary school course.
Before including more than this bare minimum it is advis-
able to ascertain whether, in the whole domain of geomet-
rical knowledge, there may not be other matter of more
,
102 MATHEMATICAL EDUCATION
educative value within the grasp of ordinary boys and girls.
Here I am passing beyond the range of my own experi-
ence, but I have, as the outcome of a maturing conviction,
made tentative experiments with individual pupils and
small classes and deem it worth while to state the result.
If, without regard to the age of the pupils, one asks
where the methods and ideals of geometry are presented
in the greatest unity, simplicity, and beauty, the answer
must be that geometry of position and projective geometry
have no rivals. Estimated on such standards Euclid's work
is dwarfed by these modern creations, and not least so in
respect of the ease and generality of their conceptions.
Must it be said, in spite of this, that they are to remain the
property of professed mathematicians since they are beyond
the grasp of the normal adolescent, or may it be that their
power and beauty can be appreciated, even if only in some
comparatively crude form ? Even those, if there be any,
who at once deny the possibility will not dispute the im-
portance of the question and the value of success if it can
be attained. My own conviction, fortified by such limited
experience as I have indicated, is that the elementary con-
cepts and methods of projective geometry can be grasped
by ordinary pupils ; that they would excite a greater inter-
est and fuller spirit of enquiry than any form of Euclidean
geometry, and that their educational value would be far
greater. The amount of knowledge required is not so great
as might be imagined. A pupil who has had some expe-
rience of three-dimensional work can grasp the relations
between a figure in one plane and its projection in another
plane, including the particular cases wherein a point or
line in either plane have their corresponding element in
EDUCATIONAL VALUE OF GEOMETRY 103
the other at infinity. The theorem of Desargues, with its
simpler consequences, is then within his grasp, and he may
so gain some idea of the extent and variety of the results
which can be deduced from the axioms of position only,
and the manner in which they unite in one statement
results which he had regarded as disconnected.
It is then natural to enquire whether there is any simple
relation between corresponding segments in the two planes.
The position ratio AP : PB, which defines the position of
a point P on a line when two of its points A, B are given,
should have become familiar in connection with earlier
work, and it is now easy to prove that corresponding
position ratios remain in a constant ratio as the points
P, P' move along their respective lines. Hence the ratio
AP AO
- : , formed from two such ratios, is unchanged by
PB QB
projection. The metric properties of quadrangles and
quadrilaterals (deduced by projection from a parallelogram),
and the simple properties of the conic regarded as the
projection of a circle, can then be investigated to such
extent as may appear possible or desirable. I have myself,
with ordinary boys of eighteen, reached Carnot's theorem
without great difficulty.
It may be suggested that this work is unnecessary, and
that when the minimum of geometry has been acquired
the pupil should proceed to trigonometry or other branches
of mathematics. As to this I can only say that this paper
is in part an attempt to justify the teaching in schools
of an amount of geometry much larger than has hitherto
been thought possible, and this without increase of the
time devoted to the subject. My own experience is that
104 MATHEMATICAL EDUCATION
an early commencement of trigonometry can and should
be made by some sacrifice of the large amount of time
now devoted to algebra by pupils who are too young to
understand the subject. Trigonometry is a valuable stimulus
to geometrical thought, but is no substitute for it.
It only remains, in considering the scheme of geometri-
cal education, to refer to the interdependence of the pos-
tulates which have been adopted. I do not believe that
any detailed or systematic discussion of this is possible or
advisable at the school age, but if towards the end of this
period examples of deduction of some postulates from
others were shown, it might be possible to lead the pupils
to realise the ideal of a geometry based on a minimum
number of assumptions concerning the nature of space.
Such considerations would then acquire more reality for
them in that they would have some acquaintance with
mechanics and physical science and could therefore con-
ceive the general ideal of a minimum of induction and
a maximum of deduction. And I feel bound to state my
conviction that every student of whatever subject, who pro-
ceeds to a university education worthy of the name, should
gain some slight idea of the nature of non- Euclidean geom-
etry. The simpler portions of the paper by Carslaw in the
Proceedings of the Edinburgh Mathematical Society for
Ipio, or the description of Poincare's well-known illustra-
tion given in Young's " Elementary Concepts of Geometry
and Algebra," already mentioned on page 60, are within
the grasp of any one who has even a slight acquaintance
with the geometry of the circle ; an appreciation of these
ideas throws a light on the space-concept in particular
and our so-called knowledge in general which can be
EDUCATIONAL VALUE OF GEOMETRY 105
gained in no other way. It may fairly be said that there
are few portions of mathematical knowledge which have
more educational value.
The nature of the current teaching of geometry in Eng-
land is best understood by reference to recent history.
Until a few years ago the use of Euclid's text in matter
and sequence was universal owing to the regulations of
examining bodies. Attempts to secure more freedom had
not been wanting. The Association for the Improvement
of Geometrical Teaching was formed with this object as
early as 1871, and undoubtedly succeeded in awakening
interest in the presentation of elementary geometry,
though no tangible result appeared. In 1901, however, a
paper read by Professor Perry before the British Associa-
tion at Glasgow aroused fresh interest, and a committee
was formed by the association to consider and report upon
the teaching of elementary mathematics. In their report,
issued a year later, they advocated a preliminary course of
practical geometry, and stated that it was in their opinion
unnecessary and undesirable that one text-book or one
order of development should be placed in a position of
authority.
Simultaneously the Mathematical Association, a develop-
ment of the association founded in 1871, had formed a
committee to consider the subject, and this body issued
a report in May, 1902. In principle it proceeds on lines
similar to the report of the British Association Committee,
but it contains a definite statement that "it is not proposed
to interfere with the logical order of Euclid's series of
theorems." In effect it simplifies the development by in-
troducing hypothetical constructions, changes the order of
io6 MATHEMATICAL EDUCATION
certain groups of propositions, and introduces an algebraic
treatment (for commensurables only) of ratio and propor-
tion. As a consequence of the work of these committees
and their supporters the examining bodies yielded one by
one, announcing that Euclid's order and methods would
not be enforced, and setting questions involving numerical
construction and calculation. Finally, the University of
Cambridge issued in 1903 a syllabus indicating the amount
of geometrical knowledge required for its entrance exam-
ination, and stated that the examiners would accept any
proof which appeared to form part of a systematic treat-
ment of the subject. This syllabus, which is still in force,
omits portions of Euclid's text and introduces no addi-
tional matter.
Taking these changes in detail, we are first concerned
with the postulates. The intention appears to have been
the increase of their number in order to simplify certain
proofs which had presented difficulty ; proofs of facts
which can be perceived by intuition are still retained,
notably in regard to congruent figures and properties of
the circle. The present scheme may therefore be not
unfairly described as Euclid's ideal of the minimum num-
ber of assumptions, tempered by consideration for the age
of the pupils a consideration which renders the whole
scheme meaningless from his point of view. It may be
doubted whether such a compromise is likely to be suc-
cessful. The broad lines of educational advance should be
based on fixed principles and not on expediency. If the
pupils can understand the development of a scheme based
on a minimum of assumption, and can also understand
this motive, let it be adopted as the highest ideal. But if
EDUCATIONAL VALUE OF GEOMETRY 107
not, as experience has shown with some certainty, the only
logical alternative is to allow all possible intuitions as pos-
tulates. The educational advantages of this course have
already been described, and it has the further advantage
of lessening the amount of time and effort required, and
so clearing the way for some study of more advanced
geometry, which is not yet represented in any of the ex-
aminations referred to. Should any doubt be felt as to the
persistence of Euclidean methods despite the abandonment
of his ideal, consideration of the fact that even the elements
of solid geometry are not included in school courses, ex-
cept in preliminary work (to which reference will be made),
may carry conviction. All examining bodies (the Civil
Service Commission excepted) appear to imply by their
schedules that sufficient training in schools can be obtained
without it, k the fact being, of course, that its omission is a
survival of Euclid's order.
Thus it may perhaps be said that what is often called
the abolition of Euclid must involve the complete abolition
of his order and methods and the construction of an-
other theory of development, and that in both respects the
changes are incomplete. The structure, always of course
fallacious in its foundations, has now been shattered and
we are groping among the fragments.
Turning to educational methods, there are two important
developments : the introduction of preliminary courses, and
the use of numerical examples. The latter only requires
brief reference ; it must give greater reality and preci-
sion to the results which it is intended to illustrate, and
there is common agreement that it has done this. The
meaning and proofs of propositions are admittedly better
108 MATHEMATICAL EDUCATION
comprehended, but the comprehension is too often of
isolated results rather than structures of reasoning.
The object of the preliminary course is to enable the
pupils to acquire some familiarity with the leading facts
and concepts of the subject. For the most part such courses
consist of exercises in measurement and construction,
coupled (in some cases at any rate) with numerical intro-
ductions to or illustrations of the axioms. So long as such
exercises are confined to the performance of constructions
of known type there is little to be said for or against them
in logic or philosophy, though they are, in the opinion of
some, as deadening to the intellect as the excessive per-
formance of algebraic simplifications. But when the angle
properties of parallels and triangles are introduced by
measurement with a protractor, instead of by turning a
rod, and when we find an example such as : " Draw a
triangle whose sides are 2, 3, and 4 inches and then
draw the perpendiculars from each vertex to the opposite
side. These lines should meet in a point; see that they
do so," the matter becomes more serious, for numerical
measurement has been substituted for intuition or demon-
stration and the impression is hard to eradicate. There is
also some general introduction dealing with space-concepts,
and here there is usually some allusion to objects in three
dimensions; beyond this, solid geometry finds as a rule
no place in such courses. Its persistent neglect by teachers,
examining bodies, and writers of text-books is one of the
most marked and regrettable features in the developments
of recent years.
Apart from this, the main criticisms, from the point of
view of this paper, to be directed against such courses are
EDUCATIONAL VALUE OF GEOMETRY 109
that they are not based on a gradual extension from previ-
ous experience and imagination, and that there is a distinct
tendency, as has been said, to relate the postulates to nu-
merical processes. Of the second I need not speak further ;
of the first it may be said that it violates the principle that
the development of the powers of imagination, abstraction,
and reasoning should be made continuously from experi-
ence and knowledge gained in daily life. To construct a
preliminary course which consists of work concerning
angles, lines, triangles, and circles, with perhaps a pass-
ing reference to a few surveying problems, is to place the
child suddenly in a new world where things are replaced
by abstractions, and to give him an occasional glimpse of
his own sphere as from behind bars bars which are
not made thinner by assigning numerical measures to the
lengths and angles with which he deals. In the alterna-
tive which I have suggested the endeavour is to lead him
gradually to this new world of abstract thought and ideal
truth, or, perhaps, to present it as an outcome of and one
with that more limited world of which alone he is at first
cognisant.
The result of this period of freedom has been summed
up in a circular published by the Board of Education
(No. 711, 1909), which contains luminous and practical
suggestions to teachers, based on the experience of the
Board's inspectors. For our present purpose the most
striking statement made therein is that the time taken to
acquire the matter of the first three books of Euclid varies
from one to three years in different schools, and that it is
where the work proceeds quickly that it is best, and nearly
always where it proceeds slowly it is poor. The difference
no MATHEMATICAL EDUCATION
is ascribed to the manner of treating the earlier part of the
work, with which the circular is mainly concerned. As to
this, I will only say that, while its suggestions are far in
advance of the matter contained in most text-books, it
perhaps hardly lays sufficient stress on the need of devel-
opment from the child's previous experience, nor does it
suggest concrete problems requiring a considerable amount
of imagination and reasoning. The importance of solid
geometry is pointed out with some force, but no very
definite suggestion is made as to the time or manner of its
introduction in a deductive course, a point on which most
teachers are in need of guidance, and especially those who
now succeed in covering a matriculation course in two or
three years.
The circular also deals with the general effect of the
changes, stating that it has been beneficial. " Unintelligent
learning by rote has practically disappeared, and classes,
for the most part, understand what they are doing, though
they often lack power of insight and have but a narrow
extent of knowledge."
Had the changes already described been the only edu-
cational changes during this period, a fairly conclusive
inference might be drawn. But it must not be forgotten
that the modern secondary school, with graduates of teach-
ing universities for its teachers (often trained) and a cur-
riculum designed to develop all the pupils rather than to
benefit the few of exceptional ability, has, during the same
time, come into being, and the issue is thus confused.
Greater comprehension, as due to improved teaching,
would have been likely even though Euclid had not been
dethroned. The older schools have for the most part been
EDUCATIONAL VALUE OF GEOMETRY ill
comparatively unaffected by such developments during this
time. Speaking as one who has some experience, both as
teacher and examiner, of these and other schools, I can
but state my opinion that the improvement in the modern
secondary schools is far greater than in the schools of
other types, including those which have adopted the
changes in geometry most fully. Devoting roughly an
equal amount of time to the subject, they obtain better
results at an earlier age. If I am right in this and I
believe that most men who have acquaintance with the
various types of school in this country would confirm the
statement it follows that most of the recent improve-
ment in modern secondary schools is due, not to any
recent changes in the syllabus of geometry, but to the ac-
quisition of teachers who not only understand the subject but
also know how to teach it. Some confirmation of this view
is given by an enquiry made some three years ago among
the professors and examiners in the modern universities.
They declared themselves as dissatisfied alike with the
results of the older and more modern methods, the major-
ity against the modern methods being the larger. The
improvement in quality of knowledge was admitted in
many cases ; it was rather the material and its co-ordina-
tion that were criticised.
Finally, then, it may be said that improvements in
teaching methods and in personnel of the teachers have
produced their natural results, and to the teachers must be
ascribed much of the admitted improvement. Schemes of
geometrical education in this country are lacking in foun-
dation, method, and extent, and this arises from the fact
that Euclid's scheme itself utterly unsuitable as an
112 MATHEMATICAL EDUCATION
introduction to the subject has been so far tampered with
that hardly any scheme remains. So long as no attempt is
made to devise a connected development based on the
many intuitions which are common to all civilised beings
before they reach maturity, so long will the subject realise
a painfully small proportion of its potential value.
I have endeavoured in this paper to interpret the quo-
tation at its head in its reference to my subject. I do not
forget that children will have to do the work of the world
and must be fitted for it, and I believe that a training such
as is here described will assist them in this. But they will
not be less fitted if their education provides them with
widening and inspiring subjects for contemplation when
they reach maturity, nor indeed is such fitness the sole
end of life.
THE PLACE OF DEDUCTION IN
ELEMENTARY MECHANICS
(Reprinted by permission from the Proceedings of the International
Congress of Mathematicians , August, 1912)
THE PLACE OF DEDUCTION IN ELEMEN-
TARY MECHANICS
A science consists of a definite class or of definite classes
of entities, a set or sets of postulates relating them, and a
series of deductions which are logical consequences of these
postulates. In the earlier stages of its evolution it may be
it usually has been that the sets of postulates contain
redundancies. Only when the logical consistence of these
sets has been investigated, and the number of independent
postulates established, can the science be termed complete,
for then only is it certain that the number of assumptions
has been reduced to a minimum, and that no one of them
conflicts with any other. Although the concept of a perfect
science was attained by Euclid in connection with geometry,
the first approximately successful presentation of a science
in this form came as late as Newton, and then in connec-
tion with mechanics ; geometry, on the other hand, has only
been completed within the last generation, and this after
struggles extending over two thousand years.
This historic distinction between geometry and mechanics
implies a corresponding didactic distinction. It is the pecul-
iarity of geometry, as opposed to other physical sciences,
that its postulates, and many of the deductions which can
be made from them, are and have long been the common
property of civilised mankind. Who doubts, whether he
has learnt geometry or not, that all right angles are equal,
that only one parallel can be drawn through a point to a
"5
Ii6 MATHEMATICAL EDUCATION
given line, or that any diameter divides a circle into two
identical parts? Most, if not all, of the postulates of
mechanics are, on the other hand, known only to those
who have consciously adopted them. Indeed, many persons,
otherwise educated, will dispute their truth. The problems
before the teacher are therefore entirely different in the
two cases. In geometry he steps into an inheritance of pre-
acquired space concepts, crude perhaps, but formed beyond
doubt ; he can develop deductions with little trouble con-
cerning postulates. But in mechanics he must set up the
entities and develop the postulates from the commence-
ment ; he steps into no such inheritance as does the teacher
of geometry.
With methods of exhibiting mechanical entities, and the
choice of postulates for use in a first treatment of the
subject, this paper is not concerned. It is assumed, how-
ever, in accordance with modern practice in most schools,
that the number of postulates is more than the minimum.
My first concern is to point out that, treat it how we may,
the direct evidence which can be brought before a boy in
support of any of these postulates is singularly narrow and
unconvincing. And it is an essential part of the argument
that this weakness should be exposed at the outset. Take,
for instance, the triangle of forces ; the pupil may fairly ask
Within what degree of accuracy has it been demonstrated ?
Is it true whatever be the body on which the forces act ?
Is it true at all places and at all times ? Is it true under
any circumstances of motion ? Is it true for forces of all
kinds electrical, magnetic, or any other ?
Or again, take the proportionality between force and
acceleration, if the subject be developed so that this is a
DEDUCTION IN MECHANICS 1 1/
postulate. The pupil may be convinced that, in his own
locality and for some substances, the statement is approx-
imately true. But does the functional relation between
force and acceleration involve no other variables, for ex-
ample, temperature ? And is its form the same for all kinds
of matter ? May not the force be proportional to the square
of the acceleration for some substances other than those
used in the experiments ? Pupils should be trained, and
trained from the outset, to question in this manner the
degree of accuracy of every measurement, and the gener-
ality of the circumstances under which each experiment is
performed. It is scientific method, and education of the
most practical and valuable character.
But, then, the pupils may say, what is the use of going
further ? Must not some better evidence be obtained ?
There are, of course, many reasons which can and should
be given for going on in faith, but one of the most illumi-
nating and interesting illustrations is contained in Faraday's
verification of Coulomb's law of electrostatic attraction. Few
boys fail to find interest in the picture of Faraday, basing
highly complex calculations on Coulomb's crude experi-
ments, testing them inside the highly charged iron box,
and emerging from it with a demonstration that the law
corresponds really closely with observed facts. The pupil
thus realises the true importance of deduction as an aid to
his very imperfect powers of observation. In place of
building complacently on a foundation whose imperfections
have been glossed over, too often with pulleys mounted on
ball bearings and other viciously misleading trivialities, he
has a sane idea of what he is doing. He sees that, on these
meagre foundations, he is to erect a structure which will
118
MATHEMATICAL DEDUCTION
come into contact with practical experience at many points,
and must be judged by its degree of accordance with such
experience at all these points.
The structure having been erected on a number of these
very dubious supports, it becomes necessary to examine
their possible interconnection, to ascertain whether the truth
or falsity of any one of these assumptions involves of logical
necessity the truth or falsity of any others. To illustrate
the process suggested, I have dealt with the postulates of
statics, but the method is equally applicable to dynamics,
or to any combination of mechanical postulates.
The customary assumptions in a first treatment of statics
suggested, and rightly so, by crude experiment are
three in number, namely, the triangle of forces, the principle
of the lever, and the principle of moments for two forces
acting along intersecting lines. All three have been adopted,
but any or all may be true or false. Thus the possibilities
are eight in number, as shown in the following table, and
among them must the truth be sought :
TRIANGLE OF
FORCES
PRINCIPLE OF
LEVER
PRINCIPLE OF
MOMENTS
I
true
true
true
2
false
true
true
3
true
false
true
4
true
true
false
5
true
false
false
6
false
true
false
7
false
false
true
8
false
false
true
Now it is possible to show, by logical deduction, that any
two of these assumptions are necessary consequences of the
DEDUCTION IN MECHANICS
119
third. 1 When this is done (and the proofs are well within
the comprehension of a boy of seventeen), the alternatives
2 to 7 disappear, and the three assumptions have become
one. The only possibility now being the simultaneous truth
or falsity of all, the rough experimental results acquire
greater import. If all three were false, it is trebly unlikely
that they should every one accord fairly well with experi-
ment, and the only alternative to the falsity of all is the
truth of all. The probability of this truth has thus been
strongly reinforced by processes purely logical in nature.
It is worthy of notice that the conventional deductive
method does not give an equal amount of strength to the
hypothesis. Postulating the triangle of forces, the principle
of the lever and the principle of moments are obtained by
logical deduction, the possible alternatives, five in number,
being left thus :
TRIANGLE OF
FORCES
PRINCIPLE OF
LEVER
PRINCIPLE OF
MOMENTS
true
true
true
false
true
true
false
true
false
false
false
true
false
false
false
The proofs that any two of these postulates can be de-
duced from the third are not given, and the full power of
deduction to reinforce assumption is not exhibited.
Treating other groups of postulates in the same manner,
the structure is seen to be based not on weak, isolated
1 If this is done, selecting any one assumption only, the consistence
of the three assumptions follows at once, and the demonstration of con-
sistence should not be overlooked.
120 MATHEMATICAL DEDUCTION
supports, but on interlinked groups of such supports, the
strength of the foundation being greatly increased by these
interconnections. And finality is reached when the sup-
ports have been interlinked into groups, between which it
can be demonstrated that no such logical interconnections
are possible. This, I conceive, is the true aspect of New-
ton's achievement in the statement of his three laws of
motion ; he stated a number of consistent and independent
hypotheses, and developed the whole subject from these
by purely logical processes. I do not imagine that Newton
had any such general concept of a science as is set out at
the beginning of this paper, but he perceived, intuitively
or sub-consciously, that mechanics could be based on three
sets of postulates, each set referring to a different class of
entities. In their statement his logic was defective, but
this is trifling compared with the greatness of his achieve-
ment; he formed an ideal for mechanics similar to that
of Euclid for geometry, and he attained practical success
in its elaboration, in contrast with Euclid's decided failure.
We have now before us three distinct didactic treatments
of mechanics the old method, the method now current,
and a development such as has here been suggested. The
old method presents the science in its complete form, with
no indication of its evolution. Three postulates are laid
down, to be accepted in blind faith, and from them the
subject is developed by logical processes ; it is a course in
applied deduction. The current method presents a number
of mechanical assumptions, based on foundations whose
strength is hardly discussed, and uses them in application
to various problems. There is little or no attempt to discuss
their logical interconnection, and certainly no suggestion
DEDUCTION IN MECHANICS 121
of its scientific meaning; it is a course in applied com-
putation. But if this current method were followed by a
logical discussion, exhibiting mechanics as based on inde-
pendent supports, each consisting of interlinked assump-
tions as has been described, I venture to suggest that it
might fairly be called a course in applied mathematics.
The tendency of modern education, as it seems to me,
is to lay undue stress on direct sensation as the one and
only basis for faith. Undoubtedly education must find its
origins, and these as widespread as possible, in direct sen-
sation ; but a false and very dangerous ideal is left, unless
finally these origins are linked together by logical process,
so as to give them their maximum strength and expose
their ultimate weakness. Final contentment with a set of
postulates which may or may not be inconsistent or re-
dundant, and for which there has appeared little real justi-
fication, is vicious ; vicious also is the attempt in a first
course to develop any science from a minimum of hypoth-
esis. The one method is an undue suppression of per-
ception, the other an undue glorification. The first step
of the teacher should be to develop a wide spirit of en-
quiry; the second should be to breed a " divine discontent"
with the imperfections of perceptual evidence. Recognis-
ing its essential nature, our inevitable bondage to it, we
may yet liberate ourselves, so far as may be, in each branch
of knowledge. It is the peculiar function of mathematics
to point the way to this freedom in each science, and it is
here that modern developments of mathematical thought
may yet find application in other sciences.
A COMPARISON OF GEOMETRY WITH
MECHANICS
(A paper read before the Liverpool Association of Teachers of
Mathematics and Physics)
A COMPARISON OF GEOMETRY WITH
MECHANICS
The subject of this paper may at first sight seem unlikely
to provide much opportunity for profitable discussion.
When it has been said that the bodies which are the con-
cern of mechanics move in the space which is the concern
of geometry, and are therefore subject to the laws of that
space, and when it is added that the same processes of
arithmetic are applied to measurement in each case, it may
be thought that the title is exhausted.
The comparison to which I invite you is not, however,
directly concerned with the matter or the domain of either
subject. My object is to compare them in their relation to
mathematical education ; to examine in how far each may
fulfil the ideals of that education, and in how far each may
supplement the deficiencies of the other. To do this, the
basis of comparison must first be assured ; that is, it is
necessary that I should explain what are the ends which
I conceive to be furthered by the teaching of mathematics.
In the conflict of educational interests which has in the
last fifteen years become so acute, mathematicians have
borne their part, but they cannot be said to have spoken
with one voice, whether in advocacy or defence. I do not
know that this is to be regretted, for no progressive devel-
opment is likely, except as the outcome of such differ-
ences ; but the fact enforces, on any who would discuss
mathematical education, a clear statement of their creed.
125
126 MATHEMATICAL EDUCATION
Even yet, however, the preliminary enquiry is unfinished.
I have just said that differing views on the aims of mathe-
matical teaching are held by those who are concerned with
the subject. For the most part they are held consciously
and can be explained at will. But an enquiry as to the
nature of mathematics itself the short question, " What
is mathematics ? " is apt to produce no immediate or
definite response. Too often, I fear, the only reply will be
that mathematics consists of algebra, geometry, trigonom-
etry, the calculus, and so on, arithmetic being, for some
unaccountable reason, omitted from the category. Now this
is a trifling with logic by just those people who ought, above
all others, to be in this respect beyond suspicion. If they
consider heat, light, sound, they can see why these are
grouped under the common term " physics " ; if they study
atoms, decomposition, elements, they can defend the one
term " chemistry " for these. But why the one term " mathe-
matics " for algebra, geometry, calculus, and all the other
branches ? What are the common elements in these sub-
jects which entitle them to a generic term ? Failure to
answer is a failure in logic, for to group entities in one
class without cognisance of common elements among them
is to offend the cardinal principles of classification and
definition.
It may, I know, be said with truth that mathematics is
concerned with reasoning pure reasoning, if you will.
But this is logic, and even those who are most uncertain
as to the definition of mathematics are equally certain that
there is some distinction between the study of mathematics
and the study of logic. A necessary preliminary to the
discussion which I have undertaken is, then, to define this
A COMPARISON WITH MECHANICS 127
distinction, and this must be my first task. After the
explanation of the nature of mathematics, its relation to
education must be discussed ; our comparison of geometry
and mechanics can then be developed.
The nature of mathematics may best be explained by
showing its relation to the physical sciences. The study of
any one of these commences by the assertion of a number
of statements, on bases more or less uncertain. The assump-
tion of laws of motion here on the earth, from observation
of the planets, is a daring one, however exactly these laws
describe the motions of the planets themselves ; the direct
evidence for the law of conservation of energy, or that
for most other laws of physics and chemistry, is weak to a
degree. To see general possibilities in a maze of results
apparently unco-ordinated, to fashion various hypotheses
to fit the facts as they are observed, is the function of the
natural philosopher ; it is no concern of the mathematician.
The process is exemplified in the lives of any of the great
natural philosophers above all, perhaps, in the life of
Newton, in that the development of his mind is known in
such detail.
The natural philosopher, having thus fulfilled his first
task, presents the hypotheses so formed to the mathemati-
cian, who accepts them for investigation without regard to
the evidence for or against them. Let us call these hypoth-
eses A, B, C, . . . . The mathematician does three things :
First, he makes deductions from the hypotheses ; that
is, he says to the natural philosopher : " If what you say is
true (and that is no concern of mine) then must certain
other statements P, Q, R, . . . also be true ; further, if cer-
tain of these be true, then must certain of A, B, C, . . . also
128 MATHEMATICAL EDUCATION
be true." The natural philosopher then tests the truth of
P, Q, R, . . . and finds his hypotheses strengthened or
destroyed as the case may be.
Next, the mathematician informs the natural philosopher
that he has examined the hypotheses A, B, C, . . . and
finds that they are consistent ; that is, that there is nothing
in any one of them to negative any other by the force of
logic. It should here be borne in mind that all measure-
ments are more or less inexact, and it is therefore possible,
from actual observations, unknowingly to frame hypotheses
which are logically contradictory one of another.
Finally, the mathematician informs the natural philos-
opher that some of his assumptions were unnecessary ;
that is, that some are logical consequences of others, and
so need not have been assumed. Or he tells him that all
were necessary, no one being deducible from some or all
of the others. That is, he investigates the possible redun-
dance of the hypotheses, and tells the natural philosopher
to how many distinct assumptions he is really committed.
Shortly, then, the mathematician receives sets of hypoth-
eses from the natural philosopher, tests their consistence,
examines how many assumptions are in fact involved, and
develops logical consequences from them. His function
is entirely impartial material truth is not his concern, but
is that of the natural philosopher ; but I would point out
to you that the really great natural philosophers, from
Archimedes downwards, have been men who in themselves
combined both these functions. In no one have they been
exemplified more nearly in their due proportions than in
Newton ; there have been greater mathematicians and there
have been men whose power of speculation was more rapid
A COMPARISON WITH MECHANICS 129
and prolific, but no other man has so balanced the one with
the other, and therein lay the secret of his pre-eminence.
I must not, however, leave an impression that the mathe-
matician has no use for his imagination and performs no
creative functions, though the slightest consideration of
any branch of the subject suffices to show the fatuity of
such an idea. The straight line at infinity, the circular
points at infinity, complex numbers, lines of force, the ether,
at once occur to the mind as commonplace instances of
creations in which the natural philosopher has had no
direct share. His entities are groups of sensations, and
relations between them are suggested by further sensa-
tions ; but the mathematician creates other entities which
would forever remain beyond the vision of the natural
philosopher, and has often, by their means, revealed unsus-
pected unities in his work. The creations of the mathe-
matician which are opposed to the suggestions of the senses
must forever rank among the most striking of all human
creations, and the shortness of this allusion to them is only
excused by their irrelevance to the matter under discussion..
It is now easy to explain the function of mathematics
in a well-balanced education. The purpose of teaching
natural science is to develop in combination the powers
of observation and speculation ; to train the pupil to use
his senses and, from the material which they afford him, to
frame hypotheses which accord with that material as nearly
as may be possible. The purpose of teaching mathematics
is to enable him to develop the consequences of these
hypotheses, to test their consistence, and to reduce them
to the minimum of pure assumption. I say to train him
in these things, but it were perhaps better to speak of setting
130 MATHEMATICAL EDUCATION
them before him as ideals for which he must strive in his
dealings with things as he finds them. A man who has in
his mind this chain of processes, observation, speculation,
proof of consistence in speculation, rejection of redundant
speculation, and finally the erection of deductions on this
foundation, is in possession of an intellectual creation
which, in beauty alone, is worthy to rank with the creations
of poetry, music, or art ; and beyond this, it is a possession
which, in so far as it guides his life, will make of him a
more efficient labourer and a better citizen.
We must now examine the relations of geometry and
mechanics to the description of mathematics which has
been given, and so ascertain which of the three processes
which have been said to pertain to the mathematician are
most clearly exemplified in each subject. First, however,
I must point out that this description of mathematics con-
signs geometry, and even arithmetic, to the domain of
applied mathematics. The one shows the application of
mathematics to number, the other to space ; in neither is
the underlying essence seen, except through illustrations
of one kind or the other. But the first thing to consider
with respect to a machine is to see what it does, rather than
to find out how it performs its functions, and we need not,
therefore, cavil at the idea that our so-called branches of
mathematics are really applications of mathematics, by
examination and contrast of which we may perceive the
underlying unity.
When a child commences geometry, what is his personal
position in regard to space ? Certainly it is far different
from his position in regard to matter when he commences
mechanics, or from his position in regard to light or heat
A COMPARISON WITH MECHANICS 131
when he commences physics. Personal experience of
S p ace his space has been forced on him from his
earliest days in his every movement, and from this experi-
ence he has formed ideas or hypotheses concerning a space
beyond, which he cannot reach with his own limbs, and
is therefore not his space. Show him a triangle ABC
cut out in cardboard, and make another by taking a tracing
of the corner A and producing its sides until they are
equal to AB and A C, and then ask him if these triangles
are an exact fit. He will not have much doubt about this,
nor will he question the equality of all right angles, if it be
similarly suggested to him. Next draw a circle, rule a
diameter, and ask him if one part of the curve so divided
will fit the other. About this also he will not have much
doubt. He will assert the truth of these things wherever
and whenever the acts are performed, not merely in his
own personal space ; that is, he asserts them for the
imagined space beyond, concerning which he has formed
ideas fashioned from experience in his own space. Now
as a matter of fact, as is well known to all of us here, the
third statement concerning the circle is a necessary and
inevitable consequence of the first two. These three asser-
tions are in truth redundant, to use our technical phrase,
as also are many others which the child will make with
equal certainty concerning this imagined space beyond his
personal space.
Thus, in commencing the study of geometry, we apply
mathematics to a subject space concerning which the
pupil is already in possession of a set of beliefs which are,
as a matter of fact, interdependent one on another. And
these beliefs are held with great tenacity ; nothing will
132 MATHEMATICAL EDUCATION
induce the pupil to doubt any of them, for they have, in
truth, become a part of his very being. It is further to be
remarked, for purposes of comparison, that we elders and
experts have not recanted any of these beliefs ; we hold
them as does the child, though we know them to be but
beliefs ; our imagined space has the same properties as his.
Next let us consider the position of a pupil in regard to
matter, when he commences the study of mechanics. It
is true that here also experience has been forced on him
from his earliest days, but it is of the narrowest kind, for
it concerns little more than the sensation of lifting, and
few assumptions are made. He will say, if two boxes are
known to be exactly alike and one feels heavier than the other,
that this one has something inside it ; and, which is a con-
sideration fully as important, if neither feels to him heavier
than the other, he will refuse to say that each is empty,
pointing out that the possible contents of one may be so
light that he does not notice them. But he is unconscious
of the concept of mass (or nearly so), of the triangle of
forces, and of any of the mechanical concepts and laws which
are to us so familiar. More, he only receives them with
difficulty ; explanation, illustration, and distinct effort are
required before the concepts are grasped and the state-
ments accepted as more or less nearly corresponding with
the results of experience. Some of these statements, as, for
example, Newton's first and third laws, are indeed received
with incredulity ; who has not heard it argued that the
horse pulls the cart more than the cart pulls the horse ?
At the outset, then, there is a sharp distinction between
geometry and mechanics. In geometry the pupil com-
mences with a number of ideas and beliefs concerning
A COMPARISON WITH MECHANICS 133
space beliefs held so tenaciously that to question them in
any way produces bewilderment. In mechanics he has but
a few crude concepts and few or no beliefs, and he greets
with disbelief some of those held by his elders. In the
terms of our earlier discussion, he has, in regard to space,
been his own natural philosopher though all uncon-
sciously and has produced a set of beliefs which are
ready for examination by the mathematician within him ;
but in regard to the matter the natural philosopher within
him has yet to play his part before the mathematician can
receive the materials for his task.
Returning to geometry, to which of the three processes
performed by the mathematician should the pupil first be
led in his study of this subject ? Shall he find whether
his beliefs concerning space are consistent with one an-
other, or enquire whether some of them may not be logical
consequences of others ; or shall he, without any such
analyses, pass on to make deductions from his spatial
creed, deferring its analysis for the time at least ? There
can, I think, be little doubt as to the answer. To analyse
the number of his beliefs, to question and examine their
foundations, is repugnant to any normal child, though
welcome to most educated adults ; we should then avoid
these processes and allow the study of geometry to centre
round deductions from the pupils' spatial beliefs. In pri-
mary education this subject can only illustrate the deductive
side of mathematics ; it can do little or nothing to show
the analytical processes which are concerned with consist-
ence and redundance.
We have said that, in commencing mechanics, the child
must play the natural philosopher before the mathematician
134 MATHEMATICAL EDUCATION
can find scope for his efforts. This subject thus provides
the first example of the methods of natural philosophy,
and the part of the teacher is one of supreme importance.
Shall he allow the child's fancy to roam whither it will,
or set him down to prescribed tasks with definite ends, or
endeavour to lead him to the examination of natural phe-
nomena by those methods which history has shown to be
most productive ?
There is, I know, a strong movement in favour of rely-
ing upon the " interest "or " play " motive in primary edu-
cation ; and the vocal organs of this movement are highly
developed. There is also, I believe, a perhaps stronger
movement, whose vocal organs are as yet rudimentary, in
favour of severe limitation of the use of these methods.
To some extent each party misjudges the other. The up-
holders of interest accuse their opponents of enforcing
meaningless drudgery, while these in their turn accuse
their opponents of allowing education to degenerate into
disconnected fripperies, and each is more or less unjust
in so doing. For myself, and speaking only in regard to
this present subject, I would say that the mere perform-
ance of prescribed tasks, which arise apparently from the
brain of the teacher or the designer of apparatus for use in
schools, can have nothing to do with the development of
the spirit of natural philosophy. On the other hand, to
allow education to be dominated by interest is to cast aside
all hope of discipline not discipline of class by teacher,
but discipline of pupil by himself. To show the exact
meaning of this statement, let me recall to you the behaviour
of young boys learning to play football or cricket. Their
interest is to toss the ball aimlessly from one to another,
A COMPARISON WITH MECHANICS 135
and to scamper about as irregularly as young horses ; as
much discipline is required to induce them to play the game
as is used in many a class-room to ensure the due perform-
ance of work. The effect of this discipline is to replace
casual interest in passing sensations by continued interest
in definite achievements ; in like manner, the effect of our
teaching should be to leave the pupils with this desire for
achievement, rather than with a mere craving for a tickling
of the fancy. I do not mean to imply that interest is not
to be considered ; on the contrary, I will assert most em-
phatically that teaching which is met by a continued lack
of interest must of necessity be at fault. But I assert
also that the mere presence of interest is insufficient as
a testimony to the value of education ; and, further, that
courses which are chosen on the basis of maximum imme-
diate interest are, in all probability, thereby grievously
at fault.
We must then, in commencing the study of mechanics,
guide our pupils to the attitude of the natural philosopher
towards the phenomena which he studies. But what is this
attitude ? How far is it concerned with matters of common
experience, and how far with the more artificial happen-
ings of the laboratory ? And is there any order of priority
as between common experience and laboratory investiga-
tion ? Here there can be little doubt, whether the question
be viewed in the light of history or of common sense. The
foundation of this work must be common experience; it
must be reviewed, questions must be asked to be an-
swered in the laboratory, and hypotheses must be made
to be tested there. Two illustrations may show my meaning
more precisely.
136 MATHEMATICAL EDUCATION
The customary experimental introduction to the triangle
of forces starts from nothing ; bodies are suspended and
balanced on pulleys without suggestion of a why or a where-
fore. The attitude which seems more natural is to give to
or draw from the pupils everyday illustrations of the balanc-
ing of two forces by a third, and to obtain from them as
much information as possible about this third force; namely,
that it always exists, that it lies between the two original
forces, and is nearer to the larger of these. Then the
question of the law of determination arises ; this is a ques-
tion to be answered in the laboratory, and the answer found
must receive every possible verification.
Next, let us consider some hypothesis which may first
be made and then be tested in the laboratory. Several such
hypotheses are possible, but perhaps the best is found in
the motion of a falling body. To surmise a rule for the
composition of two forces is beyond any child ; but specu-
lation on the actual law of acceleration of a falling body,
its dependence on distance or time, is natural from the
outset, and should be encouraged before investigation is
undertaken. It is not difficult to divide mechanical inves-
tigations into two classes ; namely, those in which initial
speculation is possible, and those in which the pupil can
only ask a question with no power of suggesting a possible
answer, and this division does much to simplify the early
treatment of mechanics.
So much, then, for the part of the natural philosopher
in mechanics ; the pupil has now a body of statements
which he is willing to accept, for the time at any rate,
though he recognises them as assumptions which may be
unwarranted. Observe that I call them statements, not
A COMPARISON WITH MECHANICS 137
beliefs, and do not refer to them as his mechanical creed ;
this because they are accepted in no such blind and un-
reasoning faith as the assumptions in geometry to which
I gave such terms.
The mathematician in the child is now presented with
these accepted statements ; what is he to do with them ? Can
any or all of his three functions here be exercised with
profit to the pupil in that he will so gain some idea of their
nature and import ? In particular, can we now develop the
two questions of consistence and redundance in speculation,
which were put on one side in the consideration of geometry?
First consider deduction. This process is possible and
necessary, but its area of application is limited in compari-
son with geometry ; the link polygon, the centre of mass
and its motion, the path of a body under gravity, are in-
stances, and others will occur to you. But there is no such
wealth of propositions and riders as is found in geometry ;
most of the problems in mechanics are applications of
principles rather than logical deductions.
Next take analysis. It must first be said that the appli-
cation of such processes to mechanics is at least legitimate
for pupils in schools. To question and investigate their
geometrical creed at this age must lead to perplexity and
boredom, as has already been said ; but the accepted state-
ments of mechanics lie under no such ban. They are
assumptions made consciously on the best evidence obtain-
able, and investigations tending to their confirmation are
at least acceptable ; this in contrast to geometry, of which
the reverse has just been said.
To discuss in detail the possibility of undertaking this
analysis for mechanics would lead me too far. Here I can
138 MATHEMATICAL EDUCATION
only say that for certain parts of mechanics, at any rate, it
is not only possible but easy and interesting. For example,
any boy can understand proofs that from any one of
the three assumptions known as the triangle of forces, the
principle of the lever, and the principle of moments, the
other two can be deduced ; he can thus see that only one
of them need be assumed, of course that one for which
the evidence is strongest; and he can see further that
there need be no fear that, at some future time, one of
these assumptions will be found in logical conflict with
another. For the meaning of such conflict there is ample
illustration in mechanics and physics.
This concludes our comparison of geometry and me-
chanics. What, in short, is the outcome ? First, we have
seen that there are three leading processes in mathematics,
deduction, analysis of consistence, and analysis of redun-
dance, these being exercised by the mathematician on
material presented to him by the natural philosopher. Next,
geometry was seen to be well suited for exercise in deduc-
tion, but not suited for illustration of the other processes ;
this is because the natural philosopher acted too early in
regard to space, and will not now brook criticism of his
results. Finally, this deficiency was seen to be remedied
by mechanics, which should provide the first deliberate
exercise in natural philosophy, and so present material
better suited for illustration of these remaining processes
of mathematics.
It may be asked whether it can be hoped that a pupil
may end his school career with these ideas fully de-
veloped. Frankly, I do not think that he can ; indeed, I
am sure that he cannot. We may regard such ideas as
A COMPARISON WITH MECHANICS 139
mountain peaks, standing far above the mists of the partic-
ular applications, two of which we have in particular been
discussing to-night. He who has scrambled longest among
the mists sees these peaks most clearly ; some indeed have
pierced the clouds and seen them in their full beauty have
even scaled them and viewed one from another. The
function of the teacher is to lead the child through the
mists by such ways as will give him glimpses, even though
they are but shadowy, of the higher ground beyond. These
will remain and develop in minds to which they are
suited minds, I am convinced, far more common than
is generally supposed.
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
AN INITIAL FINE OF 25 CENTS
WILL. BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO SO CENTS ON THE FOURTH
DAY AND TO $1.OO ON THE SEVENTH DAY
OVERDUE.
DEC 171934
ocr 13
JUN 8
FEB&71942E
MAR. 10 1947
JAN - 5 1959
FEB 2 2006
LD 21-50m-8,33
YC 04113
OF CALIFORNIA LIBRARY
