THE TEACHING OF NUMERATION 327
the first, should we say one, and on adding another, two, and so
on? The little child tends to say one about every new object which
is added—" One, one, one, one, one," rather than " One, two*
three, four, five."
The fact that with the addition of a new unit the group is
enlarged, and that there must be considered this increasing whole
which constitutes the real obstacle is met with in numeration,
when it concerns children of three and a half to four years of age.
The grouping together into one whole of units which are really
separate from one another is a mental operation beyond the child's
powers. Many small children can count, reciting from memory,
the natural series of numbers, but they are confused when dealing
with the quantities corresponding to them. Counting the fingers,
the hands and the feet certainly forms something more concrete
for the child, because he can always find the same objects,
invariably joined together as a definite quantity. He will always
know that he has two hands and two feet.
Rarely, however, will he be able to count with certainty the
fingers of one hand and when he does succeed, the difficulty is
to know why, if the hand has five fingers, he should have to say
about the same object—" One, two, three, four, five." This con-
fusion, which the rather more mature mind corrects, interferes
with numeration in the earlier years. The extreme exactness and
concreteness of the child's mind needs help which is precise and
clear. When numerical rods are in use we find out that the very
smallest children take the keenest interest in numbers.
The rods correspond to the numbers and increase in length
gradually, unit by unit, hence they give not only the absolute but
also the relative idea of number. The proportions have already
been studied in the sense-exercise; here they are determined mathe-
matically, constituting the first studies in arithmetic. These num-
bers, which can be handled and compared, lend themselves at
once to combinations and comparisons. For example, by placing
together the rod of one unit and that of two, there is produced a
length equal to that of the rod of three. From the union of rods