LATER DEVELOPMENTS IN ARITHMETIC 343
kings by the names a, b, c and writing the names of the separate
pieces according to their dependence each on his own king, it
happens that children, five years old and certainly those of six,
store up in their minds the algebraic formula of the cube of a
trinomial without looking at the material, because there is fixed
the visual memory of the disposition of the various objects. This
gives some idea of the possibilities that could be attained in practice.
All the teaching in arithmetic and in those principles of
algebra in the form of reading and other memorizing cards and
other material lead to results which might seem to be fabulous,
and which show that the teaching of arithmetic ought to be com-
pletely transformed, starting from a sense preparation of the mind
based on concrete knowledge.
It is clear that these children, sk years old, on entering one
of the ordinary schools, where they begin to count 1, 2, 3, are now
out of place, and that a radical reform of the elementary schools
is essential if there is to continue this wonderful development of
But, beyond the active method, in which there is always
operating the movement of the hand which moves objects about,
and in which the senses are so energetically employed, one must
think of the special attitudes of the child mind towards mathe-
matics. Because the children, leaving the material, very easily
come to love writing out the operation, thus doing abstract mental
work and acquiring a kind of natural and spontaneous leaning
towards mental calculations.
For example, a child, when he left a London bus along with
his mother, said: *elf everybody had spat, £34 would have been
collected." The child had noticed a card which said that spitting
in the car carried a penalty of a certain number of shillings. Then
the child had passed the time in calculating mentally the amount
involved and in turning the shillings into pounds.