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 education. 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.