“Above all, it is the old within the new, which clarifies, unfolds and transmutes itself, thus developing what is new.… We must not require of the child anything not conditioned by his previous achievements.”—P., p. 169.
No one, surely, can maintain that these words are carried into effect in e.g.:
“Could forms of knowledge (mathematical forms) be, for a child of one to three, play forms, and thus forms produced by spontaneous activity? Well, why not? Arrange the eight part-cubes together, and say, ‘One whole.’ But divide it immediately and say, ‘Two halves.’… Or, comparing and connecting and describing by song at the same time that the objects are manipulated:
‘Look here and see! One whole two halves.
One half two fourths, two halves four fourths.
One whole four fourths.
Four fourths eight eighths.
Eight eighths one whole.’”—P., p. 138.
There is certainly no “old within the child” of one to three, which can condition this achievement, nor is there any spontaneity. For the child a little older we have: