We must beware of abstractions in these forms of knowledge, and let the child see and build for himself, then lead him to express in numbers what he has seen and built. He will not call it Arithmetic, nor be troubled with any visions of mathematics as an abstract science.[56]
The cube may be divided into thirds, ninths, and twenty-sevenths, and the fact thus practically shown that whether the thirds are in one form or another, in long lines or squares, upright or flat, the contents remain the same. We may also illustrate by building, that like forms may be produced which shall have different contents, or different forms having the same contents.
Halves and quarters may be discussed and fully illustrated, and addition, subtraction, multiplication, and division may be continued as fully as the comprehension of the child will allow.
During the practice with the forms of knowledge we should frequently illustrate the lawful evolution of one form from another, as in the series moving from the parallelopiped to the hexagonal prism.
It should not be forgotten that whenever the cube is separated and divided, recombination should follow, and that the gift plays should always close with synthetic processes.
Some of the mathematical truths shown in the fifth gift were also seen in the third, but "repeated experiences," as Froebel says, "are of great profit to the child."[57]
We should allow no memorizing in any of these exercises or meaningless and sing-song repetitions of words. We must always talk enough to make the lesson a living one, but not too much, lest the child be deprived of the use of his own thoughts and abilities.
THE FIFTH GIFT B.
There is a supplemental box of blocks called in Germany the fifth gift B, which may be regarded as a combination of the second and fifth gifts, and whose place in the regular line of material is between the fifth and sixth. It was brought out in Berlin more than thirteen years ago, but has not so far been used to any extent in this country.