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]
[56] "Perceptions and recognitions which are with difficulty
gained from _words_ are easily gained from facts and deeds.
Through actual experience the child gains in a trice a total
concept, whereas the same concept expressed in words would be
only grasped in a partial manner. The rare merit, the
vivifying influence of this play-material is that, through
the representations it makes possible, concepts are
recognized at once in their wholeness and unity, whereas such
an idea of a whole can only very gradually be gained from its
verbal expression. It must, however, be added that later,
through words, the concept can be brought into higher and
clearer consciousness."--Froebel's _Pedagogics_, page 206.
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.
[57] "It is through frequent return to a subject and intense
activity upon it for short periods, that it 'soaks in' and
becomes influential in the building of character. Especially
is this true if the principles of apperception and
concentration are not forgotten by the teacher in working
upon the disciplinary subjects." (Geo. P. Brown.)