The children possess all the instinctive knowledge necessary as a
preparation for clear ideas on numeration. The idea of quantity was
inherent in all the material for the education of the senses: longer,
shorter, darker, lighter. The conception of identity and of difference
formed part of the actual technique of the education of the senses,
which began with the recognition of identical objects, and continued
with the arrangement in gradation of similar objects. I will make a
special illustration of the first exercise with the solid insets,
which can be done even by a child of two and a half. When he makes a
mistake by putting a cylinder in a hole too large for it, and so
leaves _one_ cylinder without a place, he instinctively absorbs the
idea of the absence of _one_ from a continuous series. The child's
mind is not prepared for number "by certain preliminary ideas," given
in haste by the teacher, but has been prepared for it by a process of
formation, by a slow building up of itself.
To enter directly upon the teaching of arithmetic, we must turn to the
same didactic material used for the education of the senses.
Let us look at the three sets of material which are presented after
the exercises with the solid insets, _i.e._, the material for teaching
_size_ (the pink cubes), _thickness_ (the brown prisms), and _length_
(the green rods). There is a definite relation between the ten pieces
of each series. In the material for length the shortest piece is a
_unit of measurement_ for all the rest; the second piece is double the
first, the third is three times the first, etc., and, whilst the scale
of length increases by ten centimeters for each piece, the other
dimensions remain constant (_i.e._, the rods all have the same
section).
The pieces then stand in the same relation to one another as the
natural series of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
In the second series, namely, that which shows _thickness_, whilst the
length remains constant, the square section of the prisms varies. The
result is that the sides of the square sections vary according to the
series of natural numbers, _i.e._, in the first prism, the square of
the section has sides of one centimeter, in the second of two
centimeters, in the third of three centimeters, etc., and so on until
the tenth, in which the square of the section has sides of ten
centimeters. The prisms therefore are in the same proportion to one
another as the numbers of the series of squares (1, 4, 9, etc.), for
it would take four prisms of the first size to make the second, nine
to make the third, etc. The pieces which make up the series for
teaching thickness are therefore in the following proportion: 1 : 4 :
9 : 16 : 25 : 36 : 49 : 64 : 81 : 100.
In the case of the pink cubes the edge increases according to the
numerical series, _i.e._, the first cube has an edge of one
centimeter, the second of two centimeters, the third of three
centimeters, and so on, to the tenth cube, which has an edge of ten
centimeters. Hence the relation in volume between them is that of the
cubes of the series of numbers from one to ten, _i.e._, 1 : 8: 27 :
64: 125 : 216 : 343 : 512 : 729 : 1000. In fact, to make up the
volume of the second pink cube, eight of the first little cubes would
be required; to make up the volume of the third, twenty-seven would be
required, and so on.
[Illustration:
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1
1 2
1 2 3
1 2 3 4
1 2 3 4 5
1 2 3 4 5 6
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 10