s.

The _synthetic_ function of language and the wide field of work which
it opens out for the intelligence is _demonstrated_, we might say, by
the function of the _figure_, which now can be substituted for the
concrete rods.

The use of the actual rods only would limit arithmetic to the small
operations within the ten or numbers a little higher, and, in the
construction of the mind, these operations would advance very little
farther than the limits of the first simple and elementary education
of the senses.

The figure, which is a word, a graphic sign, will permit of that
unlimited progress which the mathematical mind of man has been able to
make in the course of its evolution.

In the material there is a box containing smooth cards, on which are
gummed the figures from one to nine, cut out in sandpaper. These are
analogous to the cards on which are gummed the sandpaper letters of
the alphabet. The method of teaching is always the same. The child is
_made to touch_ the figures in the direction in which they are
written, and to name them at the same time.

In this case he does more than when he learned the letters; he is
shown how to place each figure upon the corresponding rod. When all
the figures have been learned in this way, one of the first exercises
will be to place the number cards upon the rods arranged in gradation.
So arranged, they form a succession of steps on which it is a pleasure
to place the cards, and the children remain for a long time repeating
this intelligent game.

After this exercise comes what we may call the "emancipation" of the
child. He carried his own figures with him, and now _using them_ he
will know how to group units together.

[Illustration: FIG. 41.--COUNTING BOXES.]

For this purpose we have in the didactic material a series of wooden
pegs, but in addition to these we give the children all sorts of small
objects--sticks, tiny cubes, counters, etc.

The exercise will consist in placing opposite a figure the number of
objects that it indicates. The child for this purpose can use the box
which is included in the material. (Fig. 41.) This box is divided into
compartments, above each of which is printed a figure and the child
places in the compartment the corresponding number of pegs.

Another exercise is to lay all the figures on the table and place
below them the corresponding number of cubes, counters, etc.

This is only the first step, and it would be impossible here to