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.