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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

FIG. 40.--DIAGRAM ILLUSTRATING USE OF NUMERICAL RODS.]

The children have an intuitive knowledge of this difference, for they
realize that the exercise with the pink cubes is the _easiest_ of all
three and that with the rods the most difficult. When we begin the
direct teaching of number, we choose the long rods, modifying them,
however, by dividing them into ten spaces, each ten centimeters in
length, colored alternately red and blue. For example, the rod which
is four times as long as the first is clearly seen to be composed of
four equal lengths, red and blue; and similarly with all the rest.

When the rods have been placed in order of gradation, we teach the
child the numbers: one, two, three, etc., by touching the rods in
succession, from the first up to ten. Then, to help him to gain a
clear idea of number, we proceed to the recognition of separate rods
by means of the customary lesson in three periods.

We lay the three first rods in front of the child, and pointing to
them or taking them in the hand in turn, in order to show them to him
we say: "This is _one_." "This is _two_." "This is _three_." We point
out with the finger the divisions in each rod, counting them so as to
make sure, "One, two: this is _two_." "One, two, three: this is
_three_." Then we say to the child: "Give me _two_." "Give me _one_."
"Give me _three_." Finally, pointing to a rod, we say, "What is this?"
The child answers, "Three," and we count together: "One, two, three."

In the same way we teach all the other rods in their order, adding
always one or two more according to the responsiveness of the child.

The importance of this didactic material is that it gives a clear idea
of _number_. For when a number is named it exists as an object, a
unity in itself. When we say that a man possesses a million, we mean
that he has a _fortune_ which is worth so many units of measure of
values, and these units all belong