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 to one person.
So, if we add 7 to 8 (7 + 8), we add a _number to a number_, and these
numbers for a _definite_ reason represent in themselves groups of
homogeneous units.
Again, when the child shows us the 9, he is handling a rod which is
inflexible--an object complete in itself, yet composed of _nine equal
parts_ which can be counted. And when he comes to add 8 to 2, he will
place next to one another, two rods, two objects, one of which has
eight equal lengths and the other two. When, on the other hand, in
ordinary schools, to make the calculation easier, they present the
child with different objects to count, such as beans, marbles, etc.,
and when, to take the case I have quoted (8 + 2), he takes a group of
eight marbles and adds two more marbles to it, the natural impression
in his mind is not that he has added 8 to 2, but that he has added 1
+ 1 + 1 + 1 + 1 + 1 + 1 + 1 to 1 + 1. The result is not so clear, and
the child is required to make the effort of holding in his mind the
idea of a group of eight objects as _one united whole_, corresponding
to a single number, 8.
This effort often puts the child back, and delays his understanding of
number by months or even years.
The addition and subtraction of numbers under ten are made very much
simpler by the use of the didactic material for teaching lengths. Let
the child be presented with the attractive problem of arranging the
pieces in such a way as to have a set of rods, all as long as the
longest. He first arranges the rods in their right order (the long
stair); he then takes the last rod (1) and lays it next to the 9.
Similarly, he takes the last rod but one (2) and lays it next to the
8, and so on up to the 5.
This very simple game represents the addition of numbers within the
ten: 9 + 1, 8 + 2, 7 + 3, 6 + 4. Then, when he puts the rods back in
their places, he must first take away the 4 and put it back under the
5, and then take away in their turn the 3, the 2, the 1. By this
action he has put the rods back again in their right gradation, but
he has also performed a series of arithmetical subtractions, 10 - 4,
10 - 3, 10 - 2, 10 - 1.
The teaching of the actual figures marks an advance from the rods to
the process of counting with separate units. When the figures are
known, they will serve the very purpose in the abstract which the rods
serve in the concrete; that is, they will stand for the _uniting into
one whole_ of a certain number of separate units.
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 speak
of the succeeding lessons in zero, in tens and in other arithmetical
processes--for the development of which my larger works must be
consulted. The didactic material itself, however, can give some idea.
In the box containing the pegs there is one compartment over which the
0 is printed. Inside this compartment "nothing must be put," and then
we begin with _one_.
Zero is nothing, but it is placed next to one to enable us to count
when we pass beyond 9--thus, 10.
[Illustration: FIG. 42.--ARITHMETIC FRAME.]
If, instead of the piece 1, we were to take pieces as long as the rod
10, we could count 10, 20, 30, 40, 50, 60, 70, 80, 90. In the didactic
material there are frames containing cards on which are printed such
numbers from 10 to 90. These numbers are fixed into a frame in such a
way that the figures 1 to 9 can be slipped in covering the zero. If
the zero of 10 is covered by 1 the result is 11, if with 2 it becomes
12, and so on, until the last 9. Then we pass to the twenties (the
second ten), and so on, from ten to ten. (Fig. 42.)
For the beginning of this exercise with the cards marking the tens we
can use the rods. As we begin with the first ten (10) in the frame, we
take the rod 10. We then place the small rod 1 next to rod 10, and at
the same time slip in the number 1, covering the zero of the 10. Then
we take rod 1 and figure 1 away from the frame, and put in their place
rod 2 next to rod 10, and figure 2 over the zero in the frame, and so
on, up to 9. To advance farther we should need to use two rods of 10
to make 20.
The children show much enthusiasm when learning these exercises, which
demand from them two sets of activities, and give them in their work
clearness of idea.
* * * * *
In writing and arithmetic we have gathered the fruits of a laborious
education which consisted in coordinating the movements and gaining a
first knowledge of the world. This culture comes as a natural
consequence of man's first efforts to put himself into intelligent
communication with the world.
All those early acquisitions which have brought order into the child's
mind, would be wasted were they not firmly established by means of
written language and of figures. Thus established, however, these
experiences open up an unlimited field for future education. What we
have done, therefore, is to introduce the child to a higher level--the
level of culture--and he will now be able to pass on to a _school_,
but not the school we know to-day, where, irrationally, we try to give
culture to minds not yet prepared or _educated to receive it_.
To preserve the health of their minds, which have been _exercised_ and
not _fatigued_ by the order of the work, our children must have a new
kind of school for the acquisition of culture. My experiments in the
continuation of this method for older children are already far
advanced.