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: 106 “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, 108 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.