They represent the same thing: that is to say, as numbers they are 2 × 2; but they differ in position—one has the form of a line, the other of a square. It can be seen from this that if as many bars as there are beads on a bar are placed side by side they form a square.
In the series in fact we offer squares of 3 × 3 pink beads; 4 × 4 yellow beads; 5 × 5 pale blue beads; 6 × 6 gray beads; 7 × 7 white beads; 8 × 8 lavender beads; 9 × 9 dark blue beads; and 10 × 10 orange beads; thus reproducing the same colors as were used at the beginning in counting.
For every number there are as many bars as there are beads for the number, 3 bars for the 3, 4 for the 4, etc.; in addition there is a chain consisting of an equal number of bars, 3 × 3; 4 × 4; and, as we have seen, there is a square containing another equal quantity.
The child not only can count the beads of the chains and squares, but he can reproduce them by placing the corresponding single bars either in a horizontal line or laying them side by side in the shape of a square. The number repeated as many times as the unit it contains is really the multiplication of the number by itself.
For example, taking the small square of four the child can count four beads on each side; multiplying 4 by 4 we have the number of beads in the square, 16. Multiplying one side by itself (squaring one side) we have the area of the little square.
This can be continued for 5, 8, 9, etc. The square of 10 has ten beads on each side. Multiplying 10 by 10, in other words, "squaring" one side we get the entire number of beads forming the area of the square: 100.
However, it is not the form alone which gives these results; for if the ten bars which formed the square are placed end to end in a horizontal line, we get the "hundred chain." This can be done with each square; the chain 5 × 5, like the square 5 × 5, contains the same number of beads, 25. We teach the child to write the numbers with symbol for the square: 52 = 25; 72 = 49; 102 = 100, etc.
Our material here is manufactured with reference to the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10. It is "offered" to the child, beginning with the smaller numbers. Given the material and freedom, the idea will come of itself and the child will "work" it into his consciousness on them.
In this same period we take up also the cubes of the numbers, and there is a similar material for this: that is, the chain of the cube of the number is made up of chains of the square of that number joined by several links which permit of its being folded. There are as many squares for a number as there are units in that number—four squares for number 4, six squares for 6, ten squares for 10—and a cube of the beads is formed by placing the necessary number of squares one on top of the other.
