Let us consider the cube of four. There is a chain formed by four chains each representing the square of four. They are joined by small links so that the chain can be rolled up lengthwise. The chain of the cube, when thus rolled, gives four squares similar to the separate squares which, when drawn out again, for a straight line.
Fig. 5.—This shows only part of the entire chain for 43.
The quantity is always the same: four times the square of four. 4 × 4 × 4 = 42 × 4 = 43.
The cube of four comes with the material; but it can be reproduced by placing four loose squares one on top of the other. Looking at this cube we see that it has all its edges of four. Multiplying the area of a square by the number of units contained in the side gives the volume of the cube: 42 × 4.
In this way the child receives his first intuitions of the processes necessary for finding a surface and volume.
With this material we should not try to teach a great deal but should leave the child free to ponder over his own observations—observing, experimenting, and meditating upon the easily handled and attractive material.
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Little by little we shall see the slates and copybooks filled with exercises of numbers raised to the square or cube independently of the rich series of objects which the material itself offers the child. In his exercises with the square and cube of the numbers he easily will discover that to multiply by ten it suffices to change the position of the figures—that is to say, to add a zero. Multiplying unity by ten gives 10; ten multiplied by ten is equal to 100; one hundred multiplied by ten is equal to 1,000, etc.
Before arriving at this point the child will often either have discovered this fact for himself or have learned it by observing his companions.