A similar process may be adopted with the multiplication table. With a case like the other, it is only necessary to tell the child that if he wants to know how much are three times four he has only to make heaps of four things each, take three of them and put them in the box at the intersection of the line three and the column four. If he can write the figures he will write 12, instead of gathering up the twelve objects that represent the product. When he has played at this for some time he may become acquainted with all the products up to ten times ten or beyond without having to make any abnormal effort of memory.

The idea of numeration, which is usually put off till a later period, should also be given at the beginning. Children soon understand the decimal numeration and learn to write 10 for ten, and other numbers composed of one of the nine ciphers and zero. But the fact which, however, though quite essential to know, receives very little attention is that there is nothing particular about this number ten, and that systems of numeration can be devised resting on any basis that may be taken; that the principle of every system of numeration consists in taking a certain number of units and grouping them. Take, for example, a system having five as its basis. All the numbers of such a system can be represented with the figures 1, 2, 3, and 4, the symbol 10 standing in this case for five. To construct a number we have only to group the units by fives and observe the result.

To learn decimal numeration by this process we put tens of objects into little boxes, tens of little boxes into larger ones, and so on. The child can in this way acquire an exact idea of the units of successive order in any system that may be desired.

This method of teaching was developed in a remarkable way about thirty years ago by Jean Macé in a little book entitled L'Arithmétique du Grand-Papa—Grandpa's Arithmetic—which made some impression when it appeared, but has been substantially forgotten.

In this method I attach much importance to giving these exercises a form of play. I believe that nothing in primary instruction should savor of obligation and fatigue. It would, on the other hand, be better to try to induce the child to desire himself to go on, and it would always be well to try to give him the illusion, in all stages of instruction, that he is the discoverer of the facts we wish to impress upon his mind.

We need not stop with arithmetic, but may go on and give the child a little geometry. To accomplish this we should give him the idea of geometrical objects, and to some extent their nomenclature, and this can be done without causing fatigue. To accomplish this he should be taught to draw, however rudely. He can begin with straight lines, of which he soon learns the properties; then, when he has drawn several lines side by side, he will learn that they are parallels and will never meet. He will learn, too, after he has drawn three intersecting lines, that the figure within them is called a triangle, that the figure formed by two parallel lines meeting two other parallels is a parallelogram, and he can go on to make and learn about polygons, etc (Fig. 3). All this nomenclature will get into his head without giving abstract definitions, but in such a way that when he sees a geometrical object of definite form he will recognize it at once and give it the name that belongs to it.

Fig. 3.

In the practical matter of the measurement of areas we convey immediate comprehension as to many figures without special effort, provided we do not present the demonstration in professional style, limiting ourselves to making the pupil comprehend or feel things so clearly and definitely that it shall be equivalent, as to the satisfaction of his mind, to an absolutely rigorous demonstration. At any rate, he will be better provided for the future than by rigorous demonstrations that he does not understand. Taking the parallelogram, for example, let us suppose a figure made like Fig. 4, and we saw through it along the lines A A' and B C. It does not need a very great effort of attention to recognize, experimentally if need be, that the two triangles A A' D and B B' C may be placed one upon the other and are identical. If, from the figure thus formed, we take away the right-hand triangle the parallelogram will remain; if we take away the other triangle a rectangle will be left, or a peculiar parallelogram, of which also we give the idea to the child as a figure in which the angles are formed by straight lines perpendicular to one another. Here, then, the child gains the notion of the equivalence of a parallelogram and a rectangle of the same base and height; and this notion, obtained by cutting up a piece of board or pasteboard, he will carry so seriously and firmly in his head that he will never lose it. By cutting the same parallelogram in two, along a diagonal A C, it may be easily shown that the two triangles can be placed exactly one upon the other, and that, consequently, they have equal areas. These lessons constitute a series of classical theorems in geometry which the child can try with his fingers and learn without even giving them the form of theorems. I might show the same as to the area of the trapeze and with many other theorems, but my purpose is only to present as many examples as will make my idea understood, without going into details.