The illustration at the top shows the square and the cube of 4 and of 5. That in the middle shows the arithmetic board being used for multiplication. In the photograph at the bottom a problem in division is being worked out on the arithmetic-board: 26 ÷ 4 = 6 and 2 remainder.
Now the beads must be adjusted. The two beads that are left over on the card marked with the number 6 are green but the box above this card is the gray-white one. We must therefore change the green beads into white beads, taking for each one of them a tube of ten white beads. The white beads which were left over on the other card must be brought to the card above which the white box is now placed. We have only to arrange the white beads now in rows of six while the other box of red beads is emptied on to the second board in rows of four, as in simple division.
With the material arranged in this way according to color, we proceed to the reduction, which is done by exchanging one bead of a higher denomination for ten of a lower. Thus, for example, in the present case we have twenty-three white beads distributed on the first board in rows of six, which gives a quotient of three and a remainder of five. On the second board there are six red beads distributed in rows of four, giving a quotient of one with a remainder of two. Now the work of reduction begins. This consists in taking one by one the beads from the board to the left—in this case the white—and exchanging them for ten red beads, which in turn are placed in rows of four on the other board until the quotients on the two cards are alike. What is left over is the remainder. In this case it is necessary to change only the one white bead so as to have the other quotient reach three with a remainder of four.
The same process is continued until all the boxes are used.
The final remainder is the one to be written down with the quotient.
The exercise requires great patience and exactness, but it is most interesting and might be called an excellent game of solitaire for children for home use. There is no intellectual fatigue but much movement and much intense attention. The quotients and remainders may be written on a prepared sheet of paper, so as to be verified by the teacher.
When the child has performed many of these exercises he comes spontaneously to try to foresee the result of an operation without having to make the material exchange and arrangement of the beads; hence to shorten the mechanical process. When at length he can "see" the situation at a glance, he will be able to do the most difficult division by the ordinary processes without experiencing any fatigue, or without having been obliged to endure tiring progressive lessons and humiliating corrections. Not only will he have learned how to perform long divisions but he will have become a master of their mechanism. He will realize each step, in ways that the children of ordinary secondary schools possibly never will be able to understand, when through the usual methods of rational mathematics they approach the incomprehensible operations which they have performed for several years without considering the reasons for them.