LONG DIVISION
Not only is it possible to perform long division with our bead material, but the work is so delightful that it becomes an arithmetical pastime especially adapted to the child's home activities. Using the beads clarifies the different steps of the operation, creating almost a rational arithmetic which supersedes the common empirical methods, that reduce the mechanism of abstract operations to a simple routine. For this reason, these pastimes prepare the way for the rational processes of mathematics which the child meets in the higher grades.
The bead frame will no longer suffice here. We need the square arithmetic board used for the first partial multiplications and for short division. However, we require several such boards and an adequate provision of beads. The work is too complicated to be described clearly, but in practise it is easy and most interesting.
It is sufficient here to suggest the method of procedure with the material. The units, tens, hundreds, etc., are expressed by different-colored beads: units, white; tens, green; hundreds, red. Then there are racks of different colors: white for the simple units, tens, and hundreds; gray for the thousands; black for the millions. There also are boxes, which on the outside are white, gray, or black, and on the inside white, green, or red. And for each box there is a corresponding rack containing ten tubes with ten beads in each.
Suppose we must divide 87,632 by 64. Five of the boxes are put in a row, arranged from left to right according to the value of their color, as follows: two gray boxes—one green inside and the other white—and three white boxes with the inside respectively red, green, and white. In the first box to the left we put 8 green beads; in the second box 7 white beads; in the third, 6 red beads; in the fourth 3 green beads; and in the fifth box 2 white beads. Back of each box is one of the racks with ten tubes filled with beads of corresponding colors. These beads—ten in each tube—are used in exchanging the units of a higher denomination for those of a lower.
The child here is solving a problem in long division. (A Montessori School, Barcelona, Spain.)
There are two arithmetic boards, one next to the other, placed below the row of boxes. In the one to the left, the little cardboard with the figure 6 is inserted in the slot we have described, and in the other to the right the figure 4.
Now to divide 87,632 by 64, place the first two boxes at the left (containing 8 and 7 beads respectively) above the two arithmetic boards. On the first board the eight beads are arranged in rows of six, as in the more simple division. On the second board the seven beads are arranged in rows of four, corresponding to the number indicated by the red figure. The two quotients must be reduced with reference to the quotient in the first arithmetic board. All the other is considered as a remainder. The quotient in this case is 1 and the remainders are 2 on the first board and 3 on the second.
When this is finished, the boxes are moved up one place and then the first box is out of the game, its place having been taken by the second box; so the gray-green box is no longer above the first board but the gray-white one instead, and above the second board we must place the box with the red beads.