By this time the child can easily perform operations with numbers of two or more figures, for he possesses all the materials necessary and is already prepared to make use of them.

For this work we have for the first three operations, addition, subtraction, and multiplication, a counting-frame; and for division a more complicated material which will be described later on.

ADDITION

Addition on the counting frame is a most simple operation, and therefore is very attractive. Let us take, for example, the following:

First we slide over the beads to represent the first number: 1 on the thousands-wire, 3 on the hundreds-wire, and 2 on the tens-wire. Then we place next to them the beads representing the second number: 4 on the hundreds-wire, 3 on the tens-wire, and 5 on the units-wire. Now there remains nothing to be done except to write the number shown by the beads in their present position: 1755.

When the problem is a more complicated one, the beads for any one wire amounting to more than 10, the solution is still very easy. In that case the entire ten beads would be returned to their original position and in their stead one corresponding bead of the next lower wire would be slipped over. Then the operation is continued. Take, for example:

We first place the beads representing 390: that is, 3 on the hundreds-wire and 9 on the tens-wire; or, vice versa, beginning with the units, we would first place the 9 tens and then the 3 hundreds. For the second number we place 4 beads for the hundreds and then we begin to place the 8 tens. But when we have placed only one ten, the wire is full; so the ten tens are returned to their original position and to represent them we move over another bead on the hundreds-wire; then we continue to place the beads of the tens which now, after having converted 10 of them into 1 hundred, remain but 7. Or we can begin the addition by placing the beads for the units before we place those for the hundreds; and in that case we move on the hundreds-wire first the bead representing the ten beads on the wire above, and then the 4 hundreds which must be added. Finally we write down the sum as now indicated by the position of the beads: 872.

With a larger counting-frame it is possible to perform in this manner very complicated problems in addition.

SUBTRACTION

The counting-frame lends itself equally well to problems in subtraction. Let us take, for example, the following:

We place the beads representing the first number; then from them we take the beads representing the second number The beads remaining indicate the difference between the two numbers; and this is written: 2212.

Then comes the more complicated problem where it is necessary to borrow from a higher denomination. When the beads of one wire are exhausted, we move over the entire ten and take to represent them one bead from the lower wire; then we continue the subtraction. For example:

We move the beads representing the first number; then we take 3 beads from the units. Now we begin to subtract the tens. We wish to take away 9 beads; but when we have moved five the wire is empty, and there are still four more to be moved. We take away one bead from the hundreds-wire and replace the entire ten on the tens-wire; and then we continue to move beads on the tens-wire until we have taken a total of nine—that is, we now move the other four. On the hundreds-wire there remain but 8 beads, and from them we take the 5, etc. Our final remainder is 1361.

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It is easy to see how familiar and clear to the child the technique of "borrowing" becomes.

MULTIPLICATION

When there is a number to be multiplied by more than one figure, the child not only knows the multiplication table but he easily distinguishes the units from the tens, hundreds, etc., and he is familiar with their reciprocal relations. He knows all the numbers up to a million and also their positions in relation to their value. He knows from habitual practise that a unit of a higher order can be exchanged for ten of a lower order.

To have the child attack this new difficulty successfully one need only tell him that each figure of the multiplier must multiply in turn each figure of the multiplicand and that the separate products are placed in columns and then added. The analytical processes hold the child's attention for a long period of time; and for this reason they have too great a formative value not to be made use of in the highest degree. They are the processes which lead to that inner maturation which gives a deeper realization of cognitions and which results in bursts of spontaneous synthesis and abstraction.

The children, by rapidly graduated exercises, soon become accustomed to writing the analysis of each multiplication (according to its factors) in such a way that, once the work of arranging the material is finished, nothing is left for them to do but to perform the multiplications which they already have learned in the simple multiplication table.

Here is an example of the analysis of a multiplication with three figures appearing in both the multiplicand and the multiplier: 356 × 742.

Each of the first numbers is combined with the three figures of the other number in the following manner:

When this analysis is written down, the work on the counting-frames begins. Here the operations are performed in the following manner: 2 × 6 units necessitate the bringing forward of the ten beads on the first wire. However, even those do not suffice. So they are slid back and one bead on the second wire is brought forward, to represent the ten replaced, and on the first wire two beads are brought forward (12).

Next we take 2 × 5 tens. There is already one bead on the tens-wire and to this should be added ten more, but instead we bring forward one bead on the hundreds-wire. At this point in the operation the beads are distributed on the wires in this manner:

Now comes 2 × 3 hundreds, and six beads on the corresponding wire are brought forward. When the multiplication by the units of the multiplier is finished, the beads on the frame are in the following order:

We pass now to the tens: 4 × 6 = 24 tens. We must therefore bring forward four beads on the tens-wire and two on the hundreds-wire:

4 × 5 = 20 hundreds, therefore two thousands:

4 × 3 thousands = 12 thousands; so we bring forward two beads on the thousands-wire and one on the ten-thousands-wire:

Now we take the hundreds: 7 × 6 hundreds are 42 hundreds; therefore we slide four beads on the thousands-wire and two on the hundreds-wire. But there already were nine beads on this wire, so only one remains and the other ten give us instead another bead on the thousands-wire:

5 × 7 thousands = 35 thousands, which is the same as five thousands and three ten-thousands. Three beads on the fifth wire and five on the fourth are brought forward; but on the fourth wire there already were nine beads, so we leave only four, exchanging the other ten for one bead on the fifth wire:

Finally 7 × 3 ten-thousands = 21 ten-thousands. One bead is brought forward on the fifth wire and two on the hundred-thousands-wire.

At the end of the operation the beads will be distributed as follows:

This distribution translated into figures gives the following number: 264,152. This may be written as a result right after the factors without the partial products: that is, 742 × 356 = 264,152.

Although this description may sound very complicated, the exercise on the counting-frame is an easy and most interesting arithmetic game. And this game, which contains the secret of such surprising results, not only is an exercise which makes more and more clear the decimal relations of reciprocal value and position, but also it explains the manner of procedure in abstract operations.

In fact, in the multiplication as commonly performed:

In adding the five beads to the nine beads only four remain on the fourth wire, since the other ten are substituted by a bead on the lower wire; this bead may be brought forward even after the three for the ten-thousands have been placed.

By the use of the frame the child acquires remarkable dexterity and facility in calculating, and this makes his work in multiplication much more rapid. Often one child, working out an example on paper, has finished only the first partial multiplication when another child, working at the frame, has completed the problem and knows the final product. It is interesting even among adults to watch two compete in the same problem, one at the frame and the other using the ordinary method on paper.

It is very interesting, also, not to work out on the frame the individual products in the sequence indicated in analyzing the factors, but to work them out by chance. Indeed, it does not matter whether the beads are moved in the order of their alignment or at random. The beads on the ten-thousands-wire may be moved first, then the hundreds, the units, and finally the thousands.

These exercises, which give such a deep understanding of the operations of arithmetic, would be impossible with the abstract operation which is performed only by means of figures. And it is evident that the exercises can be amplified to any extent as a pleasing game.

MULTIPLYING ON RULED PAPER

Take, for example, 8640 × 2531. We write the figures of the multiplicand one under the other but in their relative positions; this also can be written by filling in the vacant spaces with zeros.

In this way we repeat the multiplicand as many times as there are figures in the multiplier; but instead of writing beside these figures the words units, tens, etc., we indicate this with zeros, which, for the sake of clearness, we fill in till they resemble large dots.

The child already knows, from his previous exercises, that zero indicates the position of a figure and that multiplying by ten changes this position. Therefore zeros in the multiplier would cause a corresponding change of position in the figures of the multiplicand.

The accompanying figure shows clearly what it is not so easy to explain in words.

We are now ready for the usual procedure of multiplication. A child of seven years reaches this stage very easily after having done our preliminary exercises, and then it does not matter to him how many figures he has to use. Indeed, he is very fond of working with numbers of unheard of figures, as is shown in the following example—one of the usual exercises done by the children, who of themselves choose the multiplicand and the multiplier; the teacher would never think of giving such enormous numbers. They can now perform the operation

without analysis of factors and without help from the frames but by the method commonly used. This may be seen by the way in which the example is written out and then done by the child.

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.

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.

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[239]
[240] 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.