and it was necessary to add 35 thousands—five beads to the thousands-wire and three beads to the ten-thousands-wire. The three beads on the fifth wire can be brought forward without any thought as to what will happen on the wire above when the five are added to the nine. Indeed, what takes place there does not make any difference, for it is not necessary that the operation on the higher wire precede that on the lower wire.
Fig. 2. The disposition of the beads for the number 54,152; after adding 5 thousands to the number 49,152.
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.