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.
| 742 = | 2 units | |
| 4 tens | ||
| 7 hundreds |
| 356 = | 6 units | |
| 5 tens | ||
| 3 hundreds |
Each of the first numbers is combined with the three figures of the other number in the following manner: