Consider, from the same point of view, exercises like (3 × 4) + 2, (7 × 6) + 5, (9 × 4) + 6, given as a preparation for written multiplication. The work of
| 48 3 | 68 7 | 47 9 |
and the like is facilitated if the pupil has easy control of the process of getting a product, and keeping it in mind while he adds a one-place number to it. The practice with (3 × 4) + 2 and the like is also good practice intrinsically. So some teachers provide systematic preparatory drills of this type just before or along with the beginning of short multiplication.
In some cases the bonds are purely propædeutic or are formed only for later reconstruction. They then differ little from 'crutches.' The typical crutch forms a habit which has actually to be broken, whereas the purely propædeutic bond forms a habit which is left to rust out from disuse.
For example, as an introduction to long division, a pupil may be given exercises using one-figure divisors in the long form, as:—
| 773 | and 5 remainder |
| 7 ) 5416 | |
| 49 | |
| 51 | |
| 49 | |
| 26 | |
| 21 | |
| 5 |
The important recommendation concerning these purely propædeutic bonds, and bonds formed only for later reconstruction, is to be very critical of them, and not indulge in them when, by the exercise of enough ingenuity, some bond worthy of a permanent place in the individual's equipment can be devised which will do the work as well. Arithmetical teaching has done very well in this respect, tending to err by leaving out really valuable preparatory drills rather than by inserting uneconomical ones. It is in the teaching of reading that we find the formation of propædeutic bonds of dubious value (with letters, phonograms, diacritical marks, and the like) often carried to demonstrably wasteful extremes.