PROPÆDEUTIC BONDS
The formation of bonds to a limited strength because they are to be lost in their first form, being worked over in different ways in other bonds to which they are propædeutic or contributing is the most important case of low strength, or rather low permanence, in bonds.
The bond between four 5s in a column to be added and the response of thinking '10, 15, 20' is worth forming, but it is displaced later by the multiplication bond or direct connection of 'four 5s to be added' with '20.' Counting by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc., forms serial bonds which as series might well be left to disappear. Their separate steps are kept as permanent bonds for use in column addition, but their serial nature is changed from 2 (and 2) 4, (and 2) 6, (and 2) 8, etc., to two 2s = 4, three 2s = 6, four 2s = 8, etc.; after playing their part in producing the bonds whereby any multiple of 2 by 2 to 9, can be got, the original serial bonds are, as series, needed no longer. The verbal response of saying 'and' in adding, after helping to establish the bonds whereby the general set of the mind toward adding coöperates with the numbers seen or thought of to produce their sum, should disappear; or remain so slurred in inner speech as to offer no bar to speed.
The rule for such bonds is, of course, to form them strongly enough so that they work quickly and accurately for the time being and facilitate the bonds that are to replace them, but not to overlearn them. There is a difference between learning something to be held for a short time, and the same amount of energy spent in learning for long retention. The former sort of learning is, of course, appropriate with many of these propædeutic bonds.
The bonds mentioned as illustrations are not purely propædeutic, nor formed only to be transmuted into something else. Even the saying of 'and' in addition has some genuine, intrinsic value in distinguishing the process of addition, and may perhaps be usefully reviewed for a brief space during the first steps in adding common fractions. Some such propædeutic bonds may be worth while apart from their value in preparing for other bonds. Consider, for example, exercises like those shown below which are propædeutic to long division, giving the pupil some basis in experience for his selection of the quotient figures. These multiplications are intrinsically worth doing, especially the 12s and 25s. Whatever the pupil remembers of them will be to his advantage.
1. Count by 11s to 132, beginning 11, 22, 33.
2. Count by 12s to 144, beginning 12, 24, 36.
3. Count by 25s to 300, beginning 25, 50, 75.
4. State the missing numbers:—
| A. | B. | C. | D. |
|---|---|---|---|
| 3 11s = | 5 11s = | 8 ft. = .... in. | 2 dozen = |
| 4 12s = | 3 12s = | 10 ft. = .... in. | 4 dozen = |
| 5 12s = | 6 12s = | 7 ft. = .... in. | 10 dozen = |
| 6 11s = | 12 11s = | 4 ft. = .... in. | 5 dozen = |
| 9 11s = | 2 12s = | 6 ft. = .... in. | 7 dozen = |
| 7 12s = | 9 12s = | 9 ft. = .... in. | 12 dozen = |
| 8 12s = | 7 11s = | 11 ft. = .... in. | 9 dozen = |
| 11 11s = | 12 12s = | 5 ft. = .... in. | 6 dozen = |