_Other things being equal, one new sort of bonds should not be started until the previous set is fairly established, and two different sets should not be started at once._ Thus, multiplication of two- and three-place numbers by 2, 3, 4, and 5 will first use numbers such that no carrying is required, and no zero difficulties are encountered, then introduce carrying, then introduce multiplicands like 206 and 320. If other things were equal, the carrying would be split into two steps--first drills with (4 × 6) + 2, (3 × 7) + 3, (5 × 4) + 1, and the like, and second the actual use of these habits in the multiplication. The objection to this separation of the double habit is that the first part of it in isolation is too artificial--that it may be better to suffer the extra difficulty of forming the two together than to teach so rarely used habits as the (_a_ × _b_) + _c_ series. Experimental tests are needed to decide this point. _Other things being equal, bonds should be formed in such order that none will have to be broken later._ For example, there is a strong argument for teaching long division first, or very early, with remainders, letting the case of zero remainder come in as one of many. If the pupils have been familiarized with the remainder notion by the drills recommended as preparation for short division,[9] the use of remainders in long division will offer little difficulty. The exclusive use of examples without remainders may form the habit of not being exact in computation, of trusting to 'coming out even' as a sole check, and even of writing down a number to fit the final number to be divided instead of obtaining it by honest multiplication. [9] See page 76. For similar reasons additions with 2 and 3 as well as 1 to be 'carried' have much to recommend them in the very first stages of column addition with carrying. There is here the added advantage that a pupil will be more likely to remember to carry if he has to think _what_ to carry. The present common practice of using small numbers for ease in the addition itself teaches many children to think of carrying as adding one. _Other things being equal, arrange to have variety._ Thus it is probably, though not surely, wise to interrupt the monotony of learning the multiplication and division tables, by teaching the fundamentals of 'short' multiplication and perhaps of division after the 5s, 2s, 3s, and 4s are learned. This makes a break of several weeks. The facts for the 6s, 7s, 8s, and 9s can then be put to varied use as fast as learned. It is almost certainly wise to interrupt the first half-year's work with addition and subtraction, by teaching 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2, 2 × 5, later by 2 × 10, 3 × 10, 4 × 10, 5 × 10, later by 1/2 + 1/2, 1-1/2 + 1/2, 1/2 of 2, 1/2 of 4, 1/2 of 6, and at some time by certain profitable exercises wherein a pupil tells all he knows about certain numbers which may be made nuclei of important facts (say, 5, 8, 10, 12, 15, and 20).