Figures 22 and 23 show the distribution of practice on the divisions of 72, 73, 74, 75, 76, 77, 78, and 79 by either 8 or 9, and on the divisions of 81, 82 ... 89 by 9. Each tenth of an inch along the base line represents ten pages here also.
Figures 19 to 23 show no consistent plan for distributing practice. With 5 × 5 (Fig. 19) the amount of practice increases from the first treatment in grade 3 to the end of grade 6, so that the distribution would be better if the pupil began at the end and went backward! With 7 × 7 (Fig. 20) the practice is distributed rather evenly and in small doses. With 6 × 7 and 7 × 6 (Fig. 21) much of it is in very large doses. With the divisions (Figs. 22 and 23) the practice is distributed more suitably, though in Fig. 23 there is too much of it given at one time in the middle of the period.
[Illustration: FIG. 19.--Distribution of practice with 5 × 5 in the first two books of the three-book text E.]
[Illustration: FIG. 20.--Distribution of practice with 7 × 7 in the first two books of text E.]
[Illustration: FIG. 21.--Distribution of practice with 6 × 7 or 7 × 6 in the first two books of text E.]
[Illustration: FIG. 22.--Distribution of practice with 72, 73 ... 79 ÷ 8 or 9 in the first two books of text E.]
[Illustration: FIG. 23.--Distribution of practice with 81, 82 ... 89 ÷ 9 in the first two books of text E.]
Even if we knew what the best distribution of practice was for each ability of the many to be inculcated by arithmetical instruction, we could perhaps not provide it for all of them. For, in the first place, the allotments for some of them might interfere with those for others. In the second place, there are many other considerations of importance in the ordering of topics besides giving the optimal distribution of practice to each ability. Such are considerations of interest, of welding separate abilities into an integrated total ability, and of the limitations due to the school schedule with its Saturdays, Sundays, holidays, and vacations.
Improvement can, however, be made over present practice in many respects. A scientific examination of the teaching of almost any class for a year, or of many of our standard instruments of instruction, will reveal opportunities for improving the distribution of practice with no sacrifice of interest, and with an actual gain in integrated functioning arithmetical power. In particular it will reveal cases where an ability is given practice and then, never being used again, left to die of inactivity. It will reveal cases where an ability is given practice and then left so long without practice that the first effect is nearly lost. There will be cases where practice is given and reviews are given, but all in such isolation from everything else in arithmetic that the ability, though existent, does not become a part of the pupil's general working equipment. There will be cases where more practice is given in the late than the earlier periods for no apparent extrinsic advantage; and cases where the practice is put where it is for no reason that is observable save that the teacher or author in question has decided to have some drill work at that time!