CHAPTER VIII

THE DISTRIBUTION OF PRACTICE

THE PROBLEM

The same amount of practice may be distributed in various ways. Figures 7 to 10, for example, show 200 practices with division by a fraction distributed over three and a half years of 10 months in four different ways. In Fig. 7, practice is somewhat equally distributed over the whole period. In Fig. 8 the practice is distributed at haphazard. In Fig. 9 there is a first main learning period, a review after about ten weeks, a review at the beginning of the seventh grade, another review at the beginning of the eighth grade, and some casual practice rather at random. In Fig. 10 there is a main learning period, with reviews diminishing in length and separated by wider and wider intervals, with occasional practice thereafter to keep the ability alive and healthy.

Plans I and II are obviously inferior to Plans III and IV; and Plan IV gives promise of being more effective than Plan III, since there seems danger that the pupil working by Plan III might in the ten weeks lose too much of what he had gained in the initial practice, and so again in the next ten weeks.

It is not wise, however, to try now to make close decisions in the case of practice with division by a fraction; or to determine what the best distribution of practice is for that or any other ability to be improved. The facts of psychology are as yet not adequate for very close decisions, nor are the types of distribution of practice that are best adapted to different abilities even approximately worked out.

SAMPLE DISTRIBUTIONS

Let us rather examine some actual cases of distribution of practice found in school work and consider, not the attainment of the best possible distribution, but simply the avoidance of gross blunders and the attainment of reasonable, defensible procedures in this regard.

Figures 11 to 18 show the distribution of examples in multiplication with multipliers of various sorts. _X_ stands for any digit except zero. _O_ stands for 0. _XXO_ thus means a multiplier like 350 or 270 or 160; _XOX_ means multipliers like 407, 905, or 206; _XX_ means multipliers like 25, 17, 38. Each of these diagrams covers approximately 3-1/2 years of school work, or from about the middle of grade 3 to the end of grade 6. They are made from counts of four textbooks (A, B, C, and D), the count being taken for each successive 8 pages.[10] Each tenth of an inch along the base line equals 8 pages of the text in question. Each .01 sq. in. equals one example. The books, it will be observed, differ in the amount of practice given, as well as in the way in which it is distributed.

[10] At the end of a volume or part, the count may be from as few as 5 or as many as 12 pages.

These distributions are worthy of careful study; we shall note only a few salient facts about them here. Of the distributions of multiplications with multipliers of the _XX_ type, that of book D (Fig. 14) is perhaps the best. A (Fig. 11) has too much of the practice too late; B (Fig. 12) gives too little practice in the first learning; C (Fig. 13) gives too much in the first learning and in grade 6. Among the distributions of multiplication with multipliers of the _XOX_ type, that of book D (Fig. 18) is again probably the best. A, B, and C (Figs. 15, 16, and 17) have too much practice early and too long intervals between reviews. Book C (Fig. 17) by a careless oversight has one case of this very difficult process, without any explanation, weeks before the process is taught!

Figures 19, 20, 21, 22, and 23 all concern the first two books of the three-book text E.

Figure 19 shows the distribution of practice on 5 × 5 in the first two books of text E. The plan is the same as in Figs. 11 to 18, except that each tenth of an inch along the base line represents ten pages. Figure 20 shows the distribution of practice on 7 × 7; Fig. 21 shows it for 6 × 7 and 7 × 6 together. In Figs. 20 and 21 also, 0.1 inch along the base line equals ten pages.

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.

POSSIBLE IMPROVEMENTS

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!

Each ability has its peculiar needs in this matter, and no set rules are at present of much value. It will be enough for the present if we are aroused to the problem of distribution, avoid obvious follies like those just noted, and exercise what ingenuity we have.