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.
[Illustration: FIG. 7.--Plan I. 200 practices distributed somewhat
evenly over 3-1/2 years of 10 months. In Figs. 7, 8, 9, and 10,
each tenth of an inch along the base line represents one month.
Each hundredth of a square inch represents four practices, a
little square 1/20 of an inch wide and 1/20 inch high representing
one practice.]
[Illustration: FIG. 8.--Plan II. 200 practices distributed
haphazard over 3-1/2 years of 10 months.]
[Illustration: FIG. 9.--Plan III. A learning period, three reviews,
and incidental practice.]
[Illustration: FIG. 10.--Plan IV. A learning period with reviews
of decreasing length at increasing intervals.]
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!
[Illustration: FIG. 11.--Distribution of practise with multipliers
of the _XX_ type in the first two books of the three-book text A.]
[Illustration: FIG. 12.--Same as Fig. 11, but for text B. Following
this period come certain pages of computation to be used by the
teacher at her discretion, containing 24 _XX_ multiplications.]
[Illustration: FIG. 13.--Same as Fig. 11, but for text C.]
[Illustration: FIG. 14.--Same as Fig. 11, but for text D.]
[Illustration: FIG. 15.--Distribution of practice with multipliers
of the _XOX_ type in the first two books of the three-book text
A.]
[Illustration: FIG. 16.--Same as Fig. 15, but for text B. Following
this period come certain pages of computation to be used by the
teacher at her discretion, containing 17 _XOX_ multiplications.]
[Illustration: FIG. 17.--Same as Fig. 16, but for text C.]
[Illustration: FIG. 18.--Same as Fig. 16, but for text D.]
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 x 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 x 7; Fig. 21 shows it for
6 x 7 and 7 x 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 x 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 x 7 (Fig. 20) the practice is distributed rather evenly and in small
doses. With 6 x 7 and 7 x 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 x 5 in
the first two books of the three-book text E.]
[Illustration: FIG. 20.--Distribution of practice with 7 x 7 in
the first two books of text E.]
[Illustration: FIG. 21.--Distribution of practice with 6 x 7
or 7 x 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.]
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.