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to 44.

The early period of rapid progress may be due (i) to the fact that
the first elements of a new set of materials or a new set of associa-
tions may be picked up rather easily and quickly because of their
simplicity, (2) to the probability that the first stage of practice in a
new type of learning makes available various elements or activities
already in the possession of the learner, (3) to the initial zeal in
beginning a new task, (4) to the large opportunity for progress in
the beginning, (5) to the physiological limit in many types of skill
such as typewriting, mirror tracing and writing numbers for letters,
and (6) to an absolute limit of the number of bonds that the task
presents to the learner. Thus typewriting has a physiological
limit in the rapidity with which the fingers can be moved in striking
the keys. Progress cannot go on indefinitely at the original rate.
Typewriting also has an absolute limit in the number of strokes
to be learned.



144 EDUCATIONAL PSYCHOLOGY

Book explains the initial period of rapid progress thus:

"After what has been said our explanation of the general features of
our curves can be brief. The first rapid and continuous rise is due to
the fact that the learner is making progress along many different lines at
once. Rapid strides of improvement are possible and made simulta-











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Days

Fig. 39. — Progress in learning Russian. After Swift ('oS, p. 19S).

neously in every department of the work. The learner is not only forming
and perfecting letter associations but syllable, word and phrase associa-
tions as well. He is simultaneously improving his method of dealing
with every problem that the writing presents; locating the keys, directing
and controlling his fingers, 'spelling' or initiating the movements, get-
ting his copy, learning to deal with special difficulties, learning to keep
attention more closely and economically applied to the work, etc. The
curves will rise rapidly and continuously so long as many of these possibU-



THE RATE AND PROGRESS OF LEARNING



145



ities of improvement exist. As they grow less numerous the rate of gain
will likewise decline until, as still more skill is acquired, a state is reached
where most adaptations or short cuts in method have been made; fewer




12 3 4



5 6 7 8 9 10 11 12 13 14
Successive Trials



Fig. 40. — Improvement in tracing a star outline when seen in a mirror.
Continuous lines represent reduction in seconds in successive trials. Dotted
line represents reduction in errors.





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Amount of Practice in Hours

Fig. 41. — Improvement in typewriting by the sight method. After Book
('08 plate, opposite p. 21).



146



EDUCATIONAL PSYCHOLOGY



special habits remain to be developed; fewer adaptations are possible.
Those possible have become harder and harder to make, because they
must be made in the realm of higher habits where the learner has had
less experience. Every man has had experience with the first stages of
learning, but little with the later stages because {most people touch
lightly many things and are masters of nothing7^ There being now fewer
adaptations to make, and the process of finally^erfecting all the special
associations being so gradual and slow, the learning curve becomes, as



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Fig. 42. — Progress in ball-tossing. The horizontal axis represents days.
The vertical axis represents the number of balls caught. After Swift ('08, p. 1 74) .

the expert stage is approached, almost horizontal. In the later stages of
learning the sole gain must come from an occasional adaptation and
from a further perfection of the present habits and methods of work."
(Book, '08, pp. 99 f.)

Swift and Batson have each published curves based on the in-
crease in skill in ball-tossing which purport to be of the concave
t5^e. A careful examination of the original data shows that the
apparently concave form is in reality due in each case to peculiari-
ties in the method of plotting. Fortunately some of the data pub-



THE RATE AND PROGRESS OF LEARNING



147



lished by Batson permit of being plotted strictly according to the
principle laid down at the beginning of this chapter, namely, that
vertical distances represent amounts of performance and horizontal

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Successive Trials



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Days Practice



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Upper graph shows



Fig. 43. — Lower graph shows Batson's original curve,
the reconstruction of his curve as stated in the text.

distances represent equal amounts of time or practice. This method
of plotting yields the perfectly typical convex learning curve shown
in Figure 43.

The types of learning so far investigated have been for the most
part of a relatively simple sort. Other types of learning may be ex-



148



EDUCATIONAL PSYCHOLOGY



pected to bring to light curves of very different shape. This is par-
ticularly true of forms of learning which depend chiefly upon an-
alysis and selection or in cases where there is no physiological or
absolute limit within ordinary attainable bounds, such as, for ex-
ample, learning facts of history.

Two rather extensive studies on analytical types of learning are
now available. They reveal a very characteristic type of curve.
The first investigation, by Ruger ('lo), was based upon the number
of successive solutions of a given mechanical puzzle that could be



I I 1 1 — I \ 1 1 r— n \ 1 1 —

1 2 3 4 5 6 7 8 9 10 11 12
Time Spent in Practice

Fig. 44. — Curve to show the progress in solving puzzles,
as reconstructed by Thorndike ('14), III, p. 342.



After Ruger ('10).



performed by an individual within a certain period of time. Fig-
ure 44. It yields a strikingly concave curve in marked contrast to
those we have previously examined. Unfortunately, practice was
not continued long enough to reveal the complete curve of this type
of learning. A second study, by Hull, was based upon the rate of
evolution of abstract ideas as shown by their increase in ability
to function, Figure 45. The material used was an elaborate-
system of Chinese characters combined with nonsense syllables.
In this case the work was carried to the point of perfection. He
found as a consequence not only the initial concave section shown by
Ruger but a later period of diminishing returns. Taken alone, this



THE RATE AND PROGRESS OF LEARNING



149



last section strikingly resembles the learning curves of the simpler
processes and its course is doubtless determined by the same causes.
The initial plateau or period of slow progress is probably due to the
necessity of making a preliminary analysis of the material used be-
fore proceeding with the remainder of the process. Clearly the
elements common to many situations, as in Hull's investigation,
for example, must be perceived as separate elements before they
can be perceived as common elements.

It seems also quite likely that in learning facts of history or facts
of science, in which there is no physiological limit and in which

12



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12 24 36 48 60 72 84 96 108 120 132 144
Minutes of Work
Fig. 45. — Progress of two individuals in generalizing abstractions or forming
concepts. After Hull ('19).

the number of items that may be learned is practically unlimited,
the course of the curve, at least for a considerable distance, is con-
cave. This is hinted at in the results obtained with the author's
tests in geography and history. Thus in the former test, given at
the end of the school year to some 1,300 pupils, and in the latter
test given to some 2,000 pupils, the average scores for the ends of the
respective years were as follows:

















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Geography 25

American History



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38



These scores substantially mean that so many geographical or
historical items were known to the pupil. Both sets of figures show
a larger gain from the second year to the third year than from the
first to the second. These data furnish of course only fragmentary



150 EDUCATIONAL PSYCHOLOGY

portions of the curves of learning in these subjects. But so far as
any form may be inferred from them, it is probably of the con-
cave sort.

It seems probable that future experimentation will yield similar
forms of curves in other types of learning. Apparently there are
types of learning in which continued training brings increasing
returns. Thorndike states that:

"Negative acceleration (that is rapid rise or convex form) of any great
amount is far from being a general rule of learning. On the contrary,
it may well be that there are some functions, such as amount of knowledge
of history or geography, or of foreign languages, or of fiscal statistics,
where, by any justifiable score for 'amount of knowledge' the rate of
improvement in hour after hour of practice would rise, giving a pro-
nounced positive acceleration. Each item of information may, in such
cases, make the acquisition of other items easier; learning some one fact
may involve knowledge of a score of new facts in the shape of its relations
to the facts previously learned. So knowledge may roll up like a snow-
ball, its sum being, say, as the cube of the amount of time spent. What
we may call the 'knowledge functions' do, as a rule, show, to say the
least, very much less of the diminishing returns from increasing practice
than do the functions of skill in some single line of work which figure so
often in the experimental studies of practice." ('14, II, p. 257.)

Wliether or not plateaus occur universally in all types of learning,
and whether they are really unavoidable stages in the course of
learning, is an open question. They have not been found to occur
as generally as the initial rise even in curves of skill. Bryan and
Harter found periods of slow progress in three-fourths of their
subjects. Book found them in two of his three persons, and Swift
reports none. In the twenty curves obtained in the author's sub-
stitution test, eleven contained plateaus and nine did not. The
practice periods totaled 120 minutes. It is possible that some of the
curves might have shown plateaus if the practice had been con-
tinued longer.

Batson, who undertook an investigation for the purpose of study-
ing plateaus, found none in the ball-tossing curves, although the
training was continued for a long time, but found a pronounced
plateau in learning to throw shot into a pocket.

Plateaus may be caused by lagging in energy, by loss of atten-
tion, interest, and effort, by fatigue, by periods of mechanization,
and the like. Rapid progress after a plateau may be due to a re-



THE RATE AND PROGRESS OF LEARNING 151

cuperation in physical energy, in attention, interest, and effort,
to the acquisition of new methods of learning and doing the task
concerned, and to better use of the bonds which have been made au-
tomatic by the preceding practice. Bryan and Harter beheve that
the plateaus in the learning of telegraphy were due to the establish-
ment of a hierarchy of habits. During the initial period of progress,
the simple elements such as the signals for letters, were acquired
first and when these all had been learned, there came a dead level
during which the connections became automatized, and then rapid
progress was again possible by virtue of the acquisition of combi-
nations of letters into words and words into phrases. Their own
statement follows:

"A hierarchy of habits may be described in this way: (i) There is a
certain number of habits which are elementary constituents of all other
habits within the hierarchy. (2) There are habits of a higher order which,
embracing the lower as elements, are themselves in turn elements of
higher habits, and so on. (3) A habit of any order, when thoroughly ac-
quired, has physiological and, if conscious, psychological unity. The
habits of lower order which are its elements tend to lose themselves in
it, and it tends to lose itself in the habits of higher order when it appears
as an element therein.

" 2. The Order of Learning the Habits of the Telegraphic Language.

" The synchronous curves of Figure 38 and the experience of operators
agree in showing that from an early period letter, word, and higher habits
make gains (a) simultaneously, but (b) not equally.

" (a) The simultaneity in these gains is shown in Figure 38 by the fact
that from the point where the curves diverge, each continues to rise.
This is perhaps to be explained by the fact that from an early stage the
learner practises with sentences, taking them as slowly as necessary. In
this way there is incidental practice of every language unit and of every
language unit in its proper setting.

" (b) The curves of Figure 38 show also, however, that for many
months the chief gain is in the letter and word habits, that the rate of re-
ceiving sentences, is in this period, mainly determined by the rate of re-
ceiving letters and words, and that rapid gain in the higher language
habits does not begin until letter and word habits are well fi.xed. This ob-
jective result is supported by the introspective evidence of operators. In
the first days one is forced to attend to letters. In the first months one is
forced to attend to words. If the learner essays a freedom for which he is
unfit, suddenly a letter or word which is unfamiliar explodes in his ears
and leaves him wrecked. He has no useful freedom for higher language
units which he has not earned by making the lower ones automatic. The
rank and file of operators are slaves to the machinery of the telegraphic



152 EDUCATIONAL PSYCHOLOGY

language. They must copy close. They cannot attend much to the sense
of the message as it comes, but must get its form, and re-read for the
sense. Only when all the necessary habits, high and low, have become
automatic, does one rise into the freedom and speed of the expert.

"3. The Plateaus.

" We are now prepared to offer an explanation for the salient peculi-
arity of the receiving curve — its plateaus.

" A plateau in the curve means that the lower-order habits are ap-
proaching their maximum development, but are not yet sufhciently
automatic to leave the attention free to attack the higher-order habits.
The length of the plateau is a measure of the difficulty of making the
lower-order habits sufficiently automatic."

The explanation of plateaus probably depends upon the nature
of the learning process in which they occur. The theory of the
hierarchy of habits would probably not apply to such a task as
mirror tracing.

Experimenters are divided in their opinions concerning the in-
evitableness or the usefulness of plateaus even in those types of
learning in vi^hich they frequently occur. Bryan and Harter, Swift
and others believe that they serve a beneficial purpose. Swift,
for example, says:

"The real advance in the early stages of learning is made during the
periods of seeming arrest of progress. The manifest advance, that
which is revealed by the curve or by examination marks, which is the
same thing, is discouragingly brief. By far the greater part of the learning
period is spent on plateaus when both teacher and pupil, failing to under-
stand the situation, feel that they are marking time. Yet it is during
these days of retardation that the valuable and solid acquisitions are
being made. Americans who spend several years in Germany pass
through a long period of discouragement. Though they study the lan-
guage faithfully, and avail themselves of every opportunity to practice
conversation, they seem to make absolutely no progress. The length of
this plateau-period varies with different persons, but all experience its
oppressiveness. Now the most curious feature of this plateau, aside from
its overpowering monotony, is the suddenness with which it finally dis-
appears. Several have told the writer that they went to sleep one night
imable to understand anything, as it seemed to them, and utterly dis-
couraged, and awoke the following morning to find that they had mas-
tered the language, that they could understand practically everything
that was said to them. The word associations and national peculiarities
of thought sequence had been automatized during the long period when
no visible progress was being made." ('06, pp. 310 f.)



THE RATE AND PROGRESS OF LEARNING 153

Other investigators believe that plateaus are not necessary-
stages in the course of learning, but that they are due to causes
which may be avoided by introducing new stimuli or new methods
of attack in learning so that continued progress may be possible.

Plateaus are apparently not universal in all types of learning,
nor are they found in all persons in the same type of learning.
Whether they are useful stages in the learning process is a moot
question. If they are not necessary, it would be highly important
for education to prevent their occurrence in the learning of school
material (i) by removing the conditions which bring them about,
and (2) by providing stimuli at the points at which they are apt
to occur so as to continue upward in the course of learning. Fiu-ther
experimentation will have to be made to furnish a definite solution
of the problem.
.^^ Factors Affecting Progress, a. Length and DistribtUion of the'
Periods of Work. How long at a time, and how often, should the
learner work at his task in order to make the maximum progress
for the time devoted to it? Every type of learning probably has
an optimum length and frequency of periods of practice. Ebbing-
haus ('85), in his pioneer study of memory, found that it was better
in learning nonsense syllables to distribute a given amount of
time over three days than to spend it all on one day. Sixty-eight
repetitions made in immediate succession were not as advantageous
for later relearning as thirty-eight repetitions distributed over
three days. Practically all investigators who have touched upon
this phase of learning have found a principle of similar nature.
Jost ('97), also working with syllables, found, for example, that
two repetitions a day for twelve days were better than four repeti-
tions a day for three days. Some of the results of both Ebbinghaus
and Jost imply that in some instances a decreasing amount of
time on successive days would be more economical than an equal
amount on all days; that instead of distributing 24 repetitions
by having four on six successive days, it would be better to have
eight on the first day, six on the second, four on the third, three
on the fourth, two on the fifth, and one on the sixth day.

Lueba and Hyde ('05), in an experiment on learning to transcribe
English words into German script, found that of four plans of
distributing time, twenty minutes twice a day yielded the slowest
gain, while twenty minutes every third day yielded better, and
twenty minutes every day or every other day yielded the best
results.



154 EDUCATIONAL PSYCHOLOGY

Miss Munn ('09) made an investigation of practice in a sub
stitution test consisting of transcribing 4,000 letters into other
letters according to a key. Her distributions of time and results
are given in the following table:



Average Time (Seconds)


First 200




Letters


Last 200


Approximate


Letters


41 -S


13-4


57-5


17. 1


47.00


16.5


39-5


18.2


38. 5


18. s


44.00


21. 1



TABLE 38

Practice in substituting letters for other letters according to a key. After

Munn ('09)



No. of
Subjects Distribution of Work

23 200 letters a day for 20 successive days
4 800 letters a day for 5 successive days

(400 in a. m., 400 in p. m.)
4 1000 letters a day for 4 successive days
4 2000 letters a day, seven days apart

4000 letters in one day (looo at a sitting)
4 3000 letters a day (at one sitting)

The highest degree of efficiency was reached by the 20-day
group who reduced their time for the last 200 letters to 13.4 seconds.
A definite comparison is a little difficult to make owing to the
large differences in initial ability among the various groups.

In the substitution test carried out by the author, Figure 46,
ten minutes twice a day was productive of the greatest progress,
twenty minutes once a day was productive of almost as rapid
progress, forty minutes once a day was productive of considerably
less progress, while 120 minutes at one time produced scarcely
half as much progress as the ten-minute or twenty-minute periods.
The total time in all four distributions was the same.

Dearborn ('10), who reported an earlier experiment with the
same substitution test, divided the subjects into two groups work-
ing ten minutes once a day and ten minutes twice a day respectively.
He found a small advantage in favor of the former group.

Pyle ('13), working with a substitution test, reports that:

" Generally speaking, daily practice seems to give better returns than
the same number of periods distributed on alternate days or in twice-a-
day periods. However, there is some evidence that in the early stages of
habituation, the second practice on the same day gives good returns and
that, later on, alternate days may be the best distribution."



J



THE RATE AND PROGRESS OF LEARNING



155



Kirby ('13) carried out a practice experiment in addition and
division with 1,300 pupils in the third and fourth grades. The
pupils practiced addition in 22.5, 15, 6, and 2-minute periods, and
division in 20, 10, and 2-minute periods with the following gains:




34 5 6 7 8 9 10 11 i:i 13 14 15 lli 17 IS 19 20 2122 23 24

Successive Five Minute Periods

Fig. 46. — Practice in writing numbers for letters according to a key. After
Starch ('12).

10 min. curve = group working 10 min. twice a day.
20 " " = " " 20 " once " " .

40 " " = " " 40 " every other day.

120 " " = " " 120 " at one time.





TABLE


39








Addition


Division






Per Cent Gain over the




Per Cent Gain over the


'eriod


22.5-Minute Period


Period




20-Minute Period


22.5




20






IS-


'21"%


10




10.5%


6.


I %


2




77 %


2.


46.5%









The superiority of the 2-minute period is probably exaggerated,
as Thorndike has suggested, by the greater opportunity for out-
side practice and longer continuation of regular school work, since
this period was extended over a larger number of days.

Thorndike ('11) compared the improvement in multiplying



156 EDUCATIONAL PSYCHOLOGY



Online LibraryDaniel StarchEducational psychology → online text (page 12 of 41)