ARITHMETIC 40I
under various conditions. Each experiment lasted six weeks. The
time spent in drill was in all cases subtracted from the regular
class time in arithmetic. The Courtis tests, Series A, were given
before and after each experiment. In Experiment I, the drill group
was rather miscellaneous, grade 5 being drilled five minutes per
day (probably in fundamentals, though it is impossible to tell
from Wimmcr's account), grade 6E five minutes per day, three
fifths on reasoning and twofifths on fundamentals, grade 6W
fifteen minutes in a single period per week on reasoning and funda
mentals in the same ratio as 6E. In Experiment II, two sections
of the sixth grade, which had equal ability as shown by tests, were
drilled five minutes daily, one for speed and the other for accuracy.
In Experiment III, the seventh grade was drilled five minutes
daily on reasoning, while the eighth grade was drilled five minutes
daily on fundamentals. The results of all three experiments are
shown by percentages of gain in Table 119. As in the previous
experiments, drill as such is shown to have a decided value though
much more for reasoning than for fundamentals, the advantage of
the drilled groups being 32.7% and 4.4%, respectively. In Ex
periment II, drill for speed is seen to have a distinct advantage
over drill for accuracy. Here the gain is considerably more in the
fundamentals than in reasoning. In Experiment III, we find that
the class drilled on reasoning gained very largely in reasoning
alone, while the other class trained in fundamentals gained almost
exclusively in fundamentals. This presents a contrast to Brown's
experiment where, it will be remembered, drill in fundamentals
showed as much gain in reasoning as in fundamentals themselves.
402
EDUCATIONAL PSYCHOLOGY
TABLE 119
Wiinmer's results
Problem
Experiment I
Drill vs. No Drill
Experiment II
Drill for Speed vs.
Drill for Accuracy
Experiment III
Drill in Rea
soning vs.
Drill in Funda
mentals
Grades Used as Sub
5th
7th
Differ
ence IN
Favor
OF
Drill
6E
6W
Differ
7 th
8tu
jects
Number of Subjects
Type of Activity
6th
79
Drill
SXH
70
No
Drill
22
Drill
Speed
22
Drill
Accur
acy
ence IN
Favor
OF Drill
FOR
Speed
35
Drill
Reason
ing
35
Drill
Funda
mentals
All tests, attempts
and rights
333
16.2
17. 1
II .1
8.8
23
9.1
8.6
Fundamentals ....
15.6
"â€¢3
4 4
12.9
6.9
6.0
47
133
Reasoning, at
tempts and
rights
58.0
2SI
329
18.6
â€¢4
Reasoning, rights
only
68.1
354
3^7
II .
0.6
13
19. 1
8
(6) The Optimum Distribution of DriU. A number of experi
ments have been conducted to determine the most economical,
distribution of time, that is, the most economical duration of drill
periods in arithmetic. T. J. Kirby carried out such experiments
on a large scale for both addition and division. He used special
blanks for the practice in both experiments. In addition funda
mentals there was a beginning and a final test each of fifteen
minutes. Between these periods fortyfive minutes of drill were
variously distributed. The following table shows the distribution
of the periods, the median initial ability of each group in examples
correct, and the gross gains as measured by three different methods
of calculation. Seven hundred and thirtytwo fourthgrade children
were used as subjects.
TABLE 120. After Kirby
No.
Sub
jects
Initial
Test
Period
Distribution of
Intervening 45
Minutes
Final
Test
Period
Median
Initial
Ability
(Exam
ples
Correct)
Gain Due to Drill
Group
Av.
Gross
Gain of
iNDivro
UALS
Med.
Gross
Gain of
iNDivro
UALS
Av. OF
Med.
Gross
Gains
I
n ..
III. . .
I\ . ,
194
104
205
220
15 min.
15 "
15 "
15 "
2 periods 15 min.
3 " 15 "
7 " 6 "
and one 3 "
21 periods 2 "
and one 3
22J min.
15 "
15 "
15 "
22.9
25.4
21.0
25.1
11.0
13.6
10.7
16.1
9.5
11.0
9.6
12.6
10.2
9.6
9.4
13.9
ARITHMETIC
403
When accurate correction had been made for differences in
initial ability, the gains were then in proportion 100, 121, loi, and
i46_K respectively. There was no very distinct tendency observ
able here except that the short periods of two minutes yielded a
distinct advantage over the rest. Unfortunately for the consist
ency of these results the next shortest period (6 min.) yielded
nearly the least gain of all.
The same general plan was followed by Kirby in the experiment
on division. Six hundred and sLx children from the second half
of grade three and the first half of grade four were used. The
following table shows in detail the various distributions of the
drill together with the results by each method.
TABLE 121. After Kirby
No.
Sub
jects
Initial
Test
Period
Plstrtbution of
Intervening 40
Min. Of Drill
Final
Test
Period
Medlvn
Initlal
Ability
(Exam
ples)
Correct)
Gain Due to Drill
Grolt
Av.
Gross
Gain of
Individ
uals
Med.
Gross
Gain of
Individ
uals
Av. OF
Medwn
Gross
Gains of
Classes
I
11....
III. .
204
209
193
10 min.
10 "
10 "
2 20 min. periods
4 10 "
20 2 "
10 min.
10 "
10 "
38.4
33.4
41.4
25.1
25.5
42.6
22.6
23.5
40.4
20.6
25.1
44.7
When inequalities of initial ability had been removed, the gains
were found to be in the proportion of loo, iio^, and 177. Thus
we find here very consistent and decided advantage in favor of the
shorter drill periods. Unfortunately in each of these experiments
it is impossible to tell how much of the gain in the shorter practice
periods was due to spontaneous practice outside the class. It is
needless to say, that the children were not permitted to take any
of the practice cards away from the class.
Kirby's experiments as a whole both in addition and division
showed great improvement. Addition with a practice period of
60 minutes yielded an improvement of 48%, while division with
a practice period of 50 minutes yielded an improvement of 75%.
Accuracy was not disturbed in addition, but in division it improved
2.6%.
Kirby also investigated the permanence of the improvement
resulting from the drill. He found that from June to September
fourthgrade children lost 17% of the abihty possessed in June and
required 58% as much time to regain the efficiency which they pos
sessed the preceding year. In division there was a loss of 21%.
404
EDUCATIONAL PSYCHOLOGY
It required 60% as much time to recover the preceding year's
efficiency.
Hahn and Thorndike repeated the addition part of Kirby's
experiment. Each grade was divided into two sections, section B
receiving periods exactly half as long as section A. All received
a total of 90 minutes of drill, which was preceded and followed by
a 15minute test as in Kirby's investigation. Table 122 shows
the distribution of drill for the various groups, the initial ability,
the gain, and the advantage of group A according to two different
methods of scoring the results.
TABLE 122. After Hahn and Thorndike
Number
OF Sub
jects
Length of
Practice
Period
Score Based on Rights
ONLY
Score Based upon the
Rights + OneHalf thi;
Wrong Answers
Grade
Average
Initial
Score
Average
Gain
Advan
TAGE IN
Favor of
Group A
Average
Initial
Score
Average
Gain
Advan
tage IN
FLAVOR of
Group A
7 A ...
7B . ..
6A ...
6B ..
SA ...
SB...
10
16
13
12
9
12
22Â».< min.
1454 "
20
10
IS
7K "
25.9
26.0
16.3
17.4
13.5
14.8
25.7
17.5
10.7
10.7
11.4
10.0
8.2
.0
1.4
32.2
33.1
22.0
22 .2
17.2
19.5
23.8
17.6
11.3
14.7
15.3
17.6
6.2
3.4
2.3
There is no clearly defined advantage for either the long or the
short period as was the case with Kirby's experiment on drill in
adding. What little tendency there is, however, is in favor of the
longer periods.
Superintendent Wimmer, in connection with his drill experiments
previously reported, also investigated the problem of economy of
long and short periods of practice. The drill, threefifths on rea
soning and twofifths on fundamentals, was given to one group
for five minutes at the beginning of each class period. The other
group received one 15minute period per week. Drill lasted for
six weeks. The results are shown in the following table. They
are distinctly in favor of the longer drill period despite the fact that
only threefifths as much time was spent by this method.
ARITHMETIC 405
TABLE 123
After Wimmer
Problem Favorable Distribution of Time for Drill
Grades Used 6E 6W Difference is
Number of Subjects 22 22 Favor of Drill
Type of Activity 5 Min. 15 Mint. once per Week
Drill Daily Drill Weekly
All tests, attempts and rights. . . 34.3 45.1 10.8
Fundamentals 14.7 18. i 3.4
Reasoning, attempts and rights. . 67.2 83.5 16.3
Reasoning, rights only 75 . 9 102 .1 26 . 2
This is particularly the case with reasoning. The rather slight
advantage of the long periods on fundamentals in connection with
the uncertain indications of Hahn and Thorndike's results and
the opposite finding of Kirby suggests that there is little or no
advantage in the distribution of time in arithmetical drill on fun
damentals but that the longer periods are more favorable for drill
in reasoning.
(7) Special or Economical Methods of Drill. A number of
special methods of giving drill in arithmetic have been advocated
and used. That of Studebaker has already been noticed. Courtis
has also published practice pads for drill puqooses. Flora Wilbur
undertook to determine experimentally the value of this kind of
drill at the Fort Wayne, Indiana, training school. Two classes
of 14 children each were divided each into two groups of equal
ability on the basis of the Courtis tests, Series B. One section
of the fifth grade received four and onehalf minutes of drill with
the pads at the beginning of each class period, and one section
of the sixth grade received similar drill for four minutes. The
remaining sections received the regular class work. The experi
ment lasted from September to May. The results are shown in
percentages of gain in the following table. The drill was clearly
of value in both grades and in all four processes.
4o6
EDUCATIONAL PSYCHOLOGY
TABLE
124.
After Wilbur
Grade Five
Grade Six
Speed
Accuracy
Speed
Accuracy
a tj
a u
u w
M H
u
u
H H
hh
S2
a K
a d
U OS
euOi
fi^O,
P^d.
B
flnPL,
(J
<
(J
f1
fc
Z
u
H
U
Si
3
t.
z
z Â«
2
S
n
6.
Z
o
<
:; w
< g
<ti
s
'/:
P,
Oo
'^
ei
:z;
PL,
Oo
^
oi
Oo
Addition .
.30
43
13
30
37
7
43
60
17
12
20
8
Subtrac
tion . . .
27
54
27
29
49
20
28
49
21
2
21
19
Multipli
cation. .
12
14S
136
26
31
5
SI
cS8
37
18
20
2
Division. .
144
144
43
42
â€” 2
76
129
53
28
34
6
Division and multiplication profited most and addition least,
as is usual in such experiments. The real question remains, how
ever: Is drill with the Courtis practice pad more or less efficient
than drill as ordinarily given?
Kirkpatrick performed two comparative experiments to de
termine the relative economy of various methods of memorizing
multipUcation tables. His subjects were twenty normal school
men, divided into groups of equal ability. As these subjects knew
the ordinary tables, he had them learn the products of 7 multiplied
by all the prime numbers between 17 and 53. One group simply
memorized the table by rote for the first five days, then spent the
periods of the next five days in writing down the answers on a
blank with a card containing the table before them for reference.
The other section spent the periods of all ten days working on
the blank with the table for reference. The time consumed during
the five days of memorizing was about an hour. A test at the end
of the experiment showed that, in a period of two minutes, the
practiced group put down 46.2 answers while the memorizing
group put down 40.9, thus showing a distinct advantage for the
practiced group.
A second experiment was performed with two groups of normal
school students of equal abiUty, twentyfive in each group. One
group practiced with the keys and blanks as in the first experiment
while the other group spent the same amount of time multiplying
out the products as needed. The experiment extended over eight
ARITHMETIC 407
days. When tested at the end of the experiment, those using the
key put down 25.4 answers in two minutes, while the computers
wrote down 44.3 examples, showing a decided advantage for the
computation method. He concludes that tables should be learned
by use rather than by memorizing.
Conrad and Arps ('16) investigated the effect of suppressing artic
ulatory movements upon the effect of drill in rapid adding. They
divided sixtyfour high school students into equal groups of equal
ability. The students were then given eight periods of drill in
rapid addition of columns. The pupils in one group were per
mitted to add in their ordinary way which involved a great deal
of articulation or inner speech. The other group was cautioned
repeatedly and emphatically to "think results only." The former
was called the traditional method and the ktter the economical
method. The percentages of gain by the two methods were as
follows:
The traditional method gained in attempts 8.5% and in rights
2.5%. The economical method gained in attempts 34.4% and in
rights 30.9%. This gave an advantage in favor of the economical
method of 25% in attempts and of 33.4% in rights. These results
came out almost startlingly in favor of "thinking results only."
The evil effects of articulation and lip movements have been no
ticed in connection with reading (page 2S7). It is probable that
the cause is the same in both cases.
P. B. Ballard investigated the comparative efficiency of the
"equal addition" method and the "decomposition" method in
subtraction.
"In the equal addition method the compensation is made â€” accounts
arc squared â€” at the very first number dealt with after the minuend has
been disturbed. In subtracting 37 from 85, after taking 7 from 15 the
disturbed relationship of difference between minuend and subtrahend is
immediately restored by increasing the 3 tens to 4 tens. In the method
of decomposition, however, it is the 8, the second figure dealt with, that
has to be changed to restore the balance. If the minuend figure is zero,
the balancing of accounts is still longer deferred."
Ballard gave tests in fundamentals to 71 Enghsh schools of
which 23 had been taught subtraction by the method of equal
addition while the rest had been taught it by the method of de
composition. While there was little difference in the average ability
of the two groups in the other three fundamental operations,
4o8
EDUCATIONAL PSYCHOLOGY
there was a very striking superiority in the score for subtraction
of the equal addition group. At 13 years of age it amounted
to over 10% and at earUer ages it amounted to over 40%. (See
Figure 84.) Inasmuch as tlie decomposition, or less efficient
method, is the one in general use in this country, it is evident
that this matter deserves careful attention.
(38

y^
60

/ ^'*
"~
/ ^'

^ /^
~
y^ /
f /
50


~
v/ :^/
40

/ 1
~
/ 1

/ 1
30
_
/ 1
5Â».^
/ 1
1
_
/ '
_
/
20
/
10

1 1 1 1 1 1 1 1 1 1 1
84 ^% 9i 9i lOi 101 Hi HI 12J 121 13i 13
Ages
Fig. 84. â€” Showing superiority of teaching subtraction by the "equal addi
tions" method. After Ballard ('15).
Mead and Sears performed two experiments in comparative
economy of methods in arithmetic. The first was to compare the
efficiency of the ordinary "take away" method of subtraction
which involves the learning of an entire subtraction table, as
compared with "addition" subtraction which permits the use of
the addition table, thus saving the learning of an entire table.
In the first the formula is "8 minus 2 equals what?", in the second,
"2 plus what equals 8?" Two secondgrade classes of approxi
mately equal median ability as indicated by the Courtis tests were
ARITHMETIC 409
taught subtraction by the respective methods thirty minutes per
day for four months, all other factors being equalized as fully as pos
sible. Tests given periodically throughout the experiment showed
that at the outset the addition method was superior but the "take
away" method gradually overtook it until at the end of the four
months the "take away" method was superior by 4.5 points which
was nearly onetliird of the final median score made by the addition
group. Tliis difference disappeared, however, when both groups
were tested on longer examples.
The second experiment was to compare the multiplicative
method of division. The formula of the first is, " Five into twenty
how many times?", that of the latter, "Five times what equals
twenty?" Two thirdgrade classes were used as subjects in this
experiment. Other conditions were similar to the experiment on
addition. In this experiment, the final test on combinations
revealed the fact that the multiplicative class stood 4.3 points
above the "into" class, which was about onefifth of the final
score of the "into" class. This difference disappeared, however,
when the class was tested on longer examples just as that in ad
dition noted above.
J. A. Drushel investigated the relative efficiency of two methods
of determining the position of the decimal point of the quotient in
the division of decimals. The rule of method A, the older one, is:
"There are as many places in the quotient as those in the dividend
exceed those in the divisor." The rule of method B, the newer of
the two, is: "First render the divisor an integer by multiplying
both dividend and divisor by 10 or some power of ten. Then
proceed as with integral divisors." A short test in division of
decimals was given to 576 freshmen at Harris Teachers' College.
Of these, 507 had studied division of decimals by method A, while
69 had studied it by method B. The results show that the stu
dents taught by method A had the very low accuracy of 66% in
placing the points, while those taught by method B had the very
excellent accuracy of 99%. If future investigations confirm these
results, method B should be generally adopted.
(8) Speed vs. Accuracy. Thorndike investigated the relation
between speed and accuracy in simple addition. Six hundred and
seventyone students were tested apparently on two different occa
sions in a class experiment in adding columns of nine digits. The
subjects were then grouped according to speed as shown by the
following table:
4IO
EDUCATIONAL PSYCHOLOGY
TABLE 125. After Thorndike ('is)
NuNrnER OF
Individuals IN
Group
Number of Addftions pkr 100
Seconds (Counting the Time of
Writing the Answer Equal to
One Addition's Time)
Approximate Number of Errors per
1000 Additions, i. e., Wrong Answers
PER 100 TÂ£NDiGiT Additions
Early Test
Late Test
Early Test
Late Test
65
loS
86
115
109
103
65
20
150
icS
88
75
64
55
46
37
162
120
99
87
75
66
5S
46
7
9
10
12
12
12
14
17
I
3
7
6
4
5
3
6
6
8
9
9
10
14
8
5
7
3
3
5
4
With a decrease in the rate of additions there is a steady increase
in the number of errors per i,ooo additions. Thorndike concludes
"that the sort of individual who is quick in adding is more accurate
also than the one who is slow."
(9) Limits of Attainment. Since there is general agreement
that the fundamental number combinations should become auto
matic association processes, it is pertinent to ask, how high a
degree of skill should be developed in pupils? This question is
similar to the one discussed in connection with quality of hand
writing. It is reasonable to maintain that it is probably uneco
nomical to attempt to develop a degree of speed beyond a certain
point.
How great speed is practically necessary or worth while? Obvi
ously the school may devote relatively too much attention to the
development of speed in the four fundamental operations at too
great a cost of time. It would, therefore, be important to know
what degree of proficiency is needed for the practical affairs of life.
An investigation of the problem is needed,
(10) Errors. The detection and classification of errors and the
discovery of the frequency with which they occur are highly useful
facts in any School subject because such information will help to
make instruction specific. It will indicate the particular points
at which drill should be directed.
Howell made an analysis of the mistakes in division occurring
in the Courtis tests as applied to the pupils in his school. He
found the following rubrics of errors:
ARITHMETIC 41I
"i. Making the quotient figure the same as the divisor,
(a) When a difference of only one exists between the divisor and
quotient;
(b) When the quotient is commonly used as the divisor of the
given dividend.
" 2. Making some factor (other than the divisor), commonly used as
the divisor of a given dividend, the quotient figure.
"3. When dividing a digit by itself, making the quotient figure the
same.
"4. When dividing a digit by itself, making the quotient figure zero.
" 5. When dividing by one, making the quotient one.
"6. When the dividend is zero, making the quotient the same as the
divisor.
"7. Pupils whose associations are as yet feeble or become so through
fatigue or distraction are commonly observed to resort to running up the
table.
"They frequently miss count and get a quotient figure one removed
(say) from the right one.
"8. Making one of the quotient figures the quotient.
"9. Substituting multiplication for division.
"10. Unclassified."
The frequency with which the different tjpes of errors occurred
is shown in Table 126.
TABLE 126. After Howell
Showing the number of mistakes in the division tables falling into each class
Grade
a&b
3
16
12
4
13
7
5
5
7
6
4
8
7
10
6
8
9
9
tal
57
49
Kinds of
Mistakes
4
5
6
7
8
14
I
2
26
6
20
4
4
23
3
23
3
I
14
32
16
4
16
2
27
4
2
7
2
65
II
II
5
3
181
39
24
91
16
TOT.'O.
. Mistakes
10
43
124
49
133
4
62
17
99
19
78
3
120
135
616
From this table it appears that certain types of error occur
much more frequently than others. For example, Error No. 4,
"when dividing a digit by itself, making the quotient figure zero,"
occurs 181 times; whereas Error Nos. 3 and 9 occur only 12 times
each. More extensive studies of this sort are much needed. Learn
ing in any school subject is apt to be more economical the more
412 EDUCATIONAL PSYCHOLOGY
specifically the learners' attention may be directed to the particular
processes to be exercised.
A. S. Gist tabulated the errors in subtraction, multiplication,
and division of 812 arithmetic test papers from six schools in
Seattle. Table 127 shows the percentage of different types of errors
separately for each of the three processes and for each of the grades
from 4 to 8. It is noticeable that the greatest difficulty was pre
sented in subtraction by borrowing, in multiplication by the
tables and in division by the remainder. For the most part the
proportions of the various errors are fairly constant from year to
year.
Besides the prevalence of the general type of errors noted above,
the relative difficulty of the various elementary combinations is
a matter of much practical importance. C. L. Phelps attempted
to determine this for the addition combinations by finding the