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pictures similar to Lay's. The groups of dots used on the pictures
ranged in size from three to twelve. Each card was exposed for
5 seconds. The investigation was carried out with pupils in an
elementary school. Howell's results are shown in the curve of
Figure 79. This curve shows a rapid drop in errors from the first
grade to the second, and that pupils reach a high degree of cer-
tainty between the third and fourth grade in ai)prehcnding the
numbers. IIowcU also found that certain numbers arc not appre-






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Fig. 80. — Number of mistakes in perceiving numbers by pupils in various
grades. After Howell ('14, p. 21 7) .

hended as accurately as others, the most difficult ones being seven
and eleven, as shown in Figure 80.

(2) The Operations to be Learned. One of the first important
problems in the economy' of learning arithmetic, as in every school
subject, is the determination of the operations that pupils should
really master. Considerable attention has lately been given to
this question with the result that the recent texts and courses of
study have eliminated much material and have decidedly shifted
the emphasis from certain topics to others.

Two methods of attack may be employed in determining the
topics and the material to be learned in arithmetic. According to
the one method, we might obtain the consensus of many expert
opinions as to the topics that should be included and the relative
amount of emphasis on each. According to the other method, we
might proceed to gather a large number of the arithmetical prob-
lems and operations actually involved in the occupations and



ARITHMETIC



387



professions of all classes of people, and to determine on the basis
of such a collection, the sort of material that should be taught.
Some results have been obtained by both methods.




iiij 40 50 eo

Per cent



m 90 100



Fig. Si. — Percentage of superintendents in 830 cities who favor elimination
of tlie various tojjics rejiresented by checked surface and tliose who favor "less
attention" are represented by shaded surface. After Jessup and Coflman ('15).



388



EDUCATIONAL PSYCHOLOGY



Jessup ('15) and CofTman obtained an expression of opinion
from superintendents of 830 cities with a population of 4,000 or
over as to which topics should be eliminated, which should receive
less emphasis, and which should receive more emphasis than they
do at present. Their results arc shown in Figure 81 and Figure 82.




10 20 30 40 50 60 70 80 90 100
Per cent

Fig. 82. — Percentage of superintendents in 830 cities who favor giving more
attention to each of these topics. After Jessup and Coffman ('15).

As a result of these returns Jessup and Cofifman recommend the
elimination of the following topics from the elementary course
of study:

"Apothecaries' weight, alligation, aliquot parts, annual interest,
cube root, cases in percentage, compound and complex fractions of more
than two digits, compound proportion, dram, foreign money, folding



ARITHMETIC



389



paper, the long method of greatest common divisor, longitude and time,
least common multiple, metric system, progression, quarter in avoirdu-
pois table, reduction of more than two steps, troy weight, true discount,
unreal fractions."

Furthermore, they recommend greater attention to such topics
as:

"Time saved through the omission of the material mentioned in the
foregoing may be wisely devoted to the study of social, economic and
arithmetical issues involved in such facts as saving and loaning money,
taxation, public expenditure, banking, borrowing, building and loan
associations, investments, bonds and stocks, tax levies, insurance, profits,
public utilities, and the like."

G. M. Wilson ('17) collected 5,036 problems from 1,457 persons,
representing practically all varieties of occupations and profes-
sions. He then classified these problems according to the type of
operation involved, and the number of problems of each type as
shown in Table 114.



TABLE 114.
Addition i-PI

2-Pl

3-Pl

4-Pl

Over 4-PI

Total

Multiplication i-PI

2-Pl

3-PI

4-Pl

Over 4-PI

Total

Subtraction i-Pl

2-PI

3-Pl

4-Pl

Over 4-Pl

Total



After Wilson ('17)

30 Accounts 251

706 Addition of Eractions 3

748 Amount 11

193 Area i

65 Average Weiglit 14

1742 Banking 18

Board Measure 12

Cancellation 26

1660 Capacity 10

904 Circular Measure i

195 Cubic Measure 56

17 Debts 56

3 Decimals 4

2779 Discount 5

Division of Fractions i

40 Dry Measure 5

407 Exchange 5

406 Insurance 10

167 Interest 66

65 Liquid Measure 14

1085



390



EDUCATIONAL PSYCHOLOGY



TABLE 114— Continued



Division


i-Pl


334




2-Pl


31Q




3-Pl


121




4-PI


48


Over


4-Pl


5



TotaL



Fractions 2-5



6-10



10 plus
I plus

1-5
I plus

5 plus

Total

United States Money 1 PI

2-Pl

3-Pl

4-Pl
Over. . 4-PI

Total



839



534



86



60

103s

23

2982

1714

550

247

5516



Making Change 3

Measuring 21

Percentage 217

Plastering 2

Practical Measurement 79

Profit and Loss 16

Proportion 5

Receipts i

Square Measure 27

Taxes 6

Time Measure 13

Buying 3 1 28

Selling 646



"The problems solved in actual usage are brief and simple. They
chiefly require the more fundamental and more easily mastered proc-
esses.

"In actual usage, few problems of an abstract nature are encountered.
The problems are concrete and relate to business situations. They
require simple reasoning and a decision as to the processes to be em-
ployed.

"The study justifies careful consideration of the following question:
After the development of reasonable speed and accuracy in the funda-
mentals and the mastery of the simple and more useful arithmetical
processes, should the arithmetic work not be centered largely around
those problems which furnish the basis for much business information?"
(Wilson.)



ARITHMETIC 39I

W. S. Monroe ('17) has compiled the problems in four text-
books and classified them according to the types of operation
involved and then compared the frequency of these types of prob-
lems with the number of workers in the different occupations.
His preliminary report states:

"In the first place, out of a total of 1,023 types of practical problems
found in four text-books, 720, or 71%, occur in occupational activities.

"A study of the frequency with which type problems occur reveals
a significant fact; viz., the frequency ranges from one to 434.

"This wide variation in frequency shows that the authors of our text-
books are far from being in agreement on the type problems of arithmetic.
Only one author out of four has recognized 511 out of i ,023 type problems
and 140 type problems have received the recognition of only two authors
out of four." (Monroe.)

(3) Length of the Class Period. J. M. Rice made an investiga-
tion of efficiency in arithmetic after the general plan of his investi-
gation of spelling. He tested some 6,000 pupils in eighteen dif-
ferent schools in seven cities. His results are exliibited in Table
115, which

" Gives two averages for each grade as well as for each school as a
whole. Thus, the school at the top shows averages 80.0 and 83.1, and
the one at the bottom, 25.3 and 31.5. The first represents the percentage
of answers which were absolutely correct; the second shows what per
cent of the problems were correct in principle, i. e., the average that
would have been received if no mechanical errors had been made. The
difference represents the percentage of mechanical errors, which, I be-
lieve, in most instances, makes a surprisingly small appearance."



392



EDUCATIONAL PSYCHOLOGY



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ARITHMETIC



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394 EDUCATIONAL PSYCHOLOGY

With reference to the factor of time in relation to eflficiency in
arithmetic, Rice concludes thus:

"A glance at the figures will tell us at once that there is no direct
relation between time and result; that special pressure does not neces-
sarily lead to success, and, conversely, that lack of pressure does not
necessarily mean failure.

"In the first place, it is interesting to note that the amount of time
devoted to arithmetic in the school that obtained the lowest average — •
25% — ^was practically the same as it was in the one where the highest
average — 80% — was obtained. In the former the regular time for arith-
metic in all the grades was forty-five minutes a day, but some additional
time was given. In the latter the time varied in the different classes,
but averaged fifty-three minutes daily. This shows an extreme variation
in results under the same appropriation of time.

"Looking again toward the bottom of the list, we find three schools
with an average of 36%. In one of these, insufllcient pressure might be
suggested as a reason for the unsatisfactory results, only thirty minutes
daily having been devoted to arithmetic. The second school, however,
gave forty-eight, while the third gave seventy-five. This certainly seems
to indicate that a radical defect in the quality of instruction can not be
offset by an increase in quantity.

"If we now turn our attention from the three schools just mentioned
and direct it to three near the top — Schools 2, 3 and 4, City I — we find
the conditions reversed; for while the two schools that gave forty-five
minutes made averages of 64% and 67%, respectively, the school that
gave only twenty-five minutes succeeded in obtaining an average of 69%.
This would appear to indicate that while, on the one hand, nothing is
gained by an increase of time where the instruction in arithmetic is
faulty, on the other hand, nothing is lost by a decrease of time, to a
certain point, where the schools are on the right path in teaching the
subject. Perhaps the most interesting feature of the table is the fact that
the school giving twenty-five minutes a day came out within two of the
top, while the school giving seventy-five minutes daily came out prac-
tically within one of the bottom."

Stone ('oS) made a similar investigation, testing some 6,000
pupils in the 6th grade in twenty-six school systems. He reports
results practically identical with those of Rice,, namely, that while
the amount of time devoted to arithmetic in different schools
varied from 7% to 22% of the total school time, yet a comparison
of time expenditure with the efficiency attained showed, according
to his interpretation, that time plays a negligible part.

These results and inferences are interesting and valuable but



ARITHMETIC 395

they cannot be interpreted with absolute assurance. The various
factors cooperating or counteracting are so intricate that a more
careful isolation of the clTcct of the time element is necessary. In
general, the same criticism made in connection with Rice's and
Cornman's investigations of spelling applies here. Dependable
conclusions could be reached only by an experimental procedure
similar to the one there suggested.

The findings of Rice and Stone probably represent correctly
the situations in the schools examined. A possible explanation of
the fact that the schools giving more time to arithmetic did not
obtain on the whole any higher efficiency than those devoting
less time to it may, perhaps, be sought in the likelihood that the
schools giving longer periods of time may not have worked as
intensively and used their time to as good advantage as the schools
devoting less time.

(4) The Effect of Various Environmental Factors. Both Rice
and Stone massed their results with reference to ascertaining the
effect of such factors as the home environment of the pupils, size
of classes, age of pupils, the time of day of the test, amount of
home-work required of the pupils, method of teaching, teaching
ability, the course of study, the superintendent's training of the
teachers, etc. .Rice reports that none of the factors had any influ-
ential part in producing efficiency in arithmetic. The results are
open to the same criticism of complication of factors as were pointed
out previously. It seems quite improbable that these elements
played no part. It is rather a question of more rigorous isolation of
the effect of different factors. Stone, for example, found that the
correlation of excellence in the course of study, as rated by judges,
with efficiency in arithmetical reasoning was .56, and with effi-
ciency in fundamentals .13.

That environmental factors, and perhaps particularly method
and spirit of teaching, do make important differences in the attain-
ments of pupils is shown clearly in such results as those exhibited
in Table 116 which gives the distribution of class averages of the
grades in sixteen different schools as measured by the author's
Arithmetical Scale A.



396 EDUCATIONAL PSYCHOLOGY

TABLE 116

Average scores attained in various schools as measured by Arithmetical Scale A

(Starch)

Grades 3 4 5 6 7 8

City A 9.7

B School I 13 . 1

2 7.2 10.4 10.6 II. 2

C School 1 5.1 5-9 7.2 9.2

2 ,3-9 5-6 6.9

3 3-9 5-3 5-6 7-5 9-2 12.6

G School 1 9.0 10.9 II. 6 14.5

2 8.9 12.0 13.0 13.7

3 75 IO-2 9.2 10.9 II. 5

4 10.0 10.6 II. 3

I School 1 5.1 6.0

2 6.2

3 10. 10. 2 II .0

L School 1 4.6 5.8 8.5 9.8 11.9 14.0

2 4-6 7.4

3 6.9 8.5 II. 3

4 6.6 8.8

Thus we note that the best eighth grade attained an average of
14.5 as compared with the poorest one whicli attained an average
of only 9.7, Such differences would not be surprising if they
were the scores of individual pupils. They are, however, the mean
scores of whole classes. It is quite unlikely that the hereditary
differences of the groups as wholes differ so much from one another.
It seems quite probable that the environmental circumstances,
and chief among them the teacher and the attitude of the learner,
were mainly responsible for the ultimate differences in achieve-
ment.

Similar results have been reported by Judd for the fifth and
eighth grades in ninety schools in Cleveland, as measured by his
Test A in simple addition, Figure 83. The best fifth grade made
an average score nearly three times as high as the poorest fifth
grade, and the best eighth grade made a score nearly twice as
high as the poorest eighth grade.

(5) Drill in Fundamental Operations. Various methods of
drill in the fundamental operations have been devised.

Studebaker, Assistant Superintendent of Schools at Des Moines,
has prepared a scries of drill cards. The various combinations of
numbers in fundamental operations are given on one side of the



ARITHMETIC



397



card. Below each example there is an opening through the card
in which the pupil may write his answer on the sheet of paper
placed underneath the card. The pupil works as rapidly as he can
within a certain limit of time. Then he turns the card over and
places it again over the sheet of paper containing his answers so







21

21
21




23
23
23




Fifth Grades


40]






21


22


23






21


22


23






21


22


23










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22


23








20


21


22


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20


21


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23










19


20


21


22


23




26












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19


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22


23


26


27








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19


20


21


22


23






26


27






18


19


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22


23


24




26


27


28






16


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18


19


20


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22


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26


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|14|


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17


18


19


20


21


22


23


24


25


26


27


28


|30| 1 1 1341 1 1 1


38{ 40


1



Eighth Grades




Fig. 83. — Median scores of the sth grades and of the 8th grades in 90 schools
in simple addition. After Judd ('16, p. 112J.

that they can be seen in the openings of the card and compared
with the correct answers printed on that side of the card.

This plan of drill work has a number of advantages, such as an
incentive to rapid and accurate work, immediate self-checking of
the answers, and so on.



398 EDUCATIONAL PSYCHOLOGY

A considerable number of careful experimental studies on the
influence of drill are now available and without exception they
show drill to be distinctly valuable. Thorndike had nineteen
university students practice adding 48 lo-digit columns of figures
daily for seven days. While the work required on the average less
than an hour in all, there was an improvement of 29% over the
original rate.

J. C. Brown performed two elaborate comparative experiments
to determine whether children under controlled school conditions
profit more by. giving a small part of each class period to drill or
by spending the entire period in ordinary routine work in arith-
metic. In each experiment the children were first tested with the
Stone Arithmetic tests and then divided into two groups of equal
ability on the basis of their performance in the tests. One group
was given the special drill as a part of the regular class work while
the other did the class work as usual. At the conclusion of the
drill, both were tested again by the Stone tests to see which had
made the greater gain. In the first experiment 51 children from
the sixth, seventh, and eighth grades were used. They averaged
thirteen and one-half years of age. Drill on the four fundamental
operations was given to one-half of the group for the first five
minutes of each class period of twenty-five minutes. About half
the drill was oral and half was written. The drill lasted thirty
periods. In the second experiment 222 children were used and the
drill was given for twenty periods. The results of the two experi-
ments are given in parallel columns in Table 117. In each case
section I received the drill and section II received the regular class
work. The pupils did not know that any experiment was in
progress.



ARITHMETIC



399



TABLE 117



Section


Per Cent of Improvement of Sf.cond
Test over First in


Per Cent
First Experi-
ment
(51 Children)


Per Cent
Second Experi-
ment
{222 Children)




Number of problems worked
Fundamentals, Addition
Fundamentals, Subtraction
Fundamentals, Multiplication

Fundamentals, Division

Total number of points made

Number of points made on the
first six prol^lems (averaged)

Number of points made on the
first six problems (a\'eraKC(l)


21.2
9.8

33-4
II. 8

36.9
13-1
30.0

13-7
28.0

193
32.0
14.7

S-8
2.4


16.9




6.4
18. s




6.8
32.0




II. 9

24.1




10.9
34-2




iS-4
24.2




9-4


II


II. 7
-1.8



In both the experiments there- was a decided advantage in using
a part of the recitation period for drill. In the first experiment,
the drilled group gained about twice as much as the undrilled
group; while in the second experiment the drilled group improved
about two and one-half times as much as the undrilled group. The
sixth grade gained the most (35%) and the eighth the least (13.8%).
In order to determine whether group I had gained on fundamentals
at the expense of reasoning, both groups were tested in arithmet-
ical reasoning before and after the drill. Here again the drilled
group did better, making a gain of 6.3%, while the undrilled group
gained only 3.0%. This last factor is interesting in the light of
the small amount of connection between fundamentals and arith-
metical reasoning pointed out above as well as the small amount
of transfer of one arithmetical process to another (Chapter XIV).
Since the improvement in reasoning, which had not been drilled
at all was almost exactly the same proportionately as the processes
which were drilled, it suggests that the drill had a tonic effect
upon the remainder of each recitation period following the drill, to
which much of its value was due.

In order to discover the permanency of the effects of drill, Brown
tested both groups once more after a twelve-weeks vacation and



400



EDUCATIONAL PSYCHOLOGY



found that the drilled group was also superior in retention, having
lost .2% wliilc the undrilled group had lost 2.29%.

The same experiment was repeated by F. M. Phillips. He had
69 cliildren for subjects and gave to one group drill in fundamentals
and in reasoning, both oral and written, for eight weeks. Neither
teachers nor students knew the purpose of the tests. He found
that, "The improvement in fundamentals of the combined drill
groups was 15% greater than that of the non-drill groups. In
reasoning, the drill groups improved 50% more than the non-drill
groups. . . . The greatest gains were made in the sixth grade and
the least in the eighth." Almost all the gain on fundamentals
was in multiplication.

Mary A. Kerr under the direction of Haggerty reported an
experiment carried on for six weeks at Bloomington, Indiana, on
the effects of five minutes of drill in addition at the beginning of
each class period. The drill was begun by adding five three-place
numbers per column, which were gradually increased to nine three-
place numbers per column. Four hundred and twenty-three chil-
dren took the drill. Table 118 shows the average performance on
the Courtis tests, Series B, before the drill began and at its con-
clusion in June and, for comparison, the May scores of the best
twenty Indiana cities for the previous year.

TABLE 118





Attempts
Bloomington
Feb. June


Highest
Median
Scores of
20 Ind.
Cities
(May)


Rights
Bloomington
Feb. June


Highest

Median

Scores of

20 Ind.

Cities

(May)


Per Cent Accuracy


Grade


Bloomington
Feb. June


Highest

Median

Scores of 20

Ind. Cities

(May)


6B ...
6A ...
7B...
7A...
8B ...
8A ...


8.7 9-7
9.0 10. 5

9.7 10.8

9.8 II. 8
II. 4 12.0
II-5 13-7


8.9

9.4

10.3


5-5 6.9
5-3 7-5
5.6 S.I
6.0 8.4
6.9 9.3
6.3 10.4


5-6
6.4
7.2


64 72

59 71

60 75
62 71

61 78
55 76


65
68
69



The decided advance in each grade and the great superiority to
the best twenty Indiana cities in each test bears eloquent testimony
to the value of drill in addition fundamentals.

Supt. Herman Wimmer of Rochelle, Illinois, conducted a series
of comparative experiments on the effects of drill in arithmetic




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