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The Verbal Thought and Overt Behavior of Children During Their Learning of Long Division

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THE VERBAL THOUGHT AND OVERT: BEKAVXOR
°♦
' * '*
•'
OF CHILDREN DURING THEIR LEARNING;. ,
OF LONG DIVISION
by
Casper 0. Dahle
A dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy,
In the Department of Education, In the Graduate
College of the State University of Iowa
August, 1940
ProQuest N um ber: 10984089
All rights reserved
INFORMATION TO ALL USERS
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and there are missing pages, these will be noted. Also, if m aterial had to be rem oved,
a n o te will ind ica te the deletion.
uest
ProQuest 10984089
Published by ProQuest LLC(2018). C opyright of the Dissertation is held by the Author.
All rights reserved.
This work is protected against unauthorized copying under Title 17, United States C o d e
M icroform Edition © ProQuest LLC.
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P.O. Box 1346
Ann Arbor, Ml 4 8 1 0 6 - 1346
11
AQKE CWLBDQMBOTS
The writer wishes to acknowledge his
Indebtedness to Dr* F. B. Knight who suggested
this study end assisted with helpful suggestions.
He also acknowledges the very valuable assistance
of Dr. J. B* Stroud who In the absence of Dr, F.
B* Knight made the completion of the study possible#
He also wishes to express his apprecia­
tion to Dr. C# A# Huckmlck for the use of the
Motion Picture Laboratory and the Psychogalvano­
meter; to Dr* H* A. Greene for the use of the
Iowa Oral Language Machine j and to Mr. Lee Cochran
for the use of the photographic equipment#
The writer wishes at this time to ex­
press his gratitude to Superintendent I. A# C£>sta&
for permission to carry on the project at Long­
fellow School; and to Miss Marie Carrlgg whose
extreme cooperation aided materially in completing
the study.
ill
CONTENTS
I
INTRODUCTION
1*
2#
3#
4*
5#
6#
II
PROCEDURE
1*
2.
3*
III
IV
Purpose of the Investigation
Importance of the problem
Content Used
Apparatus Used
Materials Used
Subjects Used
General Procedure of the Study
Specific Procedure of Each Project
Specific Materials of Each Project
page
1
1
1
1
2
4
6
9
9
13
14
AMOUNT AND NATURE OF DATA SECURED
25
1*
2*
3.
4.
5#
25
28
28
28
39
Children1s Written Responses
Moving Picture Film
Oral Report Records (Dictaphone)
Electrodermal Response Records
Observational Records
ANALYSIS AND INTERPRETATION OF DATA
47
1.
2.
47
3*
4*
5.
6.
7.
Preliminary Statements
Scientific Status and Interpretation
of Electrodermal Response Records
Electrodermal Response Records of tills
Study and their Significance
Arithmetical Status of Subjects
Effect of Failures and Successes upon
Subsequent Work
Project 1
a. Learning Experience Prior to the
Project
b* Selected Case Studies
c* Summary Treatment of Experimental
Group
d* Summary Treatment of the Class
e. Hypotheses and Tentative Conclu­
sions
Project III
a. Learning Experiences Prior to
the Project
b. Selected Case Studies
c* Summary Treatment of Experimental
Group
48
51
60
65
71
71
76
120
131
138
142
142
144
193
iv
8.
9*
a. Summary Treatment of the Class
© * Hypotheses and Tentative Conclusions
Project VI
a* Learning Experiences Prior to the
Project
to* Selected Case Studies
c« Summary Treatment of Experimental
Group
d. Summary Treatment of the Class
G « Hypotheses and Tentative Conclusions
Project VII
a* Learning Experiences Prior to the
Project
to* Selected Case Studies
c* Summary Treatment of Experimental
Group
d. Hypotheses and Tentative Conclusions
205
211
215
215
215
278
291
298
300
300
300
561
374
LIMITATIONS AND CONCLUSIONS
377
BIBLIOGRAPHY
389
V
TABLES
Table
Title
Page
1
Summary of Examples for each Project
5
2
Test Results
6
3
Test Results for Fourth Grade
45
4
Immediate Effect of Failure and Success
upon Electrodermal Response Records
54
Effect of Failures and Successes upon
Subsequent Deflections
56
Effect of Failures and Successes upon
Subsequent Work
66
Effect of Failures with Unawareness
upon Subsequent Work
68
Effect of Failures with Unawareness
upon Subsequent Work (Errors due to
previous errors excluded)
68
Effect of Failures and Successes upon
Subsequent Work
69
Effect of Failures and Successes upon
Subsequent Work (Errors due to previous
errors excluded)
69
Class Remedial Program
75
5
6
7
8
9
10
11
for Sixteen Subjects
12
Test
Results for Selected Cases
77
13
Test
Results for Selected Case
89
14
Test
Results for Selected Case
105
15
Verbalization Analysis
129
16
Class Remedial Program
144
17
Test
Results for Selected Cases
145
18
Test
Results for Selected Case
160
19
Test
Results for Selected Case
176
vl
Title
Pag©
20
Verbalization Analysis
202
21
Test Results for Selected Cases
215
22
Test Results for Selected Case
236
23
Test Results for Selected Case
258
24
Analysis of Verbalized Solutions
289
25
Test Results for Selected Cases
301
26
Test Results for Selected Case
319
27
Test Results for Selected Case
338
28
Comparative Analysis Projects VI and
VII (Comparison In %)
366
Analysis of Verbalized Solutions
372
29
1
chapter x
INTRODUCTION
1.
Purpose of the Investigation
The primary purpose of this investigation is
to study the progressive, continuous aspects of learning;
that is, the events that take place during learning*^
What transpires during the actual learning
process Is important both from a psychological and an
educational point of view#
The writer is chiefly In­
terested in this problem from an educational standpoint.
What happens in the child while on the way to an under­
standing of a mental function matters much educationally;
and any Insight that can be gained into the Inner and
outer aspects of the learning process as such should lead
to a more practical, a more psychological approach to the
mental function In question.
The content studied In thi s Investigation Is
long division.
The writer has been Interested in getting
as complete a picture as possible of the learning process
involved as It progresses from one stage to another.
It
so happens that this mental function lends Itself nicely
to such an approach as various steps of Increasing dif­
ficulty are successively presented during a given period
of time.
There are five such major successive types of
difficulty presented In the treatment of long division.
2
By studying several subjects at each of these stages of
their learning of long division, the writer has attempted
to obtain a rather complete picture.
2.
Apparatus and Reasons for Their Us©
A preponderance of rigorous work on learning
has been don© with very "short” learning series, such as
a maze or a series of nonsense syllables, -content, the
whole extent of which the subject could survey at once.
Mo doubt the techniques, standards, conventions, and cri­
teria of appraising research in learning have been greatly
Influenced by the type of material used.
This study requir­
ed material much of which the subject knew nothing until
weeks after his initial work,
Such "long" in contrast
to "short" material necessitated radical adjustments In
technique.
Some of these may seem objectionable to the
experimentalist accustomed to a closeness of control w M ch
is possible with "short" learning material.
Direct Measurements:
a.
Moving Picture Camera
A complete picture of each child1s manual
activities, as exhibited in long division, was wanted.
Mo
amount of training on the part of an experimenter would
yield adequate powers of observation for this purpose.
The moving picture camera, however, takes a complete, con­
tinuous picture of the subjectfs overt responses.
This
picture can be studied repeatedly with the attention
3
focused on any one isolated aspect during a given presenta­
tion.
Thus an analysis was made possible which ordinary
observation could not yield.
b.
Iowa Oral Language Machine and Metronome
These instruments made it possible to record
simultaneously the subject's verbalization of solutions
and the metronome's time units on dictaphone records.
Fol­
lowing the experimental periods, these records could be stud­
ied repeatedly, insuring accurate transcription and valida­
tion*
Indirect Measurements?
a.
p sychogal vanome t er
Mental effort cannot be expended without accompany­
ing physiological changes.
Emotions or affective reactions
accompany much of mental effort, and such disturbances have
an effect upon the product of mental effort,
well stated?
As has been
“Emotions have an extensive and intensive
grip of the mind.**
Fortunately, some of these physio­
logical changes are observable and can be objectively meas­
ured, as for example, changes In skin resistance, changes
In heart beat, changes in body temperature, changes in blood
pressure, etc.
This study measured the first of these dis­
turbances, the psychogalvanometer being used for the purpose.
This Instrument can be seen on the table at the subject’s
left, figure 1, opposite page 9.
These measurements were made in order to gain
4
Information about relationships that exist between vary­
ing learning situations and physiological changes and
thus arrive at a more complete description than would
otherwise be possible*
5.
Materials Used
As previously Indicated, the material used in
this investigation is long division*
The mastery of long
division requires the understanding of five major types
of difficulty*
For purposes of this study these types are
designated as follows?
a*
Long division without remainders
Example s
2 4
3 2 ) 7 6 8
6 4
12 8
12 8
Long division with remainders
3 rem*
5 2
Example? 2 3 ) 7 3 9
6 9
4 9
4 6
c*
Long division with ssero in quotient
Example?
2 0 4
3 4 ) 6 9 3 6
6 8
13
6
1 51
5
&.
Long division with three-figure divisors
Example:
e,
3 7
7 1 2 ) 2 6 3 4 4
2 13 6
4 9 S 4
4 9 8 4
Long division with non-app&rent quotient figures
ExampleI
5 1
3 7 ) 1 8 8 7
18 5
3 7
3 7
The following table indicates the types of long
division examples presented at each project and the class­
room status of each type:
Table 1
Summary of Examples for each Project
Classroom
Status
1
Review
Projects
5
4
2
3
A
B
C
D
E
Taught
A
B
C
D
Untaught
B
C
D
E
6
7
ABODE
ABODE
In addition to the above types of long division
examples, several long division verbal problems were in­
cluded in the study.
The following is an example:
A canning factory packed 744 cans of c o m into
boxes. Each box held 24 c&na# How many boxes
were needed for these cans?
6
Table 2
1*♦
J.B.
128
%A
S.F.
3.
to
fcOtO
to
do
Woody McCall*s
pest - 5-26-36
Subjects
a ©
0 0
W to
10
© tO * 8 *. 60
a) © > 1
rH |
IH r-f & ^ pH <«4»
•cd o
a P 03
3 © CM
< 0 00 | 083 0|
O «
rH
1 GO
0003 O rH
10
dh
CAr-H 10 1
S rH aJ
3p«p1 d Pi
6
©
,d 4-5
o «
8©
®
ow
© o
23JPH! i
O0
J*
O E^ .o £h JBLEH.
Iowa El era. Lang.
Pest - 5-27-36
Teat Results For Sixteen Subjects
$ «
,w,,£H
107
106
22
25
11.8 11.2
170*
96
96
20
23
5*7
5.2
142
160
R.H*
103
82
94
15
19
5*2
6.9
131
139
4.
Al.H.
130* 105
112
22
26
8.0
9.7
172
199
5.
An.H.
114
98
108
19
23
10.1 10.1
188
176
6*
3.J.
105* 105
110
21
19
5.8
5.6
151
155
7*
R.H.
110
101
86
20
21
6.3
7.1
119
147
8?
I. 0,
107*
85
88
12
20
5.5
6.5
105
110
9.
C.P.
117
103
106
22
21
7.0
7.5
149
162
.0*
B.R.
106
90
95
16
17
6.0
6.5
116
137
.1.
B.Sc.
108
107
108
22
28
5.6
6*2. 116
94
.2«
B.Sm,
92
74
14
20
4*2
5.0
71
88
.3«
N.T.
112
91
20
21
4.9
4.6
108
117
.4*
D.T.
114
.5*
B.W.
102
97
107
20
19
5.2
5.4
129
140
.6*
R.W*
104
107
109
21
18
8.3
8.6
142
184
Medians_
109
9&,
106
20
21
6.0
6,7
131
151
Class Medians
108
97
100
19
21
6.0
6.5
116
134
93
108
dJ
•H t>
TJ h
d i
© rH
W
1
03
0P
P «0
<0
gjeo
«rl 2s*
•d CM
aJ |
0 113
od
i
0
0 P
P W
®
O E4
•CM
© rH
H
£3 I
©P
ft 03
O©
H&t
201
207
8.9
23
155
102 109 14
Standards
85
4*5 5.0
18
*I.Q. as derived from Binet-Slmon Intelligence Test.
7
These long division problems were presented along with
addition, subtraction, and multiplication problems rather
than in isolated form.
As is well known, if the stage is
set for long division, children will proceed to divide
without respect for the verbal problem*
A group of easy two-digit subtraction examples
were used as a pre-and post- experimental series in order
to obtain levels of response before and after each project
and also to aid in orienting each child to the experimental
set-up*
4*
Subjects Used
As principal of the Longfellow Elementary School,
the writer was granted permission to use public school
children for the project.
Sixteen representative subjects
from the fourth grade were selected.
Subject S. P* was
selected because of her extremely high IQ as derived from
the Binet~Simon Intelligence Test.
A second individual
intelligence examination yielded an IQ of 147 for this sub­
ject.
Aside from this one subject, an effort was made to
select representative subjects, as is evident from the in­
formation given In Table 2, opposite this page.
This table
indicates the Intelligence and achievement of each subject,
the median scores for this experimental group and for th©
entire class of thirty-nine children from which they were
chosenj and the standards for each of the tests as given
in the test manuals*
8
Figure 1
Laboratory Setting
CHAPTER IX
PROCEDURE
As previously stated, the mastery of long di­
vision requires the understanding of five major types of
difficulty*
The usual teaching procedure and that which
Obtains in the Longfellow school consists In the presents*
tion of these types*
The sixteen children selected were
studied individually at each stage of their learning of
long division and again after a summer’s vacation.
Dr.
Ruck ml ck granted the writer permission to use the moti on
picture laboratory at the university for this purpose.
As the picture opposite this page Indicates,
each child was seated at an adjustable school desk.
A
partially elevated surface on this desk made it possible
to take a motion picture of the child’s work.
The moving
picture camera, stationed on a table three feet In height,
was at a distance of six feet from the writing surface.
By means of a telephoto lens It was possible to obtain
clear pictures of the entire process at this distance, and
f
at the same time reduce the disturbance to a minimum.
The psyohogalvanometer rested on a large table
to the left of the subject.
involved little or no noise.
The operation of this instrument
The psychogalvanometer attach­
ment consisted of a two*inch elastic band to which two flat
copper electrodes were attached.
This attachment was so
10
Longfellow School
Iowa City, Iowa
Jan. 21, 1957
Dear
I plan to m^ce a special study of long di­
vision next semester. The equipment needed for this
study make a It necessary to do all of the work at the
university.
Several pupils from the present 4B room
will be helping me with this work. The study will
continue throughout the semester* From time to time,
I shall ask these children to solve © few long di­
vision examples. At auch times I shall take them to
and from the university in my car.
May X have permission to Include your child
In this group?
Yours very truly,
Principal
Yes _______________
No
_______________
Note:
_______
______ ____________
fParentfs sIgnature)
Please have your child return this letter to me.
Thank you.
11
placed on the left hand that the electrodes were In con­
tact with the palm.
The elastic hand end a special rubber
cushion Insured constant pressure at the ppints of contact.
To further insure constant pressure and minimize the effect
of movement, the left hand and arm were pieced on a pedestal
having a rotating top.
This arrangement is easily observed
In the picture.
The Iowa Oral Language Machine was stationed in
an adjoining room.
The three microphones which are visible
in the picture served to catch all verbal reports of the
subjects.
They also made possible a signaling system be­
tween the writer and the operator of the language machine.
By means of these microphones, it was possible for the
assistant not only to get the writerfs signals but also to
not© the progress of the entire experiment.
The above experimental set-up required the ser­
vices of two assistants.
The writer operated the psycho­
galvanometer, gave the necessary signals to his assistants,
and otherwise supervised the procedure.
One assistant
operated the moving picture machine; the other, the Iowa
Oral Language Machine.
Certain preliminary procedures must be described
at this time.
Written permits were received from the par­
ents of the selected children.
The form opposite this page
served the dual purpose of informing the parents about the
12
INDIVIDUAL DATA SHEET
Name __________
Pat©
Time
Conditions*
A,
Physical *
B«
Physiological*
1.
Ventilation ____ 2,
Health
a.
b*
Cm
dm
e•
f,
C*
1.
Specific ailments
Previous m e a l ___
Night1a sleep ___
Hunger __________
Temperature _ _ _
Puls© ___________
P sychologi eel*
1. Nervousness ______ _ _ _
2, Emotional attitude ____
Zm Mental alertness ___ _
4* Previous experiences
a. C l a s s ________
b* Outside
5* Anticipated experiences
a. Class
.
b* Outsl.de _______ _
6. Disturbing factors ____
D.
Special Comments
Temperature
15
study and securing permission to use their children as
subjects*
In order to secure the proper attitude on the
part of the subjects and to avoid apprehensive states which
might be caused by the rather unusual setting for the solu­
tion of examples and problems, the writer brought the sub­
jects to the laboratory and explained the entire set-up to
each child*
Each subject was then seated In the adjustable
desk, the electrode attached to the palm of the left hand,
and a preliminary set of addition and subtraction examples
worked under conditions identical with those of the actual
study*
This preliminary project served to satisfy the
natural Inquisitiveness of the subjects and to minimize any
apprehensiveness that might have been aroused by the ex­
perimental setting*
The first project followed the completion of
first Instruction In type A.
Each of the sixteen subjects
was brought to the motion picture laboratory*
The temperature
and heart beat were recorded as well as information pertaining
to general health conditions and activities*
The form oppo­
site this page was used for the recording of these data*
Having procured the necessary information, each
subject was then seated and the electrode attached*
In­
structions to the child were Identical for each project.
The
children were told to work the examples and problems presented
14
to thorn*
They were asked to do their thinking wout loud”.
The signals given served both the subjects and the assist­
ants*
The signal wget ready” Indicated that the experiment
would begin within five seconds*
experiment began*
At the signal ”gon, the
The signal ”stop” was given at the con­
clusion of each solution*
Project I included two examples of type A and
one example of type B (See Chapter 1, page 4)#
Thus each
child was confronted with two examples of a type that had
just been taught and one example of a type not yet taught.
In addition to these three examples, two verbal problems
were presented to each subject, one involving addition,
the other long division of type A*
A pro- and post-exper­
imental series of simple subtraction examples were presented*
Each child was permitted to work two minutes on these ex­
amples*
These Identical subtraction examples were present­
ed In each project for purposes of orienting the subject
to the experimental situation and giving the writer an op­
portunity to get the pro- and post-experimental levels of
electrodermal resistance*
The examples and problems of Project I were as
follows*
15
Examplesl
A~1
(Just taught)
3 2 ) 7 6 8
A-2
(Just taught)
8 3 ) 5 3 9 5
B
(Not taught)
2 3 ) 7 3 9
Verbal Problems*
1*
John had 126 marbles. His father gave
him 14 more. How many marbles did he
then have?
2.
A canning factory packed 744 cans of
corn into boxes. Each box held 24 cans.
How many boxes were needed for these
cans?
Pre- and Post-series*
Subtract*
2 9 1 5
4 1 3
1 9 3 6
7 1 2
4 8 3 7
2 1 6
3 7 5 9
5 3 6
5 4 6 5
5 2 4
etc.
Approximately three weeks elapsed between Pro­
ject I and Project II.
During this time the children
were taught how to solve long division examples with re­
mainders, designated as type B in this study.
Project II
Included one example of type A which had now been transferred
16
to tto© maintenance program, two examples of type B, which,
had just been taught and one example of type G which had
not been taught.
As In the previous project, verbal prob­
lems and subtraction examples were Included,
The examples and problems of Project II were
as follows}
Examples:
A
(Review)
3 1 ) 8 3 7
B^l
(Just taught)
5 6 )~1 9 2 1
B**2
(Just taught)
9 4 ) 6 0 3 5
C
(Hot taught)
3 4 ) 6 9 3 6
Verbal Problems}
1*
In 1930, two men flew an airplane for 552
hours without stopping, How many days
did they fly the airplane without stopping?
2,
Dick brought home a box of 135 oranges
when he returned from his vacation. He
gave 45 of them to his aunt and the others
to his friends. How many did he give to
hi s friends?
Pre- and Post- series*
(Same as In Project 1)
17
About two weeks elapsed between Project XI and
Project III*
During this time the children were taught
how to solve long division examples with zero difficulty,
designated as type G in this study.
Project III included
one example of type B which had now been transferred to
the maintenance program, two examples of type C, and one
example of type D which had not been taught*
As In previ­
ous projects, verbal problems and subtraction examples
were included*
The examples and problems of Project III were
as follows*
Examples:
B
(Review)
3 6 ) 8 69
__________
3 1 9 4 1
C-l
(Just taught)
63)
C—2
(Just taught)
____ ____-.r-8 3 ) 2 8 2 2 0
D________________________ __________
(Hot taught)
7 1 2 ) 2 6 3 4 4
Verbal problems:
1*
The population of a certain city In 1920
was 43,568. Ten years later there were
70,002 people in the city. How much did
the population Increase in the ten years?
18
2,
A coal company collected. $2,620 during the
month of February selling coal at $13 a
ton* How many tons were sold?
Pre- and post-serieas
(Same as In Project I)
Approximately three weeks elapsed between Pro­
ject III and Project IV*
During this time the children
were taught how to solve long division examples with
three-figure divisors, designated as type D in this study*
Project IV included one example of type G which had now
been transferred to the maintenance program, two examples
of type D, and one example of type E which had not been
taught*
As in previous projects, verbal problems and
subtraction examples were included*
The examples end problems of Project IV were
as follows;
Examples 5
(Review)
4 1 ) 2 0 8 6 9
D-l
(Just taught)
4 0 7 ) 1 9 9 4 3
D-2
(Just taught)
2 3 7 ) 9 8 3 5 5
E
(Hot taught)
3 7 ) 1 8 8 7
19
Verbal Problems#
1.
Last year Mr. Hogan drove his car 7,527
miles* He used 431 gallons of gasoline
during the year. How many miles did he
average on 1 gallon of gasoline?
2.
Mr. Carter was paying for hi s house by the
month. He owed $4,940* He paid $38 a
month* In how many months would he pay
what he owed?
Pre- and Post-series#
(Same as In Project I)
About three weeks elapsed between Project IV
and Project V.
During this time the children were t©tight
how to solve long division examples with non-apparent
quotients, designated in this study as type E.
Project
V included one example of type D which had now been trans­
ferred to the maintenance program, and two examples of
type B which had just been taught.
As in previous pro­
jects, verbal problems and subtraction examples were in­
cluded.
The examples and problems of Project V were as
follows#
Examples:
D
(Review)
E-l
(Just taught)
__________
8 3 6 ) 3 9 2 9 2
2 9 ) 1
4 7 9
20
B-2
(just taught)
5 8)2
7 8 9 2
Verbal ProblemsI
1*
In 1955, Settle and Fordney went up to
01,257 feet In their balloon* Stevens
went up 60,613 feet in a balloon in 1934*
How much short of the first record was
Stevens* record?
2.
A music store paid $1450 for 29 radios*
If all the radios were the same price,
what did each cost?
Pre-* and Post-seriesI
(Same as in Project I)
At the time of this project all of the major
difficulties Involved In long division had been taught*
These included*
A - Long division without remainders
B -
Longdivision with remainders
C -
Longdivision with zero difficulty
D -
Longdivision with three-figure divisors
1 -
Longdivision with non-apparent quotient figxres
Luring the three weeks following Project V the
children had a general review of the semester's work in
arithmetic*
Project VI followed this general review*
This project Included one example of each of the above
types*
Verbal problems and subtraction examples were
presented as In the previous projects,
The examples and problems of Project VI were
as follows*
21
Examplesi
A
5 6) 2
4 08
B
8 2 )~2
7 99
0
7 2 >“4 5 8
D
2 3 6 ) 5 6 6 4
E
3 9) 2
4 8
8 47
Verbal Problems!
1*
A quail was marked In the south one spring*
During the summer it was seen 1904 miles
farther north* At its fastest flying speed
(56 miles per hour) how long would it have
taken the quail to fly that far north?
2*
Birds are known to fly as long as 9 hours
without stopping, while on their way north
in the spring* A thrush to which a tag
had been tied traveled north at an average
speed of 130 miles a day during the whole
month of April# How far north did It
travel? (There are 30 days in April*)
Pre- and Post-seriess
(Same as in Project I)
Project VI was collated during the last week
In May*
Three months of vacation followed*
During the
week of Sept driber 7 - 11, 1937, twelve of the original
sixteen subjects were brought into the laboratory and
presented with the identical problems and examples as
those of Project VI#
Four subjects had moved out of Iowa
City during the summer*
With one except! on*y*arrangements
22
Teacher*s Record
of
Dally 01©ss Work
Day______ _ Length of Period
Articles Presented!
a*
Review Material
b.
New Material
e*
Practice Material
d*
Self-testing Drill
r
Written Work by Class*
a*
b.
Maintenance Work
1,
Special Exercises from Text _
_
2*
Special Exercises from Board
3,
Other Materials __________________
Remedial Work
1*
Special Exercises from Text _ _ _
2m
Special Exercises from Board _ _ _
3,
Remedial Sheets
4.
Other Materials __________________
Individual Woik t
a*
Special Exercises
b*
Remedial Sheets
c*
Pupils Receiving Help ____________
d.
Special notationsi
(As per record)
23
with each of these twelve subjects were made on the same
day that they were brought to the laboratory.
effort was made to prevent any review.
A special
In several in­
stances the cooperation of the parents was enlisted in
this regard.
The writer has every reason to believe that
the responses of these subjects represent the status of
their Mlong division knowledge” after three months of
vacation*
The discussion thus far has pertained solely
to the sixteen subjects used for special study.
During
the entire investigation, however, an attempt was made to
secure a complete record of the learning environment not
only of the sixteen subjects but also of the remaining
twenty-three members of the class.
The form opposite this
page made It possible to get a detailed account of what
transpired each day In the classroom#
The supervisory technique used In the Dongfellow school {5 ) further Insured specific Information
pertaining to the class? for example, class weaknesses,
individual weaknesses, and the specific remedial materials
used to overcome them.
Sample sets of written papers were collected
from the class several times during the semester*
These
papers included c M l d r e n fs reactions to problem scales,
self-testing drills, and regularly assigned examples and
problems.
24
In order to get a more complete basis for com­
parison, the twenty-three pupils not included In the
experiment proper were given the exact examples and prob­
lems of each project*
This was done In each case Im­
mediately after the sixteen subjects had been cared for
individually#
These examples and problems were presented
in mimeographed form, and in such a manner as to present
but one at a time#
The same signals used in the laboratory
obtained in the classroom#
2b
CHAPTER III
AMOUNT AND NATURE OF THE DATA SECURED
The investigation made available a vast amount
of data*
The actual written responses to all the examples
and problems presented were secured.
Thus written solu­
tions to 448 long division examples, 128 verbal problems
involving long division, and 96 verbal problems involving
addition* subtraction, and multiplication were obtained#
The following solutions taken from Project V are illustra­
tive reproductions of the childrenfs work*
They are re­
sponses to long division examples of type D, examples
with three-figure divisors, which were review situations
fn
P r n 1
V -
a
26
3
c
J .
S
3
*-
3
^
27
3 3
~ v-
D
m
Or
&/
28
While such solutions were being written, moving
picture records and galvanometer records were simultan­
eously and synchronously being obtained.
The moving pic­
ture camera was operated at a speed of eight frames per
second, making possible an analysis as fin© as 1/8 second
if desired.
Pictures were limited to the written responses
to the long division examples.
Some 7,000 feet of such
film was thus made available for study.
The Iowa Oral Language Machine yielded dicta­
phone records of verbal reports.
Such records were made
for all verbal responses except those to the pre- and
post-series of subtraction examples.
By means of the
metronome, it was possible to include simultaneously 5second intervals on the dictaphone records.
The accompanying electrodermal responses were
photographically obtained with 5 -second intervals auto­
matically recorded as vertical lines#
Electro dermal rec­
ords were made on sensitised paper, 3 5/8 inches wide, in­
dicating the subjects level of resistance in ohmage units
and the number and amplitude of the deflections.
Such rec­
ords were obtained for the entire experiment Including
both the pre- and post-series of subtraction examples.
The following pages indicate electrodermal, dicta­
phone, and film records for the three subjects whose written
responses were reproduced earlier in this chapter.
The
records ere presented in the form usable for analysis and
are self-explanatory.
.
„
(Continued on page 39)
29
Figure 2
Electrodermal Response Record
(Reproduction)
I
II!
i i I *
\
V.
D.
D.T.
5-11
(Review)
)
!
I I
l
30
4 7
8 3 6 ) 3 9 l 9 2
5 3 4 4
5 8 5 2
5 8 5 2
V*
D*
5-11
(Review)
5
10
10
4
15
0
20
8
25
9
4 times 3 is 1 4 --- 4 times 8 i s -- —
30
7
---- 33---4 from 5 — *4 from
35
13
40
7
45
10
50
6
7 times 5 is 25 ---------
55
6
7 times 8 is 58 ------------ -
57
3
5 thousand (laughs) 9 hunderd
and 29 p;oes--- — ----- --
9 is 5 —
4 from 12 is 8— --- 4 from 9 is 5
--- bring down the 2----- 8 inta fifty
six goes 7 times — 7 times 6 is — -- 42. _
51
Film Analysis Record
4 7
8 3 © )~3 9 2 9 2
Steps
Frame s
1#
Finds and writes 4
169
2*
Multiplies 4x836, writes3344, draws line
110
3*
fudges
4,
Subtracts 3929-3344, writes
5#
Compares
6*
Brings down 2
20
585
11
6
7m Finds and writes 7
49
Qm Multiplies 7x836, writes6852, draws line
9*
10,
Judges
Subtracts 6852-5852, writes
11* Remaining film
45
134
3
0
3
17
32
Figure 2
Electrodermal Response Record
(Reproduction)
l
D
'w'
yA
'Ay
a
V.
D.
S.J,
5-12
(Review)
33
8
V,
D.
S*J.
5-12
Seconds
Words
3
6 7)
3 9 2
3 3 4
& 8
5 8
4
9
4
5
5
7
2
2
2
0
(Review)
Verbal Report
-------- *...0 hundred and 36 into 3 thousand
9 hundred and 29
5
13
10
1
15
8
20
9
25
10
30
9
4 times 8 is 32 and 1 — —
55
2
40
14
45
15
50
17
t h r e e -- ^iudse
3 thousand 3 hundred and 44 is sm H e r than
—
3 thousand 9 hunderd
and 29 - so you draw a line and subtract—
4 from 9 is 5
-you can*t take 4 from 2 so you borrow 1 fror
-the 9 and make that a
55
0
60
6
65
0
70
6
--- 8 —
75
7
An
15
85
13
is 5 — jjuda:© — compare 5 hundred n
85 la smaller than 8 hunred 36 so you bring
down your 2
8 hundred and 36 goes into 5 thousand — 8
hunderd and fifty-
90
7
95
13
100
7
105
16
109
1
— g:oes---- ----------
L 24 -tv.4 times —
carry your 2 —
4 times
— 3 Is 12 a n d ---- 2 is 14 - carry vour 1
42 —
3 is
thirty — -
3 f r o m ---- 3 from 8
write the 2 and carry the 4 —
7 time
--- 21 and — - 21 and 4
is 25 — carry your 2 — 7 times 8 is 56
and 2 is fifty—
— eight--
-
— /
(all-right, ston.)
34
Film Analysis Record
4 7
8 3 6 )“ 3 9 6 9 2
_____ Steps_______________________________
Frames
1*
Finds and writes 4
116
2*
Multiplies 4x836, writes3344, draws line
292
3.
Judges
4.
Subtracts 3929-3344, writes
5*
CoWare s
61
6*
Brings down 2
11
13
585
278
7* Finds and writes 7
94
8*
70
Multiplies 7x836, writes5852, draws line
9« Judges
10*
Subtracts 5852-5852, writes
11* Remaining film
7
0
6
20
35
Figure 4
Electrodermal Response Record
(Reproduction)
iD
i
’i
V.
D.
B. Sin,
5-12
(Review)
36
%s
8 3*6
D.
446
) 3 9 292
3- j k 4
o 0 fe 2
B.Snw
5 0 16
5-12
Seconds
036
Words
Verbal
_______7____ ~— — ah
10
36
*536
7
15
(Review)
Report
8 hunderd and 56 divided by -
-»-»lnto — a~ 3 - - 3
thousand 1 *—
million
9 thousand
20
11
25
14
39 thousand 2 hunderd 92 — — .--- 85 wonft go
won#t go Into 39 so you will have to go Into
3 hunderd
___________
and 92 -—
4 times
35
6
*■—4 times 6 Is
ML
3.
4 ,..tlms.-3 -1
45
11
50
6
2 la 1 3
——a.
24
.12. end,
put down the 5
4- times 8 is
4 times 8 1.3 —~ a
55
32
60
and 1 is 35 — oh that*s — it's wrong:
{Whatfs wrong with It?) % have to -- {All
rl&hta do It and do It the — )___________ _
65
70
JZ5l
80
0
85
90
0
95
9
100
4 from 9 Is 5 — - 2 end 2 1B
gero (louder) 4— from 9 Is 5
37
*3
4
83*6 ) 3 9 2
S g*y
5 0
5 0
0
V.
D#
B*Sm.
4 6
92
4
5 2
16
3 6
36
836
(Review)
13
110
7
115
11
120
0
125
7
130
9
--six - put down your 6 and carry your 3
135
0
i
i
r
— and 8 hunderd and 36 won’t go Inta — -5
.hunderd and 5
105
140
5
145
7
a— 48 and —
150
5
------ 6 from 12 Is 6
155
5
2 from 5 Is 3
160
3
36 remainder-- -
165
0
167
0
?
i
ia
i
i
i
i
11
bring down your 2 — • ah
8
—
— -36 goes Into 5 thousand----- and - 5
hunderd 52
2 Is 50
38
Film Analysis Record
8 5 6 )
4 4 6
3 9 2 9 2
** *3 A* ,,
Steps__________
56
836
i
_____
Frames
1*
Finds and writes 4
312
2*
Multiplies 4x856* writes 344
302
3.
Judges and sees mistake
82
4*
Grosses 4 and writes 4
17
5*
Crosses out 344 and writes 3324
6*
Judges
20
7#
Subtracts 3929-3424* writes 505
56
8.
Compares
65
9.
Brings down 2
166
4
10*
Finds and writes 6
11*
Multiplies 6x836, writes 5016* draws line
12*
Judges
20
13*
Subtracts 5052-5016, draws line, writes 036
41
14.
Writes 36 In answer
836
Remaining film
84
15*
57
216
6
39
Certain observational reports were made by
the writer to supplement the preceding objective records.
These reports include qualitative statements concerning
attitudes and changes in attitudes, qualitative statements
concerning behavior and changes in behavior, and special
notations of points at which the stimulus key of the
psychogalvanometer was pressed.
In addition to such records as pertained solely
to the sixteen subjects were those which concerned the
entire class.
The class supervisory chart, an aspect of
the supervisory plan used in the school, gave a rather
complete picture of the learning status of the class
with specific reference to class weaknesses and remedial
materials used.
The accompanying chart indicates the
learning situation during the first six weeks of the
semester.
An explanatory chart is Included on page 62.
A special Teacher*s Bally Becord Sheet was used
as an aid to a more detailed picture of the learning en­
vironment.
typical.
The following record as of March 27, 1936, Is
The symbols (4c-6, 3e-5, etc.) refer to special
remedial sheets which had been given to the class to
meet specific weaknesses*
This feature Is part of the
supervisory plan employed.
The written reaction to the examples and prob­
lems of each of the projects was also obtained for the
twenty-eight children not included In the experiment
P.I ASS
4 A
PI ASS
DATE
Qiun/. A 7
INVENTORY OF LEARN 1NQ
W EAK. S K ILL S O F CLASS
DETECTED BY CLASS BECOBD
B A T IN G
m a in t e n a n c e
PROCESS IN ARITHMETIC
DBILLS
I2FMFDI Al
CHART NO./
BE MEDIAL
DIAGNOSTIC SCHEDULE
/o -/4-3
4
.A
/}.c - s'
//- /£/
/£-/&>?> Z3-/79 /Z-//T
4
/s- jfC,
/ — G»
hl-L
LOW PUPILS
W EAK SKILLS REMEDIAL
LOW PUPILS
WEAK SKILLS
REME DI AL
T O P IC
S T U D IE D
C LA S S
PPOgPESS
SELF - TESTING}ft 4 /o U
9
9
4,
8
P ^ L /0j~-//3
CO
er
/\
/\
/
——<
V
z
7
6>
t
I
_l
4
4
cv
o
3
3
\
_J
?
a
1
1
R E C O N S T R U C TIO N
s
£HgONlC-LOW-CASES
AV RATE
WOODY-McCALL
AV. IRATE WOODY McCALL
h<
5
5
7?Kcl4sU.°c
14- /S Ji /7 /y /? .*•>
10
7
!<£-(*
D RILLS
10
8
/£S-
11. n
C H A RT
PREVENTION
41
Teacher1a Record
of
Dally Class Work
Date.
1.
2
.
-
T
Dength of Period
Articles Presented:
a.
Review Material^
b*
New Material
c*
Practice Materiel
d.
Self-testing Drill
Written Work by Class:
a*
b«
3.
^ 7
Maintenance Work
1. Special Exercises from Text
2m
Special Exercises from Board,
3*
Other Materials
____
__
Remedial Work
lm Special Exercises from Text
2.
Special Exercises from Boards
3*
Remedial Sheets
4*
Other Materials
c - ^____ S c.- $
___
_________
Individual Work 5
a.
Special Exerci ses.
b.
Remedial Sie ets
c*
Pupils Receiving Help
-<% A >• <• ■'
Jl e— 6.
Special Notations!
__________ (As per record)
, £ <±-C ;______
J.c-C ;
2 C.-L'
.
42
proper.
Special mimeographed forms were used.
These
©xasples and problems were presented In such © way as
to present but one at a time.
Specific time limits
obtained In order that meaningful comparisons might be
made with the work of the sixteen subjects.
The two
forms following this page were used for the fifth pro­
ject.
Complete record^ of the ability and achievement
of
the entire class were also obtained.
These records
include scores made on the Kuhlm&n-Anderson Intelligence
Test, Cosqpasa Survey Test, Woody McCall Test, Gates Bead­
ing Test, and the Iowa Elementary Language Test, Table 3,
pages 45 and 46, gives this Information.
Name
7.'S 9'
8 3 6 ) 3 9 S 9'2
D
*
^
7
7 7
?-
i
&
7 M
3 8 ) 2 7 8 9 2
/
5
7
,
44
1.
In 1933, Betti© and Fordney went up 61,237 feet
In their balloon. Stevens went up 60,613 feet in
a balloon In 1934. How much short of the first
record was Stevens1 record?
7
/J
2.
A music store paid $1450 for 29 radios. If all the
radios were the same price, what did each cost?
/fiN
0 0
45
Table 5
4A
Children
« E-4
s
*
4) O
X) o
< ©
1 te
K *H
d ri
r1
*Hr©H
XJ -P
i§s
t>r1
H
oca
vi i
© rl
© 1
£Lp
i
o 0©3
©to
© CO
3 oa
tO
©
© 1
0
P,P
3 003
6
.O EH.
© t£> w to
*■> tO
rH f H 1
rH ^
rH O
0 C'l fidOJ
O 1 O 1
O rH o
m
t
1
>*
nOoPto
o©
o © p ®
.eh
..SBLfct. jp
- to
Gates Reading
Test - 1-17-56
TP---- --------- ---------- --- ----8 ©
83
0
(dates Heading
Test - 5-27-36
Test Results for Fourth Grade
.
.
ib£J
w
a <o £jtQ
© 60 aSCO
M « M 1
O
E>
*CO *N
$ i1
^ tI
©
H ©
O
r~i
rH
W 1 tt 1
0P 0P
|B (0 P «
o© W
O g©
-4
W &s
1.
E.A.
96
95
92
18
18
3.8
87
81
2.
J.B-
128
107
106
22
25 11.8 11.2
201
207
5.
V.B.
105
97
102
19
21
9.9 10 .3
146
171
4*
J.B.
85
94
101
19
16
4.4
5*4
96
114
&'
E.A.B.
113
103
100
19
24
4*5
4.8
96
10©
6.
Bl.B.
104
73
80
12
14
4.9
6.4
107
116
7*
M.B.
110
99
85
16
21
6.7
7.0
105
124
a.
A.C.
129
89
88
17
22
3.5
3.5
43
66
9.
F.D.
117
80
89
15
20
6.6
7.2
108
117
LO.
R .D*B«
107
18
22
3.4
3.9
63
90
LI.
S.F.
105 106
170*
147*3 98
96
20
23
5.7
5.2
142
160
12.
D.H*
109
102
100
18
22
7.6
8.6
170
167
13.
K.H.
103
82
94
15
19
5.2
6.9
131
139
14.
S.H.
120
15.
L.H.
115
104
96
20
19
3.1
7.8
133
178
16.
Al.B.
130* 105
112
22
26
8.0
9.7
172
m
17.
An.B.
114
98
108
19
23 10.1 10.1
188
176
18.
3.J.
105* 105
110
21
19
5.8
5.6
151
155
19.
M.J*
105
101
103
21
22
5.6
6.5
119
134
20.
B.K.
102
80
100
17
16
5.2
6.7
112
129
3.5
195
11.5
20
99
46
Test
In te llig e n c e
C h ild r e n
g u h lm a n -A n d e r s o n
T a b le
3
(C o n tin u e d )
l^<0
* JO
u
R 02
CO I
rH
w i
0
03
OS
to
TO |
TO
P iP
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w
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o en
...
co to
<*
HI 1
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to oa
q «
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|
P«-P
2
d ?p'
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M 01
© ©
O (0
o £~»... 1
- & - & I ..
0
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to
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to ca r
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HUS Hi O £H
p i
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to
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to
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♦ CQ
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ti)
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$ to
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Hi j*H
TO P
Jft TO
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Hi £H
135
137
03 Id
TO
2 1 .
B .K r *
115
92
22*
M .L .L ,
1 0 7 *
77
18
6.0
100
2 3 ,
M .M *
95
97
16
6 ,9
114
2 4 ,
R .M ,
113
100
101
18
24
5 ,8
6 .7
120
159
23*
R .A .H .
110
101
86
20
21
6 .3
7 .1
119
147
2 6 ,
I.O .
1 0 7 *
85
88
12
20
5 .5
6 .5
105
H O
2 7 ,
W .O .
109
99
106
20
23
:.0 .5
1 0 ,4
150
12 2
2 8 ,
C .P .
117
103
106
22
21
7 .0
7 .5
149
162
29*
B .R .
106
90
95
16
17
6.0
6 .5
116
137
3 0 ,
B .S c h .
108
107
108
22
22
5 .6
6*2
116
94
3 1 .
G .S *
106
69
89
16
25
4 .1
5 .5
67
69
32*
B .S m .
92
74
14
20
4 .2
5 .0
71
88
3 3 .
H .T .
112
91
20
21
4 .9
4 .6
108
117
3 4 ,
D .T .
114
3 5 ,
G .V .
83
72
72
15
16
4 .2
5 .4
82
110
3 6 .
B .W .
102
97
107
20
19
5 .2
5 .4
129
140
3 7 .
M .W .
98
46
53
6
7
6 ,7
7 ,9
109
116
3 8 .
R .W .
104
107
109
21
18
8 .3
8,6
142
184
3 9 .
S .X .
108
100
114
21
23
6 *3
7 .2
116
125
C la s s
M e d ia n s
108
97
100
19
21
6.0
6 .7
116
134
102
109
14
18
4 .5
5 .0
S ta n d a rd s
*
I.Q .
as
d e r iv e d
92
93
21
108
fro m
22
7 .7
155
8 .9
23
B in e t* * S lm o n
10.2
In te llig e n c e
85
T e s t
47
CHAPTER XV
ARALY3XS A m INTERPRETATION OF DATA
1*
Preliminary Statements
As indicated in the previous chapter, this
investigation yielded a vast amount of objective
and
subjective data pertaining to the learning process in­
volved In long division as It progressed from one stage
to another*
The writer has, throughout the study, been
desirous of getting occasional glimpses of the inner
mental processes, In so far as these are exhibited be­
havior! stlcally.
Hence, in the treatment of the accu­
mulated data, an attempt will be made wherever possible
and whenever warranted to enter into the marginal and
intimate areas of the learning process*
The writer Is fully cognizant, as he proceeds
In the treatment of these data, of the "toughness” and
the 11stubbornness” of this problem, of the resistance
offered any attempt to penetrate the Inner recesses of
the mind or to follow thought processes enroute to a goal
or end-product*
He Is cognizant also of the bald fact
the educational research has been concerned with end
products largely because of the comfortableness of work­
ing with them*
T
His boldness In attempting to penetrate
this area is accounted for by the growing conviction that
what happens during learning has tremendous educational
48
significance*
It was tills conviction that prompted the
writer to use the motion picture camera, the Iowa Oral
Language Machine, and. the psychogalvanometer as means
of getting details of what actually does happen during
the learning process*
2.
Scientific Status and Interpretation
of Blectrodermal Response Records
The records furnished by the moving picture
camera and the Iowa Oral Language Machine are of such a
nature that their Interpretation can be made with consid­
erable confidence*
This confidence, however, does not
accompany interpretation of Electroderami Response Rec­
ords*
In fact, there la still considerable question in
the minds of some psychologists as to Just what the
psychogalvanometer does measure*
For this reason a
word concerning the scientific status and Interpreta­
tion of Electrodermal Response records Is in order*
Present interpretation of the electroderm&l
response by a majority of psychologists stresses "the
emotional, or at least the affective, aspect of the re­
sponse." (21, p* 348)
The writer can do no better than
refer to some of the pertinent quotations from prominent
psychol©glsts *
If, now, the subject Is given stimula­
tion calculated to excite emotional reaction,
the bodily resistance is found to decrease;
the excursions of the galvanometer are great­
er, and It is assumed that the amount of in­
crease measures the amount of the emotional
49
excitement aroused. . . The galvanometer
technique has been found particularly use­
ful in determining the emotion-arousing
value® of different stimulus words given to
the subject in the word association test (q.v.,
Infra) $ some having claimed that it is the
most delicate index employed in that experi­
ment. And in this connection there are those
who have held that it is serviceable in bring­
ing to light Individual differences in emo­
tional stability. (6, pp. 220-221)
Much studied also has been the change
in the electrical resistance of the skin
that occurs during an affective state. A
man has applied to his skin electrodes of a
character to avoid themselves exciting elec­
tric currents by chemical reaction. A faint
current Is then Introduced and measured by
the galvanometer. Xt la found that if a
pleasant or unpleasant stimulus is given to
the man, the resistance of the skin is mark­
edly decreased. ( 1 7 / p. 272)
Bven in momentary thrills of fear, sur­
prise, embarrassment, or expectancy, the
sweat glands are stimulated to a slight
degree by their nerves, and the result is
a momentary change in the electrical condi­
tion of the skin, which can be registered
by a galvanometer and is called the psycho­
galvanic reflex. This electrical change is
perhaps the most delicate Indicator of ©mo­
tion that the laboratory has yet produced .(23,p.34^
The great majority of experimental sub­
jects manifest these sudden and definite
changes in skin resistance during emotion.
Whether these changes are specific to emo­
tion as such, or whether they are the ex­
pression of a stirred-up bodily state of
which emotion is only one aspect, is a de­
bated question, the answer to which is tied
up with the problem of exact definition of
what emotion is. It has been urged that
striving or conation rather than emotion
strictly speaking Is the basis for the
psychogalvanic reflex. It has also been
argued that bodily upset, without any
60
particular ©motional or conativ© aspects,
will produo© the same results. However,
this may be, It remains true that for most
persons In most reasonably well-controlled
laboratory experiments, emotions may be
roughly "measured" by this method.(15,pp.100-102)
From the work done at the University of Iowa
laboratory and from the investigations of many others,
a tentative general statement can be made that the E.
D. R. fais subject to the following classes of conditions;
(1) primarily physiological changes and functional pro­
cesses in the body, Ilk© changes in digestion, may ef­
fect the electrodermal response; (2) closely correlated
with this possibility Is the probability that widespread
mental and bodily exhilaration, and alertness may account
for many changes; (3) then, also Intimately associated,
states of attention, states of high and low degrees of
mental concentration on and application to problems re­
quiring varying amounts of exertion and effort may cause
changes In the electrodermal response; (4) not far re­
moved, but often also entirely dissociated from these
phenomena, electrodermal variations may occur with motor
adjustments, with skilled and unskilled movements, with
voluntary and Involuntary reactions and, In short, with
many mental phenomena called in British psychology, the
orectic, appetitive, and conative processes, or with
whet Avellng refers to as the dynamic aspect, ”the *setf
of the conscious organism towards action,” also with
51
definitely volitional acts that are traditionally re­
lated to the Jamesonian 'flat* of the self j and (5) the
electrodermal response apparently Is Influenced by the
feelings and emotions and all other evolutionary grada­
tions of the affective life.” (21, p. 263)
It is quite obvious that the above categories
are not Isolated, distinct, and Independent of one an­
other*
On the contrary, no one of the above changes
can occur without concurrently Involving elements of
the others*
In summarizing, I>r» Ruckmick states wthat
while the
electrodermal response Is linked along with
a long array of physiological and mental conditions
and processes, It shows a characteristic end pronounced
effect In relation to affective processes, especially
of the emotional sort•’(21,p.370) Further, ” • * * we can
say that, with few contradictions, there is a strong
tendency to correlate the degree of deflection with the
Intensity of the affective experience*” (21, p. 371)
3*
Electrodermal Response Records of
this Study and Their Significance
While It can be assumed with considerable
confidence that affective experiences do cause varia­
tions In eleotrodorraal responses, there Is no assurance
that stimulation caused by solving arithmetical examples
and problems Is sufficient to aroiise deflections to
52
which significance can be attached.
As a first approach, the writer mad© careful
examination of each subjects verbal reports and written
responses.
In each instance where failure occurred the
exact time Interval was noted,
Errors In computation,
pauses, hesitations, repetition of numbers and words,
unusual ejaculations, and expressed failure were con­
sidered evidences of failure.
In addition to recording
the time Interval during which each failure occurred,
notation of awareness or unawareness of failure, was
also made,
Pai ses, hesitating attacks, repetition of
numbers and words, unusual ejaculations, and expressed
failure were regarded as evidences of awareness; the
absence of these as evidences of unawareness.
Random
successes were similarly listed for each subject.
These
points of failures and successes were then numbered and
corresponding numbers assigned to these points as located
on the eleotrodeimal response record.
Following the
numbering of failures and successes on the electrodermal
response records for the seven projects of each subject,
the writer proceeded to Indicate for each numbered in­
terval on the electrodermal response record whether or
not the deflections were significant.
Significance was
based upon the number and extent of the deflections for
the given interval in comparison with the general electro­
dermal pattern for the Individual with particular reference
53
to th© record for th© pre~ and post-series of examples,
The following table gives In summary form the results
of this work*
54
Table 4
Immediate Effect of Failure and Success
Upon Electrodermal Response Records
Subd.
Failure without
Awareness
Cases SlK.Defl.
Failure with
Awareness
Gases SiR.Defl*
Random
Successes
Gases SlK.De
3
0
8
4
13
1
A1«H«
6
2
2
2
18
2
N.T.
9
0
5
2
11
3
K*W#
2
0
3
2
18
4
J.B.
5
1
4
3
20
1
S« J *
1
0
3
2
12
5
S.F.
5
0
5
2
11
2
B.W*
6
0
5
1
12
1
1* 0*
6
2
6
2
15
2
P.T.
2
1
2
0
11
0
R.N.
1
0
2
1
17
2
B.Sm.
12
5
12
8
10
4
B.Sc.
0
0
5
4
17
5
16
1
1
1
18
6
An* H*
5
0
3
1
15
2
K.H.
4
1
17
5
7
JL
13
81
% of cases
In which
significant
deflections
occurred ~~ 16 *0—
83
40
225
41
B.R.
2------
*
-----
eo
C.P*
•18.2
£
55
Th© results ©a tabulated Indicate (1) that arithmetical
reasoning and computation does give rise to fluctuating
electrodermal responses!
(2) that electrodermal responses
©re not th© same in number and extent in cases of failure
and success*
It is interesting to note that success and
fajlure unaccompanied by awareness give rise to approx­
imately the same per cent of significant deflections*
Significant deflections appear in 48*2$ of the cases of
failures accompanied by awareness in contrast to 16*0%
of the cases of failure without awareness*
The effect of failures and successes upon sub­
sequent deflections wes also studied.
Intervals of one
hundred seconds following previously determined failures
and successes were located, studied, and judged as to
whether or not significant deflections occurred*
results are herewith tabulated*
The
56
table 5
Effect of Failures end Successes
Upon Subsequent Deflections
Failures
Successes
sub 3*
Aware
S.F.
5
1
5
0
12
2
J*B.
5
2
7
2
16
3
Al.H.
5
1
6
2
17
1
B.E.
0
0
17
4
17
5
B.W.
5
3
2
0
17
7
An.H.
3
0
3
0
15
5
B.Sm.
13
11
14
13
8
5
R.H.
17
3
4
1
7
1
H.T.
6
0
11
2
14
1
B .Sc .
5
2
0
0
17
11
x.o.
6
0
5
4
14
1
R.N.
2
1
1
0
17
7
B.W.
5
0
6
2
11
1
C.P.
9
5
5
1
12
2
D.T.
2
1
2
0
11
0
S. J*
5
3
-A
0
12
8
87
33
87
31
217
60
Totals
SlR.Pefl. Unaware
37*9$
SipuDefl. Cases
55.6%
Sip;.]
27.
57
It appears from this table that deflections
during the one-hundred second Intervals following failures
are not significantly different from those following
successes.
The
electrodermal response records of the sub­
jects of this study were carefully studied.
When the
records of a given subject which had been obtained dur­
ing the seven proj©cts---ov@r a period of eight months--were placed one above the other, the writer was impressed
by the apparent fact that each subject has a specific pat­
tern of response which is different from that of others.
This is in agreement with findings of those who hold
that electrodermal response records Indicate individual
differences in emotional stability.
Two series of elec-
trodermal response record patterns are included, pages
58 and 59, to illustrate this difference in pattern.
Be­
cause of this apparent uniqueness of each Individual fs
record, interpretation of a given subjectfs record must
be made in the light of his particular electrodermal re­
sponse record pattern.
Interpretation of electrodermal response records
shall be made in the light of the above discussion of
the scientific status of electrodermal responses and the
specific findings in regard to electrodermal responses
of subjects working arithmetic examples and problems.
58
Figure 5
V.
R.W.
Electrodermal Pattern
(Subt• Example s)
(Verbal Problems]
v.p
L L 9^'
: i
: ; . i i i i j i i i i ! i i i : i ! i i : i i
L
I
i : i
ii
III.
R.W.
(Verbal Problems)
(Subtraction Example
Figure 6
III.
S. J(
59
Electrodermal Pattern
(Verbal Problems)
(Subtraction Examples)
v. p.
M
l
i
I
l
l
i
i
!
i
!
f
i
N
I I
I
V.
S. J.
___
(Verbal Problems)
(Subtraction Examples)
L
V.p
i i i i 1
L
1 1 1
9
i
i
!
a
r*
1
-/
'
-
X
-
-
-
-
-"M
\
^
X
1
s
00
4.
Arithmetical Status of Subjects
of this Study
Tli© fourth-grade children of this study have
progressed through the kindergarten and th© first three
and a half grades at Longfellow school*
During this
time th© group remained as one class with only such
changes as were caused by "drops" and "transfers".
Aside
from special teachers in art and music, one teacher has
had complete charge of the group during each semester.
(Mid-year promotions are th© rule
in th© public schools
of Iowa City.)
Arithmetic, the subject of particular interest
in this study, has been taught as a separate subject
with as much
correlation as the subject matter and in­
genuity of the teacher would permit.
The writer con­
stantly urged his teachers to cross subject-matter lines
whenever and wherever such procedure vitalized learning
or otherwise made teaching effective.
The Standard Service Arithmetics by KnightStudebaker-Kuch have been used in Iowa City public schools
for several years.
The fourth-grade children with which
this study is concerned have used these texts since enter­
ing the second grade.
The writer, as principal of Long­
fellow school, has been particularly interested in the
supervision of arithmetic.
His stpervisory plan, as
described In his M. A. thesis ( 5 ), has been effective
61
In grades 3 - 6
for five years,
The operation of this
plan has therefore directly affected these children for
three semesters.
The following supervisory chart indicates the
nature of the supervisory plan employed.
It is present­
ed here to help the reader interpret later charts which
will refer specifically to the children of this study,
A study of the chart will make it evident that
it gives in definite and concise form the status and
a perpetual summary of the "arithmetical learning” of a
given class from week to week.
The writer wishes to
use this means of summarizing the warithmetical learn­
ing11 of his subjects during their first semester in the
fourth grade.
He therefore includes Charts 2 and 5 for
thl s purpose.
Reference to the preceding explanatory
chart should answer questions pertaining to these charts.
A study of the January, 1956, test results
gives further indication of the arithmetical learning
status prior to the second semester*s work in the fourth
grade*
The results for the Compass Survey and Woody
McCall*s tests as given in Table 5, page 45, Indicates
that the
class medians are very near the standards as
given in the test manuals.
In one case the class median
Is slightly below and In the other slightly above.
Having ascertained with some degree of com­
pleteness the arithmetical learning of these children
CLASS
LA
CLASS
W E A K S K ILL S OF CLASS
DETECTED BY CLASS BECQ&D
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3
Ar
B A T IN G
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D B ILL5
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INVENTORY OF LEARNING PROCESS IN ARITHMETIC
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Legend for upper left square:
10 Indicates the drill number.
P.136 indicates the page number.
4 indicates the class rating.
5 indicates the standard rating.
REMEDIAL
G
3 c. - 2,
3d
4-A,
loAS
Legend for upper squares:
Symbols represent weak skills.
Bach square includes weak skill
for a particular week.
4
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LOW PUPILS
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The above gives the exact point of class
3 — progress. By referring to the Time w _bequence Chart it is possible to check
this progress with that set up for the
K_
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4
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WOODY-McCALL ACCELER.ATED-CASE5 ia AV. RATE WOODY McCALL
Children falling in this division
need special diagnostic tests if
their I.Q*s. warrant it. This mayresult in special work.
/3
Critical and preventive notes pertaining to the drill or to the^
text to aid the teacher when this particular drill comes during
______ ____ _____
the following semester or year.
JL
Children in this division need
special advanced work whenever j
—
possible. If they are advanced!,_
in their other subjects, they ;
are possible candidates for
double promotion.
.
*
^1
17
PREVENTION
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CHRONIC* LOW-CASES
iL
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The above names are those of the
2 lower or failing pupils. Their
W difficulties are diagnosed from
K the individual Class Record Sheet
These pupils must do this special
= remedial work besides the clas3
Remedial work.
31
u
The Compass Diagnostic
Test is administered to
/64.-3
.those pupils for whom
/7«, - 2^
/«£ a,
the self-testing drills
are not sufficient diag­
+ * -2^
Id-S
nostic instruments.
Remedial material is so designated
as to tie up with symbols for skills.
This space tells
at a glance if
(3 - 2) refers to remedial material
the class has
for skill 3 . The 2 indicates that
■made the expect-^
the material is second in difficulty,
ed progress
that it is the second remedial sheet
4
for that particular skill.
T O P IC
W
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Ad
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1
CLASS
4 &
DATE
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learning
process
W E A L S K ILL S OF CLASS
DETECTED BY CLASS SECOND
4G
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CLASS B A T IN G
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3
in arithmetic
t
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INVENTORY OF LEARNING PROCESS IN ARITHMETIC
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A
LO W P U P IL S
TESTING
D RILLS
PREVENTION
Q
65
prior to the times they entered the latter half of the
fourth grade , th© writer proposes to present first the
information drawn from all the projects and then treat
each of the projects separately*
5.
Effect of Failure and Success upon
Subsequent Work
Much space in educational literature is de­
voted to the effect of failure and success upon sub­
sequent work*
Success supposedly leads to further suc­
cess; failure to further failure.
The type of records
obtained in this study made it easily possible to dis­
tinguish between cases of failures and successes.
Verb­
alisations as obtained on dictaphone records clearly
indicated pauses, hesitating attacks, incorrect at­
tests, and egressions of defeat or victory,
These
indications of failures and successes In turn could be
substantiated by the written solutions, the motion pic­
ture film, and the electroderraal response records.
The failures and successes previously select­
ed when their effect upon electrodermal response deflec­
tions was studied were used*
Pauses, hesitating attacks,
repetition of numbers and/or words, unusual ©j&cuations,
and expressed failure had been recognized as evidences
of awareness of failure; their absence as evidences of
unawareness of failure.
The verbalization records and the written solu­
tions during a period of 100 seconds following feilur©
66
or success were used as bases for determining whether the
subsequent effect of failure and success was positive,
negative, or zero*
Increased confidence as ©pressed
or as indicated by a prompt, steady, and correct attack
was considered a positive effect; decreased confidence
as expressed or as Indicated by a hesitating, stumbling,
end incorrect attack was considered a negative effect;
and no apparent change in attack, a zero effect.
In order to get more objective evidence of
the effect of failure and success, the number of errors
that occurred during the 100 seconds following each case
of failure and success was determined.
The following
table Indicates the findings relative to the effects of
failures and successes upon subsequent work.
The symbol
M-n signifies a negative effect; rf0fl, a zero effect; and
,,+M, a positive effect.
Table 6
Effect of Failures and Successes
Upon Subsequent Work
Effect
0 +
Effect in
per cent
0
•+■
No.of
Errors
per
Errors Case
56
16 15 75,3 16.0 0.0*
156
1.8
Failure
Un­
aware
89
12
72
148
1.7
Suc­
cesses 221
20
16 185 2.9
0.0 93.4*
153
.6
# Median Percentages
5
o•
o
Failure
Aware
87
o
.
o
$
Situ­
ation Cases —
80.3
67
From this table It Is clear that the effect of
failure with awareness upon subsequent work Is very def­
inite whether in terms of negative effect as judged from
verbalizations or in terras of errors as listed for the 100
seconds following each case#
For each failure accompanied
by awareness, 1*8 errors occurred in the 100 second inter­
val that followed#
This Is three times ©a many errors
as occurred for each case of success.
The effect of successes upon subsequent work
Is likewise clear whether It is gauged in terms of pos­
itive effect ©s judged from verbalizations or in terms
of errors made*
For each instance of success, an aver­
age of *6 error per 100 second Interval followed In con­
trast to 1*8 error for each failure with awareness.
The effect upon subsequent work of feilure
without awareness approaches zero as judged from verbal­
izations but In terms of errors made following failure
this does not appear necessarily to be true.
The number
of errors following each case of failure without aware­
ness Is ©bout the same as in the case of failure with
awareness.
In order to Investigate this apparent dis­
crepancy, the writer referred to the original verbaliza­
tions and determined whether subsequent errors were ac­
companied by awareness or unawareness.
was taken.
One further step
For each error, the writer determined whether
or not It was due to a previous error.
Table 7
Effect of Failures with. Unawareness
Upon Subsequent Work
Conditlon
Wo* of Cases
Failure with
Unawareness
89
Wo* of Errors Following*
Aware
unaware
58
110
From this table it is clear that of the 148
errors following failure without awareness, 110 are er­
rors of which the subjects are not aware.
In other words,
the subjects contlnue making errors of which they are un­
aware*
Such errors cannot rightly be attributed to pre­
vious failure*
When errors due to previous errors are deducted
from the total number of errors following failure with
tanawareness, the following situation obtains?
Table 8
Effect of Failures with Unawareness
Upon Subsequent Work
{Errors due to previous errors excluded)
Gonditlon
Failure with
Unawareness
No* of Cases
No. of Errors
Aware
89
2
Followings
Unaware
75
This final table indicates more correctly per­
haps than the previous one that errors following failure
without awareness are chiefly of the unaware type and
therefor© It can b© said that the effect of failure with­
out awareness upon subsequent work approaches zero.
While obtaining the above information relative
to errors following failure without awareness, the writer
obtained the same information relative to errors follow­
ing successes and failures with awareness*
The following
table gives information relative to these two situations*
Table 9
Effect of Failures and Successes
Upon Subsequent Work
Condition
Ho* of Cases
Failure with
Awareness
Successes
Ho* and type of Errors Following:
Aware_______ Unaware_____
87
95
69
221
42
100
Table 10
Effect of Failures end Successes
Upon Subsequent Work
(Errors due to previous errors excluded)
Condition
Ho* of Cases
Failure with
Awareness
Successes
Ho* and type of Errors Following
_______Aware_______Unaware_____ _
87
20
52
221
16
64
It is interesting to note that while errors
following successes are much less frequent than errors
70
following instances of failure with awareness, i.e. 81
following 221 oases vs. 72 following 87 cases, approximate­
ly the same proportion of the errors in each instance
are of the aware type*
It appears that such students
are as sensitive to error as successful students hut
have not the power to avoid or do anything about them.
From the study of the effect of failure and
success upon subsequent work the following statements
can be made:
1.
Evidence of failure end success can be very
definitely obtained from the verbalizations
and the written solutions of subjects.
2.
Fail Tire with awareness has © negative effect
upon subsequent work as judged from verbal­
izations aisd from the number of errors that
follow.
S.
Success has a positive effect upon subsequent
work as judged from verbalizations or from
subsequent errors.
4*
Failure unaccompanied by awareness has very
little If any effect upon subsequent work.
The writer has chosen to treat Projects I, III,
VI, and VII.
Projects I, III, and VI should yield in­
formation at spaced intervals during the study.
Project
VII was carried on after the summer,s vacation and Is
Included because of the possible Information relative to
forgetting that may be found.
the following treatment*
Each project will be given
71
1#
Learning experiences prior to the project
2*
Selected case studies
3*
Summary treatment of sixteen subjects
4*
Summary treatment of entire class
5#
Hypothesis and tentative conclusions
6*
a*
Project 1
Learning Experience Prior to the Project
As the phvslclan diagnoses and interprets each
patient's condition In the light of such Information as
he Is able to gain concerning the patient1s past history
and experiences , so must the educational worker consider
his subject's condition at any time in the light of the
subject's previous learning experience*
Such an approach
as applied to the present study, however, calls for an
accurate, detailed account of what transpired arithmetic­
ally prior to each experimental setting*
By means of the
teacher's daily record, the textbook, the supervisory
charts, and the writer's observe ional reports, it was
possible to reconstruct the learning experiences of the
subjects of this study.
During the period of initial instruction in
long
division prior to the first project, the textbook
provided the following learning experiences pertaining
to long division:
72
1*
Article 98# A need Tor learning long division
was developed*
2#
Article 96* Long division steps were intro­
duced by using short division
examples*
3*
Article 97* Steps in long division were pre­
sented in detail, with an illus­
trative example.
4*
Article 98* The terms dividend, divisor* and
quotient were presented *
5*
Article 102, Instruction and practice in finding
and placing first quotient figures
waa given* Twenty-four examples
were provided.
6.
Article 103* Pour long division examples with
their solutions were presented to
emphasize the four steps in long
division*
7*
Article 106* One long division verbal problem
was included in Problem Scale
No. 7.
8*
Article 108. Special emphasis and practice on
multiplication in division was
given, by means of eighteen ex­
amples*
9.
Article 109* Twenty examples were given as
special practice in finding first
quotient figures.
10.
Article 110. Eight unfinished long division
examples were given.
11.
Article 111* Thirty preparatory examples were
givenJ for example, finding first
quotient figures. These examples
were portions of examples later
presented* This work was oral.
12.
Article 112* Eight long division examples of
type A were given.
13* Article
114, Sixteen unfinished and incorrectly
solved examples to be checked by
chilldr en were provided.
73
14*
Article 115,
An Illustrative long division
verbal problem was given, fol­
lowed by five such problems.
15.
Article 116
This article consisted of thirty
preparatory examples as of Art­
icle 111*
16.
Article 117.
Long division with carrying in­
volved in the multiplication was
presented, followed by twenty
such examples.
17.
Article 120*
This article consisted of thirty
preparatory examples as of Art­
icle 111.
18*
Article 121.
Twenty long division examples
and problems Involving harder
subtraction were given.
19.
Article 123.
This article consisted of twentyfive prep a? atory oral examples
as of Article 111.
While the text used by these children gives
unusually complete initial instruction in long division,
the teacher found It desirable to supplement the mater­
ial occasionally*
Following Article 97, which gave the
solution of an example as a means of teaching the four
steps, the instructor added another Illustrative example.
2 0 ) 8 8 0, emphasizing the steps in Its solution.
She
then presented three other examples! 4 0 ) 8 8 0,
3 0)960,
and 5 0 ) 6 7
50.
Again, following Article 103 In order that her
pupils be given an opportunity to attempt solutions to
wholly new examples and also to emphasize the four steps
in each solution, the teacher gave the following four
74
examples} 6 0 } 1 9 8 0,
3 0 I 6 9 S i),
5 0 ) 1 8 9 5 o
and 2 0 ) 8 8 4 0 .
Emphasis was again placed on the four steps In­
volved In
the solution of long division examples in
Article 110.
One example, 5 0
/ 1 7850,
not in the
text, was used to illustrate further the four steps.
Special supplementary instruction was also
given by the teacher In the matter of checking errors
in judging and con^arlng.
During the period of initial instruction in
long division, the
maintenance and remedial program was
generously provided for by the textbook and the super­
visory plan operative in the school*
The textbook pro­
vided maintenance material in the form of rapid oral
drills, self-testing drills, and problem scales*
1,
Article 99.
Bapid oral, mixed drill consist­
ing of fifteen examples involving
skills previously taught.
2*
Article 100* Self-testing Drill No. 11
5.
Article 104. Self-testing Drill No. 12
4.
Article 105. Rapid oral, mixed drill consist­
ing of twenty-five examples involv­
ing skills previously taught.
5.
Article 106* Problem Scale No* 7, consisting
of eleven problems involving
addition, subtraction, and
multiplication.
6.
Article 107* Rapid oral, mixed drill consist­
ing of forty examples.
7*
Article 113. Self-testing Drill No. 13
75
8*
Article
118* Self-testing Prill Ho* 14.
9*
Article
119* Problem Scale Ho. 8* consist­
ing of fourteen problems involv­
ing addition, subtract!cn, multi­
plication, and division*
10*
Article
122. Rapid oral, mixed drill consist­
ing of thirty examples*
In addition to this generous provision for
maintenance, the supervisory plan revealed certain def­
inite weak skills of the class as a whole and of the
slower pupils and thus made possible special remedial
work*
The following table indicates such class break­
downs end the remedial sheets which were provided.
Table 11
Class Remedial Program
Glass Weak Skills
Remedial Sheets
4c-6
4c
{Short division with remainders)
2c
(Subtraction of whole numbers,
three or more digits - $ & $)
5e
(Multiplication of whole numbers,
with zero difficulty)
3e-3
2f
(Subtraction of whole numbers,
involved In thought problems)
2f-4
5c
(Multiplication of whole numbers,
two digit multipliers)
3c—2
Id
(Addition of whole numbers,
three or more digit numbers)
ld-6
4c
(Short division with remainders)
4c-6
2c-speelal
I & i
The weak skills of the slower pupils, other
76
than the class weak skills, were similarly listed end
remedial materials provided.
The charts (pp .65 end
64) Include this information.
Thus It Is evident that the Instruction prior
to the first project was of an enlightened type.
The
textbook guaranteed a psychological, sequential treat­
ment leading children pedagogically to an understanding
of the first type of long division.
The teacher Ijitro-
duced such additional supplementary material as her
judgment Indicated,
A planned, defensible maintenance
program was provided and a tested and approved super­
visory plan employed to care for such unforseen break­
downs as occurred,
b.
Selected Case Studies
With such detailed information pertaining to
the learning experiences In long division prior to
Project I, it Is possible to view the responses of the
subjects more intelligently and arrive at a more accurate
analysis and Interpretation.
The table on the following page, derived from
Table 2, page 7, indicates the three subjects chosen
for Individual study and their relative rankings in the
experimental group and In the class as a whole.
Case 1: Al.H.
The first subject for consideration, Al.H.,
represents a superior fourth grade child.
77
Table 12
Test Results for Selected Cases
p
■p
03
0>
EH
tO
OJ
>I
0303
mI
09
a iH
&
u
o
o
Subjects
Al.R.
N.TB.Srn.
Exp.GroujpT
Medians
Class
Medians
Standards
P
03
©
EH
to
H»fcO
©
> i
§ io
K>04
0
3I
*n
OjtO
3
o
*0
o
gos
CO
a i
*o i>
03fH
©
3* I
©
43
B
as
*
©
eh <X> fctOtQ
&0to © 60
d i t-^l I
*Q
*0
t>
cdcvj g 03
o
K I fH I
03 H
03 LO
43
as
©
0)
O
P
©
p
of
o
OS
o
M
£
s
*o
§ 02
HI I
W l~O
M
26
8.0
9>7_ 172
199
20
21
4.9
4.6
108
117
14
20
4.2
6.0
71
88
106
20
21
6.0
6.7
151
151
97
100
19
21
6.0
6.5
116
134
102
109
14
18
4.5
105
112
112
91
93
92
74
109
98
108
130
4-3
as
33
Eh
M *fj
«3 tO
«•»
H 1 H I
H
H
aJ
O
o
■8
^ _I 2S I
pa
©
B to
85
His superiority over the medians of the sixteen students
as well as that of the entire class Is evident in Table
12 .
.
This subject was brought to the laboratory,
February 26, 1956, together with two others at approximate­
ly three o'clock*
Although it was 5:15 before we could
proceed with Al.H., he was eager to begin.
The suggestion
was made that he might return the next day, but this did
not meet with his approval!
His Individual data record
indicates that he was In good health, calm, interested
In the project, and confident of his abilities.
The original records obtained for this subject
have been set up In the special record forms to facilitate
Interpretation.
These special record forms are inter­
spersed In the discussion that follows.
It Is evident from the verbal report and film
analysis records that the learning experiences prior to
this project had been adequate for this subject.
He pro­
ceeded in a deliberate manner, Identifying each step un­
hesitatingly as he proceeded and responding with confidence
in the correctness of his work*
The dictaphone record in­
dicated that very often this subject audibly expressed his
mental checking and affirmation of a correct mental de­
cision by the expression "uhu".
In A-l, for examples
”— -uhu-- 32 into 1 hundred and 28 goes 4 times"; and
again, in A-2, "—
uhu—
you can subtract, so you subtract,"
The dictaphone re<&ord further indicated that the subject
judged;
—
tho 64 is smaller than the 76, so you subtract.
Els comparison Is stated thus:
(Continued on page 85)
79
Verbal Report Record
g
3 2 T 7 6
6 4
X.
12
A-l
12
Al.H*
2-26
Seconds Words
A-lt
a
8
4
8
6 8
8 3 ] 5 5 9 5
4 9 8
0 4 15
4 15
8
8
I*
A-2
Al.H,
2**26
Verbal Report
10
8
15
11
are 6 -— — ----the 64 Is smaller then the 7
20
8
25
14
30
14
35
12
so you subtract --- — 4 from 6 equal 2 ---— ----- § from 7 equal 1 ---- since the 12
Is smaller than the 32
you bring down the 8 — -uhu -— 32 into 1
hunderd an 28
goes 4 times — — — 4 times 2 are 8 ----4 times 3 are
37
1
12
A-2:
5
7
--- 83-- uhu-~ Into 5 hunderd and
10
11
39 goes -— 6 times — 6 times 3 are 18--- —
15
10
20
7
25
10
times 8 are 48-— and 1 are 49 — ----- -- unun — you can subtract, so you
subtract ------— 8 from 9 equal 1 — ---------- 13 from
9 equal 4 -----— — aught - - --- 5 from 5 equal aught
30
6
35
9
40
12
45
11
47
7
49
0
— 2 times 2 are 4 ---- 2 times 3
(all right, stop)
83 into 41 won*t go — — so you
bring down the 5 — - uhuh - 83 into 4
hunderd and 15
goes —
5 times ------5 times 3 are 15
-----5 times 8
are 40 and 1 are 41
80
Verbal Report Record
2 3
X«
B
Al.H.
2-26
Seconds
5
8
10
10
15
10
20
11
25
9
Verbal Report
— -23 Into 73 g o e s ------ three— ee
times — :
— . ...
-- 3 times 3 are 9 ---------3 times 2
are 6 —
-- you can subtract, so you doo-- 13 from
9 eauals
4 — -- -— -- 7 from 7 equals a u g h t --you bring down your 9
— -----uhuh ----- 23 into 49 goes
2 times ---
30
5
—
35
7
3 times 2 are 4 ---- ------u h u h -- y o u
40
9
can subtract, so you do------- 6
44
6
3 —
■
Words
2 times 3 are 6 ----
4 from 4 equal aught
from
9 aq
81
Film Analysis Record
Steps
_________________
Frames
1*
Finds and writes2
36
2*
Multiplies 2x32, writes64, draws line
54
3*
Judges
55
4*
Subtracts 76-64, writes
5*
Compare
22
6.
Bring down 8
14
7*
Finds and writes 4
44
8*
Multiplies 4x32, writes 128, draws line
32
9*
Remaining film
12
19
7
I. A-2
1.
Finds and writes 6
57
2»
Multiplies 6x83, writes 498, draws line
72
3.
Judges
52
4.
Subtracts 539-490, writes 041
80
5.
Compares
40
6*
Brings down 5
23
7*
Finds and writes 5
55
8#
Multiplies 5x83, writes 415, draws line
68
9*
Remaining film
13
82
Film Analysis Record
3 2
2 3 ) 7 3 $
I.
6JI
B*
Al* H*
2-26
0 4 9
4 6
3
Steps
Frames
1*
Finds and writes3
55
2*
Multiplies 3x23, writes 69, draws line
39
3,
Judges
22
4#
Subtracts 73-69, writes 04
52
5.
Compares
12
6*
Brings down 9
11
7*
Finds and writes2
48
8*
Multiplies 2x23, writes 46, draws line
54
9*
Judges
67
10. Subtracts 49-46, writes 03
22
II. Remaining film
16
£.L
84
Verbal Report Record
1*
John had 128 marbles.
His father gave him 14 more#
How many marbles did he then have?
2.
A canning factory packed 744 cans of corn into boxes.
Bach box held 24 cans.
How many boxes were needed for
the a© cans?
Seconds Words
Verbal Report
Ijl
---- John had a hunderd and 26 marbles. His
8
©r gave him 14 more — - how many marbles did
1.1
...
....
. ha
have
-■--- 1 hunderd and 2 6 ----- and you add to
is
11
it 14
-
20
___5
25
lp
5p
5
88
14
2.
11
AR
4 and 6 nr»R in
1 and 1 ar*e 2 *»*.«••■ «nd 2 ar^ 4 —
-- uhuh
1 and r*F.iTicrVit «r»«i 1 —-—
A canning factory packs 7 hunderd and 44 cans
Of a a w w In-kri
Bach box held 24 cans • how many boxes were
7
a
—
-—
55
2
AO
A
65
io
70
12
75
10
an
1T
85
8
n an
gv 1
1&
4
17
J-3 f'A'W+'tr-.’
pA-siyi Ktf ^TVffiinfiTPi W
5 tiwiAn -.-•,•••.5 times 4 is 12 *»- en'l 1 to
carry -— ~~ 3 times 2 are 6 and 1 are 7
— ------------ ,V2.......... .
Is smaller than 74 so you subtract *-------2
_
4 equal 2 --— — -- the 2 is smaller than
.♦■Via 0A
---------—
SO w© bring down the 4- uh-uh
-— 24 into 24 goes *■*— once — —
)»' ...P4
(all T»1ffhf.. sf.nr\i
_
1 times 24
85
"since th© 12 Is smaller
the 8 *M
than the 32 you bring down
Similar statements occur in examples A-2 and
B.
The multiplication and subtraction skills
have been so mastered end maintained that they functioned
even though burled in the more complex mental pattern*
At no time did these connections fail while the subject
was dealing with th© larger mental function*
The depend­
ability and smooth functioning of these earlier bonds
are quite clearly Illustrated in the verbal profile
charts *
The method of subtraction used by this subject
was interesting*
It was neither the ‘’additive" method
which he had been taught nor the "take-away" to which
he inadvertently had been exposed from time to time.
Instead, a "hybrid” method had evolved which incorporat­
ed parts of each method and which, aside from making
sense, was effective*
The first subtraction example,
Involved in the solution of A-2, illustrates this method*
5 3 9
In this example, 4 9 9 , "--- 8 from 9 equal 1 -----*-- —
13 from 9 equal 4
aught*"
---- - aught -— —
5 from 5 equal
The expression "13 from 9 equal 4" is an added
interesting feature of this example - a case of mind and
voice not functioning together 1
This subject attacked the verbal problems with
the same confidence as in the case of the long division
86
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87
Maniples•
By the time he had read the problem, the
process to be used In Its solution was determined;
hence little or no delay occurred.
hawing read the second problemi
For example, after
w
------ uh-hu
ao you divide.*1
The electrodermal response record of this
subject, Al.H., gives a general picture of affective
stability.
There are neither a large number of deflec­
tions nor sharp deflections of great magnitude.
The
record substantiates the other evidence of the mental
and emotional poise possessed by this subject as he
proceeded to solve the experimental problems.
c.
Summary of Case Study Findings
Case 1 *
A l.
H.
(A Pupil of Superior Ability)
1.
The Instruction prior to Project X results in
the formation of new bonds which function readily and
accurately.
£.
The maintenance end remedial program keep old
bonds at a functional level though the skills Involved
may be buried in more complex patterns.
3.
The learning environment prior to Project I
makes possible dependable, smooth functioning of new
88
bonds and of old bonds in new patterns
4#
The learning environment prior to Project I
makes possible the solution of examples which include new,
untaught aspects of long division*
&•
Little or no time beyond that of reading the
verbal problem is required by the subject to decide the
method that shall be used in its solution.
6#
The electrodermal response record is in agree­
ment with other records in indicating relative affect­
ive stability#
7#
Inadvertent exposure to two methods of sub­
traction results in an equally effective "hybrid"
method#
8*
Verbal responses give cues as to "judging”
and "comparing", mental disturbances, and other aspects
of the learning process which otherwise would not
appear#
9#
Verbal profile charts tend to Indicate uni­
formity of performance.
89
c&86 2# nr# x*
Table 15
Test Results for Selected Case
43
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1-21-36
|Compass Survey Test
1
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Gates Heading Test
1-17-36
[Gates Reading Test
5-27-36
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Intelligence Test
N. T.
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Class
Medians
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112
91
93
20
21
4*9
4*6
108
117
109
98
106
20
21
6.0
6.7
131
151
108
97
10 0
19
21
6*0
6.5
116
134
109
14
18
4.5
5.0
Standards
102
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This subject represents individuals of median
ability in arithmetic as found in the experimental group
and the class#
He is definitely below the median, how­
ever, in reading and language ability*
During the period prior to this project no
special individual help was given to this pupil.
The
regular classroom instruction appeared to be sufficient.
He was absent on the morning of February 10, but took it
upon himself to work the examples of the assignment that
same day#
90
Th© subject was brought to the laboratory
at approximately four o*clock on February 26, together
with two others.
project.
Great Interest was shown In the
He was anxious to know when he might see the
pictures and whon "mother11 might see them!
He was not
the least worried about whether or not he would be able
to work the examples correctly.
The verbal report and film analysis records
and the verbal profile cherts are herewith presented,
pp. 91-98.
This subject proceeded with more confidence
than caution and made an error immediately when attempt­
ing to find the first quotient figure*
He failed to
judge prior to his first subtraction? and also failed to
compare following it, as evidenced by his verbal report.
This error in finding the first quotient figure, there­
fore, was unnoticed as was also his second error when he
attempted to find the second quotient figure.
for these error® can only be conjectured.
The cause
The subject’s
test results, daily work, and subsequent computations
indicate that he possessed a mastery of the division
facts*
Two possibilities seem probable, either a tem­
porary lapse in ability occurred, due to the complex
pattern involving the division example and the experimen­
tal set-up, or temporary confusion as to method, during
which time he divided the first digit of the dividend by
(Continued on page 99)
91
Verbal Report Record
3 2
I«
A~1
N,T,
2-26
Seconds
34
6 8
9 6
8 0 8
-3*?£ 8
6 8 0
)7
Words
Verbal
5
6
-—
10
6
—
15
4
20
5
25
5
30
4
35
6
40
O
45
6
50
4
55
5
60
7
65
3
70
10
75
5
74
n
Report
32 goes Into 76 ----3 times
2 threes are 6 — -
3 threes are 9 --------------
— ~ 9 from 17 is - 8-
rir*«w « line and. subtrad
*----- - 8 from 8 is nothin#----- 2 from
----- --bring down your 1 - — — 1---the 1
becomes a 2
-— —
2 from 8 is 6
(all right f atop)
92
Verbal Report Record
8 5 J 5 3
4 9
4
4
I*
A~2
N.O?.
6 5
9 5
8
16
16
2-20
5.....
6
---- 83 goes into 5 hunderd 39 -— ...------8 times— -
-.10 -
7
16
9
20
8
26
9
30
7
36
7
40
0
45
6
50
8
are 15 —— --- - 5 eights are 40 and 1
__ _ 52
3
is 41
54
0
„
.—
3 fl^Xea nre 1ft «•«.*».» «*r» yrvn ——
— ——f
eights are 48 and 1 is 49 --- ------
—
make the 3 a 1 -— -- 9 Trom 13 is
4 a 5 ---- ----- brims: down the the 5 —
........ .
(all right, stop)
93
Verbal Report Record
2 3T
3 2
S.*.
2**SQ
Words
5
10
9
15
Verbal
Report
-23 goes into 73
times
3 threes are 9
are 6 — — --
2 threes
■ draw a Xine~~subtract
9 from ’-you
make the 3 a 1— — -■W from 13 is 4 -— then
bring down the 1— make
20
16
25
8
the 6 a 7-
30
8
_ _____-25 soos intu-zero— 49 -— --- 2
times ---— ---2 threes are 6 ----- 2 twos
are 4
35
40
bring down your 9
0
-6 from 9 is 3
45
4 from
47
4 is zero
52
0
55
0
(all right« stop)
94
Film Analysis Record
I.
A-l
tf.T.
2-26
3
3 2 J 7 6
9 6
80
12
6 8
4
8
8
8
0
Step s
Frames
1.
Finds and writes 3
60
2*
Multiplies 3x32, writes 96, draws line
63
3.
Subtracts 76-96, writes 80
99
4#
Brings down 8
56
5.
Finds and writes 4
96
6#
Multiplies 4x32, writes 128, draws line
97
7.
Subtracts 808-179, writes 680
8.
Remaining film
156
6
95
Film Analysis Record
X.
A~2
N.T.
2-20
6 5
8 3 T 5 3 9 5
4 9 8
4 15
4 15
Steps
Frames
1*
Finds and writes 6
69
2*
Multiplies 6x83, writes 498, draws line
61
5*
Subtracts 539-498, writes 41
4.
Brings down 5
27
5.
Finds and writes 5
89
6«
Multiplies 5x83, writes 415, draws line
78
7.
Remaining film
167
6
96
Film Analysis Record
3 2
2 5T
I.
B
H«T.
2-26
Steps
Frames
1.
Finds and writes 3
37
2.
Multiplies 3x23, writes 69, draws line
45
3.
Subtracts 73-69, writes 4
4.
Brings down 9
29
5.
Finds and writes 2
43
6.
Multiplies 2x23, writes 46, draws line
56
7.
Subtracts 49-46, writes 0
91
8*
Remaining film
65
108
mi
s..
Zdz
1*
John had 126 marbles.
98
His father gave him 14 more.
How many marbles did he then have?
2
A canning factory packed 744 cans of corn into
boxes*
Each box held 24 cans*
How many boxes
were needed for these cans?
u
12
.-10..
12
-lb..
7
1 hunderd and 2 6 -- and 14
5
4 and 6 Is — — - 1 2 ----
....
_
~— --John had 1— a hunderd an 26 marbles-hiS :Piatln»:r* crntrA
him 14 more marbles. How many more marbles
&___
---- -—
--20
25
.. 9 .
3Q
2*.
_ _35
_12
4-0
13
45
7
50
3
55
7
60
8
1 and 2 Is 3 and 1 is 4-— -— --
7
A canning factory packed 7 hunderd ©n 44
cans of corn
Into boxes •‘-— -each box had 24 cans--- -ho
many bnxea were needed
for all these cans — -—
-add----- — 7-— 4
4 — -— ------ twenty--- four
4 Is 6 -- -----7 from nothing is 7
0
(all rightt stop)
99
the second digit of the divisor*
This letter method
would explain both of his errors in the project.
The
method of subtraction used by this subject
is identical with that used by the previous subject.
8 0 8
The second subtraction example, ^ g 3 * illustrates this
“hybrid” methods ”8 from 8 is nothing
is 8
method; ”
-- 2 from —
10
indicates a portion of the ”tske-awa^P
bring down your 1 -- 1 —
the 1 becomes
a 2H indicates a portion of the “additive” method*
The subject’s reliance on the “additive" aspect
made it possible for him in his first subtraction ex7 6
ample, ^
g, to somehow make the 7 a 17 and proceed to
subtract "9 from 17” to get 8 without suspecting that
something might be wrong#
Since no more mistakes were made in finding
quotient figures nor in subsequent computations, this
subject’s failure to judge or compare did not result in
further difficulties*
Unlike the first subject, Al.H*, this boy was
disturbed by the fact that example B would not "come
out even".
Following his second multiplication, sane
seven seconds were spent in 11silent meditation” before
he was willing to continual
It was quite evident that
the remainder bothered this subject considerably.
He
did not see the supposed "error of his ways", but did
think that something was wrong*
100
The verbal profile cherts give further evidence
of his consistent, steady attack on the examples.
The
silence indicated during the 35-40 second interval in
both examples A-l and A-2 occurred when the subject was
finding second quotient figures; the silence during the
same interval in example B followed his second multi­
plication when the subject realized that there would be
a remainder!
The verbal problems gave this subject dif­
ficulty*
The trouble, however, was not caused by a lack
of reading comprehension.
his verbalization.
This Is q\iite evident from
In the first problem, the subject
added words as "marbles" and "more" indicating definite­
ly that he understood the situation.
His straightfor­
ward reading In the case of the second problem gives
assurance that neither the vocabulary nor the situation
described was beyond his comprehension.
In the first problem, "4 and 6 isonly be explained as a spurious error.
12", can
This subject’s
failure to name the answer reflects a tendency common
to all of these fourth grade children.
A check on prior
arithmetical learning experiences indicates a lack of
emphasis on this aspect of problem solving.
A peculiar shift in the addition process as
verbalized in the second problem is Interesting, " -7 from nothing Is 7!"
In every instance prior to this
101
the word "and" was used, e.g. "1 and 2 Is 3”, "4 end
4 Is 81**
This verbalization appears to be an instance
where mind and voice were not cooperating - the mind
adding, the voice subtracting*
102
The electrodermal response records for this
subject were incomplete.
However, the following in­
teresting record of example A-l was obtained:
Figure 8
Electrodermal Response Record
(Reproduction)
I A '*
A-l
N.T.
2-26
"i/f'
I
I
This record appears to support the other evi­
dence of the subjectfs unruffled state.
It seems that any
tension consequent to the experimental situation was less­
ened during the solution of this example.
The record
is especially interesting in that errors of which the
subject is not aware or at least is not disturbed ap­
parently do not cause extreme fluctuations.
xhis Is In
agreement witn tne findings of ur. Greenwald ( 9
,.
103
Summary of Gas© Study Findings
Caa© 2*
N. T.
(A Pupil of Average Ability)
1*
The instruction prior to Project I does not
guarantee the formation of all such new bonds as are
necessary to the solution of long division examples
of type A,
2*
The maintenance and remedial program does not
keep all old bonds at such a functional level as to
operate when such skills occur in new and more complex
mental patterns.
3.
The learning environment prior to Project I
does not make possible the dependable, smooth function­
ing of all new bonds nor all old bonds in new patterns.
4#
The learning environment prior to Project I
does not make possible the solution of examples which
Include new, untaught aspects of long division.
5.
Incomplete mastery of the four steps In long
division does not prevent a consistent, steady attack
on the examples#
6.
Inadvertent exposure to two methods of subtrac­
tion may result In an equally effective "hybrid” method.
104
7*
The "additive" aspect of the above "hybrid”
method made it possible to subtract 96 from 76 without
suspecting anything wrong#
8*
Spurious errors due to temporary lapses in
attention or other causes occur.
9.
The situations described In the vex*bal problems
are not beyond the subject’s comprehension.
10.
Understanding the situation described In a
verbal problem does not Insure a correct attack.
11.
Errors unaccompanied by awareness or mental
distress do
not give rise to extreme fluctuations in
the electrodermal response record.
105
G®-8® S#
B* Sbi*
Table 14
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97
100
19
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109
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Group
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Class
Medians
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Test
Teat Hesults for Selected Case
This subject represents one of the slower
pupils of the group as is very evident from the above
table.
The learning opportunities prior to this project,
however, were in his favor*
In addition to the regular
classroom instruction, special individual help was given
him after school, February 10, 1936, on Articles 111 and
112} and on February 18, special remedial sheet, 2f~4,
was also given the subject.
Since he was the fifteenth
106
subject and therefore brought to the laboratory on
February 28, he had further advantage over the earlier
subjects*
He had had special practice on herder problems
In long division, Article 124, finishing fourteen exaiqples; and a special test in long division, Article
126.
In view of this special aid in his favor, the
results are particularly interesting*
It is quite evident from the verbal report
and film analysis records that the le a m i n g experiences
prior to this experiment had been very Inadequate for
this particular subject*
The subject proceeded in a
hesitant, uncertain manner without any degree of confidence In his ability*
Note the hesitating, uncertain
attack and the ten second pause In the verbalization:
Seconds
Words
40
10
45
7
50
0
55
0
60
3
Verbal Report
-68 Roes 2 times --- 2 times 2 is 4 —
let1s see— ee—
2 times 3 Is 6
— — *.— -— --(What1s the matter— -Is
there anything wrong---) yes, Itf3
wron^ (Do you see what*s wrong with it
Example A-l, page 108, Illustrates nicely how
an Isolated skill fails to function when burled In a
more complex pattern*
Ordinarily the subtraction ex­
ample ^ ® offers no difficulty.
In fact, the pre-experl
107
mental series of subtraction examples involved three
and four digit numbers.
In this new situation, the
subject must attend to more factors; less attention can
be given to the isolated si btraction example, with the
result that an error is made.
Incidentally, this case
adds evidence to the "mechanicalness” of bringing down
1 to "make that a 7”, and hence the ease with which
such errors can be made.
The method of subtraction used Is a variant
of the same type used by the two previous subjects.
In
example A-l, the first subtraction example Illustrates
this "hybrid” method which involves parts of the Htake­
away” and the "additive” method.
It is very evident from the subject*s verbal­
isation that neither judging nor comparing entered Into
his solutions.
As It happened, this lack was not respons­
ible for wrong solutions.
Another very odd but interesting error in method
Is responsible for most of his troubles.
After having
completed the first four steps, with the exception of
judging and comparing, this subject proceeded to divide
the third digit of the dividend by the divisor, neglect­
ing his first remainder and the digit brought down.
When he found that 32 would not go into the third digit,
8 , he proceeded to include tie preceding digit, 6 , and
found that ”32 into — — — —
68 goes 2 times *”
(Continued on page 114)
108
Verbal Report Record
2 2
5 2 T 7 6 8
6 4
2 8
6 4
I*
A-l
B.Sm.
2-28
Words
5
10
Verbal
Report
--------------- 52 Into 76-_______________
8
--goes — --- 2 times
— -2 times 2 Is 4
15
--------- 2 times 5 Is 6
80
4 from 6 Is 2
-make that a V—
bring down your 8
25
13
50
and 7 *
8
w o n ^ ffp into 8 —
40
10
-68 goes 2 t i m e s
50
_________
7 £rom 7 Is zero— *
and 52____________________
35
45
____________
letfs see— ee—
- so 52 i n t o ----- --- 2 times 2 is 4— _____
2 times 3 is 6
0
55
6Q
•
— -- (what *3 the matter— is there any­
thing rcrong-— )yes, itfs wrong (do you see
what*s wrong with it)
65
No, X don’t --
68
(all right, stop)
109
Verbal Report Record
8 3 y
Seconds
X*
A*2
B*3m,
2-28
Words
r
Verbal
m
Report
------ --85 Into
53
go——so 83 will have to go Into 5 hunderd
15
30
--— (bow many times, does _lt gel
(what)
46
(all right*, stop)
___________________
110
Verbal Report Record
2 5 T T
4
2
2
I.
B
B»8m«
2-28
Words
5
8
10
11
15
2
3
6
7
3
4
1
9
9
9
Verbal
Report
---------2
—— 5
—75. T”T. 2 times
____
--- 2 times 5 la 6 ---- 2 times 2 times 4
Is
— I mean 2 times 2 Is 4
20
8
25
11
50
0
55
10
~-6 wonft— --6 is larger than the 5
so the 5 becomes a 15 —
4 becomes a 5 and 5 end 7 is 2 -
40
—
45
bring down the 9 — - 25 into
- into 59 **—
50
0
55
10
60
0
65
0
70
15
6 from 15 is 7
once
1 times 5 is 3 —
1 times 2 is 2 —
75
(talk right along:. Bob)
5 from 7 is 4 -— 2 from 2 is zero — bring
down the the 9
(Xs there anything wrong with this problem)
(Xs it all right)___________ _____ _
80
— no.
82
—
(what*a wrong) -oh
divided wrong
(all right, stop)
111
Film Analysis Record
2 2
Steps
_____________
A-l
1* Finds and writes 2
Frames
86
2* Multiplies 2 x 32,
writes64, draws line
72
3* Subtracts 76 - 64,
writes2
25
4* Brings down 8
65
5* Finds and writes 3
140
6# Remaining film
238
A-2
1* looks at example
342
112
Film Analysis Record
1.
2 1
2 5 TTHTS
B
B.Sc.
2-28
4 6
F? 9
2 5
4 9
Step a____________________________
Frames
1. Finds and writes 2
68
2 . Multiplies 2 x 23, writes 46, draws line
94
3« Subtracts 73 - 46
4* Brings down 9
5. Finds and writes 1
6* Multiplies 1 x 23, writes 23, draws line
7* Subtracts 279 - 23 , writes 49
184
42
105
68
r[
114
It also appears that In addition to choosing
the wrong dividend to use for the second part of his
solution, he falls to place the second product in the
proper position*
This subject was not able to proceed with
example A~2 because of his inability to find the first
quotient figure.
In attacking example B, he first failed to
find the correct quotient figure#
Then he proceeded
with his Incorrect method and incorrect placement of
digits as he had done in attempting to solve example
A-l.
His solutions to the above long division ex­
amples are marked by hesitations, pauses, repetitions,
Incorrect attacks, and expressed defeat#
This Inability
to cope with the situations resulted In a confused men­
tal state In which condition certain bonds previously
operative dropped out*
Only an awareness of something
wrong remained but not the ability to find or correct It.
The verbal profile charts serve to picture this condi­
tion quite strikingly.
A very curious and interesting phenomenon
occurred when this subject attempted to solve problems.
His verbal responses to the first problem indicate that
he did comprehend the problem and the process necessary
to its solution.
A most strange method of addition
1X5
followed*
Without writing the numbers 126 and 14 in a
vertical column* he proceeded to adds w — - 4 and 6
are 10* put -- your -—
ze —
your 10 down - you put
1 and 2 are 3 and so you bring down the 1 d o w n
*"
He added from right to left * wrote his numbers from
left to right * arriving at 1031 as the sum!
That this is not a spurious, accidental pro­
cedure is shown in his attempt to add In the second
verbal problem*
were 744 and 24*
--- 8
---
Here the numbers he attempted to add
Note his additions ff
2 and 4 are 6
let's see
and bring down your 7*”
He again added from right to left, but wrote the num­
bers from left to right, arriving at 867 as the sum!
Thus the attempt to establish proper bonds for the cor­
rect solution of long division examples had not only
failed of their purpose but had definitely interfered
with previously established bonds - a specific case of
retroactive inhibition*
In spite of the novel method of addition used,
these verbal problems did not cause him the mental dis­
tress occasioned by the earlier long division examples*
In attempting to solve long division examples, he in­
variably met with wconscious" defeat or else encountered
an wimpasse” as in example A-2*
The verbal problems
gave him an opportunity to make such mental decisions
as would permit him to progress by the route of familiar
(Continued on page 119)
Verbal Hep or t He cor&
1*
John bad 126 marbles*
His father gave him 14 more*
How
many marbles did he then have?
2*
A canning factory packed 744 cans of corn into boxes* Each
box held 24 cans# How many boxes were needed for these cans?
Seconds
TT
W
o
r
d
s
Verbal
R
e
p
r
t
____
---had 1 hunderd 26 marbles
Uis father gave him 14 more* How many marbles
did he then have?
10
13
15
8
JO
8
14 m o r e
25
4
-—
30
8
down you put 1 and 2 are 3
35
o
‘1
your
hunderd 26 and then
you
add.
1 and 6 are 10 put
ze—
your 10
and so you
37
o
40
0
45
11
A canning factory packed 7 hunderd 44
cans of corn
_____
10
Into boxes. Each box held 24
cans* How "’’
many bans— - h o w many boxes were
needed for
XT
50
50
briry^ down the 1 down
60
these cans? 24 Into
.65
7 hunderd 44 cans
70
let's see ---» S
-—
80
and bring down your 7
85
0
90
2
95
2 and 4 are 6 -
--—
— (what was it you did here?) added-(wha t did you do) added— (you added-J----and
why was it you added) .because ah
117
100
105
6
0
— shows how many boxes were needed
(end that's why yotiadded, was It--)
(all right, stop)
118
Figure 9
Electrodermal Response Record
(Reproduction)
I.
B.Sm*
Verbal
Problems
119
skills to an answer*
This is most apparent In M s
tion to the second verbal problem#
solu­
Note his response
to the writer's query as to why he added!
This Infantile
regression no doubt eased the tension under which the
subject had labored*
It Is Interesting to not© that the
electrodermal response record for the verbal problem
tends to show this same effect*
How different this
electrodermal record Is from the one shown on peg© 35*
That record was made when the subject was having great
difficulty trying to divide 59,292 by 856!
Summary of Case Study Bindings
Case 3*
B* Sm.
(A Pupil of Inferior Ability)
1*
Regular classroom Instruction supplemented by
a specific maintenance and remedial program and special
Individual aid for specific weaknesses does not Insure
formation of new bonds necessary to the mastery of
simple long division*
2*
The bonds formed In learning subtraction fail
to function when the skill appears in a more complex
pattern*
3*
The bonds formed in learning short division
fall to function when these skills appear in long
120
division examples.
4*
In spit© of careful presentation of the steps
Involved In long division and special Individual aid,
a confused method of attack results.
5*
Instruction designee! to identify end teach the
skills Involved In long division results In a confused
mental state.
6.
Instruction In long division retroactively
affects previously formed skills involved in addition?
from right
that Is, adding/to left, hut writing the figures from
left to right J
7.
When confronted with an "impasse”, Infantile
regression occurs, In which case a mental decision is
made which permits progress towards eu answer by way
of familiar and known skills.
8.
Inadvertent exposure to two methods of sub­
traction results In an equally effective "hybrid”
method.
c.
Summary Treatment of Experimental Group
An attempt will be made to treat some of the
materials secured from the sixteen subjects in a sum­
mary manner*
Unfortunately, many interesting features
121
peculiar to certain Individuals will not be brought
out by this method, but space does not permit individual
treatment of all cases.
Before proceeding to treat the written re­
sponses, It was necessary to Isolate the skills Involved
in Project I and arbitrarily assign symbols to them*
After analyzing the skills Involved in solving the exangles and studying the possibility of their appearance
in the written responses, the following skills were
isolated!
1*
Basic division combinations
a. 1st quotient figure
b* 2nd quotient figure
2*
Placement of quotient figures
a. 1st quotient figure
b* 2nd quotient figure
3*
M\iltip II cation
a* 1st product
b* 2nd product
4*
Subtraction
a. 1st difference
b. 2nd difference
The two skills, judging and comparing, were
omitted because their presence or absence is not ap­
parent In the written responses*
These skills will be
studied when the verbal response records are analyzed*
In the summary analysis which follows, the symbols shall
be Interpreted in this manner!
2b refers to placement
of 2nd quotient figure; 3a, multiplication to obtain
122
Example Analysis
Project I
-P
O
©
a> b
o
o
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imTi.TT.
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o
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u
u
a
M*3
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4*
0
H
&
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o
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i. r .w .a -i
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.. .73 A Aly*
7&+ 3A-3
f~ 3A- X.
7
C
.
-fto
6,</~ 4
/<£, 4as /4*41
t/ .?^ /4? .....1.3.Xi w/? szor - ?3-- i ...........
A -2
B
2 *A n *IL
*/
A -l
A —2
!cu
B
3 ,S .J ,
A -l
A -2
B
4 .C *P
«
A -l
S -2
4~4r
B
A —1
B
6, K
.T .
3..... ......... ..
||j>
I
to
5 *J ,B ,
... .
.._. .'/? - 4kj=
.
A -l
A —2
y
Jaj 4 ^ f-jr' 7£_l jpjl=3
7c-fC ' fo
Yo Y +yji - /
y
B
7 *A 1 .H .A -l
A -2
B
8 *1 *0 .
A —1
A-2
S
y
/
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.. /<**. 4*?
.
7t> ~ 33 '3
i/
B
9 .S .F .
A -l
A —2
B
LO.R.N. A-l /
A—2 /
"'B.
.„J__
4
jrjf- 4 ? f - 3 /
~7t> — ft - 0
123
Example Analysis
Project I
+» -P
•P
ft
Subject
o ©
©
rH
o 8 S
p h m
Errors
Exact Mature of Errors
tr*
1
CO
11*D*T* A-l
B_
12* B*W.A-1
a ~£ s
B
tjU-'
13.K.H. A-l
A-2
B
r"„ X 6Lr / y
✓ Jas 3 as
/Os. 3 O-'
/
14,B*Sm.A-l
A-&
B
✓
15.B.Sc.A-l
V7_
^i
'
JJ X .
2.= .TY A* f ~ 3JL =■ $
£-39 + ?3^3
S3 XV = 3 3C
,
7,3.+ 33 ».*... A 3
.... _ ... . .
x
s
03
4
B
16.B.R. A-l
A-2
B
Summary A-l
A-2
B
Total
s
j.£-39 -rS3 =sy_. S3 XS’= ¥4 S' ....
......
93*7=4P/: 7¥sr-¥&? -S~¥
73~¥&>-~2f- A£?-!-A3-¥; J3 X¥'SX
/CS2«S Js^J 3-J- i 7¥\rr S3 -7
|
/o-' y fl/
73^23=Ji:
i/
LI
L0
L0
311
2
1
1
4
3
5
5
13
Summary of Errors
Skills
A-l
la
5
1 3
lb
3
2
1
2a
0
0
0
0
2b
0
0
0
0
3a
1
3b
0
0
4a
3
1
1
11
Q
6
4b
Totals
A-2
2
B
Total
7
6
1
4
1
1
1
1
8
5
2
25
124
1st product; etc#
This analysis reveals some Interesting facta
©oncoming the sixteen subjects.
As the summary in­
dicates, 31 of the 48 examples were correctly solved,
13 were incomplete, and 4 complete but incorrect*
Of
the 13 incomplete solutions, 5 were attempted solutions
to examples of type B, the type which had not been taught
prior to the project.
It is interesting to note further that no
errors were made in placing first and second quotient
figures.
Of the total number of errors, 15 or more than
50 per cent Involved finding first and second quotient
figures; 5 Involved finding the first difference; 4,
the first product; 2 , the second difference; end 1 , the
second product.
Upon closer Inspection of the summary of er­
rors, additional interesting Information emerges.
Three
errors were made in finding first quotient figures in
the ease of examples A-l and A-2, whereas only one
error was mad© in finding the first quotient figure in
Bl
It therefore appears that 3 2 T 7 6 and 2 3 ) 7 3
confront the child with more difficulty than 8 3 T 5 3 9!
Examination of the exact nature of the errors gives a
possible explanation#
The three errors In the case of
A-l were: 3 2 ) 1 6 = 3j 3 2 ) 7 6 = 3; 3 2 ) 7 6 = 3.
In the ease of B, the following errors occurred:
125
2 3 I 7 5 • Sj 2 3 ) i'"z = 4} 2 3 T 7 3 = 2.
In 5 of
the 6 errors, it appears that the subjects divided the
second digit of the divisor by the first digit of the
dividend in order to get the first quotient figure*
Thus, It seems that examples A-l and B presented set­
tings which invited Incorrect functioning of the newly
learned method of finding first quotient figures.
Further examination of the exact nature of
the errors made shows that in tnree Instances incorrect
rj zs
solutions led to the example^ g.
In two of these
Instances, the subjects gave ”0" as the difference;
in the third, f,80n.
In order to understand more fully
these responses, the writer referred to the dictaphone
records*
These records Indicated that the first two
subjects recognized "something’s wrong", the one pro­
ceeding in spite of It; the other, not.
The third sub­
ject, unaware of his error, continued In his attack on
the example.
It is the writer’s opinion that the addi­
tive aspect of this subject’s method of subtraction
made It easier for him to subtract 9 from 7 without
experiencing any misgivingsJ
The verbal responses ©s recorded on dictaphone
records lose much of their qualitative value upon trans­
cription.
The dictagrephs, however, do indicate much
that could not otherwise be deduced from the written
solutions.
The verbal responses, for example, revealed
126
the following methods of subtraction:
"Take-away"
9 6
6 7
5 9
5 3 9
19 8
41
1•
2*
We cannot subtract 7 from 6 so we borrow
a ’’tens” from 9,
7 from 16 is 9 j3
from 6 is 5*
8 from 9 Is 1 * You can#t take 9 from
3, so the 3 becomes a 13* 9 from 13
is 4* Because you made tl® 3 a 13,
the 4 becomes a 5. 5 from 5 is zero*
"Additive”
(Austrian)
5 3 2
2 8 7
2 4 5
1.
7 and 8 are 12* Carry the 1 to the 8
and make that a 9* 9 and 4 are 13#
Carry the 1 to the 2 and make that a
3# 3 and 2 are 5*
7 3
6 6
— 7
2*
The 6 is larger than the 3, so you
take 1 from the 7* The 7 becomes a
6 , and the 3 a 13. 6 and 7 are 13.
In the summary analysis which follows, T-l
refers to the first "take-away” method, T~2 refers to
the second; Ad-1 refers to the first "additive" method,
Ad-2 refers to the second*
In most Instances, it was easy to determine
whether or not a given subject did judge or compare#
The symbols relative to judging and comparing as used
in the table are to be interpreted as follows:
, did
judge or did compare; o, did not judge or did not com­
pare#
Blank spaces indicate that the particular step
was not reached in the solution*
127
Dictaphone Record Analysis
©
<D
H fl O
a. *h *d
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t*
ci *o ©
©
^
©
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4J to s
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t b o o j3 O ,0 -P t3 E]
d ^ £3
Subject
(23 js M
EH W tO ££ *-D O
Special Remarks
AMsivo
2gfc^
J<i£&dZ.
2. An.H.
A-l
iW2
6. N.T.
A-l
<C^-<^.tttrizL
7. Al.H.
A-l
'iHsQf
22
9. S*F.
A-l
10. H.N.
A-l
12,8
Dictaphone Record Analysis
©
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0)
«H £ «
©
0t «H rO
*0
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«p tsO g
© & d O
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13. H.H. A-l
A-2
B
14. B.Sm.A-1
A-2
B
15.B.SC. A-l K
A-2
i \S
16. B.R. A-l
A-2
B
Q
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£1
ko & p *d *3
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11. D.T. A-l
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T -A
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4 4 -f
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129
Table 15
Verbalization Analysis
Subtraction
Methods
rH
*0
<
Solutions in
which subjects
Solutions in
which subjects did not
Cl
<
fr. £-.
Judged compared
judge
compare
R. w. L R. W. I. R. W. I. R. W. I.
2
1
2.
An.H.
i
3*
S.J.
3
4.
CUP.
5.
J.B.
6.
N.T.
7.
Al.H.
3
8.
1.0.
2
9.
S.F.
1
1
1
1
1
2
1
3
i/
3
2
1
2
1
2
1
2
1
3
3
1
2
i/
10.H.H.
3
1
3
12.B.W.
2
1
2
lb.B.Sc.
3
¥
/
15
1
2
19
3
3
3
3
3
16.B.R.
/
1
1
Sin.
3
2
3
13.H.H.
Summary
1
3
1 1 . D .T .
14«B*
2
1
4
UMJ‘
1
1
1
16
3
11
1
1
1
12
3
9
130
In order to facilitate Interpretation, the
writer has recast some of these data into a table, p. 129.
This table Indicates that of the sixteen subjects, only
three used the pur© "additive" method, the method which
had been taught at Longfellow school.
Of the remaining
thirteen subjects, only four used the pure "take-away**
method, T-l.
Of these four, two had originally been
taught the "additive” method at Longfellow school; two
had been taught the "take-away"method In other schools*
Eight subjects used a "hybrid" method, T-2, which In­
volves part of the "take-away" method and one aspect of
the "additive" method.
One subject had derived another
"hybrid" method, Ad-2, which includes part of the "addi­
tive" method and one aspect of the "take-away" method.
From this table it is evident also that six
subjects judged, ten did not; that eight subjects com­
pared, eight did not,
The table also gives information
about the solutions of these children.
The six children
who judged are credited with 15 correct, 1 incorrect,
and 2 Incomplete solutions; the ten children who did
not judge had 16 correct, 3 incorrect, and 11 Incomplete
solutions!
Failure to judge leads to situations and
difficulties which very often make It impossible to
complete solutions.
This accounts for 11 incomplete
solutions of the ten subjects who did not judge in con­
trast to 2 Incomplete solutions for the six who did
131
Judge*
The eight subjects who compared are credited
with 19 correct, 1 Incorrect, and 4 Incomplete solu­
tions, the eight subjects who did not compare are
credited with 12 correct, 3 Incorrect, and 9 incomplete
solutions.
Thus it is evident even from these few
cases that failure to judge or failure to compare great­
ly increases the number of Incorrect and Incomplete
solutions.
Other interesting details appear In the
original summary analysis.
Of the children who succeed­
ed in completing example B, a majority of them were
puzzled by the remainder.
This was particularly
evident in such sub jects as X. 0. where the verbaliza­
tion clearly indicated this disturbances
"Let*s see -—
something*s wrong her©•"
It appears that the meaning of the terms
"judge" and "compare" are not clear to some of the
children.
The first subject used the term "judge"
when he meant "compare", the tenth subject used "com­
pare" when he meant "judge".
d.
Summary of Findings Based on Sixteen Subjects
1.
are
Case study flndinga^further substantia toythe
records of the other subjects.
2.
Placing 1st and 2nd quotient figures is done
132
correctly by all the subjects*
3*
Finding 1st and 2nd quotient figures account
for more than one half of the errors#
4*
Finding the first difference account for ap­
proximately one third of the errors*
5*
Finding the first product account for one
fourth of the errors*
6*
Such settings as 3 2 j 7 6 and 2 3 ) 7 3
tend to invite incorrect functioning of the recently
learned method of finding the 1st quotient figure*
7*
The additive method of subtraction or an
aspect of it seems to permit such absurd computations
&S 9 6 su*>^r&c^ nS
from 76 without accompanying dis­
turbance or misgivings.
8*
Four types of subtraction obtain in the ex­
perimental group: T-l, T-2, Ad-1, and Ad-2.
(See
page 126)
9.
Eight subjects use the T-2 method; four, the
T-l method; four, the Ad-1 methods
10.
adequate*
Instruction in judging and comparing Is not
six subjects judged, ten did not; eight
133
subjects compared, eight did not.
11.
Failure to judge greatly increasesthenumber
of incorrect and Incomplete solutions.
12*
Failure to compare greatly increases the
number
of Incorrect and Incomplete solutions.
13*
A majority of those subjects who are able to
complete example B are puzzled by the "remainder,!*
14.
The terms "judge" and "compare" ere sometimes
confused by the subjects.
e»
Summary Treatment of the Class
As previously indicated the examples and
problems of each project were given to the remaining
fourth-grade pupils.
These examples and problems were
presented in mimeograph form, and in such a manner as
to present only one at a time.
In order that conclusions and comparisons
may be drawn, the results are given In th© form used
for the regular subjects.
134
Example Analysis
Project I
r-t
w
Subject
Errors
Exact Nature of Errors
X*P.D
A—2
$3.
~?-3 ~
3X/
&
9
-
/y
-u
? - ¥jff- AC/
k
?
•-- /'• /? .(^25*.
4 « A - l
'^3scld3
A-l
A-l
10,W*0*
A-X
A-l
A-2
3<.auuj
....
-...
^
135
Example Analysis
Project I
-p
o
©
o
o
Subject
p
o
©
&
o
o
Ma
•©P
©
r-1
n«
*
o3
o
«
M
Errors
Exact Nature of Errors
1 2 « M ,X i* A - I
~A£Stie^z*3a6sa_A.
rf-t-
/&-.
/-^
^9 -
p(?3
7 C + 32.- Y
<*(-r^l^
3X/-^j,X
X**2.___ £
14*B«Kr%A-1
A -2
16«S*Y * A-l
17 *R*D* A-l
18.D.B. A-l
19*R*M. A-l
A—2
20.Ed*A.A-l
£d2^£&l--J2&L.iJ~sJ3. .JB&ga
3
17
Total
v
(&&{ -?-
~-c£
156
Summary of Class Errors
Skills
A-l
A-2
B
Total
la
1
2
1
4
lb
1
1
2
4
2a
2
1
1
4
2b
1
0
1
2
5a
0
5
0
5
5b
0
0
2
2
4a
0
1
2
5
4b
0
0
1
1
Totals
5
8
10
25
The part of* the Example Analysis Record
that applies to the correctness, incorrectness, and
incompleteness of solutions 3hows a striking resemblance
to that for the 16 subjects.
Of a total of 65 prob­
lems presented to the class (exclusive of the 16 sub­
jects), 18 solutions were incomplete or £0 *6$; of the
48 problems presented to the subjects, 15 solutions
were incomplete or 27*1$.
Of the class incomplete
solutions, 9 or 50$ were attempted solutions to ex­
ample B; of the 16 subjects, 5 or 38$.
These figures
tend to show the representativeness of the subjects
selected on the one hand, and on the ether the fact
that responses of the class are very similar to those
137
of the e:jqperimental group.
This condition should
rightly he so if instruction was given as directed
and supervised*
The summary of class errors also shows that
the class and the subjects performed in very much the
same manner#
Finding quotient figures offered greatest
difficulty for the class as well ©s for the subjects.
Multiplication and subtraction errors occurred with
approximately the same frequency.
The class and subjects
differed with respect to placement of quotient figures.
The class members made six errors in placing quotient
figures, the experimental group, none.
The responses to example B are interesting.
This example was of a type not previously presented to
the class.
All previous long-division examples had
"come out" even.
One subject erased part of his work
when he saw there would be a remainder
incomplete solution.
leaving an
Another made 23 x 7 equal 149 in
order to get the example to "come out" even.
A third
subject did not do the final subtracting necessary in
obtaining the remainder.
A fourth, was satisfied that
23 x 2 equal 49 so as not to have © remainder.
Two
other students made adjustments because of the trouble­
some remainder that appeared.
It is very evident that
unless students are prepared for long division examples
with remainders they will have difficulty with them.
138
It Is also apparent that pupils will permit incorrect
products, Incorrect differences, or will avoid the
final subtraction in order to satisfy their desire to
have the example "come out" even#
Since they have not
been apprised of remainders, they will be disturbed by
them and proceed to arrange matters so as to avoid them.
Students of the class also made such errors
as 3 x 6 equal 9, 3 x 2 equal 9, and 3 x 6
equal 12.
As In the case of the sixteen subjects these errors
are either spurious or appear because the skills are
burled in more complex situations.
e.
Hypotheses and Tentative Conclusions
An attempt will be made on the basis of the
Individual case studies, the experimental group study,
and the examination of the written responses of class
members to set up such hypotheses end conclusions as
appear evident.
1.
Instruction prior to Project 1 Is adequate
for students of average and superior ability.
Hew skills
necessary to the solution of new type examples function
readily and accurately*
2.
Instruction prior to Project 1 supplemented
by a specific maintenance and remedial program and by
special Individual aid for specific weaknesses does not
139
Insure for the low-ability student the formation of new
bonds necessary to the mastery of simple long division*
3*
The learning environment prior to Project 1
makes possible dependable, smooth functioning of new
bonds and of old bonds in new patterns for the superior
student but not for the average and below average pupil*
4.
Instruction in long division retroactively
affects previously formed skills as in the case of the
low-ability subject who added from right to left but
wrote the figures in the sum from left to right.
5*
Spurious err or sAbrought about by retroactive
Inhibition, temporary lapses In attention, or other
causes occur.
6#
The mastery of skills and principles necessary
to the solution of examples of Type A Is adequate to
the solution of examples of Type B for the superior
pupil but not adequate for either the average or below
average student*
7*
Students able to complete examples of Type B
are puzzled by the "remainder".
8.
Students will often permit incorrect products,
Incorrect differences, and avoid the final subtraction
140
In order to avoid "remainders” •
9*
Incomplete mastery of the four steps in lor^;
division (omission of "Judging* and "comparing”) does
not prevent a consistent, steady attack on the examples.
10.
When confronted with an "impasse" Infantile
regression occurs, in which case a mental decision is
made which permits progress towards an answer by way
of familiar skills.
11#
Very little time beyond that of reading a
verbal problem is used to decide the method that shall
be used in its solution,
IS.
Electrodermal response records provide evidence
of affective stability and instability.
13.
Errors unaccompanied by awareness or mental dis­
tress do not give rise to extreme fluctuations in the
electrodermal response record14*
Instruction in the "additive" method of sub­
traction and exposure to the "take-away" method result
in four methods of subtraction, two of which are variants.
15.
The "additive” method of subtraction and its
variant make possible such subtractions as 96 from 76
without causing any misgivings or disturbances.
141
16#
Finding first and second quotient figures
account for approximately half of the errors made bj
students.
17.
Finding differences and multiplying account
for approximately the same number of errors#
18#
Placement of quotient figures rarely cause
difficulty.
19#
Instruction provided in "judging” and com­
paring is not adequate#
Verbalization records indicate
more than half of the sixteen subjects omitted these
steps*
20#
The terms "judge" and 11comp are11 are sometimes
confused by subjects and used interchangeably#
21,
Failure to judge and compare increases the
number of incorrect and incomplete solutions#
22f
Verbal responses give cues as to "judging"
and "comparing", mental disturbances, and other aspects
of the learning process which otherwise would not ap­
pear.
23.
Evidences of success and failure accompany­
ing pauses, hesitations, etc. that appear in the writ­
ten responses, the verbalizations, and the ©lectrodermal
records are substantiated by the motion picture film record*
142
7* Project III.
a. Learning Experience Prior to the Project
As In the case of the first project, the
teacher1s dally record, the textbook, the supervisory
charts, and the writer*s observational reports furnish
the Information necessary to reconstruct the learning
experiences*
During the Interval between Project I and Pro­
ject III the textbook provided the following long-divi­
sion learning experiencesi
1* Article 124
Twenty-three examples and problems
were presented to give practice in
harder division.
2. Article 125
A review test of 12 long division
examples and 3 problems was pro­
vided*
3. Article 138
Initial presentation of long divi­
sion with remainders (type B) was
ijtade *
4 . Article 159- Twenty-eight examples of long
141 division with remainders were
presented for purposes of checking
and practice.
5. Article 144
Special help was given in estimat­
ing answers and problem analysis.
Six problems were used for this
purpose.
6 . Article 147
Long division with zero In the quo­
tient (type C) was presented. Six
examples were used.
7. Article 148
Eighteen examples of Type C were
given.
143
The maintenance and remedial program was pro­
vided for by the textbook, the teacher, and the Super­
visory Plan operative in the school.
The textbook pro­
vided self-testing drills, rapid oral drills, and problem
scales,
1 . Article
2# Article
126 Self-testing Drill Ho. 15
128- Review tests In addition and sub129 traction of whole numbers.
3# Article 130- Review tests In multiplication and
131 division.
4. Article 137
Review test consisting of 57 items
Including skills and Information
taught In grade 4.
5. Article 142
Rapid oral, mixed drill consisting
of 18 examples involving all pro­
cesses •
6 . Article
143 Self-testing Drill No. 16.
7* Article
145 Problem Scale, No. 9, consisting
of 10 problems involving addition,
subtraction, and multiplication.
8 * Article 146
Rapid oral, mixed drill consisting
of 35 examples Involving all pro­
cesses.
9# Article 150
Self-testing
Drill No. 17.
The teacher found it desirable occasionally
to supplement the material provided by the text.
Fol­
lowing Article 125, a review test in long division,
special emphasis was given judging and comparing as the
class cooperatively worked a long-divislon example.
addition to the problem included in Article 144 on
In
144
problem analysis, three practical verbal problems were
placed on the board and cooperatively analyzed.
Special remedial materials were furnished
from the office as class and Individual weaknesses
appeared*
The following table indicates such class
weaknesses and the remedial sheets that were provided.
TABLE 16
Class Remedial Program
Class Weak Skills
Remedial Sheets
Id (Addition of whole numbers,
three or more digit numbers.)
Id
-6
3c (Multiplication of whole numbers,
two digit multipliers)
3c
-5
3e (Multiplication of whole numbers,
with zero difficulty)
3e
- 3
4c (Short division with remainders)
4c
-5
The weak skills of the slower pupils were
similarly listed and remedial materials provided.
With this knowledge concerning the arithmetic
experiences of the class during the Interval between
Project I and III, an attempt will be made to analyze
and interpret the subjects* responses.
b. Selected Case Studies
The table on the following page, derived from
Table 2, page 6 , indicates the three subjects chosen for
1443
Table 17
Gates Reading Test
11.2
Compass Survey
Test 5-25-36
11*8
Compass Survey
Test 1-21-36
25
Kuhlman Anderson
Intelligence Test
ow
o <d
*.&
•&P
o w
o ©
Gates Reading
Test 1-17-36
Test Results Tor Selected Cases
09
rHCO
HItO
<A t
o^
O Cl
^
l
i
t Ht
1
J.B.
128
107
106
22
D.T.
114
R.W.
Exp*Group
Medians
Class
Medians
104
107
109
21
18
8*3
8*6
142
184
109
98
106
20
21
6*0
6.7
131
151
108
97
100
19
21
6.0
6.5
116
134
109
14
18
4*5
5.0
Standards
108
102
O CM
•1 * 1
S
o ©
nC
*l
>
© CM
H t HI |
H pa to
GJP
erf P
$ © ft «1
O © O ©
EH H £-i
201
8.9
23
£
erf
>3CO Pt CO
to
to
5 -2 7 -3 6
Subject
J4*
w
01
H
r-ftO
a$ t
o to
•
U>
207
155
85
14 6
individual study and their relative rankings in the
experimental group and in the class as a whole*
Case It J. B.
This subject Is a superior fourth-grade
student, excelling in all of his school work.
This
superiority is evident In his test scores as reported
for him in Table 17.
He was brought to the laboratory, March 30,
1956, and began work at 11514.
The Individual Data
Record Indicates that he was in good health.
The special record forms for this subject
are interspersed in the discussion that follows.
It
is evident from the verbal report and film analysis
records that the Instruction and maintenance program
provided between Projects I and III was adequate.
He
proceeded cautiously and accurately in his attack on
each example.
There was no evidence of lack In knowl­
edge of procedure or mastery of processes.
The dictaphone records Indicated both ”judging”
and ’’comparing11.
Judging was expressed thus:
’’the 14
is smaller than the 56 so you can bring down the 9 — ”.
In subtraction examples this subject used
neither the ’’additive” method which he had been taught
nor the ”take-away” method to Yrhich. he had been exposed.
Instead, a ”hybrid” method which Incorporated parts of
(Continued on page 157)
147
III,
B
«r*B*
3-30
Verbal Report Record
5
2 4- 3 6
3 6 ) 8 6 9
7 2
1 4 9
1 4 4
---36. goes Into ©Icrhty—
.... 5,
... §
10
3
six
15
4
20
14
25
11
30
19
— -------2 sixes er — -— 18---— 2 sixes -— er-- 12--carry 1---- --2 thre<
are 6 and 1 is 7
—
you can subtract so you do ---- 2
from 6 equal 4
---*7 from 8 equal 1---- - the 14 is smaller
than the 36 so you brin& down the 9 — -
35
9
40
8
45
9
50
13
2----- 4 threes are 12— n 2 Is 14
you can subtract so you subtract-- 4 from 9
equal 5 -- the answer
55
7
Is twenty-four-r-— en fi-Ive— thirty-sixths
57
1
remainder — — ---
----
2 times -------
36 Roes Into a hundert en 41 — nn-
148
Verbal Report Record
5 0 7
6 3 T 3 i 9 41
3 15
44 1
44 1
-----
0-1
J*B«
3-30
Seconds
Words
Verbal
Report
5
2
10
8
15
10
20
11
25
7
30
14
-— -3-- from-m- 5 from 9 equal 4--— -— 4 ia smaller than the thirt— 63 so you
bring down the 4 «--
35
11
---63 won *t go into 44 so you— put--- zero—
goes Into 3 hundert en 19-—
5 times -
3 fives are 15— -carr? 1
- 5 sixes ara ta
and 1 is 31 -— - you can subtract,
so you do — -—
40
rjf
-—
45
8
into 4 hundert en 44 — — - 7 times
50
7
55
4
60
8
65
io
bring down your 1--- 63 goes
are 42 and 2 is 44 — — Th© answer is 5 hundert en 7 and no
ranininriftT*
149
Verbal Report Record
5
8 5 )~2 8 2
2 4 9
3 5
5 5
III.
e~2
J »B»
3-30
4 0
2 0
2
2
0
Words
Verbal
5
Report
-~Q3__&oeslnt o 2 hunderd en
82
15
5
3 times-—
20
25
3 threes are
5 eights are twenty-r- four-— 9
50
—
yu can subtract ao you do
equal 5
9 from 12
5 from 8 equal - 1--
35
9
40
13
45
10
50
6
— — 4, threes are 12 --- carry .1
55
8
3 ....—
£Q_
JB5_
70
11
8
JZ2_
10
JZ2_
__a
«.-g
2 from 2 equal zero-— 53 Is
•taller then the 85 so you bring down the
2 --- 85
gpe.s into 3 hundert en 32 —
4 times
4 elghte are 52 and —
1
la 35 — —
ah-h -— you can subtract so you subtract en you
don f,t. get anything .-- ----------------------------—
bring down the zero
en-- 85 goes
Into gero no times so y o u out up the ^ e m
150
Verbal Report Record
7X2
III.
D
J*B *
3-30
Seconds
J 2 63 44
213 6
49 84
49 84
Words
Verbal
Report
5
0
10
9
s-seven hunderd en 12 goes Intu ---thousand 6
15
4
hundert ©n thirty--- fn--un
20
2
25
5
times
30
9
35
11
«*——-3 ones are 3 --.—3
are 21------you can subtract so you do ---- — 6
from 14 equal 8
4.0
1
46
6
50
4
55
18
60
2
65
11
«--- — —
7 hundert
en 12 goes into 4 thousand 9 hundert en
8 4 ------
70
6
----- 7 times --- 2 sevens are 34 -----
75
8
80
8
81
0
2
3 twos are 8 —
4 from thirt--een equal 9
.--o
from 6 eaual 4 --------- the 4 hunderd 98 is smaller than 7
hundert en 12 so yon hn^ng Hawji the 4— —
carry 1
7 ones are 7 and 1
ia 8 --- 7 times 7 is 4 9 --------
151
1* The population of a certain city in 1920 was 45,568. Ten
years later there were 70,002 people in the city.
How
much did the population increase in the ten years?
2. A coal company collected $2,626 during the month of Feb­
ruary selling coal at $15 a ton*
Seconds
rt..
Words
Verbal
How many ton were sold?
Resort
5
.5
10
7
city —
____15. .
9
thousand-- 5 hundert en 68* Ten years la tier
....
7
in 1920 was-—
forty---- - thrfte
25
9
there was seven— n--70 thousand--en---- 2
people in the city. How much did the
pnpulatipn
50
4
Increase in ten years? -----
35
4
-- you-u---- subtra----ct--- 4 hundert — -
40
8
45
4
50
0
55
0
60
5
65
7
4--- 7 from 10 equal 5 --si--- x
70
4
from-m 10 equal 4 ------— -
75
7
80
7
7 equal 2— -population-- increased in ten
85
8
years 26 thousand 4 hunderd 34
90
8
HO.2 --— a coal company collected 2 thousan<
95
8
100
7
SO
„
-- from 70 thousand 2 --------
2*
month of February s— selling coal at ---
Verbal Report Record
Ho.2. (continued)
Seconds
105
Words
10
Verbal
Report
110
— 13 dollars a ton-— -How many tons were sold
X dU — ----------------------------------divide
----2
115
—-thou- -sand— - 6 hundert en— n
120
six
by-y
twenty
thirt
,,
—
—
*
_—
125
een dollars-------- 15
130
goes into 26 --------— -----2 t i m e s --- 2 threes are 6 ---2 ones are_______________________________ _
135
140
145
10
g ----- — -y-yu- bring d— —
____________
ah--- subtract —
you won't get anything
bring down the 2
____________________
150
8
-— 13 won’t go into 2 so you put_______ __
155
160
12
165
8
170
2 ones are 2 — - he sold exact-
exactly
two-o--hundert ©n — --- 2 tons in month
171
172
a zero bring down the 6 --—
13 goes intb 2 6 -- :- 2 times 2 threes
are 6
en__________________________
of February
0
15,3
3 6
111.
B
J.B.
5-50
Film Analysis Record
5 0 7
g 4 - S
6 3 T 3 19 4 1
8 6 9
36
3 1 5
7 8
4 4 1
14 9
4 4 1
111.
14 4
C-l
5
J •B.
3-30
Steps
Frame
1.
Finds and writes 2
2*
Multiplies 2 x 36, writes 72, draws line
3.
Judges
4.
Subtracts 86 - 72, writes 14
5.
Compares
6.
Brings down 9
7.
Finds and writes 4
68
8.
Multiplies 4 x 36, writes 14-4, draws line
77
9.
Judges
28
10.
Subtracts 149-144, writes 5
11.
Remaining film
05
118
6
52
*
U£<03
9
9
109
Ill,. C-l
1.
Finds and writes 5
88
2.
Multiplies 5 x 63, writes 315 , draws line
91
3.
Judge s
47
4.
Subtracts 319 - 315, writes 4
5#
Compares
6*
Brings down 4
37
6
7.
Finds and writes 0
43
8.
Brings down 1
49
9.
Finds and writes 7
71
10.
Multiplies 7 x 63, writes 441 , draws line
11. , Remaining film
7
145
26
154
Film Analysis Record
8 3 > 2
2
111
C-2
J.B.
3-30
.
8
4
3
3
5
2
9
3
3
4 0
2 0
.
2
2
0
111
D
J.B.
3-30
3
4
6
8
8
7 1 8 5 2 6 3
2 15
4 9
4 9
Steps
Frames
I.
Finds and writes 3
117
2m
Multiplies 3 x 83, writes 249,draws line
79
3.
Judges
38
4*
Subtracts 282 - 249, writes 33
63
5*
Compares
63
6.
Brings down 2
7*
Finds and writes 4
8*
Multiplies 4 x 332, draws line
120
9*
Brings down 0
120
10. Finds and writes 0
41
II. Remaining film
23
6
60
111» D
1.
Finds and writes 3
195
2.
Multiplies 3 x 712, writes 2136
83
3.
Judges
40
4#
Subtracts 2693-2136, writes 498
5*
Compares
6.
Brings down 4
7«
Finds and writes 7
8*
Multiplies 7 x 712,writes 4984, draws line 120
9*
Remaining film
134
83
9
111
23
7
4
4
4
,S-
SI
157
each method was used* While not making sense, the
method was effective, vis*
oQp
-249,
£rom 1® equal 5 —
------ 5 from 8 equal
i ------3.«
It was interesting to note this s\b jectfs
attack on example D, a type which had not been taught
prior to this project.
While a trifle hesitant in
making the attack and using more than the usual amount
of time in finding the first quotient figure, he pro­
ceeded cautiously and accurately In its solution .
The
skills and principles that had been taught were adequate
for this subject when confronted by this new type
example.
The subjectfs attack on the verbal problems
was likewise cautious and accurate.
In the case of each
problem the method of solution was decided upon by the
time the problem was read.
The adequacy of his knowledge was further
attested to by the promptness with which he properly
labelled his answers as ”Increase in population” and
11tons of coal”.
The electrodermal response record for this
subject conforms to his electrodermal response pattern,
Indicating no apparent significant deflections.
This
record substantiates the calm, deliberate manner evidenced
in his verbalizations.
158
*
o
»-i
©
Project III
S>
•H
P4
15 9
Summary of Case Study Findings
Case 1.
J. B*
(A Pupil of Superior
1*
Ability)
Class instruction on long division examples of
Type B and 0 is clearly adequate for the acquisition of
the new skills involved*
2*
Skills and principles involved in the solution
of examples of Type A, B, and C are adequate to the sol­
ution of examples of Type D.
5*
The maintenance and remedial program is ad­
equate.
ho breakdowns of previously acquired skills
appear*
4*
This subject ^judges11 and ,fcompares11 consist­
ently as these steps are reached*
5.
Instruction In tie ''additive" method and ex­
posure to variants of the "additive" m d the "take-away"
methods results in a "hybrid" method which includes
aspects of each.
6*
The verbalization record and derived Verbal
Profile Chart give evidence of this subjectfs steady
attack on examples A, B, and C and the more hesitant
but accurate attack on example D end verbal problems.
160
The
r e g u la r ity
F ilm
and
8 .
a
c a lm ,
C ase
2 .
A n a ly s is
Ir r e g u la r ity
Th©
com posed
D .
R e c o rd
o f
T*
re s p o n s e
w o rk *
re c o rd s
in d ic a te
fo r
18
S e le c te d
C ase
Subject
D.T.
114
Exp.Group
Medians 109
Class
Medians 108
108
possesses
g ra d e
c la s s *
d en t
In
in g ,
and
C la s s
pr"
65p
M
1
e• 03
^ *
i
p
© tO
r-i
p.;
ajP
oj P
gs w
O ©
K»
O©
S—1£H
4-1 W
cd ©
fJS.
O &4
M EH
155
21
6.0
6.7
131
151
97
100
19
21
6.0
6.5
116
1 34
102
109
14
18
e v id e n t
th a n
H is
th e
th e
ta b le
81
5.0
th a t
th is
s u b je c t
m e d ia n
a b ilit y
o f
h is
fo u rth -
a b o v e -m e d ia n
a b ilit y
Is
n o t
o n ly
b u t
a ls o
as
m e a s u re d
by
e v i­
m e n ta l,
re e d ­
te s ts .
s u rv e y
show s
fro m
_i
20
la n g u a g e
¥ ^ o rk
toto
J>
*£H J
cm
d 1
© iO
106
a r ith m e tic
A
to
d co
qj to
£o
W I
o
.03
HiH1
©
PH
98
is
m o re
•
8*9
23
«
01
Standard
I t
Woody McCall's
Test
5-26-36
1 loodv McCall's
Test
1-24-36
Compass Survey
Test
5-25-36
Compass Survey
Test -1-21-36
Kuhlman Anderson
Intelligence Test
.
|
R e s u lts
s u b je c t’ s
th e
s ta te .
T a b le
T e s t
th e
e le c tr o d e r m a l
e m o tio n a l
s u b s ta n tia te s
Gates Heading
Test
1-17-36
i
.....
. .... .
7*
o f
th a t
th e
no
T e a c h e r’ s
in d iv id u a l
R e c o rd
a id
w as
o f
D a lly
g iv e n
to
161
th is
s u b je c t*
tio n
r e c e iv e d
g iv e n
to
h is
h is
o f
ed
to
th e
w o rk
9 2 ,
o f
s u b je c t
p ro c e e d e d
In
I t
o r
he
tim e
H is
e ffe c ts
as
th a t
a s p e c ts
m o re
fig u r e *
and
X II
w as
th a t
b o th
he
te n
H is
fro m
by
a tta c k
a fte r ­
n o r m a l;
c o n fid e n t
3 ; 30
he
v e rb a l
th a t
-
p ro c e e d ­
re p o rt
fo llo w
m a rk e d
s u b je c t’ s
c o m p a re d .
th e s e
ju d g e ,
o f
o r
th is
r e g u la r ity ,
on
he
o f
a
e x a m p le
d e fin ite ly
b e fo re
th is
he
d id
as
s h o u ld
h is
used
" h y b r id ”
and
v e r b a liz a tio n s
q u ic k ly
c o m p a re
s u b tr a c tio n
cas e,
I f
s te p s
" ta k e -a w a y ”
w as
and
a c c u ra c y
a p p ro a c h e d
b u t
th e
B -2 ,
s ile n tly
th e
f ir s t
th is
n o r
done
he
th e
s o lu tio n s .
seconds
h e s ita tin g
fro m
d is p a tc h ,
th e
on
p r o je c t.
m uch
m e th o d
th e
th e
w ith
to
ta u g h t,
th a n
in s tr u c ­
w as
and
A t
and
fa ilu r e
show n
com posed
B - l,
w hen
As
v io u s ly
o f
e la p s e d
o f
o f
The
end
la b o r a to r y
wc o c k i n e s s ” .
ju d g e d
w as
The
w ith
I
te m p e ra tu re
e v id e n t
a p p e a rs
i t
ta k e n .
p e rfe c t#
A ,
h is
n e ith e r
c o m p a re
l i t t l e
o f
e x a m p le s
a c c u ra c y
th e
a p p e a re d
v e ry
re c o rd s
th a t
w as
P r o je c ts
H is
e x a m p le s
is
to
1 93 6.
p o in t
th e
As
and
b e tw e e n
b ro u g h t
3 0 ,
b e a t,
to
h im
w as
M a rc h
h e a rt
a lm o s t
by
a tte n d a n c e
c la s s .
He
noon
H is
a
had
no
is
n e g a tiv e
id e n tic a l
m e th o d
ty p e
p u z z le d ,
f ir s t
e x a m p le
w as
m o re
w o r k in g
In v o lv in g
m e th o d s .
n o t
p re ­
p a u s in g
th e
in
been
w o rk .
g iv in g
succeeded
v e ry
have
" a d d itiv e "
D ,
ju d g e
i t
fo r
q u o tie n t
c a u tio u s
c o r r e c tly .
162
The
v e rb a l
f ic u lt y
b u t
u re
c o rre c t
and
onds
w e re
in
each
fig u r e s .
h is
r e fle c tio n
p ro c e d u re
o r
on
10
th e
o f
a re
e v id e n c e s
p o s s ib le
B ,
Is
tio n
e x a m p le s
c a lm
and
B ,
g re a te r
in
th e
p a tte rn
is
and
w r itin g
p r o b le m
d e c id in g
w hat
e x p r e s s io n s
m ean
and
I t
C -l,
Ir r e g u la r ity
and
re s p o n s e
and
h is
w as
In
C -2
d iv id e
a c c e p ta n c e
h is
m ust
and
do
g iv e
w as
th a t
r is e
be
H is
m uch
to
c o n a tio n ,
such
o f
s o lu ­
th a t
e le c tr o -
o r
th a t
he
f e lt
I t
d e s ir e ,
e le c tr o d e r m a l
on
s ig ­
fa ilu r e
succeed.
in te n s e
(C o n tin u e d
h is
D e fle c tio n s
c o n s id e r e d
to
D*
a p p a r e n tly
n o rm a l
h im
In
th is
th e
success
s tr iv in g
to
e x a m p le
in d ic a te s
th a n
so
s tr iv in g
he
e x te n t
m a tte re d
a n s ly s is
p ro c e d u re
D u r in g
s ta b le .
th e re fo re
a c c e p te d
in
fro m
s itu a tio n
g e n e r a lly
o f
fa r
th a t
i t
film
re c o rd
w hen
re c o rd
p o s s ib le
about
and
r e g u la r ity
is
In te n s e ly
sec­
second
I
in te r e s tin g .
com posed,
w e re
n ific a n t.
20
In d ic a te s
The
c h e c k in g ,
p ro c e d ­
------- "
to
u sed .
c h a rts
th e
th e
e s p e c ia lly
c o n d itio n
d e rm a l
p r o file
C -2 ,
e m o tio n a l
m uch
and
th e
M u ltip ly
e le c tr o d e r m a l
s u b je c t
w as
In
d if ­
p ro p e r
p r o b le m
see
d e v o te d
be
m e n ta l
th e
f ir s t
" le t ’ s
w e re
in d ic a te
C -l,
The
o f
o f
a t
p ro c e d u re
p r o b le m *
s h o u ld
v e rb a l
c le a r ly
e x a m p le s
th e
c o n s id e r a b le
a tta c k s .
The
re c o rd s
In
upon
seconds
p ro c e s s
o ffe re d
a r r iv e d
e x p r e s s io n
" - - s o m e t h i n g ------------------------ .
"
he
d e c id in g
H is
a p p r o x im a te ly
case
s o lu tio n *
sp en t
th e
p r o b le m s
page
re s p o n s e
174)
163
Verbal Report Record
III*
B
D.T.
5-30
.... fi
-
2 4
3 6 ) 8 6 9
7 g
14 9
1 4
4
7
10
15
15
14
-20
times--- 2 times 6 Is 12-- 2 times 3 is 6
and 1 is 7
-- 2 from 6 is 4---7 from B is 1---- bring
down thfl Q
————3 Into 14——is———4 — -— -4 times 6
1 & 24,
-- write the 2 ----- 4 times 3 is 12 and 2
.
25
J2
27
- -.2
0
.
.
.
.
.
.
.
164
Verbal Report Record
5 0 7
6 3 y 3
19
3 13
III.
C-l
D.T.
41
4 41
4 41
3-30
Seconda
Words
Verbal
sixty
10
15
teen
10
20
three Into 5 hxmderd en nine
5 times
30
12
10
38
40
5 times 3 ls--15
5 times 6 Is 30 end 1 is 31
--4 remainderbring down t h e 4—
-ty three intb 44 goes hero times
bring
irit'6"'42IsT
down the 1
7 times 3
is-----White --— carry the 2— * 7
times 6
forty
_______
Six
25
Report
two and 2 Is 44
165
Verbal Report Record
III*
C-2
D#T*
3-30
8 3 ) 2 8
A.-4
3
3
3
2
9
3
3
4 0
2 0
2
2
0
5
8
10
4
15
9
3 is 9---
20
4
---- is 24 -------9
25
9
from 12 is 3-— 5 from 8 is 3 ---
30
11
35
7
40
7
45
11
47
0
— — 83 Into 2 hunderd. en 82 — _____
3 times 8 is 24 -----
--bring down the 2-- 8 into 32 goes 4 times
----- 32 and 1 is 33--bring down the zero--- 83 goes into
zero zero times
166
Verbal Report Record
7X2
III*
D.T#
3-30
Seconds
Words
J 2 6 3
2 1 S
4 9
4 9
4 4
6
8 4
6 4
Verbal
Report
5
4
10
8
15
0
20
0
25
7
30
5
35
7
---- 3 times 7 is 21------ 6
40
4
from 14 is---- 8—
45
5
50
5
-- 2 from 6 is 4 -
55
4
60
13
— bring down the 4— -— —
-- -- *7 hunderd en 12 goes into 4 thousand
9 hunderd en 84
65
6
70
11
75
8
79
1
Into 2 thousand 6 hunderd en 34--- —
----- 7 times. 7 times 2 is
14— write the 1 ---- 7 times 1 is 7
and 1
la 8---- 7 times 7 is 49
167
Verbal Report Record
1. The population of a certain city In 1920 waa 43,568. Ten
years later there were 70,002 people In the city.
How
much did the population Increase In ten years?
2, A coal company collected $2,626 during the month of Feb­
ruary selling coal at §13 a ton.
Seconds
2jl
5
10
Words
8
____ 6
How many ton were sold?
Verbal Report
'+— --The population of a certain city
In nineteen
twenty waa -*.-.-— 43 million 5----
15
8
no--- 43 thousand 5 thou— - 5 hunderd
20
4
en 68— — ---- hunderd -— -
25
7
ten years later there were 70 thousand
30
9
2 people in the city*
35
©
--population increase in the ten years?-—
40
2
-- let,s see ------ (mumbling) -----
45
0
----- (mumble)------
50
0
55
0
60
2
65
9
from 12 is 4-— 7 from lO is 3 — -
70
9
6 from 10- is 4--- 4 from 10 is
75
6
6--- 5 from 7 is 2 ------
80
0
85
8
on
2
96
O
2.
.. 100 -
10 .-
Eow much did the
--- -— -twenty — ---six thousand 4 hunderd
an 34—
-— is that the way it1s spelt---- (that1s
rii£ht---all right • go ahead) A coal co.col*
168
No.2 (continued)
--1HR
... ...110.
___________
a
12
lected— (cough) -**--2 thousand 6 hunderd en
26 devt *1ans
nu
the month of February--— selling coal at
5
1
a t«n * Ho'1®’
115
4
---- many tons were sold?---- --------■
120
4
*---1 donft know—
125
5
ISO
5
--— — 13 into 26 gofifl
155
7
-—-2 times-*— --- 2 time a ?i js 6-_
140
7
145
13
2 times 1 is 2 **.«»•*.**•»-<-.
remif:1nder bring down the 2— — 13 into 2 goes zero—
bring down the 6 — -
150
7
11
160
6
164
O
— —
— -- somethin^----
-— multiply—
—
I mean d;Iv1 de— .(nh)
**—13 inta £26t goes
times
— 2 times 3 are 6-—
2 times 1 is 2
.2...
«**Mf
mm m*
# « » w
hunderd en 2 tons — —
....... .
. . .
they spl d
Ye
171
Film Analysis Record
2 4
3 6 ) 8 6 9
III
7_2
B
14
9
D. T.
14
4
3-30
5
III
0-1
D.T.
3-30
6 3 ) 3 1 9 4 1
3 15
4 4 1
4 4 1
Steps_______________
Frames
1.
Finds end writes the 2
49
2.
Multiplies 2x38, writes 72, draws line
44
3.
Subtracts 86-72, writes 14
34
4.
Brings down 9
13
5.
Finds and writes 4
30
6.
Multiplies 4x36, writes 144, draws line
63
7,
Subtracts 149-144, writes 5
18
8.
Checks, multiplies 24x36 - 5
43
9.
Remaining film
Ill . C-l
1*
Finds and writes the 5
75
2.
Multiplies 5x63, writes 315, draws line
69
3.
Subtracts 319-315, writes 4
26
4.
Brings down the 4
16
5*
Finds and writes 0
38
6,
Brings down 1
23
7*
Finds and writes 7
27
8.
Multiplies 7x63, writes 441, draws line
99
9#
Remaining film
26
172
Film Analysis Record.
3 4 0
8 3 ) 2 8 2 2 0
XU2 4 9
C-2
3 3 2
D.T.
5 3 2
3-30
0
7 12 T2
III,
D
D.T.
3-30
6 3 4 4
2 13
Steps
6
4 9 8 4
4 9 8 4
Frames
1#
Finds and writes 3
2.
Multiplies 3x83, writes 249, draws line
5*
Subtracts 282-249, writes 33
49
4#
Brings down 2
22
5.
Finds and writes 4
30
6.
Multiplies 4x83, writes 332, draws line
7.
Brings down 0
12
8.
Writes 0 in answer
28
9*
Remaining film
35
75
102
105
III. D
1* Finds and writes 3
2.
169
Multiplies 3x712, writes 2136, draws line 115
3. Subtracts 2654-2136,
writes 498
144
4* Brings down 4
5* Finds and writes 7
53
US
6. Multiplies 7x712, writes 4984,draws line 106
7. Remaining film
38
173
Figure 11
174
records*
It seems this case Illustrates that much
emotional stress and strain may be present In students
without apparent evidence*
Summary of Case Study Findings
Case 2#
D. T.
(A Pupil of Above Average Ability)
1.
New skills necessary to the solutions of
examples of Type B anc C have been acquired through
regular classroom instruction and practice.
2.
By means of skills end principles taught
in connection with long division examples of Types A,
B, and C* this subject Is able to proceed accurately
In the solution of Type D - an untaught type of long
division.
3*
The maintenance and remedial program Is
adequate.
Bonds necessary to previously formed skills
are maintained at a functional level whether they appear
In isolation or in new combination with other bonds In
new skills.
4.
Considerable time, 10 to 25 seconds, beyond
that necessary to the reading of the verbal problems
is needed to decide upon the procedure to be followed.
5,
Instruction in the additive method of subtrac­
tion and exposure to the variants of the "additive” and
175
"take-away" method results In an effective "hybridn
method*
6*
Failure to judge or compare does not affect
this subjectfs work*
7*
The verbalization records and accompanying
Verbal Profile Charts Indicate clearly the pauses
Incident to finding first quotient figure in D and
arriving at
method of attack in each of the verbal
problems.
8*
The outer emotional calm of this subject
belles the inner stresses and strains evidenced by
his electrodermal response records.
176
Casa 3. R. W.
Table 19
Standard
«o
- tb
HI t
r-ic.O
cdoa
O 1
O to
Iowa Elem, Lang,
Test
5-27-36
m (,o
- to
rH |
|^
aJcm
OO r1
H
Gates Reading
|Test
6-27-36
R.W.
Bxp *Gro\ip
Medians
Class
Medians
*
gt w
O ©
h-iJjH
104
107
109
21
18
8*3
8.6
142
184
109
98
106
20
21
6.0
6.7
131
151
108
97
100
19
21
6*0
6.6
116
134
102
109
14
18
4.6
6.0
.
Subject
-p
«a
<D
EH
Gates Reading
Test
1-17-36
_
I Xuhlman Anderson
j Intelliaence Test
1 ...
~
Compass Survey
Test
1-21-36
-
Test Results for Selected Case
<D
>
ca
w to
w to
cd i t-s
trj-P
OjLD
£ OJ O <n o ©
o 1 o © O <D
o UO fe1
,.£h s&jitEh
S1
»
o
w
Qi
This third subject possesses average mental
ability end considering all of the test scores record*
ed for him has achieved more than the median of his
class or the experimental group*
As in the case of the second subject, no
special individual aid was given*
Regular classroom
instruction and study during the interval between Pro­
jects I and III was in the teacher1s judgment sufficient*
This subject was brought to the laboratory
at 8 o *clock on the morning of March SO, 1956.
temperature was 98*4, his pulse beat, 84.
His
He was in
(Continued on page 189)
177
Verbal Report Record
III.
B
R.W.
3-30
Seconds
2
3 6 T 8 6
7_8
14
14
Words
4
9
9
4
g
Verbal Report
36 goes Into 86
twice
2 times
write the Sunder the
10
11
15
21
1 is 7-— write it under the
8-draw a line and subtract
4 and 2
13
are 6-— 1
and 7. are 8 _
— 14 Is smal 1er than 30— -brIng down 9-thirty*---six goes 5_____________________
11
hunder~~a bunderd en 49
--4 times 6
write the 4 carry
tbe 2 -—
20
25
30
35
40
44
46
-4 times 5 is 12 and 2
5 and 4 are 9
is 14—
5 remainder
178
Verbal Report Record
III.
c-l
R.W.
6 3)5
i 9 4 1
1 5
4 4 1
4 4 1
-- 60 goes Into 5-— Into 3 hunderd 19
___5__ .. 9
times--- -5 times 3 Is 15--- write the 5— earry the 1—
..... 10
12
15
9
—-;
— 5 times 6 is 50 *>nd oarny the 1
..... SO
9
51- —
25
6
--— 4 a n d 5 a re 9 -—
-- 4
is larger than 63-— -bring down the 4-— 63
goes into 4 4 - - - - Karo times
-— bring down the 1-— -sixty--- three goes
into 4 h u n d e r d and—
_
17
— w r i t e the- — -drpw a line nnd snihtrs
55
11
40
5
45
9
times 3 is 21---- -carry the 2--- -7
50
8
times 6--— -6 times 7 is 42---
55
7
and--—
57
8
line and subtract— — ! from 1 is zero
58
O
2 Is 44---- draw a
179
Verbal Report Record
5 4 0
8 3
XU.
C-2
K.W.
3*50
Seconds
Words
Y^~Wt~To
2 4 9
3 3 2
3 3 2
o 0
Verbal
Report
3
8
«»— - Q3 goes Into 2 hunderd 82*-- --------
10
8
15
17
20
10
25
10
-— 4 *---3 time a-*-*5 times 5 is 9--______
writ*© It under the 2— — 3 times 8 is 24
—
write it under the 28*—
— draw line and subtract-— - the 2 is
smaller than 9
so it becomes e
and 9 are
12
---------------------------------------------
30
8
35
14
40
12
45
10
50
16
54
12
55
0
---bring down the X-— -- 4 becomes a 5*— -" - — -3 and 5 are 8— --------33 Is smaller
than 58-- S3-___________________
bring down the 2-— ~*83 goes*-— -In— -Into
a 3 hunderd_____________________ __
en 32-— 4 times*— *4 times 5 ia 12***
— 4 times 8 is 32 and 1 Is 33-— draw a
line and su btract
-‘— zero— bring down the zero— -83 goes Into
zero*zero times______
_ _ ______ _
180
Verbal Report Record
7 12
1X1.
D
R.W.
3-30
Seconds
Words
) 2 6
6
7
9
3 4 4
4 8
6
8 6
Verbal
Report
*ummhu<
7 hunderd
en 12
goes Into---- ah— — --
JSL
----2 thousand 6 hunderd en 3420
25
30
35
12
40
14
45
12
50
9
— -9 times
— 9 times 2-- Is 18----write the 8 under the 4 en carry the 1. 9 times
7 Is 63 and 1 are 64-— — --write It — 63
-— --draw
--- -— -4 is
a line and aibtrsct
smaller than 8 so It becomes_____ _
14--------- 6 and 8 are 14— — bring
down t h e ____
.60
JJL
1----- 4 becomes a 5-— ----— your 3 Is smaller than your 5 so It be<
comes a 13---— — 8 en 5 are.-15.-— -------
65
8
70
0
55
bring down the 1------ 6 becomes a 7----
sixt*
75
-een— — is---—
— -
m
-<—
»_•----9and7 are 16
— -----(moving picture camera stopped?-— <
(go right on donyt stop j^st because that
stopped you go right ahead)
85
90
95
0
.— •(or c a n H you go aheedT
(you can*t go a h e a d ? ) ________
181
III.
R.W*
D
Seconds
96
(continued)
Words
Verbal
Report
(all rlgbtf stop)
because we*ve never had it
■iop
Verbal Report Record
1* The population of a certain city In 1920 was 43,568* Ten
years later there were 70,002 people in the city*
How
much did the population increase In the ten years?
2* A coal company collected $2,626 during the month of FebX* ruary selling coal at $13 a ton* How many tons were sold?
R.W.
3—30
Verbal Report
No.l
— — -The population of a certain city in
5
8
nineteen
10
7
twenty was 43----thou, sand- 5 hunderd
15
6
en 68*-— -Ten years-— -later there were- —
20
3
7--- 70-— -thousand----
25
4
30
9
-- --ah-----£-~— hunderd---— people
In the city* Sow much did the population
Increase?
35
7
In ten years? Subtract---- 7— 7 thousa
40
1
and — --— 2------ -
45
5
(chuckle) umhuhu by 23 thousand
50
3
55
4
eighty -— — -- - - 2 Is smal
60
7
65
14
70
17
ler than 8 so It becomes a 12— ----4 and 8 Is 12-— bring down the 1---- zero becomes a 1— --- — zero
Is smaller than 1 so It becomes 10--- 9 an*
1 are 10----
75
4
80
4
85
10
90
11
95
16
100
4
—
so It becomes a 10 — ----4 and 6
are 10-—
bring down the 1— -— 3 becomes a 4 -*--- zero Is smaller
than 4— -so It becomes a 10— — 6 and 4 &r«
1 0 --- brlmz dowp the 1---4 becomes a 5 ------
185
III.
R.W.
Verbal problem (continued)
Seconds
Words
--- 105__ ^
No.2
.110
7
-1X&
9
———2 and 5
7--.--- 2fi
thousand 4 hunderd and 94 -— AfiOfil ^fnr^p«ny
collected 2 hund-— 2 thousand 6 hunderd
26
...120.
7
dollars during the month of February sell
12b
5
___ISO
_ io
ln& coal at $13 a ton
How many tons were s o l d ? --- — -- 13 goes
into 2 thous
_
15b
2
_
34,0
2
145
5
350
o
355
A
3ftp
35
165
14
170
15
375
IO
380
36
.
_
9
6
TQn
191
5
and 6 hunderd
13 goes In 26-— —
-
. ..
tiarioa
2 tlmost 5
is 6— — write it under the 6-— — 2 times 1
—
Is 2 wr4 tii% 4t unrltfir
the 26— — draw a line and subtract ----26
fr»A»
4 a ..._
zero--- bring down the 2 --- -13 goes into 2
— zero times--— bring down the 6
--13 goes into 26—
twice— — 2 times
3 is
.......
......
6--- 2 times 1 is 2-— -draw a line and
^nhtraet— —
6 f r o m 6 is
zero— -— *— (What is your answer that you
havA?).... 2 hunderd
en 2 tons----- (tons of what, do you suppose
— — ^ o a h — (tons of coal) in February
(in February* all right, stop)
__
184
Film Analysis Record
III*
B
R.Vn
3-50
3 6)8
7
1
1
6 9
2
4 9
4 4
5
0 7
6 3)3 1 9 4 1
3 1 5
4 4 1
III.
4 4 1
C-l
R.W.
0
Steps
Prames
1.
Finds and writes 2
36
2.
Multiplies 2x36, writes 72, draws line
91
3.
Subtract 86-72, writes 14
52
4•
Compares
28
5.
Brings down 9
6.
Finds and writes 4
7.
Multiplies 4x36, writes 144, draws line
8.
Judges
7
9.
Subtracts 149-144, writes 5
8
10. Remaining film
8
67
105
31
Ill . C-l
1.
Finds and writes 5
74
2.
Multiplies 5x63, writes 315, draws line
95
3.
Subtracts 319-315, writes 4
58
4.
Compares
28
5.
Brings down 4
6.
Finds and writes 0
20
7.
Brings down 1
22
8.
Finds and writes 7
74
9
185
Film Analysis Record
(continued)
Steps____________________
9.
Frames
Multiplies 7x63,writes441, draws line
10. Judges
11. Subtracts 441-441,writes
12. Remaining film
148
7
0
6
4
186
Film Analysis Record
_____ 5 4 0
8 3 y 2 8' 2 2 0
III
C-2:
R.
3-30
7 12
III
2 r4 9
) 2 '6 '3' 4 4
6 74 5-8
3 3 2
3 3 2
0 0
Steps
1*
Finds and writes 3
2#
Multiplies 5x83, writes
3#
Subtracts 282-249, writes 33
4.
Compares
43
5*
Brings down 2
11
6*
Finds and writes 4
59
7.
Multiplies 4x83, writes
8,
Judges
5
9*
Subtracts 332-332, writes O
5
80
249, draws line
332, draws line
79
150
65
10. Compares
8
11. Brings down 0
6
12. Finds and writes 0
15. Remaining film
19
8
III. D .
1.
Finds and writes 9
245
2.
Multiplies 9x712, writes 648, draws line
151
3.
Subtracts 263-648, writes 986
421
4.
Compares
12
s~
:s:
189
good h e a l t h a n d s p i r i t s ,
w ith
th e
in te r e s te d
In
p r o c e e d in g
w o rk *
As i s
e v id e n t
fro m
th e
v e r b a liz a t io n
pp. 177-184, h e p r o c e e d e d w i t h o u t h e s i t a t i o n
ly
s o lv e
C~2 .
e x a m p le s B , G - l ,
p re s e n te d
th u s
The ne w n ess
fa r
of
in
th is
v e r y b e g in n in g
as is
h is
E x a m p le D ,
c la s s ,
o ffe re d
e x a m p le d i s t u r b e d
e v id e n t fro m
h is
re c o rd s ,
to
c o rre c t­
a ty p e
not
d iffic u ltie s .
h im
fro m , th e
fir s t
v e r b a liz a ­
” ----------------------- 7--------------------- ummhu
tio n :
7 h u n d r e d en 12 g o e s i n t o
2 th o u s a n d 6 h u n d e rd en
34
-------
ah ------------
----------------
ft
H a v in g
fig u r e
a r r iv e d
he p ro c e e d e d
one d i g i t
In
at
h is
th e
fu rth e r
d iffic u lty
fir s t
s u b t r a c t io n .
q u o tie n t
m u lt ip lic a tio n
o f th e m u lt ip lic a n d
s till
in c o r r e c t
to
w h ic h i n v o l v e d h i n
w h e n he p r o c e e d e d
A bout th is
t im e
s u c c e s s fu l
in
to
“ you c a n ’ t
go a h e a d ? ” w as:
q u e s tio n
w e ’ ve n e v e r had i t . ”
w ith
th r e e
a s a. t y p e
d ig it
can p ro c e e d
F o r th is
d iv is o r s ,
in v o lv in g
Type D , n e e d s t o
in
its
T h e v e r b a l p r o b le m s
F a ilu r e
to
s o lu tio n .
s u b je c t lo n g
a ne w p a t t e r n
s u c c e s s fu lly
d iffic u ltie s .
h is
in
w ith h i 3
he becam e c o n v in c e d
t h a t h e w as n o t
th e
ig n o r e
o f s k ills
H is
answ er
“ because
d iv is io n
be p r e s e n t e d
b e fo re
he
s o lu tio n .
d id
n o t o ffe r
c o p y on e d i g i t
a n y s e r io u s
c o r r e c tly
19D
Figure 12
Electrodermal Response Record.
(Reproduction)
Project III
R.W.
wV
191
resulted In an eri'or but procedure and solutions were
otherwise correct.
This subject uses the pur© additive method
of subtraction taught him.
There is no indication in
his verbalization that he judges In any of bis solutions
but he does compare w
bring down 9,”
14 Is smaller than 36 — -*
His failure to judge did not affect
his solutions to examples B, 0-1# C-2 but it is very
probable that If he had judged in his solution to
example D that he would have seen the Incorrectness
of his first quotient figure and proceeded to make
the necessary corrections.
The verbal profile charts Indicate quite
strikingly points of difficulty for this subject.
'The
Film Analysis Record likewise substantiates these places
of difficulty in terms of frames of film used per pro­
cess or step.
The electrodermal response record does not
Indicate extreme variation from the subject’s usual
electrodermal response record pattern.
There Is a
slight variation during the early pert of his solu­
tion to example P which may have been caused by the
subject’s Inability to promptly proceed In Its solu­
tion.
Summary of Case Study Findings
Case 3*
R* W.
(A Pupil of Average Ability)
1*
Instruction given in the solution of examples
of Type B and C Is adequate to the acquisition of the
new skills involved.
2.
Skills learned In solving examples of type
A, B, and C are not adequate to the solution of ex­
amples of type D.
3.
equate.
The maintenance and remedial program Is ad­
Previously acquired skills function either
In isolation or within more complex patterns, Involv­
ing newly acquired skills.
4.
Very little time beyond that required In
reading the verbal problem is required by the subject
to decide the method that shall be used In its solution.
5.
Though exposed to four methods of subtrac­
tion, this subject adheres to the originally taught
"additive" method.
6.
T M s subject fails to “judge" but does “com­
pare” before bringing down digits from the dividend.
7.
The electrodermal response recoi’d does not
193
effectively substantiate other records st points of
apparent disturbance.
Q*
The verbalization record end accompanying
Verbal Profile Chart gives evidences of prompt, con­
fident, steady attack on examples A, B, end Cj evidence
of slow, hesitant attack on example D.
9.
The film analysis record substantiates the
evidences of pauses and hesitating attacks that appear
In the verbalization records.
Summary Treatment of Experimental Group
In the treatment given materials secured
from the sixteen subjects, written responses to examples
will receive first attention.
The skills and errors
involved In Project III were studied, listed, and
ass igned symbols•
1.
Basic Division combinations
a. 1st quotient figure
b* 2nd quotient figure
c. 3rd quotient figure
2.
Placement of quotient figures
a. 1st quotient figure
b. 2nd quotient figure
c. 3rd quotient figure
3.
MultIp11ca 11on
a. 1st product
b. 2nd product
194
4.
Subtraction
a. 1st difference
b. 2nd difference
5#
Miscellaneous errors
a. Fells to bring down quotient figure
b. Bring3 down wrong quotient figure
6*
Non-completion csmsed by
a* Lack of time
b. Increased difficulty
c. Sensing something wrong
d. Any of above 3kllla or errors
e. Skills not listed above
The two skills, "judging” and "comparing"
were omitted because their presence or absence was not
apparent In the written solutions#
These skills will
be Included in the study of verbal response records.
In the example analysis, symbols are to be Interpreted
in this manner: 2b, placement of 2nd quotient figure;
4b, subtraction to obtain the second difference, etc.
The summary page 196
Indicates that 46 of
the 64 solutions were correct, 11 were incorrect, and
7 incomplete.
Of the 11 Incorrect solutions, 4 were
attempted solutions to example D, a type which had not
been taught prior to
he project.
Of the 7 Incomplete
solutions, 4 were attempted solutions to example D.
Eight of the 18 unsxiccessful attempt3 can therefore
be attributed to example D.
It appears that Instruc­
tion In dividing by three-digit divisors Is necessary.
Skills and procedures incident to solutions for types
(Continued on page 197)
195
Example Analysis
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196
Kxaraple Analysis
Project III
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Summary
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197
S u m m ary
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th e
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it y
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and
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28
was
d iv id e n d .
In
to ta l
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f e llin g
2
4
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to
th e
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5
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to
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D*
s o lv e
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n o t
2
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10
tre a tm e n t,
ty p e s
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in
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down
end
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C
a b il­
does
ty p e
d ig its
not
D.
s u b tra c tio n ;
q u o tie n t
D
seven,
fig u re s ;
fro m
th e
198
Dictaphone Record Analysis
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S p e c i a l R e m a rk s
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Dictaphone Record Analysis
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Special Remarks
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200
There had been a decided shirt since Pro­
ject X in certain types of errors made.
In the first
project finding first and second-quotient figures ac­
counted for more than 50$ of all erx*ors; in this
project, 21*4$.
Subtraction accounted for 28$ of the
errors in Project X; 39*3$ in Project III.
Multi­
plication accounted for 20$ of errors in Project I;
25$ in Project III.
Examination of the exact nature of errors
made reveals that a number of them are spurious or
caused by the fact that the skills Involved are burled
In more complex patterns.
The follow3.ng errors made
by different subjects are of this natures
3x3=6;
3 x 2 = 12;
3 x 8 = 23; 2 x 3 = 8; 4 x 6 - 36*
The Dictaphone Record Analysis Indicates,
as In the case of Project I, the manner In which the
verbalizations were treated.
The four types of sub­
traction methods present In Project I also appear in
Project III and are used by the same subjects.
These
methods were:
’’Take-away11
9 8
3 7
5 9
1.
We cannot subtract 7 from 6 so we borrow
a ’’tens” from 9. 7 from 16 is 9; 3 from
8 is 5.
201
5 3 9
4 9 8
4 1
2. 8 from 9 is 1,
You canft take 9 from 3,
so the 3 becomes a 13.
9 from 13 Is
Because you made the 3 a 13, the 4 be­
comes a 5* 5 from 5 is zero.
4*
"Additive"
(Austrian)
5 3 2
2 8 7
2 4 5
1. 7 and 5 are 12.
Carry the 1 to the 8
and make that s 9#
9 and 4 are 13.
Carry the 1 to the 2 and make that a 3.
3 and 2 are 5.
7 3
6 6
7
2. The 6 is larger than the 3, so you take
1 from the 7. The 7 becomes a 6, and
the 3 a 13. 6 and 7 are 13.
Bight subjects use some one method consistently in
their solutions, eight subjects do not.
Of these
latter eight one subject actually uses an "additive"
method at one time and a "take-away" method at another and occasionally within one example will shift from
the one method to the other.
The symbols relative to "judging" and "com­
paring" as used in the table are to be interpreted as
follows:
y , did judge or did compare; o, did not judge or did
not compare.
Blank spaces Indicate that the particular
step was not reached in the solution.
Some of the data included in this analysis
record has been recast in order that they may be more
easily interpreted.
As in Project IV only three
202
Table 20
Verbalization Analysis
Subtraction
Methods
Solutions of those Solutions of those
examples in which examples in which
subjects:
subjects did not:
compared
Judge
compare
CM
1, R*W
6* N«T
2
o
16» B.R
15 1
24
2
51 13
1
22 12 1
203
subjects consistently used the pure "additive" method.
Two subjects consistently used the hybrid "additive"
method; two subjects, the pure "take-away" method;
and two, the hybrid "take-away” method.
The remaining
seven used T - 1 and T - 2 intermittently in their
solution.
Of the sixteen subjects, 4 "judged” end
"compared” , 1 "judged” only, 11 "compared” only and
5 neither "Judged” nor "compared”.
Only two of the
four subjects who "judged” and "compared” did so con­
sistently.
Others were net consistent in their "judging"
and "comp aring".
It is very evident from this table that
students who Judge are credited with more correct solu­
tions than are those who do not judge.
Thirteen solu­
tions out of 14 were correct for those who judged in
contrast to 51 out of 45 for those who did not judge.
The same is evident for those who "compare”:
of 26 correct, in contrast to 22 out of 35.
24 out
It Is
clear that students who "judge" and "compare” as they
proceed In their solutions make fewer errors.
While
this may be true because only the abler students do
"judge" and do "compare", there are several Instances
as seen on the original record analysis where failure
to judge or compare were directly responsible for the
errors that followed.
204
Th© terms "judge” end "compare” are used
by only three subjects*
Others who do judge or do
compare proceed to do so without naming the process as
e.g.
"You can subtract so you do” and ”th© 63 Is
smaller than the 72 so you bring down the 4.”
Summary of Findings Based on Sixteen Subjects
1.
Instruction in the solutions to long-
division examples of types A, B, and C does not make
possible the solution of type D.
Special Instruction
for type D Is needed*
2*
Placement of quotient figures offers
little or no difficulty*
3.
Finding first and second differences
account for 39.3$ of the errors.
4.
Finding first and second products ac­
count for 25 percent of the errors*
5.
Finding first end second quotient figures
account for 21.4$ of the errors.
6.
Several errors as e. g. 3 x 2 8 12 are
either spurious or occur because old skills appear In
different and coirqplex patterns.
7.
Four types of subtraction methods are
205
used:
T-l, T-2, Ad-1, and Ad-2.
8.
Many subjects are not consistent In their
us© of subtraction methods.
On© subject actually
uses aspects of the pure "take-away" and the pure
"additive" in the same example.
9*
Failure to "judge" Is accompanied by an In­
creased number of Incorrect and Incomplete solutions.
10.
Failure to "compare" 13 accompanied by an
Increased number of Incorrect and incomplete solu­
tions.
11.
used.
The terms "judge" and "compare" are seldom
Those students who judge and compare proceed
to do so without naming the process.
Summary Treatment of Class
The results obtained when the examples of
Project III were presented to the remaining fourthgrade pupils are given In the form used for the example
analysis for the sixteen subjects.
The symbols within
the analysis refer to the same skills and errors.
Example Analysis
An examination of the summary of correct,
incorrect, and Incomplete solutions reveals, with the
206
exception of example D, very similar performance for
class members and members of the experimental group.
Examples B, C-l, and C-2 did not offer unusual diffi­
culty*
Example D, long division with three-digit
divisors, did offer much difficulty.
Of the twenty
students attempting this example, ten were unable to
complete the solution, five were able to complete the
solution but incorrectly, and only five were able to
complete the solution correctly.
It is very evident
that special Instruction in the solution of such ex­
amples Is necessary.
Skills and principles adequate
to the solution of examples B, C-l, and C-2 are not
adequate to the solution of examples of Type D.
In the solutions and attempted solutions
to the 80 long division examples, 41 errors occurred.
Thirteen or 31*7$ of these errors were made In finding
quotient figures.
This percentage figure is somewhat
larger than that for the sixteen subjects which was
only 21.4$ but It does Indicate that finding quotient
figures is done more accurately than during the first
project when finding quotient figures accounted for
more than 50$ of all errors.
Subtraction and multiplication errors ac­
count for 26.8$ and 24.4$ of the total number of errors,
respectively.
The percentages in the case of the ex­
perimental group were 39.3 and 25.
207
E x a m p 1 e A ns. l y s i s
Project III
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Exact Nature of Error
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Example Analysis
Project X U
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Subject
9, A* C.
10. W,0.
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Errors
Exact Nature of Error
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Exarvip1e Ana 1y si s
Project III
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B 15 2 3
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210
Skills
B
0
1
H
Summary of Class Errors
C-2
D
1
1
2
1
5
5
1
5 ] 15
5 r
5J
1
1
1
1
1
1
2 I7
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la
lb
lc
1
2a
2b
2c
1
5a
5b
1
1
1
4a
4b
4c
1
1
Totals
4
Total
4^
1J
2
2
2
5
1
1
2
1
1
10
10
17
5 l 10
6/
51
4 I 11
2J
41
It Is Interesting to note that 17 errors
or 41.5$> of all errors were made by students trying
to solve example D*
Of the eleven Incomplete solu­
tions, six were attempted solutions to example D.
These figures further Indicate the need of special
Instruction In the solution of such examples*
Not
to give special Instruction actually provides for
students situations In which "error” and "failure"
with their negative effects of necessity occur.
211
Hypotheses and Tentative Conclusions
Project III
Summary findings for each case study, for
the experimental group, and for the remaining fourthgrade pupils will serve as bases for such hypotheses
and conclusions as will be listed*
1*
Instruction In the solution of long-dlvision
examples of types B and C is adequate for the superior,
the above average, and the average pupil.
2.
Skills and principles Involved in the solu­
tion of examples A, B, and C make possible the solu­
tion of examples of Type D for the superior and above
average pupil but not for the average and below-average
pupil*
Special Instruction In the solution of examples
of Type D is needed.
3*
The maintenance and remedial program Is such
that previously acquired skills function
readily
and
smoothly whether in Isolation or in combination with
new bonds or skills*
4.
used*
The terms wjudge" and "compare” are seldom
Students who judge and compare proceed to do
so without naming the process*
5*
Few subjects "judge” and "compare"; some
"judge" but do not "compare"; some "compare" but do
not "judge"; many do neither.
Those who "judge" and
212
^compare" do not
6*
do so consistently.
Failure to judgeis accompanied by an in­
creased number of Incorrect end Incomplete solutions*
7*
Failure to compare is accompanied by an In­
creased number of Incorrect and Incomplete solutions.
8.
Spurious errors as 3 x 2 equal 12 occur*
9.
Four types of subtraction methods are used.
In some Instances two different methods are used inter­
changeably by the same subjects; In at least one case,
aspects of two methods appear within one solution.
10.
Placement of quotient figures offers very
little difficulty.
11.
ficulty
12.
Finding quotient figures causes
much less dif­
than was the case in Project I.
Finding products and differences continue as
approximately equally difficult tasks.
13.
Very little time beyond that required In read­
ing verbal problems is required by students to decide
the method that shall be used in their solutions.
14.
Verbalization records and the derived profile
charts give evidence of the type of attack that is
made, I.e. whether a confident and steady or a hesitat­
ing attack is made.
15.
Film Analysis Records substantiate the reg­
ularity and irregularity of a subjectfs work as those
qualities appear In other records*
213
16*
Occasionally ©lectrodermal response records
indicate ©motional stresses and strains which ©re not
evident In the outward behavior of some subjects; at
times what appears to toe emotional instability Is not
substantiated by the records*
Froject VI
a*
Learning Ejqierlence Prior to the Project
The long division learning experiences between
Project III and VII were such as to teach the skills
necessary to solve long division examples of the re­
maining type, !•©. long division with three-digit
divisors, long division with zero In the quotient, ©nd
long division with non-apperent quotients*
As was
evident in Projects I and III the new learning ex­
periences provided by the text and the maintenance and
remedial material provided for by the textbook, teacher
and Supervisory Plan were generously ©nd intelligently
supplied.
b.
Selected Case Studies
The table on page 215 shows the three sub­
jects chosen for individual study.
Their relative
rankings in the experimental group and in the class
as © whole is evident from the test scores and med­
ians recorded*
214
Case Is
C. P#
T M s subject possesses ability above the
median of his class in all of his school work.
His
test scores In arithmetic, reading, and. language as
reported In Table 21 are definitely above the medians
for the exp erImental group and for the class.
He was brought to the laboratory on the
morning of May 27, 1936 with two other subjects.
The
Individual Data Record indicates that ids temperature
and pulse were normal, that he had had a good night’s
rest and was In good health.
He began working the
examples of the project at 7:20.
The special record forms interspersed in
the treatment of this case will serve as bases for
the discussion that follows.
The verbal report records
indicate that the instruction and maintenance program
throughout the semester had been adequate.
This subject
had complete mastery of long-division procedures but
did make two computational errors, one in subtraction,
and one In multiplication.
He had some difficulty in his solution to
example of type C.
His incorrect third quotient figure,
6, resulted in the product 432.
had made a mistake: 11—
He realized that he
---------------- no
(laughs) *---------that goes ah —
8 times’*.
*
Because
(Continued on page 232)
215
Table 21
C. P.
117
103
106
Standards
Sf*?
,
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21
7.0
7,5
14-9
162
20
23
5.7
5.2
142
160
88
12
20
5.5
G.5
105
110
98
106
20
21
6.0
6.7
131
151
97
100
19
21
6.0
6,5
116
134
102
109
14.0 18.0 4.5
5.0
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into
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98
96
102
85
109
108
S. F.
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Exp.Oroup
Medians
diass
Medians
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CP ^
!Oates Reading
|Test - 5-27-36
Compass Survey
Test - 1-12-36
O E-<
Woody McCall's
Test - 5-26-36
Subject
Euhlroan Anderson
Intelligence Test
1
j
Test Results for Selected Cases
CDtD
trjiO
ra |
St-p
<0
85
216
Verbal Report Record
5
VI.
A
C.P.
5-27
Seconds
Words
.. .«L.
10
. 10
14
.JLS....
10
20
17
25
15
50
12
55
13
40
9
41
0
6 y 24 0 8
2 2 4
16 8
16 8
Verbal Report
*56 into 2 hunderd ©nd 40 goes
— 4 times
4 sixes are 24 write the 4 and carry the 2—
4 fives
are 20 and 2— 22 canft take 4 from
naught so the naught becomes a 10 4 from 10
Is 6--because you made the naught
a 10 the 2 becomes a 3 5 from 4 is 1
you bring down
the 8 5 hunderd -— 56 Into 1 hunderd and
68
goes 5 times 3 sixes are 18 write the 8
and carry the
1
3 fives are 15 and 1 is 16
217
Verbal Report Record
3
8 2 T 2 7 9
2 4 6
3 3
5 2
VI.
B
C.P.
5-27
4
9
9
8
5
11
-— -82 Into 2 hunderd and 79— goes 3 times
10
13
3 twos are 6
15
11
3— 4 from 7 is 3
20
9
25
10
4 times -4 twos are 8
30
10
two
35
6
eight —
37
0
—
3 eights are 24
6 from 9 is
you bring down the 9
82 into 3 hunderd ©nd 39----- — goes — -
2 from 3 Is 1
4 eights are— - thlrt
1 from — - 1 from
- 8 from 9 is 1
(cough) -— -- ---- (All right, stop.)
218
Verbal Report Record
6 0 ©
7 2 } 4 3 84 8
4 S2
4 8
57 6
9~2
VI.
C
C.P.
5-27
Seconds
~~S
Words
... J5
a
- .10
___ 14
is
15
.20
16
.
_
----25...
. SO
11
,
10
Verbal
Report
72. fntp, 4 h^ndo-pd
^58
-- goes 6 times 6 twos are 12
and panny the 1
write the 2
6 sAvann »r»o 4ft cmd 1 bti® 4** 9 f m m m - — a
is 6 you bring down the 4 72 won’t go
fnto 64 fin ynu
„
put down the zero you bring down the 8
_7.2.into ..
.......
6 hunderd and 48
goes fi t f t w o s
are 12 write the 2 and carry the 1
6 sevens are
35
12
40
7
45
1
50
5
55
16
60
5
and 1- is 57
_ 65
a
6 from 8 is 2
70
5
is 9 (coughs) what’s still wrong
75
O
80
0
85
0
90
7
94
O
..
~— -no— (1 smgha)
,
- that goes ah
8 times
8 twps are 16 write the 6 and carry the 1
8 sevens ere 56
5 from 14
--- (See where there’s anything wrong, Clair
T don’t flAftTfn to f*1pd ft ^nw— «
(A11 night, atop )
.
219
Verbal Report Record
j”
2
4
2 3 6 y 6'6'6 4
VI.
i s7 s8
D
8 84
O.P.
& 4? 8
5-27
17 6
Seconds
... .....
Words
13
Verbal Report
— -- 2 hunderd and 36 into 5 thousand 6
hunderd and 63
... 10.., ..
9
goes ——— *■*2 hunderd and 36 Into 5 h^’
pden^d
..-.1.
5. ,
9
and 66— "*goes 2 times 2 sl^ea one
18- write the 8 and carry the 1 -2 threes
are 6 and 1 Is 7
.....
2 jQ
17
25
11
13
_
35
11
2 twOS are 4 —you P.an *t talre the A fr>nm e>
so the 6 becomes a 16 - 8 from 16 Is 8
- beAfiiise yon
made the 6 a 16 the 7 becomes an 8 you
you canft take 8 from 6 so the 6 becomes a
1A
4.5
13
50
.
Is 8 — because you made the 6 a 16 the
4 h(5iij.nTn«fl a
.......
5 - you brung down the 4 - 2 hunderd
end 36
Inta 8 hunderd and 84 goes 4 times
A sil-VA0
pA
write the 4 and carry your 2 - 4 threes
55
14
■60
13
65
13
70
1.0
75
9
80
18
---- 3 sixes are 18 - write your 8 and carr
your 1 - 3 threes are 9 and 1 Is 10 - write
your zero and earry y m r 1 - 3
85
io
twos are 6 and 1 Is 7
90
5
Is 6 — ---- -- l from 8
95
y
is 7
_ 98
..6_
.
are 14 - write the 4 and carry your 1
- 4 twos are are 8
and 1 Is 9 -— - that should have been
three times
-
A fnnm 74
7 from 8 is 1 ----
I didn*t get that one --(All..right f atnpj--
220
Verbal Report Record
3 9
n *
B
G*P5-27
Seconds
Words
7 3
4 V
g7 3
117
117
---
T 28
Verbal
Report
5
9
—
10
4
15
15
— — 8 times 8 nines
are *72 write the 2 and carry the 7
8 threes are 24
20
6
25
10
30
14
35
12
40
14
45
12
50
8
51
0
39 Into 2 hunderd and 84 - goes--—
-
---no, that’s wrong-— that should he
-— 7 times (laughs) sev— — V nines are
63 write the 3
and carry the 6 7 threes are 2l and 6
are 27
3 from 4 Is 1 - 7 from 8 is 1 you bring
down the 7 thr-— 39 lnta 1 hunderd n-7
goes 5 times
3 nines are — — 27 - write the 7 and
carry the 2
3 threes are 9 and 2 are 11
221
Verbal Report Record
1* A quail was marked In the south one spring.
During the
summer it was seen 1904 miles farther north.
At its
fastest flying speed (56 miles per hour) how long would
It have taken the quail to fly that far north?
VI.
C *P.
-— A quail \... s marked in the south one sprl
During the summer--(clears throat) it had been — — --9 hun
— 1 thousand 9 hunderd and 4 ml lea
5
12
10
12
15
9
20
15
25
7
that far north?
30
7
Into--- 1 thousand 9 hunderd and 4 ----
35
7
40
14
56 Into 1 hunderd and 90 ---- goes
3 times-— -- 3 sixes are 18 -write the 8 and carry the 1
45
8
50
15
55
15
60
9
65
“11
70
17
75
15
Rf>
12
farther north. At Its fastest flying sneed
miles per hour how many— -how long would It
have taken the quail to fly
You divide fifty--- six
3 fives are— 15 and 1 is 16
__
you can’t take 8 from naught so the naught
becomes a 10- 8 from 10
is 2 - because you made the naught a 10
the 1 becomes a—
the 6 becomes a 7 - 7 frnm
1»
7 from 9 Is 2 - bring down the 4
- 56
into 2 hundrd and 24 goes — 4 times
4 sixes are 24 write the 4
and carry the 2 - 4 fives are 20 and
2 is 22
(
niift
i
on
nA-x-h
ps gA
problem no.
__
2 ) ---
VI.
222
C.P
Verbal Report Record
2 , Birds are known to fly as long as 9 hours without stopping
when on their way north In the spring.
A thrush to which
a tag had been tied traveled north at an average speed of
130 miles a day during the whole month of April.
north did It travel?
Seconds
8Q
Words
12
24l
m
22.
24-
JW £ L
(There are 30 days in April)
Verbal Report_____________ ____
Birds are known to fly as long as 9 hours
without stooping
when on their way north in the spring, A
th ru s h t o wit oh a .fc&ghad been tied traveled north — at an average
speed of X ,.hunderd
30 miles a day the whole — month of April
How,,.Iter .did... it....tr avel ! _______ .____________
r-T
-rYQU --- .dlV.
T — -T y o u ,multiply.
20h
JL
2-hundcx.d,Jua 5Q-b.y .r...,3Q rr.g.t.s
220.
9
naught
are 9
215.
22£i
22.5
250
255
■
140
145
X
-
2 naughts are naught -* 3 threes
5—ones — ..fer e ...a..
6
JJl
— it traveled 3 thousand 9 hunderd
ah— lesee — I guess it*s — miles —
that he traveled
(Yes, now will you name the answer in the
first..p.rQhl.em up there?)-,--------------JS.sr-TTgt.ra!T^r-ffsrJ&st, 1 G S B
-Q-
How far
J U tM
B J e t ib a t a a u M
G.~.rvrrre-wr,r»,— .
*
i
r
u u M
— — - 34 hours (in a whisper) (34 hours?)
(A12— right,— stop.)------------------—
---
Why
i— h
%<>
30
Uy
E
7'
25
5:
:s
2:
II
:is
£05
TSr
226
Film Analysis Record
_
6 Q 14
7 2 ) 4 3 8 4 8
VI.
4 5 2
C
6 ,4 8
C.P.
f4 3 2
5-27
Steps
,/
VI
D
C.P.
5-27
3
2s 4
2 3 6 J'6'6'6 4
4i-7,r8
8 8 '4
7JL°C5*
176
___________
Frames
1.
Finds and writes 6
65
2.
Multiplies 6x72, writes 432, draws line
59
3.
Judge s
24
4.
Subtracts 438-432, draws line
7
5*
Compares
7
6*
Brings down the 4
13
7.
Finds and writes 0
41
8.
Brings down the 8
19
9.
Finds and writes 6
in
rt
x
ji
j
10. Multiplies 6x72, writes 432, draws line
119
11. Stops judging, writes 8 on top of 6
103
12. Multiplies 8x72, writes 576, draws lino
83
13. Judges
17
14. Subtracts 648-576, writes 92
54
Remaining film
15.
278
VI. D
1.
Finds and writes 2
2.
Multiplies 2x236, writes 478, drawsi line
146
83
227
F ilm
S te p s
A n a ly s is R ec o rd
(c o n tin u e d )
____________
F ra m es
3 .
Judges
32
4 .
S u b tra c ts
5*
C o m p ares
6.
B rin g s
7 .
F in d s
8.
M u ltip lie s
9*
Judges
10*
F in d s
11*
M u ltip lie s
12*
Judges
1 3 .
S u b tra c ts
8 8 4 -7 0 8 ,
1 4 .
R e m a in in g
.film
5 6 6 -4 7 8 ,
167
53
down
and
w rite s 88
4
7
w rite s
4
4x236,
55
w rite s
9 4 4 ,
d ra w s
lin e
137
31
and
w rite s
3
x
3
236,
8
w rite s
708
120
6
w rite s 176
101
52
228
Film Analysis Record
4 5
5 6 ) 2 4'0
VI
2 2j4
A
16
16
C.P.
5-27
_______ 3 4
8
8
8
O
8 2 y 2 7 9 9
VI
B
C.P.
5-27
2 4 6
5 3 9
5 2 8
1 1
Steps
1*
Finds and writes 4
2*
Multiplies 4x56, writes 224,
3.
Judges
4.
Subtracts 240-224, writes 16
5*
Compares
6.
Brings down 8
7.
Finds and writes 3
14
8.
Multiplies 3x56, writes 168, draws line
65
9.
Judges
57
draws line
10. Subtracts 168-168, Y/rIte3 0
11. Remaining film
84
53
49
12
9
2
10
9
VI. B
1.
Finds and writes 3
43
2.
Multiplies 3x82, writes 246, draws line
42
3.
Judges
15
4.
Subtracts 279-246, writes 53
23
5.
Compares
11
6.
Brings down 9
8
229
Film Anslysis Record
(continued)
Steps
Frame s
7*
Finds and writes 4
73
8.
Multiplies 4x82, writes328, draws line
55
9.
Judges
10
10.
Subtracts 339-328, writes
11. Remainingfilm.
11
56
45
230
F ilm
A n a ly s ts
R ecord
7
7 u
B
5
3 9 } 2 8 4 7
2
VI
E
1 1 7
1 1 7
C .P .
5-27
Steps
___
Frame s
1.
Finds and writes 8
90
2*
Multiplies 8x32
93
3.
Crosses out 8 , writes 7
4.
Multiplies 7x39, writes 273, draws line
5.
Judges
6.
Subtracts 284-273, writes 11
17
7.
Compares
16
8.
Brings down 7
9.
Finds and writes 3
writes 2 , judges
5
112
3
6
45
io. Multiplies 39x3, writes 117, draws line
86
11. Remaining film
16
231
Figure
13
232
of writing over numbers that had previously been
written, an error was made.
5.
The figure7waa read as
This subject was not satisfied with his solution
but was not able after some 15 seconds examination to
find liia error.
now
His report: "I don’t seem to find it
".
It appears that his awareness of having
failed to solve example C correctly may have been
responsible for the error made in example D.
multiplying 236 by 2 : "2 sixes are 1 8
In
etc".
Fhlle
completing the solution and accurately save for the
one error, he lacked confidence in his work.
His
final report: "I didnft get that one".
The verbal responses do not indicate "judging"
or "comparing".
It Is possible, however, that this
subject may have judged and compared mentally without
expressing these steps.
As evidence of this mental
checking, It appears In his solution to example C that
he realised 648-432 would give him a difference larger
than the divisor and for that reason he said, " —
—
(laughs)----------- - that goes ah —
no
8 times".
The method of subtraction used by this sub­
ject incorporated aspects of the "take-away" and the
"additive", viz.
566 , " « you can’t take the 8 from 6
478
so the 6 becomes a 16 —
8 from 16 is 8 —
you can’t
233
take 6 from 6 so
the 6 becomes a 16 etc".
was used by 9 of
the 16 subjects.
This method
The profile of the verbal responses accompany­
ing the solution
of example C clearly picture the per­
iods of silence#
In the case of E, one period of near
silence accompanied the task of finding the first quo­
tient figure and another followed the subject’s aware­
ness of not having found the correct quotient figure#
The verbal problems were readily and accurate­
ly worked.
There was a slight hesitation of approx­
imately 5 seconds in the solution of tho second verbal
problem during which time the correct procedure was
decided upon.
The Verbal Profile Charts further em­
phasize the steadiness, the consistency of the attack
on examples A, B, and D.
The film analysis record substantiates the
dictaphone records and the Inferences made.
In the
solution of example C, 103 frames of film were used
during the hesitation that followed the second multi­
plication.
This is definitely out of proportion to
the usual number of frames used during this step In a
long-dlvlsion solution.
The electrodermal response records Indicates
Irregularities or deflections greater In number and
extent than those of this subject’s usual response
pattern.
This record appears to substantiate evidences
234
Of disturbance and the lack of confidence.
The verbal
responses at the conclusion of his work In example
0 and D showed clearly that he thought he had made
a mistake.
Summary of Case Study Findings
Case 1.
C. P.
(A Pupil of Above Median Ability)
1.
The Instructional and maintenance program
during the semester Is adequate for the acquisition
of skills necessary to th© solution of all types of
long-dlvision examples.
2.
Previously acquired skills 1n multiplication
and subtraction are kept at a functional level though
they appear within complex patterns.
3*
Awareness of failure disturbs —
to the ex­
tent of causing such computational errors as: "2 sixes
are IB".
4.
Instruction in the "additive” method and ex­
posure to variants of the "additive” and "take-away"
methods result in a "hybrid" method which. Includes
aspects of each.
5.
Verbal responses do not Indicate either
"judging" or "comparing".
There is Indirect evidence
235
supporting occasional unexpressed "judging" and
"comparing".
6.
The verbalization records and derived profile
charts give evidence of this subject’s confident and
consistently steady attack and solution to examples
of type A, B, and E, of the less confident and Interrupt­
ed solutions to examples C and D.
7.
The Film Analysis Records substantiates the
regularity and irregularity of the subject’s solutions.
8.
for
The Electroderma1 Response Record indicates
thissubject increased fluctuations In number and
extent during periods of unsuccessful efforts.
This
record substantiates the evidence of disturbance that
Is present In the verbalization and computet5on.
256
Table 2 2
*
jIowa Eleitu Lang.
ITest - 5-27-36
96
20
25
5*7
5.2
142
160
98 106
20
21
6.0
6.7
151
151
97 100
19
21
6.0
6.5
116
134
102 109 14.0 18.0
4.5
5.0
!>*«£>
oj to
> i
^
H
m
i
H
3 ca
03
tn
aJ
i
3 m
o
0
O
98
m
-
rH
fH
m
o
tO
I
t.0
ca
O
1
O to
1
V
O
O
M
0
Gates Heading
Test - 1-17-36
Iowa Elem.Lang.
Test - 1-20-36
Standards
Woody McCall's
Test - 1-24-36
170*
S. F.
147**
Exp.Group
Medians 109
Class
Medians 108
Compass Survey Test
5-26-36
Subject
Kuhlman Anderson
Intelligence Test
Case 2. S. F.
Test Result for Selected Case
O
to
d
•H
i
t>
0 CM
cA 1
© lO
1
0
© -p
•P 0
(A 0
O
Eh
5
8'
Binet-Simon Individual Intelligence Test
Otis Group Intelligence Test
Case 2: S. F.
This subject was originally included in the
study because of her unusual mental ability as measured
by the Binet-Slmon Individual Intelligence Test and
the Otis Group Intelligence Test*
It is evident from
the table that this subjectfs arithmetical ability as
measured by the Compass Survey and Woody McCall,a Tests
is not above the median of the experimental group nor
257
of the class.
Scores on the Gates Reading Tests are
less than the experimental group or class medians;
scores on the language tests are definitely higherI
These scores reflect the nervous, erratic, and impulsive
performance which was so characteristic of this sub­
ject.
The Teacher^ Record of Daily Class Work shows
that only on one day, May 6 , was special help or time
beyond the class period given to this subject.
Ap­
proximately thirty minutes at the close of the school
day was spent by eight pupils completing problems In
article 183.
Some individual aid was given.
Aside
from this one instance, no time or special help was
given.
This subject was brought to the laboratory
on the afternoon of May 29th.
98*4; her pulse beat, 92.
Her temperature was
While obviously disturbed
by the laboratory situation and the individual atten­
tion received, she appeared less nervous and "high
strung" than in previous projects.
She began working
the examples of the project at 2:05.
The verbal report records that follow Indicate
an accurate though somewhat hesitant attack on example
Aj B , and C.
,
_
0 [-i7\
(Continued on page 253)
238
Verbal Report Record
4
5 6 ) 2 4 0
28 4
16
16
VI.
&
S.F.
5-29
3
6
8
8
5
8
10
7
15
5
20
7
25
7
and --2 is twenty-- two
30
7
35
12
4-— - less 4 is 6 -------- a
----- 4 less — 3 less 2 is 1 - bring
down your 8 f If—
5 inta a hunderd n sixty-— — eight goes
-— 5 time 3
40
.10
45
5
50
10
52
0
56 &oes into 2 hunderd and 40— —
_.-— uh— *4 times
—
4 times 6 is----
-4 times 6 is —
3 times — m—
18
-----
a — --
— -zero—
6 is
3 times 5 .1s 15 and 1 is 16
— (All right, stop-)
less
239
Verbal Report Record
______3^4
8 2 y 2 7 9 9
2 4 6
3 3 9
VI.
B
S.F.
3 2 8
5-29
Seconds
10
11
82
TT
Words
14
Verbal
Report
~~82 Into 2 hunderd n seventh
- nine
goes 5 times\ ■* 3 times 2 la
3 times 8
is
24
15
9 less 6 Is
20
7 less
la 3
25
bring down your
26
30
82 lnta --3 hunderd and 39 goes -
35
-— -4 times
40
45
50
10
2 is
is 32
4 times
-4...-frime.g, g .la 8.
9,.M a.9,lQ,-La —
1— -5 less 2 _is — -1___
"rrrrr (All. riglat>-StopJ.
240
Verbal Heport Record
^
6 0 9
7 2 ) 4 ! 8 4 8
4 3 2
64 8
64 8
---- -
7X •
C
S.F.
5-29
Second.3
Words
5
11
10
15
15
7
20
10
25
16
50
13
35
3
40
16
42
0
Verbal Report
- - - - - ---72 into 4 hunderd end 38
goes 6 times
6 times 2 is 12 - 8 times 7 is 32 and 1
is thir---- -42 and 1 Is 43 -— ~
8 less 2 is-— 6 -tiring down your 4
seventy —
two into 64 won*t go so you put your zero
over by the 4 sev­
ering down your 8 - 7 2 goes into 6 hunde]
and forty —
uh
—
eight — — - — —
9 times
9 times 2 is 18 -- 9 times 7 is 63
and 1 is 64
---- {All right, stoo)
241
Verbal Report Record
/
2 3
6y 566
VI,
2 3 6
*5" 3 0 4
D
S.F
5-29
Seconda
— | -|
Words
12
Verbal Report
-------2 hunderd n 36 goes inta 5 hunderd
n 66 —— — ______
______________________
— ^-uh-- — twice - 2 times —
2 times — • 6 is
10
8
15
6
12 —
20
5
-— uh
25
11
30
9
35
40
4
2 times —
3 is
2 hunderd
n-— 36 inta 5 hunderd n 66 g:oes once
1 times 6 is 6
3 - 1
-
1 times 3 is
times 2 is 2 —
6
less— —
6
---- is—
6 is zero
6
6 less
45
50
a— ------6 less 3 is--
55
5 less 2 is
60
9
65
12
5-.--*. brln& down your 4 - 2 hunderd n thirty
six into 3 hunderd n — — 3 thousand 3 hunderd
n 4 goes
70
'uh-
75
80
13
85
8
m.
95
r
100
*uh*
Once —
*"«■
1 times 6 is 6 - 1 times 3 is 3 and 1
times
2 --•is 2
4 less 6 is—
a
«.________„14 leas 6 —
»«, is -— -— .
<
-------- 8 - 1 0
less 3 is
242
VI.
D (contInued)
S.F.
105
5
110
1
114
0
'
245
Verbal Report Record
1 3
_______
3 9 T~2 8 4 7
1
2 7 3
117
117
VI
E
S.F*
5-29
5
9
..10
12
IS
3
...£G
9
25
4
30
6
35
XO
— r"——40
1
45
12
sn
12
—
———————32 inta 2 hundend -.-n-.—«.«. 24
9 times --9 times 9— Is 81
9 times 3 la
__ —————all—
o-nfi.n flhnuf.
thirty---- — nine
. jzoea
— 9 intfl 2 .b n n H r » A4
*7
-***7 times 9 Is
5
7 times 3 is 21 find fi Is
—— twenty-.—
4 leas 3 «—till**—1a -4 less 3 Is 1 — £? l^jc
7 is 1 - bring down your 7 - 37 -39 inta
5
a V>nndeT*d *■> 1 *7 gnes
7
-
65
7
seven
68
3
2 is 11 ---
3 times
-
•
—
3 .times Q Is tw§ntv-r
3 times —
3 is--- 9 n
(All rinht. ston)
244
Verbal Report Record
1 . A quail was marked In the south one spring.
During the
summer It was seen 1904 miles farther north.
At its
fastest flying speed (56 miles per hour) how long would
It have taken the quail to fly that far north?
VI.
S.F.
Seconds
Words
Verbal Report
•
quail was marked in the south
one spring; durDuring the summer---summer - had been seen
— 19— — hunderd
and 4 miles farther north. It was -- ©t 11
fast— fastest
5
10
10
9
15
11
20
8
25
12
50
2
35
0
40
9
45
4
50
8
55
5
goes —
60
5
times
65
6
70
0
75
6
80
9
85
0
90
10
95
9
100
14
flying speed 56 miles per hour how
long ---would it have taken the quail to
fly that far north?
<----- -fifty — — six -— -
-— -56 Inta—
---1 thousand 9 hunderd n 4
————5
-
3 times 6 i s ----- —
-- -•••ah— ---— 18
—
5 times 5 --Is------
10 less 8----- 10 less 8 Is 2 — S less
6 Is 2 - bring down your 4 - 5 6
— 6 Inta 2 hunderd n 24 goes 4 times
times 6 is
- 4
Verbal Report Record
245
2* Birds ar© known to fly as long as 9 ho tors without stopping,
when on their way north in the spring*
a
A thrush to which
tag had been tied traveled north at an average speed of
130 miles a day during the whole month of April.
far north did it travel?
(Th©r*e are 30 days in April)
VI*
S.F* (Ho.l continued)
Seconds Words
105
Ho* 2
110
130
o
135
o
.140
o
143
3
130
in
155
9
l«n
13
----- 34 hours
Rir>H« ct><^
to fly as long pa-- 9 hour4
without stopping-- — when they — when on
tV\«i r» way nn-pth
In the spring a thrush to which— — a tag
tied tT*av**l**•«
north at an average speed of 1 hunderd n
'2m tnU Aia ci rtci-tr
r\ry
whftlft month O#—At
How far north did It travel? There are 30
rifa-vm -tn A
19
170
IP.
_ 175
O
1ftn
n
185
n
(-- What are you going to do?) 30-- inta
"■ 1..... ...... 1 1 1 —.
— 30 Inta e hunderd n 30 goes - a----- -
j
-- r
...
" ■<* lIUnUL^Jru ft"
8
6
)
0
1
125
>
1
Birds are known to flv a a loner
i
i
t
t
I
i
......
ar© 22.
l
10
0
200
Report
9
120
“»
..J
L-cttfrx\r-195
Verbal
Jk
0
1
I
1
1
115
How
...
246
VI.
S.F*
(Mo.2 continued)
205
6
210
10
215
1
220
1
225
1
230
0
232
0
zero
- 4 times 3 Is -— -12
zero from zero Is zero
---- (What are they?)
(All right , stop•)
2 less 3 is 1
:3^c!
£ii2til
232 Cl
Si?
iLa
s::
£1
B
:z:
L'L
=£u=L
:;zv
y
:z:
jLSL
S:
s;
:s;:
*2
Z
i?
.££1
‘H-
.
H
.
249
Film Analysis Record
S 6
VI
A
S.F.
5-29
T2
4 a
4 0 a
2 2 4
16 8
16 8
VI
__3__4 11
8 2 ) 2 7 9 9
2 4 6
3 3 9
3 2 B
1 1
82
Steps
1*
Finds and writes 4
2.
Multiplies 4x56, writes 224, draws line
3.
Judges
46
4*
Subtracts 240-224, writes 16
53
5*
Compares
9
6#
Bring3 down 8
3
7*
Finds and writes 3
90
8,
Multiplies 3x56, writes 168, draws line
78
9*
Remaining film
68
76
161
VI. B
1.
Finds and writes 3
53
2*
Multiplies 3x82, writes 246, draws line
58
3.
Judges
79
4m
Subtract 279-246, writes 33
33
5*
Compares
4
6#
Brings down 9
5
7#
Finds and writes 4
8,
Multiplies 4x82, writes 328, draws line
88
9.
Judges
23
119
10* Subtracts 339-328, writes 11
24
11* Remaining film
50
250
Film Analysis Record
6 0 9
7 2 ) 4 3 8 4 8
VI
4 3 2
C
6 4 8
S.F.
6 4 8
5-29
2 l
2 3 6
VI
D
S. F.
5-29
T 5 6 6 4
X
2 3 6
3 3 0 4
2 3 6
7 8
Steps________________________
78
236
Frames
1.
Finds and writes 6
51
2.
Multiplies 6x72, writes 432,draws line
96
3.
Judges
25
4.
Subtracts 438-432, writes 0
5
5.
Compares
3
6#
Brings down 4
6
7*
Finds and writes 0
48
8*
Brings down 8
29
9.
Finds and writes 9
91
10. Multiplies 9x72, writes 643
70
11. Remaining film
10
VI. P
1.
Finds and writes 2
74
2.
Multiplies 2x236, writes 2,stops
3.
Writes 1 over 2
4.
Multiplies 1x236, writes 236, draws line
90
5.
Judges
52
6.
Subtracts 666-256, writes 330
174
190
251
Film Analysis Hecord
(continued)
S t e p s _____________________
_____
Frames
7*
Compares
8*
Brings down 4
9.
Finds and writes 1
164
10.
Multiplies 1x256, writes 236, draws line
86
11*
Judges
117
12•
Subtracts 3304-236,writes
13. Remaining film
15
4
78
36
252
Film Analysis Record
7
9 5
3
9
T
2
8
4
7
VL
V I
E
S.F.
5-29
2
7
5
1 1 7
1 1 7
Steps______________
Frames
Finds and writes 9
2.
Multiplies 9x59, writes 1, stops, writes 7 182
3.
Multiplies 7x39, writes 273, draws line
4#
Judges
54
5.
Subtracts 284-273, writes 11
14
6.
Compares
8
7.
Brings down 7
2
3«
Finds and writes 3
9.
Multiplies 3x39, writes 117, draws line
87
H
Remaining film
17
•
o
1*
49
158
110
255
Repetition and pauses occur in her solutions:
uh -- 4 times
” — -
4 times 6 i s ------- 4 times 6 is —
a ---------------------
24".
Difficulties appeared in her solution to
example D.
She found the correct first quotient fig­
ure but in the multiplication that followed she ap­
parently believed the product too large and immediate­
ly reduced the original quotient figure.
Having just
learned how to solve long division examples with nonapparent quotient figures, this was a known and con­
venient procedure.
This first error was followed im­
mediately by failure to compare and subsequently by
an incorrect second quotient figure and by an incorrect
and incomplete subtraction.
The Verbal Report Record
for example D clearly pictures her dilemne which became
increasingly impossible!
It is quite apparent from
this record that the subject was aware of the fact
that she was not progressing satisfactorily.
The solution to example E, long division
with non-apparent quotient figures, was accurate.
The
slight difficulty experienced In finding the first
quotient figure Is but normal and expected.
She moved
along cautiously and somewhat hesitatingly In her solu­
tion.
There Is no direct evidence of either "judg­
ing” or "comparing” In the verbalization that accompanied
solutions.
However, the fact that the first quotient
figure In the solution to example D was considered too
large at the time the first multiplication was nearing
completion and the further evidence In example
E
where
the first quotient figure was similarly rediiced seems
to Indicate that "judging" occurs even though not ex­
pressed*
There Is reason to believe, however, that
"comparing" does not occur.
The v/rltten solution to
example D testifies to this omission.
Tills subject used the pure "take-away" method
of subtraction*
She was one of three of the experiment­
al group who used this method In Its pure form.
She
had been taught this method in the school from which
she came.
The method was Interestingly expressed:
" ---10 less 4 is 6 -------------a------—
5 less
4 less
—
2 Is 1 Considerable hesitancy and verbal repetition
characterized this subject*s solutions to the verbal
problems.
The solution to the first problem was cor­
rect; the solution to the second problem, Incorrect.
After having read the second problem and hesitating
for fifteen seconds the writer Inquired, "What are
you going to do?"
She proceeded to divide arriving
at an Incorrect and absurd answer.
of the
verbalization seems
at the
outcome of her attack*
The latter part
toshow her dissatisfaction
255
Figure
14
256
Th,© Verbal Profile Charts indicating duration
and verbal responses for each five-second interval
graphically pictures the irregularity of this subject’s
work, particularly in the solution to examples D and E.
The electrodermal response record of this
subject does not indicate significant deflections dur­
ing points of error or seeming awareness of error#
In­
stead the record conforms to her particular pattern.
The fact that this subject whether in school, in the
laboratory, or on the playground exhibited much ner­
vousness may partially account for the lack of seeming
deviation from the normal at times of error or aware­
ness of error.
TShile the electrodermal
example D —
response
record for
the example in which errors and difficulties
arose --is not complete, it is similar in pattern to
the record for example G.
The solution to this ex­
ample was derived with accuracy and dispatch.
257
Summary of Cage Study Findings
Case 2 #
8. F.
(A Pupil of Superior Mental Ability
and of Average Arithmetical Ability)
1*
All of the skills necessary to the solu­
tion of examples of types A,B,C,D, and E have not been
acquired through regular classroom Instruction and
practice•
2.
Failure to "compare11 leads to errors and
absurd mathematical situations ai tliln the exar.ip3.e*
3.
Previously learned skills Involved in multi­
plication and subtraction function though they appear
In new patterns.
4.
Verbal reports indicate awareness of error
and dissatisfaction with, results which do not appear
In the written solutions.
5.
"Judging" appears indirectly though not
expressed.
6.
"Comparing" does not appear - directly or
indirectly.
7*
The pure "take-away" method persists though
variants of both the "additive" and "take-away" are
common*
258
S.
Considerable time is needed beyond thrt
necessary to the reading of verbal problems to decide
upon the procedure to be followed-,
9*-
The verbalization records, the derived Verbal
Profile Charts, and Film Analysis Records give a clear
picture of the progress of solutions - the accuracy,
the pauses, and significant expressions,
10.
Errors and awareness of unsatisfactory
progress do not arouse significant electrodermal de­
flections for this subject#
Table 23
Case 3# I . 0.
1.0.
S u rv e y
5 -2 5 -3 6
-
C om pass
Q rH
£3
Test
1 -2 1 -3 6
Test
03 iO
- SO
1—1 1
r-f
Oi 03
O 1
1
O ca
O <1)
gs eh
03 10
- SO
rH
r -i
1
aJ o i
O 1
0 iO
W
i—4
1
*b +3
O 03
O 0
jSc EH
to
feQcO
St t
v~\ r na «H
03 1
© r-f
«
I
oi
to
tD tO
S3 1
»H i>
'13 03
o$ 1
0 to
■— !
m
0 -P
+3 03
0} 0
cm e-i
4^> CQ
© ©
O Eh
fcO
P CO
q$ eO
£
. 03
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© rH
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3 co
ctj eO
3 *
£>
J!
a *
© to
H
n i
£
CO
o ©
H Eh
j
ect
.
i
i ..................
r
I Com pass
S u rv e y
S ub j
Kuhlman Anderson
Intelligence Test
|
Test Result for Selected Case
102
85
88
12
20
5.5
6.5
105
110
109
98
106
20
21
6.0
6,7
131
151
108
97
100
19
21
6.0
6.5
116
134
109 14.0 18.0
4.5
5.0
E x p .C r o u p
Medians
Class
Medians
Standards
102
85
259
Case
3s
X.
0.
The third subject possesses mental ability
slightly below the median of the esjperimental or class
group* His scores on the achievement tests which were
administered at the close of the first semester in­
dicate less than median ability in Arithmetic, read­
ing, and language.
A very distinct Improvement in
his arithmetic and reading is evident from his scores
on tests given at the close of the year.
The Teacher’s Dally Class Record indicates
only one Instance of individual aid for this subject.
On March 31st he was given remedial sheet 4c-6.
This
sheet provided short division examples and was assigned
after he showed a weakness in this skill.
Ho other
Individual attention was given to this subject between
Project III and VI.
This subject was brought to the laboratory
on the afternoon of June 2, 1936.
He was In good
health, affable and very cooperative.
recorded was 98.4; his pulse beat, 72.
The temperature
At approximate­
ly 2*40 he began working the examples of this project*
The verbalization records that follow show
a steady, accurate attack on each of the examples with
little hesitation at any point.
Upon discovering in
example E that his first trlal-quotlent figure, 9, was
too large he proceeded without delay to try 7.
Having
(Continued on page 275)
260
V e rb a l
H e p o rt
R e c o rd
V1‘
4 2>
5 6 ^ 2 4 08
22 4
A
16 8
1* 0 *
16 8
6-2
~-- o
5
0
10
12
16
17
20
7
25
14
50
12
55
8
40
19
45
5
48
0
_
.......... ..
5 goes into
4 times—**-4 times 6 is —
twenty—
Tour - - p u t down your 4 and carry your 2 —
5 ---4 times 5 is 20 and — 2
2 2 ----4 from 10 —
is 6
make the 2 a 5 --
bring down the 1
3 from 4 is 1
2 from 2 is zero — *■ bring down your 8---5 &o@s into
16 — 3 times —
3 times 3 ---a — — . 6
is 18 — - put down your
8 and carry yoi;
— - 3 times 5 Is 15 and 1 is 16
8 from 8 is zero — -
(all right stop)
261
Verbal Report Record
6 4
8 2 ) 2
VI.
3
1*0.
6-2
..... 3
.
,
.... 5 ..
10
8
15
.... 13
2 0 ...
io
25
io
30
7
55
.... 9 ..
c—
7 9
4 6
3 3
3 2
T
8
9
11
82
9
8
T
Into 27 «...
3 times 6 (clears throat) -- 3 times 8 Is
**24
nilt rtnwn t.ViA f.«rAw+nr___
f o u r ---6 from 9 la 3 -—
4 from 7 I s ----
--- 3 --- bring down the 9 -—
8 goes intn 33 *»~
4
times *»— •* P him a s 4 * a — _-._r.__ g
2 times 8 (clears throat) 4 times 8
— w-,—
5n
*****
12
45
n
(all rip'*'1'*'-
. -- —
262
Verbal Report Record
6
7 2 ) 4
VI.
38
09
48
4 52
C
6
6
..
6-2
1 0
48
48
0
---- ------ 7 goes Into 43 - 6 times
.6 times'--' ‘2 — tTfries 6 Is 6 ~ '"Is'"12
put down your 2
and carry your 1 — - 6 times 7 is
(clears throat)
42 and 1
5
7
10
13
15
12
20
8
is
25
7
bring down the 4 -*--7 goes in (clears thro*
30
7
-- -
35
5
40
16
7 goes Into 64 ----- -— ~~
---a — 9 times — - 2 times 9 is 18
put down your 8 and carry the 1
45
11
9 times 7 Is 63 and 1 is 64
47
1
4 3 --------- 2
6----zero
from
8
Is
6
times --- bring down your
zero
(all right stop)
263
Verbal Report Record
4
2 3 6
Vt
D
X,0*
6-2
Seconds
Words
5
Verbal
Report
10
11
-2JL
20
---------- a(clear throat) 2 Roes into 6
— — 2 times — - put down your 2 --2 times 6 Is 12
put down your 2 and carry the 1 —
2 times 3
Is 6 and 1 Is 7 2 times 2
2Q.
10
Is 4 — - 2 from 6 is 4
20
— Is 9 -— S from 5 Is zero
brings down your 4 --- 2 goes Into 9 —
4 times
4 times 6 is 34 — 24 —
put down your 4
JL4
■&jqfl-o.&rar..
£Sl
ASX
2h
Ah.
2h
7 .from 16
your 2 —
4 times 3 is 12 and 2 Is 14 -—
put down your 4______ ______________________
and carry the 1 —
4 — 2 times 4 is 8 and 1
......................
J.3--9 ( M l ) .aero
(right.a.to,p)
264
Verbal Report Record
_______ 7 3
3 9 ) 2 8
.8 7
1
1
VI.
..
1 0
6-2
___ 5
10
r ..... 5
15
12
_
20
2
£5
... 15
50
12
55
.
..-.IS.
.
T ~7
3
1 7
1 7
0
...
e l g h t -- 9 times --- put down your
J times 3 is
put down the 1 and carry the 8 - —
9 times 3 ---- i a
9
----- ---- that 9 Is too large so I am going
7 times 9 — *— Is 63 — - put down the
and carry
the 6
sev -— 3 times 7 is 21 and
14
6
45
10
50
13
bring down the 7 ------ 5 goes into 11 — a tl.
3 times 9 —
27 —
put down your 7 end
c a m your 2
55
10
3 times 3 Is 9 end 2 is ll...-.--. zern —
__
la
3
27
40
(all right stop)
265
V e rb a l
1*
A
q u a il
m er
i t
fly in g
th e
w as
w as
m a rk e d
seen
speed
q u a il
to
In
1904
R e p o rt
th e
m ile s
(5 6
m ile s
f ly
th a t
p e r
fa r
R e c o rd
s o u th
on©
fa rth e r
h o u r)
s p r in g .
n o i fc h .
how
lo n g
D u r in g
A t
it s
w o u ld
th e
sum ­
fa s te s t
i t
have
ta k e n
n o rth ?
V I.
1.0.
- (clears throat) a — - a quail was
marked in the south one spring.
During the suramer— it was seen —
nine
teen hunderd n 4 miles —
a -
5
10
10
12
15
9
20
13
25
9
30
10
35
8
40
15
45
13
_ 50
9
55
IO
RO
16
65
6
70
0
75
0
(What are those 34 — —
80
0
?
85
0
90
3
95
12
-
1Q0
..
.
farther north. At its fastest flying - speed
miles per hour — how long would it have ta­
ken the quail to fly —
that - far -- north, — — - — — well, t*m
going to take fifty —
six In —
19 hurderd n 4 ----- 5 goes into
3 times — - 3 times 6 - - - - - is
- 3 times 6 is 1 8 —
put down the
carry the 1 —
3 times
5 is 1 5 and 1 Is 1 6 — - a- 8
from 10 is 2
19
24
8 and
7 from 9 is 2 —
brin& down the 4
— — 5 -goes into twen— ty - two — - 4 times
-— 4 times 6
is — - 2 4 — - put down your 4 and carry your
2.--- 4 times 5 is 20
and 2 —
....12
.......
?
— -
is 22 --- zero --------
remainder?)
--
-- 34 hours (all right go right ahead)
(and do the neat problem) --Birds are known
to fly as long as 9 hours without stopping
when on — when on their way north —
in the
spring - — A thrush —
......
266
Ho,2 Birds are known to fly as long aa 9 hours without stopping
when on their way north in the spring.
A thrush to which
a tag had been tied traveled north at an average speed of
150 miles a day during the whole month of April.
(There
are SO days In April)
VI.
X •0«
Seconds
Words
Verbal
to
105
16
110
13
w h ic h
&
ta g
e le d n o r th a
^ 0 m ile s
a d
A p r il how f a
d id
i t
tra v e
th ro a t)
a —
115
8
120
7
In
A p r il
125
6
30
in to
ISO
6
135
8
— ze ro
------------------ ( c l e a r s
1
—
h u n d e rd
and
5 tim e s
z e ro fro m
9
9
fro m
13
145
2
—
-
— -— —
150
5
155
0
160
0
165
0
170
0
175
0
180
0
182
0
( a ll
been
— -
tie d
— —
tra v ­
t an a v e ra g e
sp eed o f 1 h u n d e rd
a y d u r in g
th e —
w h o le m o n th o f
r n o rth
l - th e re
—— - a r e - (c le a r s
30 days
140
is
Report______ _________ _
h ad -
3
th ro a t)
30
is
z e ro
I
am
ta k e
p c o ln #
- —
9
—
is
z e ro fro m
z e ro
—
is --4 — -
rl&ht stop)
— —
I c a n ’t find it
edr
y_
1~
A il
269
Film Analysis Record
4
V I
A
1 .0 .
5
6
1 4
2
6-2
4
2
1
1
0
4
6
6
5
3
8
V I
B
1 . 0 .
6 -2
8
8
0
S te p s
8
2
j
2
2
7
4
3
3
4
9 9
6
3 9
2 8
1 1
i i
82
_________ __ _______________________ F r a m e s
1»
F in d s
and
2 .
M u ltip lie s
3 .
Judges
4 *
S u b tra c ts
5.
Compares
6 .
B r in g s
7 .
F in d s
8 .
M u ltip lie s
9*
Judges
1 0 .
S u b tra c ts
1 1 .
R e m a in d e r
V I.
B
1 •
F in d s
2 .
M u ltip lie s
3*
Judges
4 .
S u b tra c ts
5 .
C o m p a re s
6*
B r in g s
w r ite s
4
4 x 5 6 ,w r it e s
78
2 2 4 ,d ra w s
lin e
94
24
2 4 0 -2 2 4 ,w r ite s
58
26
down
end
16
8
w r ite s
6
3
3 x 5 6 ,w r it e s
36
1 6 8 ,d ra w s
lin e
93
15
and
1 6 8 -1 6 8 ,w r ite s
0
4
37
w r ite s
VO
3
3 x 8 2 ,w r it e s
2 4 6 ,d ra w s
lin e
90
9
2 7 9 -2 4 6 ,d ra w s
down
lin e
36
6
9
8
270
F ilm
A n a ly s is
R e c o rd
(c o n tin u e d )
S t e p a ________________________
and
.
F ra m e s
7*
F in d s
w r ite s 4
8*
M u ltip lie s
9 .
Judges
1 0 .
S u b tra c ts
1 1 .
C o m p a re s
10
1 2 .
W r ite s
24
13*
R e m a in d e r
4 x 8 2 ,w r it e s 5 2 8 ,
46
d ra w s
lin e
84
12
3 5 9 -5 2 8 , w r ite s
11
16
7
271
Film Analysis Record
7
2
T
TC
0
4
3
4
3
6
8
9
2
8
2
2
6
6
1. 0.
0
4
4
4
6
V I
B
8
8
6-2
3
5
6 6
4
9 4 4
6-2
0
S te p s
F ra m e s
I .
F in d s
and
2*
M u ltip lie s
3 .
Judges
4*
S u b tra c ts
5*
C o m p a re s
6.
B r in g s
7*
F in d s
8,
B r in g s
9*
F in d s
10*
M u ltip lie s
I I .
Judges
12*
S u b tra c ts
1 3 .
R e m a in d e r
V I.
P
1 .
F in d s
2 .
M u ltip lie s
3*
Judges
4 .
S u b tra c ts
w r ite s
6 x 7 2 ,
6
59
w r ite s
4 3 2 ,
d ra w s
lin e
118
17
4 3 8 -4 3 2 ,
5
6
4
w r ite s
down
and
6
4
down
and
w r ite s
0
61
8
w r ite s
9 x 7 2 ,
13
86
9
w r ite s
6 4 8 ,
d ra w s
lin e
77
2
and
6 4 8 -6 4 8 ,
w r ite s
0
1
18
w r ite s
2
2 3 6 x 2 ,
70
w r ite s
4 7 2 ,
d ra w s
lin e
84
18
5 6 6 -4 7 2 ,
w r ite s
94
4
4
4 7 2
*~ T T 7
1.0
0
T
44
272
Film Analysis Record
(continued)
Steps___________________________________________ Frames
5*
Compares
40
6 m Brings down 4
9
7.
Finds and writes 4
8.
Multiplies 4x236, writes 944, draws line
9*
Judges
10* Subtracts 944-944,
11« Remainder
53
132
2
writes 0
3
10
273
Film Analysis Record
T-______ ^ -3
3 9 1 2 3 4 7
Steps
Frame
1*
Finds and writes
9
74
2.
Multiplies 9x34,
writes 1
98
3.
Stops, judges
4.
Writes 7 on top of
5.
Multiplies 7x36, writes 275, draws line
6*
Judge s
7.
Subtracts 284-273,
8.
Compares
395
9*
Brings down 7
401
204
9
350
358
drawsline
10* Finds and writes 3
11. Multiplies 3x39,
216
writes 117, draws line
381
437
519
12. Judges
321
13. Subtracts 117-117, writes 0
326
14. Remainder
341
274
TZ5
U
O
o
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± =
(D
~ r~
Figure
15
03
a
o
p,
00
<D
£
0d
o
«r4
4-5
O
P
T5
O
*H
U
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<D
Pd
•tJ
O
5=
i
Q
-*r
U
4-5
O
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B
*4
M H
>
275
just; 1 earned how to solve long-dlvi slon examples
with non—apparent quotients his discovery was not too
disturbing*
"---------------- eh I ----------- that 9
ia too large so I am going to try this 7,f.
His attack on the Verbal problems Indicated
confidence, perhaps more confidence than was warranted.
It appears his successes on previous examples had given
him assurance and also caused him to believe each
problem called for long-division.
His procedure on the
second problem was wrong and the first step in his long
division was incorrect.
The absurdity of the answer
that was evolving seemed to dawn on him?
n
--------
something is wrong in h e r e --------The subtraction method used by this subject
involves aspects of the utake-away” and ”additive”
240
methods, viz* og^.
tt
4 from 10 -- Is 6 - —
bring down
the 1 — ■- make the 2 a 3 •
-5 from 4 is 1
2 from
2 is zero” .
There is no evidence of wjudging” or “com­
paring11 in the verbalization that accompanied this
subject's solutions.
It is possible that in his at­
tempted solution to the second verbal problem mental
comparing Is partly responsible for his remark that
,!— ----------- something Is wrong In here".
The Verbal Profile Charts and the Film Analy­
sis Records emphasize and confirm the previous state-
276
ments concerning the consistent, steady attack upon
the examples and verbal problems.
The
ample B, C*
electrodermal response record for ex­
and E are given.
This record Is In
agreement with the subject's usual electrodermal pat­
tern.
There were no apparent serious disturbances.
In fact, he appeared to grow more confident with each
successful solution.
The original electrod.ermal re­
sponse record from which the section for example B,
0, D, and E was taken included the verbal problems.
It Is Interesting to note that there is no indica­
tion that his failure to solve the second problem
to his satisfaction caused any serious disturbance.
It appears that awareness of error does not evoke
strong emotional disturbances unless the consequences
of error are serious I
277
Summary of Case Study Findings
Case 3. I. 0.
(A Pupil of Less than Median Ability)
1.
The instructional and maintenance program
is adequate to the acquisition of skills necessary
for the solution of all types of long-division ex­
amples.
2.
Very little tine beyond that needed In read­
ing verbal problems Is required by the subject
to
decide upon the method of solution to be used.
3.
Exposure to variants of the “take-away”
and “additive” methods of subtraction results In an
equally effective “hybrid” method that involves as­
pects of each.
4.
Verbalized “judging” or “comparing” does
not occur.
There is some evidence that silent mental
“comparing” takes place.
5*
The Verbalization Records Indicates a
consistently steady end accurate attack on all the
examples.
6.
The Electrodermal Response Records lends sup­
port to the notion that unless the consequences of
error are serious the emotional disturbance aroused Is
276
not significant.
7*
The Verbal Profile Chart and the Film Analy­
sis Record confirm other evidences of a continued,
steady attack upon the examples and problems*
Summary Treatment of Experimental Group
As in the summary treatment of previous
projects, written responses to examples will receive
first attention.
The skills and errors involved in
Project VI have been Isolated, listed, and assigned
symbols as in the third project*
1*
Basic division combinations
a. 1st quotient figure
b. 2nd quotient figure
c. 3rd quotient figure
2.
Placement
a. 1st
b. 2nd
c. 3rd
3#
Multiplication
a* 1st product
b. 2nd product
4.
Subtraction
a. 1st difference
b. 2nd difference
5*
Miscellaneous errors
a. Falls to bring down dividend figure
b* Brings down wrong dividend figure
of quotient figures
quotient figure
quotient figure
quotient figure
279
6.
Non-corap1etion caused by
a* Lack of time
b* Increased difficulty
c* Sensing some tiling wrong
&• Any of the above skills or errors
e. Skills not listed above
The symbols are to be Interpreted in this manneri
2b, placement of second quotient figure; 4b, sub­
traction to obtain the second difference, etc.
28f
Example Analysis
Project VI
<D
Subject
&
+5
o
<D
u
o
o
(D
4->
o ©
<D rH
Is a
Sm
o o
o o
Md «
Errors
Exact Ketxire of Error
*/•/ 7X.9.^<
c7
*^+6>^
‘
3L*zk..2-.
r_c^\^— /
/</; //7
'it*-■
-
7 „
5* R.H
jsiSza^.
__
6 . B.W
^ ^
:
a^u-r^- ^
7*B.Sch. A
,? 3 6*3= Jz
*3
&*t~4rA*&r 7
281
Example Analysis
Project VI
<3)
P*
B • Sm •
U
o
o
Subject
8•
O
0
&«
A
B
C
D
E
9 .
10.
S .P .
R .N .
■p
o
CO
u
S-i
o
o
d
M
CD
-P
©
Pi
0
O
o
d
Hi
Errors
3
V ^
i J-
L/
3 4
A
B
C
D
E
A
D .T .
H .T .
1 3 .A n .H .
....
c,^Oyt33> _£>Cle < ? / ,33^,
3 30l£
i^
*/£<y 3
37, 7 ~</73.3
___ jl*A
x? - £37..........
..
A
B
C
D
E
1 2.
J?30 ~ 2 A v -- 3-6?
¥ X f 3* *7
7
X
7- A / ;
2.£7- iZ ff* /
3 {£>.
/
B
C
D
E
1 1.
Exact Mature of Error
A
B
C
D
E
A
B
C
D
3
J5
?X.3 .*. A? 3.___
______ ____
...
..
-ts__ . 3 o-- 3trl7~
.
.3 -,V-............
-
3 x ?
/y
................
//
E
1 4 .
J .B .
A
B 1
C 1
D 1
E
. a// a-' .....,
........ .
3?3 o -P- 2 3- 3 3
.
—— — --------— —------------------------------- ------------- -— ■
282
Example Analysis
Project VI
43
O
©
*P
O
<D
ft
>>
..T-I... , ■«!■-- ■"... ■.. -...
~
I
©
u
u
Jh
O
O
M
15* 1*0. A
B !s
0
D
E
16. S*J.
o
o
£
&
-p
©
ft
u
O
o
.....
_
4^
A
..
B
C
X> 1
3
E
Summary
A
B
C
D
E
15
13
12
8
11
57
2
2
3
5
4
16
1
1
1
3
1
\/C**v7tM43
f^LcC*st~e£.
/3
ji?-*— .?
*2 AdJCJ
S um m ary of E r r o r s
Skills
A
la
lb
lc
B
0
1
1
1
2a
2b
2c
3a
3b
4a
4b
1
1
D
E
2
2
2
4
1
1
1
1
1
0
1
3
2
2
2
7
5
1
2
1
1
1
5
3
2
1
0
1
5a
5b
6a
6b
6c
6d
6e
1
Totals
3
T o ta l
1
1
1
4
5
16
0
0
1
3
0
6
34
284
The summary, page 282, Indicates that 57
of the 80 solutions were correct, 16 were incorrect,
and 7 incomplete*
Eight of the 23 unsuccessful attempts
(incorrect and incomplete) are attributed to example
D, five, to example E.
Long division examples with
three-digit divisors persist in giving difficulty even
after having been taught*
Inspection of the Teacher’s
Record of Daily Class Work shows that this type of
long division was not slighted*
It had been presented,
drilled upon, tested, and retaught over a period of
six days.
Examination of the exact nature of the
errors made reveals that failure to solve the examples
correctly was not always due to difficulties inherent
In long division examples of type D but rather to spur­
ious errors, as for example, "2 sixes are 18fl, "7 and
8 are 16tf, "4 times 3 Is 12 and 2 13 13", and "236
times 2 equals 22"•
It Is very probable that these
errors occurred more frequently then usual because it
was necessary to give Increased attention to the matter
of determining quotient figures and to the placement
of quotient figures in this type of example.
Spurious errors also appeared frequently in
the solutions to examples of type Es
”7 tines 9 is
56tf, "9 and 6 Is 14*', "39 times 3 is 108", and "3 times
9 Is 18".
It Is the writer’s judgment that such errors
appeared because the subjects concentrated so much on
285
the new type of approach to the solution that minor
skills which ordinarily function automatically and
with accuracy failed to do so.
A total of 34 errors
were made by the sixteen subjects.
As was true of
Project III, finding first and second quotient figures
was not responsible for the largest number of errors.
Only six of the 34 errors can be accounted for thus.
Twelve or just twice as many errors arose In multi­
plication situations# and eight arose in subtraction
situations.
As has been evident in previous analyses,
failure to perform fundamental processes accurately
is the major cause for Incorrect solutions.
Students
know the procedures but stumble on skills previously
learned and relegated to the maintenance program.
The Dictaphone Record Analysis Indicates
that the four types of subtraction methods which ap­
peared In the earlier projects still persist.
These
types are restated In order that later statements may
be meaningfuls
"Take-away11
9 6
3 7
5 9
1*
We cannot subtract 7 from 6 so we borrow
a "tens" from 9. 7 from 16 is 9; 3 from
8 Is 5 *
5 3 9
4 9 8
4 1
2,
8 from 9 is 1. You can’t take 9 from 3,
so the 3 becomes a 13* 9 from 13 is 4*
Because you made the 3 s. 13, the 4 becomes
a 5. 5 from 5 is zero.
(Continued on page 290)
286
Dictaphone Record Analysis
Project VI
40^
Q O -P
Subject E- w
<D
__ ______ ____
AL/X
(ZvAAu-^Xa J- , ^ W
.
o
<3t*-AA
//O
_^£AAi^ihJ
o
Xa yz^t\J
AZA.At^0-3
o
HL
UL
7.B. Sc. A
37
a-
/3A:
3p4AZ-^-337Z^i&. ^ 3 a A LahJ -
287
Dictaphone Record Analysis
©
H
8<
fl %
© si b
o
d
to o o
£
Subject E-» P=S £ M
02
«h <tj
• 5 O
a
© 0
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&
e 0
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*H © £ © ;:J
EH © CQ !*&*
T -
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c>
a
77
37
13.An*H. A
4&SL
c>
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T~ f
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d
Ot
d
0
Special Remarks
288
Dictaphone Record Analysis
©
a)
h a «j
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ft«H *0 *0
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op i3
£ • O © a*
©
w o © o «P r£fco ft
ft to o O g o rQ -P*d g
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B
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D
E
16* S*J. A
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289
Table 24
Analysis of Verbalized Solutions
Subtraction
Methods
r-i oa
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Solutions of thoseSolutions of those
examples In whichexamples in which
subjects:
subjects did not:
oa judged
Subject
compared
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1
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2
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4
4
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1
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3
12.N.T.
ts' </
4
3
1
2
1
4 -—
1
1
3
1
4
1
4
1
4
4
1
4
1
3
2
3
2
1
3
14 1
---
3
4
14aJ*Ba
Summary
1
2
2
1
4
2
2
3
1
16*SaJ.
3
5
13 aAn aha
ISalaOa
R W I
2
2
11aD*Ta
compare
4
2
1
6 * BaW.
9. S.Pa
H W I
3
2* Al.H.
3* RaW.
Judge
5
1
4
1
5
5
1 40 19
1 46 15
2
1
9
4
0
290
11Additive”
(Austrian)
5 3 2
2 8 7
2 4 5
X*
7 and 5 &r© 12* Carry the 1 to the 8 and
make that a 9* 9 and 4 are 13* Carry the
1 to the 2 and make that a 3. 3 and 2
are 5.
7 3
6 6
7
2*
The 6 is larger than the 3 , so you take
1 from the 7* The 7 becomes a 6 , and the
3 a 13* 6 and 7 are 13.
The symbols relative to 11judging” and "comparing" are
to be Interpreted as In previous Dictaphone Record
Analyses:
>/, did judge or compare; 0, did not judge
or did not compare.
Blank spaces Indicate that the
particular step in the solution was not reached.
Some of the data included In t h i s
analysis
has been recast in order to facilitate interpretation.
Of the sixteen subjects, 5 judged and compared; 5 judged
only; 6 neither judged nor compared.
Only two of the
five subjects who judged and compared did so consist­
ently.
It should be stated that very likely some of
the subjects who did not indicate verbally that they
"judged" or "compared" did In actuality do so.
Changes
In quotient figures before end after multiplication
and subtraction procedures, unaccompanied by verbal
comment, Indicate that mental judging and/or comparing
did take place.
Students who judge are credited with
more correct solutions than are those who do not judge.
Fourteen solutions out of 16 were correct in contrast
291
to 40 out of 60 for those who did not judge.
The
efficacy of "judging" Is also evident In the Dictaphone
Record Analysis*
There are eleven Instances recorded
where errors were discovered through "judging".
Summary of Findings Based on Sixteen Subjects
1*
Examples of Type D and E offer greatest dif­
ficulty even though most recently taught.
2.
Errors most frequently made are not peculiar
to any type of long division.
3*
Multiplication errors account for 32.5$ of
all the errors made; subtraction errors, 23*5$.
4.
Many errors are of the spurious type, as e.g.
"2 sixes are 18”, and "7 and 8 are 16".
5.
Four methods of subtraction are used?
T-2, Ad-1, and Ad-2,
T-l,
As in previous projects, sub-
jocts are not consistent In the methods used.
6*
Verbalization records indicate that few sub­
jects "judge" and/or "compare" and very few do so
consistently.
7.
"Judging" is a very effective procedure for
detecting errors*
292
8.
Subjects who consistently "judge” are credit­
ed with more correct solutions than ere those who do
not "judge".
Summary Treatment of Class
As in previous projects the examples present­
ed to the sixteen subjects were also presented to the
remaining fourth-grade pupils.
The special Example
Analysis form that was used in summarising the results
for the experimental group will be used.
Symbols that
appear in this analysis retain the meaning previously
given to them.
Example Analysis
The performance of the remaining fourth-grade
children and the experimental group on examples D and
E are strikingly similar.
Of the twenty children at­
tempting to solve example D, long division with threedigit divisor, 7 had incorrect solutions, 5 had incom­
plete solutions.
Of the sixteen children in the experi­
mental group, 5 had incorrect solutions, 3 had incomplete
solutions.
In the solutions to example E, long division
with non-apparent quotients, only 7 of the twenty chil­
dren had correct solutions.
correct, 8 Incomplete.
Five solutions were in­
Of the sixteen experimental
children, 11 had correct solutions, 4 Incorrect solutions,
(Continued on page 297)
293
Example Analysis
Project VI
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Exact Nature of Error
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294
Ex&mp1e Analysis
Project VI
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295
Example Analysis
Project VI
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Errors
15. B.K. A
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Exact Nature of Error
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296
Summary of Errors
Skills
A
B
0
D
E
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1
2
4
8
6
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A
2a
2b
2c
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1
3a
3b
1
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4b
1
5a
5b
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2
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Totals
9
15
15
0
1
0
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4
1
1
7
6
2
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6a
6b
6c
6d
6e
Total
1
2
1
3
1
O
2
2
5
7
9
9
19
15
61
297
and 1 Incomplete *
The larger number of Incomplete
solutions In the case of the twenty children was due
to the time limit of 1 minute 15 seconds which was
set for them*
This time was not sufficient for the
solution to the example#
More than half the errors
of each group were made in attempted solutions to ex­
amples D and E.
Sixteen errors or 47$ of all the errors
made by the experimental group occurred In attempted
solutions to example D, six errors or 17*6$ of ell the
errors occurred in attempted solutions to example E#
The percentages for the non-©xperimental group were
31*1 and 24.6, respectively*
A comparison of errors made by the twenty re­
maining fourth grade children and the sixteen children
of the experimental group reveals Interesting differences.
Thirty errors or nearly 50$ of all the errors made by
the
n o n -
experimental group were due to failure in find­
ing first and second quotient figures#
Only six errors
or approximately 18$ of all the errors made by the 16
subjects related to finding first and second quotient
figures.
projects.
This difference was also evident in earlier
The percentages for Project III were 31.7 for
the non-experlmental group, 21.4 for the experimental
group #
Another marked difference between the two
groups is their performance in multiplication end
298
subtraction*
MultipiIcotion errors account for 21.3$
of all errors xne.de by the non-experlmental group,
35*3$ of all errors made by the experimental group*
Subtraction errors account for only 3.3$ of all errors
made by the non-experlmental group, but 23.5$ of all
errors made by the experimental group.
Hypotheses and Tentative Conclusions
Project VI
Such hypotheses and conclusions as will be
listed are based upon summary findings for each case
study, for the experimental group, and for the remain­
ing fourth-grade pupils.
1.
The Instructional and maintenance pro­
gram during the semester Is adequate for the acquisition
of the special skills necessary to the solution of all
types of long division examples.
2*
Examples of type D and E consistently
offer difficulty.
The errors are not as a rule peculiar
to these types but appear In such fundamental processes
as multiplication and subtraction.
3.
More than half of the experimental group
fail to 11judge” or "compare".
The omission of ”judg­
ing” and/or "comparing” leads to errors and absurd math­
ematical situations within examples.
299
4*
Exposure to variants of the "take-away”
and the "additive" methods often results in a "hybrid"
method that Includes aspects of each method.
5.
The additive method and Its variants
permits of absurd subtractions without apparent dis­
turbances to the child.
6.
"Judging" and "comparing" appears In ex­
pressed verbal form, in Implied form, and at times in
silent, mental form.
7.
Previously learned skills Involved In
multiplication and subtraction do function though they
appear in new patterns.
Awareness of failure end dis­
turbance due to Increased difficulty, however, do
cause such spurious errors as "2 sixes are 18".
8.
Verbalization records, electrodermal re­
sponse records, and film analysis records aid In depict­
ing such mental states as frustration following failure,
confidence following success, etc. which accompany the
written responses.
Verbalization records, electrodermel
response records, and film analysis records aid in pic­
turing the progress of solutions, Including accuracy,
pauses, and significant expressions.
9.
Electrodermal response records tend to
support the notion that unless the consequences of er­
rors are serious, the emotional disturbance aroused is
not significant.
300
10.
The errors made by the non-experimental
, and the experimental group ©re not the same In kind or
degree*
The non-experimental group appear weak In
finding first and second-quotient figures end in multi­
plication*
The experimental group appear weak in multi­
plication and subtraction but not in finding first end
second quotient figures*
Project VII
a.
Learning experiences prior to the project.
Three months of vacation elapsed between
Projects VI and VII.
Insofar ©s the writer could de­
termine no formal attention had been paid to the skills
Involved In solving lcng-dlvlsion examples.
It is
not likely thet vacation experiences had been such as
to provide much maintenance of the newly acquired Icngdlvlsion skills.
A special effort was made to prevent
any immediate review on the part of the subjects prior
to this project.
With one exception, arrangements with
each of the subjects were made on the same day that he
was brought to the laboratory.
In several instances
parentfs cooperation was enlisted to help prevent review.
b.
Selected Case Studies
The following table, derived from Table 2,
page 6 » lists the three subjects chosen for Individual
study and Indicates their relative ranking in the
501
experimental group and in the class as a whole.
Table 25
[
Test Results for Selected Cases
.
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85
88
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92
74
14
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88
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98
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20
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108
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Exp.Group
M e d ia n s
C la s s
M e d ia n s
S ta n d a rd s
Case 1:
85
C.P.
As indicated in Project VI, this subject
possesses ability above the median of his class In all
of Mis school work.
His achievement in arithmetic,
reading, and language as reported In terms of test
scores in Table 25 Is definitely above the medians for
the experimental group and for the class.
The Individual Data Sheet Indicates that he
was brought to the laboratory on Sept. 9th, that his
temperature was 98.6, his pulse, 68.
He appeared in
good health, was very cooperative, and Interested in
proceeding with the work which began at 10s35.
Upon checking this subjects work at the
close of the school year, June, 1936, it Is found that
he had complete mastery of the long-dlvision procedure.
In the sixth project h© successfully solved both the
long-dlvision examples and the verbal problems which
followed.
The special record forms Interspersed In the
treatment of this subject will serve as bases for state
ments that follow.
ExaminetIon of the Verbal Report
Records indicates that skills Involved in the solution
of long-dlvision examples and learned during the second
semester of the previous school year are still functScn
al, that this subject still retains complete mastery
of long division procedures.
In his solution to example E, he stated:
«— - 3 nines are —
18.”
There is little question but
that this Is a spurious error.
It may be that his firs
attempted quotient figure which, was too large may have
caused sufficient inner disturbance to cause the compu­
tational error.
While the verbal responses did not indicate
503
expressed "judging" nor "comparing", there is reason
to believe such mental checking occurred*
This can
be inferred directly from his reaction as he proceeded
to multiply•
write the 1
goes 7 times
"
~ 9 three —
whoa
.11
—
_ 9 nines are 8 1
(laughs) - 7 --------------
Such Indirect evidence of "judging”
was also apparent in his solxition to example B*
After
having decided upon 4 as the first quotient figure and
begun the multiplication, his reaction was:
twos are 9 — --- 4 eights <— are --—
(laughs)
a
-4
see -----
------ that goes 5 times."
As in Project 71, this subject used a method
of subtraction which incorporated aspects of both the
"take-eway" and the "additive” methods.
The Verbal Profile Charts clearly picture the
steadiness of the attack on examples A, C, and D*
The
one marked irregularity that appears in C was causer
by silence that occurred during the time the subject
found how many times
—
72 into 6 hunderd n forty*”
The Irregularity that appeared in the solutions to
examples B and E are also emphasized.
The film analysis records substantiate the
inferences made from the examination of the dictaphone
records*
The hesitation in example B during which time
mental "judging” was Inferred lasted 137 frames, ap­
proximately twice as much tine as usually is required
(Continued on page 317)
304
■ Verbal Report Record
4
5 6 12 40
2 2 4
16
16
VII.
A
C.P.
9-9
6
a
14
S
8
8
8
0
Into 2 hunderd n40 — •* erne
4 times — ~ 4 sixes are 24--- write the 4
ftwr! Auwn: f.Vi* 5>
4 fives are 20 and 2 Is 22 — you can’t take 4
from naught so the naught becomes a 10 — 4 from 10 Is 6
because you mad© the naught a 10 — - you
make the *.*.«* 2 a 3
15
15
20
15
25
13
50
8
3 from 4 — ------ --—
55
7
40
12
45
15
you bring down the 8 —
fifty -— six
into hunderd n 68 goes 3 times - - - 5 sixes
are 18
write the 8 and carry the 1 — - 3 fives are
IS and 1 is 16
47
1
naught
(all right stopl
3 from 4 - la - 1
305
Verbal Heport Record
/
3 4
8 2 Y 2 7 9 9
2 4 6
3 3 9
VII.
B
C.P.
9-9
5 2 8
1 1
5
10
10
3
15
8
4 twos are 8 ------ 4 eights - are ---- a
20
1
----- see -------
25
4
(laughs) ----------- that coes 3 times
30
9
3 twos are —— 6 ——— 3 eights ©re 24
35
5
6 from 9 is 3
40
10
45
11
50
8
4 twos are —
55
7
- two -—
60
4
from --- 3 is 1 -— ---
63
2
_
- goes —
a
4 from 7 is 3 -- y o u bring down the 9
eighty — - two into 3 hunderd n 39 —
goes 4 times
8 _—
8—
-4 eights are thirty
from - 9 is 1
(all right, stoo)
2
306
Verbal Report Record
VI1C
C*P.
9-9
Seconds
„_____ 6 0 9
7 2 ) 4 3 8 4 8
4 3 2
6 48
6 48
‘
o
Words
Verbal Report
Into 4 hunderd n thirty ----
5
8
10
14
15
15
20
13
25
14
30
8
*—
eight
goes - 6 times --- 6 twos are 12 --- write
2 and carry the 1
6 sevens are 42 and 1 is 43 -2 from 8 Is 6
bring down the 4 — - you can *t take 7 inta
— * 72 into 6
hunderd n 40 so you put a naught —
bring­
down the 8 — - 72
Into 6 hunderd n forty — — -- — eight
goes - a
35
2
- see —
40
6
45
12
--- - 9 times — - 9 twos are 18
write your 8 and carry your 1 -- 9 sevens
are
63
50
7
now -— -- —
and 1 are 64 —
no remainder
(all right, stop)
307
Verbal Report Record
g
2 3 6 ) 5 66
47g
94
94
711*
D
C.P.
9-9
Seconda
Words
4
4
4
4
0
Verbal Report
— 2 hunderd n 36 Into 5
hunderd n 66
goes - 2 times
2 sixes ere 12
write the 2 and carry the 1
2 threes are 6 end 1 is 7 -- 2 twos are 4
-- 2 from 6
Is 4
you can*t take 7 from 6 so the 6
becomes 16
*7 from 16 is 9 — - because you made the 6
a 16 the 4 becomes a 5
5 from 15 is naught -— you bring down the
—
2 hunderd n thirty --
5
11
10
14
15
15
20
13
25
17
30
14
35
9
six into 9 hunderd n -- 44 goes —
40
8
45
12
50
16
- see
write
are
2 are
— - 2
53
4
© —
-- 4 times — - 4 sixes ©re 24
the 4 and carry the 2 -- 4 threes
---- 12 and
14 — — write the 4 and carry the 1
fours ere 8 and 1
is 9 --- no remainder
(all ri&ht, stoo)
508
Verbal Report Record
%
7 5
5 9 T^8~4~7
2 7 3
117
10 8
9
VII,
E
C.P.
9 -9
Seconds
Verbal
W o rd s
— 39 Into 2 hunderd n — 84 goes
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-=» & ~~~-.JLetts see — - a. —
____________
„— -*»— 9 tidies --- 9 t h r e e
9
nines are 81 -— write the 1
whoa — — J lau^frg) - 7________ ________ ______________
--- -—
~— -goes 7 times — - 7
nines
jfrrtt.
65..r~r,.,Mr.lt.fi-.-tte v and, carry the.-67 threes are 21 and 6 are 27
3 from 4_____________________ _ _ _ _ _
Is 1 —
7 from 8 Is 1 —
you bring down
Jbh e....Z -r,r.-39,.lnfca___________________________
1 hunderd n 17 goes 3 times — - 3 nines
are — IB_________________________—
— —
write the 8 and carry the 1 — - 3 threes are
9. ..and. 1 ,i.a IP. ~_____________________ _______
canft take 8 f r o m
7 so the 7 becomes
-a....X.7_________________________________________
8 from 17 is 9 -— because you made the
.7 a 17____________ _______ _______________
-- the naught becomes a 1 --- 1 from naught
■t t t .1. from 1. la naught------------------ —
_T.r.-~~r—
9 x.sznalnder--------------- ----
(all right,, atop) -------------------- —
309
Verbal Keport Record
1.
A quail was marked in the south one spring*
During the
summer it was seen 1904 miles farther north*
At Its
fastest flying speed (56 miles per hour) how long would
it have taken the quail to fly that far north?
Seconds
Words
Verbal Report
A quail was mOTKSCt In
tne sou
one spring# During the summer it had
-been — ■it was-seen -- 1 ‘tTOdssn'd 9 Htih'derd
n 4 miles
■Tsrther’-- further’ T io rth .; as
"A t’i t s .
fastest flying speed
56 miles per hour how long would it take th<
quail to fly that far north.
5
14
10
11
15
9
20
17
25
6
30
11
35
12
40
13
45
14
50
12
55
13
60
9
Os65
IBA
rlrie
70
16
8
Is 22 -------— -Birds are known to fly -— -
75
1
(excuse me, what are those 34?) — -a---
80
0
-- --(give a name to the answer)---
85
0
90
0
95
0
lOO
0
--- - take fifty —
six into 1 thousand
9 hunderd n 4 — - 56 Into 1 hunderd n 90
•»...»** goes 3 tim e si ——*» 3 sixes are 18 ———
write the 8 and carry
the I — - 3 fives are 18 and 1 is 16 -you canft take
8 from naught so the naught becomes a 10
— - & from — — 10 is 2
because you made the naught a 10 the 6
becomes a 7
7 from 9 Is 2 — - bring clown the 4 -56 into 2
hunderd n 24 goes - 4 times -- 4 sixes
are 24 — - write th - 4 and carry the 2
4 fives are 20 and 2
510
Verbal Report Record
2*
Birds are known to fly as long as 9 hours without stop­
ping, when on their way north In the spring,
A thrush
to which a tag had been tied traveled north ©t an average
speed of 130 miles a day during the whole month of April,
How far did it travel?
Seconds Words
105
0
110
0
115
9
120
14
125
16
130
17
135
5
140
5
145
10
150
12
155
2
(There are 30 days in April)
Verbal Report
;
--(what ahe they) days (all right then go
right on to the next problem) -- Birds are
known to fly as long as
9 hourd without stopping toheri on t h e i f -way north in the spring. A thrush
to which a tdg had been tidd braveiea — •no:
at an average speed of 1 hunderd
n 30 mildo a day during the whoie month of
April. How far north did it travel
by 3 0 ---- -— naught ----- 3 naughts
are naught — - 3 threes -- are 9 --3 ones are 3
it traveled 5 hunderd and 90 mi
thousand 9 hunderd miles
in A p r i l ---- -- —
(all rl&ht. stop)
3
'_£.c\
515
Film Analysis Record
4 5
5 6 ) 2 4 0 8
VII*
2 2 4
A168
C *p *
16 8
9-9
0
VII.
B
C.P*
9-9
82 J 2 7
2 4
5
5
Steps
3 4
9 9
8
59
2 8
1 1
Frames
1.
Finds and writes 4
58
2.
Multiplies 4x56, writes 224, draws line
92
3.
Subtracts 240-224, writes 16
4.
Brings down 8
5.
Finds and writes 3
48
6.
Multiplies 56x3, writes 168, draws line
70
7.
Subtracts 168-168, writes 0
8*
Remaining film
152
6
3
15
VII • B
93
2*
Multiplies 4x83, writes 8 , stops, judges
3.
Writes 3 over 4
4.
Multiplies 3x82, writes 246, draws line
97
5.
Subtracts 279-246, writes 53
43
6.
Bring down 9
7.
Finds and writes 4
49
8.
Multiplies 4x82, writes 328, draws line
76
9.
Subtracts 339-328, writes 11
41
Remaining film
58
H
Finds and writes 4
O
*
1.
137
7
6
324
Film Analysis Record
6 0 9
7 2 ) 4 3 8 4 8
VII.
4 5 2
C
64 8
C.P.
6 4 8
9-9
" ~0
VII.
D
C.P.
9-9
2 4
2 3 6 ) 5 6 6 4
4 7 2
944
9 4 4
0
Step s
Frames
1.
Finds and writes 6
66
2.
Multiplies 6x72, writes 432, draws line
68
3.
Subtracts 458-432, writes 6
13
4.
Brings dov/n 4
5.
Finds and writes 0
47
6.
Brings down 8
17
7.
Finds and writes 9
8.
Multiplies 9x72, writes 648, draws line
9.
Subtracts 648-648, writes 0
10. Remaining film
7
149
84
4
16
VII., D
1.
Finds and writes 2
65
2.
Multiplies 2x236, writes 472, draws. line
73
3.
Subtracts 566-472
4.
Brings down 4
5.
Finds and writes 4
6.
Multiplies 4x236, writes 944, draws line
7.
Subtracts 944-944, writes 0
8.
Remaining film
126
3
85
131
3
14
315
Film Analysis Record
VII.
E
C.P.
9-9
7 3
3 9 ) 2 8 47
27 3
117
10 8
9
Steps_____________
___
_ _____________ Frames
1*
Finds and writes 9
2*
Multiplies 9x39, writes 1, stops, judges
85
3#
Y?rites 7 over 9
21
4«
Multiplies 7x39, writes 273, draws line
5.
Subtracts 284-273, writes 11
6*
Brings down 7
7.
Finds and writes 3
42
8.
Multiplies 3x39, writes 108, draws line
97
9.
Subtracts 117-108, writes 9
39
10* Remaining film
117
105
26
4
114
316
I
Figure
15
i
0)
<0
G
O
A
<0
©
«
H
a
§3
©
Td
o
Pi
+»
o
©
u
—*
£
O
*
H
-P
O
2
*8
*
»
OJ
^
• %
,
i
317
for the first multiplication#
In his attempted solution
to example E, the same situation occurred during the
first multiplication--—
in this instance 85 frames of
film were used#
The electrodarmal response record Indicates
a general picture of affective stability#
Such deflec­
tions as occur are neither sharp nor of great magnitude.
In the two instances, examples B and E, where hesita­
tions and shifts in the solutions occur there are no
apparent significant deflections.
Examination of the
verbal reports reveals that the subject did not con­
sider either instance serious, in fact his incorrect
attempts seemed to amuse him.
These instances appear
to further substantiate the notion that if errors are
not taken seriously, very little if any emotional dis­
turbance results#
Summary of Case Study Findings
Case 1. C. P#
(A pupil of above-median ability)
1.
Skills Involved in the solution of long-
divlsion examples and learned during the second sem­
ester of the previous school year ere still functi onal
after three months of vacation experiences.
2 .
Computational skills
plication and subtraction
as
In v o lv e d
o ccu rs
in
within
such
multi­
lo n g -d iv ls io n
318
examples operate effectively though they appear within
more complex patterns than when originally learned#
3#
Awareness of Incorrect attempts at solutions
is not accompanied by emotional disturbances.
4#
A r,hybrld,t method of subtraction which in­
corporates aspects of both the "additive” and the
"take-away” methods persists.
5.
The verbal report records give definite evidence
of "judging” and "comparing” though not expressed In
the usual manner*
6#
Verbalization Records and the derived Profile
Charts Indicate the subjectfs confident, consistent
attack on examples A, C, and D end the confident but
less steady attack on examples B and. E.
7.
The Film Analysis Records substantiate and
emphasize the variations In regularity of the subject#s
attack on examples.
8.
The electrodermal response record shows the
affective stability of this subject, the lack of ap­
parent disturbance when Inconsequential errors are
made.
This record substantiates the apparent emotional
pols© possessed by the subject, poise even at times
when errors are made.
519
Case 2* 1*0.
Table 26
H
•
O
•
Subject
*
*
ofo
tdt0
Pi 1
£>
•CO
n i
© lO
»
Cd-P
& CO
o©
CO
EDtO
d-tL'i
•r
~
rO C'l
cj 1
© ,o
1
co
©
-P 02
Cj ©
CO £i
102
85
83
12
20
5.5
6 .5
105
110
109
98
106
20
21
6 ,0
6 .7
131
151
108
97
100
19
21
6 .0
6 .5
116
102
109
1 4 .0
1 8 .0
4 .5
5 .0
V''-
hi EH
I ^
[ to
_ [1
Standards 1
_ILnnl
P‘O
ctitO
^ C1
O
•CO
a »—I
»
©
rH
PI 1
c3"P
£5 KJ
O 0
hi Eh
1
Exp.Group
Medians
Class
Medians
t>6iO
0 tO <DtO
> 1 t> *
!0
P CO
P 03
CO 1
CO 1
i—i
0}(O
Q
CO 1 02 1
oj
cci
>P
El ©
m o
d©
w
o
O £-1 O iH
02
- CO M CO
r~\tO - to
rH t rH t
«in1 rH CO
OO 0I
3 «$02
O 1
O-JLO
- i 7 1
hi
r} -h •p “p
o m
O 02
O Q> O <D
Ds Ai !SsC-i
!Gates Reading
iTest - 1-17-36
Iluhlman Anderson
Intelligence Test
Test Results of Selected Case
85
The above test scores for this subject In­
dicate mental ability slightly below the median of the
experimental or class groups.
As stated in the treat­
ment of Project VI his scores on the achievement tests
that were administered prior to this study show les3
than median ability in arithmetic, reading, and lang­
uage.
His test scores at the close of the year show
real Improvement in his arithmetic and reading.
This Individual was the 9th subject of this
projectf brought to the laboratory on the morning of
Sept. 10th.
H© was very talkative, appeared somewhat
520
excited, and was eager to proceed with his work.
His
temperature, 98.6, pulse count, 80, together with his
verbal report relative to his health gave assurance of
his physical well being.
The preliminary series of simple addition
examples evidently did not suffice to orient this indi­
vidual completely*
He appeared quite fidgety, giving
evidence of the effect of nervousness In the first stages
of his attempted solution to example
A :
M---4 times
a - 6 Is 18 --- put down the 8 and carry the 1 -— •”
A little later in subtracting 218 from 240, he failed
to make allowance for the fact that he made the zero
a ten before subtracting the 8 .
These errors were
very probably brought about by the distracting effect
of being confronted by a school and laboratory situa­
tion after three months of vacation.
The errors can­
not be attributed to a lack of knowledge--
the Iden­
tical computational difficulties appeared In example
D of this project and were then correctly solved.
The verbalization records Indicate a direct
and accurate attack on example B, long division with a
remainder.
No errors nor undue hesitation occurred.
Long division with zero In the quotient in­
volved skills which were no longer available or operat­
ive for this subject.
Illustrated.
The law of disuse was nicely
On June End this subject proceeded without
321
hesitation to solve this identical example correctly.
Upon reaching that portion of example C which involved
"zero in the quotient", he hesitated for approximately
15 seconds and then ventured that 72 went Into 64 nine
times! 1
Proceeding Immediately to multiply, he placed
the product 658 below the 64, and then just stared at
the resultant situation!
Upon the writerfs query,
"something Is wrong .. ?" he responded with a "yafl.
He was not able, however, to find his error during the
twenty seconds that were allowed.
After this unsuccessful attempt at solving
example C, he proceeded accurately to work examples D
end E.
In his solution to example E, long division
with non-apparent quotient figures, several trial quo­
tient figures were necessary but this did not prevent
him from successfully solving the example.
The progress
of his solution was very Irregular and more than the
usual amount of time was consumed.
His solution to the first verbal problem was
correct though considerable hesitancy characterized his
work.
He failed to solve the second problem correctly.
His verbalization gave evidence of a confused attack
and of extreme doubt as to the correctness of his final
answer, viz.
" -—
— -—
etc*"
-- -
a
how far —
a ------
—
As In his previous solutions during the school
(Continued on page 336)
322
Verbal Report Record
______4 6
5 6 ) 2 4 0 8
2 18
3 2 8
VII.
A
1 *0.
9-10
.5
7
10
7
0
5
.
.
.
.
..
2
0
-25
.
15
a
P4.
f
T
A
A
f
l
A.
f
.
1
1
H
AO
I «
«
■
>
A.
frnm 5 «
,
- 4 times 5 Is 20 and 1 is 21
mmmmfi.m
5
**********
*
»1
10
1K
l
t^A
- 6 Is 18 — - put down the 8 and carry
15
... 30
—35
m
m
m
mm
m
*
*
.
«
*
m
m*m*
*
»
R
48 rem.
TO
«
•
>
1U
tm
m
m
mm
mm
m P. mmw £? 'PHOTM
f*T»fvnri 4- 1fl ^
wok
5? fr>Am
P
m
m
.
.
.
.
4 a wonn
V»-r»4t»;")• rtAttm f.h/i ft__ K n'/\*ct 4infn !XO---- —
40
5
times —
45
6
-— • is - .a-— -- is 6 is 30
put down the zero and carry the
5 times 5 is 25 end 3
50
55
16
28
5 times
stertt
R
__
8
8 f*rom —— 12 *.* wo* ...............
bring down your 2 and make 1 and make this
-— •
.
.
.
u
-
m
'
1,11
*
•
«
.
*
.
3 -
4
g a
■
*a
f
r
n
n
i
is —
la
5
<
*
*
.
*
*
6
a
13
70
8 from
.
.
.-
48 is the remainder
(all right stop)
..........
323
Verbal Report Record
VII•
B
1*0 *
9~10
Seconds
3
8 2 y 2 7 9
2 46
3 3
3 2
x
Words
4
9
9
8
1
Verbal
into
10
15
------12
20
25
12
30
45
10
goes into 27
put it over your 9
3 times «
sev
twenty — Tour
n r 3 -— ‘4"'from T'Tir z
2 from 2 is zero
times
10
Report
is twenty --
35
50
3 times
^g"times 3 la 6
bring down the 9
40
11 rem
is 8
goes into 33
2 times 4
8 times 4 is 32
2 from 3 is 1
3 is zero
11 remainder
(all right, stop)
3 from
324
Verbal Heporb Record
6
7 2 } 4 5 8
4 3 2
6
6 5
9
4 8
_ _________________
VII.
C
1* 0#
9-10
Seconds
10
Words
4
8
Verbal
8
Report
* _7 times -—
put the 7 o?^r_ thi*
15
8
20
- 6 times
.g_ tim a riL
is 12 put down the 2 and carry the 1
6 times 7 is forty — ___________
two - and 1 Is 43
2 .from 8 Is 6
brln& down your 4
£§.
16
30
15
7 £Loes Into 64
40
XL
J&IL
50
M l
— — -- 9 times
2 times 9 is 18
put down your 8 and
15
g&r.£y the
M l
IL
65
6
70
0
85
..7,y.
r. la — * sixty.
JL
four and 1 la six
—
five
(something is wrong there is it)
75
80
-
ya
325
Verbal Report Record
2
2 3 6 ) 5 6 6
4 7 2
9 4
9 4
VII.
D
IeO.
9-10
Seconds
—
20
.
.
25
Verbal
7
15
—AOUQjl ——»■
B—goe g
—into—U—a
n
a
"
1.
.
.
.
—
.
2
*
*
«
•m
i
f
. 1f
. ava«i
- 2 times 6 Is 12 — un^ ecwimf t
V
»
o 1 •
is
—
—
4
& —
put down the
A
^ f4f»A« P 40
—
--
2
-
7 f r o m ----
3
55
4
40
2
45
12
_
_
_
_
_
_
50
15
55
9
50
15
55
12
end
70
10
53
_
4
*
V
»
a
2 times 3 Is 6 and 1 Is 7
14
2 from 6 —
---- 30—
Report
Q
4
.15.
.
._
4
4
0
Words
.
4
4
is ©ir
—
8
— -
no
—
___ __________ ls 9 _
_
_
_
_
_
_
_
_
--carry the — 1 down to the 4 and
make it a 5
5 from 5 is zero — - bring down the 4
2 goes into 9 —
4 times ---• 4 — 6 times 4 ia — 24
put down your 4 and carry your 2 -- fort
—
5 —— is ——
*“ 12
—
and 2 Is 14 — put down your 4
1
4
/ell
-
is
w< n«Vif
8
and 1
a4“Ain\
5 s* 9
—
ze ro
- ■■ -■
Verbal Report Record
7 a
3 9 ) 2 8 4 7
2 7 5
117
117
VII*
E
1 .0.
9-10
---- &
—
.
5
- 10.
8
—— 9 times —
15
8
** 8 times
...80.
6
8 times 9 — — is - 72
put down jour 2 — —
and carry your 7
5 times 8 Is 24
no -<— ■ 3 goes into 28 —
you
put It over the 4
____ 25
14
30
.5
55
1
-----{something wrong7) ya# (well then go el
40
0
(and change It and make It correct)-— — -— -
46
6
.
50
5
__
55
5
—
60
8
—
65
12
70
8
75
14
80
4
Into 11 — —
85
6
3 times 9 is —
90
0
95
0
100..
2
and —
7 is —
—
thirtv —
one —
It should be 3 times 7 — -
7 times.9 is — — • sixty —
three — —
aev—
put down the 3 — - carry the t
3 times 7 is 21 and 6 is
—
27
— 3 from to — 4 Is 1 — - 7 from
8 is 1 — - 2 from 2 Is zero —
bring down
the 7 — - 3 goes
3 times
--- —
Is - a — -
327
Verbal Report Record
VI ♦
'ft
..
1 0
9-ib
(continued)
105
a
110
9
112
1
put down your 7 —
and carry the —
3 times 3 Is 9 and 2 —
zero
(all right, stop)
2—
is 11 —
328
Verbal Report Record
1.
A quail was marked in the south one spring.
During the
summer it was seen 1904 miles farther north.
At its
fastest flying speed (56 miles per hour) how long would
it haV© taken the quail to fly that far north?
VII*
1.0.
Seconds Words
Verbal Report
A quail was marked in the south one spring.
During the
summer «*•«■** it was seen - 9 hunderd n 4 mil©
farther north
at Its fast — ©st flying speed — 56 miles
per hour
how long would it have taken the quail to f
that —
fly that far
5
11
10
11
15
10
20
14
25
0
30
4
- 5 goes Into 19
35
6
40
11
a — 3 times
5 times 6
*
Is -—
24 — - put down your 4 and
carry your 2
45
3
— — — no it's 18
50
9
3 times 5 Is —
55
5
60
14
8 from --.t— 10 Is 2
__
and 7 from 9 Is 2 **--* bring down the 4 -5 goes Into twenty
65
4
70
7
75
18
80
1
85
1
QA
3
95
1
100.
.
„ 0
a
4 times.
...
...
put down the 4 and carry the 2 —
ia 20 and 2 (s 29
4 times
1
1
1
1
e
a>
I
0
*
1*
j
j
two
15 and 1 Is 16
34
** no
hours
*. .....
329
Verbal Report Record
Ho* 2 *
Birds are known to fly as long as 9 h o w s without
stopping, when on their way north In the spring.
A
thrush to which a tag had been tied traveled north at
an average speed of 150 miles a day during the month
of April. How far north did it travel. (There are 50
days In April)
Seconds words
Verbal Report_____
105
3
.110
12
115
14
_ 120
17
125
12
150
6
— — -“ --Birds are known to fly as long as 9
w i t h o u t stopping
when on their way north in the spring **■* a
— thrush to whl r*h » t«g
had been tied traveled north at an average
speed o f h u n d e r d 30 ml lea a d a y
---- ...--- - during the whole month of April
_— • how f a r
n n r t h did It travel
T h e r e are
d a y s In April
—«•**.»
6
140
...
4
(What are you going to do?) —
f*1rid mit hrtinr tnn-w’tr,
A
ifin
n
155
6
150
7
165
6
-— — » zero — - 4 times 3 is 12 -...
170
4
zero —
176
O
iAn
A
1OR
n
187
0
rinii mifc h n w .nmAh it 'hFAVfilAd «*«»«»
goes Into a ~ —
-1
15 — *«* 4 times —
— 10 remainder
(what are those 4 —
—a —
how flit* «— R
fall right atoo)
)
......
Cl
ZLi.
jL'l
332
Film Analysis Fiecord
5 6
VII.
A
4 3
2 4 0 B
2 18
. .
1 0
48 rem*
3 2 8
2 8 0
9-10
4 8
VII*
B
I.Q.
9-10
3 4
8 2 J 2 7 9 9
2 4 6
3 3 9
11
3 2 8
Steps________
1 1
Frames
1.
Finds and writes 4
2*
Multiplies 4x56, writes 218, draws line
5.
Subtracts 240-218, writes 32
4.
Brings down 8
5*
Finds and writes 5
6.
Multiplies 5x56, writes 280, draws line
7*
Subtracts 328-280, writes 48
84
8.
Remainder
71
78
122
93
4
70
182
VII . B
91
1.
Finds m d writes 3
2.
Multiplies 3x82, writes 246, draws line
5.
Subtracts 279-246, writes 33
4*
Brings down 9
5.
Finds and writes 4
72
6.
Multiplies 4x82, writes 328, draws line
81
7.
Subtracts 339-328, writes 11
29
8.
Remainder
69
115
64
9
Film Analysis Record
6 9
7 2 ) 4 3 8 4 8
VII#
4 5 2
C
6 4
1.0.
6 6 8
9-10
2
2 3 6 j 5 6 6
VII,
4 7 2
D
9 4
1.0,
9 4
9-10
S t e p s _______
_________
4
"4
4
4
0
Frernes
1*
Finds and writes 6
122
2.
Multiplies 6x72, writes 432
165
3.
Subtracts 458-432, writes 6
9
4*
Brings down 4
4
5#
Finds and writes 9
180
6*
Multiplies 9x72, writes 658
363
7 • Remainder
VII» D
1*
Finds and writes 2
140
2,
Multiplies 2x236, writes 472, draws line
115
3*
Subtracts 566-472, writes 94
183
4. Brings down 4
5* Finds and writes 4
6 * Multiplies 4x236, writes 944
7* Subtracts 944-944, writes 0
8 * Remainder
6
42
153
5
16
334
Film Analysis Record
7
.__ S' 3
VII.
B
1 .0.
9-10
3 9 fS 8 4 7
27 3
TT 7
117
0
Steps
Frame s
1*
Finds and writes 8
135
2.
Multiplies 8x39, writes 313, draws line
148
3.
Judges, writes 7 over 8
139
4•
Multiplies 7x39, writes 273
198
5.
Judges
34
6.
Subtracts 284-273, writes 11
41
7.
Brings down 7
8.
Finds and writes 3
9.
Multiplies 3x39, writes 117, draws line
10. Subtracts 117-117, writes 0
11. Remainder
5
52
281
13
335
?
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a
th e
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--------------
a s p e c ts
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th e
28
tim e s * ”
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3 goes
8
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—
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C
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-------
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337
S u m m a ry
(A
1*
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P r o je c t
V I
278;
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72
6
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224
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to
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in te r e s tin g
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school
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V I
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240
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w ith
d iv is io n
c o rre c t*
num ber
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ceeded
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re a c h e d
w e ll.
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10s 15
th e
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d ir e c t
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(C o n tin u e d
be
on
and
p ro m p t
a s c e r ta in e d
page
355)
340
Verbal Heport Becerd
4 4
5
6
)
2
V II,
A
8
4
0
2
4
8
2 0 8
2 0 4
9 -1 1
5
12
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15
7
4
20
8
24
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Is
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7
s u b tra c t
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4
35
7
2
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2 ™
**«*•*
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56
40
goes
12
th e 8
n —
56
45
50
5
55
10
4
tim e s
6
is
a
60
6
4
tim e s
5
Is
20
64
4
th e
56
goes
In
tim e s
2
«*—■■*•4
tim e s
c a rry
20
and
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4
2
22
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24
is
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-------
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to
.. . _
20
341
Verbal Report Record
43
8 2 ) 3 7 9 9
2 7 if
*
9
8
B.Sm.
9-11
Seconds
..........
Words
__ 11
5
______ 1 , 0 ,
,
ta k tfi
AS
in t o
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—
27
w o n 't
go
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ID
15
9
4
20
4
25
17
is
- a - 2 7 ------------------ -- — -**8 fro m 9 Is
1 —
82
y o u b r in & dow n th e 9
30
0
35
0
40
G
45
11
50
0
tim e s
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go
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in to
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you r
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60
7
3
65
6
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6 fro m 9 is
5 -—
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_
____
2 w o n 't g o i n t a — ~ 3 s o y o u b r in g d o w n
y o u r 9 —
-
70
10
75
11
80
tim e s
8
-.-a --
0
O
90
•
0
{All
r l& h t•
s to p )
tw e n ty
..
Verbal Report Record
6
7 s J 4 3 8
4 3 8
6
36
3
*
0
B.Sm*
9-11
5
_________
ID
12
14
.
0 9
4 8
4 8
3 /8
3
wonft go Into — — 43 so you
take - a— into
58 —
I mean Into 4 hunderd n 58 -a-e---see---*- seventy-.*.
_
15
9
SO
4
S5
10
50
9
55
10
2 from 3 is 6**.-*— 72 wcm*fc go ^oto
40
ID
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ing
45
7
50
13
___ 55
Q
60
1
65
5
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1
85
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90
5
95
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42 and 1 is 43 --.-a-...... aiihtnjant
your ——a—— 4“-
__
72 ..
won *t go Into 84
s o ™ * 5™ —
— 72 won*t go into 64 so you bring down
ymi.r*
8
—
72 goes Ir»t®>———6
sixtv
.~—
-
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tbree—
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4*
.
9 times 2 is — —
w*
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—
.
. -
.
343
VII.
B.Sm*
C (continued)
Seconds
Words
Verbal
Report
105
110
18
115
11
150
9 Roee Into -
9 times; 2 la
18
-*-8 from 8 Is sero
■*
55 remainder
{ M 1, rl&fcfc* stop),
Verbal Report Record
2
) a 6 6
4 7 2
9 4
1 8 9
VII.
D
R.Sb w
9-11
Seconds
3
4
4
6
Words
5
10
10
14
15
0
so
4
25
4
30
9
—
35
6
- 7 — * 2 times :
------ - 2 is 4
40
8
45
6
50
17
55
7
goes Inta - 9—
60
3
—
65
6
- 3 times * a * 2 ia —
70
4
3 times 3 —
75
5
3 times 6
80
1
85
1
90
0
- — — (Do you know what1s the matter, Bobby-
95
0
100
4
— — -— —
(Multiply your 3 by your 236
once more — let me see you do that work)
3 times 2 Is-— —
a- 5 hunderd
— won1t go Inta 58— ~so yon take It Into
5 hunderd — ©6— -- —
------------ (I’ll get It) -----
12—
2 times 3 Is 6 and —
2 from 6 Is
1 Is
- 4— - 7 from 6
la-— —
— — 9—
2 hunderd n thirty —
3lx Won ft go Ihta 94 so you bring down your
2 hunderd n 36
hunderd n 44 —
3 times — *—
—
a ©
.— — 9
■—
345
VII*
B*Sm*
D (continued)
105
1
—
110
5
(down her© ---- --- ) 5 time© 2 Is 6 —
115
10
ISO
0
125
0
6 ~
(no that *s)
3 times 5 is 9 -- 3 times 6 is 18
(Is that the right way th multiply 236 by 3
(All right,, stop)
346
Verbal Report Record
6
4 7
7
7
5
2
9
3 9 T 2
7
6
8
8
VII *
. E
B«Sm«
9-11
Seconds
8
4
4
5
8
Words
won’t go Into 28 so you
5
11
10
6
Into 2 hunderd n 84
15
5
— —
20
5
times —
25
8
30
14
- a-— —
36 - carry the 3 u p here
I mean 9 times 3 Is -— a 27 carry the 2
up her©
35
3
9 times —
40
4
Is - a -- --------- 72
45
1
50
4
2
55
9
--- 7 from 14 is 7--- 4 from 8 is —
60
1
65
2
70
12
75
4t
80
i
85
7
90
1
95
io
— 18
— 18 —
writ© your 8 and carry your
1 up her©
100
9
— — 6 times 3 — I mean 6 times 9 Is -
take It
a— goes
ni~~n©
9 times 3 Is
9
- are 74
.. .
.**.»•*«*M
7 from
— - I mean —
6 from 12 Is 6 —
-- isoes into 6
hunderd n—
47
....... .
39
—
347
vxx*
B*Sn*
® (continued)
Seconds
Words
Verbal
Report
105
4
110
4
115
5
-— —
120
8
-2— - 4 from
125
2
130
4
135
6
— —
8
140
4
from
16
145
3
150
0
—
8 t h a t 12 wrcnst
(is that wrongs-you donrt know where your
mistake is th o u g h * do tou)
5 from 7 Is X wean 5 from
— *------- 5 from 14 Is
( a ll
14
—
rnd --6 from -- 8-~- 8
is -a------ — -
rl^ht* . s t o r )
....
....................
348
Verbal Report Record
1*
A quail was marked in the south one spring*
summer it was seen 1904 miles farther north*
During the
At its
fastest flying speed (56 miles per hour) how long
would it have taken the quail to fly that far north?
VIIB.Sm.
Seconds Words
Verbal Report
A quail was marked in the south one---5
9
south
— - one spring* During the summer - —
10
8
It was seen — -it was seen 1 hunderd n —
1 thousand
15
12
9 hunderd n ----* 4
20
4
25
12
30
8
35
9
40
5
45
7
hunderd n 19 —
1 mean 1 thousand
50
9
9 hunderd n 4—
and your 1— - 56 --—
55
0
60
9
65
5
70
12
75
19
9 Is 54— -carry the 4-*— 6 times 1 is 6
ana 2 are 8— o times 4 Is 20 — bring your
2 up here-- 5 times zero Is zero
80
12
n— -put down Your 2 - - n 5 times 9 is 45
85
11
- - a n d 2 Is-— 5 times 1 is 5 and 4 is
90
4
95
4
100
8
- 4 miles further — north— - at its-- fast— — at its fastest flying
speed —
56 miles per
Hour -------- now long would--- How long
would It -Have taken the quail — - to fly that—
far - north
. 6 times 4 Is 24 — - 6 times zero
Is zero —
6 times - - a - — -
n carry your 1 —
and 9 Is 10--------
340
Ho* 1 (continued)
2
Birds are known to fly as long as 9 hours without stopping when on their way north, in the spring.
A thrush to
which a tag had been tied traveled north at an average
speed of 130 miles a day during the whole month of April*
(There are 30 days in April)
No.l
105
0
110
Ho*2
115
0
11
120
9
— ■
— ■-— Birds are known to fly as long--as 9 hours without
stopping ---when — - when on their w a y north - in the
125
7
----in the spring— —
130
7
35
7
14,0
11
]4.5
TO
150
10
155
6
160
0
165
2
170
4
175
13
ion
£>
*IA5
2
190
1
105
O
son.
... .1
A thrush
a thrush
..to which a tag.
which — a tag
had been tied — traveled n o r t h ---- - —
at — - at
an average speed - of -- e hunderd n 30
vallaa .
— a day
....
—
a day - a day — during — the whole month
of April." How far north did it -- did it
travel
...
there are 30 days in April
~~— ---- 30 --- - 30 won’t go
into 13 so you take it into 1 hunderd n — n 30 a -------------------4 times-— ...
—
....
no remainder
(What are those 4) huh— (what are those 4-—
..
---Bobby — --- *--- ) —uhnu
352
F ilm
Analysis Record
3
______ 4_4
5 6 ) 2 4 0 8
2 2 4
VII.
2 0 8
A
2 0 4
B •Sm •
4
9 -1 1
82 T 2 7 9 9
2 7
S 9
VII.
B
B •Sm *
9 -1 1
Steps____________
Frames
1.
Finds and writes 4
116
2.
Multiplies 4x56, writes 224,draws line
164
3.
Subtracts 240-224, writes 20
4*
Bring down 8
5.
Finds and writes 4
90
6*
Multiplies 4x56, writes 204,draws line
96
7,
Subtraot 208-204, writes 4
8.
Remaining film
99
4
6
28
VII > B
1*
Finds and writes 4
104
2*
Multiplies 4x82, writes 8 , drawsline
98
3.
Subtracts 279-278, writes 1
36
4.
Brings down 9
5.
Writes 3 over 4
149
6#
Multiplies 3x82, writes 276
245
7.
Subtracts 279-276, writes 3
4
8.
Remaining film
4
119
553
F ilm Analysis Re cox'd
6 0
7 2 )4 5 8 4
4 3 id/
VII.
6 4c
B. Sm.
6 3
9-11
3 3
9
8
8
8
VII.
D
B.Sm*
9-11
2 5 6 1 6 6 6 4
o9
4 7 <
9 4 4
1 8 9 6
Steps
Frame 3
1.
Finds and writes 6
177
2.
Multiplies 6x72, writes 432, draws line
126
3.
Subtracts 438-432, writes 6
4•
Brings down 4
5.
Finds and writes 0
78
6.
Brings down 8
33
7.
Finds end writes 9
131
8.
Multiplies 9x72, vxrites 638, draws line
527
9*
Subtracts, writes 33
43
*
o
H
Remaining Film
36
57
4
VII . D
1.
Finds and writes 2
£25
2.
Multiplies 2x236, writes 472, draws line
130
3.
Subtracts 566-472, writes 94
4.
Brings down 4
5*
Finds and writes 3
6.
Multiplies 3x236, writes 1896
9.
Remaining film
92
6
95
619
*.■<i>4
Film Fna lysis Record
J/
3
VII.
9 6
9
>
2
8 4
7
7 4
K
6
B.Sm*
9-11
8 55
8 8 2
4
7
Steps_____________________________
Fra
1,
Finds and writes9
153
2.
Multiplies 9x39, writes 747, draws line
342
3«
S\tbtracts 284-747, writes 647
270
4.
Finds and writes 6
5«
Multiplies 6x39, writes 855
295
6.
Subtracts 647-855, writes 882
168
7.
Remaining film
7
355
he was not aware of any errors#
errors did occur, however:
-—
Two computational
4 from zero Is - a
zero" and 5G x 4 equals 204#
The general long
division procedures were 3till functional, his fail­
ure In a subtraction example and in a multipliestl on
example did not necessarily indicate a loss due to
vacation since he previously lacked this special multi­
plication ability#
In his attempted solution to example B he
gave further evidence of a knowledge of the general
long-divlsion procedures, even to reducing the trial
quotient figure when in his judgment it proved too
large1
A strange shift in his multiplication proced­
ure which was to persist made its first appearance
here, viz.s
8 2
3
2 7 9 9
"times 8 Is - a
a
2 7 6
—
twenty -—
6 *M
--- -—
s e v e n ---- and —
5 times 2 Is
It appears that multiplication In the long-division
setting had notbeen overleerned sufficiently to
stand the ravages of time.
It
with­
seems that both a cor­
rect and an Incorr*ect multiplication procedure was
followed, neither of which was used consistently#
This shift In multiplication procedure was
evident in his solution to example C, long division
with zero in the quotient#
The first multiplication
was correctly done; the second, incorrectly#
Interest-
356
ingly enough, this subject placed the zero in the quo­
tient correctly after the verbalization, H
go into 6 4
so
72 won’t
a — -- ."
In his attempted solution to example D, this
subject again proceeded properly and accurately until
the final quotient figure was determined.
The final
multiplication was again Incorrect, this time a further
deviation from correct procedure!
Multiplying 236 x 3
his product, 1896, while Incorrect was not Incorrect
because of Ignorance of the multiplication tables!
Kote:
11 - 3 times - a. - 2 — - 6 , 3 times 3 —
3 times 6 - Is - © --------- — --- 18 —
------ 9,
He multi­
plied from rl&ht to left, writlnp; his numbers from left
to right*
The verbal report gives evidence of much
conscious difficulty, much hesitancy.
This is particu­
larly true of his solution to example E.
cation procedure, noted above, recurs.
by 9 his product 747 was obtained thuss
— - 27,carry the 2
72 — —
tion
up here.
The multipli­
Multiplying 39
M 9 times 3 Is
9 times — - 9 is- © ----
-----... end 2
are 74.n Examina­
of-this subject’s solutions in Project I Indicates
difficulty in addition which was also directional. In
adding 126 and 14 without first placing them In a verti­
cal column he proceeded to adds
put
your —
ze
M —
4 and 6 are 10,
— your 10 down - you put 1 and 2
557
f^r© 5 and s o y o u b x :i n ^ d o w n t u o I d o w n - — • '* He added
fro m
rig h t
a r r iv in g
a p p e a rs
m e tic
is
bo l e f t ,
at
w ro te
h ■s n u m b e r s
3.051 a s t h e sum Jt
p o s s ib le
th a t
im p e d e d b y
th is
F ro m
s u b je c t* ©
a d ir e c tio n a l
fro -*
tills
le ft
to
e v id e n c e
p ro g re s s
in te r fe r e n c e
in
r ig h t,
it
a r ith ­
fa c to r*
The three month*a vacation (disuse) may very likely have
weakened former functional bonds, making possible the reemergence and interference of this directional factor#
Vvhile one cannot Infer causal relationship between his
right-left difficulty end his left handedness, it is an
interesting possibility*
It appears from the verbaII©ation record that
”comparing" was expressed as such In his solution to
example a , that It was Implied in his solutions to exam­
ples B, C, and D, and omitted in E.
However, "judging”
was not expressed nor could It be implied from the ver­
balization that accompanied the solutions.
The method of subtraction employed by this sub­
ject appears to be the pure "take-away" in contrast to
the "hybrid" method previously used.
However, In some
instances the subtraction situations within the long-division examples were of such a nature that the "hybrid”
method had no opportunity to function; in others, he
failed to subtract completely or correctly*
This subjectfs subtraction In example E illus­
trates nicely how 747 can be subtracted from 284, how
358
X1
I
7
Ui
oS
s
I
IIB® Sm<
X
Uj
Figure 17
Electrodermal Response Record
(Reproduction)
559
Q5& can be subtracted from 647 without any qualms whatso­
ever*
Such procedure indicates that the subtraction
process has become totally mechanical without any con­
sciousness of vmat is actually done.
The verbal problems were incorrectly solved as
was true in Project VI when he was confronted by the
Identical problems.
It appears the reasoning Involved
was previously and still Is too difficult for this sub­
ject.
The Verbal Profile Charts depict very strik­
ingly the prolonged and Irregular solutions of this sub­
ject.
Note particularly the Charts for his attempted
solutions to examples C, D, and E which presented him
with such difficulties!
The Film Analysis Records show by means of
frames of film used just where this subject had dif­
ficulty and spent so much time*
The multiplication
steps in examples C, D, and E consumed much more time
than any other portion of the solutions.
The electrodermal response records tend to
substantiate the conclusions that appear evident from
the verbal reports.
The subjectfs solution to example
A, Y/hile Incorrect, was solved without hesitation and
so far as can be ascertained was considered correct by
the subject.
The electrodermal response record for
this example does not Indicate undue disturbance.
The
360
record for example E, the example which, caused so much
trouble, does Indicate deflections which may be def­
initely classified as significant.
Summary of Case Study Findings
Case 3.
B. Sm.
(A pupil of bolow-medlen ability)
1.
The long-divlsion instructional and main­
tenance program was sufficiently effective for this
subject of below-median ability to moke correct pro­
cedures possible after a summerrs vacation, even in
the solution to examples'of type C, long division with
zero in the quotient, and of type E, long aivision with
non-apparent quotient figures.
2m
Such computational errors as were formerly
made tend to persist.
3.
A directional difficulty which was first
evident In this subject's addition in Project X re­
appears, this time in hi a multiplication.
Th:Ts dif­
ficulty may be described as a directional (right-left)
interference factor.
Disuse of multiplication skills
as they appear within long-dlvision example-patterns
may partially account for the weakening of these former
functional bonds.
The possibility of © causal relation
361
ship between
th is
left-handedness
4 .
be
th e
is
The
m e th o d
" ta k e -a w a y ”
m e th o d ,
by
th e
passage
o f
th re e
fo rm
6.
The
In
o f
in
Is
h is
th e
e m p lo y e d
" h y b r id ”
been
p o s s ib le
have
to
boon
w h ic h
w eakened
th e
th e
he
bonds
had
by
been
th e
did
n o t
to
C h a rts
o c c u r;
to
e x a m p le
e x a m p le s
d o p io t
r e in fo r c e
" c o m p a r in g ”
B,
A ,
C,
in
and
and
D .
g r a p h ic a lly
and
s ta tis tic a lly
th e
d if fic u lt y ,
e tc .
a lo n g
th e
a p p e a re d
th e
ro u te
s o lu tio n s .
7 .
in fo r c e
d a ta
E le c tr o d e r m a l
c o n c lu s io n s
r e la tiv e
to
S u m m a ry
As
s u b je c ts
m oved
Response
d ra w n
g e n e ra l
fro m
th e
fro m
R e c o rd s
v e rb a l
e m o tio n a l
T re a tm e n t
p r e v io u s ly
to
m e th o d
ta u g h t
th a t
a p x > c a rs
tim e .
s o lu tio n s
R e c o rd s
h e s ita n c y ,
to
a s p e c ts
s o lu tio n s
V e rb a l
A n a ly s is
p o in ts
m ay
" J u d g in g ”
Implied
i t
m o n th s '
In expressed form
and
s u b tr a c tio n
o r ig in a lly
" a d d itiv e ”
exposed
b .
o f
c o n tra s t
H a v in g
s u b s e q u e n tly
to
in
d if fic u lt y
s u g g e s te d .
u sed .
c re a te d
F ilm
a ls o
" t a k e - a w a y 11
p r e v io u s ly
p u re
r ig h t - le f t
o f
and
In
to
re ­
w r itte n
s ta b ility .
E x p e r im e n ta l
s ta te d
c it y
te n d
th is
d u r in g
th e
G ro u p
p r o je c t,
s u m m e r*
four
T h is
362
treatment must, therefore, be limited to th© twelve
remaining pupils.
Written responses to the long-
division examples will receive first attention.
The
skills and errors listed for Project VI, page 278,
suffice equally well for this project since the Identi
cal examples are involved.
One additional symbol, 6f,
indicating forgetting has been added to the causes for
non-comp1etlon•
1*
Basic division combinations
a* 1st quotient figure
b* 2nd quotient figure
c* 3rd quotient figure
2m
Placement of quotient figures
a* 1st quotient figure
b# 2nd quotient figure
c* 3rd quotient figure
3.
Multiplication
a. 1st product
b. 2nd product
4.
Subtraction
a* 1st difference
b* 2nd difference
5.
Miscellaneous errors
a. Fails to bring down dividend figure
b. Brings down wrong dividend figure
6.
Non-completIon caused by
a. Lack of time
b. Increased difficulty
c. Sensing something wrong
d. Any of the above skills or errors
e. Skills not listed above
fm Forgetting
363
The symbols are to be Interpreted In this
manner;
2b, placement of second quotient figure;
4b, subtraction to obtain the second difference, etc.
The summary, page 365, indicates that 24 of
the 60 solutions were correct, 22 Incorrect, and 14
incomplete.
Inspection of the responses to the dif­
ferent types of long division examples does not reveal
a piling up of errors for any one type but rather a
uniform distribution of correct, Incorrect, and in­
complete solutions without regard to type of example.
With the exception of the Incomplete solutions, for
which two subjects were responsible for 10 of the 14,
the remaining errors were fairly well scattered among
the ten other subjects.
Before giving attention to the specific
errors that were made, it is interesting to compare
the correct, incorrect, and incomplete responses of
these subjects after their vacation with their re­
sponses to the identical examples at the close of
school.
Table 28 presents this comparison*
The re­
sponses of the 4 subjects who moved from the city dur­
ing the summer have been taken from the results of
Project VI in order that the comparison may be mean­
ingful.
Disuse of certain skills that had been taught
and developed to a functional level has had its effect.
While 7S$> of the twelve subjects successfully worked
364
Example Analysis
Project VII
0
J
>
-p 4
0
0
+> 0 HI
0 P
P.
0
0 J
h
1§
o
O
a» ik
>> o
p
a
H
6* o M
Subject
Errors
1 *An tH« A
B
C
D
E
/c
Exact Nature of Error
^3 ^e-VCo (&V$.^7^7/^uV) ’/^?/ f
'
2 , H,T. A
B <■.
ic>~?'(oz-'7X.
3?Vo - 2 2V-- //
ft
D
£
3
i/Ou/Y
3, C*F. A ,
.
k_
.
B
C i/
D </
E
'<o73 <$%C~67JI- /7</:V,<3b
... /Q
777X V^7i3
U- /
4. S.J, A
B
C is
D is
E
&• B.W. A
B
C
D
E
VcVf <V,
rfl'" - ^ / - -- (3^^^ //
., 7 . 3 7 7 ' / J.3
// <a £ .
/ofy
<t
t,
*
r
*■
...f
^
,
,
63
r
3 ./7 ....
.
< ,
6 * D.T. A
B
C
fl
t,
‘<
a^uVisO-JL
■'
■ r< <
,
■■
*
fK
IrzY*/* ff 'f
ft r
,‘'"'r
E *■<
72</
3<l *
_.3a^
/
-Y~
/
.....
~ ? 3
X
< = ■ ? 3 (o
7
-
3
3
7 3 ' :
j
—
7
3
Hie1 /
/
3
’7 ( £
7
7
7
|
3
|
j
'■9
m
r"' <-- 1
i*-.'
|
D
E
4^'
r\
o,
ii
>
\
7■ J*B» A
B
C
7-
*-*
V
7
7
<X>~7 7 7 ^ - \x<7>
365
Ex&mp 1e Ant&Xyals
Project VII
4#
-p
o
g
f
d
o
Bubject e** o
PS
M
o
p
o
HI
B»
jS
a
o
S3
>"*4
B r i'o r a
„
4-dL.
D
U.
j£.
* 1 .0
Exact N&frure of Error t
n '>' N' m ^ iTTui If i » m i l »rrf. ■■■■.mil-
i»ih, ■/ ■, mnTWMl
Jft
L
S. «g-| ■*/^
X-^- d
A
B
£ £ Z z 2 J t & L J L £ 2 j t . 2 x ~
£S2S...„£/£JL.J’jsL
.ug_
D
w
2^*
’•Mto UM>»wl.^ iM > R jS —
f |if;' t.md
Xl*B *S c♦
B
Z £ J ^ A £ l^ ^ I^ £ Z k s U Z a ^
E
12«B«3&i» A
.^',*Vy ..J-r/-_.£
/gt-, .?go, -y4g—
B
/22^"2 -2
S u im a r y
P o tc la
2 4 2 2 14
^2Cs.
22&.i£t^£z2Lza33*&g~>-
■««.•*'
366
e x a m p le A a t
a fte r
th re e
th e
c lo s e
e x a m p le B ;
50# V s.
4167# f o r
fo r
e x a m p le E ,
est
in
th is
6 6 ,6 7 # v s .
cases o f
ty p e
because
In c lu d e
S im ila r ly
3 3 ,3 3 # f o r
s k ills
0 and E .
th e s e
tw o t y p e s
e x a m p le C ;
o
o
#2.4
VI
3 3 .5 5 #
a p p e a rs g r e a t ­
v e ry
o f l o n g - d i v i s i on
s p e c ia l,
i.e .
28
P r o je c ts
( C o m p a r is o n i n
*
75$ v e rs u s
One w o u ld e x p e c t
w h ic h a r e
C o m p a r a t iv e A n a l y s i s
O
Jt*
£tf
4 1 * 6 7 $ d id . so
o f fo r g e ttin g
T a b le
£4
o
pel
w
o
o
o n ly
e x a m p le D ; a n d 6 8 . 3 4 # v s .
The e f f e c t
d iffe r e n c e
e x a m p le s
s c h o o l,
m o n th s o f v a c a t i o n .
50# f o r
th e
of
E“»
O
3
o
o
S3
Mi
V I and V I I
%\
pal
1-4
p-4
O
o
r-H
H#!
•*
o
O
fi
O
5h
VI
V II
A
7 5 .0 0
4 1 ,.6 7
1 6 .6 7
4 1 .6 7
8 .5 3 :
1 6 • 66
B
7 5 .0 0
5 0 .0 0
1 6 .6 7
2 5 .0 0
8 .3 3
2 5 .0 0
C
6 6 .6 7
3 3 .3 3
2 5 .0 0
4 1 ,6 7
8 .3 3
2 5 .0 0
D
5 0 .0 0
4 1 *6 7
2 5 .0 0
3 3 .3 3
2 5 ,0 0
2 5 .0 0
E
5 8 .3 4
3 3 ,3 3
3 3 . 33
4 1 ,6 7
8 .3 3
2 5 .0 0
.?2 p e i
z e ro
in
th e
The v e r y
s o lu tio n s
V II
V II
VI
q u o tie n t and non -a p p a re n t q u o tie n t f ig u r e s
la r g e
g iv e s
in c r e a s e
fu rth e r
In
th e
n u m b e r o f I n c o m p le t e
te s t im o n y
to
th e
e ffe c t
of
367
d is u s e
up o n p r e v io u s ly
le a r n e d
and f u n c t io n a l
T h e S um m ary o f E r r o r s
In
its
c o m p a r a t iv e
of
th e
fo u r
tio n s
In
w ho l e f t
e rro rs
P r o je c t V I o p e ra te
q u o tie n t f ig u r e s
e rro rs
o r f o r m o re
H o w e v e r,
it
th e
r e s p o n s ib le
s e e m in g ly g r e a t e r e f f e c t .
re c t
is
in
e x c lu d e s
c ity
fo r
is
r e s p o n s ib le
th a n 2 8 $ o f a l l
r e v e a lin g .
in
fin d in g
fo r
fa ilu r e s
16 o r 3 2 *7 $ o f e r r o r s
14 o r 2 8 $ , a re
doubt but
th a t
th e
c h ie fly
a re
s u b t r a c t io n
in
th e
s o lu ­
c o r­
14 o f t h e
e rro rs
seem s t h a t f u n d a m e n t a l s k i l l s
a re
re s p o n s e s
u n s u c c e s s fu l
D iffic u lty
and s u b t r a c t io n
little
p r o j e c t w h ic h
P r o je c t V I I b u t w ith
p lic a t io n
e rro rs ;
th is
c o lu m n s i m i l a r l y
s u b je c ts
The i d e n t i c a l
fo r
s k ills .
In
49
m ade.
m u lt i­
r e s p o n s ib le
fo r
m u lt ip lic a t io n
e rro rs J i
te a c h in g
T h e re
o f lo n g
Is
d iv is io n
m u c h g r e a t e r e m p h a s is n e e d s t o
be p l a c e d u p o n m u l t i ­
p lic a t io n
in
and s u b t r a c t io n , n o t
lo n g - d iv is io n
s u b tr a c t io n
p a tte rn s .
s k ill
In
s k ill
w hen i t
p a tte rn
as
lo n g
in
lie s
d iv is io n
th e
sh o w s t h a t 10 s u b j e c t s
of
s u b t r a c t io n ,
fro m
a " h y b r id ”
s u b je c t,
w ith in
is
D ic t a p h o n e
seme r e c o r d
c o n tin u e
th e p u re
R . H . , h a v in g
one t h in g ;
e x a m p le s
see p a g e 2 8 5 .
to
is
b u r ie d
E x a m in a t io n o f t h e
a n d a c o m p a r is o n o f
b u t w ith in
M a s te ry o f m u l t i p l i c a t i o n
Is o la tio n
o f th is
Is o la tio n
to
or
m a s te ry
a m o re c o m p le x
q u ite
a n o th e r.
R e c o rd A n a l y s i s
f o r P r o je c t V I
use
th e
same m e th o d s
On© s u b j e c t made a s h i f t
" t a k e - a w a y " m e th o d ;
fo rg o tte n
how t o
p ro c e e d
a n o th e r
in
lo n g
368
d iv is io n
d id
n o t g iv e
F a ilu r e
e ith e r
in
E ven In
of
e x p re s s e d
th o s e
"c o m p a re ",
s u b je c t,
in
th e
o f th e s e
c le a r ly
w h ic h
s tu d e n ts
of
th e
o n ly 15 w e re
Is
in
46 I n
in a l
m a in te n a n c e
of
th is
" ju d g e "
" ju d g e "
d iv is io n
in
c o rre c t.
d id
s tu d e n ts
In
O f th e
7 w e re
c o r­
n o t o c c u r,
o f th e s e
e m p h a s iz e
tw o s t e p s
th e n e e d f o r
a n d " c o m p a r in g "
a d v is a b ility
re g a rd .
and
is
w h ic h
o c c u rre d ,
" c o m p a r in g "
th e
o r do
O n ly o n e
39 s o l u t i o n s
18 w e re
These r e s u lt s
w o rk i n
v o i* y e v i d e n t .
p r e v io u s p r o je c t s
lo n g
th e
T h e v a lu e
an d a ls o
do
16 s o l u t i o n s
on " ju d g in g "
in s tr u c tio n
in
" c o m p e r in g "
c o rre c t.
s tre s s
As i n
s te p s
w h ic h
is
c o n s is t e n t ly
n o t ju d g e ,
w h ic h
u n q u e s t io n e d .
g re a te r
to
c o rre c t;
d id
fo rm
m a tte r.
o r "c o m p a re "
so c o n s i s t e n t l y .
O f th e
1 1 w e re
9 s o lu tio n s
re c t;
tw o
on t h i s
" ju d g e "
s u b je c ts
h e r s o lu tio n s *
a p p a re n t.
" ju d g e d " ,
w h e re
a p p e a rs
"c o m p a re "
to
I m p lie d
do n o t do
S. J . ,
v a lu e
s u b je c ts
or
cases
th e y
a n y In fo r m a tio n
In
th e
o r ig ­
o f c o n t in u e d
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Table 29
Analysis of Verbalized Sortitions
S u b tr a c tio n
M e th o d s
r~i
S u b je c t
3
cm
i
rd
rH
1
S o lu t io n s o f th o s e S o lu t io n s o f
th o s e
e x a m p le s I n v m ic h
e x a m p le s i n v / h ic h
s u b je c ts :
s u b je c ts d id n o t :
03
1
e-t
1 * A n . II*
Judged
R
W
c o m p a re d
I
R
\7
l£ 3 E ©
I
1
2.
H .T *
i/
3.
C .P .
i/
4#
3 * J«
5.
B .W *
1
1
R
VV
I
3
1
4
1
3
2
3
2
4
1
3
5
o
1
1
3
1
7*
J .B .
8.
B .R .
9.
1 *0 .
1
1 0 .R .H .
T h i s s ib j< o t
ild
not
a t semi: >t a v r ' so li: t i e IS
1
2
1
i
1
2
0
11
l
4
1
ll.B .S c ,
I
5
2
6 # D *T .
1 2 . B .S m .
Sum m ary
W
R
2
4
1
2
2
7
2
0
2
3
4
1
4
4
5
2
3
2
2
1
1
15 24
7
2
3
18 1 8
2
3
Summary of Findings Based Upon Experimental Group
1*
Errors are distributed rather uniformly with­
out regard to type of long-division example#
2*
Three months1 vacation has definitely weakened
certain skills which formerly were maintained and op­
erated on a functional level#
3#
Forgetting affected the special types of skill
necessary to the solution of long-division examples
of type C and E more than those involved in the solu­
tion of examples of type A, B, and D*
4*
An increase in the number of incomplete solu­
tions gives further evidence of the effect of disuse*
5.
Lack of proficiency in the fundamental skills
of multiplication and subtraction as these skills ap­
pear within the long division pattern is chiefly respons
ible for incorrect and incomplete solutions*
This weak­
ness suggests greater emphasis upon these skills as
they appear In this particular long-division setting*
6*
Each subject continues to use the method of
subtraction previously employed*
7*
Only one subject consistently 11judges*1 and Mcom
pares”; very few others consistently rtjudge” or ncomp are
374
8*
The e f f ic a c y
is
c le a r ly
in
fir s t
of
wj u d g i n g ”
a p p a re n t as is
in s tr u c tio n
u p o n th e s e
tw o
s te p s
th e
and in
in
n e e d f o r g r e a t e r e m p h a s is
th e
lo n g
a n d / o r Hc o m p a r in g ”
m a in t e n a n c e p r o g r a m
d iv is io n *
H y p o th e s e s a n d T e n t a t i v e
C o n c lu s io n s
P r o je c t V I1
S u c h c o n c lu s io n ®
h a v e b e e n d e r iv e d
fro m
fro m
a c o n s id e r a tio n
th e
of
and h y p o th e s e s a s a re
in d iv id u a l case
th e
r e m a in in g
s tu d ie s
s u b je c ts
e x p e r im e n ta l g ro u p *
The o th e r fo u r th - g r a d e
w e re n o t in c lu d e d
th e
!•
in
in v o lv e d
th e
c o rre c t
o f th re e
s p e c ia l
s o lu tio n
C and E a p p e a r to
in
th e
s o lu tio n
s c h o o l y e a r a re
f u n c t i o n a l a f t e r a s u m m e rf s v a c a t i o n
C e r ta in
be
s k ills
fir s t
to
th e
sec­
s till
o f t h r e e m o n th s *
w h ic h a t o
o f lo n g - d iv is io n
th e
o f th e
p u p ils
e x a m p le s a n d l e a r n e d d u r i n g
o n d s e m e s te r o f t h e p r e v i o u s
2*
and
s e v e n th p r o je c t *
C o rr e c t p ro c e d u re s
o f lo n g - d iv is io n
lis te d
n e c e s s a ry f o r
e x a m p le s o f t y p e s
s tiffe n
fro m
a p e r io d
m o n th s 1 d is u s e .
3*
Computetlonal skills involved in such multi­
plication and subtraction situations as occur within
long-dlvislon examples function effectively for abovemedian pupils but fall to so function for below-median
subjects*
375
4*
The s te p s
b y v e r y fe w
wj u d g i n g *
s u b je c ts
and. * c o m p a r in g *
a re used
an d c o n s is t e n t ly b y o n ly
one i n d i ­
v id u a l •
5*
The e f f ic a c y o f * ju d g in g *
m e a s u re d b y c o r r e c t
6*
T h e tw o o b s e r v a t i o n s
" c o m p a r in g ”
s u g g e s t th e
fir s t
in s tr u c tio n
th e s e
tw o
7*
—- ”
s o lu tio n s
s te p s
S p u r io u s
o ccu r in
c e r ta in
9*
tio n s
is
10*
ence
of
and
g r e a t e r e m p h a s is i n
4 t im e s
o f s u b je c ts
fro m
a b o ve - to
a -
6 is
v a r y in g
18
in
b e lo w - m e d ia n *
c o m p u ta tio n a l e r r o r s p e c u l i a r
to
s u b je c ts p e r s is t *
to
in c r e a s e
fu rth e r
in
th e
e v id e n c e
S uch " h y b r id *
oped o r a p p e a re d d u r in g
11*
as "
s o lu tio n s
S p e c ific
* ju d g in g *
lo n g d i v i s i o n .
a r it h m e t ic a l a b i l i t i e s
8*
e v id e n t*
t h e m a in te n a n c e p r o g r a m u p o n
e rro rs
th e
c le a r ly
c o n c e r n in g
need f o r
and in
in
is
a n d ^ c o m p a r in g * a s
n u m b e r o f in c o m p le t e
of
th e
e ffe c t
o f d is u s e .
m e th o d s o f s u b t r a c t i o n
th e p r e v io u s
in d iv id u a l d i f f i c u l t i e s
th e
d ir e c tio n a l r i g h t - l e f t
th e
s u b je c t B .
Sm*
d iffic u lty
as d e v e l­
s e m e s te r p e r s i s t .
A s u m m e r1s v a c a t i o n m a kes p o s s i b l e
some s p e c i a l
s o lu ­
in
th e
th e
© m erg­
as e .g .
case
of
37©
12*
to
S k ill
In
s o lv in g
v e r b a l p r o b le m s
a p p e a rs n o t
hav© b e e n a f f e c t e d b y d is u s e .
13*
V e rb a l P r o f ile
C h a r ts , F ilm
e n d K le e t r o d e r m a 1 R e s p o n s e R e c o rd s a i d
c o n fir m in g ,
b e mad© f r o m
sponses*
and r e in f o r c in g
a s tu d y o f th e
A n a ly s is
in
R e c o rd s ,
s u p p le m e n t in g ,
such In te r p r e ta tio n s
w r itte n
as can
and v e rb a l r e ­
377
CHAPTER V
D IM IT A T IO H S AUD CONCLUSIONS
As s ta te d
th is
th e s is #
p ro g re s s e d
a n d a ls o
th a t
to
fro m
n o t©
tim e
in tr o d u c to r y
one
s ta g e
o f th e
th e
le a r n in g p ro c e s s
a n o th e r
some I n n e r a n d o u t e r d i s t u r b a n c e s
In
s e t up
g a in as
o f m a s te ry t o
le a r n in g *
The w r i t e r
c o n s u lta t io n
w ith
R u c k m lc k , a n d D e a n S e a s h o re p r i o r
o rd e r to
c h a p te r o f
a t t e m p t w a s m ade t o
as p o s s ib le
accom pany t h is
s id e r a b le
th ©
a s e r io u s
c o n ^ p le te a p i c t u r e
as i t
in
to
s p e n t con­
D r. K n ig h t,
th is
D r.
s tu d y in
a n a p p r o a c h w h ic h w o u ld m ake p o s s i b l e
a t t a in m e n t o f th e s e
o b je c tiv e s .
u s e d and th e p ro c e d u re s
T h e e q u ip m e n t
f o l l o w e d w e re a g r e e d u p o n a s
p r o m i s i n g m eans#
Th© a n a l y s i s
in v o lv e d
in
th is
r e la t iv e
to
th e
m o tio n p i c t u r e
a n d u s e o f th e m any r e c o r d s
s tu d y le a d s
t e c h n iq u e
t e c h n iq u e
one t o
and p ro c e d u re s
r e q u ir e d
a n d s t u d y o f a p p r o x i m a t e l y 1 0 ,0 0 0
th © w r i t e r
s till
h o ld s
c o n t in u o u s p i c t u r e
y ie ld ,
th a t
tlo n s
a ll
th e
a n a ly s is
a d v is a b ility
s u b je c ts
in
th a t
th e
a g iv e n
c o n c lu s io n s
e m p lo y e d *
th © d e t a i l e d
W h ile
t e c h n iq u e p r o v i d e s a
c o u ld n o t
c a n be s tu d ie d r e p e a t e d ly ,
can be m ade, he
s e r io u s ly
o f u s i n g m o t io n p i c t u r e s
s tu d y *
The
a n a ly s is
fe e t o f f ilm .
s u c h as u s u a l o b s e r v a tio n
th e p ic t u r e
th a t d e ta ile d
c e r ta in
and
ques-r
fo r
T h e m a jo r p u r p o s e s
378
s e rv e d b y t h is
fo r c in g
film ,
th a t o f
In t e r p r e ta tio n ®
s u p p le m e n t in g a n d r e i n ­
o f th ©
v e rb a l and w r itte n
r e s p o n s e s w o u ld h a v e b e e n a d e q u a t e ly
w r ite r * s
ju d g m e n t ,
if
s e rv e d ,
o c c a s io n a l p i c t u r e s
nu m b e r o f s tu d e n ts h a d b e en ta k e n o r I f
s e le c te d p r o je c t s
a d d itio n a l f i l m
a l l p r o je c ts
h a d b e e n ta k e n *
a® i s
a re
in
th e
o f a s e le c te d
p ic tu r e s
The v a lu e
of
of
such
s e c u re d w hen m o tio n p i c t u r e s
ta k e n d o e s n o t w a r r a n t th e
of
added e f f o r t
and exp ense *
T h e Io w a O r a l L a n g u a g e M a c h in e p r o v i d e d
t e c h n iq u e
in
fo r
g a in in g
a r it h m e t ic ,
a m in a t i o n
o f w r itte n
to
of
w r itte n
o m is s io n s
th is
re s p o n s e s ,
th e
and o th e r
w r itte n
v e a le d b y v e r b a l r e c o r d s .
iv e
th e
is
P e c u lia r itie s
w h ic h a r e n o t a p p a r e n t I n
th a t
th a t
It
is
v a lu a b le
in
d iffe r e n t
add m a te r ia lly
p ro c e d u re s ,
ty p e s o f e r r o r s
re s p o n s e s a re
th e w r l t e r * s
Io w a O r a l L a n g u a g e M a c h in e
m e ans f o r o b t a i n i n g
in v a lu a b le *
tr e a tm e n t o f th e
p ro c e d u re s ,
to p ro c e s s e s
g a in e d t h r o u g h e x ­
s t\x d y , v e r b a l r e p o r t s
re s p o n s e s .
in
r e la t iv e
b e y o n d In fo r m a tio n
A s w a s e v id e n c e d I n
p r o je c ts
in fo r m a tio n
a
Is
re ­
ju d g m e n t
a v e ry e ffe c t­
s u p p le m e n t a r y i n ­
fo r m a tio n .
C o n s id e r a b le
In
d i s a p p o i n t m e n t w as e x p e r ie n c e d
t h e u s e o f t h e p s y c h o g a lv a n o m e t e r .
d id n o t y ie ld
such s p e c ific
had been a n tic ip a te d .
The In s tr u m e n t
a d d it io n a l In fo r m a tio n
Th© © l e e t r o d e r m a l r e s p o n s e
as
379
r e c o r d s w e re u s e d f o r
th e p u rp o s e o f
in d ic a t in g
g e n e ra l way a f f e c t iv e
r e a c tio n s
accom pany le a r n in g .
W hen t h i s
th a t
In
I n s t r u m e n t ha® b e e n f u r t h e r p e r f e c t e d
a
and has
r e c e iv e d w id e r a p p r o v a l b y r e p u ta b le p s y c h o lo g is t s
m eans o f r e c o r d in g
It
e m o tio n a l l e v e l s
In
h is
th is
s tu d y ,
e v a s iv e n e s s
an d th e
a tte m p t to
o b ta in
In fo r m a tio n
and s u b je c tiv e
d a ta
fro m
le a r n in g p ro c e s s —
th ©
" s t u b b o r n n e s s 11 o f t h e p r o b le m .
to
th e
s e c u re d
t h e w r i t e r h a s b e e n im p r e s s e d w i t h
o f th e
m any In s ta n c e s
th e
"to u g h n e s s "
He h a s h a d I n
be c o n te n t w it h p u r e ly d e s c r ip t iv e
r a t h e r th a n e x p la n a to r y
th a t
s u p p le m e n t in g w r i t ­
o f w o rk .
v a s t am ount o f o b je c tiv e
in
and f lu c tu a tio n s ,
s h o u ld b e co m e a v a l u a b l e m e a n s o f
te n and v e rb a l re c o rd s
as a
I n m any In s t a n c e s
s ta te m e n ts , w it h
a re
r a t h e r th a n c o n c lu s io n s
c o n c lu s io n s
s u g g e s te d b y th e
re c o rd s
th a t have been s t a t is t ic a l l y
d e r iv e d .
In
th e
m a jo r c o n c l u s i o n s
su m m a ry t h a t f o l l o w s ,
lis te d
Is
f o llo w e d b y a s e r ie s
s u p p o r tin g
s ta te m e n ts .
r iv e d
s u c h o b s e r v a tio n s
fro m
These
a s w e r e m ade w h e n p r o j e c t s
ea ch o f th e
I,
fiv e
of
s ta te m e n ts h a ve b e e n d e ­
and te n t a t iv e
III,
V I,
c o n c lu s io n s
a n d V I I w e re
s tu d ie d .
I.
film - a n a ly s is
The w r i t t e n
re c o rd s ,
s o lu tio n s ,
v e rb a l re c o rd s ,
a n d © le e t r o d e r m s l r e s p o n s e
380
re c o rd ®
o f p u p il®
in
th is
r a t h e r c o m p le t e p i c t u r e
e v id e n c e
in g
s tu d y
o f o v e r t re s p o n s e s
o f I n n e r d is tu r b a n c e s
o f lo n g
make a v a i l a b l e
a
a n d g iv e
t h a t a c c o m p a n y th ©
some
le a r n ­
d iv is io n *
1*
in fo r c e
These re c o rd s
each o th e r in
o f s o lu tio n s ,
s u p p le m e n t a n d r e ­
p ic tu r in g
In c lu d in g
th © p r o g r e s s
a c c u ra c y , p a u se s,
a n d s i g n i f i c a n t e x p r e s s io n s .
8*
These re c o rd s
m e n ta l s t a t e s
in
as f r u s t r a t io n
a n d c o n f id e n c e
3*
a id
fo llo w in g
of
fo llo w in g
film
a c c o m p a n y in g p a u s e s , h e s i t a t i o n s ,
th e w r i t t e n
is a tio n s ,
and th e
4*
re s p o n s e s ,
and su ch
e tc *
th e
as
v e r b a l­
© le c tr o d e r m a l r e c o r d s *
V e rb a l re s p o n s e s g iv e
" ju d g in g "
fa ilu r e
s u b s ta n tia te s
success and f a ilu r e ,
®PP Gs.r i n
such
success*
T h e m o t io n p i c t u r e
e v id e n c e s
d e p ic tin g
cue s as to
a n d " c o m p a r in g " , m e n ta l d i s t u r b ­
a n c e s , and o th e r a s p e c ts
o f th e
le a r n in g
p r o c e s s w h ic h c o u l d n o t o t h e r w i s e h a v e b e e n
o b ta in e d *
5*
O f th e
o c c u r d u r in g
e x a m p le s ,
p e a r to
th e
© le c tr o d e r m a l re s p o n s e s
s o lu tio n
o f lo n g - d lv is lo n
some s i g n i f i c a n t d e f l e c t i o n s
h a v e b e en c a u se d b y th e
c o m p u t a t io n i n v o l v e d
o r to
th a t
ap­
r e a s o n in g
or
h a ve a r is e n b e c a u s e
381
o f s itu a tio n s
d u r in g
o r c o n d itio n s
th e p ro c e s s *
6*
E le c tr o d e r r o a l re s p o n s e r e c o r d s
e a c h s u b je c t s e c u re d a t
o v e r a p e r io d
re c o rd s ,
sponse re c o rd s
an e v a lu a tio n
fo r
re c o rd s ,
o f p u p ils
o f th e
s tr o n g ly
t im e s
sug-
each In d iv id u a l*
E x a m in a t io n o f w r i t t e n
filr a - a n a ly s ls
of
seven d if f e r e n t
o f s e v e n m o n th s
g e a t s u n iq u e p a t t e r n s
II*
t h a t d e v e lo p e d
s o lu tio n s , v e r b a l
a n d © le c tr o d e r m a l r e ­
o f th is
s t u d y m a ke s p o s s i b l e
I n s t r u c t io n a l m a te r ia ls
u se d and
th e p ro c e d u re s fo llo w e d *
1*
I-V I
In s tr u c tio n
Is
a d e q u a te
s p e c ia l s k i l l s
a ll
ty p e s
p u p ils
fo r
p r io r
th e
to
e a ch o f P r o je c ts
a c q u is itio n
n e c e s s a ry to
o f lo n g - d iv is io n
th e
s o lu tio n
of
e x a m p le s f o r
o f m e d ia n a n d a b o v e - m e d ia n a b i l i t y .
2*
C la s s r o o m
b y a s p e c ific
in s tr u c tio n
m a in te n a n c e
s u p p le m e n te d
a n d r e m e d ia l p r o ­
g ra m a n d b y s p e c i a l I n d i v i d u a l a i d
In s u re
o f th e
th © m a s t e r y o f s k i l l s
s o lu tio n
o f lo n g - d lv is lo n
does n o t
n e c e s s a ry to
th e
e x a m p le s f o r p u p i l s
o f b e lo w - m e d ia n a b i l i t y *
3.
In s tr u c tio n
The l e a r n i n g
a n d m a in t e n a n c e )
m a ke s p o s s i b l e
th e
e n v ir o n m e n t
d u r in g
th e
s m o o th a n d e f f i c i e n t
( fir s t
s e m e s te r
m 2
f u n c t io n in g
o f ne w s k i l l s
appearing In now and
fo r
th ©
and o f o ld
© k ill®
patterns
m o re c o m p le x
s u p e r io r p u p il b u t n o t f o r
th © •m e d ia n
a n d b e lc w - m e & Iim p u p i l *
4*
M a s te ry o f © k i l l s
n e c e s s a ry to
th ©
lo n g - d lv is io n
s o lu tio n
e x a m p le i s
o f one ty p e
of
n o t a d e q u a te
to
s o lu tio n , o f o th e r ty p e s *
tio n
tm d p r i n c i p l e ©
th e
S e q u e n t I» 1 p r e s e n t a ­
o f th © d i f f e r e n t t y p e s a p p e a r s t o
b e im ­
p e r a tiv e *
3*
In s tr u c tio n
In
th e
o f s u b t r a c t io n an d e x p o s u re
to
th e
* ta k e -a w a y "
m e th o d r e s u l t I n
f o u r m e th o d s o f s u b t r a c t i o n *
tw o o f w h ic h a r e
v a r ia n ts *
4*
In s tr u c tio n
a n d " c o m p a r in g " I s
o u t th e
III#
In s p e c tio n
sponse re c o rd s o f
f ig u r e s
" ju d g in g "
In a d e q u a te
th ro u g h ­
o f w r itte n
s o lu tio n s ,
v e rb a l
o f th is
s tu d y m akes p o s s ib le
o f p u p il s u c c e s s e s and f a i l u r e * # '
P la c e m e n t o f q u o t i e n t f i g u r e s
v e ry l i t t l e
2»
c le a r ly
In
r e c o r d s , en d © le c tr o d e r m a l r e ­
s u b je c ts
a n a ly s is
1*
p r o v id e d
s e m e s te r #
re c o rd s , fllm - a n a ly s ls
a d e ta ile d
" a d d i t i v e " m e th o d
d iffic u lty
F in d in g , f i r s t
in
o ffe rs
a n y p r o je c t*
an d s e c o n d - q u o tie n t
a c c o u n t f o r a p p r o x im a te ly h a l f th e
383
e rro rs
In
P r o je c t I ;
o f th e
e rro rs
in
fo r
F in d in g
e rro rs
in
e x a m p le s a c c o u n t f o r
h a lf
th e
e rro rs
4.
F in d in g
In
P r o je c ts
d iffe r e n c e s
e x a m p le s a c c o u n t f o r
h a lf
e rro rs
e rro rs
p r o x im a te ly
P r o je c t V I I .
in
In
lo n g -
a p p r o x im a te ly
I,
d iv is io n
o f th e
le s s
th e p ro d u c ts w it h in
d iv is io n
th ©
th a n a t h i r d
P r o je c t V I 5 and f o r
th a n a s e v e n th o f th e
3*
le s s
V I,
and V I I .
w ith in
lo n g -
a p p r o x im a te ly
P r o je c t I 5 f o r o n e - t h ir d
P r o je c t V I;
o n e -fo u rth
o f th e
and f o r ap­
e rro rs
In
P r o je c t V I I .
5.
In
A t th e
c lo s e
m u lt ip lic a t io n
fo r
th e
s k ills
s e m e s te r,
and s u b tr a c t io n
s k ills
do fu n c tio n
a b o v e - m e d ia n p u p i l e v e n t h o u g h t h e s e
appear in
6.
w ith
o f th e
P r io r
ne w a n d m o re c o m p le x p a t t e r n s .
to
m a s te ry o f lo n g
r e m a in d e r s , p u p i l s
" r e m a in d e r "
d iv is io n
a r e p u z z l e d b y th ©
and w i l l p e r m it I n c o r r e c t p r o d u c ts
and in c o r r e c t d iffe r e n c e s
In
o rd e r to
a v o id
" r e m a in d e r s " *
7.
E x a m p le s o f
ty p e D , lo n g d i v i s i o n
w ith
th r e e - d ig it d iv is o r s ,
lo n g
d iv is io n
fig u r e s ,
w ith
n o n -a p p a re n t q u o tie n t
c o n s is t e n t ly
th a n o th e r ty p e s .
end o f ty p e E ,
o f f e r m o re d i f f i c u l t y
Th© e r r o r s
a re n o t as a
384
r u le
p e c u lia r
to
th e s e
ty p e s b u t a p p e a r I n
s u c h fu n d a m e n ta l p ro c e s s e s a s m u l t i p l i c a t i o n
a n d s u b t r a c t io n *
8*
T h e " a d d i t i v e 1* m e th o d o f s u b t r a c t i o n
and i t s
tio n s
v a r i a n t m a ke s p o s s i b l e
a s 9© f r o m
m is g iv in g s ,
9.
su ch s u b tra c ­
76 w i t h o u t a n y a p p a r e n t
q u a lm s , o r d i s t u r b a n c e s .
" J u d g i n g 11 a n d " c o m p a r in g "
e x p re s s e d v e r b a l fo rm ,
a t tim e s
10.
in
In
Im p lie d
a p p e a rs
fo rm ,
In
and
s i l e n t fo rm .
The te rm s
s e ld o m u s e d *
" ju d g e "
an d "c o m p a re ”
Some s u b j e c t s u s e
th e
a re
te rm s
in te r c h a n g e a b ly •
11.
Less
th a n h a l f o f th e
and "c o m p a re ";
p a re ” }
some " j u d g e "
s u b je c ts
" ju d g e "
b u t d o n o t "c o m ­
some " c o m p a r e " b u t d o n o t " j u d g e ” ;
some d o n e i t h e r ;
an d th o s e
t h a t do " ju d g e "
a n d /o r "c o m p a re ” do n o t do so c o n s is t e n t ly .
12.
F a ilu r e
to
" ju d g e "
a n d /o r "c o m p a re "
does n o t p re v e n t a c o n s is te n t,
s te a d y a t t a c k
o n e x a m p le s .
13*
F a ilu r e
d e fin ite ly
In c r e a s e s
a n d I n c o m p le t e
14.
ta l
g ro u p
to
" ju d g e "
a n d /o r "c o m p e re "
th e num ber o f I n c o r r e c t
re s p o n s e s .
T h e e r r o r s made b y th © n o n - e x p e r lm e n and th e
e x p e r im e n ta l g ro u p
a re n o t
385
th ©
same I n
p u p lla
k in d n o r d e g re e .
w e re w e a k i n
and m u lt ip ly in g ;
tio n
fin d in g
th ©
Th© f o r m e r
q u o tie n t f ig u r e s
la tte r ,
In m u lt ip lic a ­
and s u b t r a c t io n .
15.
V e ry l i t t l e
tin ©
r e a d i n g a v e r b a l p r o b le m
th © m e th o d t h a t
beyond th a t
Is
used to
of
d e c id e
s h a l l b e e m p lo y e d i n
its
s o lu tio n .
16.
C e r ta in
n e c e s s a ry .f o r
d iv is lo n
be
to
s o lu tio n s
th e
s u b je c ts
in
th e
e x a m p le s a n d l e a r n e d
s e c o n d s e m e s te r o f
th e p r e v io u s
f u n c t io n a l f o r 10 o f
is
re a s o n to
b e lie v e
and su cce ss have a d ir e c t e f f e c t upon
sub sequent a ffe c tiv e
s ta b ilit y ,
and s u b s e q u e n t a r it h m e t ic
1.
e v id e n c e
lo n g -
th e
12
a f t e r a sum m er v a c a t i o n *
T h e re
th a t fa ilu r e
(1 )
and u p o n
im m e d ia te
(2 )
E v id e n c e o f f a i l u r e
o f a w a re n e s s o f f a i l u r e
v e r b a liz a t io n s
and
im m e d ia te
a c h ie v e m e n t .
c a n be v e r y d e f i n i t e l y
je c ts .
to
s u f f e r fro m d is u s e .
o f lo n g - d iv is io n
s c h o o l y e a r a re
IV .
c o rre c t
w h ic h a r e
C o r r e c t p ro c e d u re s in v o lv e d
s o lu tio n
d u r in g
th e
s k ills
e x a m p le s o f t y p e s C a n d E a p p e a r t o
th © f i r s t
17.
s p e c ia l
and s u c c e s s and
and succe ss
o b ta in e d fr o m
and w r it t e n
s o lu tio n s
th e
of
sub­
386
2*
S i g n i f i c a n t © le c tr o d e r m a l re s p o n s e
d e fle c tio n s
e x te n t f o r
In
a r e n o t th ©
fa ilu r e
same I n
and success#
32#3$ o f 164 " f a il u r e "
1 8 # 2% o f 2 2 5 " s u c c e s s "
fro m
th e w r it t e n
3#
s itu a tio n s #
is
4*
d e fle c tio n s
fa ilu r e s ,
6*
is
s o lu tio n s #
In
p o s itiv e
in
s ig n if ic a n t
2 7 *6 $ o f 217 c a s e s ;
3 6 *8 $ o f 174 cases#
and v e r b a l r e p o r ts *
o f success upon subsequent
9 5 *4 $ o f 221 c a s e s .
o f success*
.6
s u b s e q u e n t 1 0 0 -s e c o n d i n t e r v a l .
The e f f e c t o f f a i l u r e
7 3 .3 $ o f 8 7 eases#
th is
in g
For
e r r o r o c c u rre d
a c c o m p a n ie d b y
a w a re n e s s u p o n s u b s e q u e n t w o r k I s
In
a c c o m p a n ie d
1 0 0 -s e c o n d I n t e r v a l
th e w r i t t e n
The e f f e c t
e a c h in s ta n c e
th e
th e
a p p e a re d i n
s e le c te d fro m
in
appear In
1 6 . 0 $ o f t h e 8 1 c a s e s w h e re
s u c c e s s fu l
f o llo w in g
s e le c te d
u n a c c o m p a n ie d b y a w a r e n e s s *
D u r in g
fo llo w in g
w o rk i s
in
and in
and v e r b a l re p o rts #
S ig n if ic a n t d e fle c tio n s
b y a w a re n e s s ;
5#
a p p e a r in g
s itu a tio n s
4 8 * 2 $ o f 8 3 c a s e s w h e re f a i l u r e
fa ilu r e
num ber and
ty p e ,
1#8 e r r o r s
n e g a t iv e
F o r each f a ilu r e
o c c u rre d
in
th e
of
succeed­
1 0 0 -s e c o n d i n t e r v a l .
7*
The e f f e c t
o f fa ilu r e
u n a c c o m p a n ie d
b y a w a re n e s s u p o n s u b s e q u e n t w o r k a p p r o a c h e s
367
z e ro #
(E rro rs
w h ic h a r e
o r ig in a l f a ilu r e
V#
a b le
in
s tu d y *
th e
T
a re
d ir e c tly
caused by
n o t c o n s id e r e d * )
S e v e ra l phenom ena o f le a r n in g
le a r n in g
a c tiv itie s
o f th ©
a re
o b s e rv ­
s u b je c ts
o f th is
/
1*
The le a r n in g
r e tr o a c tiv e ly
s k ills
th e s e
a ffe c t
o f new s k i l l s
su ch p r e v io u s ly
as m u lt ip lic a t io n
s k ills
o fte n
fo rm e d
a n d s u b t r a c t io n w hen
ap pea r w ith in
th e
lo n g - d lv is lo n
e x a m p le s .
2.
When c o n f r o n t e d w i t h
in fa n tile
r e g r e s s io n
m e n ta l d e c is io n
Is
o c c u rs ,
a n 11im p a s s e * 1,
I n w h ic h c a s e a
m ade w h i c h p e r m i t s p r o g r e s s
to w a rd s a n a n s w e r b y w ay o f f a m i l i a r
3*
o ccu r in
S p u r io u s
e rro rs
s o lu tio n s
of
s k ills .
as * 4 tim e s
s u b je c ts
6 Is
18*
o f v a r y in g
a b ilitie s *
4*
C e r ta in
n e c e s s a ry f o r
s p e c ia l
s k ills
c o r r e c t s o lu tio n
w h ic h a r e
o f lo n g - d iv is io n
e x a m p le s o f t y p e s C a n d E a p p e a r t o
to
s u f f e r fro m
a p e r io d
be
th e
fir s t
o f th r e e m o n th s 1 d i s ­
use*
5*
A n in c r e a s e
s o lu tio n s
o f d is u s e *
Is
in
th e nu m ber o f
f u r t h e r e v id e n c e o f
th e
I n c o m p le t e
e ffe c t
388
6#
d is u s e ,
F o llo w in g
c o n fu ta tio n © !
m u lt ip lic a t io n
o c c u r w ith in
e ffe c tiv e ly
to
lo n g - d iv is lo n
fo r
d iffic u ltie s
as
but fa il
s u b je c ts *
o f som e s p e c i a l I n d i v i d u a l
in
th e
th e
O c c a s io n a lly
d ir e c tio n a l r ig h t -
case o f
s u b je c t B*
s tr a in s
w h ic h a r e n o t e v i d e n t
© m o t io n a l s t r e s s e s
b e h a v i o r o f some s u b j e c t s ?
In
and
th e
o u tw a rd
a t tim e s w h a t a p ­
t o b e e m o tio n a l i n s t a b i l i t y
s ta n tia te d
Is
not
sub­
b y th © r e c o r d s *
E le c tr o d e r m a l re s p o n s e r e c o r d s
s u p p o r t th e n o t io n
sequences o f e r r o r s
d is tu r b a n c e
Sm*
e le c tr o d e r m a l re s p o n s e
in d ic a t e
to
such
s itu a tio n s
b e lo w - m e d la n
as © *g *
d iffic u lty
9.
In
e x a m p le s f u n c t i o n
re c o rd s
p e a rs
In v o lv e d
A s u m m e r*s v a c a t i o n m a k e s p o s s i b l e
re -e m e rg e n c e
8*
s k ills
t h r e e m o n th s *
f o r a b o v e - m e d ia n p u p i l s
7.
le ft
of
and s u b tr a c t io n
so f u n c t io n
th ©
a p e r io d
t h a t u n le s s
a re
a ro u s e d I s
s e r io u s ,
th e
th e
te n d
con­
e m o t io n a l
n o t s ig n if ic a n t *
389
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