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6-1-1936
The relation of certain factors to achievement in
Algebra
Lillie Sirmans Weatherspool
Atlanta University
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Woodruff Library. Paper 919.
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THE RELATION OF CERTAIN FACTORS TO ACHIEVEMENT
IK ALGEBRA
A THESIS
SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF MASTER OF ARTS
BY
LILLIE SIRMANS 1EATHERSPOOL
DEPARTMENT OF EDUCATION
ATLANTA,
GEORGIA
JUNE 1936
ACKNOWLEDGMENT
The •writer wishes to aeloiowledge her appreciation for the
untiring interest and guidance in the writing of this thesis of her
adviser, Miss Hattie V. Feger, and for the cooperation in the collec
ting of date of Mr. P. R. Lampkin, Supervisor of the Columbus, Geor
gia, Public Schools, and members of his faculty at the Spencer High
School.
ii
TABLE OF CONTENTS
CHAPTER ■
I
PAGE
INTRODUCTION
Title of Study
Purpose of Study
1
1
...................
Plan of Study
Limitations of Problem
3
Previous Studies in the Field
1.
Studies Relating to Predicting Suecess in Algebra
3
4
2.
II
Studies Relating to Errors in Algebra
7
THE INVESTIGATION
13
Selection of Pupils
Description of School and Community
Method of Obtaining Data
.
IS
IS
15
Administering the Tests
Testing Movement and the Teaching of Algebra
17
......
Description of Tests Used
III
ANALYSIS AMD INTEHPRETATION OF DATA
24
SUMMARY AID CONCLUSIONS
APPENDIX
Illinois General Intelligence Scale, Form I
24
26
SO
35
36
41
42
46
51
A. Record-Sheet, Boys
B. Reoord-Sheet, Girls
C. Sims Score Card for Socio-Eeonomie Status
D.
17
19
Socio-economic Status of Pupils
Mental and Chronological Ages of- the Pupils ......
Intelligence Quotients
Algebraic Ability of Pupils
Achievement of Pupils in First Year Algebra for First
Semester
Use Made of Scores
Significance of Results of Correlations
IV
1
2
51
52
53
....
54
E. Lee Test of Algebraic Ability, Form A
55
F. Douglass Standard Survey Test for Elementary Algebra
Test I, Form A
56
BIBLIOGRAPHY
57
iii
LIST OF TABLES
TABLE
PAGE
I
Data on Validity of Lee Test of Algebraic Ability
II
Distribution of Scores Made on the Sims Score Card for
III
Distribution of Scores Made on the Sims Score Card for
I?
Distribution of Boys According to Father's Occupation
22
Socio-Economic Status, Boys
27
Socio-Eeonomic Status, Girls
28
Sims' Group Classification
29
V
Distribution of Girls According to Father's Occupation
........
29
VI
Distribution of Mental and Chronological Ages, Boys ....
31
VII
Distribution of Mental and Chronological Ages, Girls
32
VIII
Distribution of Intelligence Quotients, Boys
33
IX
Distribution of Intelligence Quotients, Girls
34
X
Pupils' Response on the Douglass Standard Survey Test for
XI
Pupils' Response on the Douglass Standard Survey Test for
Sims' Groups Classification .
...
Elementary Algebra, Boys
Elementary Algebra, Girls
37
38
CHAPTER
I
IITRODUCTIOI
- A study of the relation of certain factors to
achievement in algebra.
.QJlj>tudff.- In large numbers of the secondary schools of
the United States attempts are made to group students in the various
jects according to their ability in that particular subject.
grouped into four major groupss
namely, the superior,
sub
They are
the normal,
the
borderline, and the group that laeks ability to succeed in the subject.
One of the problems in selecting students for the different ability groups
is to determine the best basis for predicting the probable success of a
student in the subject.
Knowledge of the probable success of a
student in
algebra is most worthwhile to educators for the purpose of guidance in fu
ture mathematics.
Are there indications that a student should or should
not pursue the higher courses in mathematics?
factors that may hinder success?
Are there any measureable
If these factors can be determined and
the correct counsel given to the students, the large number who start
courses in mathematics and are forced to stop because of failure can be
lessened.
If the factor is one that is subject to remedial treatment,
this treatment can be applied and thereby help to lessen the number of
those who fail.
The purpose of this study is first, an effort to note
the number and type of errors made by a group of students on a diagnostic
test for the first semester of first year algebras an endeavor to deter
mine the relation of the following faetors to achievement in algebra:
a.
Socio-economic status.
b.
General
c.
Special aptitude
d.
Teacher's estimate.
intelligence.
for algebra.
Plan of Study.- Tests to determine the four factors mentioned
above and the achievement of a group of pupils in a semester of algebra
mere given.
The aim is to show to what extent achievement in algebra is
associated with socio-economic status, general intelligence, teacher's
estimate and special aptitude for algebra.
Robert Emmet Chaddock makes
the following assertion.
As a rule phenomena are neither absolutely independent nor de
pendent—they are associated in varying degrees. The problem is to de
termine in each ease the degree of association, as this indicates the
significance of the relationship.1
The characteristic of relationship is determined by a statisti
cal process called correlation and the quantity vnhieh indicates the degree
of relationship is called a coefficient or a coefficient of correlation.
The problem of this study can best be treated by a study of the
correlations involved.
The coefficient of correlation was found between
socio-economic status and achievement, general intelligence and achieve
ment, special aptitude and achievement, and teacher's estimate and achievement.
Correlations were made by means of the produot moment method.
The following formula ms used:*
N
(Tx
Robert Emmet Ghaddock, PrjLnolple^ajri^ethods of Statistics, Chicago}
Houghton Mifflin Company, 1825. p". 248.
^Ernest W. Teigs, Tests and Measurements fjor_Teachers_, Chicago:
Houghton llifflin Cokpany71-93T~. PP« 79-80. *""
SChaddock, op^cit., pp. 268-274.
The probable error of
r
was found by the formula:*
Specific results were then studied and compared9 thereby elimi
nating the probability of the mere stating of opinion.
compared with the teacher's estimate.
These results were
The correlation for boys and girls
•was made separately.
.- This problem is confined to the first
year algebra class for the term 1935-1936 of the Spencer High School, Co
lumbus, Georgia.
It is limited to a study of the errors made by these
pupils on a diagnostic test for the first semester of first year algebra
and to a study of the relative effect on the achievement of these pupils
for the first semester of algebra of the following:
a.
Socio-economic status*
b.
General intelligence.
c.
Special aptitude.
d.
Teacher's estimate.
previous Stttdj^8^njttie__Figld»- Several investigators have made
studies similar to or of a nature closely related to this study.
It is
not the aim in this study to review or to synthesize the results of vari
ous research studies in the field, but, for comparative purposes a brief
summary of a few of the previous studies is given.
^•Robert Emmet Chaddook, Pjrinoipl^jL _^
Houghton Mifflin Company, 1925.
p. 274.
Chicago:
4
Studies Itel^jyag_to_Predicting^ Success i
McCuen1 in hie study to determine a basis for predicting success
in algebra studied the results of data obtained at the Palo Alto High School,
California, from students who were completing their first semester in alge
bra.
These students -were divided into four groups of 32, 24, 28 and 32
students respectively.
He attempted to show the relative value in predic
ting success in algebra of the Terman Group I. Q. and scores obtained on
the revision of the Stanford Achievement Test; the arithmetic computation
section of this testj the arithmetic reasoning section of this test} Terman Group Test, number series section of this test; and the arithmetic sec
tion of the Terman Group Test,
Due to the fact that the correlations were rather low McCuen con
cluded that factors other than intelligence and arithmetical ability affect
the success of students in algebra, yet, it isas conceded that the group I.
Q. as determined by the Terman test vias the best criterion used to predict
the probable success of the students.
He
further asserts that there are
too many unmeasureable factors, such as industry, interest of student, and
attitude of the student toward the teacher, to permit an accurate predic
tion of success.
Records of pupils selected at random from one of the high schools
of Fort Worth, Texas, were used by Johnson2 in an effort to note the rela
tion that existed between success in first year algebra and success in
other subjects; in order to shoxir that success in algebra is an indicator
of probable success of pupils entering senior high school; and in order to
1Theron L. McCuen, "Predicting Success in Algebra", Journal of Educational
Research, XXI (January, 1930). pp. 62-74
label Alice Johnson, The Predictive Value of Success in girgVYgar Alge
bra, l&ster's TBoesis.
University of Colorado. 1932.
show that this information may be used in the guidance program of the
school.
The results of th© correlations made in this study indicate that
algebra marks alone are a better basis of prediction of scholastic success
than I, Q.'s alone, but algebra marks and I. Q.'s together give a better
basis of prediction.
Shewman1 in a study of intelligence and achievement of the June
1925 graduating class of the Grover Cleveland High School, St. Louis, Mis
souri, found a variation in the intelligence scores obtained from tests
given at the beginning and at the end of seven semesters of high school
work.. Because of this finding he concludes that I. Q.'s do not furnish
adequate bases for classification or prediction and that additional bases
should be considered Tshen grouping students or for predicting success in
high school subjects.
He adds that intelligence must be accompanied by
a fair degree of excellence in essential unmeasureable qualities, if suc
cess in high school work is forthcoming.
According to Condit's2 study of the prediction of success in
school work from results obtained by means of classification examinations,
predictions based on achievement tests are of noticeable value.
Diekter3 in an investigation made in the Upper Darby Junior
High School, Upper Darby, Pennsylvania, of the scores of eighty-three
pupils on the following:
Otis Group Intelligence Test, Roger's Test of
1¥. D. Sheiman, "A Study of Intelligence and Achievement of the June
Graduating Class of the Grover Cleveland High School, St. Louis, Mis
souri," School Review XXXIV (1S26) pp. 137-146} 219-226.
2Philip Condit, "The Prediction of Scholastic Success by Means of
Classification Examination," Journal of Educational Research, XIX
(May, 1929). pp. 331-335.
3M. R. Dickter, "Predicting Algebraic Success," School Review, XLI
(Ootober, 1933). pp. 604-606.
Mathematical Ability, Breslich Algebra Survey Test First Semester Form A,
and teacher's marks.
diction:
In this study seven criteria were selected for pre
namely, intelligence quotients; prognostic tests; teacher's
marks; composite of intelligence quotient and Roger's test; composite of
teacher's marks and intelligence; composite of teacher's marks and Roger's
test; composite of intelligence quotient, teacher's marks and Roger's test.
The respective correlations and the probable error of same for the seven
criteria were
.54 ± .06, .65 ± .05, .61 ± .06, 66 ± .05,
.73 ± .04, and .74 + .04.
.70 ±.04,
He concludes that the relation of intelligence
to achievement, the coefficient of correlation being .54 ±.06, is the
least significant in his study; the correlation of a composite of intelli
gence, Roger's test, and the teacher's mark is ifre most significant, the
degree of association in this case being .74 ± .04.
Burton1 attempted to find a practical method for grouping pupils
into classes of high, medium and low ability.
He selected a number of
pupils in Shortbridge High School, Indianapolis, Indiana, and distributed
them into three groups.
One group was based on intelligence, another on
teacher's marks, and the third on a combination of teacher's marks and
intelligence scores.
Coefficients of correlation were computed in a study
of the various methods of classification as measured by the marks of the
second semester in algebra.
Burton's study revealed the facts that rank
ing based on a combination of teacher's marks in algebra for the first
semester and the intelligence quotients was the most effective method of
classification; that teacher's marks are some?xhat better than gross scores;
^noch Burton, "An Experiment in Grouping Pupils According to Ability in
High School Algebra," Master's Thesis.
.Indiana University 1925. Sixth
Yearbook of the Department of Superintendence^
p. 330-331.
that the arithmetic section of the Terman test is on the -whole as good as
the entire test for grouping in algebra % and,
finally,
that the best way
to group pupils for the first semester of algebra would be to use a compos
ite of a score in eighth grade arithmetic and the intelligence quotient.
In a study of the nature
of the difficulties encountered in read
ing mathematics in the first year Junior High School at the University High
School at the University of Chicago George1 reports that reading difficul
ties caused by mathematical terminology -were most predominant.
ficulties found E3.4 per cent were
Of the dif
caused by a deficiency of the mathemat
ical vocabulary of the pupils and 18.8 per cent were caused by lack of un
derstanding of symbols and notations.
A recent experiment in the study of errors was conducted by Mac-
Rae and Uhl
in the regular class period of the ninth grade class of the
East High School, Superior, Wisconsin.
Data were collected from 104 pupijs
whose I. Q.'s, with the exception of one boy with an I. Q. of 65, ranged
from 72-134.
Upon the basis of results obtained from the Otis Self Admin
istering Tests, Form A,
containing 26, E8,
the pupils were divided into four abilii^r groups
22 and 28 pupils respectively.
Later when algebraic
ability had been discovered by means of a test involving 179 problems in
the fundamental processes of algebra the pupils were shifted into other
■^J. S. George, "Nature of Difficulties Encountered in Reading Mathe
matics," School Review Vol. XXXVII (March, 1929), pp. 217-226.
2Hargaret liacRae and Willis L. Uhl, "Types of Errors and Remedial Work
in the Fundamental Processes of Algebra," Journal otijj
search, Vol. XXVI (September, 1932), pp. 12-21
ability groups.
In the meantime, because of failure and other causes,
some of the pupils had dropped out of school.
In the part of the inves
tigation which involved a study of the four fundamentals the group con
tained 24, 28, 24 and 21 pupils and in the part involving a study of these
processes as they arose in the solution of equations there were 26, 28, 24
and 23 pupils in the respective rrsups.
Simple time tests were given to
the pupils on the errors diagnosed from the test of the 179 problems.
Pupils making less than 90 per cent were drilled and then retestedi the
others were allowed to do advanced work.
A second time test ms given
and those failing in this test were given individual attention.
The plan
followed with these pupils was practice, test, further practice, test un
til mastery \isas obtained.
In addition 97 per cent of the errors noted were coefficient
errors, the largest per cent found in thi3 process; the use of signs
ranked second, with the greatest difficulty being with the minus sign.
Multiplication and division gave the greatest amount of difficulty to
gether with a predominating number of unfinished problems in division.
An outstanding difficulty in subtraction was a misunderstanding of the
correct process when the subtrahend was larger than the minuend.
Of
the total number of problems, 7197, analyzed 24.6 per cent of the 1312
isolated errors were sign errors, for it was discovered that the greatest
difficulty in each of the four fundamental processes was in the use of
signs.
Arithmetical and coefficient errors furnished 10.6 per cent and
11.2 per cent respectively of the errors.
Other factors causing diffi
culties were illogical arrangement of polynomials, carelessness in writ
ing, and counting as zero an algebraic quantity having no coefficient.
They concluded that to arouse the curiosity and interest of the
pupils and to secure their attention together with abundant drill would
reduce to a minimum their errors, but even in the most capable pupil
errors would be likely to occur sometime.
lelson1 made a study of errors in algebra made by ninth grade
pupils in the Deerfield Shields Township School of Highland Park, Illinois,
The pupils in this school were divided into three sections, X, Y, and Z,
in the first year, but Kelson's
study was confined to the X and Y groups
as the Z group did not take algebra.
To make as many variables constant
as possible, the investigator used the two Y groups only and. one X group
that he ms teaching.
There were sixteen pupils in the X group and tvrenty-
four and twenty-five in the respective Y groups.
Mental maturity and rate of mental development were measured by
the Terraan Group Test,
Educational maturity was measured by the Iowa Al
gebra Aptitude Test, Form A, a test used to measure those abilities of
arithmetic and facility with numbers significant for success in Algebra.
Achievement was measured by the Iowa Unit Achievement Test in First Year
Algebra, I-'orms 1, 2 and 3.
The X group surpassed the others on all these
tests•
lelson found that correlation between intelligence and scores
of errors indicated that no real relationship was present, for in only
one case, that of one of the Y groups on Form 2 of the test, did V ex
ceed 4 P. E., "r" in this case being
—.49 ± .10.
A slight correlation
between achievement and aptitude was present.
In a class of pupils, who were failures in algebra, conducted
■^Stanley P. lelson, A Study of Errors of Pupils in Ninth Year Algebra,
Master's Thesis, University of Wisconsin, 1933.
10
for demonstration purposes, Spencer1 studied the causes of failure in this
group.
The pupils were unknown to him, so he had to get as much objective
evidence as possible as to the cause of failure.
He, therefore, constructed
a series of tests to determine failures.
An outstanding difficulty found was that pupils tended to omit
all problems that they thought they could not solve.
He concluded that
this refusal to work a problem presented a serious state of affairs because
it was impossible to determine an error until it ms made.
Yet, he further
concluded that diagnosis of failures is possible and in many instances eas
ily obtained.
In a diagnostic study of thirty pupils, twenty-four boys and six
girls, who had failed the first semester of algebra Suedekum2 sought to
establish positive answers to the .questions:
have any plan of work?
2.
1.
Does the failing pupil
Can the pupil's basic difficulties be located?
The author -was concerned about finding the basic trouble:
that is, the
incorrect responses made by the pupils.
Four teachers of each pupil were asked to give a personality rat
ing of each.
In addition the number of subjects other than algebra in
\>\rhieh the pupils had failed was obtained; I. Q.'s ranging from 71-117 with
an average of 97.5 were also obtained.
Each pupil was then taken one by
one into the office where he vms allowed to go through each step as he
worked the selected problems involving the fundamental processes. The inves
tigator recorded the method of work and cross-questioned the pupil when
1P. L. Spencer, "Diagnosing Cases Failures in Algebra," School Review
XXXIV (May, 1926) pp. 372-376.
2Erna Suedekum, A Diagnostic Study of Type Difficulties in Algebra, fes
ter's Thesis.
Washington University, St. Louis, Missouri. 1933.
11
there was evidence or suspicion, that the plan of solution -was not being
completely exposed.
A diagnostic chart was made for each pupil.
For the most part it was concluded that the questions raised at
the beginning of the study could be answered positively.
Further conclu
sions were that individual basic difficulties can be summarized for reme
dial treatment by use of charts; pupils learn rules in algebra but they
are not able to use these rules, that is, they have no bond between the
stimulus and the correct response; and few pupils tax their memory and
guess when they fail, but most have a specific plan of work.
Three hundred eighty-seven pupils, 200 ninth grade pupils and
187 tenth grade pupils, representing twenty classes in twelve different
schools in the State of Indiana were studied by Vance.1
Hone of the pupils
had studied algebra before but when they were tested they were beginning
the study of this subject.
Vance used the testing method to get his data,
giving the Otis Self Administering Test of Mental Ability, Forms A and Bj
Woody-MoCall's Mixed Fundamentals, Forms I and II; and the Equation and
Problem Scale of Hotz's First Year Algebra Scale, Series A.
Intelligence
was determined by the average of the scores of the tiro forms of the Otis
test and ability by the average for the two forms of the Woody-McCall
test.
All errors found on the equation and problem scales were classified
on the basis of a composite group of errors found in other studies.
There
were twenty-four distinct types of errors.
The largest per cent of errors was made in problem solving; he
concluded that this was due to inability to comprehend the problem.
1Ira If. Vance, An Analysis_of Errors in Beginning Algebra as Revealed
by Two of the Hotz's Algebra Scales, Master's Thesis, Indiana State
Teacher's College. 1932.
12
Relative to the placing of first year algebra in ninth or tenth grade he,
after noting that the errors in both classes were of the same type and
that the number ms relatively the same, reached the same conclusion as
Ward,* namely, that first year algebra can be offered in either grade.
In this
study it was found that there ms no difference in the
total errors made by boys and
girls.
However,
the boys made more errors
in the equation solTing teohnique and girls more in the comprehension of
■written problems.
Pupils with high
intelligence too, made fewer errorsj
therefore he concluded that intelligence had a decided effect on the num
ber and type of errors.
^olney Ward, "A Study to Determine lihether Beginning Algebra Should be
Taught in the Hinth or Tenth Grade," School and Society. XXXIII (Janu
ary 31, 1931). pp. 179-180.
CHAPTER
II
THE IKVESTIQATION
.- Kinety pupils,
forty-two boys and forty-
eight girls, from the first year algebra class of the Spencer High School,
Columbus, Georgia, were selected for this study.
school are sectioned into four groups s
groups.
slowr.
All pupils entering this
namely, two A groups and two B
The A groups represent the faster pupils and th© B groups,
the
In this investigation, however, this grouping has not been re
tained, but instead all boys have been grouped together and all girls
have been grouped together.
This was deemed justifiable after each divi
sion bad "been tested and the difference in scores found to be not very
significant.
In some instances an A pupil scored low while a B pupil
scored high.
These pupils were all taught by the same teacher except one of
the B groups.
Columbus.
All but twenty had attended school in no other town but
Of these twenty, four were in Columbus for their first year
which signified that all elementary work had been done elsewhere? the re
maining sixteen had spent one or more years in a school in some other
tornn or rural community.
Ho marked degree of difference was noted between
the work of these twenty and the remaining seventy, yet, it is perhaps
significant to note that the girl making the highest soor© for both groups
on the achievement test was one of the four who was in Columbus for the
first time; she ranked second in her group on the aptitude and the intel
ligence tests and had a socio-economic score that was above the average.
Description of Sohool^.and__CoijTOUiity_..- Colurabus, Georgia, the
county seat of Muscogee County, has a population of 43,151, approximately
IS
14
20,000 of whom are legroes.
The Hegro population is scattered all over
the city in both desirable and undesirable sections, consequently, idiese
pupils have come from various types of homes and surroundings.
Columbus
is located in -western Georgia on the Chattahooehee River, a river whose
mter power is great.
In Columbus this power is utilized to a large ex
tent in manufacturing, %vhich includes hardware, textiles, machinery, cloiiiing and furniture.
The woolen and cotten mills are among the largest in
the South and produce vast quantities of gingham and colored goods.
These
manufacturers give employment to large numbers of people; many of the pupils
in this investigation have come from homes where the father, some member
or members of the family are employed at some one of these factories.
Railroads furnish another source of employment for the inhabi
tants of this city, for three important lines, the Southern, the Seaboard
Air Line and the Central of Georgia, pass through it.
The fathers of
eight of the pupils in this study are employees of a railroad company.
Farming and dairying are lucrative occupations for the people
who live on the outskirts of the city.
Coming into the city from all di
rections one sees as he nears the city limits thickly settled sections
where both of these occupations are being carried on.
Many parents from
these sections and also from the more remote sections of the county take
advantage of the educational facilities in the city and send their chil
dren to the high school there.
Five pupils in this investigation have
come from the farm.
Ten miles south-east of Columbus at Fort Benning, Georgia, is
one of the largest Infantry Schools in the United States.
Children from
this fort help to augment the school population of Columbus; six of the
pupils in this study are the children of soldiers.
15
The Chattahooche© River is the dividing line between Columbus
and two towns in Alabama, Phenix City and Girard.
The proximity of these
towns make ■them seem as one and each helps the other in advancement.
In-
dxjstries in the Alabama tovms give employment to people living in Columbus
and vice versa.
Two of the pupils in this investigation live in Girard.
Spencer High School, a school accredited as an A class high school
by the Southern Association of Colleges and Secondary Schools, is located
in the extreme southern section of the city and is surrounded on all sides
by factories.
Its extreme location makes it not easily reached by all pu
pils who attend it.
The building is composed of twenty-one rooms, a spa
cious gymnasium xvhich is also used for assemblies in the absence of a
chapelt the principal's offices and a store room.
library where there is a large collection of books.
One room is used for a
For the scholastic
year 1935-1936, five hundred forty-one pupils have been enrolled in grades
seven through eleven.
Eighteen teachers are employed in the school.
Be
sides the regular academic work both manual training for the boys and do
mestic training, including both foods and cooking, for the girls are in
cluded in the curriculum of the school.
Method of Qbteining__Data_.- The data for this study were obtained
from a study of literature relative to the subject and from the scores
made by the pupils on the following tests;
Sims Score Card for Sooio-Economic Status.
Illinois General Intelligence Scale, Form I.
Lee Test of Algebraic Ability, Form A.
Douglass Standard Survey Test for Elementary Algebra,
Test I, Form A.
16
For comparative purposes the teacher's estimate of each pupil's work in
algebra was procured.
These marks were given on the basis of the marks
received at the end of the first semester.
The intelligence quotients were taken from the table1 given in
the manual of directions for the Illinois General Intelligence Scale.
On
the table the point scores ranging in multiples of five from 0 - 185 are
tabulated with the corresponding mental age given on the same horizontal
line.
No interpolation has been used but the in between scores have been
considered the same as that multiple of five to ?jhieh they were nearest.
For example, scores 71, 72, 73, and 74 have been regarded as 70, -while the
scores 76, 77, 78 and 79 have been regarded as 75.2
The chronological ages
are written across the top and bottom of the table; above or below each
chronological age is a vertical I. Q. column.
An I. Q. -was found by first
locating the mental age corresponding to a given score, and then, by means
of the chronological age of the person
making the score, locating the ver
tical column in which the I. Q. should be found.
The number found at -tfee
intersection of the horizontal and the vertical lines \»m,s taken as the I.
Q. of the pupil who made the given score.
1Hanual,_of_.pijgctigns. - Illinois..^e^rnl Jint^m^en^jcale.
2Ibid., p. IS.
p. 14.
17
A record card for ©ach pupil was made.
Below is a sample of one
of these cards.
FIGIffiE 1.- IIDIVIDUAL RECORD CARD
Score
M. M.
16
Socio-economic status
118
I. Q.
94
Teacher's estimate
112
Algebra Aptitude
21
Achievement
Mental age
16-0
Chronological age
13-7
Administering .thjj_Tgstg_.- Each pupil ms given the Sims's test
for socio-economic status rating and the Illinois intelligence test by
the investigator.
In order to determine the specific ability or aptitude
of the pupils for algebra, the Lee Test for Algebraic Ability was admin
istered by the investigator assisted by three members of the faculty of
the Spencer High School.
With the assistance of members of this same
faculty, the Douglass Survey Test was administered at the end of the first
semester to all but fifteen of the pupils; due to sickness and severe
weather this number was absent.
This group ms given the test later when
all had returned to school.
Testing Movementi_e^-J^e^eaghin^_of-^lgebra_.- More than twenty
years ago when Courtis first published his test in arithmetic the testing
movement in mathematics had its beginning.
Since that time numerous tests
18
in arithmetic, algebra and geometry have been developed.
By giving a
means by -which the work of one class or school ean be compared numerically
with that of others, these tests have helped to fix standards of achieve
ment in respective fields.
A teacher may wish to compare the status and
progress of her class in first year algebra with that of other classes in
first year algebra; for which purpose a survey test such as the Douglass
test may be used.
Tests to predict success in algebra such as the Lee
test for Algebraic Ability may be utilized for the purpose of grouping
pupils in ability groups or for the purpose of eliminating from the field
those pupils who show no significant aptitude for the subject.
Systematic diagnostic testing or the means by which a teacher
may find the specific difficulties of each pupil is an important aid to
the teacher who would improve his method of instruction and who would at
tain more satisfactory classroom results.
l'*en the difficulties are known
a plan of teaching and remedial treatment can be instituted to help the
pupil overcome the trouble.
In his mastery formula of teaching Morrison
dearly expresses the function and method of testing and corrective treat
ment for all subjects.
Tiegs makes the following assertion:
Diagnosis really begins before teaching as suchj its two prin
cipal forms are mental testing and achievement testing. A general group
test of mental ability yields a general measure of the mental level of
the class as a whole.
The results of such a test do not guarantee any
particular achievement, but experience indicates that they do constitute
evidence of capacity to do the work of the average school.
^H. C. Morrison, TjigJPractice of Teaching in the Secondary Schools,
Chicagoj
The University of Chicago Press, 1934. p. 81.
2Tiegs, op. cit., p. 110.
19
Tests have been and can be important instruments for teachers in
the improvement of their instruction, for by them they can obtain definite
date on which to base their instruction.1
It is believed by the writer
that tests may be used to measure the efficiency of a teacher of mathematics
as well as of other subjects.
Pupil failures are not always the fault of
the pupil but may be indicative of a lack of efficiency in the teacher.
A
complete testing program including a follow-up of the failures of pupils
may show the teacher that the success of the persons being instructed de
pends on a change of his method of instruction and thereby an improvement
of his efficiency.
Description of Tests UsecU- Verner K. Sims of -fee Louisiana Poly
technic Institute, Ruston, Louisiana, developed a device by which the
quantitative record of the social and cultural background of an individual
could be obtained.
By means of this record a numerical rating can be given
to the social status of an individual.
After much experimentation oarried
on at the School of Education, Yale University, this device was developed
into a Score Card to be used with the pupils in grades IV-XII, inclusive.
The Card3 requiring twenty to twenty-five minutes to administer is so de
veloped that it can be given to a group as easily as to an individual.
The Illinois General Intelligence Scale,4 an intelligence test
intended to be used with pupils in grades 1II-VII1, inclusive, consists of
seven different testss
analogies, arithmetic problems, sentence vocabulary,
substitution, verbal ingenuity, arithmetical ingenuity and synonym-antonym;
^E. R. Breslich, Thg_Technique of Teaching Secondary Mathematics^
Chicago:
The University of Chicago Press, 1931.
Chapter VII.
2Manual_^^Pireptipns - S.ims_S.oore Card for Soeio-Eeonomio Status,
See Appendix, p.
53..
%ee Appendix, p. 54.
p. 2.
20
it is published in two forms.
Sixteen minutes are required for actual
testing time.
According to Tiegs-*- there are eight desirable features that should
characterize every standardized test; namely, validity,
reliability,
tivity, ease of giving and scoring, adequacy of directions,
objec
tabulation
blanks and standards for interpretation, adequacy of alternate forms,
venience of form,
size and composition and adequacy of manual.
con
In the
Illinois test these features are found to exist in a high degree.
In the
first place its validity or the extent to which it measures x?hat it pur
ports to measure is shown by its high correlation with other similar tests,
the -correlation with the Stanford Revision of the Binet Scale being
.74 ± .02, with the probable error of tJie estimate being 1.2 years; the
correlation with the lational Intelligence Scale A, Form I being .81 ±.004,
with 10.3 points on the scale of the Illinois test as the probable error
of the estimate; the correlation vn.th the Otis Group Intelligence Test be
ing ,82i.O2 when 83 VI-A pupils were tested and .83 .£ .02 when 124 VI-B
pupils were tested, the probable error of the estimate being 6.4 in the
first case and 5.9 in the second.2
Reliability or the extent to which the test attains consistent
results has been calculated by giving both forms of the test to each of a
group of pupils and finding first, the coefficient of reliability (0.92);
secondly, the probable error of the measurement (5.30); and thirdly, the
ratio of the probable error of the measurement to the average score (0.07).
Megs, O]3u_j3it<>,»
SIbidi,
p. 21.
p. 283
El
The test is accompanied by a manual which gives the objective
of the tests,
full instructions for administering and scoring,
the time
limit for teach test and an adequate explanation on the utilization of
test results.
Attention is called to the fact that the I. Q.'s derived from
the Illinois test exhibit a wider spread
of distribution than those de
rived from the Binet-Simon scale, the "type of intelligence test through
which the concept of mental age and intelligence quotients first came in
to caramon usage.
For comparative purposes the authors of the manual in
serted the facts in Figure Z to show the difference between the interpre
tation of the results obtained from the Binet-Simon Scale and the Illinois
1
Scales
FIGURE 2.- COMPARISON OF ILLINOIS SCALE OF INTERPRETATION
OF I. Q.'S AHD THE BIHET-SIMON SCALE
Descriptive I|it©X£J>e'feiiBE,
of Correspondin
I. Q.'s
In Binet Scale
Range
In Illinois Scale
Very superior
125-139
140 and above
120-139
Superior
115-124
110-119
"Near11 genius or genius
Normal
140 and above
or Average
Dull
Borderline
Mental Deficiency
85-114
90-109
75-84
60-74
80-89
Below 60
Y/ith the increasing number of pupils
70-79
Below 70
in high school has come the
problem of finding the special abilities of the pupils so that time will
not be wasted trying to do that for which they have no aptitude.
devices to find this ability have been tried,
Various
chief among xirhich are subject
Manual of Directions - Illinois General Intelli&enoe Scale,
p.
15.
22
ability tests or prognosis tests.
Lee Test of Algebraic Ability,
four partss
An example of this type of test is the
a prognostic test in algebra consisting of
arithmetic problems, analogies, number series, and formulas.
The purpose of the test is primarily to discover the specific ability or
aptitude of a pupil for algebra.
The test possesses a reliability of
.93 ± .009 -which coefficient tries computed by correlating the odds and evens
and using the Spearman-Brovm Prophecy Formula.
o
In establishing the validity of the test the author gave the
test to a large number of high school pupils and correlated these findings
with the pupil's success in algebra.
lidity of the test.
The correlations determined the va
Table I gives these correlations together with the
number of pupils tested and the name of the schools in which the standard
ization was carried on.
TABLE I.- DATA OH VALIDITY OF THE LEE TEST OF ALGEBRAIC ABILITY3
Harvard H. S.,
School
Glendale, Calif.
lumber of cases
True validity
Correlation between Test of Algebraic
Ability and achievement test
al
gebra marks
Correlation between Test of Algebraic
Ability and achievement test
Correlation between Tost of Algebraio
Ability and algebra marks
See Appendix, p.
Burbank,
313
81
.766
.846
.713
.791
.710
.693
.631
.642
.763
.775
Correlation between achievement test
and algebra marks
John Muir Jr. H.S.,
55,
^feg»l-jof Direp-bions - Lee Test p_f,Algebraic Ability_.__p. 5
Ibid.,
p. 4.
Calif.
23
For the purpose of diagnosis in elementary algebra the Douglass
Standard Survey Test* is a short but reliable test, published in two forms,
Test I to test the work of the first semester of first year algebra, and
Test II to test the work of the second semester or the entire term.
Each
test required forty minutes for administering.
The reliability coefficients were taken separately and together
for the tests and were found to be as follows:
2
One form
Both forms
r
I
r
Test I
.78
148
.87
Test II
.74
86
.86
Both tests
.86
48
.93
The author makes the following statements regarding the scope of
the test.
The two parts of the test have been constructed with a view to
conforming to the recommendations of the Mational Committee on Mathemati
cal Requirements. Emphasis has been placed upon a solution of the types
of equations, graphs, and formulas likely to be of most use both in nonschool life and in future courses in mathematics. ... .A number of
"story" or applied problems are included. These are not artificially
segregated but are introduced in connection with topics and processes
which they involve.3
See Appendix, p.
56.
2llanual of Directions - Douglass Standard
Algebra.
SIbid.,
p. 2.
p. 1
CHAPTER III
AIALYSIS AID INTERPRETATION OF DATA
Socio-economic Status of..?upilg.«- A consultation of the record-
sheet for boys1 will reveal the fact that the highest score for socio-eco
nomic status is twenty-five and the lowest score is one.
The median is
11.3, a score corresponding somewhat closely to the median score of ten
found by Sims for his Score Card for Socio-Economic Status.2
Figure 3
shows the levels of soeio-econoraic status suggested by Sims.
FIGURE 3.- PROVISIONAL LEVELS OP SOCIO-ECONOMIC STATUS3
Score
Corresponding
Percentile
36
«
Suggested
Rating
10
«
Corresponding Levels of
Socio-Economic Status
Indeterminately High
29.2
94.5
9
Highest
24.5
88.5
8
Very High
17.6
78.8
7
High
13.2
65.5
6
Medium High
10
50
5
Medium
7.5
34.5
4
Medium Low
5.1
3.2
21.2
12.2
3
2
1.8
5.5
1
-
0
0.0
-
Low
Very Low
Lowest
Indeterminately Low
From the above figure it will be observed that the maximal pos
sible score is 36 which represents a socio-economic status that is theo
retically perfect.
see Appendix, p.
^
3Ibid.,
The numbers 1 to 10, preceding the descriptive levels,
53.
of.^irej3tions, -
p. 11.
p. 12.
24
are suggested ratings used to designated strata of homes graded from 0,
no home at all,
to 10, theoretically perfect home.
In the group being considered in this investigation there is no
home that may be
rated as no home at all nor is there one to be classed as
theoretically perfect, but the range is from lowest to the very high level.
Nineteen or forty-five per cent of the group made scores from 10-13, inclu
sive; that is, forty-five per cent of the scores of the group are near the
median or fiftieth percentile as found for the group and also the fiftieth
percentile established by Sims for his group.
See Table II, page 27, for
the distribution of the socio-economic scores for the boys.
A somewhat different picture is made by the scores of the girls
on the socio-economic test.
lowest score is three.
The highest score is twenty-eight and the
In this group the range of socio-economic status
is from very low to very high.
The median is 12.9 for this group.
Only
seven or fifteen per oent of the group made scores which fell in the range
of the median 12-13.
Table III, page 28, gives the distribution of the
scores made by the girls on the socio-economic test.
For a knowledge of
the scores mad® by each individual girl see the record-sheet.1
It will be noticed that the fiftieth percentile for the scores
of both the boys and girls is higher than that found by Sims.
However,
in the group from %vhich Sims deducted his conclusions there were 686 pupils
from the New Haven, Connecticut, schools2 and in the group from which the
writer has made her findings there are forty-two and forty-eight pupils
respectively.
It is probable that this accounts in part for the difference
Appendix, p.
52.
^Manual of Directions - Sims Score Card for. Sociq-Eopnqmic Status,
p. 11 <
26
in percentiles; and the fact that the pupils came from a different section
of the country is another probable factor causing this difference in medians.
Socio-economic status is in a large measure determined by the oc
cupation of the father.
Table IV, page 29, shows the distribution of the
boys according to the Sims group classification lshich divides the occupa
tions into five groups as followst
Group I, professional menj
Group II,
commercial service; Group III, artisan proprietors; Group IV, skilled la
borers; and
Group V, unskilled laborers.1
It will be noted -tfoat the
largest number ms found to be in Group IV while Croups III and V ranked
second and third respectively.
three pupils only.
In Groups I and II combined there were
Six could not give their fathers1 occupation because
they were either dead or their whereabouts were unknown.
The occupations of the fathers' of the girls were in very much
the same class as those of the boys; Groups III, IV and V contained the
largest per cent.
Group I and II combined contained five.
Five could not
give the occupations for the same reason as the boys who could not.
Mental and Chronologioa,l_Aggs_g£L^gLSHE^li*" The chronological
ages of the boys in this study ranged from 12-11 to 17-10 with the median
falling at 15.0 years.
13-6 and 15-0.
The majority of the pupils ranged between the ages
A different distribution was found in the mental ages, the
range in this case being from 9-0 to 17-0, with the majority of pupils
falling in the interval 11-6 to 13-0.
The median mental age was calculated
to be 13,0 years.
%anual of Directions - Sims Score Card for Socjj>-Economio. Status,
p. 9.
27
TABLE II.- DISTRIBUTION OF SCORES MADE OH THE SIMS SCORE CARD
FOR SOCIO-ECONOMIC STATUS
BOYS
Scores
Number of Pupils
24-25
1
22-23
1
20-21
2
18-19
2
16-17
3
14-15
2
12-13
5
10-11
14
8-9
7
6-7
2
4-5
2
2-3
0
0-1
1
Total
Median
42
11.5
28
TABLE III.- DISTRIBUTION OF SCORES MADE OK THE SIMS SCORE CARD
FOR SOCIO-ECOHOMIC STATUS
GIRLS
Score
lumber of Pupils
28-29
1
26-27
0
24-25
0
22-23
2
20-21
1
18-19
5
16-17
8
14-15
3
12-13
7
10-11
9
8-9
5
6-7
5
4-5
1
2-3
1
Total
48
Median
12.9
29
TABLE IV.- DISTRIBUTION OF BOYS ACCORDING TO FATHER'S OCCUPATION
SIMS1 GROUP CLASSIFICATION1
lumber of Pupils
Group
I
1
II
2
III
11
nr
12
V
10
6
Unknown
42
Total
TABLE V.- DISTRIBUTION OP GIRLS ACCORDING TO FATHER'S 0CCUPATI01
SIMS' GROUP CLASSIFICATION2
Group
lumber of Pupils
I
1
II
4
III
11
IV
16
V
11
Unknown
Total
^Manual of Directions. -
2Ibid.,
p. 9.
5
48
P*
so
Except in a few cases there ias quite a difference between the
mental and chronological ages of the pupils.
two years.
A striking case
The mean retardation was
is that of the boy, H. A., whose mental age
was 9-6 and vfhose chronological age was 17-10.
For a comparison of the
distribution of the mental and the chronological ages of the boys see
Table VI, page 31.
Table VII, page 32, shows that of the forty-eight girls thirtyone were in the chronological age range 13-6 to 15-0 and that twenty-five
were in the mental age range
11-6 to 13-0.
The chronological ages of the
girls ranged from 12-3 to 18-5 while the mental ages ranged from 8-6 to
1706.
A median of 12.8 years was computed for the mental ages and for the
chronological ages 14.8 was computed.
A consultation of the record-sheet
will show the fact that the difference in mental age and chronological
age is small in most cases, the smaller mental age predominating.
group also, the retardation -was two years.
In this
The girl, C. S., is an example
where there is a larger difference between mental and chronological age,
for in this ease the mental age is 8-6 and the chronological age is 16-6.
^,-
Intelligence quotients were calculated
for eaoh group and were found to range from 60-118 for the boys and 56-121
for the girls.
The distribution of these scores is shown on Table VIII,
page 33, for the boys and Table IX, page 34, for the girlsj the per cent
of pupils falling in each interval is also shown in these tables.
In
31
TABLE VI.- DISTRIBUTION OF MENTAL AND CHRONOLOGICAL AGES
BOYS
Number of Pupils
Years
Mental Age
Chronological Age
17.5-18
0
2
16.5-17
1
4
15.5-16
1
8
14.5-15
3
14
13.5-14
11
6
12.5-13
10
8
11.5-12
8
0
10.5-11
3
0
9.5-10
2
0
8.5-9
3
0
Total
42
42
13.0
15.0
Median
32
TABLE VII.- DISTRIBUTION OF MENTAL AND CHRONOLOGICAL AGES
GIRLS
Number of Pupils
Years
Mental Age
Chronological Age
17.5-18
1
3
16.5-17
0
6
15.5-16
5
5
14.5-15
0
14
13.5-14
10
17
12.5-13
12
2
11.5-12
13
1
10.5-11
5
0
9.5-10
1
0
8.5-9
1
0
Total
48
48
12.8
14.8
Median
S3
TABLE VIII.- DISTRIBUTION OF IHTELLIGEICE QUOTIENTS
BOYS
I. Q.
Number of Pupils
Per cent of Pupils
115-119
1
2.4
110-114
3
7.1
105-109
3
7.1
100-104
4
9.5
95-99
1
2.4
90-94
1
11.9
85-89
8
19.0
80-84
6
14.3
75-79
4
9.5
70-74
2
4.8
65-69
1
2.4
60-64
4
9.5
Total
42
Median
88.8
34
TABLE IX.- DISTRIBUTION OF INTELLIGENCE OUOTIEHTS
GIRLS
I. Q.
Per cent of Pupils
Number of Pupils
120-124
1
2.0
115-119
3
6.3
110-114
1
2.0
105-109
2
4.2
100-104
2
4.2
95-99
7
14.5
4
8.3
85-89
11
22.9
80-84
3
6.3
75-79
10
20.8
70-74
1
2.0
65-69
2
4.2
60-64
0
0
55-59
1
2.0
Total
48
90-94
Median
88.
35
Table VIII it is revealed that a little more than half or 57 per cent of
the boys had I. Q.'s ranging from 85-114, 23.8 per oent ranging from 75-84,
16.9 per cent ranging from 60-74, and only 2.4 per cent above 114} thus,
according to the Illinois scale1 the range of intelligence in this group
is from the borderline of mental deficiency to superior, with the largest
per cent, 57 per cent, in the average class.
division was found to be 88.8.
The median I. Q. for this
Table IX indicates that 54.S per cent of
•toe girls are in the normal or average range, 85-114, and 8.3 per cent are
in the superior range of intelligence.
in three classes:
Belovir the average the pupils fall
dull 27.1 per cent, borderline deficient 6.2 per cent
and mentally deficient 2 per cent.
A median of 88.2 was computed for this
group.
Algebraic Ability of Pupils.- From results obtained vAxen the Lee
Test of Algebraic Ability ms given it is evident that there are pupils in
the group who possess a very significant aptitude for algebra while there
are others whose scores would place them in a class whose aptitude for al
gebra was not as pronounced and still others whose scores would place them
in a class where aptitude was completely lacking.
A score of 126 and a
score of 27 mark the highest and lowest level of the boys -while the highest
and lowest level for the girls are 114 and 15 respectively, the median
score being 76.7 for the boys and that for the girls 66.4.
For a record
of the scores made see the record-sheets.2
On the basis of a study of the performance of 446 unselected
pupils in two schools Lee took the score of 60 as that score to determine
" Illinois General Intelligence Scale,
2See Appendix,
pp. 51,52
p. 15.
36
the borderline of those who should take and those who should not take al
gebra. 1
He suggests that pupils making below 60 be counseled against
taking algebra and that those making a round 60 be carefully watched.
An
acceptance of this score as the determinant would place 15 of the boys and
17 of the girls in the group who should be advised not to take the subject;
2 boys and 11 girls could be allowed to pursue the course with careful
mtohing.
The median found by Lee was eighiyone'* which is higher than those
for the two groups being considered, but it must be borne in mind that the
latter groups are composed of about one-tenth the number of pupils that
the Lee investigation contained.
The number of oases as well as other fac
tors such as the difference in locality of the pupils studied, the famil
iarity of the pupils with such tests, the attitude toward the examiner,
and the cultural and social background of the pupils will no doubt affect
the results of a test to the extent that the applicability of a set of
norms may be destroyed, but the purpose and usefulness of the test will
not be destroyed.
Sometimes it may become advisable for a school to es
tablish its own norms.
Achievement of Pupils in FirstL Yeari.
A careful study of Tables X, page 37, and XI, page 38, will disclose the
type and number of responses given on the Douglass Survey Test for Elemen
tary Algebra by each pupil in the investigation.
It tjIII be observed that
only 490 or 47 per cent of the 1050 possible correct responses were made
^•Manual of_jjirgetisas. - Lee Test of Algebraic. Ability,
2Ibid.,
p. 12.
p. 10
37
TABLE X.- PUPILS1 RESFOMSE 01 THE DOUGLASS
STANDARD SURVEY
Right /
TEST FOR ELEMENTARY ALGEBRA
krone *■
Omitted O
BOYS
Question 1
2
3
4
5
6
7 8
9
10 11 12 1
14 15 16 17 18 19 20 21 22 23 2
2
H. A.
•
E. A.
y y •
W. A.
y y • y a 0
N. B.
X
A. B.
.X
X O
X
(7
1/ X
*/ y
X
X X X
X X X
J. G.
y
V
0
y
E. C.
y
1. D.
y
B. D.
y
A. D.
y
E. D.
y
A. G.
X X
J. H.
X
•
C. H.
y
• • A 0
J.
•
B. K.
y
y
R. L.
A. M.
V V
X * X X X X
y y
•
V
A
.1/ y y
y
y
X
X
y
x
•
X
X
•
•
x
X
y
X
X X 6
y
X y
y
I/1
X.
y y
A
,x y •
% X
X
0 y y y y
C. P.
X X. X
y
V
G. P.
(X
G. P.
•
J. R.
• *x
y
•
L. S.
X
X
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X X
X
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• y
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\x
y
y
y X • •
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X
TIT. T.
*
H. T.
X
X x
L. T.
•
•
L. VI.
«x
X X
o
X 0
•
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S X X 0
X y X X X
P. W.
y
• X
34 26 IS 13
y
V
y
X"
x X
y
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V
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\y •
y y
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X y
y
X
y y
11
r"
o
a
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X
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0
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X
o
17
X y X
0
IX
0
y
y y y y
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9
11
X
c
10
6
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X X
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X
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u
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X
17
y
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y iX
9 -5
0
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0
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Total
Right
Pupil
X o
17
o O
o
9
,/•
f) o
18
2
490
y tX
14 15 13
2
2
TABLE XI.- PUPILS1 RESPONSE ON THE DOUGLASS STANDARD SURVEY
TEST FOR ELEMENTARY ALGEBRA
Right y
GIRLS
Question
1
2
3
4
s
6
7
8
Wrong X
Omitted 9
9 10 H 12 13 14 15 16 17 L8 19 20 21 22 23 24 25
Pupil
M. A.
X
H. A.
M.
X
•
B.
R. B.
C. B.
X
x
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y
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y
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x
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s
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n o
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y
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y x
y
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v/
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y X •
y X y X y X y
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x •
x n
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39 28 26 18 31 43 34 21 3 7 85 17 18 40 38 30
13
11
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21
12
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12
13
15
15
12
V
o o 0
6
14
6
7
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15
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y ts
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8
16
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12
11
16
n o 0
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x;
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X X
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16
X
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x
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o o a X X
X y X X • €> O o f) 0
y
r> o O n •
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X X x. X X y X y K X X
y
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16
11
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•
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G. P.
M. R.
•
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y
X •
y
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M. M.
A. R.
•
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X
•
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y
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•
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•
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y y
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i.
Total
Right
0 29 2 26 20 20 16 19 1 3
7
6
10
10
15
15
11
12
4
14
571
39
by the boys and 571 or 49 per cent of the 1200 possible correct responses
were mad© by the girls, the girls hairing surpassed the boys by a very small
margin.
The tables also reveal the fact that for both boys and girls the
largest number of omissions and incorrect responses was to be found in the
problem solving exercises.
Exercise 16, which Involved the writing of an
equation for a problem without solving, was either incorrect or omitted by
each pupil.
Problem 18 ms omitted by 19 boys and 19 girlss 21 boys and
27 girls gave incorrect responses.
Problem 23 was omitted by eleven boys
and ten girlsj IS boys and 19 girls gave incorrect answers.
Twenty boys
and 21 girls omitted problem 24 while 20 boys and 26 girls tried to solve
it but failed.
Twenty-four, 10 boys and 14 girls, who failed to do this
problem failed because they did not express correctly the perimeter of a
rectangle.
Thirty-four girls and 32 boys omitted the twenty-fifth problem;
eleven girls and 8 boys gave incorrect responses.
The high per cent of
failures to solve problems is no doubt in a large measure due to inability
to translate the written problem into algebraic symbols, as is shown by
the fact that no one was able to do the sixteenth, the first problem sit
uation on the test.
This is in agreement with the finding of Vance,x
namely, that problem solving caused the largest number of failures.
Below is a list of the errors which caused failures in other
than the problem exercises.
1.
Combination of unlike terms.
2.
Addition of exponents when adding or subtracting.
3.
Sign errors in the fundamental processes
^•Tance, pp. .pit.,
p. 72.
40
4.
Coefficient errors.
5.
Sign errors in transposing terms
6.
Incorrect operation.
7.
Miscopying.
8.
Arithmetical errors.
9.
Failure to differentiate between the minuend and
of an equation.
the subtrahend.
10. Indications only of division and multiplication in place of answer.
11. Unclassified errors.
The largest number of errors, 82 by the boys and 65 by the girls,
was made in the use of incorrect signs in the fundamental processes.
Coef
ficient errors ranked second, the boys making 60 and the girls making 70.
In the study by MacRae and Uhl sign errors ranked first also.*
Due to the
fact that in many instances the pupils did not transfer the solution of
the test from the sheet of scratch paper, but recorded only the answer ob
tained, it was impossible to account for the incorrect answers;
such er
rors could not be classified.
T%venty-five was the highest possible
score that could be made on
this test as each of the exercises was marked either right or iwong,
total rights giving the score.
the
The highest score for the boys -was 18 and
the highest for the girls was 21 while the lowest scores were two and four
respectively.
Computation of the median gave 10.2 for the scores of the
boys and 10.6 for those of the girls.
There is a close correspondence of
these norms to 9.4 established as the tentative median for Test II of the
series, but there is a decided difference between them and 20.3 established
MaoRae and Uhl, loc.». cit..,
p. 21.
41
as Hhe median for Test I.1
In considering the difference in the medians found for differ
ent groups it must be borne in mind that no set of norms can be taken as
a final measure of all groups, and yet,
in view of the fact that the
norms established for the Douglass test were calculated from results ob
tained from a large number of freshmen high school pupils in various parts
of the country»
p
the two groups being considered
in this study are far be
low the standard set by other groups i«ho had had a semester of algebra.
T»hat are the factors th&t have affected the achievement of these groups
is the question to be answered.
Use jfade of Scores•- In a statistical analysis and
interpreta
tion of data the investigation of the association or relation between two
or more series of facts is often profitable in predicting future results
when data are taken under similar conditions.
For the two groups involved
the degree of association or the coefficient of correlation (r) between
achievement for the first semester of algebra and each of the following
socio-economic rating,
general intelligence, special aptitude for algebra
and the teacher's estimate of the pupils was computed.
The probable error
(P. E.) of (r) was also computed.
The relationship of achievement to socio-economic status was
found to be .05 dt .10 for the boys' group and .06 ± .07 for the girls'
group.
A relationship of .04 ± .10 for the boys and .08 ± .07 for the
girls was found between achievement and intelligence.
For achievement
and aptitude the relationship was computed to be .10 d: .10 for the boys'
^Manual of Directions - Dougla,sg Standard .Survgy_Tegt_
Algebra«
2Ibid..,
p. 3.
p. 3.
42
group and .07 ± .07 for the girls.
and the achievement of the pupils
The association of teacher's estimate
on the Douglass test was
found to be
• IE i .10 for the boys and .13 ;fc .09 for the girls.
Significance pf Rejj^lts^f C^rj^elationg.-» The findings of this
study indicated that the degree of association between achievement and
the factors under considerations
namely,
socio-economic
status, general
intelligence, special aptitude for algebra and the teacher's estimate is
very lew and is,
therefore,
of doubtful significance.
other than socio-economic status,
Evidently factors
intelligence, special ability or apti
tude and the teacher's estimate affected the pupil's success in algebra.
As pointed out by McCuen
there are many unmeasurable factors that will
not allow one to predict accurately a pupil's success in algebra.
He
mentioned such factors as the interest on the part of the pupil and his
attitude toward the teacher.
To these may be added certain other measure-
able factors such as the pupil's method of studying, teacher's efficiency,
the physical equipment of the sohool plant, the health of the pupils, the
location of the school building, the grade placement of material and the
presence of material for drill and remedial work.
In the Atlanta Public
Schools, both white and colored, the work in arithmetic for the past two
years has been pushed up a whole grade.
The authorities felt that the
work as outlined was too difficult for the grade.
In conclusion one must consider these factors which may affect
a pupil's success, for often a pupil who is perhaps interested in his
work is hindered in achievement by poor methods of studying, inefficient
TfcCuen,
loo, cit.,
p. 4.
43
teaching, poor health, retardation, an undesirable location of the school
and inadequate school equipment and supplies.
The correlations between aptitude and achievement concur with
the findings of Nelson* who found a slight correlation in these respects.
He also found no real relationship to be present between intelligence and
scores of errors.
Like an average, the coefficient of correlation represents one
phase of the data only.
It is a number which may be used as an index of
the association between two or more series of data; its range is between
zero and unity,
lot only the degree of relationship between two series
may be measured by it but also the effect of some other factor to which
they are both related.
Chaddock2 makes the following assertion:
The coefficient is an index of relationship, not a proof of
causal dependence. Like other statistical coefficients and constants
it is computed for the purpose of clarifying the interpretation of com
plex masses of data.
Conclusive evidence of the degree of association is not ob
tained from the computation of a single random sample, but consideration
must be given to the probable variation or error which may be due to cer
tain uncontrolled conditions of sampling.
The unreliability of the co
efficient is measured by its probable error which shows the range of var
iation measured plus and minus from the computed value of
r .
A knowl
edge of the probable error of
r
sets a limit to the chance fluctuations
of
r
can then be determined from the relation
r j
the unreliability of
kelson, op. cit.,
2Chaddoek, pp.._cita.»
p. 9.
p. 304.
44
of the probable error to the size of
r •
Chaddock* makes the assertion
that
Conservative statistical practice in interpreting
r
requires
that the size of
r
should be 4 P. E. before it becomes indicative of
any significant degree of association.
The significance of a coefficient, therefore, depends upon the
relative value of its probable error.
Experimentation has shown that the
size of the probable error and the degree of unreliability of
creases as the number of cases decreases.
On the other hand
r
r
in
increases
in significance as the probable error decreases and as the number of cases
increases, the significance increasing in proportion to the square root
of the number of cases and not directly as this number increases.
significant an
r
To be
of low value must be based on many more cases than one
of high value.2
In no case in this investigation did
r
equal 4 P. E.
The high
est degree of association for both groups was found between achievement
and the teacher's estimates aptitude and achievement ranked second for
the boys while intelligence and achievement ranked second for the girls;
socio-economic status and achievement ranked third for the boys while the
third ranke for the girls was between aptitude and achievement; the least
significant for the boys was between intelligence and achievement; for the
girls it was between socio-economic status and achievement.
Despite the
fact that there are these differences in ranks of associations in this
study no marked difference was found in the results of the study of these
Chaddock, op. pit.,
2Ibid.,
p. 275.
p.
275.
45
factors on the achievement of the boys and. that of the girls in the first
semester of algebra.
CHAPTER
17
SUMMARY AID CONCLUSIONS
This investigation involved a study of the scores made by ninety
pupils, forty-two boys and forty-eight girls, from the first year algebra
class of the Spencer High School, Columbus, Georgia, on the following tests:
Sims Score Card for Soeio-Economie Status.
Illinois General Intelligence Scale, Form I.
Lee Test of Algebraic Ability, Form A.
Douglass Standard Survey Test for Elementary Algebra,
Test I, Form A.
Separate studj.es of the results
boys and girls were made.
obtained from the
scores of the
For comparative purposes the estimate of the
teacher was procured for each pupil.
The purpose of this
study was first, to note the type and number
of errors made by the pupils on the survey testj
secondly,
lation of the pupil's achievement to each of the following:
status,
to find the re
socio-economic
intelligence, teacher's estimate, and special aptitude for algebra.
The relationship betsreen the factors here under consideration was expressed
by the coefficient of correlation computed between achievement and these
factors.
Findings and__C_cmcJLusjLon8_.1.
The median socio-economic
for the girls,
score ¥*as 11.3 for the boys and 12,9
scores somewhat higher than the median established by Sims
when standardizing the test.
Nineteen or forty-five per cmt of the boys
and seven or fifteen per cent of the girls made scores which fell in the
Tflanual of Directions - Sims Score Card for Socio-Eoonomic Status,
p. 11.
47
immediate or near immediate range of the median.
scores for the boys -were twenty-five and one;
The highest and lowest
the highest and
lowest scores
for the girls were twenty-eight and three.
2.
A chronological age range of 12-11 to 17-10 and 12-3 to 18-5 -wag
found for the boys and girls respectively.
The median C. A. for boys was
15.0, for girls 14.8.
3.
The mental age of the boys ranged from 9-0 to 17-0,
from 8-6 to 17-6.
of the girls
For the boys the median M. A. was "13.0, for the girls
12.8.
4.
The mean retardation was two years for both groups.
5.
I, Q.'s for boys were found to range from 60-118, for girls from
56-121.
6.
girls.
The median I. Q. for boys was 88.8, for girls 88.2.
The median ability rating was 76.7 for the boys,
65.4 for the
One hundred twenty-six marked the highest and lowest levels of
the ability rating of the boys, 114 and 15 for the girls.
7.
Of the 1050 possible correct responses, 490 or 47 per cent were
given by the boys and 591 or 49 per cent of the 1200 possible correct re
sponses were given by the girls on the Douglass test.
8.
Problem solving caused the greatest number of difficulties on
the Douglass test.
This is in accordance with the findings of Vance1 and
was evidently due in a large measure to the inability to translate the
written problem into algebraic symbols.
9.
In a list of specific errors noted,
sign errors in the fundamen-
tal processes, in accordance with MacRae and Uhl,
ranked first, 82 for
the boys, 65 for the girls; coefficient errors ranked second, 60 for the
boys, 70 for the girls.
Vance3 found little difference in total errors
for boys and girls.
•'•Vance, op. cit.,
p. 11.
%aeRae and Uhl, loc. cit.,
3Vance, op. cit.,
p. 11.
p. 7.
48
10. Coefficients of correlation,
computed by means of the Pearsonian
formula or the product-moment method, between achievement and each of the
following,
socio-economic status,
intelligence,teacher's estimate and ap
titude Trere found to be as follows:
Boys
Girls
Socio-economic status
.05 A .10
.06 ± .07
Intelligence
.04 ± .10
.08 £ .07
Aptitude
.10 ±.10
.07 db .07
Teacher's estimate
.12 ±.10
.13 ± .09
11. From the low correlations found in this investigation it is con
cluded that factors other than socio-economic status, intelligence, teach
er's estimate and aptitude affected the success of the pupils in algebra.
12. The teacher's estimate gave the best, though not very significant
degree of association for both groups.
findings of Burton.1
This is in accordance with the
A variation -was found in the ranks of association
of the other factors for each group, yet no significant difference in the
results for each group was fouhd.
In the recent trends of education there is a call for a readjust
ment in teaching procedure and a readjustment in school room activities in
general.
High school mathematics has received its share of criticism, the
contention being that there are some factors which cause the large number,
of failures in this field each year.
The high per cent of errors on the achievement or survey test
given the two groups in this study indicates that something is wrong.
The
majority of the pupils were found to have at least a normal amount of in
telligence; it does seem that any person with a normal mind would be able
to do successfully the work covered in elementary algebra.
^Burton, op. cit.,
p. 6.
Could this
large number of incorrect responses be due to the development of faulty
habits of response, improper method of attacking a test, failure to apply
one's self to the test, poor health, undesirable location of the school
or some other factor?
It is suggested that an individual diagnostic test
ing program could profitably be carried on by the classroom teacher or
some o-Uier investigator.
The individual difficulties could thus be lo
cated and a remedial program planned to help the pupil overcome the diffi
culties.
The need for further investigation of the factors affecting suc
cess in algebra is evident.
1.
It is further suggested:
That other studies involving the same factors as this study
be carried on mth other pupils in this section of the country.
2.
That studies involving other factors, such as the pupils'
method of studying, adequate and proper school facilities, prep
aration of teachers, method of teaching, quality of supervision
and grade placement of content be made also.
APPENDIX
51
RECORD-SHEET
BOYS
Pupils
Ages
M. A.
C. A.
economic
Tea t Scores
Aptitude
I. Q.
29
Achievement
Estimate
65
11
61
111
94
118
14-10
10
100
17-3
10
68
8
88
71
51
81
18
61
37
8
14
76
52
14
64
5
70
46
11
70
IS
93
89
12
10
75
57
70
89
13-10
5
15-11
10
96
84
12-6
14-9
10
E. H.
17-0
16-6
C. J.
9-0
14-8
N. B.
9-6
15-0
14-0
16-0
A. B.
14-6
J. C.
R. C.
10-6
E. C.
13-0
9-0
15-0
15-2
H. D.
11-6
16-0
B. D.
A. D.
13-6
13-6
15-7
14-10
E. D.
11-6
16-8
A. G.
13-0
J. H.
G. H.
12-6
H. A.
E. A.
'3. A.
17-10
13-6
15-6
13-6
11
19
9
57
6
11
83
71
7
73
11
80
2
12
60
12
84
40
78
66
50
13
13
86
102
105
11
18
1
111
109
14
84
IS
62
27
13
65
75
80
90
B. K.
11-0
13-11
17
82
91
16
80
w
12-0
14-0
14-9
13-3
11
10
83
69
108
82
11
9
80
78
14-6
9-0
13-4
15-7
8
111
94
11
84
J. McC.
8
60
27
3
70
J. McN.
14-0
15-7
20
94
108
12
80
G. P.
14-0
11-6
13-5
15-0
10
108
116
10
78
8
78
40
6
80
12-6
13-0
14-7
15-3
17
86
11
80
9
88
35
78
12
90
13-0
13-6
15-6
12-11
12
87
94
17
84
25
108
8
75
13-6
14-10
15
13-4
15-7
13-0
15-6
23
10
18
84
94
10
9
84
92
80
58
13
14
17
77
J. S.
13-6
12-6
12-0
12-0
93
104
126
91
105
39
126
W.
XT
R. L.
A. M.
C. P.
J. P.
G. P.
G. P.
J. n.
T. R.
L. S.
J. S.
J. 3.
12
88
84
T.
14-0
13-8
17
104
73
10
75
H. T.
L. T.
12-0
14-5
11
86
91
80
12-0
17-9
6
77
L. '..•
10-0
12-11
7
80
0. W.
12-6
14-8
86
W. 1.
13-6
13-4
12
21
39
61
106
11
12
104
w. w, •
F. TV.
10-6
13-0
15-5
15-1
10
10
60
12
60
15
64
79
17
86
71
72
9
60
88
100
18
96
52
RECORD-SHEET
GIRLS
Pupils
11. A.
,G. A.
M. A.
14-0
14-10
H. A.
13-0
14-5
M. B.
13-6
15-7
E. B.
11-6
15-6
C. B.
12-6
12-6
14-0
14-7
15-4
13-11
R. B.
S. B.
J. R.
12-0
12-6
F. B.
9-6
T. B.
A.
C.
Test Scores
Ages
11-0
economic
I. Q.
Aptitude
10
75
94
87
16
80
12
14
90
86
11
55
77
62
13
80
11
86
44
9
75
70
85
8
97
8
85
67
16
12
104
67
17
89
71
12
78
31
11
63
85
13-7
15-11
11
13
84
14-5
17
68
70
16
72
49
3
74
16
9
84
16-6
12
13-6
14-6
14-0
15-10
19
85
19
23
76
89
84
114
E. F.
11-6
11-0
12-6
13-0
9
88
61
C
13-6
14-7
18
93
14
77
9
E. C.
E. E.
G.
H. II.
M. H.
A. H.
11-6
15-6
13-6
75
16
84
48
15
80
21
10
74
111
29
6
60
96
73
13
80
9
83
76
80
13
85
7
96
100
85
17
6
14
4
86
53
6
72
18-5
6
77
75
13
75
14-9
7
42
13-9
37
11
10
78
10
79
82
107
111
14
80
70
15-7
14-5
14-2
12-0
14-10
H. H.
11-6
12-3
I. H.
B. H.
13-6
M. H.
S. H.
12-0
11-6
71
7
E. H.
13-0
Teacher's
Estimate
69
14
80
E. D.
Achievement
13-10
16-5
17
67
89
70
R. J.
H. L.
11-0
15-6
14-8
19
E. L.
K. L.
11-6
15-3
10
10
78
62
7
85
57
11
58
13
90
38
70
12-6
13-0
15-0
14-6
13-7
14-4
16
118
112
1. 11.
16-0
13-6
13
21
17
96
62
12
80
0. N.
11-6
17-8
16
79
26
14
67
G. P.
12-0
16-9
17
78
77
12
80
13-4
13-7
96
69
13
75
28
15
80
15-2
10
115
95
90
A. R.
■12-6
15-6
14-0
17
84
15
84
E. R.
12-0
17-8
77
67
11-0
15-5
75
49
52
12
P. S.
V. S.
6
12
7
70
10-6
16-9
20
69
15
6
40
c. s.
8-6
16-6
11
56
61
10
60
M. S.
16-6
13-0
10
85
37
10
108
80
10
15
70
T. \J.
13-0
14-0
i. :i.
17-6
14-0
22
121
77
15
91
v. w.
12-6
14-5
19
89
36
11
60
D.
vi.
16-0
13-6
23
118
111
12
78
M. W.
12-0
13-6
7
89
68
4
70
D.
13-6
14-4
16
96
60
14
85
B. Me.
M. M.
M. P.
M. R.
'!.
94
83
53
_
_
.€"
Published btj the
_/*
Copyright 1927 by the
SI'S-c^ TimSdmlPammq& — —«5as
Printed in U. S. A.
SIMS SCORE CARD FOR SOCIO-ECONOMIC STATUS
Form C
Score
1.
Name
2. Age
3.
.....Years and
Grade
_
Months
Date
4. Have you spent two years in any grade?
If so, what grades?..—
5. Have you skipped any grades?.....
If so, what grades?
6.
Home address: City
State
7.
How many years have you lived in this town?
8. Have you attended schools in any other towns ?
If so, name
them
9.
Name of your School...
_
Don't answer any of the questions below until you are told what to do.
If you have brothers or sisters in this school, write their names and
grades on these lines:
Name
Grade
Name
Grade
In the Following Questions Underline the Correct Answer:
Are you a Boy?
a Girl?
(Underline correct answer)
Are you living at home with your parents?
Yes
No
Are you living in the home of someone else, such as a rela
tive, adopted parent, guardian, etc. ?
Yes
No
Are you living in an institution, such as an orphan asylum
or a home for children?
-
-
Yes
No
78-4©
Underline the Right Answer
1.
Have you a telephone in your home?
2. Is your home heated by a furnace in the basement ?
Yes
No
Yes
No
Yes
No
3. Do you have a bathroom that is used by your family
alone?
4.
Do you have a bank account in your own name?
Yes
No
5.
Did your father go to college?
Yes
No
6.
Did your mother go to college ?
Yes
No
7.
Did your father go to high school ?
Yes
No
8.
Did your mother go to high school ?
Y,es
No
Yes
No
9. Does your mother (or the lady of the home in which you
live) regularly attend any lecture courses of which you
know?
_
10.
Do you have your own room in which to study?
Yes
No
11.
Do you take private lessons in music?
Yes
No
12.
Do you take private lessons in dancing?
Yes
No
13.
Does your mother belong to any clubs or organizations
Yes
No
Yes
No
of which you know ?
If you know of any, write the name of one of them on
this line (
14.
)
Do you belong to any organizations or clubs where you
have to pay dues?
If you do, write the names of the organizations that you
belong to on these lines (
15.
Does your family attend concerts?
Never
16.
Occasionally
Where do you regularly spend your summers?
At Home
17.
Frequently
Away from Home
How often do you have dental work done ?
Never
When Needed
(Underline only one)
Once a Year
Oftener
18.
How many servants, such as a cook, a housekeeper, a chauffeur,
or a maid, do you have in your home?
None
19.
One Part Time
One or More All the Time
Does your family own an auto which is not a truck?
None
One
Two or More
If your family does own an auto, write the make of the auto on
this line (
20.
)
How many magazines are regularly taken in your home?
None
One
Two
Three or More
If any are taken, write the names of three of them—or as many
as are taken—on these lines (
21.
About how many books are in your home ?
(Be very careful with
this one. A row of books three feet long would not have more
than twenty-five books in it.)
None
22.
1 to 25
26 to 125
126 to 500
More
How many rooms does your family occupy?
2
3
4
5
6
7
8
9
10
11
12
More
10
11
12
More
How many persons occupy these rooms?
2
23.
3
4
5
6
7
8
9
Write your father's occupation on this line (
Does he own
Part
All
None
)
of his business? (Underline)
Does he have any title, such as president, manager, fore
man, boss, etc. ?
Yes
If he does have such a title, write it on this line (
No
)
How many persons work for him ? (Underline the right number)
None
Total Credits
1 to 5
5 to 10
-f- No. Answered
More than 10
= Score.
.6*
Illinois
Intelligence Scale
Published by the
PUBLIC SCHOOL PUBLISHING CO.
FORM 1
BLOOMINGTON, ILLINOIS
Printed in U. S. A.
ILLINOIS GENERAL INTELLIGENCE SCALE, FORM 1
(Directions Revised, 1926, by Guy M. Whipple
and Mrs. Helen D. Whipple)
Name
Boy or Girl.
Age last birthday....
Grade
Next birthday ■will b©.
Date
City-
School
..19..
.... State....
Teacher..
Write Pupil's Scores here.
General
Intelligence
1
.
General Directions
Score
Test
.
This booklet contains a number of tests.
will be shown them one at a time.
2
3
what to do with each one of them when
4
You
I will tell you
we come
to it, and I will tell you when to start and stop each
5
6
one.
7
right.
Total
Mental
Age
Work as fast as you can, but try to get them
Remember,
do
not turn any page until
I
tell
you to.
I.Q.
Directions for Test No.. 1—ANALOGIES
(a) sky—blue:.-grass—table
(b) fish—swims:: man—paper
(c)
day—night::white—red
green
time
clear
warm
big
girl
walks
black
pure
Do not turn the page until you are told to.
S-8p
Test No. 1—ANALOGIES
No. Bight..._
T
eat—bread:: drink—water
2
3
4
finger—hand::toe—bo& foot doll coat
shoe—foot:: hat—kitten bead knife penny
dress—women:: feathers—bird n,eck feet bill
5
dog—puppy:: cat—kitten
iron
dog
lead
.
stones
tiger
.
1
2
3
4
house
5
6
7
8
9
10
sit—chair:: sleep—book tree bed see
foot—man:: hoof—corn tree cow hoe
handle—hammer:: knob—key room shut door
chew—teeth::smell—sweet stink odor nose.
bird—song::man—speech woman boy work
6
7
8
9
10
11
sailor—navy:: soldier—gun
private army fight
legs—frog:: wings—eat
swim
bird
nest
man—home:: bird—fly insect worm nest
camp—safe:: battle—win
dangerous field fight
water—fish:: air—spark man blame sleep
11
12
13
14
15
16
17
18
19
20
pan—tin:: table—chair wood legs dishes
tiger—wild::cat—dog mouse tame pig
hospital—patient:: prison—cell criminal bar jail
floor—ceiling:: ground—earth sky hill grass
feather—float::rock—ages hill sink break...
16
17
18
19
20
21
airplane—air:: submarine—dive
cream
engine
frost
ship
..
water
22
cold—heat:: ice—steam
23
24
framework—house:: skeleton—bones
carpenter—house:: shoemaker—hatmaker
wax
25
pretty—ugly:: attract—fine
draw
26
hour—day::day—night
repel
12
13
14
15
21
refrigerator
skull
nice
grace
sho,e
week hour noon
clothes—man:: hair—horse comb beard hat.
28
29
darkness—stillness::light—moonlight sound sun
blow—anger:: caress—woman
kiss
child
love
30
imitate—copy::invent—study
machine
awl...
Mary had 5 apples and gave 2 to her brother.
24
26
27
window..
28
29
originate...
30
Directions for Test No. 2—ARITHMETIC PROBLEMS
many had she left?
23
25
27
Edison
22
body.....
How
Answer (
Do not tuna the page until you are told to.
Test No. 2—ARITHMETIC PROBLEMS
No. Eight
1
2
If one boy has 10 fingers, how many fingers have
six boys?
Answer (
There are 15 children in our class.
How many are girls?
5 of them are boys.
Answer (
)
Answer (
)
3
We learn 2 words a day in our class.
4
Jack is 42 inches tall and Fred is S inches taller.
5
How many do we learn in 8 days?
How tall is Fred?
«
)
Answer (
)
Mr. Gray sold ten bags of flour last Saturday at 2 dollars
a bag.
How many dollars did he get for the flour?.. .Answer (
)
6
Anna, Lizzie, Sarah, and Carrie shared 20 plums equally.
Answer (
)
7
After giving 9 cents for some candy, Helen had 2 dimes re
maining. How many cents did she have at first? Answer (
)
8
A baseball team took 12 players on a trip. The trip
cost the team $36. How much was that for each
How many plums did each get?
player?
9
Answer (
At the rate of a mile in two minutes, it takes 30 minutes
to run from one station to another. How many miles
apart are the stations?.
)
Answer (
)
10
Ned sold his rabbit for 30 cents. This was % of what he
paid. What did he pay for the rabbit?
Answer (
)
11
In a trolley car there were 29 people.
12
At the first stop 8
got off and five got on; at the second stop 13 got off and
ten got on. How many were in the car then?.. .Answer (
How many cakes at seven for 10 cents can I buy with
half a dollar?
)
Answer (
)
13
Albert had $1.50. He spent % of it for a bat
How much money had he left?
Answer (
)
14
Oil was bought for 10c a gallon and sold for 3c a quart.
15
Books were marked $1 each. Later the price was re
duced 30 cents. Find the cost of 5 books at the reduced
Find the gain on 32 gallons
Answer (
price
16
Answer (
A merchant buys y2 dozen handsaws at $16 a dozen.
How much must he receive for the lot in order to gain
fifty cents apiece?
Answer (
Directions for Test No. 3—8ENTENCK VOCABULARY
(a) Apples grow on vines
roots
grass
trees
(b) People can see through wood glass stone iron
(c) The ear is a part of the legs arms head feet
(d) Deserts are crossed by horses camels mules elephants
D© mot turn the page until you are told to
)
)
)
Test Mo. 3—SENTENCE VOCABULARY
No. Eight
1
A gown is a
2
Haste is
string
hurry
3
To tap is to
4
A dungeon is
5
Majesty refers to
6
7
animal
dress
little
sweet.
red
run
fall
open
knock
bright
dresses
Nerves are found in the
Plumbing is made of
plant.
smile.
heavy
kings
ground
rubber
dark.
countries
sky
glass
climates.
skin
fruit
hair.
pipes.
8
9
10
A man is afloat in a mine tower boat hospital.
Pork comes from pigs sheep cows calves.
A guitar is used to make toys glass music furniture.
11
A reception is a
12
To snip is to
show
cut
party
sew
13
Staves are used in
14
To regard is to
15
Skill is
16
Disproportionate amounts are
17
Mars is a
18
A selectman is a
19
Coinage refers to
20
A forfeit is a
21
To bewail is to
22
A fen is a
scales
barrels
painting
neglect
understand
23
To tolerate is to
24
To be sapient is to be
25
A milksop is a
26
The lotus is a
27
To drabble is to
anger
grief
country
seignior
rough
applaud
lament
waste
tax
multiply
flirt
lout
soil
nostrum
unequal
lawyer.
coincidence.
find.
beware
laugh.
pudding
record
wise
permit.
sardonic
prude
water-lily
excite
pigment
crowd
questionable.
mollycoddle.
bird.
twaddle.
28
Ochre is a
stone
monster.
29
Ambergris is used in
candles
fishing
medicine
30
A harpy is a
monster
litany
hobby
equal.
marsh.
savory
poison
fair
conspirator
currency
valley
fairy
consider.
mark.
officer
bonds
gift
golf.
expertness.
goddess
confederate
penalty
fish
sleep.
tie.
magnify
keenness
planet
game
paste
perfumery.
harpist.
Directions for Test No. 4—SUBSTITUTION
Do not turn the page until you are told to.
No. 4—SUBSTITUTION
No. Eight...
3
J
J
1
J
3 7
J
1
3 d
1 J
7
J J
1
J
7
7
J
1
3
3
1
7
1
J
3
J
7
d
d
d
d
7
1 J
d 1
1 l
3 d
73
J 7
i
7
d J 7
d 3 7
1 J d
J 3
J 7
d
1
1
d
d
d
7
d
7
7
3
3
7 7
3 1 7d 7
d
d
7 3 7 d
d
7 1
7
d 7
d 7 7 -^
7 7 d 3
3 d 1 d
d 7 7 7
3 3 7 7
d d 7
J
J
d 1
7 7
J d 3
d
1
3 7 d
3 3 1
j/
7
^
7
7
Directions for Test No. S—VERBAL INGENUITY
(a) see a I man «m
(b) knife chair the sharp ia
(c) John broken window trees has the.
Do S0£ turn th@ pag@ isntil you are told to.
Test No. 5—VERBAL INGENUITY
No. Bfigkt
1
the cat at see.
2
boy was sky the sick.
3
Bread sweep will the kitchen I.
4
are going yesterday to-morrow we.
5
me mine give my straw hat.
6
brown the horse come is.
7
my suit dollars wear twenty cost new.
8
know ice most boys how swim to.
9
their soldiers for fight gun country.
10
teacher me from gave a pencil my.
11
brother lamp is my than I older much.
12
dusty road the is hot and miles.
13
in the chalk he brightest is boy class our.
14
house hard to is climb very the hill.
15
broke his robin the flew little poor wing.
16
gave me candy brother my of knife a box.
17
the flood roaring valley came bridge the down.
18
the song birds flown during the to have south.
_„
19
boy gold watch brightest over get the will a.
20
I not Monday do to bag like go to school on.
21
watch summer the man stole is jail who the in.
22
old back only the chair legs has three.
23
told girl I I the to would her with home walk.
24
man whom the hat saw is you uncle my me with.
25
do not boy the I like who me school in sits desk behind.
Directions for Test No. 6—ARITHMETICAL INGENUITY
(a)
2
4
6
8
$
10
12
(b)
7
6
5
1
4
3
2
(c)
1
3
S
7
2
9
11
(d)
1
2
4
8
16
17
D© not torn the page until you are told to.
Test No. 6—ARITHMETICAL INGENUITY
No. Eight...
12
3
9
4
2
4
6
7
8
9
8
7
6
5
11
10
8
5
7
10
3
6
9
19
4
16
2
1
4
10
2
4
8
9
14
2
3
6
17
13
18
9
5
9
1
3
9
27
3
9
9
13
18
54
1
18
9
65
21
12
12
7
63
56
81
70
15
6
17
14
36
9
33
27
17
77
6
29
5
72
9
32
24
84
3
12
3
27
3
15
11
24
11
22
10
16
12
36
18
9
19
21
14
24
24
19
7
15
20
2
21
5
4
13
16
3
3
25
15
16
16
24
2
12
14
8
1
4
20
17
8
27
6
12
4
2
15
11
18
8
5
6
24
24
27
48
Directions for Test No. 7—SYNONYM-ANTONYM
(a)
good—bad
same—opposite
(b)
little—small
same—opposite
(c)
rich—poor ..
same—opposite
Do not turn the page until you are told to.
Test No. 7—SYNONYM-ANTONYM
No. Eight _
No. Wrong
Difference _
1
2
3
4
high—low
go—leave
large—great
bitter—sweet
5
begin—commence
6
7
8
9
accept—take
find—lose
expand—contract
shrill—sharp
same—opposite
same—-opposite
same—opposite
same—opposite
1
2
3
4
same—opposite
■. same—opposite
same—opposite
same—opposite
same—opposite
6
7
8
9
10
11
12
13
14
15
same—opposite
5
10
fault—virtue
11
12
13
14
15
command—obey
tease—plague
similar—different
delicate—tender
careless—anxious
same—opposite
same—opposite
same—opposite
same;—opposite
same—opposite
16
17
18
19
20
diligent—industrious
masculine—feminine
concede—deny
linger—loiter
accept—reject
same—opposite
same—opposite
same—opposite
same—opposite
same—opposite
16
17
18
19
20
21
22
23
24
25
vanity—conceit
appeal—beseech
docile—refractory
knave—villain
confer—grant
same—opposite
same—opposite
same—opposite
same—opposite
same—opposite
21
22
23
24
25
26
27
28
29
30
acquire—lose
compute—calculate
repress—restrain
depressed—elated
hoax—deception
same—opposite
same—opposite
same—opposite
same—opposite
same—opposite
26
27
28
29
30
31
32
33
34
35
reverence—veneration
same—opposite
vilify—praise
same—opposite
accumulate—dissipate
same—opposite
apathy—indifference
same—opposite
contradict—corroborate .... same—opposite
31
32
33
34
35
36
37
38
39
40
comprehensive—restricted . .same—opposite
assiduous—diligent
same—opposite
amenable—tractable
same—opposite
suavity—asperity
same—opposite
encomium—eulogy. .......same—opposite
36
37
38
39
40
TEST 1 — ARITHMETIC PROBLEMS*
Write the answers to these problems on the lines in the column to the right.
V
1. What will 4 eight-cent stamps and 1 three-cent stamp cost?
Answer.
(1)
• (3)
Answer.
.(2)
2. What does a pound of candy cost when you pay 10 cents for a quarter of a pound?
Answer.
How long is it from seven o'clock in the morning to two o'clock in the afternoon?
4.
How much longer is 100 minutes than an hour?
3.
The sum of two numbers is 40.
number?
One of the numbers is 14.
.(5)
What is the other
Answer.
• (4)
Answer.
6. What number minus 16 equals 20 ?
Answer.
20 = % X'
14.
Answer....
20 = how many times 12 ?
13.
Answer....
How much more is the sum of 3% and 414 than the sum of 2% and 3^?
12.
A man spent two-thirds of his money and had $8 left.
first?
11.
Answer.
At the rate of $2.25 per week how long will it take to save $90.00 ?
10.
Answer.
12 is % X t
9.
A man bought land for $400.
many acres were there ?
8.
12 is % of ?
7.
15. 8 is 1% X
16.
Answer.
He sold it for $445, gaining $15 an acre.
How
Answer.
How much had he at
Answer.
Answer
?
Answer
.(6)
• (7)
• (8)
• O)
• (10)
.(11)
.(12)
.(13)
.(14)
• (15)
.(17)
Answer....
.(16)
A watch was set correctly at noon Wednesday. At 6 P. M. on Thursday it was 15
seconds fast. At that rate how much will it gain in half an hour?
Answer....
17. Five-sixths equals how many thirds?
STOP HERE!
DO NOT TURN THE PAGE UNTIL TOLD TO DO SO!
'Used by permission of E. L. Thorndike.
0.0
Score
0
No. Right
1
1.5
2
3.0
3
4
4.5
5
6.0
7.5
10.5
9.0
7
6
13.5
12.0
9
8
10
15.0
11
16.5
13
12
18.0
19.5
14
21.0
15
22.5
16
24.0
17
25.5
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looqog
patpn^s noA* aA«q sjajsamas Aweva mojj"
JI9 P
1VXOI.
(f «WX 33S)
V WHOd
aioDg a3y
Ainiav Divnaaoiv jo isai 331
f
e
t •■ -a
z
1
V WHOJ
aaoos
99
TEST 2 — ANALOGIES
Find a number which has the same relation to the third number as the second number has to the first.
Look at the sample:
3 — 9 : : 5—
9 is 3 times 3.
swer 15.
30
15
20
23
....15....
By applying this same relation to the figure 5, you would multiply 3 times 5 and get the an
15 is one of the four possible answers, and "15" has been placed in the blank to the right.
There are
other relations between 3 and 9, but these relations could not be applied to the figure 5, as the answer thus pro
duced would not be one of the four possible answers.
8 —17:: 7—
9
12
16
20
....16....
In this case, the correct answer is 16, as 8 plus 9 is 17 and 7 plus 9 is 16.
Do the following exercises in the same way. Eemember that you are to put the correct number in the blank
space at the right of each exercise and that you must select one of the four possible answers given.
1.
3 —6 : : 6—
10
12
16
18
2.
2 — 8 : : 4—
12
20
8
16
3. 12 — 4 : : 6—
1
2
3
4
4. 15 — 30 :: 40 —
60
65
80
70
5.
4 —12:: 6—
18
16
2
24
6. 15 — 5 : : 30—
25
15
6
10
2000
1000
400
20,000
8. 25 —150 : : 8 —
32
50
48
40
9.
4 —16:: 5—
10
40
15
25
10.
y8 — %::9—
27
3
%
18
11-
% — %::*—
3
8
16
6
8
7
13
29
13. 17 —68 :: 15—
75
60
45
30
14
33 —38:: 47—
43
61
54
52
15.
3 — 27 : : 2—
12
24
16
8
7. 20 —200 :: 200—
12.
36 —6::49—
STOP HEBE!
No. Right
Score
DO NOT TURN THE PAGE UNTIL TOLD TO DO SO!
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.0
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
13.5
15.0
16.5
18.0
19.5
21.0
22.5
TEST 3 — NUMBER SERIES**
Look at each row of numbers below, and on the two dotted lines write the two numbers that should come
next, as in the samples.
Samples:
2
4
6
8
10
12
....14....
....16....
9
8
7
6
5
4
3....
2....
2
2
3
3
4
4
5....
5
1
7
2
7
3
7
4....
7....
Do the following exercises in the same way:
1.
3
4
5
6
7
8
2.
10
15
20
25
30
35
3.
5
9
13
17
21
25
4.
27
27
23
23
19
19
5.
1
2
4
8
16
32
6.
25
24
22
21
19
18
7.
2
3
5
8
12
17
8.
11
13
12
14
13
15
9.
29
28
26
23
19
14
10.
81
27
9
3
1
11.
16
12
15
11
14
12.
16
8
4
2
1
13.
15
16
14
17
13
18
14.
1
4
9
16
25
36
15.
3
4
6
9
13
18
16.
10
12
15
17
17.
4
8
22
44
20
10
13/l6
19.
14
18
13
17
12
16
21
19
18
17
15
14
20.
23
22
STOP HERB!
I'/ie
.
10
18.
1
.
13/
1%
DO NOT TURN THE PAGE UNTIL TOLD TO DO SO!
**TJsed by permission oi P. M. Symonds.
No. Right
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Score
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
TEST 4 — FORMULAS
In a formula, letters stand for numbers, and numbers may be substituted for them.
Substitute the given values in place of the letters, and then do what the arithmetical signs tell you to do.
Note: ab means a X &
VALUE FOUND
GIVEN VALUES
FORMULA
Sample 1:
o = 10,
A—
6=8
4 = 10X8 = 80 =40
Since A = 40, you put 40 on the blank line.
Do it now!
Sample 2:
C=
Since C = 9%, you put 9% on the blank line.
Do it now!
Fill in the value of each letter in the third column of the following table:
FORMULA
VALUES FOUND
GIVEN VALUES
4—
1
A = lw
Z—6
2
A = bh
6 = 11,
ft = 10
4 = 45,
w=5
I—
4.
4 = 63,
6=9
~h •' -
5. p = 2l-\-2w
J = 7,
3.
w;
to — 4
4—
w=10
6. P = 2l-\-2w
7.
bh
24
8,
9.
6 = 12,
A~Y
ft = 8
4 = 28,
6
4=
6=8
s= 7
4 = sa
4—
^ = 2%,
10.
r = 56
11. V = lhw
7=
7 = 140,
» = 7,
w=5
w=m
7 = 360,
ft = 8,
Z=9
w—
C
J? —121,
B — 6.
r—5
C—
12.
I——
hw
13.
14.
V
B+r
T = 2%,
15
ffa
16
i
j,2
ca
17.
4—
o = 6,
5=9
c2 —
c = 8,
6=6
a2 —
4srrs
18.
19.
r= 6
7—
3
0—
C = % (F —32)
20.
c
r —
-
STOP HERE!
No. Right
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Score
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
72
76
80
56
Score
TEST I
Form A
DOUGLASS STANDARD SURVEY TEST FOR ELEMENTARY ALGEBRA
HARL B. DOUGLASS, Ph.D.
Professor of Education and Director of the University High School, University of Oregon.
Published by C. A. Gregory Co., 345 Calhoun Street, Cincinnati, Ohio.
Name of Pupil
Name of School
Name of Teacher.
Period class meets
1.
Allow 40 minutes.
Date
Add: —5y
3y
2.
3.
1.
Answer:
2.
Answer:
3.
Answer:
4.
Answer:
5.
Answer:
6.
Answer:
ya and 3x2 — 2x — 3y2
Add: 4x
Subtract: —5ab2
3ab2
4.
5.
6.
Subtract: 9s2 + 4r2 — 3t from 6s2 — 7t + 4r2
Simplify by collecting terms:
10x2 + 5xy2 — 6mn2 — 13x2 — (4xy2 — 12x2)
Multiply:
4by
7.
8.
9.
10.
11.
12.
13.
14.
Multiply: 6ab (2 — 7ab2)
Multiply: a2 — ax + x2
Divide: 8y2
by
Divide: SaxSy3
by
7.
Answer:
8.
Answer:
9.
Answer:
10.
Answer:
11.
Answer:
12.
Answer:
13.
Answer:
14.
Answer:
a + x
—2y
by
4axy
Divide: 16a8 — 24a2 + 32a
Divide: a3 — 4a2 — 13a — 56
by
—8a
by
a— 7
Solve for a:
4a =—20
Solve for x:
4x — 6 = 50 — 4x
15.
Solve for x:
5x —x8 = x(2 — x) +3
15.
16.
Answer:
I paid $525 for an advertisement of 7 lines. The rate was 10 cents a line for the first five times the adver
tisement was printed, and 5 cents a line after that. How many times did the ad appear? Write the equa
tion but do not solve.
16.
17.
Using p for principal, r for rate of interest, and i for the Interest and t for time, express the following fact
in a formula: the interest may be found by multiplying the principal by the rate of interest by the time:
17.
18.
19.
Answer:
Answer:
If twice a certain number exceeds a third of that number by 10, what ia that number?
18.
Answer:
19.
Answer:
In the formula d = rt, solve for r when d - 120, and t = 20.
20.
21.
22.
23.
Factor: 6aa —20a
20.
Answer:
21.
Answer:
22.
Answer:
Factor: ma —n2
Factor: ma + 2ma + aa
A board 14 V£ feet long is to be cut into two pieces such that one will be one and one-half feet longer than
the other. How long will the short piece be?
23.
24.
The Lincoln Memorial at Washington stands on a rectangular platform whose perimeter ("distance around)
is 680 feet. The length of the platform exceeds the width by 70 feet. Find the length of the platform.
24.
25.
Answer:
Answer:
A grocer has some $.90 tea and some $.40 tea. How many; pounds of each must he take to form a mixture
of 100 pounds which he can sell for $.60 a pound? Write in the answer blank the number of pounds of 90
cent tea he will put in the mixture.
25.
Answer:
57
BIBLIOGRAPHY
ARTICLES
Ayers, C H. "Predicting Success in Algebra."
Vol. XXXIX (January), pp. 17-18.
School and Society,
Condit, Philip, "The Prediction of Scholastic Success by Means of Classi
fication Examinations."
(May, 1929), pp. 331-335.
Journal of gducatipnaj. Jleseajgh, Vol. XIX
Dickter, 1. R. "Predicting Algebraic Success."
(October, 1933), pp. 604-606.
School Review, Vol. XLI
George, J. S. "Nature of Difficulties Encountered in Reading Mathematics."
School. Review, Vol. XXXVII (March, 1929), pp. 217-226.
Judd, Charles, "A Psychological Explanation of Failures in High School
Mathematics."
Mathematics Teacher, Vol. XXV.
MacRae, Margaret and Uhl, Vfillis L. "Types of Errors and Remedial llork
in the Fundamental Processes of Algebra." Journal of Educational
Research, Vol. XXVI (September, 1932), pp. 12-21.
McCuen, Theron L. "Predicting Success in Algebra."
Research, Vol. XXI (January, 1930), pp. 72-74.
Journal of Educational
Pyle, W. H. ''The Relation of Ability to Achievement."
Vol. XXII (September 26, 1925), pp. 406-408.
§chool_and.£ocie^,
Shewraan, W. D. "A Study of Intelligence and Achievement of the June Grad
uating Class of the Grover Cleveland High School, St. Louis, Missouri."
School Review, Vol. XXXIV (1926), pp. 137-146.
Spencer, P. L. "Diagnosing Causes of Failure in Algebra."
Vol. XXXIV {May, 1926), pp. 372-376.
School Review,
Ward, Volney, "A Study to Determine Whether Beginning Algebra Should be
School and Society, Vol. XXXIII
Taught in Hinth or Tenth Grade."
(January 31, 1931), pp. 179-180.
BOOKS
Breslich, E. R.
The Administration of Mathematics in the Secondary Schools.
Chicago. 1933.
Breslich, E. R.
The Technique of_ T?&c>hlMj$eQon&&T%J£&JfrgJEa^JSJ.•
Chicago.
1931.
Chaddock, Robert E.
Morrison, H. C.
Principles and Methods, .of Statistics.
Chicago.
The Practice of Teaching in the Secondary Schools. Chicago.
1934.
Teigs, Ernest W.
1925.
Tests. and. Measujremejats_for__Tgache_rs.
Chicago. 1931.
58
MISCELLANEOUS MATERIAL
Burton, Enoch. An Experiment in Grouping Pup.ils According .to_£bilijty__in
Hifih School Algebra. Master's Thesis 1925. Sixth Yearbook of the
Department of Superintendence.
Manual. P.f_PirgotioS§. - Douglass S
Manual of Directions. - Illinois, general Intelliggnce_Soale.
Manual of Directions. - Lee Test of Algebraic Ability.
Manual of Directions - Sims Score
THESES
Johnson, Mabel Alice, Thei Predictive Value of Success in First Year Algebra,
Master's Thesis.
University of Colorado.
193E.
Kelson, Stanley,
A Study of Errors_
y
Master's Thesis.
Vance, Ira IT.
University of Vfisconsin. 1933.
An Analysis..pf_ Errors. in,.£2££2££^MsgIEJb-JZgSJlO
Two of the,.Hotg_'s^.Alggbra_Scales.
Teacher's College"! 1932.""*
Master's Thesis.
Indiana State