AJ T INVESTIGATION OF TIHE COMLTEIJCS iN ARllTIJ1IC P&~AlSOITiiG
OF THE STUDENTS mROLI I TiD IT2TIGE. SCHOOLS FOR
fEGROES IN ORANGE COUNTY, FLORIDA
A Thesis
Presented to
the Graduate Committee of the
Florida Agricultural and Iechanical College
In Partial Fulfillment
of the requirements for the Degree
waster of Science
By
Annie Nitchell Felder
August 1951
AN INVESTIGATION OF THE COMPETENCE IN ARITHMETIC REASONING
OF THE STUDENTS ENROLLED IN THE HIGH SCHOOLS FOR
NEGROES IN ORANGE COUNTY, FLORIDA
A Thesis
Presented to
the Graduate Committee of the
Florida Agricultural and Mechanical College
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Approved
by
Annie Mitchell Felder
August 1951
222
R2003
TABIM OF CONfTJS
CHAPTER I PA g
I. IH*TRODUCTION . . . 1
The Problem . . 2
Statement of the Problem . 2
Background of the Study .. . 4
The Delimitations . 8
The Basic Assumptions . . 9
The Definitions of the Terms Used 10
The Source of Data . . 10
The Organization of the Study .. 11
II. eELATEI LITERATUREE . . . 12
III, ANALYSIS OF RSUTS .. 18
IV. SUMMARY AND CONCLWTSSION0S . 28
Recommendations . *. 51
BIBLIOGRAPEY . . . . . 34
APPENDIX ,,............... .. 37
LIST OF TABLES
TABLE PAGE
I. Total: Population of Jones High and
Hungerford Schools Who Participated in
the Arithmetic Reasoning Test, May 1617,
1951 By Grade and Sex . 19
II. Percent of the Total Possible Scores
Attempted and Solved by Grades . 23
III. A Frequency Distribution Showing the
Scores Made by Ninth, Tenth, Eleventh
and Twelfth Grades Students on the Otis
Arithmetic Reasoning Test, May 1951 . 24
LIST OF FIGURES
FIGURE PAGE
I. Arithmetic and Chronological Ages of
Students in the Grades Tested . 21
"The power to think is the
educational Kingdom of Heaven; if
we seek it persistently, other
things will be added to us".
B. H. Bode
ACKNOWLEDGMENT
Grateful acknowledgment is made to the following:
To the chairman of my Graduate Thesis Committee,
Dr. Leon Steele, the writer is deeply appreciative for his
continuous interest and encouragement. To Dr. Melvin 0.
Alston for his guidance and his many valuable suggestions
dealing with the writer's field of study and to Dr. R. S.
Cobb, Jr. for his constructive criticisms. To Mrs. Beatrice
FlemingClarke, who aided in the selection of the test used
in this investigation, and to Dr. Wade Ellis and Mr. W. N.
Smith for their immeasurable assistance.
To the Principals, Mr. C. W. Banks and Mr. F. M. Otey,
Jones High and Hungerford Schools, respectively, and the
teachers and students who were so generous in their coopera
tion.
To the Librarians at the Rollins College, Winter
Park, Florida and of the Florida Agricultural and
Mechanical College, Tallahassee, Florida.
To my family for words of encouragement and untiring
sacrifices.
CHAPTER I
INTRODUCTION
The demands of our modern living make mathematics one
of our basic essentials. There is scarcely any job which does
not require some knowledge and use of numbers. A worker is
usually interested in "How long will it take to finish this
job?" "How much will I receive an hour for the day or the
week?" "How far is it from my home?" "How much can I afford
for transportation, for lunch?" To answer these questions
the worker employs a system of quantitative thinking or as we
know it, arithmetic. The employer used mathematics to deter
mine how many workers can he afford; how much can he pay them;
what equipment he can afford; how much he can spend for ad
vertisement; how often he can offer the merchandise at a
reduced price; and how much profit he is desirous of making.
Our social and cultural activities are most arithmetical.
The homemakers as well as the career girls need to know how to
budget the income; whether to buy in quantity or for quality;
whether to buy life or endowment insurance; how to read time
schedules; how to shop and cook for two or twenty; how to
read the thermometer and the other guages used in and around
the house, car and office.
Every citizen should consider reading the newspaper as
one of the privileges of living in a democracy yet few read
' 0 0~ 9 Q
s~u D &
2
it with understanding. The lack of mathematical skills and
concepts hinder the interpretation of the amount of money
spent for war, the number of men who give their lives daily
because of war, the value of property destroyed because of
carelessness, the speed of the modern machines, the cost and
size of buildings, the baseball batting averages and all the
other interesting information found on the printed page.
Mathematics plays an important part in all of our daily
activities; therefore most educators desire that students ac
quire skills in handling numbers and use these skills as aids
to living a full and enriched life.
A. THE PROBLEM
An investigation of the competence in arithmetic
reasoning of the students enrolled in the high schools for
Negroes in Orange County, Florida.
Statement of ~e Problem. The purpose of this study
was to determine the extent to which:
1. Reasoning ability is evident in the twelfth grade
of students who have had a course in basic mathematics three
years previously in the ninth grade.
2. Reasoning ability is evident in the eleventh grade
of students who have had a course in basic mathematics two
years previously in the ninth grade.
SIL~~_ ...~:.i r. 1
3. Reasoning ability is evident in the tenth grade
of students who have had a course in basic mathematics one
year previously in the ninth grade.
It is believed that there may be a significant dif
ference among the arithmetic reasoning ability of students
of various grade categories from year to year after they
have had the basic course in mathematics.
This study is suggestive and not conclusive. The
writer is aware that much empirical data must be amassed
to make conclusive statements; nevertheless, the findings
in this study may point the way to improve teaching, and to
general trends found in the reasoning ability of students of
Jones High and Hungerford Schools.
In order that adequate light may be thrown on this
study the following subproblems seem significant:
1. What is the arithmetic reasoning ability of
students in grades nine, ten, eleven and twelve of Jones
High and Hungerford Schools?
2. Is the variability in reasoning ability greater
among students of the tenth? eleventh? twelfth grade than
among students of the ninth grade?
3. What is the relationship between chronological age
and reasoning ability for each grade (ninth, tenth, eleventh
and twelfth grades)?
4
4. What is the percent of variation in items com
pleted on the Otis Arithmetic Reasoning Test by students in
grades nine, ten, eleven and twelve?
5. How do students of the Jones High and the Hunger
ford Schools compare with the test norms on reasoning ability?
The Background of the Study. The students used in this
study are residents of Orange County, Florida and were enroll
ed in the Jones High and Hungerford Schools in Orlando and
Eatonville, respectively. Orlando, "The City Beautiful", is
located in the orange grove and lake section of central
Florida. It is the governmental seat of Orange County; a
hundred miles northeast of Tampa; a hundred and fifty miles
south of Jacksonville; and adjoins Winter Park. Several
smaller communities are dependent upon Orlando for medical,
social, religious, recreational and educational services.
Thru train and bus services from the north are provided
by the Atlantic Coast Line Railway and Florida Motor Lines,
Airway transportation by Eastern and National Airlines;
Federal Highways No. 1792 and State Highways provide highway
connections to all parts of Florida. Transit bus service is
maintained between Orlando and the other Orange County Com
munities.
Five years after Spain ceded Florida to the Unites States
(1824), the Legislative Council divided South Florida into two
counties  Mosquito and Monroe. In 1845, according to History
5
of Florida Past and Present, the name Mosquito was changed
to Orange County because of "its unusual fertility and
potential productiveness." This large tract of land has
since then been subdivided into several counties.
There are several stories of how Orlando got its name.
The most probable is that it was named for Orlando Reeves,
a soldier killed by a poisoned arrow in a running fight be
tween the Indian and white defenders in 1857.
From the descriptive slogan "The Beautiful City" and
a casual glance at the business and the outlying residential
districts, one would be led to believe that the beauty is in
evidence in all parts of the city. The colored section of
town has its share of freeholders and renters. The houses
for rent usually are small, close together and in most cases
painted alike. The quarters are known by such names as "Buck
Alley" and "Black Bottom."
The growth of the city in recent years has made the
restricted zone for Negroes very small, therefore, additional
schools, housing developments and government projects are
being constructed to relieve the crowded conditions.
There are several beautiful churches in the community.
These institutions provide spiritual guidance as well as
some recreational and educational activities for the citizens.
A Negro Chamber of Commerce, Welfare Planning Board,
several social and civic clubs, lodges, veterans organizations,
6
a base ball club, and faternal organizations contribute to
the city's recreational program.
The Jones High School is the educational institution
for Negro students of high school age. Not only does it serve
the students of Orlando but those from Winter Garden, Taft,
Oakland and Windermere. For the last three school terms, due
to increased school population, there has been a triple
session schedule to care for the 850 students in grades 8 to
12. (This situation will be quite different in 195253 as
a new building is under construction at this time.)
Last year a ParentOccupational Survey was made and less
than ten percent of the Negro students reported their parents
or one of their parents as professional or skilled workers.
The chief sources of income reported were: grove work, domestic
work, and miscellaneous jobs created by influx of tourists.
The parents reported as professional or skilled workers were:
welfare workers, ministers, morticians, tailors, printers,
recreational supervisors, beauticians, barbers, doctors,
dentists, nurses, and teachers.
In 1885, Lake Maitland was incorporated as a town with
J. C. Eaton as mayor for whom the colored section of Maitland
Eatonville was named four years later.
Eatonville, a town owned and controlled by colored men,
was incorporated and immediately they started a school and
named it the Robert Hungerford School, a memorial to a young
physician who was stricken with typhoid fever while attending
7
a Negro youth. In 1899 Edward C. Hungerford of Chester,
Conn., father of Robert Hungerford, gave the Hungerford
School 160 acres of land which enabled the school to offer
a wide variety of agricultural courses.
Hungerford was, until two years ago, a private board
ing high school supported by donations and supervised by the
Presbyterian Church at Winter Park. Day students were allowed
to attend and the County School Board supplied apportionate
teachers. Now the County School Board has leased the physi
cal plant. High school students from Apopka, Forest City,
Zellwood, Tangerine, Plymouth and Winter Park are transported
daily.
Eatonville is not a self sustaining community. Except
for sundries and basic foods everything must be purchased in
Orlando or Winter Park.
Evidence of homeownership is noted by the upkeep of
the homes. Several families have lake front homes, large
orange groves and are spoken of as 'goodlivers'. Negro law
enforcement officers are on duty within the confines of the
town.
Both schools are continuously faced with dropout and
absenteeism. Records of studentmortality reveal that ohly
fifty percent of the students who have been registered in
the seventh grade graduate from high school.
8
Jones High and Hungerford Schools aim to teach the
students to live in our democratic society. On each campus
there are many interest or hobby clubs related to the courses
offered. Students campaign and vote by secret ballot for the
Student Council officers; elect by popular vote the young lady
to wear the title of "Miss Jones High" or "Miss Hungerford",
and choose class representatives to serve on the Patrol Group
either as an activeor observantparticipant. Every student
is encouraged to engage in one or more of the planned activities.
The HiY and the TriHiY are two of the strongest organizations
on the Jones High campus. The clubs have as their motto the
development of the 'hands, hearts and the heads' of the members.
These activities, along with content courses such as
Agriculture, Industrial Arts, Commercial Subjects, Homemaking,
and Science courses are means by which the students are made
aware of the need of the knowledge of numerical computation
in their daily, social, economic and cultural activities.
Delimitations. In order for a study to have signifi
cance it must carry with it certain delimitations; therefore,
this study is delimited as follows:
1. To a study of arithmetic reasoning ability of 518
students enrolled in the Jones and Hungerford High Schools,
Orange County, Florida.
2. To a study of arithmetic reasoning as measured by
Otis Arithmetic Reasoning Test on May 1617, 1951.
9
3. To a study of arithmetic reasoning ability in
grades nine, ten, eleven and twelve.
4. To a study of arithmetic reasoning ability as meas
ured by Otis Arithmetic Reasoning Test ,of the ninth grade as
compared with the tenth grade; ninth grade as compared with
the eleventh grade; ninth grade as compared with the twelfth
grade.
5. To a study of arithmetic reasoning ability as com
pared with the chronological age of the ninth, tenth, eleventh
and twelfth grades students.
Inherent or Basic Assumptions. In this study the follow
ing assumptions are basic:
1. In order to improve any learning situation one must
have adequate objective knowledge of the present competence
and skills in the learning situation to be investigated, there
fore, there is a genuine need for determining the competence
of students enrolled in the Jones and the Hungerford Schools
in arithmetic reasoning.
2. Tests are the most objective instruments available
to the educators for determining competence of reasoning
abilities. The best tests are high in validity and reliability.
From the validity and reliability of the Otis Arithmetic
Reasoning Test it is safe to assume that the Otis Arithmetic
Reasoning Test is a valid index or measure of pupils reasoning
ability in arithmetic.
10
3. Tests are objective criteria for determining com
petences. An investigation of the arithmetic reasoning ability
will aid in improving the instructional techniques in mathe
matics.
Definitions of Terms
Investigation a careful examination, inquiry, or
search.
Competence fitness; capability; ability.
Arithmetic Reasoning 
Arithmetic the science of numbers.
Reasoning the process of reaching conclusions by
careful and connected thinking; arithmetic reasoning then
is the mental processes of a child solving a verbal
problem.
Source oq Data. The Otis Reasoning Test: Form A (Test
5 of the Otis Group Intelligence Scale: Advanced Examination)
by Arthur S. Otis (formerly Development Specialist with Advisory
Board, General Staff, United States War Department) was used to
secure data for this investigation.
The Otis Arithmetic Reasoning Test was selected for the
following reasons, (1) the problems were verbal rather than
computational, (2) the problems on the test were similar to
the problems taught in the ninth grades at the Jones High and
Hungerford Schools and, (3) the Otis Arithmetic Reasoning Test
is on the approved list of the National Council of Teachers of
11
Mathematics as a standard test for testing pupils performance
in problemsolving.
Organization f the Study. This study consists of four
chapters. The introductory chapter will consist of the state
ment of the problem; the background of the study; the delimita
tions; the basic assumptions, the definitions of terms used;
source of data and the organization of the study. In Chapter
II, a review of the literature and findings of related studies
are presented and discussed; in Chapter III, the analysis of
the results. In Chapter IV the summary and conclusion, follow
ed by the bibliography and appendix.
CHAPTER II
RELATED LITERATURE IN THE FIELD OF STUDY
Every teacher of mathematics has at some time or
another found that students can compute exampletype
exercise yet when the same numbers are given in a verbal
problem students can not respond. The truth of this
statement has been substantiated by several investigators.
SURVEY AND FINDINGS
The .Jint Commission Report of 1940 and the reports
of the Commission on PostWar Plari of the National Council
of Teachers of Mathematics made a study of the importance
of mathematics in general education and reached the con
clusion that mathematics is important because the average
citizen of today needs considerable mathematical knowledge
in the activities and experiences of everyday life.1
These studies were made because of the large number
of those who enter the services or civilian line of work
could not apply the arithmetic fundamentals which they
were supposed to have learned in the elementary courses.
1 Paul Monroe, editor, A Cyclopedia of Education
(New York: The Macmillan Company, 1960).
13
The Joint Commision made provisions for individual
differences, also curriculum differences among students.
There must be correlation, integration and continuity
in all the grades so that the student will not only learn
how to compute accurately and quickly but will learn how
to interpret the verbal problems.
Mallory's2 investigation revealed that ninth grade
students with IQ's of 109 or under did better on arithme
tic skills than concepts. In verbal problems students
did not understand or comprehend the problem, could not
determine what was given now could they translate the
written information into arithmetic symbolism,
Arthur3 in his study, Diagnosis of Disabilities in
Arithmetic Essentials concluded (1) that many high school
pupils do not have adequate understanding of nor ability
to work many of the problems which are considered essen
tial by the report "Essential Mathematics for Minimum Army
Needs", and pupils are weakest in the interpretation of
verbal problem situations.
Monroe4 concluded that a large percent of the pupils
2 Virgil S. Mallory, "Activity in MathematicsThe
SlowMoving Pupil," The Mathematics Teacher, 29, 1936,
pp. 2326.
3 Lee A Arthur, "Diagnosis of Disabilities in Arith
metic Essentials", The Mathematics Teacher, May 1950, .
pp. 197202.
4 W. S. Monroe, "How Pupils Solve Problems in Arith
metic" (Bureau of Educational Research Bulletin, Vol. 26,
No. 23, Urbana, Illinois: University of Illinois, 1929).
14
did not reason in attempting to solve verbal problems.
Instead, many of them appear to perform almost
random calculations upon the numbers given. When
they do solve a problem correctly, the response
seems to be determined largely by habit. If the
problem is stated in the terminology with which
they are familiar and if there are no irrelevant
data, their response is likely to be correct. On
the other hand, if the problem is expressed in
unfamiliar terminology or if it is a new one
relatively few pupils appear to reason. They
either do not attempt to solve it or else give an
incorrect solution.
Bradford5 reported a study in which a group of
students were given a series of arithmetic problems
impossible of solution. The purpose of the experiment
was to determine how many of the children would show
genuine critical thought in discovering that the problems
could not be solved, and how many of them would simply go
through the formal manipulations without recognizing that
the problems were impossible,
On the basis of his experiment, Bradford concluded
that many right answers are obtained under ordinary class
room conditions not as the result of genuine critical
thought, but as the resultc suggestion. When the pupils
were told that some of the problems were impossible of
solution and were required to make critical reactions to
5 E J. G. Bradford, "Suggestions, Reasoning, and
Arithmetic," Forum of Education, III., February 1925 (Ex
cerpts found in Third YearBook of the National Council
of Teachers of Mathematics,TI p. 240241T
16
them, the number of correct answers showed a substantial
decrease. The experiment is of particular interest in
view of the improvement in critical thinkingor reasoning
which is commonly supposed to result from solving problems
in arithmetic.
Ohlsen6 found in his study of students enrolled in
grades ten, eleven and twelve of fortythree Iowa High
Schools that the control of Fundamental Mathematical Skills
and Concepts by high school students was low as compared
with the mathematical concepts and skills defined by the
Joint Commission of the Mathematical Association of America
and the National Council of Teachers of Mathematics.
An analysis of the errors made by the students indicat
ed that the most common errors occurred as follows: (1)
Through a lack of understanding of a correct method for
solving the problem; (2) in confusing related mathematical
terms; (3) in selecting the incorrect data for the specified
solution, and (4) errors in computation.
Other investigators cite reading as one of the
principal reasons for errors in problem solving. Osburn7
in his study Reading Difficulties in Arithmetic listsniinF
6 Merle M. Ohlsen, "Control of Fundamental Skills and
Concepts by High School Students", The Mathematics Teacher,
December 1946, .pp. 365371.
7 W. L. Osburn, Reading Difficulties in Arithmetic,
(State Department oe Public Instruction, Madison, Wisconsin,
1925).
16
causes of misunderstandings:
1. Lack of vocabulary.
2. Failure to read or see all the elements in the
problem.
3. Failure to resist the disturbance caused by
preconceived ideas.
4. Inability to read between the lines.
5. Failure to understand fundamental relations,
particularly those of the inverse type.
6. Failure to make a quick change of mental act.
7. Failure to generalize or transfer meanings.
8. Failure to interpret cues correctly.
9. Response to irrelevant elements.
Lessenger8 made a study of the effect of difficul
ties in reading as related to problemsolving and found
that in a mixedfundamentals test many children made
errors because they did not read accurately the directions
indicating the nature of the processes. He conducted an
experiment with sixtyseven children and concluded that
when special training in reading arithmetic problems was
given, the amount of error due to faulty reading habits
was greatly reduced.
8 W. E. Lessenger, Reading Difficulties in Arithme
tical Computation, (Journal of Educational ReseFceh,
April, 1925, Bloomington, Illinois.)
11
Stevenson,9 in his investigation, "Difficulties in
ProblemSolving", offered the following six causes of
failure in problemsolving:
1. Physical defects.
2. Lack of mentality.
3. Lack of skills in fundamentals.
4. Inability to read, which of necessity affects
the ability to read arithmetic problems.
5. Lack of general andtechnical vocabulary.
6. Lack of proper methods or techniques for
attacking problems.
SUMMARY
The researchers have concluded that the mastery
of the four fundamentalsaddition, subtraction, multipli
cation and divisionare not the ultimate end of arithme
tical instruction. Our democratic way of life demands
the ability to interpret, comprehend, analyze and solve
the quantitative concrete situations that are ever present.
Thepe abilities aid in developing an appreciation of the
cultural value of mathematics, and of its usefulness as a
9 P. R. Stevenson, Difficulties in ProblemSolving
(Journal of Educational Research, Vol. XT, February 1925.
Public School Publishing Company, Bloomington, Illinois.)
mode of thinking or reasoning and as a means of interpret
ing and appreciating the world about us.
CHAPTER III
ANALYSIS OF RESULTS
Procedure. The procedure involved in this study
included the following: (1) The selection of the Otis
Arithmetic Reasoning Test to measure the arithmetic
reasoning of the students enrolled in the Hones High and
the Hungerford Schools; (2) the selection of the high
school students enrolled in the Jones High and the Hunger
ford Schools; (3) the administration of the test under
necessary.testing conditions; (4) the acquisition and
tabulation of the chronological ages of the students; and
(5) the scoring and interpretation of the arithmetic reason
ing test results and the chronological age.
The writer made interpolation of the data shown in
Table II, of the Manuel of Directions.1 This table shows
that there was a consistent increase of two score points
from grade four to grade eight; therefore, the same number
of score points were consistently increased by the investi
gator for the ninth, tenth, eleventh and twelfth grades.
1 See Appendix: Otis Arithmetic Reasoning Test, for
Directions, Key andClass Record. Table II.
From the findings in this study Table I shows the
enrollment of Jones High and the Hungerford Schools by
grade and sex.
TABLE I
TOTAL POPULATION OF JONES HIGH AND HUNGERFORD SCHOOLS
WHO PARTICIPATED IN THE ARITHMETIC REASONING TEST,
MAY 1617, 1951 BY GRADE AND SEX
School Grades Total.
Ninth Tenth Eleventh Twelfth
Boy rs Boys Girls Boys Girls Boys Girls
Joaes 47 69 25 50 33 39 33 44 340
Hunger
ford 25 30 22 32 21 16 12 20 178
Both
Schools 78 99 47 82 54 55 45 64 518
From this table it is shown that involved in this
study were 518 students of whbhli 340 were enrolled at
Jones High School and 178 at the Hungerford School. The
distribution of these students was as follows: From the
Jones High School there were 116 (boys 47, girls 69)
students enrolled in the ninth grade; 75 (boys 25, girls
50) students enrolled in the tenth grade; 72 (boys 33,
girls 39) in the eleventh grade; and 77 (boys 33, girls
44) in the twelfth grade, or a total enrollment from the
Jones High School 340 (boys 138, girls 202). From the
Hungerford School there were 55 (boys 25, girls 30)
students enrolled in the ninth grade; 54 (boys 22, girls
32) students enrolled in the tenth grade; 37 (boys 21,
girls 16) in the eleventh grade; and 32 (boys 12, girls
20) students in the twelfth grade, or a total enrollment
from the Hungerford School of 178 (boys 80, girls 98).
The total population in this study was 171 (boys 72, girls
99) ninth graders; 129 (boys 47, girls 82) tenth graders;
109 (boys 54, girls 55) eleventh graders and 109 (boys 45,
girls 64) twelfth graders.
The test was given by class sections. Each group
was allowed six minutes after completing the personal data
and reading the sample problem found on the front page.
The directions for scoring tests were carefully
followed and the scores recorded on the individual test
papers. The arithmetic age was found on the Directions,
Key, and Class Record opposite the score made. A score of
1 equals 7 years 8 months or 92 months and each additional
score merits an increase of 8 months. The students
chDonological age was then expressed in months and divided
into the arithmetic age. The results were expressed without
a decimal point and called the arithmetic quotient.
Figure 1, shows the relationship of the arithmetic
age and the chronological age of the students of both
schools by class groups.
FIGURE I
ARITHMETIC AND CHRONOLOGICAL AGES
OF STUDENTS IN THE GRADES TESTED
19I Arithmetic Age
SChronological Age
181
i ii
The histogram gives the average arithmetic and
chronological ages which existed among the students of the
Jones High and the Hungerford schools in May, 1951. From
this figure it is evident that the ninth grade class had
an average chronological age of 15 years 7 months (this
number represents an accurate average chronological age of
the students taking the Otis Arithmetic Reasoning Test)
22
and an average arithmetic age of 11 years 7 months (this
number represents an interpretation from the norms given
on the Otis Arithmetic Reasoning Test FormA.) The tenth
grade average chronological age was 16 years 6 months and
an average arithmetic age of 12 years 6 months. The
eleventh grade average chronological age was 17 years 1
month and the average arithmetic age was 12 years 9 months.
The twelfth grade average chronological age was 18 years
2 months and the average arithmetic age was 13 years.
From this figure it is evident that there was a
difference of 4 years 2 months in the average chronological
age (actual ages of students) and the average arithmetic
age (reasoning age scored by students on the Otis Arithmetic
Reasoning Test) of students in the ninth grade; a difference
of 4 years in the average chronological age and the average
arithmetic age of the students in the tenth grade; a
difference of 4 years 4 months in the average chronological
age and the average arithmetic age of the eleventh grade
and a difference of 5 years 2 months in the average chrono
logical age and the average arithmetic age of the twelfth
grade.
The test results were then analyzed in respect to
the area of arithmetic reasoning. The 171 ninth graders
attempted 2,205 or 64.5%, and solved 1,278 or 37.4% of the
3,420 possible answers; the 129 tenth graders attempted
1,824 or 70.7% and solved 913 or 35.4% of the 2,580 possible
answers; the 109 eleventh graders attempted 1,524 or 69.9%
and solved 987 or 45.3% of the 2,180 possible answers; and
the 109 twelfth graders attempted 1,603 or 73.5% and
solved 1,010 or 46.3% of the 2,180 possible answers.
Table II shows the possible number of scores, the
number of problems attempted, the number of problems
solved, the percent of the total possible problems
attempted, and the percent of the total possible problems
solved.
TABLE II
PERCENT OF TOTAL POSSIBLE SCORES ATTEMPTED AND
SOLVED BY GRADES
Total Number Number Percent Percent
Grades Possible Attempt Solved Attempt Solved
Scores ed ed
Ninth 3420 2205 1278 64.5 37.4
Tenth 2580 1824 913 70.7 35.4
Eleventh 2180 1524 987 69.9 45.3
Twelfth 2180 1603 1010 73.6 46 .
Total 10360 7156 4188 69 40
Table III, gives a frequency distribution of the
scores made by students on the Otis Arithmetic Test. The
scores were grouped into a frequency distribution with
class intervals of 2, and the average or arithmetic mean
computed by the use of a guessed mean.
TABLE III
A FREQUENCY DISTRIBUTION SHOWING THE SCORES MADE BY NINTH,
TENTH, ELEVENTH, AND TWELFTH GRADES STUDENTS ON THE
OTIS ARITHMETIC REASONING TEST IN MAY 1951
Class Interval Ninth Tenth Eleventh Twelfth Total
18 19 1 1
16 17 1 1 7 9
14 15 6 3 10 7 26
12 13 9 12 9 17 47
10 11 18 27 26 20 91
8 9 37 34 26 20 117
6 7 64 36 26 19 145
4 5 28 13 11 16 68
2 3 9 2 3 14
Total 171 1299 10 109 518
Mean 7.41 8.52 9.05 9.29 8.57
Standard
Deviation 2.70 2.76 2.90 3.62 3.04
25
From this table it is seem that in the ninth grade
there were 6 students with the arithmetic age of tenth
grade students; 6 students with arithmetic age of ninth
grade; 18 students with arithmetic age of eighth grade;
37 students with arithmetic age of seventh grade; 64
students with arithmetic age of sixth grade; 28 students
with arithmetic age of fifth grade and 9 students with
arithmetic age of fourth grade.
The mean reasoning ability score of the ninth grade
was 7.4 or comparable to 56 grade.
In Column II under tenth grade: 1 student had an
arithmetic age of twelfth grade; 1 student with an arith
metic age of eleventh grade, 3 students with an arithmetic
age of tenth grade; 12 students with an arithmetic age of
eighth grade; 34 students with an arithmetic age of
seventh grade; 36 students with arithmetic age of sixth
grade; 13 students with an arithmetic age of fifth and
2 students with an arithmetic age of fourth grade.
The mean reasoning ability score of the tenth grade
was 9.5 or equivalent to 63 grade.
The eleventh grade class had the following distribu
tion as to arithmetic age: 1 student had the arithmetic
age of eleventh grade; 10 students, the arithmetic age of
tenth grade; 9 students with arithmetic age of ninth grade;
26 students with arithmetic age of eighth grade; 26 students
had arithmetic age of seventh grade; 26 students had the
26
arithmetic age of sixth grade; and 11 students had the
arithmetic age of fifth grade.
The mean reasoning ability score of the eleventh
grade was 9.0 or equivalent to 66 grade.
The students of the twelfth grade had the following
arithmetic ages in May, 1951, as interpreted by the
directions on the Otis Arithmetic Reasoning Test. There
were 7 students with the arithmetic ages of eleventh
grade; 7 with arithmetic age of tenth grade; 17 with arith
metic age of ninth grade; 20 with arithmetic age of eighth
grade; 20 with arithmetic age of seventh grade 19 with
arithmetic age of sixth grade; 16 with arithmetic age of
fifth grade and 3 with arithmetic age of fourth grade.
The mean reasoning ability score for the twelfth
grade was 9.3 or 68.
The frequency distribution table reveals that 1
student among the 518 students of Jones High School and
Hungerford School tested in May, 1951 had the arithmetic
age of twelfth grade; 9 had the arithmetic age of eleventh
grade; 26 had the arithmetic age of tenth grade; 47 had
the arithmetic age of ninth grade; 91 had the arithmetic
age of eighth grade; 117 had the arithmetic age of seventh
grade; 145 had the arithmetic age of sixth grade; 68 had
the arithmetic age of fifth grade and 14 had the arithmetic
age of fourth grade.
27
The mean reasoning ability score of the total students
tested was 8.6 or 66 grade.
SUMMARY
There were 518 students involved in this study of
the competence in arithmetic reasoning in the high schools
for Negroes, Orange County, Florida, 1951. Of this number,
340 were enrolled in the Jones High School, Orlando, Florida,
and 178 were enrolled in the Hungerford School, Eatonville,
Florida.
The following facts were found:
1. That the average arithmetic age is far below
the average chronological age in all grades.
2. That the mean reasoning ability score for all
students tested was that of an advanced sixth grade pupil.
3. That the reasoning ability of the tenth grade
students was 6 months above that of the ninth grade stu
dents; that the eleventh grade students were only 9 months
or 1 school year above the ninth grade students, and only
3 months above the tenth grade students; and the twelfth
grade students were 11 months above the ninth grade students.
CHAPTER IV
SUMMARY AND CONCLUSIONS
The purpose of this study was to
extent to which:
1. Reasoning ability is evident
grade of students who have had a course
tics three years previously.
S. Reasoning ability is evident
grade of students who have had a course
matics two years previously.
3. Reasoning ability is evident
grade of students who have had a course
tics one year previously.
determine the
in the twelfth
in basic mathema
in the eleventh
in basic mathe
in the tenth
in basic mathema
The reader will note that this study was delimited
to the students of grades nine ten, eleven, and twelve
of Jones High and the Hungerford Schools, Orange County,
Florida; to a study of arithmetic reasoning ability as
measured by the Otis Arithmetic Reasoning Test on May 1617,
1951; to a study of arithmetic reasoning ability as compared
with the chronological age of the ninth, tenth, eleventh,
and twelfth grades' students.
This study records results obtained from the Otis
Arithmetic Reasoning Test, Form A, which was selected to
measure the arithmetic reasoning of 340 students enrolled
29&
at the Hungerford School, Eatonville, Florida.
Test results were analyzed (1) in respect to the
relationship between the average arithmetic age and the
average chronological age of the classes tested; (2) the
percent of variation in items completed by students of the
grades tested; (3) the arithmetic reasoning ability of the
students tested; (4) the variability in reasoning ability
among students of grades nine, ten, eleven and twelve;
and (5) the comparison of the students of Jones High and
the Hungerford Schools with the test norms on reasoning
ability.
Figure I, page 21, indicated that the arithmetic
quotient of the ninth grade was 73; of the tenth grade
was 76; of the eleventh grade was 75 and of the twelfth
grade was 71.
It has been pointed out that the total number of
problems solved were 4188 out of the 7154 problems
attempted. The number solved was 58.5% of the number
attempted. The total number solved is significant only
when compared with the total possible number of answers
which was 10,360. The students in grade nine solved
57.9% of the number attempted; the students in grade ten
solved 50% of the number attempted; the students in grade
eleven solved 64.8% of the problems attempted and the
students in grade twelve solved 62.4% of the number of
30
problems attempted.
In Table III, page 24, the scores were compiled
according to frequency and the central tendency of measure,
on the arithmetic mean, and the degree of variability or
the standard deviation were computed. The mean reasoning
ability score for the ninth grade was 7.4 and the standard
deviation was 2.70 which indicated a range of scores from
10.14.7; the arithmetic mean reasoning score for the
tenth grade was 8.5 and the standard deviation which indi
cated a range of scores from 11.35.8; the arithmetic
mean for the eleventh grade was 9.0 and the standard
deviation was 2.90 which indicated a range of scores from
126.2; and the arithmetic mean of the twelfth grade was
9.3 and the standard deviation was 3.62 which indicated
a range of scores from 12.95.7.
The reader then will note that the data revealed
that over ninetyeight percent of the boys and girls
tested at the Jones High School and Hungerford School
made arithmetic scores which were below the norms for
grades in which they were enrolled. These facts reveal
very clearly that there may be grave shortcomings in the
instructional program in both of these schools as far as
arithmetic reasoning is concerned for it is assumed that
these boys and girls tested have normal intelligence.
31
RECOMMENDATIONS
The writer makes the following recommendations on
the basis of the frequency of errors found in the study.
I. That remedial instruction be placed into opera
tion with emphasis on problems involving arithmetic
reasoning.
The fact that 7156 problems were attempted and only
4188 were correct indicated that the students have not
formed the habit of thinking or reasoning while attempting
to solve verbal problems.
II. That special remedial instruction be placed upon
technical vocabulary.
On the basis of some of the answers given to the
attempted problem the investigator assumes that the student
did not have the necessary understanding of the printed
word. With each unit of instruction the teacher should
present new words with definitions, diagrams or illustra
tions of the uses in every day life; then let the students
relate experienced connected with these words. In this
investigation it was noted that unless the problems were
stated in familiar terminology pupils were not able to
determine the operations to be performed. The writer
suggests that teachers of mathematics should insist upon
the students using the correct terms on all occasions when
52
they are discussing examples or problem situations. A
continued use of words is the only sure means of estab
lishing those words or terms as a part of the students'
vocabulary.
Arithmetic should be thop'1gt of as a subject,
interesting, challenging, practical and applicable to
everybody's life, therefore, the writer recommends:
III. That teachers have pupils realize the impor
tance of accuracy in numbers.
In the Florida Times Union dated July 8, 1950 an
article titled "Mathematics is Applied toPupils' Lives"
related that "the new State Department of Education Bulletin
for Teachers is based chiefly on life in Florida as the
pupil is living it during his school days, and as he
grows up and takes a job". With this in mind, the inves
tigator offers the fourth recommendations:
IV. That teachers constantly use real life situation
problems.
In our school's immediate vicinity the fruit pickers
congregate to board the trucks that take them to their jobs.
There are several types of real life situation problems
which may be introduced dealing with distance, rate of
speed, time required to complete tasks; hours worked;
amount of fruit picked; wages earned.
A unit dealing with Homelife situations can be
centered around the average incomes of the orange
pickers with the text book information as a guide. The
students can make budgets and suggestions as to how these
workers can get the most out of his income. Take a survey
of the class to see if any one has purchased anything on
installment plan or if they, or their parents, have bought
articles for cash which could have been bought on install
ment plan. If so, continue the suggested unit to show
advantages and disadvantages of buying on installment plan.
These and other suggestions are invaluable in presenting
the need, which is the inVentive for developing independent
thought on the part of the student.
V. That a valid and reliable testing program be
given regularly in order to determine levels of achievement.
VI. That other studies of mathematical competence
be made so that the information may be used to improve
instruction in mathematics.
I
34
BIBLIOGRAPHY
BIBLIOGRAPHY
A. BOOKS
Butler, Charles H., and Lynwood F. Wren, *The Teaching
of Secondary Mathematics, New York: iMGrafw Hil
Company, 1941.
Monroe, Paul, Cyclopedia of Education, New York: The
Macmillan Company, Publishers, 1950.
Reeve, William D., A Diagnostic Study of the Teaching
Problems in Hig School MathemaTcs, Boston:
Ginn and Company, 1926.
B. PERIODICALS
Arthur, Lee A., "Diagnosis of the Disabilities in
Arithmetic Essentials", The Mathematics Teacher,
63: 197202, May 1950.
Bradford, E. J., "Suggestions, Reasoning, and Arithmetic",
Forum of Education, III, February 1925.
Lessenger, W. E., "Reading Difficulties in Arithmetical
Computation", Journal of Educational Research,
April 1925.
Mallory, Virgil S., "Activity in MathematicsThe Slow
Moving Pupil", The Mathematics Teacher, 29: 2326,
1936.
Monroe, W. S., How Pupils Solve Problems in Arithmetic,
Bureau of Educational Research Bulletin, 26: No.23.
Ohlson, Merle M., Control of Fundamental Skills and
Concepts by High School Students, The Mathematics
Teacher December 1946.
Osburn, W. J., "Reading Difficulties in Arithmetics",
State Department of Education, (Madison, Wisconsin)
1925.
36
Stevenson, P. R., "Difficulties in ProblemSolving",
Journal of Educational Research, Vol. II,
February, 1925.
C. NEWSPAPER
"Mathematics is Applied to Pupils' Lives", News item.
The Florida Times Union, July 8, 1950.
APPENDIX
APPENDIX
I. Formulas Used
AA = AQ
CA
S= AM / fd i
N
 = i I
Sfd2 ( xfd 2
N N
II. Symbols Used:
A.
A.
A.
C.
A.
M.
Q,
A.
d.
f.
i.
n.
Arithmetic age
Assumed Mean
Arithmetic Quotient
Chronological Age
deviation of each score from the arithmetic
mean
frequency of a class interval
class interval
total number of students in a sample
sum of, summation
sigma, standard deviation
Arithmetic mean
4m
so
F
~~
I
OTIS ARITHMETIC REASONING TEST
(Test S of Otis Group Intelligence Scale: Advanced Examination)
By ARTHUR S. OTIS, PH.D.
Formerly Development Specialist with Advisory Board, General Staff, United States War Department
DIRECTIONS, KEY, AND CLASS RECORD
DIRECTIONS FOR ADMINISTERING
Form A of the Otis Arithmetic Reasoning Test
is Test 5 of the Otis Group Intelligence Scale:
Advanced Examination, Form A. Form B of
this test is Test 5 of the Otis Group Intelligence
Scale: Advanced Examination, Form B.
To administer either Form A or Form B,
say: "We are going to give you a test in arith
metic reasoning. As soon as you receive a
paper write your name, age, etc., in the blank
spaces. Do not turn the p.per over."
Have the papers passed, one to each pupil,
right side up (side bearing name of test). See
that the monitors understand which side up.
When all have filled the blanks, say: Look at
the directions below where you have been writ
ing. They say: On the other side of this sheet
there are 20 problems in arithmetic. You are
to write the answer to each problem in the
blank space after the problem as shown in the
following sample. Sample problem: If a boy
had 6 marbles but lost 1 marble, how many
marbles did he have left? Answer: (5) marbles.
Notice where the answer is put.' (Pause.) You
will be given 6 minutes for the test. See how
many problems you can get right in that time.
You may use the margin of the paper to figure on
if you need to. Do not stop to erase your figur
ing. Wait until you are told to turn the paper.'
Now turn over and begin." Note the ex
act time to the second. At the end of exactly 6
minutes, say: Stop; turn your papers over."
Have the monitors collect the papers.
DIRECTIONS FOR SCORING
To score the test, place the appropriate key
(see over) beside the column of responses and
put a check mark after each correct answer or
a cross after each incorrect or omitted answer,
or both checks and crosses. The score in the
test is the number of correct answers.
INTERPRETATION OF RESULTS
It is becoming a prevalent custom to convert
scores into ages in order to find educational
quotients as well ab intelligence quotients. In
order to find the arithmetic age corresponding
to any score, consult Table i.
TABLE 1
ARITHMETIC AGES
Arithmetic Age Arithmetic Age
Score Score
Yrs. & Mos. Months.. Yrs. & Mos. Months
1 78 92 11 i4I 169
2 83 99 .2z 149 177
3 8I 107 13 I54 184
4 97 15 14 i6o I92
5 io3 123 s5 i68 200
6 ioio 130 i6 174 208
7 I6 138 i7 17iI 215
8 122 146 I8 187 223
9 .12o. I54.. .9 93 231
10 135 161 20 19II 239
The correspondence
hypothetical only. In
15 the scores are not
at the two extremes is
fact, above the age of
norms. The norm for
adults is only about 15 points. These upper
arithmetic ages are fictitious and are used for
convenience in finding arithmetic quotients.,
To find an arithmetic quotient, divide the
pupil's arithmetic age by his chronological age
when both are expressed in months. Thus a
pupil of 12 years making a score of io has an
arithmetic quotient of +i = 112 (decimal point
dropped).
Grade norms are shown in Table 2.
TABLE 2
GRADE NORMS (January)
Grade... 4 5 6 7 8
Norm... 4 6 8 1o I
Published by World Book Company, YonkersonHudson, New York, and z126 Prairie Avenue, Chicago
Copyright i9z8 by Arthur S. Otis. Copyright 199, I922, by World Book Company. Copyright in Great Britain. Al rights resrmd. OAt: DoaI
Printd In U.S.A.
CLASS RECORD
Grade ................ School ...................................................
Date............................... Examiner...........................................
Name Age Score Arith. Arith.
Yrs. Mos. Age Quot.
Class Medians
SP o
09v v1
I fI
ott II
5 01
6 tr
01
9F
or I
gE
KEY
Form A
I 15
2 48
3 15
4 8
5 io
6 200
7 o1
8 8o
9 20
10 3
II 250
12 25
13 2
14 450
15 45
16 5c
17 44
18 450
19 15
20 12
'FRI 7W
: .
*T ws
OTI A I41'IC
(Tets sd! Ot aw itdllgaM 8lea: Aasremdhamiation)
By ARTHUR S.' Ons, Pa.D.
Formerly Development Specialist with Advisory Board, General Staff, United States War Department
TEST: FORM A
Read this page.
Do what it tells you to do.
Do not open this paper, or turn it over, until you are told to do so. Fill these blanks, giving your
name, age, birthday, etc. Write plainly.
Name. ...................................................Age last birthday......years
First name, initial, and last name
Birthday ................ Teacher ........................... Date. ........... 19...
Month Day
Grade............. School. .' ........ ................. .......City..................
On the other side of this sheet there are 20 problems in arithmetic. You are to write the answer
to each problem in the blank space after the problem as shown in the following sample.,
Sample problem:
If a boy had 6 marbles but lost I marble, how many marbles did
he have left?................................................ ..Answer: ( 5 ) marbles
Notice where the answer is put. You will be given 6 minutes for the test. See how many prob
lems you can get right in that time. You may use the margin of the paper to figure on if you need
to. Wait until you are told to turn the paper.
Publ(sbe by World Book Company, YonkersonHudson, New York, and 2a26 Prairie Avenue, Chicm
Copyright, 192s, by World Book Company. Copyright in Great Britain. Al rights resawL. ozr: : &26
P M r O.Lr .. A.
TMJest is copyrighted. The repro auction of any art of it by mimeograph, hectograqh, or i any mher
rw iaplaether the reproductions are sSd or ftniuhed free for ise, is a vWolation of the copyright lawr,
J *. .t
'~gSI,:~r'~~~~
SCORE
ARITHMETIC
AGE
ARITHMETIC
QUOTIENT
; '. I
I
qo
"; ... 1
' '
;.'
a'
a**' aS
*4
,f
.a
i '*
r d
i l .,,
,'',4 ,'
*".
 ',,
Otis Arith. Reas.: A
Arithmetic
DIRECTIONS. Place the answer to each problem in the parenthesis after
Do any figuring you wish on the margin of the page.
i. If a boy had io cents and earned 5 cents, how much money did lie have
then?....................... ............................. ..... (
2. At 4 cents each, how much will 12 pencils cost? ....................(
3. If a man had $25 and spent $1o, how much money did he have left?... (
4. At 6 cents each, how many pencils can be bought for 48 cents? .......(
5. A boy spent 20 cents and then earned 30 cents. How much more
money did he have than at first? ................................. (
6. How far can a train go in 5 hours at the rate of 40 miles per hour? ....(
7. How long will it take a glacier to move iooo feet at the rate of ioo feet
a year ? . .............. (
a year?............... ;......... .................. ............(
8. If 2 yards of cloth cost 20 cents, what will io yards cost? ...........(
9. If 2 pencils cost 5 cents, how many pencils can be bought for 50 cents?(
xo. If a man walks east from his home 7 blocks and then walks west 4 blocks,
how far is he from his home ? .................................. (
ii. If a boy can run at the rate of 5 feet in I of a second, how far can he
run in io seconds? ............ .. ........... (
12. A ship has provisions enough to last a crew of 20 men 50 days. How
long would they last a crew of 40 men?. .......................(
13. One schoolroom has 7 rows of seats with 8 seats in each row, and
another schoolroom has 6 rows of seats with 9 seats in each row. How
many more seats does one room have than the other ?........... ...(
14. If io boxes full of oranges weigh 500 pounds, and each box when
empty weighs 5 pounds, what do all the oranges weigh? ..............(
z5. Town X is 30 miles north of Town Y. Town Y is 15 miles north of
Town Z. How far is Town Z from Town X? .......................(
I6. If 3^ yards of cloth.cost 70 cents, what will 2 yards cost? ...........(
17. If a strip of cloth 36 inches long will shrink to 33 inches when washed,
how long will a 48inch strip be after shrinking? .....................(
18. If Frank can ride a bicycle 300 feet while George runs 200 feet, how
far can Frank ride while George runs 300 oofeet? .....................(
19. A hotel serves a mixture of 3 parts cream and 2 parts milk. How
many pints of cream will it take to make 25 pints'of the mixture? .....(
20. If a wire 20 inches long is to be cut so that one piece is 2 as long as the
other piece, how long must the longer piece be? ....................(
the problem.
) cents I
) cents 2
) dollars 3
) pencils 4
) cents 5
)miles 6
) years 7
) cents 8
) pencils 9
) blocks io
) feet ii
) days 12
) seats 13
) pounds 14
) miles
) cents
) inches 17
) feet 18
) pints 19
) inches 20
TEST
OTIS ARITHMETIC REASONING TEST
(Test 5 of Otis Group Intelligence Scale: Advanced Examination)
By ARTHUR S. OTIS, PH.D.
Formerly Development Specialist with Advisory Board, General Staff, United States War Department
DIRECTIONS, KEY, AND CLASS RECORD
DIRECTIONS FOR ADMINISTERING
Form A of the Otis Arithmetic Reasoning Test
is Test 5 of the Otis Group Intelligence Scale:
Advanced Examination, Form A. Form B of
this test is Test 5 of the Otis Group Intelligence
Scale: Advanced Examination, Form B.
To administer either Form A or Form B,
say: We are going to give you a test in arith
metic reasoning. As soon as you receive a
paper write your name, age, etc., in the blank
spaces. Do not turn the piper over."
Have the papers passed, one to each pupil,
right side up (side bearing name of test). See
that the monitors understand which side up.
When all have filled the blanks, say: Look at
the directions below where you have been writ
ing. They say: On the other side of this sheet
there are 20 problems in arithmetic. You are
to write the answer to each problem in the
blank space after the problem as shown in the
following sample. Sample problem: If a boy
had 6 marbles but lost 1 marble, how many
marbles did he have left? Answer: (5) marbles.
Notice where the answer is put.' (Pause.) You
will be given 6 minutes for the test. See how
many problems you can get right in that time.
You may use the margin of the paper to figure on
if you need to. Do not stop to erase your figur
ing. Wait until you are told to turn the paper.'
"Now turn over and begin." Note the ex
act time to the second. At the end of exactly 6
minutes, say: Stop; turn your papers over."
Have the monitors collect the papers.
DIRECTIONS FOR SCORING
To score the test, place the appropriate key
(see over) beside the column of responses and
put a check mark after each correct answer or
a cross after each incorrect or omitted answer,
or both checks and crosses. The score in the
test is the number of correct answers.
INTERPRETATION OF RESULTS
It is becoming a prevalent custom to convert
scores .into ages in order to find educational
quotients as well as intelligence quotients. In
order to find the arithmetic age corresponding
to any score, consult Table i.
TABLE 1
ARrITHTIC AGES
Arithmetic Age Arithmetic Age
Score Score
Yrs. & Mos. Months.. Yrs. & Mas. Months
I 78 92 II 141r 69
2 83 99. 2 i x 49 ... 177
3 81 I107 13 IS4 184
4 97 115 14 i6o 192
5 103 r. 23 ~ ."I i8 oo
6 ozo o 130 6 174 208
7 z6 138 17 7 izi 215
8 122 146 18 187 223
9 1210o .154 19 93 3
1o 135 i6i 20 19lI 239
The correspondence at the two extremes is
hypothetical only. In fact, above the age of
15 the scores are not norms. The norm for
adults is only about i5 points. These upper
arithmetic ages are fictitious and are used for
convenience in finding arithmetic quotients..
To find an arithmetic quotient, divide the
pupil's arithmetic age by his chronological age
when both are expressed in months. Thus a
pupil of 12 years making a score of io has an
arithmetic quotient of ii = 112 (decimal point
dropped).
Grade norms are shown in Table 2.
TABLE 2
GRADE NORMS (Jan uary)
Grade... 4 5 6 7 8
Norm  12
Norm... 4 6 8 io i
Published by World Book Company, YonkersonHudson, New York, and 2126 Prairie Avenue, Chicago
Copyright 19x8 by Arthur S. Otis. Copyright 19zg, 922, by World Book Company. Copyright in Great Britain. AU rigts rured. oA r: DaaI
Printed In U.S.A.
__
I ,L

CLASS RECORD
Grade.. ..........School......... ... .. ................................... ....
Date................... .Examiner ........... ...
Age Arith. Arith.
Name ...:Score
Yrs. Mos. Age Quot.
Y s M s. ,, .. . ,_ ..... .. ...
L J ,, ,.!i. J
p
i I
r ,
u~ r;r jl
Ctss Medians
I
13 ;In r i
~~ ,ii: 1
C\ '' ~~s: ~
i..' i I
., .~ .
,...,. ,,
I / K
FORMI
i!Prim
~. MOST
* >
\f ~> t
2'." A.
SRead ths page. Do what it tells you to do. ";
S* .. '. ,
.":: .^ '.t
Do not open this paper, or turn it over, until you are told.to do so. Fill these blanks, giving ytiI
name, age, birthday, etc. Write plainly.
Name. .................................. ....Age last birthday. .. .. .fed
First name, initial, apd last name .
I 'A, t
Birthday. ............... Teacher ...... .. ..... .......... Date. .... ..
Month Day
Grade .. .. .. .School. .... ... .. .. ....... ... .. City. ...... ..
.... ,,. ,,*t /\*
On the other side of this sheet there are 2o problems in arithmetic. You are to write the answer i
to each problem in the blank space after the problem as shown in the following sample. '
Sampl problem:
If a boy had 6 marbles but lost i marble, how many marbles did
he have left?.. .... .............. .. .. .. ....Answer: ( 5 ) marble.
Notice where the answer is put. You will be given 6 minutes for the test. See how many prop
lems you can get right in that time. You may use the Alargin of the paper to figure on if you need
to. Wait until you are told to turn the paper. .
.'~*(
i ~,t
~~Ir~'
rr
r 1
~
Q
'1 d
~h7i
'
I~ .,
I 'C .i
~~~i
!1
I~ ~f~ij~
I~ .
:
.
r. ~.li :
,.
:i
,,,
.I .
':I~
.
'tl
:"~~
.I /.
:
:If,rsr
L r
'r CI*~C':r:3:
d c.~
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b~
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i
i' .. i.
c owiy'Xb* 6,qnZdoiNWTi
WPM 404 4,P"LIJPI P~~)
lie
4.4j 4.I .I .~ ,
I* I'
let
*? ~ >' '4A "
4 4.f
SCORE.
ARITHMETIC
AGE
ARrrHMETIC
QUOTIENT
4''..
1.
'P1
1.*~ '14.i~
~ '
~C Cc r~'L li C1
b:;i .. ~e r i~F ;4'5CI~i~:f~I ;IY_ 
: 1. .i. .
":,,iR, '"' *a
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B~::

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`iC 4
tit L(rl:P :lil~j' ~I I 1.
I .. ~ ''C
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.,P a r. L
'' 5 r ~ \
I
v '~'
i. .'~ ; ~ I~ r, c.
rI It v P' ~LC:~ r
1 C i. r I,
~1 '
I ~' '. ..~. "'
~''p
u\ '
.
I "' t' .~ ii ;;
..
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:. ... ? P .jl
~rsr. .~
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~1 ~t~'. ~ ,
I
h' ... ~j
i.
r'a ,
'""'

i
If`
TEST
Arithmetic
DIRECTIONS. Place the answer to each problem in the parenthesis after
Do any figuring you wish on the margin of the page.
Otis Arith. Reas.: A
the problem.
I. If a boy had io cents and earned 5 cents, how much money did l4e have
then ?................... ................................ ..... (
2. At 4 cents each, how much will 12 pencils cost? .................... (
3. If a man had $25 and spent $1o, how much money did he have left?... (
4. At 6 cents each, how many pencils can be bought for 48 cents? .......(
5. A boy spent 20 cents and then earned 30 cents. How much more
money did he have than'at first ?.... ..............................(
6. How far can a train go in 5 hours at the rate of 40 miles per hour? .... (
S 7. How long will it take a glacier to move iooo feet at the rate of ioo feet
a year?, .. ............. .. . .. . (
8. If 2j yards of cloth cost 20 cents, what will io yards cost? ..........(
S 9. If 2 pencils cost 5 cents, how many pencils can be bought for 50 cents?(
1i Io If a man walks east from his home 7 blocks and then walks west 4 blocks,
how far is he from his home? .................. ...... .........(
ii. If a boy can run at the rate of 5 feet in of a second, how far can he
run in ,i seconds?.... ......... ................ ........... ..(
12. A ship has provisions enough to last a crew of 2o men 50 days. How
long would they last a crew of 40 men?7........ ...................(
13. One schoolroom has 7 rows of seats with 8 seats ineach row, and
another schoolroom has 6 rows of seats with 9 seats in each row. How
many, more seats does one room have than the other?............(
14. If io boxes full of oranges weigh' 500 pounds, and each box when
empty weighs 5 pounds, what do all the oranges weigh? .............(
15. Town X is 30 miles north of Town Y. Town Y is 15 miles north of
Town Z. How far is Town Z from Town X? ................. .....(
S16. If 3 yards of clothcost 70 cents, what will 2 yards cost? ...........(
17. If a strip of cloth 36 inches long will shrink to 33 inches when washed,
how long will a 48inch strip be after shrinking? ..................... (
8. If Frank can ride a bicycle 300 feet while George runs 200 feet, how
far can Frank ride while George runs 300 feet? 7... ..............(
19. A hotel serves, a mixture of 3 parts cream and 2 parts milk. How
many pints of cream will it take to make 25 pints'of the mixture? .....(
20. If a wire 20 inches long is to be cut so that one piece is % as long as the
other piece, how long must the longer piece be ? ...... ............ (
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