• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Table of Contents
 List of Tables
 List of Figures
 Quotation
 Acknowledgement
 Introduction
 Related literature
 Analysis of results
 Summary and conclusions
 Bibliography
 Appendix






Title: Investigation of the Competencies in Arithmetic Reasoning of the Students Enrolled in the High Schools for Negroes in Orange County, Florida
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 Material Information
Title: Investigation of the Competencies in Arithmetic Reasoning of the Students Enrolled in the High Schools for Negroes in Orange County, Florida
Physical Description: Book
Language: English
Creator: Felder, Annie Mitchell
Affiliation: Florida Agricultural and Mechanical College
Publisher: Florida Agricultural and Mechanical College
Publication Date: 1951
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Bibliographic ID: AM00000038
Volume ID: VID00001
Source Institution: Florida A&M University (FAMU)
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notis - ABV5545

Table of Contents
    Front Cover
        Page i
    Title Page
        Page ii
    Table of Contents
        Page iii
    List of Tables
        Page iv
    List of Figures
        Page v
    Quotation
        Page vi
    Acknowledgement
        Page vii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
    Related literature
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
    Analysis of results
        Page 18-a
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
    Summary and conclusions
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
    Bibliography
        Page 34
        Page 35
        Page 36
    Appendix
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
Full Text









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 16-17,

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

Fleming-Clarke, 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 sub-problems 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. 17-92 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 1952-53 as

a new building is under construction at this time.)
Last year a Parent-Occupational 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 t-own 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 home-ownership is noted by the up-keep of

the homes. Several families have lake front homes, large

orange groves and are spoken of as 'good-livers'. Negro law-

enforcement officers are on duty within the confines of the

town.

Both schools are continuously faced with drop-out and

absenteeism. Records of student-mortality 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 active--or observant--participant. Every student

is encouraged to engage in one or more of the planned activities.

The Hi-Y and the Tri-Hi-Y 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 16-17, 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 problem-solving.

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 example-type

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 Post-War 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 Mathematics--The
Slow-Moving Pupil," The Mathematics Teacher, 29, 1936,
pp. 23-26.

3 Lee A Arthur, "Diagnosis of Disabilities in Arith-
metic Essentials", The Mathematics Teacher, May 1950, .
pp. 197-202.

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 Year-Book of the National Council
of Teachers of Mathematics,TI -p. 240-241T







16


them, the number of correct answers showed a substantial

decrease. The experiment is of particular interest in

view of the improvement in critical thinking--or 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 forty-three 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 lists-niinF


6 Merle M. Ohlsen, "Control of Fundamental Skills and
Concepts by High School Students", The Mathematics Teacher,
December 1946, .pp. 365-371.

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 problem-solving and found

that in a mixed-fundamentals test many children made

errors because they did not read accurately the directions

indicating the nature of the processes. He conducted an

experiment with sixty-seven 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

Problem-Solving", offered the following six causes of

failure in problem-solving:

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 fundamentals--addition, subtraction, multipli-

cation and division--are 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 Problem-Solving
(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 and-Class 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 16-17, 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 Form-A.) 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 5-6 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 6-3 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 6-6 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 6-8.

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 6-6 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 16-17,

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

10.1-4.7; the arithmetic mean reasoning score for the

tenth grade was 8.5 and the standard deviation which indi-

cated a range of scores from 11.3-5.8; the arithmetic

mean for the eleventh grade was 9.0 and the standard

deviation was 2.90 which indicated a range of scores from

12-6.2; 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 12.9-5.7.

The reader then will note that the data revealed

that over ninety-eight 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 Home-life 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 MathemaT-cs, Boston:
Ginn and Company, 1926.


B. PERIODICALS

Arthur, Lee A., "Diagnosis of the Disabilities in
Arithmetic Essentials", The Mathematics Teacher,
63: 197-202, 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 Mathematics--The Slow-
Moving Pupil", The Mathematics Teacher, 29: 23-26,
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 Problem-Solving",
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 7-8 92 11 i4-I 169
2 8-3 99- .2z 14-9- 177
3 8-I 107 13 I5-4 184
4 9-7 15 14 i6-o I92
5 io-3 123 s5 i6-8 200
6 io-io 130 i6 17-4 208
7 I--6 138 i7 17-iI 215
8 12-2 146 I8 18-7 223
9 .12-o. I54.. .9 9-3 231
10 13-5 161 20 19-II 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, Yonkers-on-Hudson, 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: Doa-I
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, Yonkers-on-Hudson, 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 48-inch 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 7-8 92 II 14-1r 69
2 8-3 99-.- 2 i x 4-9- ... 177
3 8-1 I107 13 IS-4 184
4 9-7 115 14 i6-o 192
5 10-3 r. 23 ~ ."I i-8 oo
6 o-zo o 130 6 17-4 208
7 z-6 138 17 7 -izi 215
8 12-2 146 18 18-7 223
9 12-10o .154- 19 9-3 3
1o 13-5 i6i 20 19-lI 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 i-i = 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, Yonkers-on-Hudson, 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: Daa-I
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


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SRead ths page. Do what it tells you to do. -";
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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 .
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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. .


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ARITHMETIC
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ARrrHMETIC
QUOTIENT


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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 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? .............(


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 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 48-inch 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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) cents

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) miles



) years

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) pencils


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5

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7

8


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)feet i:



) days 12


) seats


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) pounds 14


) miles


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) cents 16



)inches 17

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) feet 18



)pints 19



) inches 20


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