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An analysis of the integration of arithmetic and algebra in eighth grade mathematics textbooks
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An analysis of the integration of arithmetic and algebra in eighth grade mathematics textbooks
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Content
AN ANALYSIS OF THE INTEGRATION OF ARITHMETIC AND ALGY~RA
IN EIGHTH GRADE MATHEMATICS TEXTBOOKS
A Thesis
Presented to
the Faculty of the School of Education
The University of Southern California
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Education
by
David Randolph Foster
June 1951
This thesis, written under the dire tion of the
Chairman of the candidate's Guidance Committee
and approved by all members of tlze Committee,
has been presented to and accepted by the Faculty
of the School of Education of the University of
Southern California in partial fulfillrnent of tlze
requirements for the degree of Master of Science
in Education.
June, 1951
Date ..................................................... .
Dean
Guidance Committee
TABLE OF CONTENTS
CHAPTER
I. THE PROBLEM AND ITS IMPORTANCE.
The problem
• • • • • • • • •
• • • • • • •
• • • • • • •
Statement of the problem
Importance of the problem
Eighth grade algebra •.
• • • • • • • • •
• • • • • • • •
• • • • • • • • •
Definitions of terms used
• • • • • • • • •
PAGE
1
1
1
1
4
6
General mathematics . • • . . • . . . . • 6
Ar thmetic al ebraic in nature . . . . . . 7
Organization of chapters. . . . . . . . . . 7
II. REVIEW OF EIGHTH GRADE MATHEMATICS AND RELATED
III.
LITERATURE . •
• • • • • • • • • • • • • • •
Trends in eighth grade mathematics .
• • •
Objectives of the Nineteenth Century ...
8
8
Objectives of the Twentieth Century . • . 10
Specific topics in eighth grade mathe-
matics
• • • • • • • • • • • • • • • • •
Review of related literature ..... .
• •
Literature on eighth grade mathematics
textbooks .•..•......••
• •
Summary
PROCEDURE
• •
• • •
• • • •
• • • •
Selection of textbooks
• • • • • • • • • •
• • • • • • • • • • •
• • • • • • • • • • •
13
17
18
20
22
22
111
CHAPTER PAGE
IV.
Current texts . . . . . . . . . . . . . . 22
Arithmetic material algebraic in nature . • 22
Quantity of material . . . . . . . . . . . 23
Distribution of material .
• •
The unit introducing algebra ..
• • • • • •
• • • • • •
24
25
Importance of the unit . . . . . . . . • . 25
Relationship of the unit to arithmetic . . 26
Quantity of fundamental processes
ARITHMETIC MATERIAL ALGEBRAIC IN NATURE
• • • •
• • •
Quantity of material
• • • • • • • • • • • •
Formula
Graph
• • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • • • •
Ratio-proportion •.
• • • • • • • • • • •
Square root •.........
• • • • •
Algebra ...•........
• • • • •
Amount of total material •.....•
Limitation in the present data .••.
• •
• •
Distribution of algebraic material •
• • • •
Textbook A ...•....••..
• • • •
Textbook B
• • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • •
• • • • • • • • • • • • • • •
Textbook C •
Textbook D.
Textbook E.
• • • • • • • • • • • • • • •
27
29
29
29
29
32
32
32
33
33
35
35
37
37
37
41
CHAPTER
Total distribution ••
• • • • • • • • • •
Summary . • . . • • . . . . • . . . • •
The unit introducing algebra .•..••..
iv
PAGE
41
4
44
Importance of algebra unit . . . . • • . . 44
Summary . . . . . • . • • . . . • . . . . 46
Relationship of equations to arithmetic
•
Summary . . . . . . . . . • • . • . . . •
Quantity of fundamental processes •...
Summary . . • . . . . . • • . •
• • • • •
v. SUMMARY AND CONCLUSIONS ..•.•.
• • • • •
Summary . • . • . . . • . • . • .
• • • • •
Quantity of algebraic material •
• • • • •
Distribution of algebraic material .
• • •
Unit introducing algebra
• • • • • • • • •
Conclusions
• • • • • • • • • • • • • • • •
BIBLIOGRAPHY.
APPENDIX . . .
• • • • •
• • • • •
• • • •
• • • •
• • • • • • • • • •
• • • • • • • • • •
47
49
49
5
52
52
~2
53
53
54
55
65
TABLE
I.
LIST OF TABLES
Pages of Material Algebraic in Nature ..
• •
II. The Percentage Distribution of Arithmetic
Material Algebraic in Nature
• • • • • • •
III. Rank Order Distribution of Textbooks
• • • •
IV. Chapter Distribution of Algebraic Material in
Textbook A
• • • • • • • • • • • • • • • •
v. Chapter Distribution of Algebraic Material in
Textbook B
• • • • • • • • • • • • • • • •
VI. Chapter Distribution of Algebraic Material in
Textbook C
• • • • • • • • • • • • • • • •
VII. Chapter Distribution of Algebraic Material in
Textbook D
• • • • •
• • • • • • • • • • •
VIII. Chapter Distribution of Algebraic Material in
Textbook E
• • • • • • • • • • • • • • • •
IX. Percentage of Chapter Distribution in Text-
books • • •
• • • • • • • • • • • • • • • •
x. Importance of Algebra Unit
• • • • • • • • •
XI. Relationship of Equations to Arithmetic •
• •
XII. Quantity of Fundamental Processes •.•
• • •
PAGE
30
32
34
36
38
39
40
42
43
45
48
50
CHAPTER I
THE PROBLEM AND ITS IMPORTANCE
As a teaching device in most subjects the textbook
occupies a unique place and performs a unique function.
It is an extremely important feature of the American educa
tional plan because it largely determines the content and
organization of the courses of study in many subjects.
This is particularly true of the courses in mathematics.
"It is scarcely an exaggeration to say that in most cases
the mathematics book is the course of study."
1
I. THE PROBLEM
Statement of the problem. The purpose of this study
was to determine to what extent eighth grade mathematic
textbooks are preparing and motivating students for the
study of algebra by the integration of arithmetic and alge
bra.
Importance of the eroblem. The world we live in
today has reached a point where it is essentially
1 c. H. Butler and F. L. Wren, The Teachin! of
Secondary Mathematics (New York: McGraw-Hill Boo -
Company, Inc., l941J, p. 87.
mathematical in nature. The use of mathematics in
industry, research and everyday living is steadily grow
ing. In spite of this growing demand for trained mathe
maticians the enrollment figures in mathematics have
continued to decrease in proportion to the increase in
school attendance. Jessen,2 in 1934, reported that only
slightly more than 50 per cent of the students in the
last four years of high school were enrolled in some form
of mathematics. In 1948, a survey of 336 schools, pub
lished in the Mathematics Teacher,3 showed only 44 per
cent of the students in the last four years of high
school enrolled in mathematics. The enrollment figures
for 1948 included all courses in mathematics and the
greater percentage of these students were enrolled in
general mathematics.
2
World War II brought a charge of failure against
the public schools to meet their mathematical responsi
bilities. Educators have long been aware that students
have been completing high school without the basic knowl
edge of the fundamentals of mathematics. In 1931, Raleigh
Carl A. Jessen, "Offerings and Registration in
High-School Subjects," Bulletin 6, Office of Education
(Washington: Government Printing Office, 1938), p. 5.
3 Raleigh Schorling, "What's Going On In Your
School?" Mathematics Teacher, 41:147-53, April, 1948.
3
Schorling gave a test consisting of one hundred simple
arithmetic tasks to 3,545 children in grades five through
twelve. In summarizing the inefficiency of these children
in his study, Schorling states:
1. Although all these processes have been
taught in most schools, the degree of mastery
at the beginning of the eighth grade is very
low and there is little subsequent increase in
ability in computation.
2. Even in the senior high school, mastery
is very low. In the tenth grade there are only
51 things out of 100 to which half of the child
ren responded correctly; half of the children
know half of the tasks. In the eleventh grade
we get 60 tests at the 50 per cent level and in
the twelfth grade we get 73 things. That is, at
the end of the senior high school half of these
children know three-fourths of the material.
3. One of the crucial problems of the senior
high school, though little is said about it, ap
pears to be that of computation.4
The results of a contemporaneous but independent
study reported by Bridges5 strongly supported the findings
of Schorling. A more recent study also substantiates
these f1nd1ngs.6
4 Raleigh Schorling, "The Need for Being Definite
with Respect to Achievement Standards," Mathematics
Teacher, 24:311.
5 w. A. Bridges, "Mathematical Ability of Pupils
Entering Junior High School," (unpublished Master' thesi,
George Peabody College for Teachers, Nashville, 1931).
6
B. A. Sueltz and J. w. Bendeck, "Need for Extend
ing Arithmetic Learnings," Mathematics Teacher, 43:69-73,
February, 1950.
4
The curriculum of the eighth grade mathematics
course has possibly contributed to these failures and with
the security of the nation at stake, any investigation that
might contribute to the improvement of the mathematics
curriculum seems justifiable.
Eighth grade algebra. The eighth grade mathematics
curriculum presents a logical starting point for increasing
enrollments in mathematics, particularly algebra, and there
by motivating better mathematical learning by students.
The contribution that the study of algebra can make in im
proving the arithmetic deficiencies of students has been
discussed in several recent articles. J. A. Nyberg,
presents illustrations of this fact in an article and
states:
When evaluating algebraic expressions more
drill with fractions and decimals is possible.
And these are only a few of the instances for
arithmetic review.7
Schaef8 presents four areas in which a generalized
treatment of arithmetic might result in a better achieve
ment in algebra. He relates the possibility of going from
7 J. A. Nyberg, "Notes from a Mathematics Class
Room," School, Science and Mathematics, 48:54-6, January,
1948.
8 w. L. Schaer, "Arithmetic Taught as a Basis for
Later Mathematics," School, Science and Mathematics, 46:
413-23, May, 1946.
the simple to the complex in these general areas: (1)
the nature of numbers; (2) the nature of operations; (3)
thP, nature of relationship; and (4) the language used in
mathematics.
5
For proponents of algebra there is a fertile field
for the improvement of arithmetic by the study of algebra.
Braveman9 pre ente an interesting study on the improvement
made in arithmetic by one hundred seventy students after
a year's study of algebra. His study revealed a median
increase o the group from 30.9 to 37.0 with a standard
deviation of 6.4 for the group before the study of algebra
and 4.7 after the study of algebra. Several serious
objections can be raised from the scientific point of
view to this study as Mr. Braverman points out:
The improvement shown might be attributed
to the greater maturity of the pupils tested.
Also the results in the second administration
of the test were obtained after immediate ex
posure to some kind of quantitative course of
study, whereas the first administration of the
test occurred when some of the students had
been away from arithmetic one year and most of
them a half year.10
9 B. Braverman, "Does a Years Exposure to Algebra
Improve a Pupils' Ability in Arithmetic?'' Mathematics
Teacher, 32:301-12, November, 1939.
10 Loe. cit.
6
Wrightstonel l made a comparison of student achieve
ments in a general mathematics course and in a standard
type course. His study shows a difference in favor of alge
bra but this could be accounted for by the difference in
the type of students enrolled.
With these points of view in mind and the fact that
of the schools reported by Schorling
1
2 in 1948, 100 per
cent reported general mathematics being taught in the
eighth grade, a survey of mathematics for the eighth grade
should be made.
II. DEFINITIONS OF TERMS USED
General mathematics. The term general mathematics
as used throughout this thesis is meant to be a non
compartmental course. That is, a course where the various
topics of mathematics are fused in such a way as to give
the students a survey of each topic while at the same time
relating the topics of arithmetic, algebra and geometry
together as well as to life situations.
11 J. w. Wrightstone, "Comparison of Varied Cur
riculu~ Practices in Mathematics,'' School, Science and
Mathematics, 35:377-81, April, 1935.
12 Schorling, ..2.E.· cit., pp. 147-53-
Arithmetic algebraic in nature. The expression,
algebraic in nature, refers to arithmetic content in
eighth grade textbooks considered directly related to
algebra.
III. ORGANIZATION OF CHAPTERS
The present chapter has presented the problem and
its importance. The following chapter will review the
trends in the development of eighth grade mathematics and
related literature.
7
The third chapter will develop the criteria against
which the textbooks are to be evaluated, the procedure,
method and limitations of the analysis and the selection
of textbooks to be used.
Chapter IV will report the results of the analysis
in charts and a discussion of their merits.
Chapter V will include a summary and the conclusions
drawn from the analysis of the textbooks.
CHAPTER II
REVIEW OF EIGHTH GRADE MATHEMATICS
AND RELATED LITERATURE
Trends in eighth grade mathematics. From the time
of the first curriculum of the public schools, mathematics
has occupied a place of importance in both elementary and
secondary programs. Its prominence has fluctuated consid
erably and textbooks, by their very nature, reflect the
presiding trend. The objectives of the textbook in eighth
grade mathematics have been influenced by the aims of
mathematics curriculums in general and, because of its
unique position in the school system, the eighth grade
textbook has at times been subjected to specific influence&
One can hardly come to understand or appreciate the pres
ent status of eighth grade mathematics, or its function as
one of the avenues and instruments of general education,
without having a picture of the evolution of the mathe
matics program in public schools.
Objectives of the Nineteenth Century. The prevalent
objective of early textbooks was one of "mental discipline J'
This idea exerted a profound influence on the few text
book writers of the early nineteenth century and was well
stated by Joseph Ray•
The object of the study of mathematics is
twofold--the acquisition of useful knowledge
and the cultivation and discipline of the
mental powers. A parent often inquires "Why
should my son study mathematics? I do not
expect him to be a surveyor, an engineer, or
an astronomer." Yet the parent is very desir
ous that his son should be able to reason cor
rectly, and to exercise, in all his relations
in life, the energies of a cultivated and dis
ciplined mind. This is, indeed, of more value
than the mere attainment of a branch of knowl
edge.l
9
In the last quarter of the nineteenth century, dis
satisfaction and complaints against the mathematics cur
riculum in the public schools began to arise. Large per
centages of failures became noticeable, business men began
to question the opportunity for the application of high
school mathematics and colleges raised their voices against
the poor mathematical training of entering freshmen. To
meet this criticism, "The Committee of Ten,"
2
met in 1894
and recommended certain changes. The recommendations of
the Committee affecting the eighth grade were the sugges
tions that; (1) a concrete geometry course be introduced
in the grammar schools, and (2) that systematic algebra be
1 Joseph Ray, New Elementa1 Al~ebra (New York:
American Book Company, 1848), p. I 1.
2 Report of the Committee of Ten on Secondar~
School Subjects "(lfew York: Americin BooKCompany,894),
pp. 105-106.
10
started at the age of fcurteen. A "Committee of College
Entrance Requirements," in 1899, recommended the following
courses of study for the seventh and eighth grades:
Seventh Grade: Concrete Geometry and In
troduction to Algebra.
Eighth Grade: Introduction to Demonstrative
Geometry and Algebra.3
The content of such courses to be as follows:
Algebra for the seventh and eighth grades was
to begin with literal arithmetic which was to be
followed by simple polynomials and fractional ex
pressions, equations of the first degree with nu
merical coefficients in one and two unknowns, the
four fundamental operations for rational algebraic
expressions and simple factoring. 0~e-half of the
time of the seventh grade was to be devoted to con
crete geometry, while in the eighth grade one-half
of the time was to be spent in demonstrative geometry.4
Objectives of the Twentieth Century. The beginning
of the twentieth century found the trend for a new type
of mathematics course being expressed. In an address be
fore the National Education Association in 1902, Charles
Newhall,5 expressed the hope that the time would com e when
3 A. F. Nightingale, Report of the Committee on
Collefe Entrance Requirements, Procei'dlngs and Addresies
of Na Iona! Education Association, 38th Annual Meeting,
1899, pp. 648.
4 Loe. cit.
5 Charles w. Newhall, Correlation of Mathematical
Studies in Secondary Schools, ProceedingsOr the National
Educational Association (1902), pp. 488-492.
11
mathematics would not be limited by artificial boundaries
as was the case in the study of arithmetic, algebra and
geometry. The American Commissioners of the International
Commission expressed one of the marked tendencies of this
period for a change in curriculum and method to attach
greater importance to the utilitarian possibilities of
mathematics.6 The hope of combining subject matter and
the desire for utility found expression in what is now
referred to as general mathematics courses. The growth
and popularity of general mathematics courses paralleled
the development of the junior high school . The junior
high school with no traditional curriculum and its objective
to pave the way between elementary and secondary schools
was a good proving ground for general mathematics. Tn
1929, McCormick summarized the conclusions drawn from
numerous experiments made to determine the status of this
new organization of subject matter:
1. There is no very clear or definite
agreement among mathematicians and general
educators as to what constitutes general
mathematics.
2. General mathematics is gradually re
placing the traditional type in the seventh,
eighth and ninth grades.
6 Report of the American Commissioners of the Inter
national Commlssion on the Teaching of Mathematics, Bul
letin 14, Office of Eaucation (Was ington: Government
Printing Office, 1912), pp. 29-31.
3. General mathematics provides training
for college mathematics that is as good as,
and perhaps better than, that of mathematics
of the traditional type.
4. The indications are that general
mathematics creates more interest in the
subject than does traditional mathematics.7
The growth of general mathematics did not develop
without severe criticism but the fact that two of the
fundamental services of the junior high schools were to
provide for exploration and for contact with the minimum
essentials helped to concentrate interest in general
mathematics as a unit in its curriculum. Genera mathe
matics became a part of the junior high school and the
eighth grade. The general objectives of the junior high
school mat.-hematics courses between 1924 and 1938 are
summarized in a study by Redd:
Changes in the aims of mathematics courses
have been the greatest at the junior high
school level. An examination of the state
ments of aims and objectives as found in text
books, courses of study, reports of education
associations, and periodical articles reveals
that the aims are composed of cultural, prac
tical, and disciplinary phases. Particular
7 Clarence McCormick, Teaching of General Mathe
matics in Secondary Schools (New York:-Bureau of
Publications, T achers College, Columbia Univ rsity,
1929) , p. 162.
2
13
emphasis is now be1ng
8
placed upon the first
two phases mentioned.
The extent to which these general objectives have
remained until today are revealed by the outline of
objectives for eighth grade mathematics reported by the
Commission of Post-War Plans,9 in 1945.
Specific topics in eighth grade mathematics. The
growth of general mathematics at the eighth grade level
necessitated a change in the grade placement of specific
topics. Much of the published material on grade place
ment of specific topics recognize the seventh and eighth
grades as a unit. In this study they were separated and
only the algebraic material of the eighth grade was con
sidered. It is at this grade level that an introduction
to the elementary aspects of algebra are encouraged.
The weight of evidence, however, seems to
indicate rather clearly that, while certain
work with simple formulas may be done satis
factorily in the seventh grade, it is better
to defer most of the algebraic work until the
8 John Redd, "Current Trends in the Teaching of
Algebra in the Junior High School,'' (unpublished Master's
thesis, The University of Southern California, Los
Angeles, 1938), p. 115.
9"Second Report of the Commission on Post-War
Plans," Mathematics Teacher, 38:195-221, May, 1945.
14
eighth grade.10
The types of material and extent to which algebra
has been introduced in the eighth grade has varied
throughout the years.
The influence of the suggestions made in
1923 by the National Committee On Mathematical
Requirements gave rise to many new textbooks
and courses of study in mathematics. Some of
these, in their efforts to effect a thorough
going redistribution of material for the junior
high-school grades, attempted a more ambitious
program of algebraic work for the seventh and
eighth grades than now seems justifiect.11
The unit introducing e l ementary algebra is of
major importance and its main objectives should be the
development of elementary skills and motivation for
further study in algebra. One authors says of the early
introduction of algebra:
There was a time when the traditional school
program provided no algebra beyond work with
formulas until after the course in elementary
arithmetic had been completed. Arithmetic was
an elementary school subject; algebra was a
high school subject. Algebra was thought to
be a difficult subject. When it was taught by
a method which was based upon the drill theory,
it was difficult. Its reputation for difficulty
spread until the arithmetic pupils in the
10 c. N. Butler and F. L. Wren, The Teaching of
Secondary Mathematics (New York: McGraw-Hill Book
Company, Inc., 1941), p. 271.
11 Ibid., p. 269.
15
elementary school heard about it. The word
'algebra,' meant nothing to the uninitiated
except a future experience which would
probably be unpleasant.12
An excellent criterion for determining the ap
propriateness of algebraic material for the eighth grade
has been proposed by Barber:
Any algebra which may be introduced into
these grades should be subjected to three
tests. Is it interesting? Is it useful?
Is it thought-provoking? And to these there
may be added a fourth: Does it prepare for
the new algebra of grade nine?l3
In reference to the inclusion of elementary
algebra in the eighth grade, the Joint Commission of
the Mathematical Association of America stated:
Algebra that has been successfully in
troduced into grades 7 and 8 up to the pre
sent time has been limited largely to the
understanding of the basic concepts, to the
evaluation of formulas, and the solution of
very simple equations. It seems possible
and also desirable to include other algebraic
material: but, if it 1s to prove effective,
the work should be carefully planned and
should be so organized as to be significant
in itself as well as designed to furnish a
good foundation for later algebraic study.14
12 R. L. Morton, Teaching Arithmetic in the
Elementary School {New York: Silver BurdettCompany,
l939J, p. 405.
1
3 H. c. Barber, Teaching Junior Hig¥ School Mathe
matics (Boston: Houghton Mifflin Company, 924), p. 88.
14
Joint Commission of the Mathematical Associa
tion of America, Inc ., and the National Council of
Teachers of Mathematics, "The Place of Mathematics in
The Second Report of the Commission on Post-War
Plans, in 1945, listed as one of four points to be con
sidered in the eighth grade: "An introduction to the
essentials of elementary algebra (formula and equa
tion)."15
16
As revealed by these reports, textbooks and prac
tices of schools, the formula and simple equation have
occupied a prominent place in the eighth grade. Certain
other topics have been suggested such as: graphs, square
root, literal numbers and signed numbers. That these
topics can undoubtedly be made meaningful and interesting
to young students is illustrated by one author as follows:
Literal numbers: The transition from the
arithmetic to the algebraic solution will
present little difficulty. Pupils will be
quick to appreciate the fact that the use of
literal numbers, such as x, makes for concise
ness and ease in pursuing the train of thought.
If both methods are employed in a number of
problems, the one 111 clarify the other and
pupils will have an opportunity to
6
observe the
effectiveness of literal numbers.!
14 (Continued).
Secondary Education,' Fifteenth Yearbook of the National
Council of Teachers of Mathematics {New York: Bureau of
Publications, Teacher's College, Columbia University, 1940),
p. 80.
1
5 Second eport of the Commission on Post-War
Plans, loc. cit.
16 A. R. Jerbert, "Algebra and Arithmetic," School
Science and Mathematics, 45:528-40, June, 1945.
17
At this stage in their mathematical education,
students stand at the cross-roads and the direction many
of them take may be greatly influenced by this unit.
I. REVIEW OF RELATED LITERATURE
In relation to the importance attached to the
textbook in public schools there is relatively little
material published as to their evaluation, selection,
or use. A possible explanation for this lack of
literature is twofold: (1) Educational theory and
practices are constantly changing and textbooks are con
tinually changing because of their high sensitivity, and
(2) the lack of a valid and accepted body of definite
criteria which takes into consideration the fundamental
issues involved. The literature on the evaluation of
textbooks is well summed up by Butler:
There have been various attempts to set up
such a list of principles or criteria for the
evaluation of mathematics textbooks. In the
main these have been useful in focusing atten
tion on the need for objective means of evalua
tion and comparison and in providing patterns
of analysis. The great variation among these
patterns, however, makes it clear that differ
ent people, presumably all of high competence,
may have very different ideas a to the rela tive importance of various elements, and that
they may also have very different ideas as t o
what elements should be included in such
18
analytical comparisons.17
Literature on eighth grade mathematics textbooks.
Robert Williams,18 made a study of junior high school
mathematics books in 1931. He evaluated the amount of
material presented in a series of books against the objec
tives outlined by the North Central Association of Col
leges and Secondary Schools. Mr. Williams did not divide
his study into arithmetic, algebra and geometry but such
divisions as measurements, graphs, formulas and problems
were used. As a result of his evaluation it was shown
that the authors and Association agreed as to the kind of
material to be presented but they disagreed as to the
amount.
From a study of fourteen courses of study and
eight series of textbooks, F. L. Wren and Ruth Moncreiff,19
1
7 c. H. Butler and F. L. Wren, The Teaching of
Secondarl Mathematics (New York: McGraw-Hill Book -
Company, Inc., 19~1), p. 89.
18 R. L. Williams, "The Selection of Mathematics
Texts in the Junior High School," School, Science and
Mathematics, 31:284-91, March, 1931.
19 F. L. Wren and Ruth Moncreiff, "A Suggested
Course of Study for the Junior High School Mathematics,'
School, Science, and Mathematics, 34:724-32, October,
1934.
9
developed a course of study for the junior high school
mathematics curriculum. They proposed that 41.6 per cent
of the time in the eighth grade be devoted to the study
of material algebraic in nature. In the breakdown of
topics, the form la was not listed separately as a topic
to be covered, but four weeks were to be spent studying
algebraic equations.
In 1942, F. F. Novinger,20 made an analysis of
twenty-three ninth grade general mathematics textbooks
for non-academic pupils, non-academic pupils being those
enrolled in a general mathematics type of course prior
to entering algebra or terminating their study of mathe
matics. Novinger reported five general conclusions
from his study:
1 . Subject matter fell into five major
categories; arithmetic, commercial arithmetic,
algebra, social uses and consumer uses .
2. All of these categories were not in
cluded by all the authors. They agreed only
on arithmetic and tables.
3. Arithmetic received the greatest em
phasis although the authors showed little
agreement concerning the amount to be pre
sented as they varied from four to 218 pages.
20 F. F. Novinger, "Distribution of Content in
Twenty-three Ninth Grade Mathematics Textbooks for Non
academic Pupils,'' Mathematics Teacher, April, 1942.
4. There was a great diversity of opinion
among the authors as to algebra. Seven of the
authors omitted it completely.
5. There was closer agreement as to the
amount of geometry with an average of 100
pagea.21
There were certain limitations to this study.
The investigation was made from the table of contents
of the books only and no appraisal was made of the
quality of the material, therefore it is unknown just
how much of the arithmetic material could have been
algebraic in nature. The textbook containing little
or no algebraic material, as recorded in the table of
contents, may have resulted from the aim of the author
to present the algebraic material as an integrated part
of the arithmetic content.
Summary. In Chapter I enrollments and achieve
ments in mathematics was presented along with the
growing demand of society and industry for an improve
ment of this critical situation. A survey of enroll
ments indicated that the eighth grade ls the last year
in which a major portion of the students are taking
mathematics. It is at this grade that motivation for
better achievement and further study should begin.
Chapter II presents a brief historical develop
ment of general mathematics and surveys of textbooks
20
21
in the field of general mathematics. The widespread
acceptance of general mathematics indicate its popularity.
In view of the demand for more and better mathematical
training there is relative little published material on
the evaluation of the effectiveness of this type course.
CHAPTER III
PROCEDURE
I. SELECTION OF TEXTBOOKS
Current texts. An effort was made to choose
those texts which are in wide use at the present time.
To obtain a list of currently used textbooks, corres
pondence with nineteen scattered school districts over
the nation was made. Eight of these districts represented
areas with a state adopted or recommended textbook system.
Sixteen districts replied and from their replies five
texts that were most frequently used and available were
selected. For this study, each of the books was given a
letter as A, B, c, D, E and hereafter is referred to by
such designation. (See appendix for key to texts used.)
II. ARITHMETIC MATERIAL ALGEBRAIC IN NATURE
Arithmetic material algebraic in nature, by
previous definition, refers to all textbook content
directly related to the future study of algebra. In
this study three analyses of such material is made:
quantity of material, distribution of materia within
the book, and quality of the unit on elementary algebra.
23
Quantity of material. The criterion for determin
ing the quantity of material re l ated to the study of
algebra were compiled from a study of textbooks on the
teaching of arithmetic and courses of study in elementary
algebra. Five topics were developed for the purpose of
classifying material algebraic in nature.
1. Formula. To be classified under this head
ing the formula had to be introduced in the text as a
method of procedure for the solution of ensuing prob
lems.
2 . Graph. Both the interpretation of graphs
and the construction of graphs were tabulated under
this heading.
3. Ratio-proportion. Under this heading was
classified all material relating to ratio, proportion
and indirect measurement.
4. Square root. Under this heading was classi
fied all material on square root, table of squares and
the Rule of Pythagoras.
5. Algebra. Under this heading was classified
all material designated as an introduction to elementary
algebra by the author or the material introduced was
elementary algebraic topics.
Each book was analyzed page by page according to
24
these criterion. If as much as one-fourth of a page was
considered algebraic in nature, full credit for the page
was given. Because of the overlapping of some of the
material, credit was given to the content being stressed
by the author. That is, if the study of graphs was being
stressed by the author and formulas were also used, the
material was tabulated under graph. In deriving the total
number of pages in each textbook, those pages pertaining
to remedial drill at the end of the texts were omitted.
Indices, table of contents and pages of answers if included
were eliminated from the total.
Distribution of material. The distribution of
------ - ----
arithmetic material, algebraic in nature, was classified
as to its utilization by chapters or units in each text
book. By this classification the percentage of each
chapter using the defined material was determined. This
distribution per chapter gave an indication as to the
extent each chapter provided closer correlation between
the arithmetic and algebraic material.
To determine the distribution of this material
throughout the texts, the page numbers classified as
algebraic in nature were recorded. These page numbers
were then tabulated according to their respective chapters
in the text. This procedure gave the number of pages,
classified as algebraic in nature, for each chapter.
III. THE UNIT INTRODUCING ALGEBRA
The unit in eighth grade textbooks, introducing
elementary algebra, provides an ideal place for the
integration of arithmetic and algebra. The analysis of
this unit had as its objective the determination of:
(1) the importance of the unit, (2) the relationship of
the unit to arithmetic, and (3) the quantity of the
fundamental processes involved.
25
Importance of the unit. The importance attributed
----- - -- ---
to this unit by the texts was evaluated according to the
position of the unit in the text, the number of pages
devoted to the unit, and the specific topics introduced
by the unit. The position of the unit and the number of
pages presented are controlling aspects of the coverage
given the material by the teacher. The different topics
introduced by the unit provides an indication of the
textbooks intent in correlating arithmetic and algebra.
The total number of pages in this unit were
determined by counting all the pages in the unit con
taining the material on elementary a l gebra. The position
of the unit in respect to the entire book was derived
26
by dividing the textbook into three parts; first, middle
and last. The unit on algebra was then recorded in its
appropriate position. Tabulation of pages devoted to
specific topics in the unit were recorded under one of
the following headings; formula, graph, equation, signed
numbers, like terms, and orientation.
Relationship of the unit to arithmetic. To
determine the possible relationship between arithmetic
and elementary algebra in this unit the problems devoted
to equations were analyzed. Only those problems pertain
ing to equations were utilized because of the practical
adaptation of equations to arithmetical problems.
For the analysis of problems on equations a prob
lem was defined as, any exercise requiring the student
to formulate an answer. The problems were then divided
into two types, drill and written. Drill problems re
ferred to those that require only the reading and writing
of answers or the manipulation of given symbols to arrive
at an answer. Written problems referred to those
exercises in which a statement of facts is given and
the student 1s required to set up an equation before
solving for an answer. To evaluate these problems as
to their relationship to arithmetic they were analyzed
against the following criteria:
27
Related. To be classified as related the problems
had to utilize formulas or refer to previous or future
types of arithmetic problems contained in the texts.
Examples of this type problem:
Written. Mr. Smith, paid his income tax
in quarterly payments. If each payment was
$34.48, what was his total income?
Drill. In the form la A= l/2hb , find
b if, A - 9 and H - 4.
Unrelated. Those problems classified as unrelated
made no reference to recognized ar thmetic work. Examples
of this type problem:
Written. Five times a number increased by
6 is 21.
Drill. 4n = 16.
The illustrations on how to solve the various
problems were also tabulated and recorded as either
arithmetic, if they showed relationship, or algebra, if
they showed no relationship.
Quantity of fundamental processes. Analysis of
each problem was made to determine the extent to which
the four fundamental processes of solution were made.
The purpose was to determine the probable preparation
for the future study of algebra. The problem were
tabulated according to their method of solution; addition,
subtraction, multiplication, division, and combination.
Only those problems that required the manipulation of
symbols were analyzed.
28
CHAPTER IV
ARITHMETIC MATERIAL ALGEBRAIC IN NATURE
I . QUANTITY OF MATERIAL
Formula. From Table I it is seen that the average
number of pages utilizing the formula is 56. The range
between the textbook with the highest number of pages,
textbook C with 69, and the textbook with the lowest num
ber of pages, textbook A with 46, is 23 pages. Table II
gives the percentage distribution of the material. The
average percentage is 16.3 and textbook C is highest with
21.1; textbook E is lowest with 13.9, the range between
the two is 7.2. All of the textbooks devoted more of their
content to formulas than any other topic classified.
Graph. From Table I it is seen that the average
number of pages utilizing the graph is 16. The range
between the textbook with the highest number of pages,
textbook A with 25 pages, and the textbook with the lowest
number of pages, textbook E with 8 pages, is 17 pages.
Table II gives the percentage distribution of the material .
The average percentage is 4.8 and textbook A is highest
with 8.5 and textbook E lowest with 2.2, the range being
6.3.
30
TABLE I
PAGES OF MATERIAL ALGEBRAIC IN NATURE
Textbooks A B C D E Average
Total pages
293 434 327 313 360 345
Formula 46
65 69 50 50 56
Graph
25
22 14 12 8 16
Ratio-proportion 17
46
17 15 34 26
Square root
9
0 6
5
16 6
Algebra 20 46 14 26 41
29
Total 117 179
120 108
149 135
This table should be read as follows: 46 pages in text
book A pertain to formulas, 65 pages in textbook B pertain
to formulas, etc.
TABLE II
THE PERCENTAGE DISTRIBUTION OF ARITHMETIC
MATERIAL ALGEBRAIC IN NATURE
31
Textbook A B C D E Average
Formula 15.7 14.9 21.1 16.0
13.9 16.3
Graph 8.5 5.1 4.3 3.8 2.2 4.8
Ratio-proportion 5.8 10.6 5.2 4.8 9.4 7.1
Square root 3.1 o.o 1.8 1.6 4.4 2.2
Algebra 6.8 10.6 4.3 8.3 11.4
8.3
Total
This table should be read as follows: 15.7% of textbook A
pertains to formulas, 14.9% of textbook B pertains to
formulas, etc.
32
Ratio-proportion. From Table I it is seen that
the average number of pages utilizing ratio-proportion
1s 26. The range between the textbook with the highest
number of pages, textbook B with 46 pages, and the text
book with the lowest number of pages, textbook D with 15
pages, is 31 pages. Table II gives the percentage
distribution of this material. The average percentage
is 7.1 and textbook Bis highest with 10.6 and textbook
D lowest with 4.8, the range being 5.8.
Square root. From Table I it 1s seen that the
average number of pages utilizing the square root is six.
The range between the textbook with the highest number of
pages, textbook E with 16 pages, and the textbook with the
lowest number of pages, textbook B with no pages, 1s 16
pages. Table II gives the percentage of distribution of
this material. The average percentage 1s 2.2 and textbook
Eis highest with 4.4 and textbook Bis lowest with o.o,
the range being 4.4.
Algebra. From Table I it is seen that the average
number of pages utilizing a unit of algebra is 29. The
range betw en the textbook with the highest number of
pages, textbook B with 46 pages, and the textbook with
the lowest number of pages, textbook C with 14 pages, 1s
33
32 pages. Table II gives the percentage of distribution
of this material. The average percentage is 8.3 and
textbook Eis highest with 11.4 and textbook C is lowest
with 4.3, the range being 7.1.
Amount of total material. The relationship of the
textbooks in the total amount of material, algebraic in
nature, to be presented is indicated by the total per
centages in Table II. The mean average for the group is
38.7 per cent. The range between the text with the high
est percentage, textbook E with 41.3, and the text with
the lowest percentage, textbook D with 34.5, is 6.8 per
cent.
Table III presents the rank order of each text
book on the percentage basis. It 1s evident that no one
text ranked consistently high or low . A summation of the
rank order, disregarding probable importance of topics,
gives textbook A a one point edge over textbooks Band E.
Limitation in the present data. It is recognized
that the failure to analyze the probable integration of
these topics among themselves accounts for some of the
differences tabulated in each topic. As the total amount
of material, algebraic in nature, was the objective, this
limitation does not seriously affect the purpose of this
TABLE III
RANK ORDER DISTRIBUTION OF TEXTBOOKS
Textbook
Formula
Graph
Ratio-proportion
Square root
Algebra
Weight
A
3
l
3
2
4
13
B
4
2
1
5
2
14
C
1
3
4
4
5
17
D
2
4
5
3
3
17
E
5
5
2
1
....
1
14
This table should be read as follows:
textbook A ranked 3 in percentage of
content, textbook B ranked 4 in per
centage of content, etc .
34
35
study.
Summary.
1. The texts generally agreed as to the types of
material to be introduced at the eighth grade level .
2. The texts generally agreed as to the total
percentage of material, algebraic in nature, to be pre
sented.
3. Every text devoted more pages to formulas than
they other topic, indicating agreement as to its
recognized importance at this grade level .
4. Every texts included a unit on the introduc
tion of elementary topics in algebra.
5. In general the texts considered the study of
square root least applicable at this grade level and one
book omitted it entirely.
6. No one text consistently ranked higher in the
correlation of arithmetic and algebraic material.
II. DISTRIBUTION OF ALGEBRAIC MATERIAL
Textbook A. Table IV gives the distribution of
the algebraic material in textbook A in relation to the
chapters in the text. The textbook was divided into
twelve chapters with the number of pages in each chapter
Chapters
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
36
TABLE IV
CHAPTER DISTRIBUTION OF ALGEBRAIC
MATERIAL IN TEXTBOOK A
No. of pages
in unit
33
22
33
15
20
30
25
23
20
25
27
20
No. of pages
algebraic
1
22
4
14
8
2
2
4
0
14
26
20
Per cent of
unit algebraic
3.0
100.0
12.1
93.3
40 . 0
6.7
8.o
17.4
o.o
56.0
96.3
100.0
This table should be read as follows: Chapter I contains
33 pages of which 1 page is algebraic and the percentage
of the unit algebraic is 3.0.
37
ranging from 15 to 33. Every chapter but one contained
algebraic material. Five chapters consisted of more than
50 per cent algebraic material and four chapters contained
less than 10 per cent.
Textbook B. Table V gives the distribution of the
algebraic material in textbook Bin relation to the
chapters in the text. The textbook was divided into ten
chapters with the number of pages in each chapter ranging
from 34 to 50. Every chapter contained pages of algebraic
mater al. Three chapters consisted of more than 50 per
cent algebraic material and two chapters contained less
than 10 per cent.
Textbook c.
-----
Table VI gives the distribution of
the algebraic material in textbook C in relation to the
chapters in the text. The textbook was divided into
eleven chapters with the number of pages in each chapter
ranging from 14 to 57. Every ch pter contained pages of
algebraic material. Four chapters consisted of more than
50 per cent algebraic material and two chapter c ntained
less than 10 per cent.
Textbook D. Table VII gives the distribution of
the algebraic material in textbook Din relation to the
chapters in the text. The textbook was divided into
38
TABLE V
CHAPTER DISTRIBUTION OF ALGEBRAIC
MATERIAL IN TEXTBOOK B
Chapters No. of pages
in unit
I 48
II 48
III 50
IV 36
V 34
VI 44
VII 34
VIII 54
IX 40
X 46
No. of pages
algebraic
17
16
8
6
3
37
11
5
30
46
Per cent of
unit algebraic
35.4
33.3
16.0
16.6
8.8
84.1
32.2
9.3
75.0
100.0
This table should be read as follows: Chapter I textbook
B contains 48 pages of which 17 are algebraic in nature.
The percentage of the unit algebraic is 35.4.
Chapters
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
39
TABLE VI
CHAPTER DISTRIBUTION OF ALGEBRAIC
MATERIAL IN TEXTBOOK C
No. of pages
in unit
57
46
32
36
26
18
24
24
24
26
14
No. of pages
algebraic
2
5
28
7
25
7
9
18
3
2
14
Per cent of
unit algebraic
3.5
10.9
87.5
19.4
96.2
38.9
37.5
75.0
12.5
7.7
100.0
This table should be read as follows: Chapter I of text
book C contains 57 pages of which 2 are algebraic in
nature. The percentage of the unit algebraic is 3.5.
Chapters
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
40
TABLE VII
CHAPTER DISTRIBUTION OF ALGEBRAIC
MATERIAL IN TEXTBOOK D
No. of pages
in unit
27
26
26
26
28
24
26
26
26
26
26
26
No. of pages
algebraic
1
3
4
3
6
5
10
26
26
3
5
16
Per cent of
unit algebraic
3.7
11.5
15.4
11.5
21.4
20.8
38.5
100.0
100.0
11.5
19.2
61.5
This table should be read as follows: Chapter I of
textbook D contains 27 pages of which one is algebraic
in nature. The percentage of the unit algebraic is 3.7.
41
twelve chapters with the number of pages ranging from 24
to 28. Every chapter contained pages of algebraic
material. Three chapters consisted of more than 50 per
cent algebraic material and one chapter contained less
than 10 per cent.
Textbook E. Table VIII gives the distribution of
the algebraic material in textbook E in relation to the
chapters in the text. The textbook was divided into ten
chapters with the number of pages in each chapter ranging
from 22 to 56. Two chapters did not contain material
classified as algebraic in nature. Three chapters con
sisted of more than 50 per cent algebraic material and
four chapters contained less than 10 per cent.
Total distribution. Table IX gives the relation
ship of the texts to each other in percentage of distri-
bution. Of a total of 55 chapters in the five textbooks,
27 chapters were made up of less than 25 per cent alge
braic material. Sixteen of the 55 chapters contained
more than 75 per cent algebraic material. The remaining
12 chapters contained between 25 and 75 per cent alge
braic material.
Summary.
1. The textbooks generally restricted the
42
TABLE VIII
CHAPTER DISTRIBUTION OF ALGEBRAIC
MATERIAL IN TEXTBOOK E
Chapters No. of pages
in unit
I 44
II 42
III 22
IV 42
V 56
VI 40
VII 26
VIII 32
IX 28
X 28
No. of pages
algebraic
2
3
0
41
25
19
25
0
27
7
Per cent of
unit algebraic
4.5
7.1
o.o
97.6
44.6
47.5
96.2
o.o
96.4
25.0
This table should be read as follows: Chapter I of
textbook E contains 44 pages of which two are algebraic
in nature. The percentage of the unit algebraic is 4.5.
TABLE IX
PERCENTAGE OF CHAPTER DISTRI
BUTION IN TEXTBOOKS
Textbook A B C D E Total
CYJ,- 25%
6 4
5
8 4
27
25
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Asset Metadata
Creator
Foster, David Randolph
(author)
Core Title
An analysis of the integration of arithmetic and algebra in eighth grade mathematics textbooks
School
College of Letters, Arts and Sciences
Degree
Master of Science
Degree Program
Education
Degree Conferral Date
1951-06
Publication Date
06/01/1951
Defense Date
06/01/1951
Publisher
University of Southern California
(original),
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