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#441

ABSTRACT

AN INVESTIGATION OF CHILDREN'S LEARNING OF
SOME CONCEPTS AND PRINCIPLES WHICH
ENABLE TED! TO PERFORM EXAMPLES
OF ADDITION OF COMMON FRACTIONS

by Marjorie Pickering

This study investigated the order in which
children learned some concepts and principles which
enabled them to perform examples of addition of
common fractions. To delineate the concepts and
principles, a hierarchy was developed which.had as
its base the understanding of operations with whole
numbers and as its apex the performance of the
example 3 3/10 + 2 5/6. Two classes were instructed
in the concepts of fractions and tested at regular
intervals. One of the classes used commercial
materials and was instructed as a group with.every-
one working on the same material at the same time
(Treatment A). The other class used a specially
developed set of materials which maximized individ-
ual work and allowed some free choice of the order
in which.certain of the principles were studied
(Treatment B). The test results were analysed to

determine invariances in the order in which students

Marjorie Pickering

developed an understanding of the concepts and prin-
ciples of the hierarchy. The data was examined for
patterns of learning, the relative performances of
the two classes were compared, and contrasts in the
two methods were reported.

The main criterion for determining order
were eight surveys, each containing 26 examples, one
for each concept or principle of the hierarchy. The
items from the surveys were considered in pairs
(a,b). For each class the number of students who
performed a on an earlier survey than b, who per-
formed b on an earlier survey than a, and who
performed a and b simultaneously were tabulated.
This tabulation was analyzed and where applicable an
order a< b or b< a was established. The results for
each class were compiled into a projected hierarchy.
Comparison of the two hierarchies indicated that
with the exception of the concepts of least common
multiple and the principle of multiplication of
fractions all of the orders under Treatment B also
applied under Treatment A. It appeared that pre-
scribing the order of instruction has a direct
effect upon the order of learning. More students
seemed to have an understanding of the partition
and the rational number interpretation of fraction

if the partition interpretation was taught first

Marjorie Pickering

and drill provided. More students seemed to have an
understanding of the addition of fractions having
the same denominators if they approached addition
through.the rational number interpretation of frac-
tion using concrete aids than if they approached
addition through the partition interpretation.

The average pretest-posttest gain of students
who understood the partition interpretation of frac-
tion at the time of the pretest was greater than the
class average. Several students individual histories
showed that they performed only examples which.had
easy algorithms involving whole numbers. An under-
standing of equivalent fractions was acquired under
Treatment A without an understanding of multiplica-
tion of fractions or of fractional names for one.

An understanding of mixed numeral seemed to aid in
the understanding of fractional names for one.

Although the difference in gains was not
statistically significant, students receiving Treat-
ment B showed greater gains in performance and
better retention than students who used the textbook
materials. Informal observation in later mathematics
lessons seemed to indicate that the students who had
received Treatment B were more enthusiastic than
students who had received Treatment A.

It is feasible to employ a method of

Marjorie Pickering

individualized instruction to a study of fractions.
The use of concrete aids manipulated by students
appears to promote better understanding of the
process of addition of fractions and more enthusiasm

on the part of the students.

AN INVESTIGATION OF CHILDREN '8 LEARNING OF
SOME CONCEPTS AND PRINCIPLES WHICH
ENABLE THEM TO PERFORM EXAMPLES
OF ADDITION OF COMMON FRACTIONS

B;

‘ '{1-r}“‘;"}/‘

Mar jorie( fPickering

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Education

1968

TABLE OF CONTENTS
Page
DOCTORAL COHHITTEE............................. iv
ACKNOWLEDGEMENT................................ v
LIST OF TABLES................................. vi

Chapter
I. GHERAI‘ PROBLEHOOOOOOOO00.000.00.00...

1
Historical Bases for the Study..... 2
Fractions and Rational Numbers..... 6
Design of the Study................ 7
Purposes of this Study............. 8

II. BACKGROUND FOR FRACTIONS.............. 10

The Concept of Fraction............ lO

Interpretations of Fractions....... 15
The Partition Interpretation.... 16
Fractions as Operators.......... 16
The Division Interpretation..... 17
The Element of a Mathematical
System.Interpretation........... 18

III. HEATED LITERATUMOOOOOOOOOOOOOOCOO... 20

Methods of Teaching................ 20
Histories of Class Performance..... 22
The Collection and Analysis of Data
for the Early Elementary Grades.... 2h
The Collection and Analysis of Data
for L‘t.’ Grad..OOOOOOOOO0.0.0.0... 25

IV. RESEARCH DESIGN AND DESCRIPTIVE DNTA.. 29

Development of the Hierarchy....... 29
Construction of the Tests and
Revision of the Hierarchy.,........ 30
The Materials, Treatment A......... 31
Outline of Haterials, Treatment
‘OOOOOOOOOOOOOOOOOOOOOOO0000.... 36
The Materials, Treatment B......... 37
Outline of Materials, Treatment hl
3......OOOOOOOOOOOOOOOOOOOOOOOOO
The Population Sample.............. uz
Initial Characteristics......... ha
Teaching the Units................. us
Daily Report‘OOOOOOOOOOOOOO.COO. 1"?
The Intermediate Testing........ RB
Some General Classroom Observa-
tion.‘00.0.0.0...OOOOOOOOOOOOOOO 55
smryOOOOOOOOOOOOOO000.0000... 58

H

yen.
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9..
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3
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s

s I ’ ' O O
. . .

r e e 0 v 0 e
t 0 e s '

' O I O

Q m C . ‘

TABLE OF CONTENTS

Chapter
V. ANALYSIS OF DAT-A00...OOOOOOOOOOOOOOOOOO

Resu1ts or Initial TeStingeeeeeeeeee

The Survey TeStseeeeeeeeeeeeeeeeeoee

The Order Of Performance....oo......
The Order of Performance Under
Treatment Beeeeeeeeeeeeeeeeeeeeee
The Order of Performance Under
Treatment Aeeeeeeeeeeeeeeeeeeeeee

VI. SUMMARY AND CONCLUSIONS................

Summary of the Investigation.....
Findings and Conclusions............
HYPOth881S leeeeeeeeeeeeeeeeeeeee
Findingseeeeeeeeeeeeeeeeeeeeeeeee
CODClUSiOnseeeeeeeeeeeeeeeeeeeeee
Hyp0th881332eeeeeeeeeeeeeeeeeeeee
Findingseeeeeeeeeeeeeeeeeeeeeeeee
ConCIUSIOnSeeeeeeeeeeeeeeeeeeeeee
Hypothesis BeeeeeeeeeeeeeeOOOeeeo
FindingSeeeeeeeeeeeeeeeeeeeeeeeee
COHClUSionSeeeeeeeeeeeeeeeeeeeeee
HYPOLhGSiS 4eeeeeeeeeeeeeeeeeeeee
FindingSeeeeeeeeeeeeeeeeeeeeeeeee
CODClUSiOnSeeeeeeeeeeeeeeeeeeeeee
Limitations of This Study........
Suggestions for Further Research....

APPEBIDIX A.OOOOOIOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
APPENDIX BOO...OOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOO

APPENDIX C.OOOOOOOOOOOOOOOOOOOOOOO...00.0.0.0...

BIBLIOGRAPHYOOOOOOOOOO.0......OOOOOOOOOOOOOOOOOO

iii

Page

130
139

Doctoral Committee

Dr. John Wagner, Chairman
Dr. Calhoun Collier

Dr. W. Eugene Deskins

Dr. William Fitzgerald
Dr. Robert W. Houston

iv

ACKNOWLEDGEMENT

The writer wishes to express her thanks
to John Vaughn, Curriculum Director of East
Lansing Public Schools and to Mrs. Margaret Love,
Mrs. Catherine Ferree,and their students without
whose kind cooperation this investigation would
have been impossible.

Gratitude is also due to the many members
of the staff at Michigan State University and the
many persons involved in mathematics education
whose support has meant so much to the writer in
the completion of this study.

The writer wishes especially to express
her heartfelt appreciation to Dr. John Wagner for
his confidence, kindness, forbearance and guidance
throughout the construction of this thesis; and to
her three children for their patience when this

project claimed all available time.

Table

II

III

IV

VI
VII

VIII

IX

XI

XII

XIII

XIV

LIST OF TABLES

Concepts and Mistaken Concepts of
Fractions Displayed in Early
Elementary GradeSOOOOOOOOOOOOOO0.00..

Hierarchy of Principles and Concepts
Necessary for Addition of Fractions..

The Hierarchy, By Example From Survey

TeStOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.

The Correspondence of Materials to the
HierarChy, Treatment Aeeeeeeeeeeeeeee

Summary Of Initial TeStSeeeeeeeeeeeeeeee
Summary Of Survey TBStSeeeeeeeeeeeeeeeee

The Order in Which Examples Were
Performed, Treatment A...............

The Order in Which Examples were
Performed, Treatment Beeeeeeeeeeeeeee

Analysis of the Skills With Fractions,
Pretest, Posttest, and Retention
Test by Approach Treatment...........

Number of Students Performing g on an
Earlier Survey Test than b, Treatment

AOOOOOOOOOOOOOOOOOOO.OOOOOOOOOOOOOOOO

Number of Students Performing g on the
Same Survey as b, Treatment A........

Number of Students Performing §_on an
Earlier Survey Test than b, Treatment

BOOOOOOOOOOOCOOCOOOOOOOCOOOOOOOOIOOOO

Number of Students Performing g_on the
Same Survey as b, Treatment B........

Probabilities That g Will Occur Before

b by Chance in the Quantities
TabLIIated by Table Xeeeeeeeeeeeeeeeee

vi

Page

26

32

33

44
46

49

SO

62

66

67

68

69

Table
XV

LIST OF TABLES

Probabilities That Q Would Occur Before
b by Chance in the Quantities
Tabulated by Table XII...............

Percent of Students Performing Each
Example by Percent and Number........

Projected Hierarchy Under Treatment B,
By ExampleOOOOOOOOOOOOOOOIOOOOOOOOOOO

Projected Hierarchy Under Treatment A,
By Exampleeeeeeeeeeeeeeeeeeeeeeeeeeee

vii

Page

70

72

79

CHAPTER I
General Problem

 

The general question "What mathematics
do children need to learn?" has no commonly accepted
answer. There is almost unanimous agreement that
every child should be taught to add common fractions.
The purpose of this paper is to investigate children's
learning of some concepts and principles which enable
them to perform examples of addition of common frac-
tions.

A search of the literature concerned with
fractions reveals a conspicuous absence of detailed
studies that report the actual learning history of
pupils during the period of time in which learning
is taking place. Studies tend to base their conclu-
sions on a pair of tests. a pretest and a posttest,
(Howard, 1950; Aftreth, 1958; Fincher, 1963; Pigge,
1964)1, or on posttests alone (Morton, 1924;, Hayes,
1927; Brueckner, 1928; Polkinghorn, 1935; Guiler,
1945). A history of learning would be expected to
show variations in the rate at which pupils learn,
variations in the accomplishments of learners at
different stages during the learning process, variation

1Names and dates in parenthesis refer to
listings in the bibliography.

in the total amount of learning, variation in the
manner in which learning takes place, and variation
in the nature of the difficulties which pupils
encounter during learning (Edwards, 1932). Further,
such a study should provide a variety of learning
experiences so that any invariances displayed are
not solely the result of the use of a single set of
materials. In studies of this type investigators
have generally ignored the field of fractions.

Historical Bases for the Study

Early educational research on the addition
of fractions seems to reflect the prevailing psy-
chology of the time in which it was conducted. With
the emphasis on "drill" during the mental discipline
era of the early twentieth century a profusion of
error-analysis studies were made. If the causes of
the errors were detected, it was thought, drill of
the proper type could be provided and the errors
could be eliminated (Hayes, 1927, p. 130). The re-
sults of these studies, however, indicated that a
great many of the errors in addition were due to a
lack of understanding of processes with fractions
(Brueckner, 1928; Searle 1927; Morton, 1924). In
a later study Sebold (1947, p. 71) reports that

Although approximately two thirds of the
pupils in grades five to seven who were

interviewed could add simple similar and
unlike fractions, most of them relied on

3

mechanical procedures which they had

acquired. Very few could tell why

unlike fractions had to be changed to

similar ones before adding. "That is

how I learned it," was the common

response to the question, "Why do you

change these fractions to fractions

having a common denominator?"

One consistent conclusion from error studies is that
the subjects tested showed "incompetency" in addition
of fractions (Guiler, 1945a; Guiler, 1945b; Sebold,
1947). None of the error studies, however, yielded
information concerning those students who do become
competent.

In the transition to "social" arithmetic
in the late 1920's, 1930's, and the early 1940's,
emphasis turned to instruction with only those com-
mon fractions which were socially useful (Wilson
and Dalrymple, 1937). Social utility was again
argued by Johnson (1956) who indicated that since
adult usage favors decimals, only the most common
of the common fractions should be taught and that
the place value principle should be extended at an
earlier age to include decimal fractions which are
inherently easier.

During this same period of time leaders in
mathematics education such as William A. Brownell
and C. L. Thiele promoted the idea that more "meaning"
must occur in the teaching of arithmetic (Brownell,

1935; Thiele, 1941). Investigations have shown that

students performed arithmetic computations signifi-
cantly better when specific efforts were made to
assist the pupil in understanding (Steele, 1940;
Reward, 1947; Pigge, 1964). Howard showed that

the use of concrete materials in develOping meaning
significantly improved the performance level.

Pigge‘s study showed that a combination devoting

50% or 75% of the time to developmental-meaningful
activities enabled the pupils to perform significantly
better than did pupils who had been exposed only 25%
of the class time to developmental-meaningful activities
with the remainder in each case being devoted to a
drill.

Textbook changes seemed to reflect both the
social utility arguments and the plea for the incor-
poration of more "meaning". Dooley (1950) reports
that research resulted in the elimination of awkward,
unrealistic fractions as well as the increased
utilization of illustrations as visual aids.

Just as the aforementioned studies of the
first half of this century reflect the psychology
and philosophy of that time, so must a study done
today reflect those of the present time. Some of
todays thinking is indicated by the following:

One thing we can all be quite certain of:

Wherever in the vast realm of human learn-

ing we wish to look for individual

differences, we surely will find them.
...Arthur Jensen (1967, p. 117)

At the present time it seems fair to say
that we know considerably more about
learning, its varieties and conditions,
than we did ten years ago. But we do not
know much more about individual differences
in learning than we did thirty years ago.

...Robert M. Gagne (1967, p. xi)
Approaches to teaching can be better de-
signed only when we better understand how
people learn mathematics.
...E. Glenadine Gibb (1968, p. 434)
At the turn of the century, the treatments
became less axiomatic, and presentations
were geared to what was believed to be a
child 8 level of understanding. This theory
caused a "low" as far as axiomatic insights
into arithmetic were concerned, and the sit-
uation existed for a period of at least
fifty years. Since the mid-fifties, however,
textbook presentations have been based on
the axiomatic understanding of the structure
of the number system.
eee818ter A. Me Sibilia (1959’ P0 207)
Instead of reporting the problems of the
groups of students who have not learned to add frac-
tions, this study will report on the successes of
those individuals who are learning to add fractions.
Instead of emphasizing drill as a technique for pro-
moting learning, it will emphasize pattern and
definition as a technique for discovering the proces-
ses involved in addition of fractions. Instead of
considering the social utility of the material being
learned, it will consider the overall structure of
number systems. Instead of looking only at grouped

data on a pretest and/or posttests, it will look at

the progress made by individuals at regular intervals
during the learning process.
Fractions and Rational Numbers
Although some texts define a fraction to be
a name for a rational number, there is a growing
tendency to allow a fraction to be a number as well
(Hill, 1967). In this study the latter definition
will be used.
(1) A fraction is an ordered pair of
natural numbers a and b which is usually
named by the symbol "a/b". Two fractions
a/b and c/d are said to be equivalent if
a x d = c x b.
(2) A rational number is a class of
ordered pairs of integers. The ordered pairs
are written in the form m/n, with the restric-

tion that "n" is never 0.2

When an ordered
pair for which m and n are both positive inte-
gers is chosen from the set to represent the
rational number, we call this ordered pair a
fraction. When two fractions are equivalent,
they represent the same rational number.

A more complete development of the concept

of fractions is given in Chapter II.

 

2Peterson and Hashisaki, Theory of Arithmetic
Second Edition, John Wiley & Sons, 9 5, p. 172.

Design of the Study

By analyzing the processes involved in the
addition of fractions, a list of concepts and prin-
ciples deemed necessary for the performance of an
example was established. These principles were
arranged in a logical hierarchy with the goal repre-
sented by the example 3 3/10 + 2 5/6. The individual
elements of the hierarchy were likewise represented
in mathematical terms so that a measure of pupil
understanding could be made. This was the first
component of the study.

The second component was the development
of achievement and diagnostic instruments to measure
the student's achievement of the concepts and prin-
ciples of the hierarchy. One test form was designated
to be used as a pretest and a posttest. Five other
forms were designated to be used at regular intervals
between the pre and posttests. A seventh form was
designated to be used as a retention test two months
after the learning period.

The third component of this investigation
was the materials. One set of materials A, con-
sisted of the last two chapters of the Addison-Wesley
Fourth Grade textboooksB, pencils and paper. The

3Eicholz, et a1., Elementary School Mathe-
matics, Book 4, Addison-Wesley Publishing 00., Inc.,
Reading, Massachusetts, 196%.

second set of materials B, were units specially
prepared for this experiment. These materials
began witha.number line approach to rational numbers
and permitted students to find the sums of fractions
using concrete aids after two days of instruction.

The last component of this investigation
was two classes of fourth graders; the classes being
chosen at random from all fourth grades at 7 elemen-
tary schools in the East Lansing Public Schools.
One of these classes, A, was assigned the commercial
materials and instructed as a total group with every-
one in the class working on the same material at the
same time. The other class, B, which was not as far
along in the textbook material as the first class,
was assigned the second set of materials which max-
imized individual work and also allowed some free
choice of the order in which certain of the principles
were studied. Both of the classes were conducted
by the writer for the 20 day period of the investi-
gation.

Purposes of this Study

The purposes of this study were (1) to
develop an instructional unit on the addition of
fractions which is designed for individual instruc-
tion, (2) to compare the effectiveness of this unit
with that of a standard textbook unit on the same

material presented on a class basis, and (3) to

determine invariances in the order in which students
develop an understanding of the principles involved
in adding fractions.

The primary hypothesis related to these

purposes was:

(1) The order in which items from the
hierarchy are mastered does not differ from
one class to the other.

Three secondary hypothesis were also con-

sidered:

(2) The order in which items from the
hierarchy are mastered supports the logical
order as indicated on the hierarchy.

(3) Students using the experimental mater-
ial will make no greater change in performance
than students who used the textbook material.

(4) Students who already understand some
basic concepts of fractions can progress further

in the hierarchy than those who do not.

CHAPTER II
BACKGROUND FOR FRACTIONS

The Concept of Fraqggu;

Contemporary literature exhibits a variety
of interpretations of fraction and related concepts.

Peterson and Hashisaki (1965) describe four
interpretations of fraction with Hill (1967) and
Fehr (1968) offering a fifth. Botts (1968) explains
three uses of the word "fraction" extending the list
of two offered by SMSG (1962). Still others (Brumfiel,
et. a1, 1961) equate "fraction" to "rational number"
in certain circumstances. To appreciate the full
scape of the concept of fraction, each of these points
of view needs to be considered.

Landau (1960) defined a fraction, developed
the fraction as an element of a mathematical system,
and then defined a rational number in terms of frac-
tions. Key steps in his exposition are

Definition 7: By a fraction 2} (read "xl over x2")
x
2

is meant the pair of natural numbers x1, x2 (in
this order).
Definition 8: x1 y1

“N..—

x2 y2

(nu to be read "equivalent") if

xlye = ylx2 °

10

ll

, X1 yl w u
Definition 13. By __H*__. (+ to be read plus )
x2 y2

x y + y x
is meant the fraction 1 2 1 2 o

x2V2
x1 y1
It is called the sum of ___ and ___,
x2 y2

or the fraction obtained by the addition of

y X
_1’ to _£ .

Definition 16: By a rational number, we mean
the set of fractions which are equivalent to
some fixed fraction.

Definition 17: X = Y
(z to be read "equals") if the two sets consist
of the same fractions. Otherwise,

x g Y

(£ to be read "is not equal to").
Let X and Y be integers, say X = x and Y = y.

Then by Theorem 114, the rational number E

determined by Definitions 26 and 27 stands for
the class to which the fraction % (in the earlier
sense) belongs.

Along with the above key definitions Landau
proves theorems displaying the rules that apply to
fractions under the operations of addition and mul-
tiplication. Fractions, the operations defined on
fractions, and the rules governing these Operations
form a mathematical system. Are fractions in this

context numbers?

13

Botts (1968) makes a distinction between
fraction as a number and fraction as a pair of
numbers. In regards to the latter he writes (p. 218)

Now here we may be sure that we are not
speaking of fractions as numbers, for the
numbers 3/2 and 6/4 are the same, and that is
what we assert when we write 3 2: 6/4. In
this usage the term "fraction applies to
something whose essential feature is a pair
of numbers, a numerator and a denominator....
Such a numerator-denominator pair does, to be
sure, define or determine a number in a conven-
tional way, namely the number that results
from dividing the numerator by the denominator.
But the fraction, in this usage, is really the
numerator-denominator pair, not the number we
get by dividing.

Hence, two interpretations of fraction are
suggested: fraction as an element of a mathematical
system vs. a fraction as a quotient. However, what
is the nature of the number that is obtained by
dividing?

Some authors would call this number a frac-
tion as well. Gibb, et. al, (1959. p. 29) wrote

Fractions were invented to deal with parts
of things and to make division always possible.

The authors of SMSG (1962) chose to use
fractional number when they were talking about the
number, although later in the unit the term fraction
was used, relying on the context to make clear what
was meant.

Fouch and Nichols (1959, p.334) explain

Thus, a fractional numeral is a symbol
naming a fractional number. We use the phrase

"fractional number" to be synonymous with

l4

"rational number". Since in common usage the
word "fraction" is used to refer to a number,
we may abbreviate and also use "fraction" to
be synonymous with "fractional number" or
"rational number".

Other authors use the desirability of a
system to be closed under division or to have a root
of the equation nx = m to motivate the axiomatic
development of the rational numbers.

Sibilia (1959, p. 161) summarizes

A modern approach to the study of fractions

is based on the axiomatic construction of the
rational number system. This provides an inter-
pretation of fractions as elements of the
rational number system.

The definition of fraction as a name for
a rational number is one which cannot go unnoticed.
SMSG (1962) defined fraction to be a numeral. As
such it had a numerator and a denominator. Landau
(1960, p. 42) used the same symbol x/y to refer
both to the rational number and to the fraction
which belongs to the rational number. In one sense,
he was using a fraction from the set to represent
the rational number.

The symbol for a fraction is a pair of
numerals (Gibb, et. a1, 1959. p. 29). Since the
same symbol can be thought of as naming a rational

number, the symbol and the fraction are often con-

fused (of. Mueller, 1961).

15

In summary the word "fraction" is used to
denote many different ideas:
(1) An ordered pair of natural numbers
as an element of a mathematical system. (Landau,
1960, p. 19).
(2) An ordered pair of numerals which name
a rational number (Mueller, p. 196) or a frac-
tion (in the sense of (1) above) (Gibb, 1959,
p. 29).
(3) A rational number. (Fouch and Nichols,
1958. Do 334).
In this study, the context will dictate in
which way the word "fraction" is used.
Interpretations of Fractions
Several interpretations of fractions are
common in the elementary school. In contemporary
textbooks (Cf. Eicholz, 1964; SMSG, 1962) fraction
as a partition is generally developed first. Equiv-
alent fractions are defined as those which name the
same partition. Although some texts (Brueckner,
et. a1, 1963, p. 316) discuss addition in terms of
partitions, others introduce rational numbers and
develOp addition in terms of the number line.
(Eicholz, et. al, 1964, p. 222).
For each set of equivalent fractions, think
of one rational number and one point on the
number line. Any fraction from a set of equiv-

alent fractions can be used to name the rational
number for that set.

16

The Partition Interpretation

In a fraction a/b, the denominator, b, tells
the number of equal parts an object or set is to be
divided into and the numerator, a, tells the number
of the parts which are to be considered. This inter-
pretation is used in answering questions such as
the following:

What part of this
circle is shaded?

(Ana. 6/16 0r 3/8)e

Shade 3/4 of the
squareSe

 

What is another fraction which tells the part of
the total number of squares you have shaded?
(Ans. 6/8).

Historically, fractions owe their creation
to the transition from counting to measuring. (Gibb,
et. a1., 1959. p. 29). As a measure "2/3" may be
conceived of as (1) naming the number property of a
set of_2 elements each of which is 1/3 of some unit.
Fractiong as Operators

Although somewhat similar to the interpre-
tation of fraction as a partition, the numerator and
the denominator are considered in the opposite order.
For the fraction a/b, the a is a stretcher (it mag-
nifies a quantity a-times) and b is a shrinker which

17

has the inverse effect to b. (Fehr, 1968). A
fraction as an operator is used extensively in the
UICSM Materials. (Braunfeld and Wolfe, 1966;
Braunfeld, et. a1., 1967).
Example: Find 2/3 of 12.
The 2 operates on 12, doubling its value.

Nexg the 3 operates on 24, shrinking it
to .

A line segment 12 units long would be
used to represent 12.

The Division Interpretation
A fraction a/b indicates the quotient a f b.

 

This interpretation is used in answering questions
such as the following:
Find 15/3 (Ans. 5)
What is 3 {- 2 7 (Ans. 1 1/2 or 3/2).
The Ratio Inteppretation
Ratio denotes a relative comparison of quan-
tities and as such is a pair’of numbers. (Van Engen,
1960) calls such comparisons of quantities rate pairs
and cautions that the addition of rate pairs does
not follow the usual fraction rules. Fbr this reason
he does not call a rate pair a fraction. He writes
It is now apparent why fractions and rate
pairs (usually called ratios) are often con-
fused. Both have in common the following
prOperties:
a. The test for equivalence;
b. Membership in only one equivalent set.

Here the similarity ends. Pairs of numbers
used as fractions can be added according to the

18

usual rules of arithmetic, but pairs of numbers
used as a rate pair are not added according to
the usual arithmetic rules.

Bidwell (1966). however, points out that if we
consider fractions to be added as those which
represent a ratio of disjoint sets to a given set
C, then the sum of the fractions will be the ratio
of the union of the disjoint sets to the given set.

A A U B B

At
a + b

C

 

.9
0

Old

 

Some examples that can be performed using
the ratio interpretation are (Johnson, 1948)
What part of 27 is 9?

Ans. For each unit in 9, there are
3 units in 27, hence 9 is 1/3 of 27.

Find 1/4 of 24.

Ans. We need to find a cardinal number
such that each element in its set can
be made to correspond to 4 elements of

a set of 24. Since 6 x 4 = 24, the
answer is 6.

The Element‘g§_a Mathematical_§ystem Interpretation
By a number system is meant a pgp of

numbers, Operations defined on the numbers in the

set, and pplgg governing these operations. (Peterson

and Hashisaki, 1967, p. 74.) The system of fractions

19

consists of the set of fractions with binary

Operations + and X possessing the properties

Closure for + and .

+ and x are commutative.

+ and ‘X are associative.

There exist identities for + and for X .

Fer every element of the set there exists
an additive inverse and a multiplicative
inverse (with the exception of O).

X distributes over +.

The conditions for equivalence of fractions.
Order.

Several formal developments of fraction

and of rational number are given by Hill (1967).

CHAPTER III
RELATED LITERATURE

The literature provides few results which
have direct bearing on this study. Related inves-
tigations fall into three general categories: the
comparison of methods by which students learn, the
histories of class performance during the learning
process, and the collection and analysis of data at
specified times during the learning process.

Methods of Teaching

In an investigation comparing two methods
of teaching Lankford and Pattishall (1956) found a
significant difference in favor of an experimental
method with two important features:

(a) Ideas and rules of arithmetic are
developed inductively through pupil partici-
pation.

(b) Pupils are encouraged to learn
arithmetic thoughtfully and ppdepepdgnp_1.

To this end we encourage mental arithmetic
and varied approaches.

In an initial pilot study they had found
(1956, p. 3) that many pupils had learned very
little from the conventional teaching of arithmetic
other than a list of processes which were apparently
used in a highly mechanical manner with little
thinking and often still getting incorrect answers.
Another impression received during the pilot study

was the unexpected indication that the bright pupils

20

21

interviewed followed both literally and uniformly
the conventional algorisms in arithmetic as did
the pupils with lower ability. Regarding this
impression Lankford and Pattishall wrote (p. 25)

They get more correct answers because their
memories were better and they were more careful
workers. Their attention spans were also longer
than those of the dull pupils. There was little
indication in these interviews held prior to
the experiment that bright pupils learn from
conventional teaching to do arithmetic more
independently, with more originality, or more
thought than do duller pupils.

Lankford and Pattishall concluded (p. 67)

Perhaps the most important fact demonstrated
by this study is that it is a sound procedure
to allow pupils to use as much freedom and explor-
ation as they require to understand fully working
with fractions.

Using a pretest, a posttest, and a retention
test, Fincher (1963) found that the use of programmed
materials is more effective than the use of conven-
tional textbook approach in the teaching Of addition
and subtraction of fractions and no less effective
than the conventional method for recall after a four-
week interval.

In comparing learning by drill to learning
with extensive use of audio-visual aids and consid-
erable emphasis being placed on meaning (Howard,
1950, p. 29), no significant results were found in
computation with fractions at the end of the initial

learning period. HOwever, when the same students

were tested the following September, the results

22

favored those who had made use of audio-visuals
(either part or all of the time).

Souder (1943, p. 134) found that the use
Of the diagnostic readiness test for instructional
purposes differentially affects the learning of
pupils.

In a study to determine the extent to which
the identification and correction of errors in sets
of examples in addition and subtraction of fractions
affected learning, no statistically significant dif-
ferences were found to favor either the experimental
or the control group. (Aftreth, 1958).

Anderson (1966) found no significant differ-
ence in teaching either of two procedures for the
addition of fractions: that of setting up rows of
equivalent fractions or that of factoring the denom-
inators.

Histories of Class Performance

In 1932, Edwards investigated how students
differ in learning about fractions. Using a plan
of instruction which allowed each student to move
step by step through the same materials at his own
rate. he found large differences in the amount of
progress made during identical periods of time.
Pupils who ranked high in general arithmetical ability
and mental ability required less attention from the

teacher and attained a more complete mastery of the

23

processes taught. In attempting to develop a
regression equation from which to predict student
success, he found the equations little better than
a guess. Students in his study developed some
ability to solve problems upon which no previous
instruction or drill was given.

In another sequence of units designed for
individual instruction, Brooks (1937) confirmed
that there are very great individual differences in
the time needed by pupils for the completion of a
unit of learning. He studied the workbooks of the
individual students as well as the pretest-posttest
analysis. He found that units presented different
degrees of difficulty to different students but that
certain units of work were more difficult than others
to the group as a whole. In the individual scores
no pupil showed a steady and consistent gain from
unit to unit of learning and none high in the early
units drOpped off greatly at the end. The data pro-
vided by Brooks is by units or by class. He did not
study the question of sequencing problems for indi-
vidual students. Each student follows the same
pre-designed curriculum and no data is given indica-
ting which Of the concepts or principles of each

unit are attained.

24

The Colle_ction__and Analysis of Data f9;
the Early Elementarngrades

In investigations by Gunderson (1958) and
by Gunderson and Gunderson (1957), it was found
that seven year Old children are interested in,
like to, and are able to work with fractions when
teaching is done without use of the fraction symbols
and with the use of manipulative materials. The
problems solved by seven-year-olds (1958, p. 237)
involved addition of fractions, subtraction of
fractions, and fractional equivalents. Both studies
concluded that there is a need for a long acquain-
tance period between the child's first introduction
to fractions and the time he is expected to work
with fractions using algorithms and symbols.

By means of a single interview, each of
266 children from the kindergarten, first, second,
and third grades were tested to discover when, what,
and how concepts of fractions are acquired naturally
by children (Polkinghorne, 1935). It was found that
the acquisition of concepts without formal teaching
is a continual process and a direct result of
experience with fractions in daily living. Kinder-
garten and first grade children showed understanding
of unit fractions only. In grades two and three
other proper fractions, improper fractions, and the

identification of fractions were known. No evidence

25

was displayed for an understanding of equivalent
fractions.
Preliminary to her investigation,Sebold
(see next paragraph) found indications that concepts
of fractions were known by some of the students prior
to formal teaching. She also found several miscon-
ceptions (See Table I, page 26).
The Collection and Analysis of Data for Later Grades
Sebold (1946) used individual testing or
interviews, group testing, and individual instruction
in efforts to discover the mental processes through
the development of which pupils arrive at an under-
standing of the basic concepts in fractions. She
concluded that (p. 79)

There is no uniformity in learning among
the children. Not all children in a given
grade are at the same level of learning in
respect to all the concepts. Not every child
is at the same stage in respect to all concepts,
nor do all children traverse the same series
of stages preliminary to final, meaningful
understanding.

In general, the learning of the basic
concepts in fractions progresses through the
following levels of understanding:

(a) No knowledge of the concept.

(b) Erroneous ideas of the concept.

(0) Confusion of the concept with
others, expecially with those which have
been only partially learned. But in the
confused ideas there is often an element
of correctness.

(d) Partially correct but vague ideas.

(e) Knowledge that an incorrect pre-
sentation or illustration of the concept
is not true.

26
TABLE I

CONCEPTS AND MISTAKEN CONCEPTS OF FRACTIONS
DISPLAYED IN EARLY ELEMENTARY GRADES

First
and
Second
Grades

Third
Grade

Fourth
Grade

Concept Displayed

Name a few fractions

Recognize the fractional
parts of figures or
objects which have been
divided

Unequal divisions of
figures cannot be
designated as fractional
parts (1/ 4 of pupils)

Name the fractional part
of a figure that was
equally divided

Finds fractional parts of
groups of figures with
respect to unit and other
proper fractions (1/ 3 of
pupils)

Understands the above (Far
less than 1/3 of pupils)

One half >one fourth
(60% of upils)

Can add 1 2 + 1/2

Fraction is one or more
of the equal parts of
a group and of a number.

Find fractional parts of
/gg)ups (357 o>f plO. )

1 3 1 6,1 2> 16,
Nil/47 1/2, 2/3> 1/3,

Add a few fractions.

Mistaken Concepts
Displayed

Any part of a figure

is one half or some
other fraction.
Name the fractional
part according to
the number of parts
into which the
figure was divided,
irrespective of
equal or unequal
divisions.
1/6>1/3, 1/671/2

The word fraction
connotes "one-
half" (50% of
pupils in third
and fourth grades)

* Based on data given by Sebold (1946, pp. 26-28).

27

(f) Recourse to the visualization

of a previously acquired model with

descriptive phrases explaining it.

(g) Memorized information on the
concept with or without understanding.

(h Partial understanding .

(1 Full understanding.

In an analysis, by means of class tests,
of children's mental processes in multiplying frac-
tions in the fifth grade, Collier (1922) found that
the child‘s mental processes related to whole numbers
and not to fractions.

Knight and Setzafandt (1926) studied the
problem of the extent to which training in the addi-
tion of fractions involving the denominators
2,3,6,8,10,12,16 and 24 transfers to the ability to
handle the addition of fractions in which the numbers
3,5,7,9,l4,15,16,21,28 and 30 are used as denominators.
A substantial amount of transfer was shown to occur.

Hayes (1927) found that the same type of
errors tend to appear with the same relative fre-
quency throughout the grades. Gundlach (1936)
observed that there is great variation in the ability
of individuals within each grade for each of the four
Operations in fractions. He found a great variation
in the ability in fractions between pupils within
each group representing a different level of capacity
but that the ability of those in a group of greatest
capacity is less variable than those in the group of

least capacity. In addition he computed that the

28

curves of growth in ability in the operations with
fractions for the three levels of capacity are
similar to the curve of the entire group, the dif-
ference being in level of performance.

13 m: There appears to be little difference
in the patterns of development followed by the more
able student as compared to the less able. Meaning-
ful materials on an individual basis and materials
using concrete aids have met with relative success
but other differential approaches have had little
effect. Learning, as measured by existing testing
devices and evidenced by grouped data, appears to
progress smoothly rather than.in.an irregular
fashion producing sharp changes when new concepts

or principles are encountered.

 

CHAPTER IV
RESEARCH DESIGN AND DESCRIPTIVE DATA

Development of the Hierarchy
Lists of concepts and skills related to

the addition of fractions (Cf. Becker, 1940; Sebold,
1947; Howard, 1948, p.24) have been compiled.

These, however, were designed to include the entire
spectrum of possibilities rather than simply those

necessary for the attainment of a particular objec-

tive.

Gagne (1965) offers a totally different
approach in develOping a hierarchy:

As described previously ( ...) the method
employed was to ask the question of the final
task, 'What would the learner have to know how
to do in order to attain this final performance
when given only instructions?’ In this case,
the question applied to the final task yielded
the identification of five subordinate know-
ledges. When applied in turn to these
subordinate classes of tasks, and then success-
ively to the additional tasks so identified,

the analysis yieldeda hierarchy of subordinate
knowledges,..

..., each successive step in the analysis
yielded one or more subordinate knowledge
entities that are progressively simpler and
more general as one proceeds downward in the
hierarchy. The basic set of hypotheses gener-
ated by his 'knowledge structure' is the
following: (1) the attainment of each entity
of knowledge (measureable in each case as a
particular performance) is dependent upon posi-
tive transfer of training from the next lower
subordinate knowledge connected to it by an
arrow; and (2) such transfer requires the high
recallability of all the next lower subordinate
knowledges (connected to it by arrows).

29

30

The ultimate goal set in this experiment
was the principle of adding fractions, i.e.

2 5/6 + 3 3/10. The attainment of this goal is
dependent upon first learning some other principles
such as the associative and commutative properties
of addition, the definition 3 3/1o = 3 + 3/10, the
principle of adding two fractions when they have

the same denominators, etc. These principles in
turn depend upon knowing the concepts of 3/10, 1/10,
5/6, l/6, 2, 3, etc. The logical organization of
knowledge so developed was represented in a hier-
archy of principles and concepts.

Since there are many interpretations of
fraction, the three considered most appr0priate for
fourth grade were included, relying on later testing
to determine which of these are prerequisites to
addition. Fractions as ratios or operators were not
included since interviews conducted by the writer
with fifth graders who could perform the indicated
final task have shown that the fifth graders did not
understand these two interpretations. Hence, they
could not be prerequisite knowledge.

Construction of the Tests and
Revision of the Hierarchy

Using the hierarchy, several exercises
were constructed to test each item. Six fourth

graders from the Wardcliff School, Okemos Public

31

Schools were asked to work the exercises as best

each could and the results were recorded. Trial
materials were developed and the six fourth graders
were given instruction with fractions using these
materials and being tutored by the writer during

five one hour periods. At the close of instruction,
each was asked to again work the exercises. On the
basis of the testing and instruction experience, two
items were found to have been overlooked in the
original development of the hierarchy and other rela-
tionships between items were discovered. The
hierarchy was revised resulting in that shown in
Table II, page 32. The revision included no deletions
from the original but did include a reorganization

of the order of items and the inclusion of the two
additional items.

For each item on the revised hierarchy, one
exercise was chosen (See Table III, page 33 ). These
were composed to form the final survey test which
was used for both pretest and posttest (See Appendix
A). Exercises similar to those on the final test
were written for each of the other six test required.
The trial materials were redeveloped in their final
form (See Appendix B).

The Materials, Treatment A

The materials for Group A consist of the

Addison-Wesley Fourth Grade Text (196%), pencil and

32

TABLE II

HIERARCHY OF PRINCIPLES AND CONCEPTS

NECESSARY FOR ADDITION OF FRACTIONS

 

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TABLE III
THE HIERARCHY, BY EXAMPLE FROM SURVEY TEST

 

 

 

 

 

 

 

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34

paper. Although some paper folding was used for
demonstrations at two different occasions, no
manipulative aids were employed by the students.
The text made liberal use of diagrams and illustra-
tions. The students were asked to respond orally
to some pages and in written form to others. All
pages which pertained to sections of the hierarchy
were used. A few pages on the formal definition of
equivalent fractions were omitted as well as pages
unrelated to fractions.4 An outline of the topics
studied in the order in which they occured follows.
Table IV illustrates how the materials related to
the hierarchy, the labels correspond to the topics
in the outline.

 

“Pages were assigned in this order: 240-255,
257. 258, 262, 263, 265, 269 (B parts), 282, 283,
285-289, 328 (Set 42), 295 - 2%é , 298, 299. 278,
300-3245 306, 310, 311. Extra 29 , 292, 297. 328
Set 3 .

35
TABLE IV

CORRESPONDENCE OF MATERIALS
TO THE HIERARCHY, TREATMENT A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

36

Outline of Material, Treatment A

I.
A

II.
A

QHCUQHJWU Ow

Fractions

Partition interpretation of fractions (3/4,
2/7, 4/6, etc.)

Fractions as pairs of numbers

Fractions of rectangles

A fraction a/b means a of b EQUAL parts
Fractions of segments

Fractions comparing a part of a set with
the whole set

Fractions and parts of an object

Sets of equivalent fractions (partitions)
Sets of equivalent fractions (patterns)
Fractions with numerator ) or = to the denom-
inator

Fractions with zero numerators

Review

Rational Numbers

Rational numbers as sets of equivalent fractions
1. Think of one rational number

2. One point on the number line

Practice with the concept in A

Equivalent fractions name the same rational
number

Inequalities

Rational numbers greater than one

Addition of fractions using the number line
Fractions which name whole numbers

Nuxed numerals

Addition of fractions using parts of wholes
Review

37

The Materials,_Treatment E

Before developing the materials for Group B,
the current literature on teaching and learning was
surveyed. Educational psychologists were consulted.
A fourth grade class at the Warddmff School in
Okemos, Michigan was observed and tested to determine
their understanding of fractions, as well as their
general mathematics background. The units were tested
in sequential order with a group of 6 children,
revised, and put into final form for use in the exper-
imental situation.

Jerome Bruner and Helen Kenney (1965, p. 51)
described an experiment in representation and math-
ematics learning in which eight year old children
were given instruction in various mathematical acti-
vities:

Each child had available a series of graded
problem cards to go through at his own pace.
...the problem sequences were designed to pro-
vide, first, an appreciation of mathematical
ideas through concrete constructions using
materials of various kinds for these construc-
tions. From these, the child was encouraged
to form perceptual images of the mathematical
idea in terms of the forms that had been con-
structed. The child was then further encouraged
to develop or adopt a notation to describe his
construction...

A second experiment was also described in

this paper (p. 57) in which a group of 10 nine-year-
olds were instructed in the elements of group theory.

Again the approach was one in which the children

38

first worked with physical manuevers and later
deVeloped notation and ability to work with the
symbols.

In the concluding paragraph of this paper
Bruner and Kenney (p. 59) wrote:

We would suggest that learning mathematics
may be viewed as a microcosm of intellectual
development. It begins with instrumental acti-
vity, a kind of definition of things by doing.
Such Operations become represented and summar-
ized in the form of particular images. Finally,
and with the help of a symbolic notation that
remains invariant across transformations in
imagery, the learner comes to grasp the formal
or abstract properties of the things he is
dealing with. But while, once abstraction is
achieved, the learner becomes free in a certain
measure of the surface appearance of things,
he nonetheless continues to rely upon the stock
of imagery he has built en route to abstract
mastery. It is this stock of imagery that per-
mits him to work at the level of heuristic,
through convenient and non-rigorous, means of
exploring problems and relating them to problems
already mastered.

It is in accord with the above lines of
thinking that set E of materials on the addition of
fractions developed. The instrument in this case
is an expanded concept of the number line in which
students in the early elementary grades are, in
general, very familiar. A unit of measure was repre-
sented both by a rectangle (in the materials provided
this is a rectangle 1%" x 10%") considered to be 1
unit in length and by a point on the number line that
distance from zero. Rectangles 1 unit in length were

divided into fractional parts of the unit. The

39

children were to learn how the pieces could be
constructed and used by constructing and using them.
The standard fractional notation is used and the
process of adding fractions is interpreted as the
summing of lengths of rectangles. Students were to
be encouraged to discover the algorithms for finding
equivalent fractions and for adding fractions so
that the concrete materials would be gradually neg-
lected. This section of the materials was designed
to be used for approximately five days of activity:
two days for orientation to the materials and con-
struction of them in groups of two and three days
for manipulation with them on an individual basis.
The instructions were provided on 4 groups of 5 X 8
cards (19 in all) and the exercises for practice on
four 8 X 10 dittoed sheets upon which the answers
could be written. The first five days of activity
were to provide the active stage.

For the iconic state, 8 sequences leading
to generalizations about the abstract processes with
fractions were developed. Any of the eight could
be chosen to be worked at any time during this stage
and each was expected to require from 1 to 3 instruc-
tional periods for completion.

The regularly given tests provided an
opportunity to develop at the symbolic level. The
tests also served as a challenge to stimulate the

discovery by the student of algorithms for solution;

40

the concrete materials provided verification of a

correct solution. No algorithms were provided.

41

Outline of Materials, Treatment B

I.

NQH
00

II.

Construction and manipulation of concrete aids

Construction of number lines with whole
number designation

Construction of number lines with fraction
designations (i's and fi's)

Agreement to use 1 unit equal to the lengths
provided (10%")

Construction of a number line to the a reed
upon scale (l/2's, 1/3's, 1/4's, 1/6'8§
Construction of rectangles of 1, 2, and 3 units
of length

Construction of rectangles of lengths a fraction
of 1 unit (1/2'3, 1/3'3 1/4'3, l/6's, l/7's,
l/l4's)

Finding different names for the same sized
rectangles (Equivalent fractions)

Finding I name for a rectangle equal to the
sum of two rectangles (Addition of fractions)
Names for lengths greater than 1 unit
(Definition 1 1/6)

Comparison of lengths of rectangles

Practice materials

Sequences for development of generalizations

Common multiples and least common multiples
Equivalent fractions (By partitions)

Names for

1. Equivalent fractions (by multiplication)
2. Equivalent fractions (by number line)
Adding using equivalent fractions
Interpretation of fraction as an indicated
division

Unit fractions using partitions

Other fractions using partitions

The associative and commutative properties

42

The Popplation Samplg

The population sample was drawn from the
East Lansing Public Schools, East Lansing, Michigan.
It is a system of about 5048 students in grades
kindergarten through twelve, contiguous to Lansing,
and composed mostly of well-educated middle class
residentsa The two classes of fourth graders used
in this study were chosen at random from seven
elementary schools and 12 fourth grade classes.

One of the classes was assigned the com-
mercial materials (Treatment A) and the other class
was assigned the experimental materials (Treatment B).
The choice was conditioned by the fact that one group
(A) had been accustomed to traditional methods of
teacher controlled instruction, whereas the second
of the classes had been introduced to individual work
two weeks prior to the initiation of the scheduled
instruction period. Both classes were conducted by
the writer during the period of the experiment.

In each of the groups there were a few
students who had skipped either part or all of the
third grade arithmetic and/or part of the fourth
grade work. These students had not yet developed
skill in the multiplication of whole numbers. In
Group A, the majority of the class had completed
the fourth grade work on multiplication and were

working with a section on approximation which

43

precedes work in long division. This group, with
the exception of those four who had been moved
ahead, displayed mastery of multiplication with
whole numbers less than ten. In Group B, however,
the majority of the students had not yet mastered
the multiplication facts. This might have served
to hinder their discovery of patterns in working
with fraction which depended on this knowledge.
Although these students had been working on the
same textbook as Group A, they had notprogressed
as far.

Fourth graders were chosen for the experi-
ment because it was felt that these students had
limited instruction in fractions. In the planned
curriculum for the East Lansing Schools, a unit of
fractions first appears in the last quarter of the
fourth grade.

Initial Characteristics

Initial characteristics of each student
were obtained from three different sources. These
measurements were his score on the mathematics sec-
tion of the Sequential Tests of Educational Progress,
a 26 item test on basic number facts of addition,
subtraction, multiplication, and division, and his
score on the pretest. The initial data is presented

in Table V.

44

TABLE V

SUMMARY OF INITIAL TESTS

 

Treatment Test Number Mean Standard
Received Tested Deviation
A STEP 1/68 22 249.6 9.4

Number Facts 22 21.1 5.4
Pretest 22 3.6 2.4

B STEP 5/68 22 249.8 11.4
Number Facts 22 20.9 4.8

Pretest 22 1.6 1.3

 

 

45

Teaching the Units

On February 1, 1968, the two regular teachers
administered the pretest to their classes. Over the
next five weeks, for 20 sessions, the writer conducted
the two classes, administering the posttest during
the twentieth session. The retention tests were given
the last week in May, 1968 and each regular teacher
said that she had not assigned any work with fractions
in the interim. The results of the tests appear in
Table VI, page 46.

The lessons for Group B were restricted to
50-55 minute periods daily; the lessons for Group A
averaged 55 minutes in length. Both the regular
teachers and the writer were available to answer any
questions the students had during the mathematics
period. Each day in Group A, the written work from
the previous day was returned and discussed, some
oral drill on material in the text was conducted,
the new material was introduced and oral responses
solicited for discussion questions, and a written
assignment was made and supervised.

In Group B, two students worked together to
construct the concrete materials, then each worked
individually through the practice sheets using the
concrete aids. The student was allowed to proceed
to some other section of the hierarchy according to

his choice and the availability of material. Each

46

TABLE VI

SUMMARY OF SURVEY TESTS

 

 

Treatment Pretest Posttest Retention
Received Mean Mean Mean
A 3.55 12.86 10.55
B 1.64 12.14 11.41

Respective Standard Deviations

 

 

A 2.42 4.23 4.88
B 1.26 6.56 5.92
Total Number Pretest- Posttest
of Exercises Posttest Retention Test
Performed, Gain, Gain,
Mean Mean Mean
A 15.41 9.32 -l.85
B 15e41 lOeSO -0073

Respective Standard Deviations
A 4.17 3.47 3.42
B 6.64 6.13 3044

47

student had as his goals, the completion of the
short units and trying to acquire enough concepts
and principles to be able to perform the examples
on the surveys when given. No direct teaching of
how to do addition examples abstractly was provided.
Daily Reports

In Class A, daily reports of class pro-
cedure and material studied were prepared by the
teacher. Time allowed for the lesson, times of
testing, and other pertinent observations were
recorded. Three sample reports for each group are
included in Appendix C.

In Class B with each student operating
individually daily reports of general procedures
being followed, time allowed for the lessons, times
of testing, and other pertinent observations were
recorded. In addition a chart was kept on which a
record was kept of the date on which each student
finished a section of the work. A second blank was
filled when the teacher felt that the work had been
understood by the student, i.e. if the answers were
90% correct or if, upon questioning the student
displayed understanding.

The Intgrmediate Testing

In addition to the pretest, posttest, reten-

tion tests, and the daily class records, alternate

forms of the pretest were administered regularly

48

beginning the sixth day. As each type of example
was performed correctly it was eliminated from
future tests for that individual.

Students in both classes were thus motivated
to discover how those he had not yet solved could be
achieved. Tables VII, VIIa, VIII and VIIIa show the
order in which each individual student performed the
exercises, the numbers used correspond to those numbers
of boxes on the hierarchy.

Some Individual Higtgrigg

In this section two students from each class
who performed few examples on the pretest are chosen
for a more thorough analysis.

Of students receiving Treatment A, C2 is an
interesting example. 02 was to be a third grader but
was instead placed in the fourth grade class because
she was exceptionally able. Her background in
multiplication and division were particularly weak.
On the pretest l, 7 were accomplished. After the
instruction on the partition interpretation of frac-
tion 2 and 16 were mastered. Next 12b, then 11
because of their relationship to whole numbers.

After discussions on equivalent fractions 10a was
performed, then 12a. On the posttest she was able

to project to 5, 6, 10b, 15 and 19. Other students
weak in multiplication in the class performed the same

examples, some adding 24 to the list or 8 (Cf. M4, L2,

49

 

 

 

 

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53

Cl, etc.). Several of the students who were better
prepared at the onset performed the same examples on
the pretest (Cf. H, N1, and N2).

Another student who was able in multiplica-
tion made stronger gains under treatment A. $2 began
with the ability to do some basic whole number Oper-
ations (l, 7, 8). After the introduction of and
practice with the partition interpretation of fraction,
2, 5, 10a, 12a, 12b, and 16 were added to the list.

As his understanding was strengthened with practice,

6 and 14, then 4, 15, and 24 were incorporated into
his rapport. But it was not until after the addition
of fractions was discussed that he was able to incor-
porate the firm conceptual background into the process
of addition so that he performed 3, 9, ll, l7, 19,

20, 21, and 23 on the posttest which he had not per-
formed previously. Fourteen other students in the
class also performed their first addition of fractions
after brief discussion prior to Survey VI with 5 more
accomplishing these after classroom practice had been
provided.

Of students receiving Treatment B patterns
are not as easy to find. Surges are much more common
as are lack of further progress. J2 worked steadily
and consistently throughout the period of experimen-
tation. Performing only example 1 on the pretest,

she showed greater strength with whole numbers on

54

Survey II (7, 8) and with some familiarity with frac-
tions (5, 6) she was able to also do 12a, and 12b.
By Survey III she was able to accomplish as much as
many under Treatment A completed in the entire exper-
imental period, adding 10b, 15 and 19. After 2 surveys
showing no progress, she developed the partition inter-
pretation of fractions (2) and the division interpretation
(3, 4) and was able to again begin building (9,14,17,
20,24). J2 was still growing in her understanding of
fractions a week after the retention test when she
joyously showed the teacher that she had figured out
how to do examples which she had found in a more ad-
vanced arithmetic book and which were similar to
example 22.

Other students also made steady gains and
1’ T2)’
Several made expecially rapid gains at the beginning

retained well on the retention test (M1, P, T

but either lost interest or had reached a point that
they needed actual instruction and practice in the
algorithms for addition rather than in the concepts
(Of. 3, K, Mg).

E had a very difficult time understanding
the concept of fraction in any of its manifestations.
Her development shows that each thing she did related
the fraction work back to an algorithm regarding whole
numbers. 1, 8, 7, her first successes were whole

number operations. 3, 10b, 15, 12a, 12b, 19 were

55
performed by relating to the whole numbers involved
and to a pattern she discovered in them. It was not
until Survey V that some fraction concept began to
develop (16) and this was retained. 5 and 16 were
both performed on the retention test.

Others, too, seemed to have trouble with the
concept of fraction. B2, N, J3, R still had not
developed them for the retention test. Some had so
much trouble developing them that they retained little
else. (B1, J1, L).

Some General Classroom Observations

The two classroom presentations place in
direct confrontation two distinctly different methods:
that of teaching the class as a whole and that of
providing individual instruction. Of the two, teaching
the class as a whole from a standard textbook is by
far the less taxing on the teacher's time. A diffi-
culty with discipline can occur when several students
finish their written work ahead of the others and
need to be directed to some individual project while
the others finish. The faster students can be en-
couraged to look beyond the present work by setting
a long range goal for them (as was done by giving the
inventories fairly often so that they could be applying
what they had learned to discovering how to solve
future problems) or by providing additional problems

of the challenge variety which help them to discover

56

principles relating to what is likely to come up in
the near future. On the whole, however, the faster
students are bored with the in-class explanations
and the classroom drill which is essential to push
the slower student along with the rest. This time,
for the more able student, might be better spent in
some other way.

The individual instruction method demands
extra time of the classroom teacher both in prepara-
tion of the materials (which are not available
commercially) and in the burden of correcting many
diversified types of papers each day. Without the
convenience of oral drill, more work is done on paper,
and the faster students turn in work at a much higher
rate than they would under the classroom plan.
However, after an initial organizational period, the
students do keep busy during the entire time pre-
scribed for arithmetic and often request extra time.

The individual plan attempts to provide the
administrative machinery whereby the pupil is per-
mitted to learn at his own rate, to receive help from
the teacher only when he needs it, and only upon his
own individual difficulties.

In this study an additional contrast was
provided since extensive practice on the examples em-
ploying the concepts and principles studied was
provided in the classroom group, but little practice

57

was provided to those receiving Treatment B. If the
unit itself were to be used as an instructional tool,
many exercise sets should be developed to provide
practice with each concept and skill which was devel-
oped in any particular unit. Developing the concept
or principle does not of itself insure being able to
use it again later. It is thought that using the
concepts or principles after developing them will
facilitate their later use and their application to
the next level of difficulty.

Contrast in enthusiasm of the two groups
was very noticeable. Those students in group A who
were academically minded, i.e. who verbalized that
they thought it was fun to learn, were mildly pleased
to see the experimental teacher at the time she made
a return visit. The girl who make the lowest scores,
on the other hand, made a point to complain with some
support from others who did not like being pushed to
accomplish a great deal of work.

In group B, however, students jumped out of
their seats and asked if the experimental teacher
could come back and teach for a few days. Even when
threatened with even harder work than ever before,
they agreed that they would do it. This group had
continued working as individuals using their textbooks

after the experiment had been completed.

After the conclusion of the experimental

58

period and after some time had passed, the teacher
of those students receiving Treatment B reported
that her students were solving their exercises in
division by writing the remainders as fractions rather
than simply R. __, This transfer indicates some degree
of understanding of fraction concepts beyond that nor-
mally encountered in a fourth grade class.
Summary

In an effort to present the analysis of data
in a comprehensive fashion this chapter first statis-
tically compared the two treatments for resulting
level of performance and statistically analyzed the
order in which the concepts and principles were under-
stood. Some individual histories which demonstrated
certain patterns were described and some contrasts
of the two methods of teaching were described. Some
conclusions which may be drawn from this analysis are

presented in Chapter V.

CHAPTER V
ANALYSIS OF DATA

Introduction

The data are analysed both with regard to
order and with regard to performance for each of the
approaches: (A) use of calmercial materials, and
(3) use of experimental materials. In addition,
individual progress and support for the hierarchy
are considered.

To test the comparability of the two treat-
ments, previous achievement (STEP) and level of
performance (Pretest) are analysed. Then, Statis-
tical significance of the change in performance
(Pretest-Posttest) is measured using the t-test.

The main criteria for the determining order
were Surveys (I-VIII) . The items frcm: each of the
Surveys were considered in pairs (_a_,_b_) . For each
class the number of students who performed 5 on an
earlier survey than _b_, who performed 2 on an earlier
survey than a, and who performed _s‘ and p simultan—
eously were tabulated. This tabulation was analyzed
using the binomial test and where applicable an
ordering ; < p or p < _s; was established. The results
for each class were compiled into a projected hier-
archy. .

At the end of this chapter observations are

made, leading into the summary, conclusions, and

S9

60

recommendations presented in Chapter VI.
Results of Ipigial Testing

Level of arithmetic achievement as measured
by the Sequential Test of Educational Progress,
Mathematics, was used as an independent variable to
judge whether or not there were significantdiffer-
ences in the mathematical ability of the two groups.
Approach A had a mean achievement of 249.60: approach
B had a mean achievement of 249.77. The difference
was less than 1 point and a t-test showed a nonsigni-
ficant t = ;95 with 3g degrees of freedom. However,
since the STEP was given to Group A in January and
to Group B in May, the lack of significant difference
indicates only that Group B did notachieve signifi-
cantly better than Group A. Both of the means fall
above the average school means quoted in the STEP
Manual for Fall testing of the fifth grade.

On the test of 26 number facts in addition,
subtraction, multiplication of single digit numbers
and division by single digit numbers, Group A per-
formed more accurately than Group B. With means of
21.10 (A) and 20.82 (B), this difference was not
significant (t = .002).

The Survey of Skills in Fractions was used
as a pretest and a posttest as the critical measure
of the effect of a particular treatment on the two

groups. The next section analyzes the two adminis-

61

trations of this test.
The Survey Tests

The main criteria for measuring performance
was the Survey Test, administered as a pretest, a
posttest, and a retention test. Table IX, page 62
reports the analysis of these tests.

Using the t—test with 42 degrees of freedom,
no significant differences were found except on the
pretest. The pretest indicates that the students
receiving Treatment A had a lead on the students
receiving Treatment B at the beginning of the study
although this lead was completely eliminated by the
time the retention test was administered. Pretest-
Retention test results showed the mean of individual
net gain to be 9.77 in Group B and 7.35 in Group A
although 2 persons in Group A did not take the reten-
tion test. The data is not sufficient for conclusion
that the net gain of students receiving Treatment B
was significantly greater than that of Group A
although data indicates that greater gains were made.

On all 3 tests, Pretest, Posttest,and Reten-
tion Test the variance of scores under Treatment B
were 1/5-3/4 again as great as the variance of scores

under Treatment A.

62

TABLE IX
ANALYSIS OF THE SURVEY OF SKILLS WITH FRACTIONS,
PRETEST, POSTTEST, AND RETENTION TEST BY
APPROACH TREATMENT

_—

Analysis by t-test

Group N Mean SD t p

A 22 3.55 2.4
Pretest 3.03 .005
B 22 1.64 1.3

A 22 12.86 4.2
Posttest .04 NS
B 22 12.14 6.6

Retention A 22 10.55 4.9
Test .05 NS
B 22 11.41 5.9

Pre-POSt A 22 9e32 3e5
Gain .71 NS
B 22 10.50 6.1

 

POSt- A 20 -1e85 Bea

Retention 1.06 NS
B 22 -e73 Bea

Total

Problems A 22 15.41 4.6

Mastered .00 NS

by Each B 22 15.41 6.6

Individual

 

The Ordgg of Performance

The hypothesis, Hi, to be tested is that the
order predicted by the task analysis diagram is indeed
the order in which each student performed each task;
i.e. if.g precedes p.0n the diagram, that the student
would perform example a prior to performing example
.b. The null hypothesis, HO would indicate that there
was no difference between the probability (p1) of
performing example.g first and p,second and the
probability (p2) of performing p,first and.g second.
Hl implies that pl< p2.

The binomial test was chosen because the
data was in two discrete categories and the design
for each class was of the one-sample type. Since
under the null hypothesis there was no reason to
think that a.should be learned prior to h" P = Q = A.

The significance level chosen was 6‘ = .001.
N, the number of cases, is the number of persons
performing example g,prior to example h, plus the
number of persons performing example p_prior to example
'3, plus the number of persons performing a.and p_both

satisfactorily for the first time on the same test.
x

The sampling distribution given by 2:61) PIQN"1 was
:0

obtained from Siegel, Nonparametric Statistics for
the Behavioral Sciences, Table D, p. 250. For

probabilities that g,would occur by chance in the

64

tabulated relationship to b_of less than .001, the
conclusion is that the learning of 2.13 prerequisite
to the learning of b, The region of rejection of

HO consisted of values of x (where x : the number of
subjects who performed example h.prior to example a)
which are so small that the probability associated
with their occurence under H0 is equal to or less
than = .001. Since the direction of the differ-
ence was predicted in advance, the region of
rejection is one-tailed. Table X to XIII, pp. 65-68
give the tabulations of the number of students
performing g,and p.0n the same survey test and items
and the tabulations of the number of students per-
forming §.on an earlier survey than 2, These
tabulations are given separately for students under
Treatment A and under Treatment B. Tables XIV and
XV, pp. 69-70, give the probabilities that the items
would be performed as tabulated under the null
hypothesis, H

O
The Order of Performance Under_Treatment B

Table XV, p. 70 gives the probabilities that
g,would occur before h_in the quantities tabulated
by Table XII, p. 67. In the event that the probabil-
ity is less than or equal to .001 the conclusion is
that g,is prerequisite to h.

Since examples 1 and 7 are in general pre-

requisite to all other examples, and since neither

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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m azmza<mme .p 24mg emma wm>mpm
magma 5 20 a afizmammm BESS mo 35,82

HHN mflmda

TABLE XIII

SAME SURVEY AS 2, TREATMENT—B

NUMBER OF STUDHTS PERFORMING I ON THE

 

 

b 1 2 3 u S 6 7 8 9 10a 10b 11 12a 12b 13 1h 15 16 17 19 20 21 23

 

68

O
N

HH

In:

:Hd-d’

H HHH

N O OO

41' Fir-{Him
:Ld’N .d‘ M-d'
OMMNHM :m
N: :NH: NN
Omm NNNm mm

d—d’MMN—d't-IMO NN
NNHMNSH N HM-d'

1%

6.0 C
OOHN
NMJMONQO‘HHHH

O
m:

gm;-
:mmO
NHHNm

H MHNNM

H
.d‘NHHHNdl-n

\ONN NJNOO‘
H

LAO—3H OMS—3m

mgmH Nmzdm

H HHHHHH

mNm xoMmNN

MNNH MNNMM

N Flu-4H ..‘d'NHH

N‘Lndm MNNMN

mm H H
NMMH NNH r-l
mmm N HM
:NNN Nmmmm
mNam-O (Ugh-IMO
NMOHHMMNHM

N H

MfiMONQOOHMfi
H HHHHHNNN

 

 

69

 

 

 

oo oo cc po cm
oo oo po mwvguco acauwwwcmHWcoc macawvcw mxcupm mm
xguco coma mwvoumgg o.

NN
co co co Pm
co oo oo om

mp oo Po Po oo ._o mo co co mm -_o mm mp
mm mp

co mm mm up

No oo -Fo _o oo -Po mp co em m_ -.o em O_
mo oo -_o -Fo oo ._o so 00 -Po ON -Fo NO NO mp
00 m_ oo mp oo cc «—

mp

mm co co Po oo -—o am oo _o mm -_o mm -_c amp
mp co co _o oo -Fo mm oo _o mm -Fo mm VmNF
_o mm mo No HP

No co No mm oo oo oo mm _o co,
oo oo oo mop

No No om m

mm co oP oF oo mm Nm oo oo No .0 w
oo oo -Fo oo -_o oo as we _o ._o oo NO ON .90 -Po -Po co mm N5 me -_o -_o Po N
mp co co NH oo Po co cm _H O
we oo Po oo oo —o oo mm we __ m
mm oo mm oo we co e
mm co co co co co Nu F_ m
-Po oo _o oo ._o -_o _N 00 mp mm -_o N
oo oo oo oo -po oo 00 No .—o co co mo me _o Lpo -Fo oo _o No me -_o _ -po -_o h

 

 

mm mm Hm om mp OH NH OH mp op mp nm— mNH Hp pop mop m m n O m e m N — a

HHx NAm<N no mmHNHNz<zo amN<42m<N mIN zH muz<zu >m.m mmommm mauuo ogzoz.m N<IN >NH4Hm<momm

>x m4m<N

'70

 

 

 

 

 

No no co ~N mo -No ‘moNLuco ucmuvyvcm—mcoc mucuNO:N mxcopm
O— Nuucu Oumo mouauwgq o.
ON
Ho co Nc oo as No -No No O0
00 No oo No oo co oo oo oo NO u—o No No mo co -No NO
ON No as _o -No oo oo NO oo No No mm
No N— _o NF Oo
No No oo —o oo oo oo Ho oo Om No co Oo
No No oo No -No co co No co No oo Om
No No oo No -No co No No -No No No
NN No oo _o -No co co Oo co __ No On
No No 00 No co oo _o No -No No -.o Oo
we we \NO
No No oo No oo oc oo _o as NO No mm mm co -No
oo No oo No oo oo oo oo oo ON uNo No -No -No Do uNo oo
_o No oo No oo co so No oo _o so No
Oo No as No co so No No 00 No 00 No
co mm oo mm om
MN co m— MN co m— Hp
co No oo No oo oo oo oo so No -Fo -Fo
co .0 as ._o oo No oo oo oo mo -No No No -No oo -No oo
em mN mm NN om ON NF «N mp amp NP a O O c m N

>Hx m4m<N

x m4m<N >m omN<42m<N mmHNHNz<30 NIN zH moz<zu >m.m mmommm mzouo 44H3.m N<IN mmHNHgHm<momm

cm
mm
NN
FN
ow
m—
OF
N—
O—
m—
ep
mp

ONH
ONH

_H
sop
mop

0‘

71
example involves fractions per se, it is indicated
that these two examples be included among the basic
arithmetic of whole numbers studied before fractions.
Examples 13, 18, and 22 are in general those requiring
all others for mastery. Hence, it is indicated that
these three types be the last types studied. The
other prerequisites under Treatment B are

2 < 9, 17, 21, 23

5 <: 17, 21
6 < 17
8 < 9
10b < 9, 17
12a < 11, 17, 20
12b < 9, 11, 17, 20
15 < 9, 11, 17, 20, 21
16 < 9, 17, 20, 21

19 < 9, 17, 20, 21

The resulting suggested hierarchy appears
1n.Table XVII, p.73 . The Table has been adjusted
to reflect the number of students who performed each
example correctly at any time during the experiment
as given in Table XVI, p.72 .

Some relations of the hierarchy that are
not supported by the data collected from the group
receiving Treatment B are those involving 2, 5, 6,
16, 15, and 19. One would anticipate that unit

fractions need to be understood before those with

72

TABLE XVI

PERCENT OF STUDENTS PERFORMING EACH EXAMPLE

BY PERCENT AND NUMBER

!

 

Percent Examples Performed Examples Performed
and by the Percent by the Percent
Number Treatment A Treatment B
lOO%-22 1,2,7 1,7,15
95%-21 6,11,15,16 l2a,12b,19
91%-2O 5,10a,12a,12b,19 2
86%-19 8
82%-18 8,16
77%-17 10b
73%-16 5,10b
68%-15 3.6
64%-14
59%-13 24 14,24
55%-12 3 4.20
50%‘11 103,23
45%-1o 14 11
41%—9 9
36%-8 A 17.21
32%-7 9
27%-6 20,23
23%-5
18%-4 13
14%-3 21
9%-2 17 13,18
5%-1
O%-O 18,22 22

73
TABLE XVII
PROJECTED HIERARCHY UNDER
TREATMENT B, BY EXAMPLE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9
ll
{23]
I 20 I
E]
lmll 5|
We
__ .
[2|
19 15

 

 

 

 

 

 

 

I Basic operations with whole numbers
? including 1 and 7.

*3,4,10a,14,24 do not occur in this diagram.

understood that they require 1 & 7 and are prerequi-
site to 13,18, and 22. But no other relationships

were well defined.

74
non-unit numerators and also before one learns to
add fractions. However, there was no strong indi-
cation that this was true. Students may have
developed a pattern for addition which depended on
the concrete aids without understanding fully how
the concrete aids were deve10ped.

The number of simultaneous solutions of 15
and 19 indicate that the commutative and associative
laws are used by the students to the point that they
do not interfere with the performance on example 19
once example 15 is understood. The early solutions
of 12a and 12b indicate an easy association of
fractions with fractions and whole numbers with whole
numbers. There appears to be much more difficulty
in relating whole numbers to fractions as is indicated
by the lower number of students performing examples
14, 20, and 23 compared to the number performing 19.

Examples 4 and 14 and 15 are closely inter-
related. The tendency was for 15 to be solved before
either 4 or 14 but this might be accounted for because
of the number of students who did not receive instruc-
tion in the definition of a fraction as an indicated
division whereas all students had eXperience with
material similar to example 15 either through the
use of fractions on the number line or fractions as
partitions.

Example 11 was included in the diagram to

75
provide one route to the understanding of equivalent
fractions. There is no indication that being able
to perform 11 in any way aided further performance
of examples.

One would not anticipate that 2, 10b, 12b,
15, 16, and 19 would be prerequisite to 9 since the
concepts involved in 9 relate only to whole numbers.
Perhaps, since the processes of multiplication were
not fully developed by the students receiving Treat-
ment B, 9 proved so much more difficult than the
others as to influence the authenticity of the statis-
tical test. Similarly, the difficulty with which

students relate whole numbers to fractions as was
apparent in the relationships of 19 to 14, 20, and
21, may have caused the unexpected result of showing
some items prerequisite to 17 and 20.

The types of errors made by students under
Treatment B seem to progress in a set pattern. At
first students seem to add all four numbers together
and arrive at a single whole number answer. Next the
numerators were added together and the denominators
were added together giving a fraction for an-answer
(similar to the addition of ratios). Following this
the students became able to correctly perform those
having common denominators but merely substituted a
convenient number in the denominator of the other sums

such number being the sum of the denominators or the

76

product of the denominators or one of the denomin-
ators. The fourth stage was one in which a common
denominator was determined but it was not understood
how to deal with the numerators. Correct addition
of easy combinations which could be analyzed though
diagrams occured next with successful performance
coming last.

Order of Performance Under Treatment A

Table XIV gives the probabilities that.g
will occur before b according to the tabulations in
Tables X and XI.

It is apparent that examples 1 and 7 were
performed satisfactorily prior to the satisfactory
performance of most other examples. Since neither
of these involve knowledge of fractions they could
be considered part of the basic arithmetic prerequi-
site to understanding fractions.

Of the other examples, 16 shows the highest
number of examples to which it can be considered
prerequisite. On Survey II all except one of the
students receiving Treatment A performed satisfac-
torily on this item. This predominance of success
may result from the fact that the first instruction
received introduced the concept necessary for the
performance of this item and provided practice with
similar examples. Only 2 students failed to retain

the concept well enough to perform item 16 on the

77

posttest or retention test.

The next most required item was number 8
which may indicate that the students receiving
Treatment A had develOped their concept of multipli-
cation to a higher degree than that of the students
receiving Treatment B. This example required no
understanding of fractions.

At the other end of the scale, items 13, 17,
18, and 22 require the greater number of prerequisite
understandings with 20, 21, 23, and 24 followed by
4 and 9 requiring the next most. Since 18 and 22 were
performed by no student receiving Treatment A and 13,
21, and 17 by only 4, 3, and 2 respectively, we chart
these at the top of the hierarchy. Other prerequisites
are listed here:

2 3.4.9.14,19,2o,23,24
5 3,4,9,14,2o,23
6 3,4,9,14,20,23,24
8 3,4,9,20,23,24
lOa 4,9,1A,2o,23,24
10b 20,23,24
ll 4,9,20,23
12a 4,9,14,20,23.24
12b 4,9,14,20,23,24
15 4,9,20,23
l6 3,4,9,lOa,ll,14,15,l9,20,23,24
l9 9,20,23

78

Table XVIII, page'fi; gives a projected hier-
archy for these relationships. One striking feature
of the projected hierarchy is the fact that those
which were performed by the most students were also
performed on earlier tests and were prerequisite to
most of the others. Of items performed by all of the
group 1 & 7 are prerequisite to at least 19 items,

2 to 14 items. Of items performed by all but 1
student in the group, 16 was prerequisite to 16 items,
6 to 12 items, and 11 and 15 each to 9 items. Of
items performed by less than 17 persons, none is pre-
requisite to more than 2 others at the .001
significance.

Another phenomenon is the performance of an
item for the first time on approximately the same
test for all students. Examples of this are 16 and
2 (on Survey II), lOa (on Survey III), 1 and 7 (on
Survey I), 19 and 24 (on Surveys VI and VII).

The processes of multiplication may not have
been developed by fourth graders sufficiently to
enable 9 to have been performed in a natural order.
However, in the teaching of material similar to
example 10a, students showed relative ease in assim-
ilating this concept which also relied on multiplication.
No teaching was directed toward examples similar to 9.

Another unexpected order was the position

which 3 and 4 take in the hierarchy. Instead of the

79
TABLE XVIII

PROJECTED HIERARCHY UNDER TREATMENT A,

BY EXAMPLE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Basic knowledge gfdghole gumbers 7]
I no u ng

80

concept of fraction as indicated division being a
building block, it is an application of other concepts.
Perhaps this was the result of no formal instruction
in the use of the bar to indicate division. Those
who performed 3 and 4 may have done so through con-
siderations of equivalent fractions.

Seemingly unrelated on the hierarchy are
10a and 10b. 10a asks for an equivalent name for a
rational number less than one and 10b asks for an
equivalent fraction name for one. Only 2 persons
performed this for the first time on the same test;
others were randomly split, some performing lOa first
and some performing 10b first.

Exercise 2 required the student to divide
a figure into equal parts and to name a unit fraction
for the shaded part. Exercise 16 was already marked
in equal parts, the student was required to write a
fraction for the shaded part which involved the use
of a non-unit numerator. No relation between the
two appeared in the data. 16 was performed on the

average on Survey II, 2 on the average on Survey IV.

CHAPTER VI
SUMMARY AND CONCLUSIONS

Summarypof the Investigation

This study investigated the order in which
children learned some concepts and principles which
enabled them to perform examples of addition of
common fractions. To delineate the concepts and prin-
ciples necessary to the addition of common fractions
a hierarchy was developed which had as its base the
understanding of operations with whole numbers and
as its apex the performance of the example 3 3/10 +
2 5/6. Two classes were instructed in the concepts
of fractions and tested at regular intervals. One
of the classes used commercial materials and was in-
structed as a total group with everyone working on
the same material at the same time (Treatment A).
The other class used a specially developed set of
materials which maximized individual work and allowed
some free choice of the order in which certain of the
principles were studied (Treatment B). The test
results were analyzed to determine invariances in the
order in which students develop an understanding of
the concepts and principles of the hierarchy. The
data was examined for patterns of learning. The rela-
tive performances of two classes were compared.

Contrasts in the two methods were reported.

81

82

Consideration of the hypotheses, the rele-
vant findings of each, and the conclusions follow.
Suggestions for further research in teaching frac-
tions conclude the chapter and the dissertation.

Findings and Conclusions
Hypothesis 1

The order in which items from the hierarchy
are mastered does not differ from one class to the
other.

Findings
The main criteria for determining order were
the Surveys (I-VIII). The items from the surveys
were considered in pairs (a,b). Fbr each class the
number of students who performed a on an earlier survey
than 2, who performed p on an earlier survey 3, and
who performed a_and h simultaneously were tabulated.
This tabulation was analyzed and where applicable an
order a<b or b (a was established. The results for
each class were compiled into a projected hierarchy.
Comparisons of the two hierarchies indicates
that:
(i) There were more significant pairings
under Treatment A than under Treatment B.

(ii) In general all of the prerequisites
under Treatment B are also prerequisites under
Treatment A. Two exceptions are those items

prerequisite to understanding the concept of

83

least common multiple (9) and the principle of
multiplication of fractions (11). One explana-
tion may be that since the students receiving
Treatment B had not received as much practice
in whole number multiplication as those receiv-
ing Treatment A, these two items proved more
difficult to them and were delayed by reason of
their difficulty.

(iii) Students receiving either treatment
seem to find the same items easy and the same
items difficult. The items performed by the
most students in each group are considered the
easy items and are 1 and 7, then 2, 12a, 12b,
15, 19 and 16. The items considered difficult
are those which few performed: 13, 17, 18 or
22. There were noticeable differences, how-
ever, in the number of students of each class
performing 10a and 11. Ebre of the students
receiving Treatment A could name rational numbers
on the number line even though direct instruction
was provided during the first week of Treatment
B.

(iv) Even though most all students under
either treatment performed the addition examples
involving fractions with the same denominators,
there was a noticeable difference in the survey

on which these items were first performed. Under

84

Treatment A, 2 performed both on the pretest,

15 did not perform either of the examples until

Survey VI and 5 not until Survey VII. Only 4

(of 20) retained both principles and 2 others

retained one. At the time Survey V was given,

all except one receiving Treatment B had per-

formed both of the items, that one had performed

only one. Only 5 of the students failed to

perform one of the items on the retention test.
Conclusion

Evidence indicates that with the exception
of the two items cited above (9 and 11), those orders
established under Treatment B are invariant. It also
appears that prescribing the order of instructions
has a direct effect upon the order of learning. more
students seem to have an understanding of the parti-
tion and the rational interpretation is taught first
and drill provided. Mere students seem to have an
understanding of the addition of fractions having
the same denominators if they approach addition through
the rational number interpretation of fractions using
concrete aids than if they approach addition through
the partition interpretation.
Hypothesi§_g
The order in which items from the hierarchy

are mastered supports the logical order as indicated

on the hierarchy.

85

Findings
The hierarchy indicated that n/n should be

recognized as a name for 1 (item 24) before p/q can
be renamed by np/nq (item 10a). The basis for this
inclusion was that students could see the pattern

in equivalent fractions through a multiplication by

l : n/n. (Item 11 tests understanding of the prin-
ciples of multiplication of fractions.) Students
receiving Treatment A, however, learned principles
for renaming fractions, p/q : np/nq prior to learning
that n/n : 1. This is the only contradiction to

the logical order as indicated on the hierarchy.

There are some other relationships, however,
that are indicated by the data which are not indicated
on the hierarchy. One of these is the paralleling
deveIOpment of the concept of unit fraction as com-
pared to non-unit fractions as embodied in examples
2 and 16. Results indicated that 16 would be more
likely to precede 2 rather that the other way around
as would be logically expected.

Under both treatments 9 and 11 required many
elements not indicated as prerequisite by the hier-
archy. This may be because they are more difficult
concepts or because no direct teaching of either item
was provided. Items 14 (a + bbé = 1 +(b/a) and 24
also seemed to be more difficult than anticipated

and required 12a and 12b for performance under

86

Treatment A. Items 3 and 4, which can be performed
using the division interpretation of fractions, did
not prove to be as basic as indicated on the hier-
archy.
Conclusions

Understanding equivalent fractions can be
acquired without understanding of the multiplication
of fractions or of fractional names for one. An
understanding of mixed numerals may aid in the under-
standing of fractional names for one or in renaming
an imprOper fraction with a mixed numeral. The
division interpretation of fraction was not utilized
by either group to aid in the develOpment of the
principles of addition of fractions. Direct teaching
of least common multiple may be necessary for the
understanding of this item.
Hypothesis 3

Students using the experimental material
will make gains in performance no greater than students
who used the textbook material.
Findings

Students receiving Treatment B did make
greater gains in performance than students who used
the textbook materials and retained them better.
However, these gains were not statistically signifi-
cant.

Although the mean number of problems performed

87

by each individual did not differ under the two treat-
ments, the standard deviation under Treatment B was
1% times that under Treatment A. There was greater
diversity in the order in which problems were per-
formed under Treatment B than under Treatment A.
Informal observation in later mathematics lessons
seemed to indicate that the students who had received
Treatment B were more enthusiastic than students who
had received Treatment A.
Conclusions

It is feasible to employ a method of indi-
vidualized instruction to a study of fractions. The
use of concrete aids manipulated by students appears
to promote better understanding of the process of
addition of fractions and more enthusiasm on the
part of the students.
Hypothe§;§_fl

Students who already understand some basic
concepts of fractions can progress further in the
hierarchy than those who do not.
Findings

For those 6 students who understood the
partition interpretation of fraction at the time of
the pretest (Item 16), the average pretest-posttest
gain was 12.7 as compared to the average class
pretest-posttest gains of 9.32 and 10,50. One

phenomenon which might account for this difference

88

would be the translation of Operations with fractions
into algorithms with whole numbers by those recently
introduced to fraction concepts. Several students
individual histories showed that they performed only
examples which had easy algorithms involving whole
numbers.
Conclusigg

Students who have some previous understanding
of the concept of fraction seem to make greater gains
than those who do not.

MW

It is recognized that most students in this
study had average or above average mathematics back-
grounds and came from a suburban middle class
neighborhood. By using materials related to the last
two chapters of a textbook in the middle of the year,
some of the whole number backgrounds necessary for
the study of fractions may have been missing. In
other situations different results may have been ob-
tained. It is this writers belief, however, that
in situations where the mathematics background is
not as strong, students will show even greater gains
with the concrete materials than with regular text-
book materials and that their use need not be limited
to the fourth grade. It is the individual classroom
teacher, in her individual situation, with each

89

individual child, who will determine whether the
results of this study have implications for her.
It is hOped that the study will provide some basic
ideas from which she can draw.

Suggestions for Fprther Rgsgarch

This study has investigated the order in
which some of the concepts and principles related
to the addition of common fractions are learned.
Introducing the partition interpretation of fraction
first followed by appropriate drill seemed to pro-
mote greater understanding than introducing fractions
as represented by points on the number line. Would
the results have differed significantly if the
division interpretation of fraction had been intro-
duced first?

Another of the findings was that the ear-
lier introduction to fractions on the number line
seemed to facilitate understanding and retention of
addition of fractions. If the concept of fraction
were introduced using the partition or division
interpretation followed by the use of fractions to
represent points on the number line before any other
concepts or principles, would the result prove sig-
nificantly different?

Do plateaus of learning exist? Knowledge

of the associative and commutative properties and

90

of the multiplicative identity seemed to precede
knowledge of other concepts. Another level appeared
to exist which included the partition and rational
number interpretations of fraction, the addition of
fractions having common denominators and the under-
standing of common multiples along with mixed
numerals. This was approximately the average per-
formance level. This level seemed to be accompanied
by an understanding of fraction in terms of its
number components and often is accompanied with

1/6 )1/4 or similar misconceptions. What is the
nature of the transition from thinking with whole
numbers to thinking with fractions? Is the tran-
sition enhanced by the introduction of the concept
of fraction as early as the first grade?

The present investigation involved fourth
graders. Discussions pertaining to the proper
placement of fractions range from beginning in the
first grade without using symbols to waiting until
the latest possible time. At what states of child
development should each of the concepts and prin-
ciples be introduced? Does the answer to this
question depend upon mathematical,cultural, or
economic backgrounds? Does it depend on the child's
level of ability or upon something else? Should
the concept of fraction initially be introduced

91

without symbolism?

The present study placed in direct con-
frontation two distinctly different methods. Will
a mixture of teaching the class as a whole with
short sequences of individual instruction promote
better understanding with fractions than having the
entire unit taught by one method or the other?

Do students in the individual situation
develop attitudes, habits, ability to discover or
other abilities that are not developed by those in
the class dominated by the teacher?

Whereas the two methods were equally suc-
cessful with the two groups, there is no guarantee
that the same results would have been achieved had
the methods been reversed. What past learning
experiences of the students make one method more
efficient for them than another? Does having
participated in individualized instruction affect
the rate or kind of learning that takes place under

other treatments?

APPENDIX A

Surveypl

 

 

 

 

 

1) Find 3) Find
Whatpart
2+9+3+8+l+7 Of the 15,
figure is 3

Show your work. shaded?

4) Find 5) Name the point 6) Name the
indicated by a dot point indicated

15. on this line. by a dot on
2 this line.
C l O

7) What is 8) Find a number 9) Find the
which can be found least number

3,679,215 x l? by multiplying a which is a
number by 10 or by multi 1e of
multiplying another both and 10.
number by .

10) 11) 12)

2:9. 2,9,- 3+1:

5 15 3 5’ 10

- 2
l’g 3§+2=

|-'
U

 

93

 

94

 

 

 

 

 

Men
13) 14) 15)

1 1 l2 2

2 + S = II = 7 + g):
16) What part of 17) 18)
this figure is
shaded? l 1

9+6: he:
19) 2o) 21)
1
3 k + 2 2 _ g + = 7 5 + 9 } =

22) 23) 24)

3 Q5 + 2 2 =

 

3;+6$:

 

owm
u

Basic Number

9 +

72

36

24

35

54

63

95

Facts

 

13

15

14

APPENDIX B

Pack 1 Card 1

1.

2.

3.

Take a strip of cardboard which has a line drawn
annit. Choose a point on the line and label it
0 (zero). Choose a point 1 unit from 0 and
label it '1" (one).

What point should you label 2? Label it.
What point should you label 3? Label it.

The line you are constructing is called a number
line. Label two more points on this line.

 

 

Materials needed: Several strips of heavy paper,

1%" wide and approximately 9' long. Several
short strips of heavy paper 1%" wide and
approximately 2" long labeled "1 unit".

 

 

 

Pack 1 Card 2

1.

2.

3.

Take a short strip which is labeled "1 unit".
Place this piece so that it fits between 0 and
l on your number line. Did you put 1 in the
right place?

Place another short strip, “1 unit", between

1 and 2 on your number line. It's left edge
should be at 1. Where is it's right edge? What
is another name for 1+1?

Place a third piece to the right of the first
two. Where should its right edge lie? What
is another name for l+l+l?

97

98

Pack 1 Card 3

Make another number line. This time place the
point 0 near the left edge of your strip. Be
accurate.

 

 

Example:
A J J J _1 1
o 1 2 3 4

 

Take a one unit piece and fold it in the middle;
fold it in 2 pieces each the same length. Cut the
piece on this fold. Mark a point on your number
line which is the distance from O.

 

 

 

 

 

 

 

, .02 fold
[ 1 unit _; l
Pack 1 Card 4
L _ IA I 43 ' Ac L AD J E
o 1 2 3 4

1. If we label point A, "A", and point B, "it",
how should we label point C? - -

2. How should we label points D and E?

3. Fold a piece 1 unit long into 4 equal pieces.
Cut on the folds. Mark the corresponding
points on your number line. Label these points.
Mark and label all of your line.

‘A A

7 7 7
1 1 .L

_h

1’

L

J
4

0L

_1 _L_ L21?
-%- 2 3

44h.)-
H h
H
wu- r-
..n
{JP
...1 )-
$9».

.1.
4

 

99

 

Pack 1 Card 5 Materials needed: A long strip

1.

2.

3.

of paper approximately 1 yard
long. 2 or 3 strips of card-
board 105" long by 1%" wide, one
of them marked off into thirds..

 

 

 

Take a long strip of paper and a shorter strip
of cardboard marked 1 unit. Construct a number
line with O, l, 2 and 3 labeled on it.

Fold the cardboard "1 unit" into 2 e ual parts
and cut it. Mark the points "t", "1' ,"23"
on your number line.

Fold each piece of the cardboard into 2 e ual
pa ms and cut it. Mark the poi ts "%", "3"
Of ’ "1% "1%." "1%." 2101,012 n’ "12$ ’ on

your number line.

Take another 3strip of cardboard 1 unit long.
Cut it into 3equa1 parts. Mark and label the
points "3L", "a"...1§',"1%", 02$", "2’" on
your number line. Save your number line.

Pack 1 Card 6

How can we label the points marked on each of these
number lines? Label them. Then take this card to
your teacher to be checked.

r—F—l

boo
_-
4

O
H

 

 

 

 

 

 

[J I 1 I 34! 43—1 Li I 2;; L: I

Lo 1 2
(2314.12.1333
L o 2

 

 

100

Pack 2 Card 1

1. Find a 1 unit strip which is marked off in
3 equal parts. Cut on the marks. Label
each piece "3".

2. Find a 1 unit strip which is marked off in
2 equal parts. Cut on the marks. Label each
piece " '.

3. Find a 1 unit strip which is marked off in 7
equal parts. Cut on the marks. Label each
piece '4".

4. Make some "fi"‘s and some "i"'s. Label them.

 

Materials needed: Number line made on Card 5
of Pack 1. Several strips of cardboard 1 unit
long marked off in 2,3,4,6 and 7 equal parts.
(Strips should measure 10%" long by 1%" wide).
Box to keep pieces in.

 

 

 

* ‘

Pack 2 Card 2

1. Find a 1 unit strip which is marked off in 3
equal parts. Cut off 1 of these parts. Label
it "j”. Label the other piece "33.

2. Find a 1 unit strip which is marked off in 4
equal parts. Cut off 1 of these parts. Label
it "i". Label the other piece "=32

3. Find a 1 unit strip which is marked off in 4
equal parts. Cut off 2 of these parts. Label
each of these "i". Label the other piece "2y.

4. Sort out all of the pieces you have made by
looking at the number under the bar. Put the
5' s in one pile, the " "'s in another and so
on.

101

Pack 2 Card 3

1. Find a 1 unit strip which is marked off in 7
equal parts. Cut 1 piece. Label it "1/7".
Label the other piece "6/7".

2. Find a 1 unit strip which is marked off in 7
equal parts. Cut a piece 2 parts lon . Label
it "2/7". Label the other piece "5/7 .

3. Find a 1 unit strip which is marked off in 7
equal parts. Cut a piece 3 parts long. Label
it "3/7". Label the other piece "4 7 .

4. Put all the pieces which have 7 under the bar
together in one pile.

k k ‘ ‘

Pack 2 Card 4
1 . 2. 4.
1. Make pieces the correct lengths for 3, 6' E

g: g. and label them.

2. Make pieces the correct lengths
l l 2 12 ll 4 10 6
TE, fit 1?, It, &9 1?, In! it, 1%! &9 It,

3. The numeral under the bar is called the
denominator. Put all the pieces which have a
6 in the denominator together. Place all the
pieces which have a 14 in the denominator
together. You should have 6 piles.

102
Pack 3 Card 1

1. Find a piece which is labeled 1 whole. The
length of this piece is chosen to be 1 unit.

2. Find a piece which is 2 units long.
3. Find a piece which is 3 units long.

4. Can you show how long a piece would be that
was 5 units long?

 

Materials needed: A strip 10%" long and 1;"
wide of heavy paper labeled "1 whole". Other
pieces 1%" wide of approximate lengths labeled:

.L L A l. 2 l .L l. I

3-939192939323’13-g‘é‘9‘3'90962909an

, 2: j: %; g; g: I 1 .1 .L l 1 1L :2

Evita: v¢9o97e797v797t797v7a
r

-§79%,’5,é7"

 

 

 

 

Pack 3 Card 2

1. Find two pieces of the same length which together
make a piece 1 unit long. What is the name on
each of these two pieces? [3+ [:3 :35: l.

2. Find three pieces the same size which together
make a piece 1 unit long. What is the name on
these 3 pieces? )3 + [:l + [j =3: 1.

3. Find 6 pieces the same size which together make
a piece 1 unit long. What is the name on each
of these pieces?

4. How would pieces be labeled if 4 of them ma e
a length 1 unit long? (:3 + [:1 + {:1 + [j =7».- l.

A- . _Q._
5. -—77-_ 1 , £5 _ 1.

Have the teacher check your answers.

103

Pack 3 Card 3

l.

2.

3.

5.

Take two pieces each labeled "1/7" and place
them end to end. Can you find one piece the
same length as the two of these together?
How is it labeled?

Take three pieces each labeled "1/7" and place
them end to end. Can you find one piece equal
in length to the sum of these three? How is
it labeled?

Can you find one piece equal in length to 4
pieces labeled "1/7" ? How is it labeled?

HOw would you label a piece the same length
as 6 pieces together, each labeled "1/7" ?

What is another name for 1/7 + 1/7 ?
1/7 + 1/7 + 1/7 + 1/7 7 1/7 + 1/7 + 1/7 +
1/7 + 1/7 + 1/7 ?

 

Pack 3 Card 4

1.

3.

Take 2 pieces each labeled "1/3" and place them
end to end. Can you find a piece equal in length
to both of these together? How is it labeled?

How would you label a piece the length of 3 pieces
labeled "1/3" ? Can you think of two names for
this piece?

How would you label a piece the length of 5 pieces
labeled "1/6" ? HOw would you label a piece the
same length as 6 pieces labeled "1/6" ? Can you
think of 2 names for this piece?

Find two names for 1/3 + 1/3 + 1/3,
1/6 + 1/6 + 1/6 + 1 6 + 1/6 + 1/6, 1/2 + 1/2,
1/7 + 1/7 + 1/7 + 1/7 + 1/7 + 1/7 + 1/7 .

104
Pack 3 Card 5

1. Find several pieces labeled "1/6". Find a piece
the same length as 5 of these pieces. How is it
labeled?

2. How would you label a piece equal in length to
4 pieces each labeled 'é" ? Find this piece.
Can you find another piece the same length? How
is it labeled?

3. How would you label a piece equal in length to
three pieces labeled "1/6" ? Find this piece.
Can you find another piece this same length?

4. Write two names for 1/6 + l/6 + 1/6

 

’—

5. Write two names for l/6 + 1/6 + 1/6 + 1/6

 

 

8

6. Write a name for 1/6 + l/6 + 1/6 + l/6 + 1/6

 

 

.9

Pack 3 Card 6

1. Find a piece labeled "1". Find two pieces that
can be placed end to end to equal "1'. (These
two pieces do not have to be equal to one
another.)

2. List all the sets you can find which equal 1
unit when placed end to end.

E :
xample %_+ 2.: 1

Have your teacher check Cards 5 and 6.

105
Pack 4 Card 1

1. Find a piece labeled "2/6". Find another piece
the same length as "2/6". Find two pieces which
together make "2/6". Complete the following:

a? = A+A=e

2. Find a piece labeled "2". Find another piece
the same length as "t". Find two pieces which
together make "i". Find three pieces which
together make " ". Complete the following:

29's- ‘ 0+0 =2
%=-% =5+Qaa

3. Can you write another equation?

 

Pack 4 Card 2

1. Place a "l" and a "1/6" end to end. A piece
equal in length to "l" + "1/6" is "l + 1/6" or
"1 1/6". Find some pairs of pieces which make
"1 1/6" when placed end to end:

1 + 1/6 = 1 1/6

 

5/6 + 2/6 = 1 1/6

 

2. Complete the following:
1 + 1/4 = 3/4 + 2/4 =

1 + 1/3

2/3 + 2/3

 

Check cards 1 and 2 with your teacher.

Practice Sheet 1

Place the sign > , = , or (.in the space provided
so that the following statements will be true

1 1 4 6
r a 3 7 a f
2 6
%__3’ 7__; ?__E
1 g .1. 2 2 2
2____ 3___€ 3____7
'3? ‘3" *

Find a fraction which will name the same number as

%+% %+%
1 2 2
7*? 8+3
5‘. 1 3. 2.
7+7 7*?
* 41* 4t
1 2 1
3+3 2+3
M
1 2 4
3+2 6+6
4(- * ‘N'
2 2 2
3+3 3+5
2 .‘i 9. 2
7*? 7+7
1 1 1 1 1 1
§+§+§ 3+§+§

/ CHI/10 7

Practice Sheet 2

108

Place the sign > , .-: , or < in the space provided,
so that the following statements will be true.

Find

1
2

NIH

NIH

2
K'

8

_it

_1_
3

UIH

NIH

own

his

cvo

;_ 6
2___'1'4'
.2.

3 A
2 13.
3 21

a fraction which names the same number as

l.
2
2
6
1 2
3+3
1 .1.
3+6
2
I4
1 2
7+1?
1 1
7+1?

+E

 

2
E'

KIM NIH

Egox

$1"

NIH

Ova are

one

'17?

Nun

 

109

Practice Sheet 3

Place the sign , = or in the space provided so
that the following statements will be true.

 

% .‘l l 2
1 3 l3 7 1

6 2 i I
6 3 3 1 7 l

1
6 16 6% 1 I 1
it- 'I' it

2 2- _

7+7‘ 1+6-

3 5.- l-
7*7-_____ 1*5‘___

2 1 _

3+3: 2+;.

2 2.- 1. .1.-

3“'3"’ 23*3‘

#- * i-

2 1.. .2. 1..
17”27-___ 33"3-____

1 2 2 l-
3g+46:——_ 33+3_-_-

1 1 2..

ll 3 + 7 3|: 5 2 + 4’-

2+9= M:

110

Find another name for

Practice Sheet 4

._
1.4 : :

2 170 1.2 :
+ + + z ._ 1
1.2 1.3 1.7 1.2 56 1”.
l 1 2 3 2
+ + +
a. a. 8—
1.2 2.3 6.7 56
: 3 6 1 3
— _ _ : _
1m
._ a. 1..
.. _. 2 ..
1.4
1.4 1.6 1 + 1.3
+ + + 1.7 + =
1.2 1.3 1.2 3 1.2 7? ._ = : =
a. 2.3 5.6 3.7 1.. 1.3
+ + + +
_ _ _ ".41_ _ _ 1.2 1.3 2.3 1.4
l l 2

: : .. .. : ..
2.4 2.6 7.”... 1.7 3.6 2

111

Box A Card 1

Do the work from this card on a separate sheet of
paper.

1.

7.

9.

10.

11.

12.

13.

14.

Multiply 10 by 1. Multiply 10 by 2. Multiply
10 by 3, by 4. Ybur answers are called multiples
of 10.

List seven multiples of 10.

Do you notice anything special about all multiples
of 10?

Multiply 4 by l. Mu1tiply 4 by 2. Multiply 4
by 3, by 4. Ybur answers are multiples of .

List seven multiples of 4.

Do you notice anything special about all multiples
of ?

Look at your answers to exercises 2 and 5 above.
Are there any answers in 2 that are also in 5?
Are there any numbers that are multiples of 10
that are also multiples of 4? List two.

Multiply 6 by 1. Multiply 6 by 2, by 3, by 4.
Your answers are called multiples of 6.

List seven multiples of 6.

Are there any multiples of 6 that are also mul-
tiples of 4? List 3 of them.

List seven multiples.of 3.

Are there any multiples of 3 that are also mul-
tiples of 6? List three of them.

Are there any multiples of 3 that are also
multiples of 4? List three of them.

Find a number which is a multiple of 3, 4, and 6.

112
Box A Card 2

Do the work from this card on a separate sheet of
paper.

1. List 6 multiples of 8.
11:,8X2:
8x5=,8x6=).

2. List 8 multiples of 6.
(6 x l = , 6 x'2 = , 6 x 3 = , 6 x 4 = , etc.)

8x3: ,8X4:,

3. Can you find two numbers that are multiples of
both 6 and 8?

4. What is the least number that is a multiple of
both 6 and 8? This is called the least common
multiple of 6 and 8.

5. list 8 multiples of 2.

‘6. List 8 multiples of 3.

7. What numbers are multiples of both 2 and 32

8. What is the least common multiple of 2 and 3?

9. List 6 multiples of 7.

10. What numbers are multiples of both 2 and 7?
11. What is the least common multiple of 2 and 7?
12. What is the least common multiple of 3 and 7?

Complete the chart showing the least common multiple
of each of the pairs of numbers.

 

 

 

 

 

 

 

1.9.m. 2 _3, 17 4 5 8
2 _g A-
3 6 3 6
7 __ 21 7 56
__ 4 28 4
6 13 o
8 8 24 8

 

 

 

 

 

 

113

Box A Card 3

1.

2.

3.

4.

5.

What is the least common
What is the least common
What is the least common
Do you see a pattern?

What is the lpp§p_common
What is the lgppp common
What is the M common
Does the pattern noticed

Can you describe how you
common multiple of three
and 9 as an example.

multiple

multiple

multiple

multiple

multiple

multiple

of 2

of 2

of 3

of 4

of 6

of 4

and

and

and

and

and

and

in 4 still hold?

3?

7?

7?

6?

8?

8?

would find the least
Use 4, 6,

numbers?

Have the teacher check your chart from card 2 and
problem 9 on this page.

114

Pack A Card 1

2

1. Take a large sheet of paper. (Notebook size
is O.K.) Fold it in half. Fbld it in half
again. Open the sheet. The folds should
divide it into 4 equally sized parts. Label
each part i. Shade 3 of the parts grey with
your pencil. What fraction of the whole sheet
is shaded?

2. Fbld the sheet back into the fourths. Fold it
once more. How many pieces is the sheet folded
into now? Haw many of these are shaded? What
fraction of the whole sheet is shaded? (Open
up the sheet and check.)

3. Was the same part of the sheet shaded for both
questions 1 and 2? We say 3/4 and 6/8 are
equivalent fractions because they name the same
part of the whole sheet.

BX :6

4X :8

4. How can we obtain 6/8 as an equivalent fraction
to 3/4? When 3 parts of the 4 parts are shaded,
we make each part into ? pieces.

115

Pack A Card 2

1.

2.

3.

5.

2

Take a large sheet of paper folded into 4 parts
with 3 of the 4 parts shaded. What fraction of
the whole sheet is shaded?

With a dark pencil or ballpoint, draw lines
dividing each of the four parts into 3 pieces.
How many pieces do you have all together? How
many of them are shaded? What fraction of the
whole sheet is shaded?

0n card 1 we found that 3/4 and 6/8 were
equivalent fractions. What is another fraction
equivalent to 3/4?

3x =9
4X :12
How can we find 9/12 as an equivalent fraction

to 3/4?

116

Pack A2 Card 3

1. On cards 1 and 2 we found that the fractions
3/4, 6/8, and 9/12 were equivalent.

%= i—x-é i%=2—3§-%

Nl'x
H

6

’8'

£3.
0

2. Is 12/16 equivalent to 3/4?

24:24
#

3. Name some other fractions equivalent to 3/4.

, , and ______ .

 

4. Find 3 more fractions for this set.
{3/4,6/8,9/12,12/16,15/2o,18/21,_,__,__,...} .

5. Find 3 fractions equivalent to 3.
6. Find 3 fractions equivalent to 1/3.

7. Find 3 fractions equivalent to 2/3.

Box

117

B Card 1

Materials: Cuisenaire rods: 6 white (W), 3 red (R),
2 lightgreen (G), 2 purple (P), 2 yellow (Y) and
l darkgreen (D).

 

 

 

1. Take 5 whites (W) and place them in a row. Which
one rod is the same length as these 5 whites?
Complete the following: W x 5 = .

2. How many whites (W) end to end is the same length
as l lightgreen (G)? Fill in E’x __= G.

3. Fill in the blanks: R x 3 = __, R x __ = P,
Git—:13, GX3:G+G+G:_+G.

Box B Card 2

1. Take 1 white (W). Complete the following:
WX1=__O

2. Take 1 purple (F). Complete the following:
PX1::___.

3. If any color could be filled into the box, what
is [::]x 1 ?

4. let the darkgreen (D) be 2. How long then is
the lightgreen (G)? How long then is the W?

How long then is the R?

5. Complete the following: 2 x l = __, l x l = __,
1/3 x l : __, 2/3 x 1 : ___.

Box B Card 3

1. If the white is 1/3, how long is the red (R)?

How long is the lightgreen?

2. In problem 1 you probably said red was 1 3 + l 3
or 2/3. = w + w + w. So G = 1/3 + 1 3 + 1 3.
Did you call lightgreen 3/3 or did you call it 1?
Is 1 = 3/3 ?

D Q. D E;

3. Complete the following: 1 = E: l = 5; l = 3, l : lO.

2 a 1 3

4. Fill in the blanks: 1 x 1 = 3 x 1 = 3 x I': 3.

2 2 2X0 6
1X1:§XB=2X3=6:1.

118

Box B Card 4

1. How many 1/2 does it take to make 1? How many
l/3 does it take to make 1?
1 ::E!; 1 : 9?; l x l : E!x =
2

6
2 3 6

bflo

2. Fill in the blanks:

lxlzgxd: DXA:8:O
2 ‘E 2“E‘E

Islxlzl?

3. Find 1/2 of l.

X12D

l.

2
01' X1: Ig:O. DOeSlz2 7
2 i I

l.
2 2

1
2’

EEEEE—‘\\) (a) Into how many parts
iz’ is each figure

divided?
(b) How many of these
parts are shaded?

(c) Write a fraction to represent the shaded area of
98011..

(d) Does 1/2 = 2/4?
(e) Find some other names for 1/2.

Box B Card 5

 

 

 

 

119

Box B Card 6

1. On card 5 we found that:

1 2 6 1+ .
§=z=fi=e=§

Complete the following:

1.1. -1 E' V
5-2X1-2X2:E
l-l. ..l- 2-
2'2XO“2X3'5
15:1: 1:; ---
2 2 x 2 x 6 _
1.; ..l i-
2 _ 2 x 1 - 2 xzala 3
Notice: 2 = 2 x 1 : 3.x 2.:
3 3 3 2
Find: 2 _ a _ g. E!—
3‘3XO‘3xD‘

01¢-

\olm

120

Pack B Card 1

2 Materials: A numberline having 1/2's, 1/3's, 1/A's,
l/6's, l/7's, 1/14's, 1/21's, l/l2's.

2

 

 

1.

Look at the number line. What are some other
fractions that name the same number as l/2?
List these.

What are some other fractions that name the same
number as 1/3? List these.

What are some other fractions that name the same
number as 2/3? List these.

Can you find a pattern?

Pack B Card 2

 

 

 

2
1. Try your pattern on these fractions. %-= 3';
i=-‘2=1‘§
Check your answers with the number line. Does
your pattern work?
2. If your pattern did not work, see if you can find
another pattern. Try your pattern on these fractions:
1;: .1..- 01—- Cg-—
7 13" 7 - 2T ' 7 - II" 7 ~ 21
Check your answers with the number line. Does
your pattern work?
Pack B2 Card 3
l. Add:3_+g_
7 7
2. Did you find %’+ 2.: 3,? What do the fractions
2 and g.have in common?
7 7
3. Complete the statement: When two fractions have
a common , you can add the numbers by
4+g—itg——o
7' 7 “ 7 ~ 7
4. Can you add 1/2 + l/3 by adding the numerators?

Why not?

121

Pack B Card 4

 

 

2

1. 1,: CI. ;___C>. ;,+ 1,_ :1+ 0 _ [3 +<3 : A ,

2 6" 3 7 6 ' 2 3 - 6 6" "6" " 6
2. (1) above shows how 1/2 and 1/3 might be added.

Try this same method with 1/2 + 1/7.

1:0 .1...
2 15’ 7 ’ IE
3. Add 1 + 1 . Add 1 + 1
3 6” E 6

4. Do problem 3 again using 6 and 12 as denominators.
PaCk B2 Card 5
1. Find at least 6 other names for 1/2, 1/3, 1/7.

1/2: : : : : = .

1/3:_-=__-:—-=__:_—:__0

1/7:____::____:___:___:____:____.
2. If you were going to add 1/3 and 1/2, what denom-

inators could you choose?
3. If you were going to add 1/3 and 1/7, what denom-

inators could you use? Add these.
4. If you were going to add 2/3 and l 7, what denom-

inators would you use? What is 2 3 + 1/7?
Pack B2 Card 6

Going backwards: I 1 l
1. Fill in the blanks. 5 + 6': 6'+ g-z 6" Can you

find another name for 4/6 which has a smaller'

denominator?
2. Find other names for each of these with smaller

denominators: (Use the number line.)

4 , l2 , 14
6’1‘% 12: at

3. Check your pattern on these exercises:

10 _ _ , 1 _,. 8 .._.. 2.-
II 7 21 7 ’ 12 ‘ 3 ’ ‘

2

Box C Card 1
C?
Definition: When we writetz we are indicating the
quotient when U is divided evenly by A .
Example: 21 is another name for 9, and 6 is another
2
name for 3.

1. Tell what whole number is named by the following:
13. 6.6.2.48.8.1.
5 7 9 9 ‘6'8 7

2. List two or three fractions which name the same
number as: 4, 7, 9, ll, 1 .

Box C Card 2

1. Divide 9 sticks between 2 peOple in your group.
How many whole sticks may each have? How many
are left over? If we wanted to divide this
between the 2 people how might we do it?

2. 9 3 2 = R 1 or 2 = 4%. Why do you think
° 2

we write % to mean 1 stick divided between 2
people?

__1 —¥ u _ — ..—
— __— —

Box C Card 3

1. Divide l9 sticks between 3 people. How many
whole sticks will each one receive? How many
sticks must be broken to make this come out even?
Into how many pieces should this be broken?

2. Fill in the blanks:

123

Box C Card 4

1. Divide l7 sticks between 2 peOple. Write the
problem mathematically.

2. Repeat the same process for 22 sticks divided
between 2 people. Divide l7 sticks between
4 people. Divide 26 sticks between 5 people.
Divide 37 sticks between 6 pe0ple. (It is dif-
ficult to break sticks into these small parts.
Maybe)you can write the answer without breaking
them.

Box C Card 5
1. Divide l4 sticks between 4 people. Can you do
this by only breaking two sticks?

2. Divide 2l sticks between 6 people. Can you do
this by only breaking 3 sticks? Can you do this
by making only 2 breaks?

Box C Card 6

Answers to cards 1-5

Card 13liz-j; 56:8; @227; 2:1
-5 7 9 9
Lizflzézlgzlézggzg‘lt

1 2 3 T 5 '6'
48:8,8:1,Z:1
77 6 7

Card3:1=3+%=3%; 19§3=6R1;12=6+1/3=61/3
2 3

; 1%,: 4%; %g = 5

Card 5: l4 : 32 (Put 3 sticks on each pile. There
are 2 left over. Break each of these in 3.)
21 f 6 = 3 R3. 21 = 3 + s = 3%.

17

124

WOrksheet D1

John has a candy bar. He wants to divide it with
Bill so that each will have an equal amount. How
big a piece should each one have?

You can probably answer this question easily. (The
answer is i, one half.) But have you thought about
what the fraction 2 means? In the example of this
problem 2 is the name of each part when one thing
is divided into two equal parts. If the rectangle
shown below represents one whole, what part repre-
sents 2? You should answer, "The shaded part".

I W

Beside each of the figures below is a fraction.
Shade the part of the whole figure which repre-
sents that fractional part of the figure.

 

 

emu

 

 

"K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

owe

 

UTIH

01H

Hf

125

Beside each of the figures below write the fraction
which represents which fractional part of the figure
is shaded.

 

 

 

 

 

 

 

 

 

 

 

/

126

Worksheet D2

John has a candy bar. He wants to divide it with
Bill so that he will have twice as much as Bill.
He decided that if he divided it into three pieces
and gives Bill only only one piece, he will then
haVe two pieces or twice as much as Bill. How big
a piece of the whole candy bar does each one have?

Since the candy bar was diveded into three pieces,
each piece is 1/3 (one-third) of the entire candy
bar. Bill has one of these pieces so Bill has 1/3
(one-third) of the bar. John has two of these pieces
so John has 1/3 + 1/3 or two-thirds of the candy bar.
"Two thirds" can be written "2/3".

%%%%%% 1

The above rectangle is separated into three parts
of the same size and two of these parts are shaded.
In terms of the size of the whole figure as a unit,
what number tells the size of the shaded part of
this figure? (Answer 2/3 or two-thirds). How are
the 2 and 3 in the fraction 2/3 related to the rec-
tangle shown? (The 3 tells the number of parts of
the same size into which the figure is separated,
and the 2 tells the number of arts of this size in
the shaded part of the figure.§

 

Answer each of these three questions about the fol-
lowing figures:

(3) Into how many parts of the same size is
the figure separated?
(b) How many parts of this size are shaded?
(c) What fractional part of the whole is shaded?

sea-II W

323“

 

 

 

127

Werksheet D2

Beside each of the figures below is a fraction.
Shade the part of the whole figure which repre-
sents that fractional part of the figure.

OS
A
HI 3

\/

 

 

 

 

 

 

 

Write the fraction:
two thirds
one tenth
three fourths

seven eighths

Challenge questions:

1. Is 1/2 of two different things the same?

2. In figures of the same size and shape,
which fractional part is bigger? (e.g

1/2 1/3. 1/4 1/5. or 1/2. 2/3. 3/4 85/8).
3. How does three fourths differ from three fours?

4. In how many ways can a figure be divided
into fractional parts, e. g. 1/4‘ 3 ?

128

 

Box B Card 1

1. In the kit you are provided (cuisenaire rods, 2
of each color) let W 2 white, R : red, G = light
green, P : purple, Y = yellow, D : dark green,

2. Make a train by putting end to end a lightgreen
(G) for the engine and a red (R) for the caboose.
We can name this train G + R (with G first).

3. Make a second train by putting end to end a red
for the engine and a lightgreen for the caboose.
We can name this train R + G (with R first).

4. Are these two trains the same length? We can
write this by saying G + R : R + G.

Box E Card 2

1. Make a train by puttin first the yellow (Y)
and then the purple (P . Make a second train
by putting first a purple (P) and then a yellow
(17). Is Y+P=P+Y?

20 IS G+D=D+G?

3. Can you find two rods for which the length is
different when the order is changed?

4. If we say the lightgreen (G) is 1 unit long, how

long is the dark green (D) rod? Is 1 + 2 = 2 + 1?
How long is W + R? Is W + R.: R + W?

129

Box E Card 3

1.

2.

The switchman is hooking up some longer trains.
First he hooks the lightgreen (G) engine onto

the yellow (Y) passenger car. Then he hooks on
the red (R) caboose. He has the train (G+Y)+R.
The "( )" show that these two were hooked together
first. Make the train (G+Y)+R.

Make the train G+(Y+R). In this train you should
hook the yellow and red together first with the
red at the end. Then the lightgreen engine should
be hooked on afterwards.

Are the two trains different because they were
hooked together in a different order? Is
(G+Y)+R = G+(Y+R) ?

Box B Card 4

1.

Make the train (G+D)+E. Then make the train
D+(E+G). Are these trains the same length?

If light green (G) is 1 unit long, how long is
the dark green? How long is the blue?

Find (1+2)+3. Then find 2+(3+1).
Does (1+2)+3 = 2+(3+1) ?

Find 4+(7+11)+2. Then find (4+11)+(7+2). Did
you get 2 different answers? Why or why not?

APPENDIX C

Group A
Lesson 3
February 8, 1968
Time - 55 minutes

The papers were returned at the beginning
of the period. Exercises 2, 3 and h were discussed,
p. 2&3 with.the correct answers written on the board.
The words numerator and denominator were introduced.
The discussion exercises at the bottom.of the page
were discussed using this socabulary. The discussion
exercises on page Zhh were answered in class. The
top of page 2A6 was discussed in class.

The new assignment was given and instruc-
tions were given as to how to set up the homework
paper. The assignment was page 2&5 (1-10) and
pass 246 (1-3).

The challenge proble-.was solved on the
board by dividing the 6 circles into 3 groups and
shading those circles in 2 of the groups. A new
challenge problem was given.

04. 04
(>4 O<I
(>4 (>4
<><I O<I
Shade B/h of the fish.

131

132

Students worked at their seats with very
few questions. Exercise 2 page 2116 needed to have
the directions explained. It was suggested that
students who finished early read page 2117.

For drill on multiplication tables, a Isath

down was held, giving everyone at least 2 questions.

133

Group A
Lesson 9
February 19, 1968
Time - 60 minutes

Those papers that were turned in on Thurs-
day were returned. The answers for page 257 were
given by volunteers so that these who did not turn
theirs in on Thursday could correct theirs.

The top of page 258 was read by various
students in the class. we had previously discussed
numerator and denominator so this was not new. All
of page 262 and the top of page 263 was discussed
by having a student read the explanation, calling
on students to respond to the questions asked in
the book, asking additional questions where it seemed
necessary. Exercises 1 and 2 on page 265 were dis-
cussed. Each student in the room had at least the
chances to recite.

The written assignment was made: page
258 (1,2), page 263 (1-6) and page 265 (3-8), Since
this work could be accomplished by most in 15 min-
utes and 30 minutes remained in the hour, page 269,
any (8) part was assigned as a challenge.

The last 5 minutes of the period were
devoted to drill in multiplication by 3 and.h via
the "mathsdown" technique.

 

134

Group A

Lesson 1h
February 26, 1968
Time - 60 minutes

Papers from Lesson 12 were returned on
Set h2. Etch.student in the class was given a
chance to work at the board, either to write and
complete an exercise or to find a point named by
a given fraction on the number line. These at
their seats were also involved in the process by
describing what it was necessary for each to do
to locate a point or to decide whidh rational
number is greater Or to decide which sign might
be used.

Pages 298-299 were assigned to be written
beginning with 2 (B). Exercise 1 was done orally.
This exercise was somewhat confusing because the
capital letters (A), (B) etc. were used for two
different purposes.

The surveys were passed out and it was
requested that these be completed before the
assignment. Each student finished this and exercise
2 and part of exercise 3. However, very few fin-

ished the entire assignment.

135

Group B
Lesson 3
February 7, 1968

Boxes were distributed and students began
where they had left off the day before. After
answering initial questions, the teacher worked
with.3 girls reviewing the construction of the
number line and the concept that when 1 thing is
divided into 6 parts, each part is called 1/6, 2
cf them.are aalled 2/6, etc. Also that if, e.g.,
2 is a whole number point on the number line, 2
plus 1/6 is written 2 1/6.

lext 20 minutes of the period were spent
answering miscellaneous questions and passing out
new materials as students became ready for them.

When [10 minutes of the period were up,
students were instructed to write requests for
help or supplies on the inside of the top of their
bones, put everything in.their boxes, and be ready
for a game.

The function game was played using the
sum of the number plus the next larger, the square
of a number, and multiplication by 7.

136

Group B
Lesson 9
February 15, 1968
Time - 55 minutes

Progress of each individual was recorded
on a newly posted chart by shading in those packs,
etc., that had been completed by each. Separate
shadings were made for marking when the teacher
had checked a written exercise and for when the
students had made the appropriate corrections.

A tag system was instigated allowing each
student who needed help to take a number and then
see the teacher in that order. This worked better
than the system.of having the teacher circulate,
since fewer students ask questions - merely because
it is easier to stop the teacher and ask her than
to figure it out themselves.

Students worked on materials individually
(or in twoe for Packs l-h). The materials labeled
Box A, B, 32, 0, D1, D2, E are available for choo-
sing, whenever the practice sheets (to be used with
concrete materials) are completed. Each student
makes a written response to each.of these and turns
it in to be graded by the teacher.

During the last 10 minutes of the period
the ”methadown” of drill with.mu1tiplicaticn was

137

continued. About 2/3 of the students give good

responses. The others Just do not seem to even

be able to figure the products when given suffi-
cient time.

138

Group B

Lesson 1h
February 22, 1968
Time - 55 minutes

In an effort to motivate children some-
what to making an effort to accomplish.a little
more and to also provide the recognition they
each seem to need, the following was done: each
child's name was called and he was asked what he
was working on, how much he has done, and what
he was doing this period. If he needed help he
was told to take a number. If he hadn't turned
in any work lately, he was encouraged to complete
something today.

This same process was used at the close
of the period to provide direction for the follow-
ing day's work.

Seven children of the 16 who took number
tags (to be helped in the order of the numbers)
were helped. Seven others were helped in two

groups, one on Pack A, and one on Pack D2.

BI BLIO GRAPHY

BIBLIOGRAPHY

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lhO

lhl

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1H2

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