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METHODS AND MATERIALS OF A MATHEMATICS
PROGRAM FOR THE DISADVANTAGED AND
UNDERACHIEVING CHILD

ThesEs for the Degree of Ed. D.
MICHIGAN STATE UNIVERSITY
E. Leona Hall
1966

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THESIS

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This is to certify that the

thesis entitled

METHODS AND MATERIALS OF A MATHEMATICS
PROGRAM FOR THE DISADVANTAGED AND
UNDERACHEVING CHILD

presented by

E. Leona Hall

has been accepted towards fulfillment
of the requirements for

. Ed.D. degree inmon

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ABSTRACT

METHODS AND MATERIALS OF A MATHEMATICS PROGRAM
FOR THE DISADVANTAGED AND UNDERACHIEVING CHILD

by E. Leona Hall

Statement of the Problem

In general terms the purpose of this study was to
learn more about the disadvantaged child. Specifically the
purposes were: (1) to determine if the ”concept" method of
instruction is effective in mathematics with disadvantaged
and underachiving children; (2) to learn if-attitudes toward
mathematics could be significantly changed in a positive
direction; and (3) to explore the relationship between

achievement in and attitudes toward mathematics.

Procedures

Ninety-seven children from the greater Saginaw area
were identified as disadvantaged and underachieving by their
classroom teachers in the spring of 1965. These children
were given a summer camp experience through the cooperative
efforts of the Michigan Department of Classroom Teachers,
the Michigan Education Association, and the Organization of
.Economic Opportunity. The camp was held at St. Mary's Lake

at Battle Creek, Michigan, and extended for five weeks dc

2 E. Leona,Hall

the summer of 1965. The dates of the experience were from
June 27 to July 31. The ages of the children included those
8 through 12. I ,

For the mathematics experience, uponwhich this
study is based, the children were divided into four groups--
two fourth-grade groups and two fifth-grade groups. The \
writer served as instructor for all four groups. Classes
were held daily five days per week and were one hour in
length. The first two days of class time were spent on
a testing prOgram. The last two days were spent in follow-
up testing. -Thus the actual instruction covered one full
month or four weeks, which represented twenty hours of ’
actual class time.

The children were given the Peabody Picture Vocabu-
lary Test to determine their mental ages and IQ's. The
whole battery of the California Achievement Test was also
administered. The Iowa Basic Skills, Form I, A-1 and A-2,
was used as the pre-test tool for this study to determine
levels of ability in those areas that A-1 and A-2 measure;
namely, arithmetic concepts and arithmetic problem solving
ability. Button-Adams Attitude Scale was administered to
determine the individual attitudes toward numbers. A card
file was kept to permit compilation of pertinent data such
as--least liked subjects, number of siblings, status of

the home (such as broken), etc.

3 E. Leona Hall

On the third day of the camp experience actual
instruction began. The children made their own textbooks
as the class progressed and instruction followed the "con—
cept" method insofar as possible. As each concept was
presented a follow-up activity was planned to reinforce
that concept. Extensive use was made of models and aids
as well as of the total camp environment in the arithmetic
program.

In the final two days, devoted to follow-up test-
ing, the Iowa Basic Skills, Form III, A-1 and A—2 was
given to the total group. The same Button-Adams Attitude
Scale was administered to measure any changes in attitude.

In March of 1966, after the children had been in
their regular classrooms for six months, a random sample
of 30 children was drawn from a random table and these
pupils were again tested. Form I of the Iowa was repeated
and also the attitude scale. The random population repre-
sented thirty-seven per cent of the total group as the
actual number in the experiment was eighty-two. The re-
:maining fifteen had to be rejected because of severe re-
‘tardation, sickness, early departure from camp, and other
:reasons less pertinent to this study.

Of the total group in this study thirty-seven were
Negroes, thirty-two were Caucasians, and thirteen were of

Lflaxican-Spanish descent. Forty-two were below the twenty-:

4 E. Leona Hall

percentile rank in arithmetic; sixty-three were below grade
level; and IQ's ranged from 68 to 1&5.

Findings

The specific findings of this study--those related
to the specific objectives--were interpreted in terms of
the following hypotheses:

H1 Disadvantaged and underachieving children

will respond in a positive manner to the
”concept" method of instruction in mathe-
matics as evidenced by gains on an achieve-
ment test. ‘

H2 Disadvantaged and underachieving children

will tend to show a positive change in
attitude as a result of the influence of
careful attention to method (concept) and
materials as evidenced by a positive change
on an attitude scale.

H3 Disadvantaged and underachieving children
will tend to show a positive relationship
between attitude toward mathematics and
achievement in mathematics.

Other findings in this study were simply conclusions
reached after careful examination of the data. These data
related to the writer's desire to learn more about the dis-
advantaged child and may be classified as general informa-
tion.

The first hypothesis proved true for the fifth
grade both in immediate gains and long term'gains. The
concept method did not appear to be effective for the
fourth graders in short term goals; however, the data sugw

gest a possible benefit after several months.

5 E. Leona Hall

The second hypothesis, dealing with attitudes,
proved valid with both fourth-and fifth-graders. After
having returned to their regular classrooms for six months,
hh% of the fourth graders ranked themselves higher than
they had previously on the attitude scale; 12% held their
scores to the last testing level; and th% rated themselves
at a lower level, indicating a less positive feeling or
liking for arithmetic. Of the fifth graders, 57% continued
to improve in attitude after being in regular classrooms;
lh% held at the last testing level or end of camp experi-
ence; and 29% regressed to a lower level of rating attitude.

The third and last hypothesis, dealing with the
relationship between attitudes and achievement, did not
appear valid. The fifth graders showed a substantial
correlation from the pre-test to the post-test; however,
this did not prove true for the other testing situations.
The fourth graders showed a moderately low correlation
between attitudes and achievement from the pre-test to the
post-test; but again, this did not hold true for the other
testing situations. A

It would appear from these data that the concept
method is more effective with the older children, in this
case fifth graders as opposed to fourth graders, in terms
41f immediate gains. It was further demonstrated that

attitudes can be significantly changed regarding liking

6 E. Leona Hall

arithmetic when careful attention is given to methods and
materials. And finally, there appeared to be little corre-
lation between attitudes and achievement among disadvantaged
and underachieving children at the fourth and fifth grade
level.

The fact that this particular camp group was com-
posed of multi-problem children definitely appeared to
affect scores and attitudes adversely. The evidence
seemed to indicate that more discriminating instruments

are needed to evaluate economically deprived and disad-
vantaged children.

METHODS AND MATERIALS OF A MATHEMATICS PROGRAM
FOR THE DISADVANTAGED AND UNDERACHIEVING CHILD

By

E. Leona Hall

A THESIS

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

DOCTOR OF EDUCATION
School of Education

1966

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@Copyright by
E. LEONA HALL

1967

ACKNOWLEDGMENTS

The writer is indebted to a number of persons with-
out whose help this study could not have been executed or
brought to fruition. The writer wishes to express her appre-
ciation to Dr. Calhoun C. Collier, chairman of her Guidance
Committee, for his help and encouragement throughout the
years as well as his help and interest in the planning of
this study. Special gratitude is also due to the other mem-
bers of her guidance committee: Dr. W. Vernon Hicks, Dr.
William J. Walsh, and Dr. Jay‘w. Artis, who were most gen-
erous with their time and advice.

She_also expresses thanks to Hr. Paul Oberle,
Director of the Summer Camp, who asked the writer to join
the staff and permitted her the latitude and freedom nec-
essary for this project. Mr. Robert Bowers and Mrs. Agnes
Beer, guidance counselors at camp, supervised the testing
programs, and rendered a valuable service.

The writer wishes to acknowledge appreciation to
.Dru Helen Tewes, Director of Elementary Education for the
Saginaw Schools, and to the principals and teachers of

Saginaw for permitting her access to the pupils for testing
sand interviewing. Their cooperation and friendliness were

most appreciated. '

ii

Grateful acknowledgment is accorded Dr. Jean LePere
who made helpful recommendations regarding treatment of
the data.

Special recognition is acknowledged to all children
who served as subjects for this study. They were the real
teachers and from them the writer learned much.

The writer is indeed grateful to The Iowa Testing
Service and to Dr. Hieronymus, a member of its staff. The
Testing Service furnished all testing booklets for the
program. Dr. Hieronymus reviewed the data and made many
helpful suggestions.

The acknowledgments would not be complete without
expressing a debt of gratitude to the writer's parents. They
are responsible in large measure for the values and attitudes
the writer holds. These same attitudes and values (so prized
in our middle-class society) made it possible and acceptable
to her to work with less fortunate peOple--the children
involved in this study.

iii

Table of Contents

Chapter

I.

II.

III.

V.
VI.

INTRODUCTION . . . .

Purpose or need for the study

Statement of the problem . .

Limitations to the study . . .

Definition of terms

REVIEW OF THE LITERATURE-—PART I

Review of the literature pertaining to

Attitudes toward self, others, and school

Recommendations 0 o o o e o o o o o o o
ovefliSWoooooooooooooooo

REVIEW OF THE LITERATURE-~PART II

Underachievers and low achievers . . . .
Concepts, conceptual development . . . .

Attitudes and achievement in mathematics
METHOD OF THE INVESTIGATION

A description of the students in the

A description of the camp facilities .

Method-of instructi

Typical lesson plan

Summary of teaching techniques

Materials and activities . . . .
Instruments used in the study
Testing sequence . . . . . . . .

DATA AND RESULTS . . . . . . . . .

SUMMARY, CONCLUSIONS

Summary . . .
Conclusions .
Implications .

BIBLIOGRAPHY . . . . . . .

Appendices o o o o o o o o

Page

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0.0000000... 6
00000.. 11

poverty 12

28

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. . 35

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. . 52

. . 57

000000000 61‘
study . 64

co. 65
on0000000000. 65
0000000000. 69
used . . . 70

0000.0 71
.000... 72
000.00 73

. . . . . . 75

AND IMPLICATIONS . . . . 9h
00000000 000 91:}
0000 00000000 9‘}
00.00.... 000 95
000000.000... 100
0000......... 108

iv

Table
I.
II.
III.

V.

VI.

VII.

VIII.

IX.

X.

XI.

XII.

XIII.

LIST OF TABLES
Page
Achievement in mean grades-—total group . . . 77
Mean achievement in percentile ranks . . . . 77
Achievement in mean grades for random group . 78
Attitudinal changes . . . . . . . . . . . . . 82
Relationship between attitude and achievement 85

Relationship between attitude and achievement,
Fourth grade random sample Test A . . . . . 86

Relationship between attitude and achievement,
Fourth grade random sample Test B . . . . . 87

Relationship between attitude and achievement,
Fourth grade random sample Test C . . . . . 88

Relationship between attitude and achievement, 89
Fourth and fifth grade total from Test A and B

Relationship between attitude and achievement,
Fifth grade random sample Test A . . . . . 90

Relationship between attitude and achievement,
Fifth grade random sample Test B . . . . . 91

Relationship between attitude and achievement,
Fifth grade random sample Test C . . . . . 92

Individual analysis of relationship between
achievement and attitude 0 o o o o o o o o 93

LIST OF APPENDICES
Appendix ‘ Page
A. Characteristics and statistical tables . . 109

Be Scattergrams o o o o o o o o o o o o o e o 118
C. Corespondence and data from the University
Of Iowa 0 o o o o o o o o o o o o o o o 121..

D. Sample materials and teaching aids . . . . 135

vi

UMPERI
II-I'I‘RODUCTION

This study was prompted by recent trends in our
society such as: increased concern with the problem of
school drogouts; more emphasis on new_mathematig§; govern-
mental war on_poverty; and the recognition that a large
proportion of the nation's children were neglected. At the
time these topics were very much in the news the writer was
invited to work directly with a group of neglected or disad-
vantaged children. The challenge was accepted, and work
began on the study in the summer of 1965.

Burpgse or need for the study. In the book E.-
cellence, John Gardner wrote:

For every talent that poverty has stimulated it has
blighted a hundred.

The relevancy of this statement becomes more clear when one
considers the magnitude of the problem.

In the families of the poor there are 12,000,000
children. These are the hostages to poverty.2

 

1John W. Gardner, Excellence Cag_y§“Be_§gyagg§gd Ex-
cellent Too! (New York: Harper and Row, Publishers, 1951),
p. 99.

 

 

2Ben H. Bagdikian, In the Midst of £1393 (New York:

American Library, Signet Book, 1951], p. 1A0.

In discussing the effects of poverty on children Bagdikian
cautions:

This is one characteristic of the poor that the
United States ignores at its peril.3

The investigator found various estimates in sheer
numbers of those in our society who were directly effected
by poverty. However, there was rather general agreement
in at least two facets to the problem of poverty. These
were: children suffer its ravages most and education holds
the most promise.

In the past decade many programs have been instituted
to alleviate problems associated with the deprived child.
These bear various labels indicating objectives such as:
firemedial," "preventative," "enriching," and "compensatory."
The Higher Horizons and The Great Cities Projects are probably
the better known. In the majority of these existing programs
reading and the language arts have received greatest emphasis.
On the other hand, arithmetic has received scant attention.
The bulletin entitled, "Programs for the Educationally Dis—
advantaged," summarizing state and city projects, supports
this.A Passow, in writing on "Education in Depressed Areas,"

states:

 

3Ibid.

“U. s. Department of Health, Education and welfare,
Egograms for the Edgpationally Disadvantaged, A Report of a
Conference on Teaching Children and Youth who are Educatic'
ally Disadvantaged, Bulletin No. 17 (Nashington: Goverrr-
Printing Office, 1963).

3

Because success in reading and other language arts

' constitutes the key to academic progress, most pro-
grams stress method, materials, special personnel,
and other audio-visual and guidance services to im-
prove as the verbal and other basic skills.5

From the same source the author says:

Efforts to improve school programs in depressed

areas vary considerably in scope and comprehensive-

ness. Almost all focus in some way on the improve-

ment of reading and language skills, for academic

success depends largely on ability to read.6

Two recent conferences were held which stress the

need for more emphasis and research in the area of mathematics
for the low achiever and for the disadvantaged child. One
was held in 1965 and was a joint effort by the U. 3. Office
of Education and the National Council of Teachers of Mathe-
matics. The other was sponsored by the School Mathematics
Study Group in 1964. The following excerpts were taken from
the reports of these two conferences:

From the very beginning SMSG recognizes perfectly

well that we were doing something for only part of

the school population. We have made a remarkable

amount of prOgress, but we are far enough along to

realize that the rest of the school population,

the students who are not doing well in mathematics,
must be given attention.7

 

5A. Harry Passow (ed.) Education in De ressed Areas
(New York: Teachers College Press, I953). P. 335.

61b1do, p0 2800 ‘

7School Mathematics Study Group, Conferenc on Mathe-
matics Education for Below Average Achievers (California:

3 and tan ord unior University, 9 l. , p. l.

At all levels, however, the emphasis and attention

have been directed toward the above-average mathe—

matics achiever.8
Not only is it evident that a large segment of the nation's
children has been by-passed by curriculum changes in mathe-
matics but it is now realized that our modern technological
society demands that ways and means be explored to reach
these children.

Like it or not, we have suddenly awakened in a world

which revolves around science, and it in turn rests

on mathematics.9

We believe it is now time to show some consideration
for the low achievers.lO

From the above sources, as well as others listed in
the bibliography, it appeared to the investigator that a
real need existed in the area of arithmetic for the deprived
child. Hence the purpose of this study was to explore ways
and means which might prove successful for the under-
achieving and disadvantaged child in the area of arithmetic.
Success was to be equated with better achievement and under-
standing in mathematics as evidenced by scores on standardized

tests. ioreover, it was the intent of the writer to make

 

 

*— -..

8

U. S. Department of Health, Education and Welfare,
.jjge Low Achiever in Mathematics, Report of a conference by
time U. 8. Office of Education and the National Council of
Theachers of Mathematics, Bulletin No. 31 (flashington: Govern-
ment Printing Office, 1965), p. 2.

91bid., p. 1.
10Ibid., p. 2.

every effort to influence attitudes toward arithmetic by
either reversing negative attitudes or encouraging more
positive attitudes toward arithmetic.

Statement of thegproblem. The purpose of this study
was to try out innovations in mathematical materials and
methods for a summer program for fourth and fifth graders
that would: (1) provide disadvantaged and underachieving
children with motivation; (2) combine concrete and conceptual
experiences; (3) yield positive changes in attitudes toward
arithmetic and achievement in arithmetic; (h) be compensatory
for those pupils who lacked basic understanding of simple
mathematical concepts; (5) provide opportunities for success
for all students; (6) utilize the camp environment rather
than just the classroom for instruction; (7) be feasible in
five weeks with one hour daily instruction sessions.

It was hypothesized that: (l) attitudes toward
arithmetic would be chanred in a positive manner; (2) achieve-
ment in arithmetic would be enhanced; (3) there would be a
positive correlation between attitudes and achievement. Fur-
thermore, it was felt that useful information regarding dis-
advantaged children could be gained by opyortunities, unique
in a camp setting, for establishing rapport. The inherent

iJiformality within daily living milieu, together with class-
.rcnbm teacher-learning environment, should yield pertinent

*werceptions.

D
o

The problem simply stated is: in terms of goals and
limiting factors will the major hypotheses (regarding achieve—
ment and attitudes) prove valid? Also the question is raised,
can such a program yield serendipitous information that can
contribute to greater understanding of the problem of dis—
advantagement?

Limitations tg_the_study. The study makes no attempt

 

to evaluate the following: (1) the effect of a camp environ—
ment on learning; (2) the superiority of any standardized
test over another; (3) the effectiveness of concrete experi—
ences isolated from conceptual experiences; (a) the adVis-
ability of smaller or larger groups; (5) the effect upon
learning of having junior counselors with the pupils during
much of the class time.
Definition of ter 3. Terms used in this thesis are
_those that are generally in common usage today. A few of
the definitions are discussed in_some detail further on in
the paper. For the sake of clarity however, key terms, those
providing appropriate orientation, are defined here. These
terms are: disadvantaged, underachieving, concept, conce,t
znethod, arithmetic, junior counselor, and compensatory.
Disadvantaged, as used in this study, referred pri-
nuarily to a condition of economic poverty. The children
irivolved in this study were chosen from homes in which the
ftunily income was less than three thousand dollars. Dove"

has: attending ills that cause embarrassment and deprivatf

therefore, when the term disadvantaged was used it was
with the wish to convey that for the most part these
children lacked much in a material sense that the typical
child in our affluent society enjoys. Also, the term
was meant to convey a broader meaning involving depri-
vation of experiences and opportunities. Qgpgizgg was
used interchangeably with disadvantaged.

Concept has been used profusely in recent lit-
erature, particularly in mathematical literature, but
the investigator did not find the term defined. Egb-
ggpr's New Collegiate Dictionary defines concept as
“a mental image of a thing formed by generalization,,
from particulars”; also, ”an idea of what a thing in
general should be." As the term concept was introduced
to the children in the study, the words ids; or gags;-
stahding were substituted. Conceptual learning is
directed to categories and generalizations concerning
these categories. This interpretation permits not only
identification and classification of mathematical sym-
bols, processes, etc., but also provides for drawing
apprOpriate conclusions that are related to these
categories. Hence both the category idea and the gen-

eralization notion are essential to the definition of

the term concept .

goggegggggthgd was considered an approach to teach-
ing which stressed the ideas and basic understandings which
underlie our number system. Teaching direction was away
from computation and drill and toward seeking relationships,
patterns, and structure. For examp 2: addition is bringing
together two or more groups of objects forminm one larger
group; subtraction is the undoing of addition or forming
smaller groups or sets from one larger group. If a child has
3.
7 ' £5 = 4, h * [l = 7. 3 + C3 = 7, and C3 + 1k 7 would

have real meaning for the child as in each case he would

formed such concepts then equations like 7 - C]

be applying a learned concept. Thus the concept method

 

was an approach to teaching arithmetic which would enhance
understanding and give meaning to the child's world of
numbers. In practice the method was accompanied by many
activities and a lavish utilization of tangible aids and/or
models that were intended to promote learning and make the
experience an enjoyable one. Concept method was chosen as

the term to describe where the major emphasis was plac d

(D

(i.e. greater understanding rather than on computation and
‘memorization) and should not be interpreted to mean that
concrete experiences were neglected.

Arithmetic and mathematics were used interchangeably

it) this paper to mean a skill or tool subject consisting of

synnbols and numbers which aid man in describing and keepin“

I 1

account of his physical world. It ansm: rs such questions as:
how many? how big? how much? and other quantitative conc rns
of man. In all instances it has applied only to elementary
school arithmetic.

Junior counselors were high school seniors or col-
lege students who planned to become future teachers. ”ach
junior counselor was assigned to a group of camp children

usually on a ratio of one to five or six. They lived in

the cabins with the children and often followed them to classe

q
‘ D

to aid the instructor with group activities.

Compensator' is frequently used in current literature

 

dealing with disadvantaged children and describes a type of
education. Bloom, Davis and Hess refer to it as "system
of compensatory education which can prevent or overcome

earlier def iciencie

0)

."11 As used in this paper compensatory
teaching was designed to supple e t instruction or to fill
an individual need. Most ideas presented to pupils were
those to which they had already been exposed in previous
school experiences. However, in many cases the understand—

ing was weak, faulty, or non-existent. Thus much of the

'teaching would fall into the category of compensation for

 

i

:1:

11Benjamin 3. Bloom, Allison Davis, and Robert Hess,
cnnce ns itory Educat ion for Cultural Deprivation (New York:

(311,,1' nehart and dinston, Inc., l§657, p. C.

O

10

a lack of knowledge and understanding or supplementation
to existing understanding. The program took place during
summer and outside of the child's regular school and so in
this sense too the summer camp experience may be thoug.t

of as compensatory or supplemental.

CHAPTER II
REVIEH OF THS LITSRATUR£--PART I

The survey of the literature was divided into two

general tOpics. These were: (1) Poverty; and (2) Research

H.

n Mathematics. Each of these will be treated in separate

0

y
.xap

(‘9‘
(9

rs. hapter II is a review of the literature pertain-

()

p.

ng t

O

poverty, disadvanta red children, and arpropriat e su‘s-

tOpics of disadvantagw ent. Chapter III contains a review

of the research in mathematics pertaining to disadvantaged

children, underachievement, modern mathematics, and con~

centual learning. The writer considered these areas would

provide a background and understanding of the problems of

the disadvantaged child as well as an overview of mathematical
research relevant to the present stud'.

In the main, readings were confined to those written
in the past two decades. Before this time the problems of
the disadvantaged child did not appear clearly identified
or defined. dven at present the terms "disadvantaged" or
"deprived" do not app er in the Michigan State University
library card files. Throughout our history sporadic con-
cern for the problems of poverty has been limited to a de-
pression period or some minority group. It would seem that
this myOpia has prompted several writers to speak of the

"invisibility" f the poor.
ll

12

It follows that little could be done in way of re—
search in mathematics with disadvantaged children until the
need for such was realized.

Review o{_the literature pertaining to_pgverty.
"Deprived" or "disadvantaged" have already been defined in
terms of an economic condition called poverty. Poverty is
probably as old as recorded history. 80th the Old and New
Biblical Testaments contain numerous references to the poor.
Luke even included the poor in a Beatitude. However, in
terms of today's life it is difficult to see his rationale;
for the poor seem anything but "blessed." In Matthew's
version it is the poor in spirit that are blessed. At the
risk of sounding iconoclastic it would seem that the poor
economically are also the poor in spirit.

Galbraith in The Affluent Society presented poverty

 

as something less than a major problem in America. Poverty
in modern times has been reduced to a "case poverty" and

an "insular poverty," according to Galbraith, and it is only
because our roots extend back to a time when poverty was
prevalent that we still think in terms of poverty being

part of America.

Poverty was the all-pervasive fact of that world.
Obviously it is not ours.1

 

lJohn Kenneth Galbraith, The Affluent Society (Boston:
Houghton Mifflin Company, 1958), p. 2.

 

13

As a general affliction, it was ended by increased
output which however imperfectly it may have been dis~
tributed, nevertheless accrued in substantial amount
to those who worked for a living. The result was to
reduce poverty from the problem of the majority to
that of a minority. It ceased to be a general case
and became a special case.2
Harrington's approach was that poverty, while a massive and
serious fact, was invisible.

That the poor are invisible is one of the most im—
portant things about them. They are not simply
eglected and forgotten as in the old rhetoric of
reform; what is much worse, they are not seen.3
According to Harrington poverty moreover forms a culture in
America.

The literature presented a spectrum to the approach
of poverty. Usually these fell into three major categories:
(I) an economic or coldly statistical approach that provided
numbers and income figures, (2) a cultural approach, dealing
with cultural deprivation relative to middle—class values;
and (3) a social angle often a concern of social class.

A

Bag ikian, in his book In the Midst of Plenty, used the

 

case—history approach. In all there was agreement upon who
constituted the poor. Generally they were presented as

clusters of peOple throughout the United states presenting

 

21bid., p. 323.

 

3Michael Harrington, The Other America (New York:
Penguin Books Inc., 1963), p. 14.

 

z‘Ben H. Bagdikian, In the Midst of Plenty (New York:
The New American Library, 19047.

lb

pockets of poverty and including the following groups: min-
ority peOple defining racial and ethnic groups, the aged,
the migrants, the slum dwellers, certain rural groups such
as are found on run-down farms and in Appalachia. 1he Negro
is often considered separately because his color compounds
his poverty problems.

Bagdikian gives a rather definitive description of
the poor.

There are about 8,000,000 in rural areas, but some

of these are also among the 8,000,000 aged poor.

About 7,000,000 are "unskilled" workers. About half
are in households where a man is out of work. Many
others are dependent on wages that can't raise the
family out of poverty. . . . There are special groups--
the 500,000 American Indians and the few hundred thou-
sand derelicts of Skid Row. . . . Most notable today
is the Negro, once concentrated in the rural South

but now two-thirds in the cities, North and South.

. . . But poverty is not limited to Negroes. One
calculation shows Negroes to constitute 22 per cent

of the poor.5

Slums or depressed city areas have received con-
siderable attention by many writers. Harrington presents
them as follows:

And there is a new poverty that is becoming more

and more important, a consequence of the revolution
taking place in American agriculture. In Detroit,
Cincinnati, St. Louis, Oakland and other cities of
the United States, one finds the rural poor in the
urban slums, the hill folks, the Oakies who failed,
the wgr workers from the forties who never went back
home.

 

C
/

Ibid., p. 139.

 

6Harrington, op. cit., p. 83-

m

Passow on the same theme states:

Typically, t,e depressed area poFulation te ds to be

a stratified group of predominantly unskilled
skilled workers, largely in—migrant, who have

or semi-
moved

to the city from a rural region. The ethnic and racial

composition tends to be primarily from the so-
minority groups-—southern Negro, Puerto Rican,
achian white, American Indian, Mexican, and mo
cently, Cuban.7

Poverty has many facets and many definitio
of these follow:

Poverty should be defined in te ms of those wh
denied the minimal levels of health, housing,
anal education that our present stage of scient
knowledge specifies as necessary for life as i
now lived in the United States.8

Poverty should be defined psychologically in t

called
Appal-
st re—

'W
ns. COYT‘LB

o are
food,
ific
t is

erms

of those whose place in the society is such that they

are interned exiles who, almost inevitably, de
attitudes of defeat and pessinism and who are

fore excluded from taking aci vanta 3e of new OFF
tunities.9

Poverty is a rersistent gap between "what is"
"what ougr t to be" as viewed sub ectively by t
individual himself, objectively by sc1 ence, or
according to the standards of society.10

In the final analysis most writers conclude that a

and realistic guide to poverty is a,financia1 one.

'3
(U

 

1! ‘ .' r.‘. I 0 . \ u
n ry _ ted.; .t1;q;1cn 1n Ler
.'.".‘ v ' ‘. '7 ‘ ‘ . ‘ 4-. 1 . .
(new fork: leacnezs tulle

gharrington, on. cit., p. 175.

M

O O Q
/Iblao

 

1C Wfargaret I. Liston, "Profiles of Poverty
M0 fig 1, LVIII (Cctober,l 1964) , p. la.

*'

velop
th ere-
OI"

and
he

definitive

There

0)
w
n.
3
(u
D
UJ

16

are various income guidelines by which poverty is measured.

is

hese change with the economy and are based on price indice
and estimated cost of living. The most common figures cur-
rently are dl,000 for an individual and $3,000 for a family.

It is this figure that the government presently use as a

(.0

guideline on the "War on Poverty." It was this figure which
was used as the cut-off point for the children chosen for
the sumuer camp experience who thus became the subjects for
this project.

Caudill listed four causes for present-day American
poverty. They were:

. . . the frontier, the waste entailed in settle—
ment, slavery, and the great waves of immigration.11

In following his develOpment of these factors one can form
a more knowledgeable view of the poor. The poor are often
thought of as a shiftless group who lack the initiative to

12

better their own lots. Rodman speaks of this prejudice

)

against the lower classes (which comprise th» poor) and

I.

(

uses such terms as: "dirty," "lazy," "irresponsible," etc.
to demonstrate the feelings often reflected by upper and

middle-class persons toward the poor. However, Caudill's

11Harry Caudill, "Reflections on Poverty," in flew
FersLectives on Poverty, eds. Arthur Shostak and fiilliam
GombergfiTNew Jersey: Prentice-Hall, Inc., 1965), pp. 3-9.

\ ,
2Hyman Rodman, "The Lower Class and thexhegroes;
Implications for Intellectuals," eds. Arthur dhostak and
William Gomberg (New Jersty: Prentice-Hall, Inc., 1965},
p. 168.

1?

historical treatment shows the antecedent conditions that
have caused poverty and thus serves to prompt a more char-
itable consideration.

In terms of scope, in 1937 according to Roosevelt
the poor comprised one-third of the nation.13 More recently
Kennedy in a letter to Johnson stated that one-sixth of the
people lived--"below minimal levels of health, housing, food
and education."lh Harrington placed the figure "somewhere
between 20 and 25 per cent."15 President Johnson currently
designates it as one-fifth of the nation.16 Whatever figure
is used to delineate the poor most accurately must be a
moot question. The important facts are that poverty is a
mass phenomenon; it tends to perpetuate itself; and its
presence should weigh heavily on the American conscience.

Jhether there are 20, 000, 000 poor or 54, 000, 000
(taken from various income tables), either represent
more degradation, suffering, and social blight than
the American ethic can tolerate.

Against this backdrOp of poverty the disadvantaged

child emerged. In our race with the Russians, our desire

13Robert 3. will and Harold G. ‘Jatter (eds. ), Poverty
in Affluence (New Yorla Harcourt, Brace a Jorld,1965,,
14 (Second Inaugural Address).

“\
1‘3
‘Kl-a J.

lhBagdikian, o>. cit., pr

l: .
l’h'arrington, op. cit., p. 178.

(D

16Hill and Vatter, op. cit_., Po 16°

W

l7Bagdikian,o mp cit., pp. 138, 139.

for excellence, and increased concern over the high rate of
school dropouts, delinquency and juvenile crime, some sig-
nificant facts have been observed. A relationship betwee.
o 4 Q
income and education has been well documented by bexton.1~
Dropouts and delinquents were proportionately higher among
low income groups. Also, a large percentage of our youth
was denied equality of Opportunity which brought untold
wasted potential that was sorely needed to keep America strong.
It was these concerns that caused President Johnson to de-
clare war on poverty. It was these concerns that caused
Congress to pass the dconomic Opportunity Act of 1964.

Because it is right, ecause it is wise, and be—

cause, for the first time in our history, it is

possible to conquer poverty, I submit, for the con-

sideration of the Congress and the country, the

Economic prortunity Act of 1964.

Our fight against poverty will be an investment in

the most valuable of our resources—~the skills, and

strength of our peOple.2O

Review of the literature gertaininr to disadvantaged

children. Of the many labels given to the disadvantaged
child most terms arplied either oversimplify the problem

or emrhcsize the negative aspects.

 

l8Patrica C. Sexton, fiducation and_
The Viking Press, 196a), pp. 15, lb, 132.

Q 70 a
131111 and Vatter, OFE,£1£-. p. 17.

-r-
I
J

ncome (New York:

 

ZOIbido, p. 160

19

The terms "deprived, " "handicapped," "underpriv—
ilege ," and "disadvantaged," unfortunately emphasize
environmental limitations and ignore the positive
efforts of low-income individuals to cone with their
environment.21

Riessman used the above terms interchangeably and states,

"any term connotes inadequacy. "22

Terms such as "culturally different" or "less-
privileged" offer no improvement in being definitive or fair.

Thus the writer chose to use either disad anta Led or de-

 

rrived and hoped to convey both the economic plight and
other poverty-related derrivations which cause these children
to fall short of their potential. Karlan summarizes this
idea in the following

Whether we choose to call these purils disadvantaged,

culturally deprived, or economically impoverished,

they usually exhibit two characteristics; they are

from the lower socio-e conomic grouts in the community

and they are notably deficient in cultural and aca-

demic strengths.23

Two exc wtts from Ioverty in Affluence give a more compre-

H-

13133

I

ve picture of the difficulties these children face.

—-.--.o ---.—---- ——-.

lFrar.k Bies ssman, "The Culturally Derrived Child:
A Ne ew View, " Prqg_ams for the Sducationally;Disadvant taced,

41"

Jashingron: Government Printing Office, 19597: p. 3.

22Frank Riessman, The Culturally Derrived Child
(Hew York: Harrer and BrOthers Publishers, ~19t2) p. .112.

23Bernard A. Kaplan, "Issues in fiducating the Cul-
turally Disadvantaged, " Vital Issues in American Education,
eds. Alice and Lester Crow (dew York: Bantam Books, 19tA7,
n. 98.

A

 

 

20

Hence, being born into the educationally derrived
home and community, be it city slum, marginal farm,
or deserted Aypalachian mining town, entails a firm
inheritance that even a Horatio Alger hero could
not overcome.2h

Poor parents cannot give their children the Oppor-

tunities for better health and education needed to

improve their lot. Lack f motivation, hone, and

incentive is a more subtle but no less powerful

barrier than lack of financial means. Thus the cruel _

legacy of poverty is passed from parents to children.2§
It has already been determined: (1) disadvantaged children
are products of poverty-~the children of the poor; (2) var-
ious terms are used to refer to deprived children; (3) any
term or label used to refer to these children fails to
convey a precise meaning that is fair to the child; and
(A) disadvantagement is a massive problem--there are approx-

’imately 12,000,000 children who belong to th's group.
9

For a more complete understanding of the disadvantaged

9'.

child at least three aspects should be given minimal con-
sideration. These are: (1) home environment, (2} learning,
both barriers to and capacity for, and (3) attitudes toward
self, school, and others. In reality these are interrelated
and present an almost circular kind of interaction.

Envircnment. Much has been written in recent years

 

regarding the effects of environment on learning. The

m--u-- -O“--.’--.—

. 2“Robert Hill and Harold Vatter (eds.), Poverty in
;£££l£§g£§ (New York: Harcourt, Brace & Norld, Inc., 1905?,
Pro 191,192.

25;§;g., p. 193.

length of this paper does not permit a detailed account of
the old debate of nature vs. nurture. McCandless wrote:

This controversy has for the most part been reduced
to its common-sense merits.2

Kelley maintains that for over thirty years we have known
the effects of environment on intelligence:

Dullness is therefore more an achievement than a
gift.27

A look at what the environment provides or fails to provide

in the impoverished home gives some insight into the prob-
lems of the disadvantaged.

These living conditions are characterized by great
overcrowding in substandard housing, often lacking
adequate sanitary and other facilities. . . . In
addition, there are likely to be large numbers of
siblings and half-siblings, again with their being
little Opportunity for individuation.2

In the child's home there is a scarcity of objects of
of all tyres s, but especially of books, toys, puzzles,
pencils, and scribbling pa er.29

26 /‘\ fl 0 \ '\
Boyd R. Mc andless, Children anc Ach33ggnt se-
hav'o apd Develorment (New York: Holt, Rinehart and dinston,

-----o.-."-vo--v-

27Earl Kelley, In Defensg of YoutA (New Jersey:
Prentice-hall Inc., 1962}, p. 129.

28Martin Deutsch, "The Disadvantaged Chili an; the
Learr ng Process," Edlcat ion in Degressed Areas, ed. A.
ggrry Passow (New Yor Teachers College Press, 1963) , p.
7

22

Undeniably a deprived Child has many stimuli in his home
environment; however, it is resardin; the type and variety
that most writers are concerned.

A child from any circumstance who has as deprived
of a substantial {no ion of the variety of stimuli
Mb ch he is maturationally catable of responding to
is likely to be deficient in the equipment required
for learning.30

Those in our soc1ety who feel that poverty makes

the man, or that it is the wult of shiftles sness, ha

.1;
<1)

overlooked or failed to appreciate fully the role environ-

(g
H:

‘iJo

t clays and the he lrles sness o the child to overcome t

{D

«Ale

Culturally deprived children can do little to alle—

viate the devastating effect their physical surround-

ings have on them.31

Macrosc01ic environmental factors that one can see

or smell, or those that are conspicuous because of their
abse1ce, constitute only one phase to the problem of dis-
advantagement. Sociological an d rsychological ramifications
are just as real as the rhysical manifestations mentioned
above. Taken altogether the home and community milieu can
not be overemphasized in a study of the di sadvantaged o. Nor

O

can their effects be overstated when considering the limi 1ng

 

Ibid., p. 168.

 

31Augus t Kerbe r and dilfred Smith (eds.),
r Changinc Society (Michigan: Wayne Sta

Edmi aticnal
i a te
r833, 1902), p. 1.570

oanuI‘ 3.:1

 

 

 

23

factors to learning and the formation of attitudes.

Q)
()

hievement and learning pose real problems for the
teacher and comprised one of two major concerns for the
present study. A wealth of literature exists and a multitude
of studies and/or programs are currently under way which
should yield interesting findings. Evaluation, however,
appears to be one of the weaknesses of most programs. There
are at least three reasons for this. A number of the exist-
frograms are still in their infancy and their sponsors
are hesitant to divulge this type of information until find—
ings are complete. The Higher Horizons Program exemplifies
this inasmuch as its evaluator, the Bureau of Educational
Research, is waiting until the findinbs are complete before
publicizing tne full report. Another reason appears to
be that often well-meaning groups have attemnted to do too
much at one time without carefully isolating factors that
could be measured or evaluated.
But as they are generally organized, there is no way
of discovering which of the modifications and what
combination are most effective. It is possible that
one or two new approaches do as much as a whole host
of changes. But how is one to know, except by testi-
monial-«which is at best a questionable technique--

which procedure really effected a change in the
pupils.32

3ZMirian Goldberg, "Factors Affecting Educational
.Attainment in Depressed Urban Areas," Education in Depressed
Arveas, ed. A. Harry Passow (New York: Teachers College
PFC-338, 1963), p. 970

 

24

A third reason for weakne ses in evaluation lies in the fact

that there has been a scarcity of research on deprived chil~
. I . .

dren. Both Goldberg,33 and Lanaers3+ confirm this.

Initially most programs for the disadvantaged fol—
lowed the same route that was used with middle-class chil-
dren. This more—of-the-same idea was partially due to a
lack of know-how and partially due to the idea that what
worked well :ith one group should work well with another.
Goldberg states it thusly:

The various efforts now under way . . . have operated
qenerally on the theory that what the present situation
:alls for is an exnansion of services which had proved
effective in past generatior s or with middle-class
children who had learning difficulties. Character-
istics of these prog rams are increases in such ser—

vices as guidance, remedial instruction 'individual
nW33c01031cal Heat ng, and counseling."§5

W

‘

(

f

H

One reason for the difficulty tie deprived child
has in school and that effects his achievement appears to
be directly related to his home environment.

Undesirable conditions at home can cripyle a child's
ability to learn;a child who is deprived, whether of

physical necessities or of sympathetic understanding
or of both, cannot function at top capacity.36

 

33Ibid., pg. 89, 90.
34Jaoob Lenders, "The Higher Horizons Program in New

York City, " Programs for the Ed ducationally_Disadvantaged
LJashington: Government Trinting Office, 1963), p. 56.

35Goldberg, OE. cit., p. 69.

36Henen R. Nieber, "From Rich Homes or Poor," use
Jcnirnal, LIV (October, 1965), p. t5.

The culturally deprived youngster's early eXperience
may put him at both a cognitive and emotional disad-
vantage in "achieving" in the school's torms.37
Operation Headstart, with its desire to get children young
so that some of the effect of 1roover1ohments can be pre—
vented and/or overco..e, is the best testimony to the effect
of environment on learning.
ew peOple believed that a six to eight week experi-
nce in a Headstart Child Develo; me nt Center would
or could) counter all the forces impinging upon the
conomically disadvantage c.ild, but educators azreed
hat the programs would be a favorable beginning.
Of the twelve stated goals for Headstart, which included

the whole child, three relate directly to learning. One of

To imr rove and expan the child's mental processes,

aiming at expanding ability to think, reason, and

spea< clearly. 39

Much has been written about the hidden IQ of deprived

children, Riessman devotes a whole chapter to it in his
book, Culturallyiflisadvantaaed Child. Clark, Goldberg, and
Deutsch, in Education in Depre sed Areas, eXpr ess the notion
that the deprived child has hidden potential for learning
that does not show up on standardized IQ tests. If their

_— v*~—--‘-—-—.—w-—-

37Passow,o OE. cit., p. 286.

38Milly Cowles, "One Front in the War on Poverty,"
SLAUN Journal, LIX (October, 1965), p. lt.

391b1d., p. 15.

26

contentions are true (and there is much evidence to support
them) then despite the fact that a deprived child is a slow
learner, it's a false premise that he is not capable of
achieving.
Most disadvantaged children are relatively slow in
performing intellectual tasks. This slowness is an
important feature of their mental style, and it needs
to be carefully evaluated.h0

The assumption that the slow pupil is not bright
functions, I think, as a self-fulfilling prOphecy.‘*l

New Optimism is currently being exhibited both as to the
reversibility of the environmental handicap and as to the
potential for the deprived to achieve. In speaking of the
Higher Horizons Program, Riessman draws this conclusion:

It [the Higher Horizons Project] demonstrated con-

vincingly that supposedly uneducable children from

lower socio-economic backgrounds can successfully

learn and progress in a reorganized school environ-

ment.
From a report on the Detroit Great Cities Project, Marburgerh3

also affirms the ability of deprived children to perform

 

hoFrank Riessman, "The Culturally Deprived Child: A
New View," Pro rams for the Educationall Disadvanta ed, HEW
(Washington: Eavernnent Printing Office, I963), p. A.

“11b1d., p. 5.

“ZFrank Riessman, The Culturall De rived Child (New
York: Harper and Brothers Publishers, {962), p. 93.

“BCarl Marburger, "Working Toward More Effective
Education," Programs for the Educationally Disadvantaged, HEW
(Washington: Government Printing Office, 1963), p. 71.

27

providing the school can make work interesting. Clark also
makes the point that deprived children have greater potential
for learning than has been formerly thought:
The evidence is now overwhelming that high intellec-
ual potential exists in a larger percentage of in-
dividuals from lower status groups than was previously
discovered, stimulated and trained for socially bene-
ficial purposes.hh
According to Sexton if deprived children were given the same
opportunities that middle and upper-class children en-
Joy, they too could excel. ‘
If lower-income groups were afforded the same educa-
tional advantages as upper-income groups would their
children be just as "gifted"? They might be, and
there is evidence that they would be.“
The concluding remarks of the conference for disadvantaged
children as reported by Cummings include the following:
The most important discovery about the pupil from a
disadvantaged home is that he has a capacity for
learning, even as other pupils do.h
The failure to recognize the potential of the deprived child
has been costly on two levels.
The actual costs to the government are immersed in

the expenditures for criminality delinquency, un-
employment, ill health, mental illness, and social

 

““Kenneth B. Clark "Educational Stimulation of

Racially Disadvantaged Children," Education in De r ed Areas,
ed. A. Harry Passow (New York: Teachers College gress, I555),
p. 161.

Assenon’ OE. Cite, p. 61.

“éHoward H. Cummings, ”Conclusion,” Programs for the

Educatfiogallf Disadv%taged HEW (Washington: Government“
r at g ce, , p. iOl.

disorganization. The cost of undiscovered, unused,

and abused human beings who could have done so much

for society can only be guessed, but the evidence of

the unused potential is painfully obvious and en-

ormous.h7

From the literature it seems evident that though there

are limiting factors that adversely effect learning for the
disadvantaged child he has the capacity to learn and is
canable of doing so when conditions are favorable.

Attitudes_towa§d self, others, and school. McCandless

 

defines self-concept as "a set of eXpectancies, plus evalu-
ations of the areas or behaviors with reference to which

these expectancies are held." He elaborates by:

Q

ur illustration also suggests that the self- concept
holds preperties in common with drive in that one

elects some deve lormental, recreational, and avo—
cational areas as a function of certain charact .ristics
of the self- concept, and rejects other.AC

(I)

In discussing the self—concept and school achievement the
same author makes this statement.

It might be predicted that poor self-concepts, 'mply-
ing, as they so often do, a lack of confidence in
facing and mastering the environment, might accompany
deficiency in one of the most vital of the child's
areas of accomplishment--his performance in school.49

McCandless cites studies in support of his sition regardin;

time self-concert:

—-‘—- -—.m-—.«- .-.

L7Kerber and Smith, op, cit., p. 156.

thcCandless, op. cit., Po 176'

“91b1d., ph. 185, 186.

A

 

29

Change in self-concept is, of course, required by the
yrocess of maturing and is central to such activitiec‘
as counseling, psychotherapy, and remedial teaching.50

All of the present discussion of the self concept

has been based on the assumption that it is learned.

It seems legical to think that the self-concept,

based as it is on attitudes and values held about

the self, has much in common with general social
attitudes and personal be lie efs and values. Any theory
or research, then, that relates to chancing attitudes
should have relevance for changes in the self-concept.)l

McGandless further contends that there is a relationship
between self-acceptance and how the child accepts others; and
that parents play a vital role insomuch as their attitudes
strongly shape the child's self-concept. The ideas of the
above author that were relevan to this study are listed
below:

self- conce pts are leazred

self- -conce pts can be modified

parents influence self—concerts

self-concepts are based on attitude

achievement is effected oy self- con

self-concerts are central to remedial
drives and self-concepts are related

0 U)

n?
U
.—
‘—

1’ O}

11-,
a~nixg

Recent research by Brookover, LePere, et al., at
michigan state University, supports some of the above ideas
and expands on them:

The basic postulate is that academic behavior or

school learning is limited by the student's self-
conce it of his ability in these areas. We further

SOLbido. p. 198.

 

30

postulate that self-concept results from the expec-

tations and evaluation held by significant others as
perceived by the student.52

Of nine statements that were accepted, as a result of this
study one was of particular relevance.

Parents are perceived by more than 90% of the stu—
dents as acadgmic significant others in all grades.
(Junior High) 3

AprOpos to the disadvantaged child, as well as the societal
need to develop all talent to its maximum, the following
statements by this research team are germane:

There is, moreover, sufficient evidence to warrant the
position that enhancement of self-concept of academic
ability should be a crucial concern to educators striv~
ing to assist students to achieve at the highest level
of achievement possible.

Perhaps the most important implication of this investi-
gation concerns a theme in educational literature that
only a limited number of students are able to learn
mathematics, languages, science, and other school sub-
jects to the extent required by our advance technolog—
ical society.55 (emphasis mine)

Most literature on the disadvantaged at some point

refers to the negative self-concept, warped self-image, aa-

L.
gressive behavior, alienation, et_£§tera. Classroom prob-

lems resulting in part from such characteristics are of

#. -- —- ---~-fi

F.

5‘W. B. Brockover, J. LeFere, D. Hamachek, S. Thomas,
E2. Erickson, Self—Ccn‘ept of Abilit and School Achievement
’vi"h. . M.“k A qr h T o v . ,"i oo- o
(11 c 1gan. 1cnigan ovate Jnixer31ty, 190)), pp. 111, 1v.

     

II

C?
J 0v ‘-
JIb19., p. 208.

m

5LIbi’o, pp. 208,209.
55

LL.

 

Ibid., p. 210.

31

vital concern to educators and other personnel working with
deprived children.

So often, administrators and teachers say, they are
children who are "curious," "cute," "affectionate,"
"warm,“ and independent in the kindergarten and first
grade, but who so often become "alienated," "with-
drawn," "angry," "passive," "apathetic," or just
"trouble-makers" by the fifth and sixth grades.56

The negative feelings of these children are stressed by

Shostak:

Deprived youngsters by the millions are presently
failing in school and drOpping out. They do not
learn to read preperly; they fall farther and farther
behind; some begin to hate themselves and the system
that makes failure public; many withdraw from compe-
tition, and the gap between the slum and the suburb
widens.57

The attitudinal problems are further brought out by Korn-
berg:

But in our schools we have seen that the "very diffi-
cult children" are almost a norm among the culturally
disadvantaged youngsters, and those who are given the
special help often return to the classroom as diffi-
cult as ever. There is no overnight cure for the
attitudes, fears, defenses, deficiencies in children
that grew so early, over so many years.53

 

56Martin Deutsch, "The Disadvantaged Child and the
LLearnin Process," Education in Depressed Areas ed. A. Harry
Ilassow %New York: TeaChers College Press, 19635, p. 165.

57Arthur B. Shostak, "Educational Reforms and Poverty,"
hkaw Pers ectives on Povert , eds. Shostak and Gomberg (New
Jersey: Prentice-Hall, Inc., 1965), p. 61..

58Leonard Kornberg, "Meaningful Teachers for Alienated
Cluilxiren," Education in Depressed Areas, ed. A. Harry Passow
(dVevv York: Teachers College Press,1953), p. 263.

32

To me the salient characteristic about th
in a classroom is their aliengtion. To u
gon, they are not "with it. "57

se children
3 their jar-

Riessman60 and Narburgerél both have written of the aliena-

tion, indifference, and poor attitudes of the deprived chil—

62

dren. Kaplan, in discussing programs for the d sadvantag ed,

3

e inasizes the important role that attitudes and motivation

C

play. his is furthw suppo ed oy Clar

It is clear that a fundamental task of the school
stimulating academic achievement in disadvantaged
children is to provide the conditions necessary for
building in them positive images of themselves--
building in these children a positive self-esteem to
supplant the feelings of inferiority and sense of
hopelessr ess which are supported by an all-too-
pervasive pat tern of social realities.63

It is noteworthy that of the twelve goals of the
h eadstart Program referred to earlier at least five were in
some way related to self-concept and attitudes.

'w'ritings abound with information regarding indiffer-

(D

nee and/or hostility dominant among disadvantaged children.

 

59Ibid., p. 27g.

éoFrank Ri an, "The Culturally Deprived Child:
I Ilew View, " Provr

essm
{imgmgor tQ:_; dutatiorally Disadvantaged,
ILSN' (.Jashing on: Gove

rdagfit Primnting Office, lgEBF, p. 7.
61Carl Iiarburger,‘ w’orking Toward Kore jffect
jchlcation, " Programs for the Edml ationally Iisadvanta
f"
I

've
31, H

q.
of

(.4;
t

T r".

(Lflashington: Governxm ant Printing Office, 1903),?

62

Kaplan, on. cit., p. 107.

HD; H

63Clark, CE. (3112., P0 1570

In respect to how these attitudes toward self, others and
school effect achievement much is theorized but concrete evi-
dence is not abundant. In any case a healthy concern is
now evident.
In summary, as a compilation derived from the liter—

ature cited and from other eadings in the bibliography, a
list of traits or characteristics of disadvantaged children
are presented. It should be clear that when one considers

particular child any specific trait in the list may or

may not apply to this child, and if so it may be to any de-

ree.

UQ

product of a broken home
ember of a large family
aiti- -intellectual
has poor attitudes
is a physical learner
has short-term goals
has short attention span
poor reader
lacks time consciousness
has poor auditory habits
has verbal inadequacies
craves respect not love from school
needs structure
likes action
is a slow task performer in academic matters
has high mobility history
evidences low self-esteem
low achiever academically
posseSScs weak motivation

Despite tie fact that only the more negative traits
inrve been considered here, the writer is aware that dis-
czdvuantage children also have many positive attributes.

Rie:ssman, in the Culturally_De:rived Child, and in a cha apt er

ed these. A partial

(D
U]

a
O

in New Perspectives on Poverty str

list f these positive traits taken from Riessman and others

has welld eloped informal verbal saills

respects physical strength

is creative and improvisational

has strong peer ties

has strong sibling ties

has a well developed sense of humor

enjoys music, games a, sports

has preserved his ethnic traditions

values education

is free from self- blame and parental overirotection

Recommendations. Recommendations for working with

 

deprive ed children usually stress catitalizing on the positive
strengths or attributes already cited. Other recommendations
include providing Opportunities for success, promoting
better self- -image, utilizing more physical activities such
as role playing and games, building on the child's interests,
adjusting instruction to life—oriented, here-and~now kind
of experience, and encouraging closer ties with home and
community. Wilkows}1i6“ refers to respect for children,
patie e, ability to accept children fo or what they are, more
concrete materials or aids, changing pace more frequently,
13nd use of praise.

Many recommendations fall outside the scope of this

3naper and would include such concerns as more state and

 

“Genevieve fiilkowski, "Teachers of Culturally Dis-
advzintaged Children," Michigan Education Jogrnal, XLII (May,
1965), Pp. lib-l7.

 

\AJ
w

federal surrort, more comnunit3 invo vemert more counselors
and srecialists working directly with the disadvantaged
child, better facilities and a host of reforms that are be—
yond specific recommend1tions that could help the teacher

in the day-to-day classroom situation.

the problems of lower-income children and their education.
A sample of these is presented:'

use of I; test should be s’opped

the e should be warm encouragement of parents for
greacer school invol‘ement

tion of class size
extra help, rather than failing marks
more fie d trirs
more males in the classrooms
more research dealing with lower ~income groups
Overview

After the Russians troved that man was no longer
earth~bound a wave of criticism was launched at the American
public schools. Claims were made that the United States
needed more mathematicians and scientists and that these
could be re cru ed only among the gifted. Thus in the late
fifties and early sixties the gifted child received more
:attention in our so 0013. More recently there has been an

irhcreasing realization that the future of America will de-

Icerud upon pr1Hs nt development of human resources not only

.

 

-n. '-

6533

xton, 0p. cit., pp. 267—276.

among the academically elite few but also among the less
talented many.

The advent of automation brought a new set of nrob-
lems that had to be faced by the American peeple and the
schools. The questions of how to absorb the unskilled
wOrkers-—those displaced by the machine, and how to prevent
unemployment had to be answered. These concerns plus others
such as the drOpout problem, juvenile crime, and delinquency
broucht educators and politicians into a concerted effort
to fight against poverty and allied problems. Consequently
today we are in the midst of an era of genuine and massive
concern for those in our society who have until recently
been shamefully neglected.

One cannot point to any single event or publication
as having had the greatest impact uoon or having been most
significant in bringing about this realization. There were
a number of influential landmarks. Patricia Sexton's book,
Education andglncome, was noteworthy in this respect. Conant':
work, Slum and duburbs, Riessman's, Culturally Denrived Child,

and Harrinpton's The Other America-~all served to arouse

 

gamblic interest. The Anti—segregation Law of 1954 and the

(Byrortunity Act of 1964 were important events in connection
WiiLh the new awareness and they portend to accomnlish much

in away of alleviating inequalities and rreventing wasted

hunuan assets.

37

The investigator found two contemrorary writers
who were particularly cognizant of the educational froblems
related to socio-economic status at a much earlier date
than most. Frank Riessman in his introductory remarks in
a 1962 conference report stated:
I have been interested in the problems of lower
socio—economic grouns for about 15 ,eare, during most
of which time there has been a lack of concern for

the educational rroblems of children from lower-
income families. 9

tn
0.
C.
( )
,3
l'+
’L.‘
'o
‘5
b

Allison Davis, writing in Contegroragy;American

 

expressed a sinilar concert:
Nearly 15 years aho, in an address to the general
session of the American Association of School Ad—
ministrators, I pointed out that our efficiency as
a Nation and the preservation of our 3osition v’s-
a-vis the communist powers depended larcely upon
learning how to motivate and teach the socio—economic
groups in our schools.©7
There have always been a few concerned individuals
who snoke out in defense of noverty's victims. Jane Addams,
of Chicaro's famed Hull House, was one of those actively
concerned. In the late nineties, sneaking at a National
Educational Association convention Miss Addams nortraved
9 1 .

the rlicht of the immigrant child in large cities and schools.

She noted these children lac

is r
U]
(7
:7
O
O
H
.
If)
:7
(U
"3
O
H-
:3
F,
\V
Cl.
C
'5
L;

and frequently dronped out o

.—.—-—. ----“—-- -..—..-.-

éfifiiessman, on. cit., p. 3.
'3
6’Allison Davis, Contemporary American~§ducaiign,
dss. DrOplin, Full and Schwartz (New York: T.e Macmillan
(N

r.“ 1965 . r. 543.

 

38

the schools had failed:
Has anything been done up to this time, has even a
beginning been made, to give him a consciousness
of his social value ?68
In the introduction to this article (the eproduction of
Miss Addams' speech) the editor made the following comment:
The depth of understanding that pervades her testi-
mony has marked relevance to the current struggle
in schools with culturally disadvantageds tudents.69

Further evidence that the problems of disadvantaged

children are not new is t.e book, fhe iducation of .he

 

I‘Ee'er—do-«fiI ell, published in 1916. Therein Mr. Dooley was

 

concerned with the rie e of the factory system, the demise
of tne arprenticeship, and the concomitant effects upon
many children. If one would substitute "automation" for
"factory system" and' disaa ‘vantaged" for the ".e'er-do-well,"
the ideas presented would sound quite contemporary. There
are many interesting parallels in this book that demonstrate
ratner well the apyalling lag in both sociological and edu-
cational reforms. The current Great Cities Projects ”ad
their counterpart in Kinneanolis, Chicago, and New York
during the early part of the cent ry. Er. Dooley's pleas

for adult education, and vocational education is echoed in

’m‘---*-- 0-“-

 

" yaw“ fl ”3' w ‘ .- r .: f‘ “
t all u,. -, .en enn, rar" [krerican .nwlcctifr‘ eds.
'C , 17 v, a W‘r '
Urwbtlii, l‘ll 1rd -‘nvarcz (new 1J.l: ~ L iii .
o '10

 

39

Conant's Slums and Suburgs. Among other recomrendations
in the Ne'er-Do—Rell were

greater cOOperation be t ..een schools and industry

trade union involvement

holding power for drapouts

new school design including showers, gardens, roof
playgrounds

inductive approach to teaching

increasing the school day and year

Yr. Uoole y's greatest concern was for the unskilled, unem—
ployed, and unr otivated youth-—the ne'erw WI-h ll of his day.

Jaste is repugnant to us today, yet we have failed

to provide for the great majority of the boys and

girls.70

The literature shows that neither roverty nor its

children is a new rncno.3zcn in the United States. History
is replete with token innovations to help tne derrived child.
Included among t.ese were compulsory ed ' cation laws, child
labor laws, hot lunch programs, etc. Today we are witnessing
innovations at local, state, and national levels to overcome

problems that have been building up for years. Perhaps it

u)

can soon be said, le sse d are the poor--forT hey shall be

helped.

' /Ofiilliam H. Dooley, ID.) Lduc at.ion of the
:fifiL; (Boston: Houghton Li ffli n Compar;y, 1915?, F0

 

 

 

RLVISU OF THS LITdRATURi—~FAET II

According to Goodlad, the current curriculum reform
dates back to the years following florld dar II; however, he

t:tes it is usually linked more recently with the Russian

U)
3
'—r

ellite of 1957.

0)
m
(or

This spectacular event set off blasts of charges and
counter-charge 3 regarding the effectiveness of our
schools and stimulated curriculum revision, notably
in mathematics and the n hysical sciences.

The above author continues bya alyzinr the strengths and
weaknesses of the mathematics programs that were part of
the revised curricula. Some of these criticisms are rele—
vant to the present study.

Tens of thousands of schools have been scarcely touched,
or touched not at all, especially in areas of very
"tarse or very dense porulations. Tens of thousands

of teachers have had little onnortunity to come to
grins with what advances in knowledge and change in
subject fields means for them. . . . The gap between

the haves and have-nets persists and, in some ways,

is accentuated.2

Goodlad contended that, although there was great uniformity

among the various crOJ ects and/or grog rams, objectives were

- --- -..

 

 

lJohn Goodlad, "Changing Curriculum of America's
:Schools," Saturday Review, XLVI (November 16, 1963}, p. of.

 

211

m-

/
i9. , Pt. ()6.

Ll

"vague," ”not stated," or had a‘mystical quality." He felt
this w as particularly true regarding structure and concept-
attainment.

Ho and would be served to describe each of the major
math nrcgrams that came into existence as a result of cur-
riculum reform. These are well known and each group has

nublished a wealth of materials. ‘It might be fruitful though
to examine certain aspects. In 1963 the national Council

of Teachers of Nathematics published a bulletin, An Analysis

of dew hathematics Programs, in which an overall analysis

 

 

was made of eight of the better known nrog rams. The follow-
ing criteria were used in the evaluation:

Social Applications
Placement

Structure
Vocabulary

Methods

Concepts vs. Skills
Proof

Eval‘ation

A resume followed each of these tonics--usually in the form
of questions. None of"these were defined. The members of
the committee noted that mathematics is in a state of flux
and they leave it to the reader to decide his own position
on each of the above topics. One example should demon 1strate

'

this lack of a stance by tne committee.

It vs. Skills

nha relationship should exist in the mathematics
programs between the function of develoring concezts
and that of developing skill in the manipulation of

Con

(D

(*U)

ix:

r
138

symbols? some rersons
the full meaning of a conc
the level of automatic
cert. There is also the
skill is ontimum for our
to the changing concerts
1’ ures?3

1

F3

iven though the committee did n
position they did consider each

to a mathematics program.

The 1nv sti 3atcr made a

eight programs included in the

whereas each program was unique

t
b

were evident. Among he

quently were: concent orient dd

 

thinking, and rea ing;

0)

 

principles. The language of ma

important and thus a precise vo

nt was introd

earlier grade—wise than is dict

stressed tr e disc oxer y method,

teachers, develonment of aids,
This is by no means a thorough
considered a "new mathematics"

C
h)

erve as a guide as to what is

—----.‘-—..'_

3National Courm il of Te

response in his us
question:
present

bulletin a.nd reco niz

latter that

search

A

hat a student
only when he

+
V
at

:3
v

what level of
society as Oppose
sulting from changing cul

ot take a more definit1

a a
\J .‘
H

of the above torics crucial

comrarative study of the

73d that-

neverthelec commonalities

appeared mos+ fre-

stress upon understandinr

9

 

for pat terns, deas nd

 

thematics was consider ed
cabulary
uced;
ated by tradition.

in—service training
films and materials, etc.

presentation of wh .at may he
program;however, it would

deemed important by those

 

achers of l’athematic 3, An
3e43l1§l§w0f. New_ Dabh°matl s Procrams (u'ashington: 19637,
C. 3,

(>-
\,. )

who have develored such programs. It should be nosed that
all eight pr05rams were designed for the typical child or
the more able learner. The in hool Kathe atics Study Group
recognized this and have more recently made an effort to
include the deprived and underachievers.

It should be noted that much of the success of the
so-called new mathematics has been demonstrated with
middle-class children.h

Conference members from the joint meeting of the U.

F‘

3. Office of Education and the National Council of Teachers
of Mathematics recognized this neglect too:

Our intent here is to consider the nathematical need
and proper instruction in mathematics for that cate-
gory of youth referred to by Dr. Conant as "social
dynamite"—~t ose who possess no slzill, who are unem-
{loyable and unschooled. . . . Our range of interest
will include mathematics for those students who are
potential dropouts, as well as for those who remain

in school, but who, for one reason or another, exhibit
a rattern of low achievement in mathematics. >

Reaso sfor undating the math curriculum appear in
much of the literature and include the following taken from
Eront iers U; Ijathemat cs Education, published by the Tic . igan
‘erartment of Public Instruction in 1961. These afrear in

summary form:

 

“School Mathematics Study Grout! Conference on Blathe-
cs ducation for Below Average Achievers (California:

—.——-_-—

 

me all
ela nd Stanford Junior University, 19C77:'}C 15.

HQ

5U

J
O

S. Department of ealth, dducation and u'elfare,
The Low apniever in Mathematics ’uashincton: Government
Printing Office, lvi T, . l.

 

 

 

 

 

(‘1
\II

’73

l. Kathematics mus’ :row and change to meet the
demands of a Man ing world.

2. More mathematicians are needed.

3. Other fields of knowledge are na“1ng 1ncreas d
use of mathematics.

A. wxperiments indicate the value of earlier intro-
duction of selected ccrceIts and skills.

5. Inability to ore ict the skills needed in the

future.

'7

New develonmen; in chi
understanVing of
elimination of rote

drill.

ld
the learnin:
'n 3g10‘""‘

rsyehol ozy and a greater
process require
tion and meaningless

IJ

 

 

 

 

 

7. Larre num‘ H8 5 of our rorulation have been inade~
quately r e ared in nachenao'-s.

8. Greater realization that the development of per-
sons oroficient in mathematics begins in the
elementary school.

9. Sc.ools must rr daze more highly trained technical
peo;le and also informed and literate citizens
in mather watics to enable them to understand
the 1r ten2hnolocical world .5

These same reasons could be advanced for u1-detin; the cur-
ric ulum for the under achieter, the slow learner or the dis-
adlantage d. in the book Conrensatorgiiducation for Culture.
Degrivation, the authors review the social change and forces
that necessitate educational chants s for the deprived.

A centralf ctor in the entire problem of edutsat ion

and "ultural ‘wj‘i":tion is the "‘ijTy «3:1'f“”

6The Denartmen of Public Instruction, Frontiers in

gathemggics_§duggtion (hic igan: LDSA-BlO, 19 Cl), pp. 6, T.

\ll

‘1‘.

-en which requi as more
ire nonulat ion. It is
ch force charges in
onlems of cultural de-
r'

economy and iob-distri bution s
and better education for the e
this new set of requirements w.1
education to meet the specie 1r
privation in various groufs of o

Germane to the changing economy and job-dis tribu tion system,

which are realities of moiern life, are the remarks by an

Those of us in industrial math matics see very cle '1rly
that the mathematical requirements are changing. One
consequence, of course, is that miny workers are being
technologically rezlaced. Their jobs have chansed,
they must be retrained, and mathematiics is often part
of the retraining. . . . lroblems of very substantial

mathematical nature may come to be involved in jobs
which were (and in many rear cts mill are) of very
routine nature.

It is clear that these changes will efiect the underprivileged
groups.

fie will cer ainly continue to require more ani bette
mathematical t raining in all facets of our activity.r
For us today, the point is that many of these activ-
ities will involve peonle from the groujs we ar
talk n: about educatin:.9

In the "Gardner R-ero. , setting forth national

A

 

.o- ~~~--.

Benjamin Bloom, Allison Davis, and Robert Hess,

Compensatory fiducation for Cultural Derrivation (3 av York:
r. u g ‘. ’ v. _. 5 r\ r,-

hfl , ninenart and Jinston, inc., 1903 y. 5.

 

 

e on Hathematics aducation
an iron Panel Discussion,
’ "c

0/4 4,] p0 lU/o

 

53. H. Colvin, Conferenc
for Belou Average Ac .hiev vers, t

\ ’1

a?
Shea Conf ere1ce (Cal iforn 1a: 19

9

 

_’

Ibid., p. 101.

n
-. . . . 0-.) v '1 ‘Q 1
i 1 '1 ”\‘vn V?" L '3‘? " faflfi' 1.".1‘” 1'“ 1.‘ " 51"- ” (A
L‘ .- " .".' .4 54.: , ~1"‘« ' ’50-‘and -“" ‘lr" 5-D. -‘ A) " '- , \
I

. . v v

r - \V ; 1»- r , sh ' v’ "A ,..$ ~- ‘_'_;'. w I
1!. , ) {9‘ C' s ’ kl 5A; 4 1 ~. J... ‘ ’ - ~ 9-! b ‘-

-—-~‘ -———.‘_~. 0.- -~~.— -1._-

‘:~.
0“

5..)

H
b]
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L.

goals for education, mathematigf was 1' ,3 second only

to reading as a friority subject in the elementary school.

Paul Hosenbloornll equated the two (readin? and mathemat ice}

as "key sub acts, for making low-ability children emrloyable.”
t is apparent from :he literature that: (l) a

kno owledie of mathematics is uxg :ently needed by all members

of our societ"; (2)r esear ch and curricular innov tion in

mathematics for the low achiever and disadvantaged has been

lected, (3) a large percentage of the low achievers and

slow learners belongs to the group labeled disadvantaged.

Underac hievers and low achiev rs. Through the lit-

 

erature the writer found various terms applied to low—
achieving children. Often they were referred to as slow
learners, educationally har dicarped or retarded. Desrite
some confusion of terms, when these were defined and used
consistently they were based on either IQ tests or achieve-
ment tests or both. According to j. Paul Torrance12 this
traditional concert of underachieving is no longer valid.
he quotes Getzels and Jac nson in surport of this:

In det~rmininc ove rachieveme.t and underachievewx nt,
3 rs usually fail to take into account the

nbloom, "Implications of Psytnolo.1tai
'er_in hathemat cs, H£N(nas“1nton:

Reseach," The Low 1 1
fice, 19057, p. 25.

,
1
Government Printin n5

 

 

relationship between the caracities and needs of the
individual ar .d the ability of the envi: onm-en: to pro-
vide outlets for these capacities and needs.13

In poetic terms Torrance describes t e underachieve,:

A sco1ned imarination, an ur used memory, tabooed sen—
sations, an interrunted thou ght, a rejected qiestion,
a forbidden day dream, an unexpressed idea, an un-
~ought judrment, an un ainted nicture an unsung song,
a safel,lfidden noem, unused talents. . . . These make
an underachiever.l’+

0‘

Low achievers are defined in the ’fi1oG conference bulletin
as:

1. Those children who have intellectual deficie
2. Those children who have cultural definc iencies.
3. Those children who have both intellectual and cul-
tural deficiencies.
Perhaps a qualitatively different group are the
ildren who are neither culturally nor intellectuallv
deficient but who are slower earning.l§

'0

From the Low Achiever in mathematics the following descrir-

 

tion emerges:

The low achievers can best be characteriz d as st‘—
dents having an intellectual potential at the low-
avera3e level 1, who are generally about one or more

years halo; 3rade in arithmetic and one or Mcrg
years retarded in readin3.lé

Disadvanta3ed children are often thouaht of as slow learners
D K

r ‘. fl, . . ‘V , t ‘ ‘. '1 -1 ’- t" 3* . ‘. 1“ . Pu“ .
etause tney as a rule are gnysicai 1eazhe1s. A distinction

13

Eido’ pr}. 15, 160

 

 

‘SBOrge B. Grain, "fiesronsibilities of Jcrcol Ad-
ministrators,' Lhe Low 1chiever, hsJ (”ashin ton: Governmen‘
o v " - [a 1’ ."r‘ r ,
Printing Office, lacy}, pr. 39, ,b.

should be made between the slow learner as a physical learner,
and the slow learner as a less capable learner, albeit dif-
ficult in some cases.

\lthou ;h it is true that many disadvantaged children
are slow learners this position tends to overlook
some distinctions. One group of slow learners con-
sists of those with limited genetic endowment, whom
the nest of instruction would profit in only limited
ways. Disadvantaged children are not slow learners
in t1e same sense.l7

The functional slow learner, as orposed to the less capable
learner is a troduct of environmental factors that have
failed to provide the child with the variety of experiences
and stimuli n-ec essary for full de dormant and normal achieve-
ment. Leiderman's remarks substantiate this view:

The major thesis of this paper, derived from develOp-
mental apyroaches, is that the slow-learning group
consists of some children, rerhars a majority, who
are r1tarde d in their me ental deveIOtme ent bec1use con-
tact with their ,hysical and social environment was
deficient in their early years. This thesis makes
rerearc h data on the culturally deprived groups rel-
evant. 1"

V

O
(
U
(

There is a cons ensus among many that the number of children
who as a result of a deprived background are doing poorly
in school generally, and in mathematics in particular, is

a sizeable group.

 

'7
lfHarry Beilin and Lassar Gotkin, "Psychological
Issues in t1e Deve 10pment of Mathematics Curricula for
Socially Disadvantaged Children," Conference on blathematics
Education for Below \vera"e Achievers (California: SKSG,
19647, pp. 15, 16?

18Gloria F. Leiderman, ""ental Development and Learn
inc of 13thema*1Cs in Slow- Learning Children," 'onference c
“:33 Sducation for Below Average Ac hie ers (Califo:
. n. 47.

It seems reasonably clear that early deprivation in
the child's background will result in less ability
to abstract; that his lessened verbal develoyment is
related to difficulty in learning mathematics; and
that a larger proportion of children from disadvantaged
grouts in the bonulation contribute to the slow-
learning and low: -achieving groups in our schools.l>’
Presently there seems to be little concern for cate-
gorizing those who are disadvantaged into neat groups of
ability levels as has been done with other children. Prob-
ably this is because diagnostic instruments that would yield
reliable results are lacking. Frequently one finds terms
used interchangeably such as "undera chiever" and "slow
learner" that would have connoted more than a subtle differ-
ence in the past framework. Most disadvantaged groups con—
tain a wide range of ability levels as well as a wide range
of maturational and de"elopmental levels. Yet they are
lumred tonether in many prog ams. Or the low achievers
or slow learners are grouped together which would undoubtedly
include some disadvantaged children as well as those less
carable genet 1cally of ac hievin“. It follows that this
would depend to some de gree unon the area be ing served;

a . .
however, daVighurst‘O found as many as 805 of slow learners

to be from low socio-economic backgrounds. The ability to

2OLloyd I-h Dunn,p "The Slow Learner,‘ NBA Jogggal,
KLVIII (Cctober, l959), p. 21.

identify talent or lack of it among disadvantaged children

presently can not be done with much predictability.

The developmental approach to teaching the disad—
vantaged child,regardless of what his potential might be,
appears to o fer promise.

If we consider the slow-learning child as one with
limited abilities and potential for learning, then
our emphasis will be on creating learning experiences
suitable for his circumscribed capacity. However, if
our approach assumes that the child is a changing
organism whose development is affected by both ex-
ternal and internal factors which may expedite or
impede his development, our thinking and planning

for his education will consider his motivation for

le earning as well as his cultural environment.21

This v’ew is a charitable one and certainly for deprived
children has many strong points. Until much more is known
about the deprived learrer it offers hope for *he greatest

number of children in all academic areas including arithmetic.

:‘i

From cited de init ions, given for underachieving,

loweachieving, and slow—learning,it should be apparent that
whatever term or criterion is used it fails to connote a pre-
cise, well understood framework when applied to disadvantaged
children. Even so, the focal point of interest is a group

of children wr o are called disadvant a d and, for reasons

(3"

not fully understood, are not ach1 ving in school to the

(D

degree that available evidence indicates they should. It

“-‘-’-‘-~-"~O-n'-

°l . . ,
“ Leiderman, op. EAEI: p. 5.

51

is rsCOgnized, however, that many factors impinge upon the
learning abilities of these children. Among the identifi-
able factors that adversely affsct learning and/or achieve-
ment are:
limited background of experiences
limited background of objects, books, toys etc.
lack of adult models to stimulate and facilitate
early learning
poor attitudes and disinterest resulting from the
child's perception of the real world and his
opportunity to excel in it
history of repeated failures
poor self-image
object of discrimination
poorly developed verbal and auditory skills
unmet basic needs--nutrition, medical, security, love
As a result of poor physical environment combined with the
adverse psychological, sociological and physiological influ-
ences the child is ill equipped as a learner. Thus Leider-
man used the term "cumulative deficit" to denote the "in-
creasing discrepancy between achievement and expected learn-
ing."22 It is in this orientation that one can speak, with
a more complete understanding, of what underachievement
means in respect to deprived children.
A study by R03323 on underachievers in arithmetic,

taking a case study approach, yielded important data relevant

 

221bid., pp. 55, 56.

23Ramon Royal Ross, "A Case Study Description of
Underachievers in Arithmetic" (unpublished Doctoral thesis,
An Abstract, University of Oregon, 1962).

52

to this thesis. The pertinent generalizations taken from
the abstract follow:

1. Arithmetic underachievement did not become apparent
until the fourth grade.

2. Subjects tended to be from home environments which
provided little intellectual stimulation.

3. Subjects characteristically were withdrawn and
defeated in their attitude toward school.

4. Subjects were underachieving in school subjects
other than arithmetic.

5. Parents tended to be from low socio-economic
classes.

Concepts, conceptual development. The mathematical
literature reflects profuse usage of the term concept; but
it is assumed the meaning is understood for it is not de-
fined. It was necessary to turn to psychology for enlighten-
ment. Bruner, pp_§l. present two views of concepts:

There are those who urge that a concept, psycholog-
ically, is defined by the common elements shared by
an array of objects and that arriving at a concept
inductively is much like ”arriving at" a composite
photograph by superimposing instances on a common
photographic plate until all that is idiosyncratic
is washed out and all that is common emerges. A
second school of thought holds that a concept is not
the common elements in an array, but rather a rela-
tional thing, a relationship between constituent
part processes.

The same authors add that such a controversy is fruitless

and present another definition:

 

2“Jerome Bruner, Jacqueline Goodnow and George

Austin A Study in Thinking (New York: John Wiley & Sons,
Inc., $95 , p0 211.190

53

The workable definition of a concept is the network
of inferences that are or may be set into play by an
act of categorization.25

Concepts appear to be related to both a category and a think-
ing process:

Concept attainment is to be sure, an aspect of what
is conventionally called thinking. . . . But we also
have urged a broader view: that virtually all cog-
nitive activity involves and is dependent on the
process of categorizing.26

The concept or category is, basically, this "rule of
grouping" and it is such rules that one constructs in
forming and attaining concepts.27

Hunt disagrees with these authors on some points. According
to him:

Arbitrary categorization is not concept learning,

yet it sometimes appears in the psychological liter-
ature under this name. . . . Furthermore, the learner
can use his classifying rule only for the stimuli he
has previously experienced. This does not seem con-
sistent with the dictionary definition of a concept;
a concept should be generalizable beyond our imme-
diate experience.2

Regarding simple categorization or classification, Hunt
supports Church's conclusion:

Church reasoned that a name has two prOperties, its
meaning or concept and its denotation. The denotation

 

251mm, p. 2th.
26Ibid., p. 246.

271b1d., p. 45.

28Earl E. Hunt, Conce t Learnin (New York: John
Wiley and Sons, Inc., 19 2 , p. 4.

5b

is the set of objects to which the name can be applied.

The concept is a statement of structure in the descrip-

tion of the objects to which the name applies.29
As an example of this Hunt uses dog as an object belonging
to a set. Set is equivalent to denotation. The concept is
the rule or statement of structure based on the descriptions
of dog and determines whether or not this object should be
relegated to the set called dog. In this way the name (or
dog in this case) is both a category and an idea.

Hunt was aware of the ambiguities in the literature
tangential to the use of the terms concepts and concept form-
ation, as well as the failure on the part of many writers to
define the term. Mathematical writers are no exception.

Concepts are essentially definitions in symbolic logic.
Therefore, their role in logic should be considered.
In the psychological literature especially, very little
attention has been given to a formal definition of con-
cepts and concept learning. Several authors in mathe-
matical logic, however have considered the role of
concepts and names at length. At a less abstract level,
others have considered the problem of recovering the
definition of a particular concept from Examples of
objects to which a name can be applied.3
For an understanding of inherent meaning in most of the math-
ematical writingathat employ concept and concept learning
and/or teaching as part of the terminology a synthesis of

Hunt and Bruner could serve as a useful guide. Both authors

 

29Ibid., p. 29.

3OIbid., p. 8.

55

appear to agree that a concept is of two orders. The first
order according to Bruner is the simple category. Hunt refers
to this type as a set or denotation. Examples of these are
squares, circles, fourness, or any object or event that can be
classified. The second order, and the more sephisticated one,
involves meaning associated with the object. Bruner defines
this idea of concept in terms of inferences that are set into
play as part of the classifying act. Hunt stresses the mean-
ing behind the object, not as it may appear to the learner but
as it should be. Both authors would agree that concept for-
mation involves a thinking process. Examples of the second
order of concept would include generalizations such as those
pertaining to the cardinal use of number, addition being the
opposite process of subtraction, etc. It is in this second
frame of reference that the term concept is most frequently
used. Although there is a subtle difference between the con-
structs of these two authors, both add to a better insight.
It appears in the mathematical literature that the term con-
cept is used with the above connotations.

The Greater Cleveland Mathematics Program is a 399-

cept-oriented modern mathematics program in which the

{flimazx’iffifié‘iiiiafifiinbee¥a§i2§°€h§§°3n%§%i-§£a%%2%%fi;l

0 3

responses to standard situations. The child is continu-
ously encouraged to investigate how and why things hap-

pen in mathematics. He is led to make eneralizations,

to test these generalizations, and to find new applica—

tions for them.31 [Italics mine]

 

31National Council of Teachers of Mathematics, An

Anal%sis of the New Mathematics Programs (Washington: i§63),
p. 0 ~

56

The investigator is aware that the presentation on
concepts is an oversimplification of a very complex topic.

In moving from a purely psychological orientation to a mathe-
matical one it was necessary to establish a framework that
was tenable in the mathematical sphere.

A research bulletin published by the U. 8. Depart-
ment of Health, Education, and Welfare covering elementary
mathematics raised the following question pertinent to the
present tapic:

What extent can mathematics concepts be deve10ped in
the elementary grades?32

The authors concluded after reviewing recent studies "that
children can learn considerably more mathematics than the
present programs include." None of the studies cited in
support of this conclusion yielded any information that added
insight or support to the present study. Those reviewed in-
volved primary children, talented children or use of SMSG
materials.

One study was found that had marked relevance to
the present tOpic. This was a doctoral dissertation on a
comparison between a conceptual approach to teaching fifth

and sixth grade arithmetic and a more traditional approach.

W"-----”--—o

32Kenneth Brown and Theodore Abell, Analysis of Re-

sea ch in the Teachin of Mathematigs (Washington: U. S.
Dept. of HEW, Government Printing Office, 1965), p. 2.

57

The author, Patricia Spross,33 established that children

evidenced essentially the same progress with the conceptual
approach as with the traditional approach in the area of
fundamentals as measured by the California Achievement Test
Form BB. There was a significant difference in favor of
the group taught by the conceptual approach in achievement
as measured by other instruments. The author concluded:
On the basis of these results, it may be assumed that
it is possible to meet the arithmetic curriculum require-
ment in grades five and six by using a tangible and con-
ceptualized presentation which explains arithmetic
rather than by rote-teaching methods.34

No study was found that employed a conceptual approach
to teaching disadvantaged children.

Attitudes ang_achievement_in mathematipp. In Chap-
ter II attitudes were discussed at some length. Kaplan,
McCandless, Clark and others place a concern for attitudes
in a paramount position. Most current mathematical liter-
ature also reflectsaasimilar concern. Numerous studies have
been done to determine the relationship between achievement

and attitudes. Most of these have yielded a positive corre-

lation. However, the results differ considerably and the

 

33Patricia McNitt Spross, "A Study of the Effect of
a Tangible and Conceptualized Presentation of Arithmetic on
Achievement in the Fifth and Sixth Grades" (unpublished Doc-
toral thesis, Michigan State University, E. Lansing, Michigan,

1962).
3['Ibid” p. 67.

58

question is still an Open one. The 1965 bulletin published
by the Department of Health, Education and Welfare, dealing
with recent research, contains the question: "Does pupil
attitude affect achievement?"35 In response to this ques-
tion the authors cited a number of studies that added to
the existing knowledge. Brief summaries of these,together
with other studies the investigator found, follow.

Faust,36 in a study examining the relationship be-
tween attitudes and achievement in selected school subjects,
found a higher correlation in respect to attitudes and arith-
metic achievement than for other school subjects. Two
separate studies by Shapiro and Solomon, both of whom in-
vestigated the relationship between achievement and attitudes
associated with intermediate grade school children, found
a positive correlation.

Mean scores between groups of sixth-grade children

who liked arithmetic and groups who disliked it
proved significant in all areas.37

 

35Brown and Abell, o . cit., p. 3.

36Claire Edward Faust, "A Study of the Relationship
Between Attitude and Achievement in Selected Elementary
School Subjects" (an Abstract of an un ublished Doctoral
thesis, State University of Iowa, 1962).

37Esther Winkler Shapiro, "Attitudes Toward Arith-
metic Among Public School Children in the Intermediate Grades"
(an Abstract of an unpublished Doctoral thesis, University
of Denver, Colo., 1961).

59

The general ability to learn was found to be asso-

ciated with the children's liking for arithmetic.38
It was further learned from these investigations that fifth
grade boys and girls were influenced more by peer attitudes
than pupils at any other grade levels. Methods which failed
to promote success in arithmetic caused children to dislike
both the subject and the teacher. Parental attitudes re-
lated more closely to children's attitudes then they did to
achievement.

Bassham, Murphy, and Murphy studied the relation-
ship between achievement and attitudes in arithmetic when
reading and mental ability were held constant.

In the sample of 159 pupils (sixth graders) over
four times as many pupils with a poor attitude toward
arithmetic were classified as .65 below expected
achievement as were classified as .65 above expected
achievement. Almost three times as many high-attitude
pupils over-achieved .65 grade as underachieved that
amount.39
The authors caution against predicting achievement on the
basis of attitudes since there appeared wide variation in
achievement at both extremes of the distribution of attitudes.

The complex nature of attitudes is recognized as well as the

role of self-perception in changing attitudes.

 

38

Nellie Ollivene Solomon, "Factors Associated with
Children's Attitudes Toward Arithmetic," Anal sis of Research
in the Teachin of Mathematics based on doctoral study (Wash-
ington: HEW, €965), p. 36.

39Harrell Bassham, Michael Murphy, and Katherine

Murphy ”Attitude and Achievement in Arithmetic," The Arith-
pgtic Teacher, XI (February, 196A), pp. 66-73.

 

6O

Attempts to favorably influence attitudes toward
mastery of materials require changing the pupil's per-
ception of himself in relation to that material. He
must see himself as being able to master the materials,
as being able to use it after mastery, as succeeding
not only in his own eyes, but in the eyes of others.
Unlike interest, attitude does not respond so readily
to mere verbal appeal; the predisposition to expect
failure is often quite resistant to change.40

Larch“1 noted changes in more than one—half of the
attitudes in two groups of children. He felt the key lay
in both the teacher's attitude and in his ability to adjust
instruction to individual differences.

In The Argghmetic Teacher, March 1963, a study of
particular interest and similar to this writer's study,
finds the authors concerned with attitudes, achievement and
methods. The study by Lyda and Morse“2 involved fourth
grade pupils though the size of the group was not specified
nor were they deprived children. The writers quoted Paul A.
Witty and William Ragan to support the theory that attitudes
are important in learning; they noted other sources to demon-
strate the role of the teacher and methods used in order to
build positive attitudes and make learning meaningful. The

evaluating tools used in this study were similar to the one

 

.t

hOIbid., p. 71.

hlflarold H. Lerch, ”Arithmetic Instruction Changes
Pupil's Attitudes," The ArithmeticfiTeacher, VIII (March, 1961),
Pp. 117‘1200

h2Wesley Lyda and Evelyn Morse, "Attitudes, Teaching
Methods, and Arithmetic Achievement," The Arithmetic Teacher,
X (March, 1963), pp. 136-139.

61

under investigation. The children had twenty-one instruction
periods of forty minutes each. Concept of numbers, the
understanding of the numeration system, and place value were
stressed. The IQ mean of the group was 92; the range was
73—12A. The conclusions follow:

1. When meaningful methods of teaching arithmetic
are used, changes in attitude toward arithmetic
take place. Negative attitudes become positive,
and the intensity of positive attitudes becomes
enhanced.

2. Associated with meaningful methods of teaching
arithmetic and changes in attitude are significant
gains in arithmetic achievement, that is in
arithmetical computation and reasoning.43

Abregohh did not find a relationship between attitudes
-and arithmetic achievement with either traditional or modern
mathematics. Steven345 explored the relationship between
the attitudes of high achievers and low achievers and found
a significant difference only between the two top groups of
achievers. Students had been grouped into ability levels
of high, average and low or remedial. There was no signifi-
cant difference found between the average and remedial and
little difference between the high achievers and remedial

groups in attitudes.

 

it31b1d., p. 138.

hhMildred Brown Abrego, "Children's Attitudes Toward
Arithmetic," The Arithmetic Teacher, XIII (March, 1966), pp.
20 -209.

hSLois Stevens, "Comparison of Attitudes and Achieve-
ment Among Junior High School Mathematics Classes," The Arith-
metic Teacher, VII (November, 1960), pp. 351-357.

62

One interesting finding by Stright‘r6 was that as
children progressed upward through the grades positive
attitudes toward arithmetic diminished. At the third grade
‘level 63% of the children liked arithmetic, at fourth grade
59% expressed a liking for it and at the sixth grade level
only 53% said they liked arithmetic. Sister Josephina47
found arithmetic to be the least liked subject among ele-
mentary children.

In the majority of research findings there appeared
a positive correlation between attitude and achievement in
arithmetic. But research in this area is still in its in-
fancy and results are not always clear-cut. Moreover, re-
search has been confined overwhelmingly to the typical
child. Late recognition of the deprived child has precluded
substantial research in this area.

Finally, some important facts which have emerged
from recent literature and are pertinent to this study are:

1. There is a relationship between socio-economic
level and achievement in arithmetic.h

h6Virginia M. Stright, "A Study of Attitudes Toward
Arithmetic of Students and Teachers in the Third, Fourth,
and Sixth Grades," The Arithmetic Teacher, VII (October, 1960),
pp. 280’2870

h7Lyda and Morse, 0p. cit., p. 136.

48Alwin W. Rose and Helen C. Rose, "Intelligence,
Sibling Position, and Sociocultural Background as Factors in
Arithmetic Performance," The Arithmetic Teacher, VIII (Feb-
ruary, 1961), pp. 50-57.

63

2. IQ scores are poor predictors of success with lower-
socio-economic children in arithmetic.h9

3. Number concepts are less developed among disadvantaged
children.50

A. Attitudes toward arithmetic develop as early as
third grade; Dutton maintains that grades four
through eight are crucial years in this respect.51

5. Teaching techniques should vary in both degree and
kind from methods employed with the typical learner.52

6. Disadvantaged children are handicapped in their
ability to handle abstractions.53

 

thbid.

50M. E. Dunkley, ”Some Number Concepts of Disadvantaged
Ogildren,” The Arithmetic Teacher, XII (May, 1962), pp. 359-
3 2.

SlAbrego, op. cit., p. 206.

52Billy J. Paschal, "Teaching the Culturally Disad-
vanta 3d Child," The Arithmetic Teacheg, XIII (May, 1966),
Pp- 3 ’37h-

53Harry Beilin and Lassar Gotkin, 0p. cit., pp. 12-14.

CHAPTER IV
THE METHOD OF THE INVESTIGATION

The general design of this chapter will include:
(1) a description of the students in the study; (2) descrip-
tion of the camp environment relating to teaching facilities;
(3) method of instruction: (A) materials and activities used in
instruction: (5) instruments used in the study; and (6) test-

ing sequence.

A description of the ptudents in the ptudz. Stu-
dents ranged in ages from eight through twelve and were

composed of fourth and fifth graders. Thirty-seven were
Negroes, 32 were white and 13 were of Mexican-Spanish de-
scent. There were 4h fourth graders and 38 fifth graders.
Forty-two had come from broken homes and averaged six sib-
lings. IQ's ranged from 68 to 145 with a mean of 96. Fifteen
had repeated one or more grades and 25 rated arithmetic as
the least liked subject. Sixty-one were below grade level

in reading and 63 were below in arithmetic. All came from
the greater Saginaw area and from a low-socio-economic back-
ground. Each subject had been identified as an underachiever
and as disadvantaged by his regular classroom teacher. The
Appendix contains a more complete listing of characteristics

and a scattergram depicting the dispersion of these traits.

6h

65

A desgription‘pf_the camp_facilities. The children
involved in this study were attending a summer camp at St.
Mary's Lake. Each child had been assigned to a cabin and
a junior counselor for the five weeks duration of camp ex-
perience. The classroom assigned to the writer for arith-
metic was not ideal for such an experience. It was small,
contained no shelves and only one very small and battle-
scarred chalk board. The room was on the water front which
posed a real problem as there were children swimming and
indulging in water sports throughout the days. Visual as
well as auditory distractions were a constant source of
competing stimuli.

Camp facilities included several acres of woodland,
a small lake, a modest main lodge, six dormitory-like cabins,
and a picnic area. These facilities became part of the ex-

tended classroom for instructional purposes.

Method_9f instrugpgpp. As stated earlier each class
period was one hour daily. The concept approach to teaching,
as already discussed, utilized a methodology which stressed
main ideas, generalizations, structure, understanding, and
relationships in a mathematical setting. The children were
guided to an understanding of meaningful mathematical con-
cepts or ideas.

Objectives were: first, to reverse negative attitudes

and encourage more positive feelings toward mathematics; and

66

secondly, to help the children develop a better understand-
ing of mathematics. To accomplish these goals a procedure
was instituted that permitted flexibility, large and small
group activities, utilization of camp environment, frequent
changes in pace, numerous concrete aids and models, and in-
volvement of junior counselors.

A routine was established to give the pupils an es-
tablished framework within which they could operate and yet
enjoy a degree of freedom. Class always began on time and
started out in the regular classroom.

No commercial textbook was used in this experiment.
Each day one general topic was introduced and explored in
search for meaningful concepts (samples of these appear in
the Appendix). After these had been deve10ped in the class-
room by using aids, demonstrations, and discussion, the group
(20 to 26 pupils) would move out of the classroom to engage
in an activity pre-planned to reinforce these concepts. At
the end of each period the class members would help the in-
structor summarize what they had learned. The writer each
evening typed these statemenmsup on a manuscript typewriter,
cut a stencil, mimeographed enough capies for each child,
out the c0pies to book-size and further prepared them for
assembly into the child's personal arithmetic book. Each
day these were passed out and served as a review. Once a

week the children placed these pages into their books. In

67

c00peration with the arts and crafts director the book covers
were designed and constructed by the pupils.

The classroom became a veritable laboratory in which
the children could construct, manipulate, measure, and ex—
plore mathematical ideas and participate in demonstrations.
A minimal amount of time was devoted to paper and pencil
work. There was no stress on drill, computation, or memory
work. Every effort was made to make the class enjoyable for
the children and to give them experiences in success. Praise
was used frequently whenever a child made some gain. There
were no homework assignments except when a child volunteered
to attack a problem presented in class as a challenge and
which always related to some aspect of camp living.

In an attempt to make arithmetic relevant and de-
sirable to the pupils, considerable time was spent on find-
ing out from each child what he thought he would like to
be when he grew up. Then in group discussion the children
were guided into the realization that, whatever choice they
made, they would need to know a great deal about numbers
or arithmetic. Time also was spent on the notion that
mathematics is not truly difficult; that anyone can master
it--it really only involves four simple Operations and
there is more than one way to solve a problem. The majority
of these children had experienced so many failures that

both of these time consuming activities appeared justified

68

to the investigator to serve as motivation and to help the
child gain confidence.

A lesson plan follows in this thesis which is con-
sidered a typical class session. No two days were exactly
alike but the same general format was adhered to. A sum-
mary sheet portraying uniqueness of the program in terms of
techniques and emphases, and contrasted with a more tra—
ditional approach, appears following the lesson plan in
this chapter. Examples of extended classroom activities

appear in the Appendix.

69
TYPICAL LESSON PLAN

Beginning of the hour:
1. Pass out new book page and review previous day's work.
2. Introduce new tapic--Zero

Demonstrate with egg carton and ping-pong balls, abacus,
chalk-board, and place value charts.

Guide pupils to understand the importance of zero in
our number system; uses of zero; what happens to a num-

ber when we add or subtract zero objects. List con-
cepts of zero such as:

a. zero is an important part of our number system

b. zero tells us how many just as the other numerals do

c. the empty place on the place value chart or abacus
represents zero number

d. when zero is added to a number the sum is that number

e. when zero is subtracted from a number the remainder
is that number

f. there are many places we use zero, such as on ther-
mometers, scales, telephone dials, etc.

20 to 30 minutes past the hour:

Assign A or 5 students to a junior counselor and provide
pass-out sheets and pencils for outside activity. This
activity involves finding all the places where zero is
used at camp-—the telephone, clock, license plates, ther-
mometer, etc.

10 or 15 minutes before the end of the hour:

Reassemble the class and learn from them how many instances
of zero they found, the function zero served in each case, etc.

Clear up any questions or misunderstanding, have children
give information regarding zero they want printed for a

book page.

List on chalk-board concepts and other pertinent information
from children's prompting.

Tell children they will be dismissed as soon as they whisper
the password to teacher or junior counselor. This must be
something regarding zero. Example: I caught zero fish—-
meaning I caught no fish, 204 means there are no tens, etc.

Prepare for next group.

70

Summary Of Teaching Techniques Used in Summer Program
for Disadvantaged Children Contrasted with Regular Classes

Experimental

Considerable time spent on
motivation

Children made own textbooks

Frequent use of games and
role playing

Minimal amount of paper and
pencil work

Multiple arithmetic aids used

Classroom approximated lab-
oratory for construction
and exploration

Classroom extended to entire
camp grounds

Frequent small group activ-
ities utilizing jr. coun-
selors

Mathematics related to daily
living and realities Of life

Departmentalized, no grouping

Many Opportunities for success

No homework or grades

Daily review

Frequent change of pace and
activities

Considerable class discussion
and participation

Emphasis on mathematical
concepts

Regular

Minimal if any time spent on
motivation

Commercial texts used

Limited use of games and
role playing

Paper work stressed

Limited use Of aids
Little or no provision for
construction and exploration

Teaching confined to classroom

Aides rarely available

Textbook followed

Self-contained, Often ability
grouping used

Success often limited to few

Homework commonplace, grades
given

Infrequent reviews

Limited activities

Limited discussion
ticipation

Emphasis on drill,
.solving

and par-

problem

71

 

Experimental Regplar
Sessions one hour daily Sessions vary from 20 minutes
to 30 or 40 minutes
Concerned with attitudes Little or no concern for
attitudes

 

One experience that extended over several days and
appeared to be very successful in terms of student interest
may add further insight to techniques and procedures used
in the program. Each child was given a foot-length of wood
molding. When measurement was introduced each child made
his own ruler, calibrating it from a commercial ruler and
dividing it into quarter, half, and full inches. After
the rulers were checked for accuracy each child marked the
calibrations with a fine felt pen. The finished products
were repeatedly used throughout the remaining classes in
direct measurement, as number lines and for work with frac—

tions.

Materials and activipigp. Some of the materials have
already been mentioned and a complete list appears in the
Appendix. Despite the limiting physical facilities of the
classroom, every possible effort was made to approximate a
mathematical laboratory. An adequate portable chalkboard
was obtained; walls were covered with charts; racks construc-
ted of pegboard served to hold a variety of manipulative

aids; number lines, abaci, counting frames, etc. A draw-strir~

72

bag with counting sticks and discs was provided for each
child. The total camp ground was utilized for observations,
collections, and for problem solving. Treasure hunts were
designed for both reinforcing ideas and for evaluating pur-
poses. Role playing was introduced and games frequently
used--domino-bingo, ten pins with oatmeal boxes and a soft
ball, and numerous other games. Children kept scores,
solved problems and generally used numbers in countless

ways as part of the total experience. Every effort was made
to have available representative models or aids to represent
the real world of numbers. In this respect the teaching-
learning situation evolved into very tangible experiences,
with emphasis always upon understanding underlying concepts.
Hence both materials and activities were used in the process
as vehicles for developing the more abstract or conceptual
ends stated as objectives, i.e. greater understanding of and

more positive attitudes toward arithmetic.

Instruments used in the study. The Peabody Picture
Test was used to determine mental maturity and IQ. The

arithmetic portion of the Iowa Test of Basic Skills, Form I

 

was administered to assess achievement level prior to the camp ex—
perience and £939 III was used for evaluating progress. Thqugli—
fornia Arithmetic Achievement Test was used for comparative
purposes in arithmetic and to determine reading level. The

Dutton-Adams Attitude Scale was used to ascertain how the

73

child viewed arithmetic and to measure any changes in his
feelings toward arithmetic.

No attempt will be made to justify any of these
instruments. For the most part the guidance counselors made
the choice as to what test instrument would be used. Actually
so little is known about evaluating disadvantaged children,
and so little research has been done to develop instruments
designed specifically for the disadvantaged, that any instru-
ment used would leave something to be desired. The writer
confesses that availability was the main criterion in some

instances.

Testing sequeppp. The first two days of class time
were devoted to pre-testing. At this time the Iowa Basic
Skills Form I achievement tests were given; also, the Dutton-
Adams Attitude Scale. Within the first week, but not during
arithmetic class periods, the Peabody Picture Test and the
California Achievement test were administered. The last
two days of camp were used for follow-up testing designated
the post-test. During this time the Iowa Basic Skills Test
Form III for arithmetic and the Dutton-Adams Attitude Scale
were given.

After the children had been back in their regular
classrooms (in Saginaw) for six months, approximately seven
months after the camp experience, a random group of 30 was

interviewed and tested. This testing situation has been

74

designated as the post-post-test. At this time (the last
week in February and the first week in March) the Dutton-Adams
Attitude Scale was given for the third time and the Iowa
Arithmetic Achievement Test Form I was given for the second
time. Personal interviews were also conducted during this
interval.

Testing, with the exception of the Peabody Picture
Test and the random testing, was done on a group basis. For
the random or post-post-test the situation demanded testing
be done on an individual basis. This was necessary as the
children represented many schools throughout the Saginaw
area and rarely were two children to be found in any one
school.

Individual files were kept on each child and pertinent
information was accumulated throughout the five weeks. These
contained data regarding number of siblings, status of the

home, least-liked subject, etc.

CHAPTER V
DATA AND RESULTS

Achievement was evaluated by the results of the
standardized tests noted in Chapter IV. Raw scores from the
arithmetic portion of the Iowa Basic Skills Test were con—
verted to grade scores using the manual that accompanied the
test booklets. Percentile ranks were determined by using
the "Beginning of the Year" percentile norms for the pre-
and post-tests and the "End of the Year" percentile norms
for post—post-tests. Mean gains were determined for the
fourth and fifth grades in (1) concepts, (2) problem solv-
ing, and (3) total test scores. The same information was
computed for the random groups. The sign test was applied
to determine significant gains and to reveal which sub-groups
responded best to the concept method. According to Siegell
the sign test is useful when small samples are involved and
the population is heterogeneous in respect to IQ, sex, etc.
To establish that two pairs are significantly different, in
this case.the achievers and nonwachievers, the sign test is

applicable.

 

lSidney Siegel, Nonparametric Statistics for thefiBe-
havioral Sciences (New York: McGraw-Rill Book Company, Inc.,
1956): PP. 63: 69°

75

76

Actual numbers of students who made gains were com-
puted. Percentages of pupils who achieved were determined.

The data and analyses supplied by the Iowa Testing
Service, which were based on estimated grade equivalent
scores for the state of Iowa, were utilized to determine
achievement gains and losses and served as verification of
the investigator's data.

The Spearman Rank Difference formula was used to
determine the relationship between reading and arithmetic
and to correlate the Iowa and California arithmetic tests.
The Spearman Rank Difference Method according to Noll2 is
an apprOpriate technique when small groups are involved
and a comparison is to be made between two sets of scores
such as arithmetic and reading.

By the above procedures the first hypothesis was

Hl Disadvantaged and underachieving children will
respond in a positive manner to the concept
method of instruction in mathematics as evi-
denced by gains on an achievement test.

Table I shows achievement in mean grades for the
total groups of fourth and fifth graders from Test A (pre-
test) to Test B (post-teat). Table II shows mean percentile
rank gains for the same groups and same tests. Table III
shows the mean grade gains for the random groups in all

three tests--A, B, and C (pre, post, and post—post).

 

2Victor H. Noll, Introduction to Educational Measurg-
a: Houghton Mifflin Company, 1957). PP0 453. “5

77

Table I.--Achievement in mean grades--total group*

 

 

 

 

Test A-l A-Z A-T

Fourth Opade

Pre-test, Iowa Form I 3.41 3.74 3.58

Post—test, Form III 3.48 3.28 3.38

Gains from A to B .07 -.46 -.20
Fifth Grade

Pre-test, Iowa Form I 4.19 3.82 4.01

Post-test, Form III 4.23 4.22 4.22

Gains from A to B .04 .40 .21

 

Table II.--Mean achievement in percentile ranks

 

 

Test Rank Gain
Fourth Grade
Pre-test, Iowa Form I 33.2
Post-test, Form III 27.7 -5.5
Fifth Grade
Pre-test, Iowa Form I 22.6
Post-test, Form III 27.8 5.2

 

A-l - Arithmetic concepts
A-2 = Problem solving
A-T = Total tests

*Na 82 (44 fourth graders) and (38 fifth graders)

78

Table III.--Achievement in mean grades for random group*

 

 

T

 

 

 

 

 

Test A-l A-2 A-T

Fourth Grade_

Pre—test, Iowa Form I 3.55 3.65 3.60

Post-test, Form III 3.32 3.32 3.32

Gains from A to B -.23 -.33 -.28

Post-post-test, Form I 4.17 4.08 4.12

Gains B to C .85 .76 .80

Gains A to C .62 .43 .52
Fifth Grade

Pre—test, Iowa Form I 4.29 3.92 4.11
'Post-test, Form III 4.42 4.15 4.27

Gains A to B .13 .23 ..16

Post-post-test, Form I 4.79 4.59 4.69

Gains B to C .37 .44 .42

Gains A to C .50 .67 .58

A-l = Arithmetic concepts

A—2 = Problem solving

A-T = Total tests

*N = 30 (16 fourth graders) and (14 fifth graders)

79

According to the data supplied by A. N. Hieronymus,3
representing the Iowa Testing Service, the mean gains were
as follows: fourth graders achieved a .68 gain in concepts
and a -.84 in total score. Fifth graders achieved a .47
gain in concepts and a +2.16 in total score. Although these
results differ markedly from those of the investigator a
similar pattern emerges. Meanwhile it must be remembered
that these results were based on estimated grade equivalents
for the state of Iowa.

The information below presents actual numbers and
percentages of pupils who made gains in achievement.

Fourth grade--ll out of 44 students, or 25%, gained
from the pre-test to the post-test; 14 out of 16, or 88%,
gained from the post-test to the post-post-test (given to
the random sample), and 11 out of 16, or 69%, gained from
the pre-test to the post-post-test.

Fifth grade-~23 out of 38, or 61%, made gains from
the pre-test to the post-test; 8 out of 14, or 57%, gained
from the post-test to the post-post-test (given to the
random sample); 11 out of 14, or 79%, gained in achievement

from the pre-test to the post-post-test.

 

3A. N. Hieronymus, in a letter to the investigator
found in the Appendix.

80

From the results of the Sign Test, fourth graders
achieved significantly (at a .05 significant level) from
test B to C and from A to C. Fifth graders achieved sig-
nificantly from A to C. There was a marked difference be-
tween the sexes in achievement among the fourth graders,
the boys showing a higher incidence of regression. When
IQ's were considered the low-medium group and the high group
achieved with greater frequency. The complete information
on the Sign Test appears in the Appendix.

As the total results approximate an occurrence of
chance in many respects a correlation was computed between
the Iowa and California arithmetic tests to determine how
reliable the test results on the main evaluating tool were.
The Spearman Rank Correlation method was used for this pur—
pose. There was a correlation of .58 between the two tests
for fifth graders and a correlation of .71 for fourth graders.
These results would signify that the pupils were not guess-
ing and that the data were reasonably realiable.

It was mentioned earlier that there were almost as
many underachievers in reading as there were in arithmetic.
Therefore, the Spearman Rank Correlation method was applied
between the reading scores and the Iowa arithmetic scores
in order to determine the relationship between these two
variables. Fifth graders evidenced a correlation of .49 and
fourth graders showed a correlation of .55 between reading

.}*“fiCo

81

From the data it appears that in terms of immediate
goals the hypothesis for achievement did not prove valid
for fourth graders. However, in long term goals the concept
method may have had a marked positive effect. The hypothesis
proved valid for the fifth graders as evidenced by gains in
mean grades and percentile ranks. This proved true for
both immediate and long term results.
Evaluations regarding attitudes toward arithmetic
were made by comparisons between scores on the Dutton-
Adams Attitude Scale, considering both the self-rating re-
sults and the actual score results. These findings were
submitted to the Sign Test to determine what could be con-
sidered significant changes by eliminating chance occurrence.
The hypothesis dealing with attitudes states:
H2 Disadvantaged and underachieving children will
tend to show a positive change in attitude as
a result of the influence of careful attention
to method and materials as evidenced by a posi-
tive change on an attitude scale.
It was found that the subjects in the experiment tended
to mark the extremes on the attitude scale. It was there-
fore decided that the actual score would be more meaningful
in interpreting how the child felt toward arithmetic. There
was a mean difference of 2.7 between the actual scores and
the self-rating scores which is considerable on an eleven

point scale. Using the actual scores (averages of marked

items) results obtained are reflected in Table IV.

82

Table IV.--Attitudinal changes

 

 

 

 

Improve- Regres- No
Test ment sion Change Improvement

Fourth Grade

A to B, Total 25 15 4 57

B to C, Random 7 7 2 45

A to C, Random 13 2 1 81
Fifth Grade

A to B, Total 26 10 2 69

B to C, Random 8 4 2 57

A to C, Random 12 l 1 86

Fogpth and FifthrGrades
A to B 51 25 6 62

 

The Appendix contains scattergrams for both fourth
and fifth grades depicting dispersions of the attitude changes.
The Sign Test dealing with attitudes may be found there also.
A summary of the findings follows.

The Sign Test reflected a significant number of
changes in attitudes toward arithmetic within both fourth
and fifth grades to warrant the conclusion that the hypothesis
was true. In considering the sexes separately it was learned
that fifth grade girls responded better. In considering IQ's
the high-mediums (109-118) evidenced the greatest changes.

83

The low-mediums (89-98) were second in significance. It

was also observed that, after having returned to the regular
classrooms for six months, one-half of those who changed

in attitude among the fourth grade random group regressed

to a less positive feeling toward arithmetic. One-third

of the changers among the fifth grade random group also
regressed. However, a substantial number within both grade
levels continued to improve or held constant.

In summary, it appeared that attitudes of disadvantaged
and underachieving children can be changed in a positive
manner at the fourth and fifth grade levels. This would
seem clear on the basis of: (1) actual numbers--51 out of
82 showed improvement on the Dutton-Adams Attitude Scale,

(2) percentage--62% made positive changes, and (3) the
results of the Sign Test.

To determine the relationship between attitude and
achievement in arithmetic the Phi Correlation technique
utilizing a four-fold table was used. According to Garrett,4
when statistical data fall into dichotomies such as true
or false or in this case a postive or a negative relation

ship the phi correlation is an appropriate measure of

 

“Henry E. Garrett (New York: David McKay Company,
Inc., 1964), pp. 388-389.

84

correlation. The Pearson Product-Moment Correlation method
was applied to the random groups using the formula by Noll.5
This method reveals the extent of relationship between two
variables by determining the deviation from the mean of each
individual in respect to both variables. The third and last
hypothesis was tested by these procedures.

H3 Disadvantaged and underachieving children will
tend to show a positive relationship between
attitude toward mathematics and achievement in
mathematics.

It was not possible to arrive at a definitive con-
clusion regarding the relationship between attitude and
achievement on the basis of these data. Using the results
of the Pearson Product Moment method for the fourth graders
it appeared that there was no correlation between these two
variables. However, the Phi correlation results would indi-
cate a moderately low relationship. The fifth grade results
on both procedures would indicate no correlation except
for Test B with the random group which showed a substantial
correlation between attitude and achievement. The results
are summarized on the following page.

Some of the discrepancy is accounted for by the
fact that use of the phi correlation gives no consideration
to those individuals who make no change on either one or

the other of the two variables. Eighteen showed a positive

relationship, 15 showed a negative relationship, and 11

 

5Noll, o . cit., p. 410.

, 85
Table V.--Re1ationship between attitude and achievement

 

w —— ~— w

P. P. Moment'k Correlation Test A Test B Test C

 

 

Random Groups

 

Fourth Grade .05 -.O3 -.02
Fifth Grade " o 20 o [*3 o 07
Phi Correlation from Test A to Test B

 

Total Groups
Fourth Grade .22
Fifth Grade .08

 

*Pearson Product-Moment Method
failed to change from Test A to B in one of the two areas

at fourth grade level. At the fifth grade level there were
19 reflecting positive relationship, 13 negatives and 6 no
changes. _

Out of 90 possible changes, combining both random
groups on all three test situations, 40,or 45% showed a
positive correlation. Thirty, 33%, showed a negative re-
lationship and 20, or 22%, showed no change in one or the
other factor.

From all the available information it appeared that
a significant relationship between attitude toward arithmetic
and achievement in arithmetic did not exist with fourth and
fifth grade disadvantaged children.

86
The complete data dealing with attitudes and achieve-
ment appear within the following tables.
Table VI.--Re1ationship between attitude and achievement

Fourth grade random sample Test A

 

 

Pearson Product-Moment Correlation

r- 2 g:
V XZy2

 

X

r g 1002
V 1.9.51 x 8.39

 

V 415.3889

Ar
5.

1 02 . 1002

Y
X Achieve-

'Attitude ment x y x2 y2 xy
#06 #02 ’106 .6 2.56 036 -096
70h 208 102 ‘08 10h O6“ “096
706 305 10“ '01 109 001 '01“ .
“02 306 -200 0 “000 " -‘
7.6 4.8 1.4 1.2 1.96 1.44 1.68
600 206 ' 02 «.100 .0 1.00 020
7.6 3.8 1.2 .2 1.9 .04 .28
306 30k ’20 '02 6.76 00“ 052
8.0 4.7 1.8 1.1 3.24 1.21 1.98
707 208 105 '08 2025 06 -1020
706 “.2 lo“ 06 1.96 03 .08h
303 2.6 -209 -100 8. 1 1000 2.90
#03 be“ -109 08 30 1 o6 -1052
4.0 4.0 ‘ -202 ch #08 . 01 ' 088
706 30a 10‘ ‘02 1.9 00‘ - 028
7.8 207 106 ‘09 2:56 o§1 “lghk

49.51 8.39 1.02

Mean

6.2 3.6

87
Table VII.--Relationship between attitude and achievement

Fourth grade random sample Test B

-—~.--— _

   

I Achieve-

 

Attitude ment x y x2 y2 xy
607 305 '05 01 .25 .01 -.05
800 203 O8 "09 064 .81 -072
700 20 '02 ’ 05 .0 025 010
3.6 208 -306 ‘ 05 1209 025 1.80
7.6 4.0 .4 . 16 .49 28
505 308 -107 01 2089 001 ‘017
7.3 3. .1 .5 .01 .25 .05
403 304 "209 01 Sohl 001 -029
8.0 3.“ 08 01 o6 .01 .08
8.8 3.6 1.6 03 205 009 .48
8.8 4.2 1.6 .9 2.56 .81 1.44
806 203 10“ “100 1096 1000 -1.40
60L “.6 ' 08 103 .64 1069 -l.04
808 2.9 106 ‘ oh 2.56 ' 016 ' O6“
707 301 05 ' .2 025 00“ ' 010
802 300 100 ' 03 1:00 :22 - 020

37.53 5-97 - .48

Mean

7.2 3.3

Pearson Product-Mbment Correlation

1‘

 

 

 

 

NZ 1:2 X Z Y2
-.48
’Vamsa x 5.97
. -.48
/V 224.0541

88
Table VIII.--Relationship between attitude and achievement

Fourth grade random sample Test C

================T=E===========E=I=g===========================
X Achieve-

 

Attitude ment x y x2 y2 xy
he“ a.“ -300 03 900 009 ‘090
506 405 “108 oh 302‘ 016 -072
709 303 05 '08 .25 06‘ -.20
800 300 o6 -101 036 1021 '0 6
7.6 306 .2 ’05 o 025 -010
800 30k .6 ’07 03 o 9 -.42
8.8 4.9 1.4 .8 1.96 . 4 1.12
606 “so '08 '01 06h 001 .08
802 he? O8 O6 .6 036 .48
808 #06 102 05 109 .25 070
8.0 4.8 . .7 .36 .49 .42
7.9 #02 05 01 025 001 605
609 he? - 05 06 025 .36 -030
6.9 4.1 - .5 -- .25 -- --
7.2 3.2 ‘ 02 '09 00“ 081 018
800 Les .6 oh 036 016 ' egg

19.96 5.93 -.23

Mean

70“ “.1

Pearson Product-Moment Correlation

r ‘02} . -02

 

r . -002

89
Table IX.--Relationship between attitude and achievement

Fourth and fifth grade total from Test A to B
Attitude and Achievement '
Phi Correlation*

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ +
y P
-4» ++ —+ , 4+
13 8 7 15
B A B A
.. x+ - x+
.. +- .. +-
10 2 1, 6
1) c D 0
Fourth Grade Fifth Grade
¢ ' Jfll;ilfli W
4/ (A + B) (C + D)(?+ D) (A + C)"
. d - 80-26 or 55, H y - 0-523: 1. .8__
M21) (12) (23) (10) M22) (10) (11) (21)
or 57,960 or 50,820
2% 18
0 255055
e - .22 (f - .08

*Adapted from Garrett, p. 389.

90
Table X.--Re1ationship between attitude and achievement

Fifth grade random sample Test A

 

 

 

A Achieve-
ttitude ment
X I x y x2 y2 xy
500 506 '106 105 2.56 2025 -ZOAO
5.7 has ’ 09 0‘ .81 016 ’ .36
502 he“ “10% 03 1096 009 ' .42
8.2 503 10 1.2 2056 lath 1092
7.6 5.4 1.0 1.3 1.00 1. 9 1.30
402 306 ’20 ’ 05 5076 025 1020
6.0 3.6 - . - .5 .3 .25 .30
707 30k 101 ' O7 1021 0‘9 . ' 077
702 305 o ’ o6 .36 036 ‘ e36
800 202 10k “109 1.96 3.61 -2066
503 308 -103 ' 03 1069 . 009 039
7.7 3.0 1.1 -l.l - 1.21 1.21 -l.21
800 306 10h ‘ 05 1096 025 - 070
606 5.8 -' 1.7 " 2:82 "
23.140 . 15003 -3077

Mean

6.6 4.1

Pearson Product-Moment Correlation

r - .23 £1

 

 

 

 

 

’Vfifxz XIZCYZ
r ' -3077
1V 23.40 X 15.03
1" ' “3377 - "' 0
”V5 351.7020 ‘ 5

r . “020

91
Table XI.--Relationship between attitude and achievement

Fifth grade random sample Test B

EOE-{.6 V8-

Attitude ment

 

X I x y x2 y2 xy
7.7 5.2 .3 .9 .09 .81 .27
6.9 “06 '05 03 025 009 -015
7.7 5.2 .3 .9 .09 .81 .27
8.2 5.6 .8 1.3 .64 1.69 1.04
7.9 5.4 .5 1.1 .25 . 1.21 .55
#05 3.6 -209 '07 8ohl o 9 2.03
702 3.0 ’ .2 ”103 00 10 9 02
8.0 4.0 .6 ' 03 03 o '018
609 3.0 - 05 -103 025 1069 065
607 306 ’ 07 ’ 07 0&9 0&9 0&9
505 302 ”109 “101 3061 1021 2009
808 208 101} ‘105 1096 2025 '2010
808 #08 10‘ .5 1096 025 070
8.4 5.8 1.0 1.5 1.00 2.25 1,50

19.40 15.02 7.42
Mean
7.4 4.3

Pearson Product-Moment Correlation

r . _Z.42
AV 19.40 x 15.02

r = 62.42 ‘ = ;,%2

4/291.3880

 

 

 

 

r . oh}

92

Table III.--Relationship between attitude and achievement

Fifth grade random sample Test C

 

 

X Y x y Y2 xy
9.0 5.8 1.2 1.2 1.44 1.44 1.44
70‘ #02 ‘ eh - oh 016 016 01
508 206 “2.0 -’ 4.00 -- --
706 .1 ' 02 105 00h 2025 ' .30
7.9 5.8 .1 1.2 .01 1.44 .12
50‘ #01 '20h “.05 5076 025 1.20
8.0 300 02 “106 00h 2.56 ' 032
900 4.0 102 - o 104“ 036 ' 072
808 #03 1.0 ‘ 03 1000 009 - 030
800 306 02 -1.0 00“ 1000 ' 020
800 303 02 “103 00‘ 1069 ' 026
8.0 #06 02 " 00“ i-“ "
808 309 100 ‘ 07 1000 0‘9 - .70
8.2 7.3 .4 2.7 .16 2,22 1,08

15.17 19.02 1.20
Mean
7.8 4.6

Pearson Product-Moment Correlation

r - IETxY

’162Ex2 11552

 

r ' 1020
4/15.17 x 19.02

 

 

 

r . 1’201_h_1 . 020
Al 283.5334" " I5°93

r . 007

93

Table XIII.--Individual Analysis of
Relationship between achievement and attitude
(Based on Random Sample)

W

 

 

 

Fourth Grade Fifth Grade
Iden. ‘A'to 57B to A to Iden. A to B’to A to
No. B C C No. B C C
l - - - 6 - + +
3 - - - 9 + - -
5 + + - 15 + + +
8 + + - 16 0 - -
13 0 0 0 24 0 0 +
18 - 0 + 25 0 + +
19 0 + + 33 - o ..
22 0 + + 36 . o .
37 0 0 39 + +
42 + O + 45 - 0 0
57 0 - + 52 - * -
60 - - + 58 - - +
61 + 4- + 75 4’ O 9’
64 - - + 77 0 - +
68 - - +
62 + - +
N 16 N 14
90 Total possible changes
40 showed positive correlation 45% (+) correlation
30 showed us ative correlation ' 33% -) correlation
20 showed no change in one or the other

factor 22% (O) correlation

CHAPTER VI
SUMMARY, CONCLUSIONS AND IMPLICATIONS

Summary

The purposes of this dissertation were to explore
the literature to learn what is known about the disadvantaged
child; to survey what has been accomplished in the area of
mathematics with deprived children; and to present a syn-
thesis of these. A prime intention of the writer was to
report the experiences and results of one program designed
specifically to promote better attitudes and achievement in

mathematics on the part of disadvantaged youth.

Conclusions

Analysis of the data indicated that modest gains
in achievement were evidenced by the fifth graders by em-
ploying the concept method of instruction and other inno-
vations in materials and activities. The results revealed
both immediate and long term gains at the fifth grade level.
Fourth graders appeared to benefit less from the camp ex-
perience in terms of achievement. However, considerable
benefits may have accrued in better performance at a later

date. .

94

95

In terms of attitudinal changes the experiment ap-
peared to be effective with both fourth and fifth graders.
It was further learned that after an interval of seven
months, which represented six months for the child to be
in his regular classroom, fifty per cent continued to im-
prove in attitude toward arithmetic. Thirty-seven per cent
regressed and thirteen per cent held constant at the post-
test level.

The relationship between attitude and achievement
in arithmetic was not consistently significant. There was
some evidence of a correlation but the over-all results
proved erratic and did not lend genuine support to the

hypothesis.

Implications

The instruments used for evaluative purposes in
the experimental program left something to be desired. Even
though the scale used to measure attitudes had been modified
the children had difficulty interpreting it. Also, the
apparent desire to please in many instances was evident by
the child's marking the highest rating possible. His actual
score computed by averaging separate items was at variance
with his self-rating score. Even within the separate items
there were many inconsistences. For example, many children
marked item number 4 which states, "I have never liked arith-

metic"; but also marked item number 13 which states, ”I have

96

always liked arithmetic because it has presented me with a
challenge.” By careful examination of each individual score
sheet it appeared this was not a tool that reflected reliable
results.

Regarding the standardized tests measuring achieve-
ment it appeared that Form 111 was more difficult for this
group than Form I. However, according to the Iowa Testing
Service these are purported to be comparable forms. In
searching for possible reasons why the results were so dif-
ferent in comparing fourth grade achievement with fifth
grade achievement one explanation appeared to be in the area
of reading. As was noted, there was a greater correlation
between reading andarithmetic with the fourth grade group
than with the fifth grade group. It was learned that of
the 61 children who were below grade level in reading 36
of these were fourth graders. Of the 36, 25 were more than
one-half year below grade level in reading. Thus it seemed
apparent that reading played an important role in the cause
of lower arithmetic achievement.

In searching for other possible clues the writer did
considerable reading about Piaget's1 experiments on concept

formation and develOpmental stages related to attainment of

 

1See the Arithmetic Teacher Journals, November, 1963,
January and November, 1964, and May 1966; also the SMSG
Conference Report.

97

certain concepts. Typical children, according to Piaget,
cannot handle certain.concepts of measurement and area, nor
master ideas of invariance and conservation of number until
the second and third stages of deveIOpment. Deprived chil-
dren often lag behind in learning certain concepts and gen-
erally have more difficulty dealing with abstractions. If
substantial evidence reveals late arrival at certain devel-
opmental stages, then it's quite possible that the eight
and nine year olds in this study were not ready for many

of the experiences that were part of this arithmetic program.
Implications are clear that both reading retardation and
developmental lag deserve careful study in any arithmetic
program for deprived children.

In a study by Jarvis2 it was demonstrated that
fourth, fifth, and sixth grade children who were given
sixty minute periods of instruction in arithmetic excelled
over those in the control group who had 35 to 45 minutes
of daily instruction. Jarvis maintained that time allot-
ments for arithmetic have received scant attention in the
research. Thus what is considered an Optimum amount of
time for elementary arithmetic is a moot point.) At the

summer camp experience, the investigator chose the maximum

L
I

2Oscar T. Jarvis, "Time Allotment Relationships to
Pupil Achievement in Arithmetic,” The Arithmetic Teacher, X
”43st 1963), PP. 2148-500

98

available amount of time with the desire to help children

as much as was possible within five weeks. A full hour
daily would have been too long had not the sessions been
planned to include a varied number of activities. The
attention spans of these children were exceedingly short

and a frequent change of pace was necessary to hold interest
to any degree.

One most encouraging aspect of the total experiment
proved to be the terminal interviews. The investigator
posed four questions to the random group:

1. Are you doing as well in arithmetic this year
as you did last?

2. Are you doing better this year than you did last?

3. Are you doing worse this year than last year?

4. How do you feel you are doing in arithmetic this
year?

Of the 30 children 26 indicated that they were
doing better this year--at least they thought they were.
Only one child said he was doing worse than he did a year
~ ago. Three professed they did not know how they were doing.
It appeared from the follow-up testing and from the inter-
views a gratifying percentage of the children had either
internalized or were in the process of internalizing a

higher confidence level.

 

 

II1III()\|'II "

99

In the final analysis it appears that no one knows
what is an acceptable standard relative to achievement with
deprived children. In this study 34 benefited immediately.

A larger preportion appeared to benefit over a period of time.
Fifty-one improved in attitudes by the end of the camp ex-
perience. Inasmuch as the Sign Test is a rather conservative
tool as it demands a large proportion of either positives or
negatives before significant gains are reflected, some other
technique of analyzing data might well reflect better results.

From both the objective and subjective evaluation it
appeared to the investigator that a modest degree of success
could be claimed. This premise can only be verified by ad-
ditional research replicating this type of program. Additional
research is also needed to ascertain if these results would
be altered appreciably if the children were in their home set-
ting rather than in a camp situation. The high incidence of
regression in children's attitudes after they returned to
regular classrooms raises questions regarding methodology and
teacher influence. This should be an area of concern for all
in the teaching profession.

In this type of program it is practically impossible
to equate numbers with success. What eventually happens to
these children in terms of academic accomplishments and of
becoming responsible members of society will determine the
real value of these efforts. Thus the final burden of proof

rests with the children in this program.

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APPENDICES

108

APPENDIX A

Characteristics of Total Group

Proportion of Students Who Changed in Achievement
Preportion of Students Who Changed 1L Attitudes
Relationship between Achievement and Attitude

Fourth Grade Reading and Arithmetic Grade Scores and

Class Ranks
Fifth Grade Reading and Arithmetic Grade Scores and

Class Ranks

109

110

Characteristics of Total Group

(82 pupils)

I.Q. range . . . . . . . . . . . . . . . . . 68 to 145
Mean I.Q. . . . . . . . . . . . . . . . . . . . 96
Mean I.Q. of random sample (30 pupils) . . . . . 99
Range of random sample . . . . . . . . . . . 71 to 145
Age range . . . . . . . . . . . . . . . . . 8 to 12
Number of Negroes in total group . . . . . . . . 37
Number of Spanish-Mexican in total group . . . . 13
Number of white in total group . . . . . . . . . 32
Number of fourth graders . . . . . . . . . . . . 44
Number of fifth graders . . . . . . . . . . . . 38
Number of males . . . . . . . . c . . . . . . . 49
Number of females . . . . . . . . . . . . . . . 33
Number falling below 25th percentile rank in

arithmetic . . . . . . . . . . . . . . . . . . . 42
Number below grade level in arithmetic achievemegg 56
Number listing arithmetic as least liked subject . 25
Number that are members of a broken home . . . . . 42
Number that have repeated one or more grades . . . 15
Average number of siblings . . . . . . . . . . . . 6

Range of number of children in family

Reading range, fourth grade

0 O O O O O O O

oesoeltOllf
1.0 to 4.7

111

000 301+
Reading range, fifth grade . . . . . . . . . . 2.0 to 6.9

Mean reading level, fourth grade . . . . . .

Mean reading level, fifth grade 0 e e a e e a e e e [+07

Total number reading below grade level . . . . . . . 61

Permitting 4 months as standard error.

Proportion of Students Who Changed in Achievement
(Sign Test)*

 

 

112

 

 

 

 

 

 

 

 

Direction
of Change No. of Sig. Larger
Variables + - Changers Level Propor.

Fourth Grade-—Random

A to B 4 3 9 13 .133 -

B to C 14 1 l 15 .001* +

A to C 11 l 4 15 .059* +
Fifth Grade--Random

A to B 6 3 5 11 .500 +

B to C 8 3 3 11 .113 +

A to C 11 O 3 14 .029* +
Fourth and_Fifth

A to B 10 6 14 24 .271 -

B to C 22 4 4 26 .001* +

A to C 22 1 7 29 .001* +
Boys--Total A_t9 B

Fourth Grade 6 6 17 23 .017 -

Fifth Grade 11 4 5 16 .105 +
Q;r13--Total A to B

Fourth Grade 5 1 9 14 .212 . -

Fifth Grade 12 0 6 18 .119 +
1232

Very low (68-78) 3 0 4 7 .500 -

Low (79-88) 4 3 9 13 .133 -

Low-medium (89-98) 14 2 7 21 .095 +

High-medium (99-108) 11 3 7 18 .240 +

High (109-118) 1 l 8 9 .002 -

Very high (119-->) 1 2 2 3 —-— -

 

*Significant at the .05 level

Figures taken from table by Siegle, p. 250.

113

Proportion of Students Who Changed in Attitudes

(Sign Test)*

 

 

 

 

 

 

 

 

 

_Direction
of Change No. of Sig. Larger
Variables + O - Changers Level Propor.

Fourth Grade-~Random

A to B 10 2 4 14 .090 +

B to C 7 2 7 14 .605 -——

A to C 12 l 3 15 .018* +
Fifth Grade-~Random

A to B 11 l 2 13 .011* +

A to C 12 l l 13 .002* +
Fourth and Fifth

A to B 21 3 6 27 .001* +

B to C 15 4 ll 27 .300 +

A to C 24 2 4 28 .001* +
Boys : A to B

Fourth, Total 15 4 10 25 .212 +

Fifth, Total 12 1 7 19 .180 +
girls -A to B

Fourth, Total 10 0 5 15 .151 a

Fifth, Total 14 l 3 17 .006* +
IQL§_;_12221

Very low (68-78) 5 0 2 7 .227 +

Low (79-88) 8 0 8 16 .598 -—-

Low-medium (89-98) 15 1 7 22 .067 +

High-medium (99-108)12 3 6 18 .119 +

High (109-118) 8 l 1 9 .020* +

Very High (119-->) 3 1 1 4 -—— +

 

*Significant at the .05 level
Figures from table by Siegle, p. 250.

114

Relationship between Achievement and Attitude

Achievement

Fourth Grade

Fifth Grade

wwon

11
0

3

Total::Fourth and Fifth

10
6

14

22

(Based on Random Sample)

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

N = 16

N = 14

Change A to B

Changed B to C

Change

Change

Change

Change

Grades
Change

Change

A to

A to

B to

A to

A to

B to

30

C)

\JN'Q

#1600

15

Attitude

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained
no change
regressed

gained

4 no change

11

regressed

 

115

Achievement Attitude
Change A to C
22 gained 24 gained

1 no change 2 no change
7 regressed 4 regressed

116

Fourth Grade Reading and Arithmetic

Grade Scores and Class Ranks
N = 43

Arith. Arith. Calif.

 

 

l 3.8 25 4.2 9.5 3.1 33
2 3.6 30 3.0 33 3.6 20
3 3.2 40 2.8 37 3.9 10
4 4.0 19.5 3.4 26 3.3 27
5 4.1 15 3.5 21.5 2.4 42
7 4.4 6 4.6 5-5 3.9 10
8 2.7 42 3.6 18 3.0 36.5
10 3.4 37.5 3.6 18 3.7 15
11 3.6 30 4.0 12 3.4 24
13 4.4 6 4.8 2.5 4.4 2
14 3.8 25 3.0 33 3.7 15
17 3-7 27.5 3-0 33 3.5 23
18 3.2 40 2.6 40.5 3.2 30.5
19 4.5 3.5 3.8 14.5 3.7 15
20 3.6 30 3.2 29 2.8 40.5
22 4.3 9 3.4 26 3.7 15
23 3-5 34 3-0 33 3-0 36-5
27 4.0 19.5 3.7 16 3.2 30.5
29 4.2 11.5 3.8 14.5 3.6 20
34 3-5 34 3.0 33 2.8 40.5
35 4.7 1-5 5-7 1 h-7 1
37 4.1 15 4.7 A 3.7 15
38 4.7 1.5 4.6 5.5 4.1 6
41 3.9 22.5 3.6 18 3.6 20
42 3.5 34 2.8 37 3-3 27
49 3.2 40 2.8 37 3.3 27
55 1 4.0 19.5 3.4 26 3.0 36.5
56 4.3 9 4.5 7 4.3 3
57 4.2 11.5 4.2 9.5 3.6 20
59 4.3 9 4.0 12 4.0 7.5
60 2.5 43 2.6 40.5 2.2 43
61 4.4 6 4.4 8 4.0 7.5
62 4.5 3.5 4.8 2.5 4.2 4.5
64 3.5 34 4.0 12 3.1 33
66 4.1 15 3.5 21.5 3.3 27
68 4.1 15 3.4 26 3.6 20
69 4.0 19.5 2.7 39 3.0 36.5
70 3.8 25 2.4 42.5 2.9 39
72 3.4 37.5 2.4 42.5 3.8 12
73 (4.1 15 3.5 21.5 4.2 4.5
74 3.9 22.5 3.5 21.5 3.9 10
30 3.5 3A 3.1 30 3.3 27
£7 -._1~7 27.5 3.4 26 3.1g. 33_
3 w/Ca1. r = .71 Iowa correlated w/readf.
7ference Method rho = 1- 645'D2

‘72: 'Nz' ‘Ii "

117

 

 

 

Fifth Grade Reading and Arithmetic
Grade Scores and Class Ranks

No. Calif. Rank Iowa Rank Reading» Rank

6 5.9 7 5.6 2 4.6 13.5

9 4.3 33 4-5 9-5 3-8 28.5
12 4.9 21.5 4.5 9.5 6.5 2.5
15 5.1 18.5 4.4 11 5.2 7
16 6.1 2.5 5.3 4 6.9 l
21 4.2 34 ‘ 3-3 33 3.3 31
24 5-9 7 5.4 3 4.1 23
25 5.2 16 3.6 26 4.2 21.5
26 4.6 25 3.0 35.5 4.4 17
28 5.9 7 5.2 5.5 6.5 2.5
30 6.1 2.5 4.6 8 4.5 15
31 4.5 29.5 2.5 37 3.1 34
32 4.5 29.5 3.1 34 4-4 17
33 5.0 20 3.6 26 3.1 34
36 5.1 18.5 3.4 32 5.0 10
39 4.5 29.5 3.5 30 4.3 19-5
40 5.9 7 5.2 5.5 5.5 6
43 5.3 12.5 4.2 14 4-7 12
44 5.3 12.5 4.2 14 5.1 8
45 2.6 38 2.2 38 2.0 38
46 5.9 7 4.8 7‘, 6.2 4
47 4.9 21-5 3.9 19.5 4.0 .25
48 5.3 12.5 3.5 30 4.3 19.5
51 4.7 23 4.2 14 3.1 34
52 3.8 37 3.8 22 3.8 28.5
53 4.5 29.5 3.8 22 2.9 36
54 4.0 35.5 3.8 22 3.5 30
58 6.0 4 3.0 35.5 2.4 . 37
63 4.0 35.5 3.5 30 5-0 10
65 4.5 29.5 4-0 17-5 4.4 17
67 5.6 10 4.0 17.5 4.0 25
71 5.2 16 3.6 26 5.0 10
75 4.6 25 3.6 26 4.6 13.5
76 4-5 29.5 4.2 14 3.9 27
77 6.3 1 5.8 l 5.9 5
78 4.6 25 3.6 26 4.2 21.5
79 5.2 16 4.2 14 3.2 32
81 5;3 12.5 3.9 19.5 84.0 .25

 

Iowa correlated with Calif. r = .58
Iowa correlated with Reading r = .49

Spearman Rank Difference Method

APPENDIX B

Scattergram Showing Dispersion of Characteristics
Fourth rade Achievement Scattergram

Fifth Grade Achievement Scattergram

Fourth Grade Scattergram, Total and Random Attitudes
Fifth Grade Scattergram, Total and Random Attitudes

118

119

 

 

 

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A

APPENDIX 0

Letter, May 13, 1966, from A. N. Hieronymus, Professor
Education and Psychology, University of Iowa,
Iowa City, Iowa, stating his Opinion and analysis
of data. - ,

Analysis forms, "Report of Grade-Equivalent Scores on the
Iowa Test of Basic Skills," giving his statistical
analysis of my data.

." J

124

' 125

THE UNIVERSITY OF IOWA

IOWA CITY, IOWA 82240

 

 

Coll: Education
”of May 13, 1966

Miss E. Leona Hall
131 First Street
Breckenridge, Michigan 48615

Dear Muss Hall:

I again must apologize for my tardy reply. <However, this time it
was because I have worked almost continuously on your data when I could
find time.

I am afraid I do not know much more than I did before, but possibly
some of the additional analysis might be of help in one way or another.

I know it is not of much consolation to you but yours is not the
only project of this kind which has yielded results which were difficult
to interpret and understand. For example, in one extremely large-scale
literacy project in which forms were counterbalanced to eliminate form
effect as a possible influence, the "gains" over several months were
negative. And these were projects of several different types in different
states. The only conclusion that I could come to in consultation with
those responsible for evaluating the project was that some people were
helped a great deal by the training, but that many gained next to nothing,
or "lost" because of lack of interest in the tests (the general attitude
was that no matter how hard they tried they still would not exactly dis-
tinguish themselves!) In programs of this type it sometimes seems to be
as much a matter of trying to determine how many can be helped at all,
instead of determining average gain by averaging in the few who actually
made substantial gains and were reliably measured with those whose scores
were of questionnable validity.

In this project, the difficulty of the tests was by no means ideal
for evaluating the project. I did not determine national percentile ranks
but at one stage of analysis I did get Iowa PR's. The distributions of
these for the post-tests in Grade 4, for example, were as fellows:

A.- l A - 2
90-99 ‘
80-89
70-79 2
60-69

50-59

-2-
40-49 1 1
30-39 2
20-29 3 4
10-19 8 10
o- 9 28 29 ,I, ,
Irau"

On test A-I‘I (36 items) Qn expected 'chance" score would be 9.
21 out of your 44 final test scores were at or below this score.
This does not mean at all that these people "guessed." To the con-
trary, every child might have very carefully selected every item he
marked. But the point is that either 1) many children did not take
the tests seriously or 2) the tests were much too difficult. In
this case I would expect both to be involved to some extent. And
remember, 9 is the expected score. Half the children guessing should
be expected to do better! '

I suppose a case could be made for the proper use of tests at
-this difficulty to identify children with special problems but once
they have been identified, an instrument should be employed which
‘will measure and make fine discriminations within the group.

With respect to validity, from the description you sent of your
program, I would expect some improvement on Test Arl but not on
Test A-Z. .All of the emphasis was on concepts measured in Test A-l.
I would expect Test A-2 scores to be affected very little immediately.
Of course, in the long run improved conceptual abilities should re-
sult in improved problemasolving ability.

I tried several ways of analyzing your data but am enclosing only
. the one I believe to be best suited to your purposes. The results still
do not "look good" but I believe they are about as accurate as I can
come up with. I have no reason to doubt the analysis you originally
did. However, the Forms Lee-3 equating which lies behind the national
norms is probably not perfectly accurate (it never is).

We have many Title I projects in Iowa which we are evaluating
somewhat in reverse of your procedure (that is, Form 3'was administered
first, Form 1 second). To help schools with their evaluations we are
using what we believe to be comparable GE's for the two forms. However,
these are Iowa 63' 8. Performance in Iowa (and Mdchigan.') is, of course,
cpnsiderably higher than in the nation as a.whole, so the GE' s appear
somewhat lower than your originals.

up.

‘ Asyou can ..., the average gains or losses were as follows'when
computed by these methods.

1127

-3-
Grade 4 Grade 5
A - l A.- 2 A.-.T A - 1 A - 2 A - T

+368 -2.36 -.84 +.47 +3.89 '+2.16

I do’not know how much change you expected to get. One way to
compute this would be to figure that 25 hours might be approximately
,_22 of a year's instruction time in arithmetic. This would be about
120 2 points on the GE scale if average learners were involved. But
with "educationally disadvantaged" children'who learn slowly, a reason-
able goal mdght be something less. And it is probably not reasonable
to expect everyone to achieve this goal. '

It is here I bog down in trying to interpret results because I
do not know the'children or what happened during the course of the
study.

In Grade 4, on Test.A-1, the gains were distributed as follows:

15 '1 6 2 -3 2 ~12 -21
14 5 l -4 4 -13 -22 1
13 4 4 -5 1 -14

12 3 2 -6 l -15

11 2 3 -7 -16

10 l 6 -8 2 -17

9 3 0 4 -9 -18

8 -1 4 -10 -19

7 2 -2 1 -ll -20

There are several "unreasonable" gains or losses which means
probably that one or both scores were inaccurate. I would be as "suspicious"
of the 15 and 9-point gains as of the 22-point loss. (Incidentally, if
the child who lost 22 points had neither gained or lost, your average
would have jumped from .68 to 1.2!)

This has been quite rambling, but I wanted to make as much of a
case as I could for subjective considerations. If only 8 or 10 of the
24 gains listed above are genuine, I would think you could claim consider-
able success with your program. At least, judgments of this kind are
necessary in research after all the evidence is in and the results tallied.

I believe the results as you initially analyzed themmwere essentially
correct. These mdght be just'a little more accurate. I know this is
not what you had heped for but I hepe it will be of some help.

. Sincerely yours,

     

 

Professor, ucstion and Psychology

 
 

cc: Mr. John Sommer
Dr. Edward Drshosal

 

 

 

 

 

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APPENDIX D

A Study of Attitude Toward Arithmetic (Dutton-Adams Scale)

A Study of Attitude Toward Arithmetic (Modified version)

Arithmetic Treasure Hunt (Addition and Subtraction)

Arithmetic Treasure Hunt (Multiplication and Division)

Arithmetic Treasure Hunt (Measurement)

Sample Games

TOpics Covered in Summer Camp Arithmetic Program for
Fourth and Fifth Graders

Aids and Materials Used in the Summer Camp Arithmetic
Program

Sample pages from workbook

135

136

A STUDY OF ATTITUDE TOWARD ARITHMETIC

Check (X) only the statements which express
your feeling toward arithmetic

l. I feel arithmetic is an important part of Egg school
curriculum.

2. Arithmetic is something you have to do even though it
is not enjoyable.

3. Working with numbers is fun.
A. I have never liked arithmetic.

5. Arithmetic thrills me and I like it better than any
other subject.

6. I get no satisfaction from studying arithmetic.
7. I like arithmetic because the procedures are logical.

8. I am afraid of doing word problems.
9. I like working all types of arithmetic problems.
10. I detest arithmetic and avoid using it at all times.

11. I hays agzgwigg appnegiggigg of arithmetic through
understan ing its values, applications and proces .

12. I am completely indifferent to arithmetic.

13. I have always liked arithmetic because it has presented
me with a challenge.

14. I like arithmetic but I like other subjects just as well.

15. The completion and proof of accuracy in arithmetic gave
me satisfaction and feelin s of accom lishment.

)HHI HI)

I )

 

Before scoring your attitude scale place an (X) on the
line below to indicate where you think your general
feeling toward arithmetic might be.

 

 

ll 10 9 8 6 5 A 3 2 l
Strongly —fiZutral Strongly
favor against

Taken from ArithmeticT for Teachers by Dutton and Adams. All
underlined words were changed to prevent vocabulary diffi-
culties, pp. 360, 361.

137
Modified Version Dutton-Adams Attitude Scale

A STUDY OF ATTITUDE TOWARD ARITHHETIC

Cheek (X) only the statenents which express your feelings toward
arithmetic.

Q7
7, [#1.

3,)" ___2§_2.

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4.

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6.

 

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12.

w.)

7 5:443.

14.

7:5) _)Q15.
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I feel arithmetic is an impertant part of the school.

Arithmetic is something you have to do even though it
is not enjoy able.

Working with numbers is fun.
I have never liked arithmetic.

Arithmetic thrills me and I like it better than any
other subject.

I get no satisfaction from studying arithmetic.
I like arithmetic because it is logical.

I am afraid of doing ward problems.

I lik e working all tyL see of arithmetic problems.

I hate arithmetic and avoid using it at all times.

I J.il:e arithinetic better through undeistandlng its
values. and knowing how and where to use it.

I am co“olo ely indiff :erent to arithmetic.

I have always liked arithmetic because it has presented
me with a challenge.

I like arithmetic but I like other subjects just as well.

The completion and proof of accuracy in arithmetic gave
me a 9906 feeling.

Before scoring you: att.it ede scale. place an (X) on the line
below to indie ate there you think your general feeling toward

$2, arithmetic might be:

2%.?

2 3 4 5 6 7 8 9

 

110391 Y

“'1irct Neutral Strongly favor

l.
2.
3.
h.

5.
6.

7.
8.

9.

10.

11.

12.

13.

14.
15.

138

ARITHMETIC TREASURE HUNT*
(Addition and Subtraction)

Number of steps leading to water front. _¥
Number of individual buildings on camp ground.

Number of window panes in classroom.

 

Something in main lodge room that shows the problem 36
plus 52 equals 88. __**
How many numerals on the clock plus the letters?

Bring me a leaf with five fingers.

Bring me ____ stones plus ______ sticks to equal 13.
Bring me one-half dozen acorns.

Number of steps leading to basement minus your age.

If you ate three hotdogs last night and had to pay 15

cents for each, how much would it have cost?

 

Mr. Bower's license number is #296, Miss Moleski's is 5721.

How much is the sum of these two?

 

Mr. Squirrel stored 32 acorns for winter. Mr. Bluejay took

all but 11. How many did he take?

___

Number of picnic benchesminus number of picnic tables.

Bring me 17 acorns minus 15 plus 2.

 

Number of numerals on clock minus number of numerals on

phone dial.

 

*sample of reinforcing activities produced in manuscript.

**Piano keyboard.

1.

2.

3.

L.

6.

7.

8.

9.

139

ARITHMETIC TREASURE HUNT*
(Multiplication and Division)

How many people would be at the picnic tables if each

table had six peeple for a cook-out?

Number of steps leading to basement multiplied by number

of pianos at camp.

 

w—

There were 95 children at dinner last night and they
ate 279 hot-dogs. How many is this for each child?

 

Number of pages in phone book divided by three.
The temperature last week for each day was 82°, 64°,

75°, 83°, and 85°. What was the average temperature?

 

If each child in group 4 (2h) caught an average of three
fish each. How many fish would that be in all?

 

If each child in camp drinks nine pints of milk a day

how many pints will each child drink in a week?

 

‘V

If you paid five cents a pint how much would it cost for

one week?

 

v—w —

There are five sleeping cabins. If each cabin held the
same number of campers, how many would there be in each

cabin? (There are 93 campers now.)

 

*Sample of reinforcing activities produced in manuscript.

lhO

10. You go down and up the basement steps for each meal. How

 

many steps would this be for one day?
ll. Bring me enough pine cones to show the problem, 21 divided

by 7.
12. If the campers drink 64 quarts of milk each day, how

 

many gallons is this?

 

9.
10.
ll.
12.
13.

1h.

15.

141

ARITHMETIC TREASURE HUNT*
(Measurement)

What time is it by the clock in the lodge?

 

Temperature from the thermometer on porch.

 

Temperature from the thermometer on west window.

How large are the tile squares in the floor of lodge?

 

How long is the piano keyboard?

 

Size of one section of the south gate (feet and inches)?

 

Number of quarter inches on your ruler.

 

Size of storage doors on the outside grill (L and W).

 

w—

Size of bench by horseshoe game (L and W).

 

Size of window panes in your classroom.

 

Size of the tables in the classroom(L and W).

 

How many half inches on your ruler?

 

How many degrees warmer does the window thermometer show
than the porch thermometer?

 

How many gallons does the fire extinguisher hold?
(the one by the phone booth)

How many minutes have passed since you started this
treasure hunt?

 

*Sample of reinforcing activities. These were produced
on manuscript typewriter.

1h2

Sample Games

D mino Bin 0 (Used for both addition and multiplication.)

ar 3 contain domino shapes and include numerals. A
child may cover the prOper rectangle when he has added or
multiplied the two sets of dots on his dominoes correctly.
Otherwise played as regular bingo.

Ten Pins

Played as regular bowling game by rolling soft-ball
into oatmea boxes. Child must keep running scores using

the values of the boxes knocked over. The values are changed
grequently. Variations to this include putting problems on
oxes.

I Went Sho in '
Played in rotation, each child naming an item and

price. The next child must include all items named before him
and add another and total the prices. When the addition breaks
down, game starts over. Example: I went shopping and I

bought an ice cream cone it cost five cents. The next child
would repeat the first child and add another item and price,

it cost 8 cents, altogether they cost 13 cents, etc.

Miniature Golf

8 ng Open milk cartons lying flat on floor, hit base-
ball into a carton, trying to choose the one labeled with the
largest distance numeral. Score is kept of yardage by adding
together the values indicated on boxes.

I'm Thinking of a Number
imi ar to py.” If ten were added to my number

the answer would be 15, what is my number? The child can use
any method to identify his number. Example: My number can

be divided into L0 an e ual number of times but it is not ten.
What is it? It may be , 5, 2, etc. -

Find The Problem

An ob act is pre-chosen from a specified area which
represents a problem. T he problem can be any of the four
arithmetical operations. The child searches until he has
found it but doesn't tell the others. Example: There is an
addition problem in the recreation room which represents 6
plus 8 equals lb. Answer: 6 ping-pong mallets and 8 pool
sticks. Or find 18 triangles in the main lodge. Answer:
the roof rafters form 18 triangles. '

TOpics Covered in Summer Camp Arithmetic Program for
Fourth and Fifth Graders

Numeral and number

cardinal use
ordinal use

Our number system
Place value

Zero

Addition
Subtraction
Multiplication
Division

Story problems
Measurement
Fractions

Simple geometric shapes

Samples of Major Concepts Explored

Addition:

it is a way of grouping

it is bringing together two or more groups forming one
larger group

it is the Opposite of subtraction

it is related to multiplication

zero is the identity number for addition

3 + h is the same as h + 3, the order does not change the answer

(5 + h) + 6 is the same as 5 + (A + 6), we can associate dif-
ferent numbers

Subtraction:

it is the Opposite of addition

it is regrouping a larger group into smaller groups

it is related to division

zero is the identity number

we can show proof by adding our answer to the number we took
away

Fractions:

a fraction is part of a whole

a fraction can be expressed in different ways

the numeral over the line tells us how many parts we are
talking about

the numeral below the line tells us the size of the parts

the larger the bottom numeral is--the smaller the parts

the larger the top numeral is--the more parts we have

therefore l/lO is less than 1/5 of the same object and 2/17
is more than 1/10

a fraction can also express part of a group as 1/2 dev-

lhh

Aids and Materials Used in the Summer Camp
Arithmetic Program

Number line
Wall charts
fraction
place value
matrices
Flannel board
flannel symbols
animal cut-outs
fractional parts
geometric shapes
Magnetic board
peg board, rubber bands, and golf tees
Abaci
Number frames
Draw-string bags
COunting discs
craft sticks
Dominoes
modified bingo cards, to accommodate dominoes
Place value canisters
Egg cartons
ping—pong balls
Milk cartons and other containers
Wood molding
for rulers
Tape measures
10'
50'
Rulers and yard sticks
Oatmeal boxes and soft ball, for ten pins
Models
showing relationship between square and circle
modification of cuisenaire rods
thermometers
Graph paper
Tile squares

11.5

1.;
In arithmetic we do four

operations. They are:

. addition

. multiplication
. subtraction

. division‘

..boaNW—e

Anithmetic answers suCh
questions as these:

How many? How big? How far?
How much?

We need these answers every-
day. We believe in five (5)
weeks we can learn much ‘ 9
about numbers.

11.6

‘

 

1“"

-11

»»~‘~:

n7
2.'
Today we learned a new word,
the word is concept. It
means an idea or understand-
ing. Each day we will ex-
plore a new concept. In this
way we will have a ketter
understanding of numbers.
ill s we learned the differ-
ence tetween nurnber and
numerali

 

 

 

“Humeral means a Sign, symbol
Ol" naliieo

 

E,Ae”*1e“: 5,V,3, etc., we
use the nun erals O,l,2,5,4,
5,6,7,8,9, to express ideas.

Number means an idea or con-
cept. It is something we
think.

Examples: five (5) cows,
three (5) apples, etc.

 

 

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