:3...

vxx.

3:: I!
t

.z I. i:

5.5% w".

2.3

. x

1‘;

£3.35}. I

.w

p“: ....\a
(2:3 5.?!
i .

1k. 0.12.97.

i) 1.31? .tqip rt :
(it: . , 3.....025653J
2. a): t 1‘ ~

,3
in)! "33.! Ira; w

1:31:22: . a ..

ate... {\2‘. huhmm
.1 {Eu 1.... a ..

KHZ. «LE5

’1‘ x! o:
.5: at: I .2 A31...
$9355.13? 5.3!...
i ﬁlial-ya!

‘ntii‘?

I: .s!...

.7911.

r:

s ...... ﬁx

_ (25.5%?
. 3

l

.le .. Iv .

2“.
I: n

i... I.)
):n§15.3t
A :35:

.1 {A952

. Y) sift?“
.vl-.ul OKVuilt.) \|

 

 

 

12v. ; :.

 

.. .. 5......1915.

 

TAT NIVERSITY LIBRARIES

llllllllllllllllllHl‘ll l 1 ll

3 1293 010441859 r_‘

LIBRARY
Mlchigan State
University

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is to certify that the
dissertation entitled

AN ANALYSIS OF THE MATHEMATICS ACHIEVEMENT OF AFRO-AMERICAN
AND EURO-AMERICAN FEMALES IN GRADES 3, 4, AND 5 BEFORE
AND AFTER THE APPLICATION OF NONTRADITIONAL

MATHEMATICS INSTRUCTION
presented by

Janys Roberson

has been accepted towards fulﬁllment
of the requirements for

Ph.D. degree in Teacher Education

_ Major professor

 

 

Dam July 1994

MSU is an Afﬁrmative Action/Equal Opportunity Institution 0- 12771

 

' PLACE N RETURN BOX to remove thle dieokout from your record.
! TOAVOID FINESremmonorbeforedetedue.

     

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSUIeAnNﬂmdverthE O lnetltwon
quel mommy

m1

AN ANALYSIS OF THE MATHEMATICS ACHIEVEMENT OF
AF RO-AMERICAN AND EURO-AMERICAN FEMALES IN
GRADES 3, 4, AND 5 BEFORE AND AFTER THE
APPLICATION OF NONTRADITIONAL

MATHEMATICS INSTRUCTION

By

Janys Roberson

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Teacher Education

1 994

ABSTRACT

AN ANALYSIS OF THE MATHEMATICS ACHIEVEMENT OF
AFRO-AMERICAN AND EURO-AMERICAN FEMALES IN
GRADES 3, 4, AND 5 BEFORE AND AFTER THE
APPLICATION OF NONTRADITIONAL
MATHEMATICS INSTRUCTION
By

Janys Roberson

The primary purpose of this study was to investigate the underachievement
in mathematics concepts and problem solving Of Afro-American females in an
elementary school in a small midwestern town. A comparison was made of the
mathematics mean scores of Afro-American and Euro-American females in
grades 3, 4, and 5’before and afterthe application Of "nontraditional" mathematics
instruction. In addition, 12 subjects from the total sample were interviewed
regarding their attitudes and perceptions toward mathematics. In response to an
identiﬁed need, the elementary teachers in this school, after extensive training,
implemented an innovative "hands-on," conceptually based method of
mathematics instruction.

A basic 2 x 4 format was used to analyze the mathematics achievement
scores of Afro-American and Euro-American females in grades 3, 4, and 5 two

years before and two years after the application of the "nontraditional"

Janys Roberson
mathematics. An F-test was used to compare the two subject groups, with a p
level of signiﬁcant at .05. Results indicated that there was a Slight but signiﬁcant
difference (.04) in the mean mathematics scores of the total sample between Year
1 and Year 4 Of the study. The comparison of between-group variability in
mathematics mean scores indicated a strong main effect for race. Afro-American
females continued to underachieve in mathematics using a "traditional“ or
"nontraditional" approach to instruction.

Twelve volunteers from the sample group were interviewed by local college
students about their perceptions and attitudes about mathematics. Results
seemed to indicate that the Six Afro-American interviewees did not like
mathematics, found it difﬁcult and impractical in their daily lives, and tended to
avoid it. The six Euro-American interviewees indicated that they liked
mathematics, thought it was important, and denied their low achievement in
mathematics.

Possible confounding variables in this study included the small sample
size, adequacy Of the test to measure innovative curriculum, the relationship of
reading abilityto mathematics achievement, and cultural/gender bias affecting the
place of Afro-American females in desegregated Classrooms. The attempt to
solve a complex, systemic problem with a single methodologicalapproach was

not successful.

ACKNOWLEDGMENTS

Sincere appreciation is extended to Dr. Lois Bader, my committee
chairperson, and 'Drs. Roger Niemeyer, Charles Blackman, Norman Bell, and Bill
Cole, members of my doctoral committee, fortheir guidance and support. Thanks
are also extended to the excellent professors at Michigan State University who
encouraged me and challenged me to see this project through to completion.

A Special thank-you is extended to Dr. John Chapman, at the Michigan
Department of Education, whose mentoring has encouraged many educators to

pursue a doctorate.

Finally, a heart-felt thank-you to Keith, Johannah, Colette, and Jonathan

for their sustaining accompaniment on this journey.

TABLE OF CONTENTS

LIST OF TABLES ............................................. vii

LIST OF FIGURES ............................................ viii
Chapter

I. THE PROBLEM ..................................... 1

Introduction ................................... 1

The Study Setting .............................. 3

Purpose of the Study ........................... 5

Hypotheses .................................. 9

Research Questions ............................ 9

Need for the Study ........................... 1O

Assumptions and Delimitations .................. 11

Deﬁnition of Terms ............................ 12

Overview ................................... 13

II. REVIEW OF RELATED LITERATURE .................. 15

Introduction .................................. 15

Culture and Learning .......................... 17

Cultural Factors Affecting Achievement in School . . . . 25
Afro-American Females in American Classrooms . . . . 32
Improving the School Success of Afro-American

Children ................................... 38
Understanding How Children Across All
Cultures Learn Math ......................... 42
Chapter Summary ............................ 49
Ill.‘ METHODOLOGY.................; ................ 52
Introduction ................... ' .............. 5 2
The Problem and Purposes of the Study ........... 52
The Study Sample ............................ 55

V

Instrumentation ............................... 56

Data-Collection and Data-Analysis Procedures ...... 58

Description of the Nontraditional Approach to
Mathematics Instruction ...................... 62
A Typical Day in a "Nontraditional" Classroom ....... 64

A Typical Day in a "Traditional" Classroom

Before the Instructional Change ................ 66
Chapter Summary ............................ 68
IV. RESULTS OF THE DATA ANALYSES .................. 69
Introduction .................................. 69
Results of the Quantitative Analyses .............. 69
Results of the Qualitative Analyses ............... 75
Chapter Summary ............................ 91

V. SUMMARY, MAJOR RESULTS AND DISCUSSION,
REFLECTIONS, AND RECOMMENDATIONS ........... 103
Summary .................................. 103
Major Results and Discussion ............... . . . 104
Reﬂections and Recommendations for Practice ..... 116
Recommendations for Further Research .......... 117

APPENDICES

A. PARENTAL PERMISSION LETTER ................... 120
B. INTERVIEW SCHEDULE ........................... 121
REFERENCES .............................................. 124

vi

LIST OF TABLES

Table Page
2.1 Patterns of Selected Mental Abilities Among Four
Ethnic Groups ..................................... 19
3.1 Mean Mathematics Scores of the Total Sample ........... 60
4.1 Mean CAT Mathematics Scores of the Total Sample,
1990 Through 1993 ................................. 70
4.2 Mean CAT Mathematics Scores Of the Afro-American
and Euro—American Subjects, 1990 Through 1993 ......... 72
4.3 Mean Mathematics and Reading Scores of the 12
Interviewees ...................... . ................ 77
4.4 Summary of the Interview Responses .................. 92

vii

LIST OF FIGURES

Figure Page

4.1 Mean CAT Mathematics Scores of the Total Sample,

1990 Through 1993 ...... ' ........................... 71
4.2 Mean CAT Mathematics Scores of the Afro-American

and Euro-American Subjects, 1990 Through 1993 ......... 73
4.3 Mean CAT Mathematics Scores of the Two Subject

Groups Over the Four-Year Period, 1990Through

1993 ............................................ 74

viii

CHAPTER I

THE PROBLEM

InlLQdthjQu

In the 19805, it became apparent that many American youngsters had
major deﬁciencies in and a lack of understanding of basic mathematics concepts
and procedures, as well as basic mathematical problem-solving skills
(Mathematical Sciences Education Board, 1990; National Council for Teachers
Of Mathematics [NCTM], 1980, 1981, 1989, 1991; National Educational
Assessment Program [NEAPL 1980; National Research Council, 1989;
Schoenfeld, 1987). Comparisons of the mathematics achievement Of numerous
students in the United States and those from other countries revealed that many
American students were ill-prepared for the mathematical demands of a
technological society.

A Closer look at the mathematics achievement of American youngsters
revealed not only deﬁciencies in basic mathematics understanding and problem
solving, but also serious gender and ethnic inequities. Lightfoot’s research in
1976 documented that girls tested either higher than boys (In general

intelligence, verbal ability, and numerical ability) or the same as boys (in spatial

2
ability, analytic ability, and creativity) in their preschool years. However, by the
middle of elementary school, boys scored consistently higher on all these tests
(except grammar, spelling, and word ﬂuency).

In their comprehensive study of more than 530 ﬁrst graders enrolled in
public schools in Baltimore, Maryland, Entwisle and Alexander (1989) found that
ethnic differences in children’s skills in mathematics and verbal composition at
the point of school entry were very small to nonexistent but that signiﬁcant
differences by race on verbal skill, mathematics computation, and reasoning
began to appear early in the schooling process. By the end of ﬁrst grade, Afro-
American boys equaled Euro-American boys and girls in terms of mathematics
concepts on the California Achievement Test (CAT), but Afro-Am erican girls were
about one-quarter of a standard deviation lower than the other boys and girls in
terms of mathematics concepts on the CAT (Ginsberg 8 Russell, 1981 ). By the
end of high school, the difference in mathematics achievement that separated
Afro-American females from the others amounted to one grade level (Dorsey,
Mullis, Lindquist, & Chambers, 1988; Entwisle & Alexander, 1988).

If Afro-American and Euro-American youngsters begin school with virtually
no measurable differences in mathematical ability, then to what can the steady
decline in the mathematics achievement ofAfro-American females be attributed?
What elements are present in the schooling process and in the attitudes and
perceptions ofAfro-American females and others toward mathematics that might

contribute to this underachievement?

3
We

The setting for this study was an elementary school with an enrollment of
approximately 425 students in preschool through grade 5. East Elementary
School (a pseudonym) is located in a small town in south-central Michigan with
a population of about 10,000 citizens. Before the 19803, this community had a
thriving economy, which was dependent primarily on the oil industry and several
foundries related to the automobile industry. Between the 1980 census and that
Of1990, however, the community lost 4,000 of its residents and fell upon hard
economic times.

At the time of this study, East Elementary School reﬂected these changes
in the community’s economy. Approximately 72% Of the school population
received free or reduced—cost lunches. The ethnic composition of the total school

population was as follows:

Euro-Americans 55%
Afro-Americans 41%
Others . 4%

A large number of the children enrolled at East Elementary School were from
single-parent or step-parent homes.

The need for this study became apparent in the 1990-91 school year
when, as part of the state-mandated school improvement process and as a
necessary component of Public Act 25 in Michigan, a disaggregation of CAT data

for students at East Elementary School revealed signiﬁcant disparities between

4

the mathematics achievement of Afro-American females and that of Euro-
American females, Euro-American males, and Afro-American males in the
school. The disparity in these students” scores on mathematics concepts and
applications in 1990-91, before the implementation of the nontraditional

mathematics instruction, was as follows:

Afro-American females 75% below the 50th percentile
Euro-American males 53% below the 501h percentile-
Euro-American females 49% below the 50th percentile
Afro-American males 45% below the 50th percentile

Staff members questioned why an identiﬁable group, representing one-sixth of
thetotal school population, exposed to the same teachers, teaching methods and
materials, and school environment, Should achieve signiﬁcantly lower than their
classmates. The need to understand this phenomenon and seek ways to rectify
the problem became important to school personnel.

When the school district changed its focus from traditional mathematics
instruction to nontraditional, conceptually based instruction, the superintendent
gave the researcher permission to initiate this study. Both school-district and
school-building personnel were interested in determining whether the
nontraditional method of mathematics instruction would enhance the
mathematics achievement of all female students, but particularly Afro-American

females, at East Elementary School.

5
W

Disaggregation of mathematics achievement data in this particular school
setting revealed the underachievement of Afro-American females in mathematics
concepts and problem solving. In this study, one dimension Of schooling--an
innovative mathematics curriculum--was examined to seek a better
understanding of how the schooling process and other factors contribute to the
underachievement of Afro-American females. The study ﬁndings might be
helpful to educators attempting to raise the achievement levels of these
youngsters through appropriate mathematics instructional practice.

The method used to better understand this underachievement was to
investigate and compare the achievement In mathematics concepts and problem
solving Of Afro-American and Euro-American females in grades 3, 4, and 5 in a
particular school setting before and after the application of “nontraditional"
mathematics instruction. In addition to the analysis of mathematics achievement
scores, the attitudes and perceptions toward mathematics Of selected subjects
from the total sample were explored through structured interviews.

Many studies have been conducted in which Euro- and Afro-American
students’ learning styles and other cognitive as well as social factors affecting
learning have been examined and analyzed (Akbar, 1976; Comer, 1980; Entwisle
8 Alexander, 1989; Kerckhoff, 1989; Slavin 8 Karwelt, 1989). However, even
though numerous educational studies on Afro-American youngsters were

initiated as early as the 19503, little speciﬁc research has been conducted on

6

Afro-American females as an identiﬁable group with particular characteristics and
unique attitudes and perceptions. Afro-American females remain, as Lightfoot
noted in 1976, ”an ignored and invisible population" (p. 1) in educational
research, practice, and policy making. It is unusual for educational researchers
to examine differential academic outcomes of the genders within and between
racial and ethnic groups in this country (Educational Testing Service [ETS],
1992). However, the underachievement of Afro-American females needs to be
addressed in a technological society that is dependent on the full development
of all its members.

Some educators believe that changes in mathematics instruction and
curriculum may positively affect the mathematics achievement Of Afro-American
youngsters (Akbar, 1976; Gilbert 8 Gay, 1986; Hale-Benson, 1986; Morgan,
1980; Renzulli, 1973; Romberg, 1984; Slaven 8 Karwelt, 1989). In this study, the
writer measured the effects of a nontraditional method of mathematics instruction
on Afro-American females in the third, fourth, and ﬁfth grades. This approach
was conceptually based and used "hands-on" mathematical activities for
learning. However, this researcher went beyond the single "treatment" approach
as being the sole factor in the improvement of mathematics achievement for
Afro-American females. The case study approach also was used to investigate
whether additional social factors might affect the mathematics achievement of

young females, particularly Afro-American females.

7

The researcher examined the achievement of 19 Afro-American and 28
Euro-American females in the third, fourth, and ﬁfth grades using a "traditional"
method of mathematics instruction. The traditional method of mathematics
instruction is consistent with the absorption theory, in which mathematical
knowledge is impressed on the brain from without, using a collection of facts
learned by memorization and practice with drill (Baroody, 1987). A common
analogy for this approach to learning is to consider the youngster’s mind as an
empty slate upon which the teacher imprints selected knowledge. The learner’s
role is that of a passive receiver. This method does not rely primarily on the prior
knowledge and experience of the learner, but rather on the teacher's ability to
bring the learning to the learner. With the traditional method of instruction,
youngsters are required to sit for long periods of time, concentrate alone, and
observe and value organized time allotments and schedules.

In contrast, the nontraditional method of mathematics instruction, as it
applies to this study, is based on cognitive theory, in which knowledge is
constructed from within the learner’s mind, using experience and insight, hands-
on activities, and problem-solving process to build understanding. In this study,
the researcher attempted to determine whether the nontraditional method of
mathematics instruction would be more successful than the traditional approach
with Afro-American females because it is more closely aligned with instructional
strategies that have been found to be effective with Afro-American youngsters.

I-luman interaction, an active environment, the use of bodily motion in the

8

learning process, a loosely constructed environment, cooperation, informality,
and involvement in the instructional process have been documented as being
effective instructional strategies for many Afro-American students (Comer, 1980;
Entwisle 8 Alexander, 1988; Gilbert 8 Gay, 1985; Hale-Benson, 1986; Kerckoff
8 Campbell, 1977).

8 However, the researcher did not intend merely to determine the cognitive
outcomes of mathematics instruction forfemales by comparing two ethnic groups
with a single instructional treatment (e.g., the hands-on approach). Selected
aspects of the affective and social environments of these young females also
were studied, in order to understand why many competent Afro—American
females are underachieving. Comer (1980) suggested using this approach to
address human problems that are the result of multidetermined, interrelated
' factors. Thus, an in—depth case study Ofsix Afro-American females and six Euro-
American females from the study sample was conducted in the second portion
of this research, to seek to discover additional variables that might be related to
the subjects’ attitudes and perceptions about mathematics and their achievement
in the subject. These subjects had signiﬁcant discrepancies between their
mathematics scores and their reading scores; speciﬁcally, they had average to
high reading scores but signiﬁcantly lower mathematics problem-solving scores.
The researcher sought to determine what might have contributed to this

discrepancy.

9
Hypotheses
The following null hypotheses were formulated to analyze the data

collected in this study:

HypothesisJ: There will be no signiﬁcant difference in the mean
mathematics scores of the total sample before and after the application
Of a nontraditional, conceptually based method of mathematics instruction.
mm: There will be no difference between the Afro-American and

Euro-American subjects in terms of their mean mathematics scores over
the four-year period of study.

Beseambﬂuestions

The following research questions guided the collection of data for the
qualitative or descriptive portion of this study. The purpose Of this part of the
study was to discover additional variables and/or factors that may affect the
mathematics achievement of young Afro-American females.

1. What are the perceptions and attitudes of young Afro-American
females toward mathematics achievement?

2. What are the perceptions and attitudes of young Euro-American
females toward mathematics achievement?

3. What perceptions and attitudes toward mathematics achievement
do young Afro- and Euro-American females share?

4. What additional variables, perceptions, and attitudes revealed in

the interviews affect, either positively or negatively, the mathematics

achievement of young Afro-American females?

10

NW

This study is needed and important for the following reasons:

1. Many studies have been conducted on mathematics achievement
using the traditional method Of mathematics instruction. However, little research
has been conducted on the effectiveness of the nontraditional method of
mathematics instruction (referred to herein as conceptually based mathematics
instruction) with female learners. The effectiveness of this method of
mathematics Instruction with a particular segment of the school population-
females in grades 3, 4, and 5—was analyzed in this study.

2. There is little information on the academic outcomes of females
from various racial and ethnic groups in this country. This research project
focused on the underachievement of Afro-American females in a particular
school setting in mathematics concepts and problem solving and examined the
effectiveness of an innovative instructional approach to mathematics instruction.
The researcher also sought to investigate the attitudes and perceptions about
mathematics of selected female subjects in this school setting.

3. Little is known, as well, about the unique social factors and cultural
messages that may affectthe mathematics achievement of young Afro-American
females. Educators need to know how Afro-American females learn
mathematics, in order to raise the achievement levels Of these youngsters. The

ﬁndings from this study can have signiﬁcant implications for policy and

11
instructional practice regarding the mathematics education of both Afro- and

Euro-American females.

The statistical information for this study was obtained from the permanent
cumulative record ﬁles (usually referred to by school personnel as CA-60 ﬁles)
and general student information ﬁles at East Elementary School. The researcher
assumed that the information contained in these ﬁles was accurate and current.
Additional information on the 12 participants in the qualitative portion of the study
was obtained through structured interviews. Because students at East
Elementary School frequently are involved in studies originating at the small
private college located in the community, the researcher assumed that the
students were familiar and comfortable with participating in a study of this nature.

No attempt was made to study the long-range effects of the conceptually
based mathematics curriculum on the mathematics achievement Of the subjects.
However, the researcher did examine in depth a particular method of instruction
that was applied to a particular student population during a four-year period--
1990-91 through 1993-94.

Subjects’ scores on the mathematics concepts and applications subtest
of the CAT were used in this study. The study was delimited in this way because
the disaggregation of test data revealed that these students had signiﬁcantly
lower scores on mathematics problem solving than they did on mathematics

computation.

12
Although the phenomenon of underachievement of Afro-American and
Euro-American females in mathematics may occur in other settings, the study
ﬁndings may not be generalizable to other communities. However, it is
anticipated that the ﬁndings will provide insight into and help educators improve
the mathematics achievement scores of Afro-American females in similar

communities.

Deﬂnjlignﬂleims

The following terms are deﬁned in the context in which they are used in
this dissertation:

W. An American Citizen who traces his or her ancestry to an
African nation or nations.

QaﬂfgmﬁAmievememlesuCAD. A standardized, norm-referenced
achievement test with batteries for each subject area at all levels from
kindergarten through grade 12.

M. An American citizen who traces his or her ancestry to a
European nation or nations.

WWW. As used in this study,
a method of teaching mathematics that is based on the concept that the student
constructs his or her own mathematical learning by connecting concepts, skills,
insight, and previous learning to concrete experiences such as using
manipulatives and relating mathematical thinking and reasoning to "real life"

activities.

 

 

13

MW. As used in this study, a
method of teaching mathematics that is characterized by direct instruction by the
teacher to a student who practices learning with repetitive drill and worksheets
and demonstrates understanding through rote memory.

WWW. One of several types of scores
used in reporting results on the CAT. The NCE score scale ranges from 1
through 99 and coincides with the national percentile scale at 1, 50, and 99.
NCE scores are similar to percentile ranks but can also be used to compare
scores of different groups and different subject areas.

W. A permanent school record folder that
contains certain physical and educational data about a particular student. These
data are arranged in chronological order and maintained by the school district
throughout the student’s K-12 school years.

WM. An analysis of attitudes and perceptions gathered
through interviews with selected members of the study sample.

Quantitatmnalvsis. Statistical analysis of numerical data, in this case

the achievement test scores of members of the study sample.

QALQDLISM
Chapter I contained an introduction to the problem under investigation, a
description of the study setting, the purpose of the study, the hypotheses and
research questions, need for the study, assumptions and limitations, and

deﬁnitions of key terms.

14

A review of literature and research related to the study is contained in
Chapter II. Topics include: Culture and Learning, Cultural Factors Affecting
Achievement in School, The Current "Place" or "Role" of the Afro-American
Female in American Classrooms, Improving the Success in School of Afro-
American Children, and Understanding How All Children Across All Cultures
Learn Mathematics.

The research design and methodology used in carrying out the study are
described in Chapter III. The data collected and analyzed in the research are
presented in Chapter IV. The Chapter is divided into two sections: quantitative
and qualitative. The quantitative section contains the results of the statistical
analyses of the math achievement scores of the subjects, and the qualitative
section contains the information obtained from the case studies of 12 students
selected from the study sample. A summary of the study, major results and
discussion, implications, and recommendations for further research are

presented in Chapter V.

CHAPTER II

REVIEW OF RELATED LITERATURE

IDILQdILQIIQIl

The researcher’s primary purpose in this study was to investigate and
compare the achievement in mathematics ofAfro-American and Euro-American
females in grades 3, 4, and 5 in a particular school setting before and after the
application of nontraditional mathematics instruction. A second purpose was to
investigate selected subjects’ attitudes and perceptions toward mathematics.

The phenomenon of underachieving Afro-American females is highly
complex, subtle, and underresearched. Although many studies have been
conducted on the achievement and cognitive learning styles of Afro-American
and Euro-American children (Ames, 1967; Blau, 1981), little research has been
done on the achievement and cognitive learning styles of young Afro-American
females. Similarly, images of Afro-American females have not been presented
in the literature on teachers and children in classrooms. After conducting an
extensive literature review, Lightfoot(1976) stated that young Afro-American girls
are an ignored and invisible population.

In 1992, Nelson-LeGall, writing forthe Educational Testing Service, stated

that, in America, being female puts one at risk for gender discrimination, and

15

16
belonging to a racial minority group puts one at risk for racial discrimination.
Thus, in this society, race, gender, and cultural factors must be included in any
meaningful examination of educational opportunity. Blau (1981) provided strong
evidence that the differences in academic competence among children are
predominantly social in origin.

Differences in the mathematics skills of children from various ethnic
groups at the time they enter school are small (Ginsberg 8 Russell, 1981).
Entwisle and Alexander (1989) found no signiﬁcant differences in the
mathematics and verbal composite scores of Afro- and Euro-American ﬁrst
graders. Yet, by the end of ﬁrst grade, the scores of these children differed
signiﬁcantly (especially the mathematics scores of Afro-American females), and
the discrepancy by race had begun to develop. Thus, the rationale for the
present study is the demonstrated need to understand and respond to the
underachievement of a portion of the school population.

This chapter contains a review of current literature and research related
to the following topics: Culture and Learning, Cultural Factors Affecting
Achievement in School, Afro-American Females in American Classrooms,
Improving the School Success of Afro-American Children, and Understanding

l-low All Children Across All Cultures Learn Mathematics.

17
Cultureandteaming

Culture inﬂuences learning as well as achievement (Ames, 1967; Banks,
1978; Beady, 1981; Blau, 1981; Hale-Benson, 1986; Jones, 1981; Ogbu, 1981).
In America, if one’s cultural style and cognitive style match those of the dominant
middle-class Euro-American culture, that individual’s cultural bonds will have a
positive inﬂuence on his or her achievement. However, if one’s cultural style and
cognitive style differ greatly from those of the dominant middle-class Euro-
American culture, the individual will experience some difﬁculties in academic
achievement.

Rashid (1982) stated that children from lower-socioeconomiC-status non-
European cultural groups are at a disadvantage in American schools. This
situation exists because the American educational system has evolved from a
European philosophical, theoretical, and pedagogical context.

Historically, Thomas Jefferson envisioned the American educational
system as being the vehicle for maintaining an informed electorate, which he
considered a necessity for the survival of democracy. In reality, the American
educational system became a kind Of factory that "Americanized" huge groups
of immigrants arriving from countries throughout the world. This factory was well
tuned to select and sort those who accepted the value system of the dominant
Euro-American class.

Today, more than 40% of the youngsters in America are at serious risk

academically and are outside the cultural mainstream (Natriello, McDill, 8 Pallas,

18 -

1989). These children may experience learning difﬁculties because of
differences between the culture in which they develop and learn and the culture
they encounter in the public schools (Hale-Benson, 1986). Race and ethnicity,
poverty status, family composition, mother’s educational level, and language
background are considered the ﬁve primary indicators of "educationally
disadvantaged" children in America (Natriello et al., 1989). Any one of these
factors, as well as combinations of the factors, can lead to serious learning
deﬁciencies, primarily because of cultural differences.

Schools are inﬂuenced by, and help to maintain, the values and norms of
the total society; one of these norms is sexism (Lightfoot, 1976). Females from
all cultural groups must deal with gender bias at all levels of American society,
including the schools.

Havighurst (1976) found that social class and ethnicity inﬂuence the
shaping of human behavior. Each social class has a set of behaviors and
attitudes that deﬁne it and separate it from other social classes. Each ethnic
group forms a subculture with its own attitudes and behaviors. Cultural patterns
are transmitted within all sOcial classes and ethnic groups through the same
mechanisms: family, peers, common literature, formal associations, in-group
marriage, and residential segregation.

Sarason (1973) maintained that it is impossible to understand and
evaluate the intellectual performance of particular groups without taking into

account each group’s attitude toward such performance. His point remains valid

19

when one investigates the relationship between the intellectual performance of
particular groups and their historically rooted attitudes. People participate in a
coherent culture that shapes their cognitive development and affects the way
they approach academic tasks and behave in traditional academic settings.
Stodolsky and Lesser (1967) examined the rank-order patterns of various
mental abilities among six- and seven-year—old youngsters from different social

classes and ethnic backgrounds. Their major ﬁndings are shown in Table 2.1.

Table 2.1: Patterns of selected mental abilities among four ethnic groups.

 

 

 

 

Afro-American Chinese Jewish Puerto Rican
Verbal ability Space Verbal ability Space
conceptualization conceptualization
Reasoning Numerical ability Numerical ability Numerical ability
Space Reasoning Reasoning Reasoning

conceptualization

 

Numerical ability Verbal ability Space Verbal ability

 

 

 

 

 

conceptualization

 

Source: Stodolsky and Lesser (1967).

As can be seen in the table, subjects from the four ethnic groups differed
markedly in the rank-order patterns of their mental abilities. Each culture shaped
the cognitive development of its youngsters and, through its values, set the

sequence of priority in the development of cognitive abilities.

20

Witkin (1967) deﬁned cognitive style as the characteristic self-consistent
modes of functioning found pervasively throughout an individual’s cognitive,
perceptual, and intellectual activities. After studying various ethnic and cultural
groups in America, Cohen (1969) surmised that basically two styles of learning
are present across all ethnic groups: analytical and/or relational. The analytical
learner is a "splitter," or an individual who thinks that the attributes of a stimulus
have signiﬁcance in themselves. The relational learner is a "lumper," or an
individual who thinks that the attributes of a stimulus are signiﬁcant only in
reference to some total context. Cohen maintained that current school practice
uses only the analytical approach to learning. Requirements that the pupil learn
to sit for increasingly long periods of time, to concentrate alone on impersonal
learning stimuli, and to observe and value organized time-allotment schedules
favor the analytical learner. Hence, it becomes necessary for relational learners
continually to seek coherence from the learning environment on their own.

Cohen (1969) suggested that children develop their cognitive styles
through the socialization they receive in their families and friendship groups. He
discovered a relationship between structured families in "formal" styles of group
functions and the analytical style. Fluid families that had the characteristic of
"shared function“ in their primary groups were identiﬁed with the relational style.
Cohen’s research indicated that the use of either style was not conﬁned solely

to any one ethnic group or Class but that the analytical style was generally

21

associated with Eu rO-American families and the relational style was characteristic
of the functioning of most Afro-American families.

Hilliard (1976) listed the following general characteristics of the Afro-
American cultural style:

1. Response to the ”whole“ rather than the "parts.“

2. Preference for inferential reasoning rather than deductive or
inductive reasoning.

3. Atendency to approximate space, numbers, and time rather than

to be precisely accurate.

4. Preference for focusing on people and their activities rather than
on things.

5. A tendency to have a keen sense of justice.

6. A tendency toward altruism.

7. Preference for novelty, freedom, and personal distinctiveness.

8. Good at nonverbal communication; not word dependent.

The preceding Characteristics most commonly considered to reﬂect the Afro-
American cultural style are compatible with Cohen’s interpretation of the
relational cognitive style.

The strengths of the Afro-American family, according to Hill (1972),
include sound kinship bonds, strong achievement orientation, adaptability Of
family roles, solid religious orientation, and strong work orientation. Hill

contrasted the Afro- and Euro-American cultural styles in the following way:

22

Afro-American Ethos Euro-American Ethos

Survival of the tribe Survival Of the ﬁttest
Oneness with nature Control over nature
Cooperation Competition
Interdependence Individualism
Collective responsibility Independence

Benson-Hale (1986) projected that a Euro-American family’s motto might
be “I think, therefore, I am”; in contrast, an Afro-American family’s motto might
be "I feel, therefore, I think, therefore, I am." There is an inherent conﬂict in these
two points of view. As stated earlier, the Euro-American value system is that
which is considered to be "American" and is the one by which other systems are
measured. Because Afro-American families want their children to be successful,
they need to prepare their youngsters to live and function among Euro-American
people mtbouLbecomingjuLCL-Ameucah. The Afro-American family must
mediate the "cultural conﬂict" that occurs when the values of the family differ from
those of the dominant culture. The family must prepare its Children to function
in the culture in which they live and to assimilate those behaviors that are
necessary to be upwardly mobile in the dominant culture.

Afro-Americans are the product of their own heritage and culture, yet they
are shaped by the demands of the Euro-American culture. For this reason,
DuBois (1903) described Afro-Americans in America as having "two warring
souls.”

The family group is the primary "teacher," and it is the family that strongly

shapes the cognitive development of children and the way they approach

23

academic tasks. "Cultural difference" occurs when children have not had
experiences that provide them with the kind of information that they need in
school. "Cultural conﬂict" occurs when children have a learning style that differs
from the analytical and behavioral styles reinforced in school (Cohen, 1969).
The results of Jones’s (1981) study suggest that cultural conﬂict plays a
signiﬁcant role in the low school achievement of Afro-American students.
Cultural conﬂict results in a serious mismatch between teachers’ and students’
cognitive styles and negatively inﬂuences the school achievement of Afro-
American learners. Not only does the young Afro-American female face the
cultural conﬂict of being outside the mainstream of the cultural majority, but she
also must face gender bias and cultural messages, mythical and real, about
being an Afro-American female in a Euro-American, male-centric society.
Hale-Benson (1986) asserted that the socialization of young Afro-
American females begins early in their development with a strong motherhood
orientation. Facial beauty is deemphasized in the Afro-American family, whereas
there is a strong emphasis on personal uniqueness. The Afro-American family
values and gives prestige to young females who demonstrate verbal ability,
personality, wit, strength, intelligence, speed, and so on. There is an emphasis
on the development of sexuality, personal style, and distinctiveness. An
authoritarian child-rearing style often serves to develop toughness and self-
sufﬁciency in Afro-American females, who may not interpret this style as

restrictive or rejecting but rather as nurturing and caring. Afro-American females

24
are expected to contribute to the support of the family, and hence they develop
a strong work orientation early in their lives. Hale-Benson further described the
socialization process for young Afro-American females as an intense
interpersonal interaction in the family, with a strong mother-child relationship.
Lightfoot (1976) described the Afro-American female as being strong, patient,

and enduring.

Summanr

The literature reviewed in this section focused primarily on the inﬂuence
of culture on learning and development. There is substantial evidence that
culture does, indeed, shape learning as well as achievement (Ames, 1967;
Banks, 1978; Beady, 1981; Benson-Hale, 1986; Blau, 1981; Jones, 1981; Ogbu,
1981). In America today, more than 40% of the youngsters may be at serious
risk academically and are outside the cultural mainstream as deﬁned by the
Euro-American majority (Natriello et al., 1989).

Schools are inﬂuenced by, and help to maintain, the values and norms Of
the total society. People participate in a coherent society that shapes their
cognitive development and inﬂuences the way they approach academic tasks.
Cohen (1969) suggested that, although cultural inﬂuences may vary, most people
are either analytical or relational in their learning styles. He maintained that
current schooling practices tend to favor the analytical learner, who prefers
formal and traditional learning situations. The relational learner is most

comfortable in nontraditional, interactive, and informal learning situations.

25

Cohen’s data suggest that most Euro-Americans prefer the analytical learning
style, whereas most Afro-Americans tend to prefer the relational style of learning.

Cultural conﬂict occurs when a child has a learning style that differs from
the analytical and behavioral Styles being reinforced in school (Cohen, 1969).
Afro-American children often experience serious cultural conﬂict in their
schooling, which negatively affects their achievement (Benson-Hale, 1986;
Gilbert 8 Gay, 1985; Jones, 1981).

Afro-American families are. under tremendous pressure to maintain the
strengths and Characteristics of their own culture and to value these
characteristics in their children, while at the same time preparing their children
to function in the Euro-American school culture, which may or may not value
those same Characteristics. It is within the family that Afro-American females
initially receive the cultural messages of socialization before entering school.
After the age of ﬁve, Afro-American females often receive conﬂicting cultural

messages in the school setting.

hi v
Although there is no precise mechanism for measuring or comparing the
cultural factors that inﬂuence the school achievement ofAfro— and Eu ro-American
students, numerous researchers have provided evidence to explain and/or
describe disparities in achievement between Afro- and Euro-American
youngsters (Ames, 1967; Banks, 1978; Beady, 1981; Blau, 1981; Cadigan,

Entwisle, 8 Alexander, in press; Jones, .1981; Ogbu, 1981; Pallas, Entwisle, 8

26

Alexander, 1987; Rosser, 1980; Weikart, Bond, 8 McNeil, 1971). These
investigators attempted to discover factors that are characteristic of orfrequently
are found among particular subgroups. These factors are not deﬁnitive but rather
serve as generalizations that can aid in the understanding of a certain
phenomenon.

According to Blau (1981), Observed disparities in the measured
intelligence of Afro-American and Euro-American children of each gender were
not due to inherent racial differences. Rather, they were the result of social
factors and complex interactions among these factors.

It is not until the ﬁrst year of school and during the process of schooling
that Afro-American children begin to underachieve (Ames 8 Ames, 1967;
Entwisle 8Alexander, 1989). Many minority-group children do not proﬁt as much
from school as majority-group students do. Likewise, minority-group children’s
test scores do not increase as much throughout the school year as do those of
majority-group children (Entwisle 8 Alexander, 1989).

Coleman-Burns (1989) stated that the dual purpose of schooling is to
foster cognitive development and to promote affective and social development.
She concluded that schools bring little inﬂuence to bear on a child’s achievement
that is independent Of his or her background and general social context.
According to Hale-Benson (1986), the primary emphasis of traditional public

school education in America for Afro-Americans is to mold and shape these

27
children sothat they can ﬁt into an educational process designed for middle—class
Euro-American children.

One group of variables that inﬂuence a child’s achievement and success
in school comprises the individual and background factors that are present when
the child starts school. These factors include gender, race, various parental and
cultural inﬂuences, the Child’s own expectations, personal maturity, and/or
special concerns that would apply just to that child (such as physical/mental/
social problems).

Entwisle and Alexander (1980) studied more than 500 ﬁrst graders in
Baltimore, Maryland. They found that Afro-American and Euro-American
students were virtually equivalent in terms of computational and verbal skills, as
measured by the California Achievement Test (CAT), at the start of ﬁrst grade.
Afro-American males equaled Euro-American males and females in terms of
mathematics concepts (reasoning skills), but Afro-American females began ﬁrst
grade about one standard deviation lower than the others in terms of
mathematical concepts on the CAT. In a similar study, Ginsberg and Russell
(1981) found that ethnic differences in children’s mathematics skills at the point
of school entry were small to nonexistent. In her study of Afro- and Euro-
American Children, Blau (1981) stated that, when the full set of structural
variables of the two cultures was taken into account, the difference between the

races was reduced to 3.5 intelligence quotient (IQ) points.

28

Other researchers have found that few measurable differences exist in the
cognitive development Of children when they enter school (Rohwer, 1971). As
a result of his research, Piaget (1952) veriﬁed that constants in language,
maturation, mental operation, and universal interactions were more signiﬁcant in
a child’s development than were transmissions of cultural lessons, which would
vary from culture to culture. Piaget conceded, however, that the culture of the
Child will inﬂuence the chronology of the developmental stages but not the
inevitability of the development.

The second phase of the child’s cognitive development is inﬂuenced
primarily by the youngster’s school experiences. By the ﬁrst or second grade,
children as well as their teachers know how they measure up to their Classmates
in terms of their scholastic performance, even in the absence of IQ tests (Blau,
1981). Such factors as ﬁrst-'marking-period grades, popularity with peers,
regularity of attendance, and conduct grades in the third marking period of the
ﬁrst year of school signiﬁcantly inﬂuence the child’s perception of success
(Entwisle8Alexander, 1989). Once this generalized orientation toward theworld
is established, whatever its structural origins, it frequently operates as a self-
fulﬁlling prophecy (Merton, 1948).

A disproportional lag in the development of verbal skills and abilities has
been found among working-class Euro-American students and most Afro-
American students. This lag increases as the youngsters advance from one

grade to the next. As the curriculum content expands and spirals, that which the

29

children are expected to master becomes more difﬁcult, and deﬁciencies in basic
skills become an even greater impediment to learning. Often, the child’s sense
of efﬁcacy is undermined, and discouragement and alienation follow (Blau,
1981). By the time they reach ﬁfth grade, many Afro—American children have
lower achievement scores than their second-grade IQ scores would predict. On
the average, these students measure only at the fourth-grade level on their ﬁfth-
grade achievement tests. The data also suggest that underachievement stems,
not from lack of competence, but from the failure of parents and teachers to
provide effective instruction, encouragement, and support as children are
required to master progressively more complex concepts and skills (Blau, 1981).

For Euro-American children, differences in achievement levels, as
measured by standardized achievement tests, have been attributed to the
following:

1. The parents’ educational level.

2. The demographic origins of the family.

3. The parents’ occupational status.
In contrast, differences in achievement levels of Afro-American children, as
measured by standardized achievement tests, have been attributed to the
following:

1. The social milieu of the parents (the mean educational level of the
mother’s three closest friends, and the social characteristics of neighbors).

2. The parents' educational level.

30

3. Differences in religious composition and religiosity (Blau [1981]
established a negative relationship between fundamentalist religious afﬁliation
and IQ development).

As a rule, parents who are economically and educationally advantaged
have higher educational aspirations and expectations for their children than do
parents who are less advantaged. In turn, these children average higher scores
on all manner of school tests involving verbal and numerical skills. The
youngsters’ success reinforces their own and their parents’ aspirations and
expectations with regard to their future educational attainments (Gordon, 1972;
Hyman, 1953; Kahl, 1957; Rosser, 1956, 1959; 1961; Simpson, 1962).

As noted earlier, the achievement test scores of many minority-group
children do not increase as much over the school year as do those of majority-
group children (Entwisle 8 Alexander, 1989). Critics of most standardized
achievement tests have Claimed that most of these tests:

1. Are highly loaded with items based on middle-class Euro-American
values and experiences (Williams, 1970).

2. Penalize children with linguistic styles that differ from that of the
dominant culture.

3. Sample cognitive styles that are different from those found in many
children from low-income families (Cohen, 1969) or culturally diverse groups

(Kleinfeld, 1973).

31

4. Have fostered the creation of a dual educational system by means
of classifying and excluding some children from regular education programs
(Winnikur 8 Wohle, 1984; Ysseldyke, 1983).

5. Are of little or no use in formulating instructional programs for
children (Heller, Haltzman, 8 Messick, 1982; Winnikur 8 Wohle, 1984).

The most common criticism of standardized achievement tests isthat they
are culturally biased so as to discriminate against racial and ethnic minorities of
low socioeconomic status (SES) (Jensen, 1980) and may endanger these
individuals’ futures (Williams, 1974). These tests are normed for the majority
population and cannot measure the different talents that are derived from the

unique cultural backgrounds of minority groups (Williams, 1974).

Summanc

This section focused on cultural factors affecting the achievement ofAfro-
and Euro-American children in school. According to Blau (1981), observed
disparities in the measured intelligence of Afro- and Euro-American boys and
girls were not due to inherent racial differences but were the result of social
factors and the complex interactions among them. Entwisle and Alexander
(1989) found that Afro-American and Euro-American students were equivalent
in computational and verbal skills when they entered school. Other researchers
have further demonstrated that differences among ethnic groups in terms of
mathematics and intelligence test scores at the onset of schooling are small to

nonexistent (Blau, 1981; Entwisle 8 Alexander, 1989; Ginsberg 8 Russell, 1981;

32

Rohwer, 1971). Rather, it is within the ﬁrst year of school that Afro-American
children begin to underachieve. Blau suggested that this underachievement
stems, not from lack of competence but rather from the failure of parents and
teachers to provide effective instruction, encouragement, and support as children
are require to master progressively more complex skills and concepts. It is
important to remember that the achievement of youngsters in American public
schools is measured through the use of standardized achievement tests.
Various writers have asserted that standardized achievement tests, as presently
used, are culturally biased, discriminating against racial and ethnic minorities and

those from low-SE8 families.

r - ' l ' ' l r

Schools are inﬂuenced by, and help to maintain, the values and norms of
the total society; one of these norms is sexism. Structures and behaviors that
reinforce traditional gender roles pervade the educational system, from preschool
through the postdoctoral level.

During the preschool years, females test either higher than boys (in
general intelligence, verbal ability, and numerical ability) or the same as boys (in
spatial ability, analytical ability, and creativity). However, by the middle of
elementary school, boys consistently score higher on all of these tests except
grammar, spelling, and word ﬂuency, and this trend continues throughout the

school years (Lightfoot, 1976).

33

Until recently, Afro-American females were expected to assume a
subordinate role to Euro-American males and females of all ages, as well as to
Afro-American males. Similarly, Euro-American females were expected to play
a subordinate role as "helpmates" to Euro-American males.

In the past, basicformal education for Afro-American females was thought
to be a necessity only for producing disciplined domestics and blue-collar
workers (Coleman-Burns, 1989). For Afro-American females, the process Of
socialization into the roles of womanhood begins at age seven or eight. It is the
peer group that seems to be the most important agent in this process (Ladner,
1971)

The responsibilities assumed by adolescent Afro-American females are
often those taken on by adult females in other social groups, as well (Ladner,
1970). In the past, most Afro-American females subtly were encouraged to
assume stereotypical roles of Afro-American women in American society rather
than strive for alternatives (Grant, 1984).

Afro-American children of both genders are socialized in theirfamilies for
assertiveness, self-conﬁdence, independence, and achievement (Lewis, 1975;
Nelson-LeGall, 1989; Scott-Jones 8 Clark, 1986). These children are raised to
be people-oriented (Benson-Hale, 1986).

Afro-American females tend to meet or exceed the attainment level of
Afro-American males at all educational levels (Coleman-Burns, 1989). There is

no ﬁrm evidence, however, that Afro-American parents encourage their

34
daughters’ education at their sons’ expense (Smith, 1982). Rather, the bulk of
the research has indicated that Afro-American families are less likely to use
gender as a basis for educational and occupational aspirations than are Euro-
American families (Nelson-LeGall, 1991).

Nelson-LeGall (1 991 ) stated that Afro-American families and communities
now place an emphasis on education for Afro-American females, knowing that:

1. The Afro-American child’s social status is determined by the
mother’s educational level.

2. Educational level is a critical factor in procuring employment.

3. Employment Opportunities are more favorable for Afro-American
females than for Afro-American males.

4. Afro-American females must work forthe economic survival oftheir
families, not self-fulﬁllment.

Schooling does not have the same economic payoff for Afro-American
females as it does for other racial/gender groups, despite the positive association
between years of schooling completed and earning levels. According to data
collected by the US. Department of Labor, Bureau of Labor Statistics (1989),
Afro-American females earned 69% Of what Euro-American males earned, and
Euro-American females earned 74% of what Euro-American males earned. The
wages of Afro-American females were the lowest of any of the gender and
racial/ethnic groups included in the comparison (National Center for Education

Statistics, 1991 ).

35

Outcomes Of schooling differ systematically across various racial and
gender groups (Hare, 1985; Trent, 1984). One reason for this difference is that
educators respond to children of various racial and gender groups in different
ways.

Grant (1984, 1985) found that Afro-American females’ experiences in
school differed from those of other racial/gender groups, but that the differences
could not easily be predicted. ln Grant’s study, teachers rated Afro-American
females’ academic skills and classroom performance more favorably than those
of Afro-American males, less favorably than those of Euro-American females,
and lower than but similar to those of Euro-American males. Euro-American
teachers gave less attention to academic criteria when assessing Afro-American
females’ abilities than when rating the abilities of any other group. Teachers
called on Afro-American females more than any other group for nonacademic
tasks, whereas they offered reinforcement and recognition of intellectual
competence to Euro-American females. The pattern Of teachers’ interaction with
Afro-American females in desegregated classrooms was found to be relatively
parallel to the pattern of treatment of Euro-American females in a segregated
elementary school classroom (Dweck, Davidson, Nelson, 8 Enna, 1978).

Afro-American females have less contact than Euro-American females
with the teacher, and such contact occurs only when necessary. These contacts
are briefer, more task related, and more often on behalf of a peer than are those

of Euro-American females. Euro-American females provide more help to Euro-

36

American males than they receive in return, whereas Afro-American females and
males are mutually helpful to each other in academic and nonacademic ways
(Grant, 1984; Nelson-LeGall 8 DeCooke, 1987; Nelson-LeGall 8 Glo-Scheib,
1985). In desegregated classrooms, Afro-American females are the most
frequent targets of racist and sexist remarks, which are generally made by Euro-
American males (Grant, 1984). Afro-American females in desegregated schools
often demonstrate lower self-esteem than Afro-American females in segregated
schools (Powell, 1970; Rosenberg 8 Simeon, 1972).

Afro-American females in desegregated schools usually have higher
levels of academic performance than do Afro-American males (Crain 8 Mahard,
1978; Hare, 1978). However, the academic performance of Afro-American
females is lower than that of Euro-American students (Crain 8 Mahard, 1978).
Hare and Castenell (1981) found that Afro-American females had lower self-
estimations of academic Skills than did Afro-American males, in spite of higher

performance.

Summanr

This section was concerned primarily with the experiences of many Afro-
American females in American Classrooms. In preschool, females of all ethnic
groups test higher than or the same as males on general intelligence, verbal
ability, and numerical ability. However, by the middle of elementary school,
males score higher than females in these areas. Thus, it would appear that the

schooling process is not beneﬁting all females. Until recently, Afro-American

37

females were expected to assume a subordinate role to Euro-American males
and females Of all ages, as well as to Afro-American males. In the past, most
Afro-American females were encouraged to assume the stereotypical (usually
blue collar) roles of Afro-American women in society, rather than strive for
alternatives (Grant, 1984). Afro-American females’ experience in school seems
to differ from that of other racial/gender groups. Euro-American teachers give
less attention to academic criteria when assessing Afro-American females’
abilities and tend to offer them conditional praise for academic work (Dweck et
al., 1978). Afro-American females have less contact with the teacher than do
others and seem to go to the teacher only when it is absolutely necessary (Grant,
1984; Nelson-LeGall 8 DeCooke, 1987; Nelson-LeGall 8 Glo—Scheib, 1985). In
desegregated classrooms, Afro-American females are the most frequent targets
of racist and sexist remarks, which generally are made by Euro-American males
(Grant, 1984).

Public schools are considered to be critical agencies in the transmission
of social-status arrangements from one generation to the next. Educators
respond to children of different races and genders in ways that reinforce
differentiation. Hence, Afro-American females must endure strong cultural

messages that indicate both gender and racial discrimination in school settings.

The key to improving the school success of Afro-American children is to
modify the means used to achieve the outcomes (Gilbert 8 Gay, 1985). Gilbert
and Gay, as well as other writers, did not recommend altering the level of
expected outcomes, but rather Changing the means of achieving these outcomes
so that the relational-style learner can be accommodated.

In 1976, on the basis of his study of Afro-American children, Akbar

developed the following proﬁle of the Afro-American learner. This learner:

1. Is highly affective.
2. Uses language with numerous interjections.
3. Uses considerable body language in communication.
4. Uses expressions with connotated meanings.
5. Uses nuances, intonation, and body language.
6. Prefers aural-oral learning communication.
7. Is highly sensitive to others’ nonverbal cues.
8. Is people oriented.
9. Is sociocentric.

10. Uses internal cues for problem solving.

11. Is highly empathetic.

12. Is spontaneous.

13. Adapts readily to novel stimuli.

39

Effective instructional strategies for Afro-Americans include opportunities
for human interaction in the learning process. Afro-American students tend to
function better in cooperative, informal, and loosely structured environments.
Positive teacher rapport, in an environment that focuses on working together to
achieve common goals, is important to Afro-American youngsters. The effective
learning environment for these students has an authority ﬁgure with high
expectations and a sense of fairness. Gilbert and Gay (1985) described the
learning style of Afro-American children as relational and ﬁeld-dependent, with
an active pace and a cooperative climate.

Gilbert and Gay (1985) also described the "stage setting" that precedes
Afro-Americans’ performance of a task as being atypical to the traditional school
setting. Stage setting refers to the activities and/or behaviors that Afro-
Americans need to perform to establish the proper context or setting before they
can begin the task. These behaviors might include looking over the whole
assignment before beginning; rearranging posture; elaborately checking pencils,
paper, and writing space; asking the teacher to repeat directions; and checking
the perceptions of nearby students. Although such activities seem to be
necessary in order to begin a task, they often consume valuable work time and
might account for a substantial cumulative loss of time on task.

For most Afro-American youngsters, spoken language is the primary
mode of communication. They learn by listening and demonstrate their learning

by speaking. The style of delivery of Afro-American communication is often

40
described as artistic and dramatic, with nonverbal nuances, word placement, and
speech rhythm.

Afro-American Children are more apt to talk simultaneously and to engage
in several different activities at once than are their Euro-American peers. Most
Afro-American youngsters are comfortable with this multimodal, multidimensional
involvement. They tend to like to involve their total selves in the learning—
cognitively, affectively, and physically. When they are unable to use this "whole
self" in the learning process, many Afro-American youngsters experience
withdrawal and frustration. An orderly environment may seem dull, stagnant, and
unstimulating to these students. Afro-American students learn best when they
are immersed in a system of interrelated learning encounters that minimize
conﬂicts between the values and expectations of the school and those of the
Afro-American culture (Benson-Hale, 1986; Gilbert 8 Gay, 1985).

Slaven, Karwelt, and Wasik (1993) stated that it is Clear that all children
must successfully negotiate key developmental hurdles in their ﬁrst decade of
life. Educators can ensure that virtually all children can overcome these
obstacles by providing:

1. High-quality preschool learning experiences, such as those found
in high-scope preschools.

2. One-on-one tutoring for serious problems in ﬁrst grade.

3. Continued curriculum improvement and long-term family support

services.

41

If learning-climate conditions that favor both the relational and the analytic
learner are included as a fourth provision, then it might be found that the active,
cooperative type of learning environment that seems to work best for Afro-
American students is also beneﬁcial for many other students, as well.

Throughout their school years, Afro-American youngsters are assessed
using standardized achievement tests, which have long been suspected of being
culturally biased, discriminating against racial and ethnic minorities. Thus, it is
imperative that Afro-American youngsters be taught systematically to be "test

wise" (Henry 8 Henry, 1990).

Summanr

This section focused on ways to improve the school success of Afro-
American Children. Gilbert and Gay (1985) believed that the key to improving
Afro-American children’s success in school is to modify the means of achieving
outcomes so as to reﬂect the cognitive style of the learner. Benson-Hale (1986)
referred to this modiﬁcation as adding "soulfulness" to the instructional process.
Keeping in mind the proﬁle of the Afro-American learner developed by Akbar
(1976), teachers should implement instructional strategies that provide these
students with opportunities for human interaction during the learning process.
Providing Afro-American learners with multimodal, multidimensional involvement
during the learning process, and using the "whole self" in the process, may help
minimize conﬂicts between the values and expectations of the school and those

of the Afro-American culture (Benson-Hale, 1986; Gilbert 8 Gay, 1985). By using

42

information about how all youngsters learn, as well as what is known about how
Afro-American youngsters learn, educators may be able to bring about higher
achievement among this group of students.
I! I I I' II CHI 9
AﬂﬁuﬂumeamMam

Piaget and others have demonstrated that learning is a complex, creative,
and intellectual process in which the learner literally constructs or builds
knowledge based on what he or she already knows. All youngsters continually
assimilate new information, which enters the frame of reference in their minds.
With this information they form connections and seek relationships. Children
invent strategies for making these connections, and the strategies may change
as the concept grows. When such a connection is made. the structure will
Change to accommodate the new learning into the whole idea. (An example is
the conceptual realization that whenever one multiplies any number by zero, the
answer will be zero. Acceptance of this concept frees youngsters from
memorizing a large number of facts.)

These connections are based on where the child is developmentally and
experientially. In his studies of youngsters, Piaget (1952) identiﬁed the
developmental stages young learners must progress through in order to reach
the formal operational stage (maturity). These stages, or periods, in which the
child’s thinking reﬂects a particular mental structure, are not predictably

chronological but sequential and developmental. Various cultural factors can

43
inﬂuence the timing of the developmental stages but not their sequencing and
inevitability.

From birth to about two years of age, youngsters are in the sensi-motor
developmental stage. That is, they learn entirely from sensi-motor input. Reﬂex
actions gradually are integrated into exploratory and experimental activities.
Even infants have some numerical sense at this early stage. They can
distinguish between some Classes Of amount, such as "big" and "little," but they
cannot arrange things in order. Number words such as "more" and "another"
appear in their language, and they form some intuitive, informal notions about
addition and subtraction (Baroody, 1987). Even infants are capable of perceiving
numerical differences between small quantities (Starkey, Spelke, 8 Gelman,
1980). They can perceive differences in number and see correlations among
different numbers of events.

The ages from two to seven are referred to as the preoperationa/ stage.
During this stage, children learn to use image and language. They are very
centered and ﬁnd it difﬁcult to take another’s point Of view and/or accept that
another’s way of life is different. They cannot conserve; that is, they cannot
accept the notion that amount may not change when the arrangement changes.

In an experiment with his four-year-old daughter, Binet (1969) observed
that she could count no higher than three, but she could identify the greater of

two sets. She always Chose the set that took up more space as the greater one.

44

Children in the preoperational stage continue to develop quantitative
concepts such as "more," "less," "order," and "same." They can begin to count
but do not understand one-to-one correspondence. At about four years Old,
children can learn the number words and count up to nine objects. At about ﬁve,
children can count about ten objects; they may be able to write the numbers but
do not really understand the abstraction of the symbols. Many six year Olds can
count up to 28, can write the numbers, and have some understanding of the
concept but not the ability to explain it. By the age of seven, most youngsters
can count, write the numerals, and explain the connection with place and value
(Ginsberg, 1987).

Piaget (1952) described children from ages 7 to 12 as being in the
concrete operations stage of development. Upon reaching this stage, children
can think logically and apply such operations as Classiﬁcation, ordering,
reversibility, and conservation. They can de-center but have not yet reached
formal operational thought, which deals fully with abstractions and hypotheses
that have no direct connection to the real world (Ginsberg, 1987).

Baroody (1989) found that an understanding of mathematics grows from
concrete experiences with objects. "Real mathematics," he stated, is
representational and highly abstract. Children’s mathematics is the necessary
ﬁrst step in the direction of abstraction. Abstract ideas cannot be taught; they

must emerge from concrete experiences.

45

The traditional method of teaching mathematics provides few opportunities
for youngsters actively to explore, experiment, and discover representational
relationships. Writing and memorizing number facts are abstract processes and
are irrelevant to most young learners.

In the 19503, Piaget recommended that teachers provide Children with
Objects and ideas to manipulate that would help them become more conscious
of problem solving and would encourage them to ﬁnd answers for themselves.
Piaget thought that teachers should be less providers of information and more
organizers of engaging, problematic situations. Young learners must essentially
rediscover and reinvent mathematical knowledge for themselves. They can do
so in guided opportunities of discovery and exploration.

Typically, mathematics instruction tends to be a process based on the
"absorption theory." That is, the valued knowledge is impressed on the student
from outside by means of the teacher. The learner is considered to be an "empty
vessel" to be ﬁlled with a collection of facts. This process depends on repetitive
drill and practice until the student has memorized the desired level of information.
The thinking process involved is associative; that is, mastery of a particular level
depends on successful mastery of the previous level. The learner is both
passive and receptive in this process. The teacher serves as the external
controller, punishing and correcting students for wrong answers and rewarding
them for correct ones. The role of the teacher is to transmit information in a

"lock-step" manner. Strategies of the teacher include direct instruction using a

46

textbook, workbook, and/or worksheet. Students are expected to listen, learn,
and practice decoding the abstract symbols of mathematics with large amounts
of drill, homework, seatwork, and worksheets (Baroody, 1987).

With the traditional method of mathematics instruction, little or no
credence is given to learners’ informal understanding of mathematics. Indeed,
learners are not expected to use any of their natural strategies and/or intuitions
in the learning process. Teachers rarely ask students how they are thinking but
seem to be interested only in the "right" answer. This way of learning
mathematics is predictable and efﬁcient uLtilthe learner runs into a problem that
outstrips his or her capacity for memorization. (This predicament is especially
difﬁcult for girls, who use their verbal skills to memorize and compute but are
later bewildered by the application of problem-solving and geometric
relationships in algebra and geometry [Clement, 1989].) These teaching and
learning practices may not be based on sound developmental and cognitive
learning theory and may not serve the needs of many youngsters.

In 1980, the National Educational Assessment Program stated that
American students were experiencing major deﬁciencies in mathematics
understanding and achievement. Speciﬁcally, they lacked understanding of
basic mathematics concepts and procedures, as well as basic problem-solving
skills. In international comparisons as well, American students made a poor
showing. In response to this grim report, the National Council of Mathematics

Teachers recommended that teachers change their Instructional practice and

47

teach mathematics so that students would be actively engaged in learning.
Teachers were encouraged to provide opportunities for students to experiment,
explore, and communicate about math. Also, teachers were asked to create an
environment that would encourage problem solving and risk taking. As a result,
the hands-on or nontraditional method of mathematics instruction, which focuses
on conceptual development based on concrete experiences, has become more
evident in elementary school classrooms across America.

This nontraditional method of mathematics instruction is based on the
cognitive theory that knowledge must be worked out from within the learner,
using insight, previous learning, and/or understanding and building relationships
with what is being learned. The essence of knowledge essentially is that which
is formed when elements of mathematics information are connected by
relationships to form an organized and meaningful whole concept. This active
construction of knowledge by the learner is based on the discovery of the
relationships of mathematics concepts. The role Of the teacher is that of
facilitator/guide and creator of an environment that is fertile with opportunities for
youngsters to experiment and to discover these important mathematical
relationships (sorting, predicting, counting, grouping, graphing, adding, building,
subtracting, Classifying, ordering, and so on). The teacher may provide games
and concrete materials in a setting that is active with small-group, large—group,
and one-on-one opportunities for interaction and exchange of ideas (Baroody,

1987). Using this method of Instruction, the classroom teacher is challenged to

48
consider how all children learn (cognitive) and what they need and feel (affective)

so that they can be successful in mathematical problem solving.

Summanr

This section focused on understanding how all children learn
mathematics, as well as the two methods Of mathematics instruction that are
used in American schools today: traditional and nontraditional. Learning
mathematics is a complex, creative, and intellectual process in which the learner
literally constructs or builds knowledge based on what he or she already knows.
Mathematics connections occur developmentally and experientially. Children of
all cultures must progress through speciﬁc sequential developmental stages that
reﬂect a particular mental structure. Piaget (1952) identiﬁed these stages as
sensi-motor, preoperational, concrete operations, and formal Operations.
Instructional strategies provide opportunities for youngsters to reinvent or
rediscover numerical relationships. Traditional mathematics instruction tends to
be a formal process, in which the valued knowledge is transmitted to the child
through such activities as repetitive drill, memorization, and rote learning.
Nontraditional mathematics instruction, which was developed in response to a
public outcry over Children’s poor achievement in mathematics, is in Closer
alignment with Piaget’s developmental stages and relies more heavily on the
learner’s natural numerical sense and experience. It is anticipated that this
closer alignment of mathematics instructional opportunities and natural learning

will enhance children’s achievement in mathematics.

49

ChapteLSIImmaDL

The phenomenon of underachievement among Afro-American females is
highly complex, subtle, and underresearched. Nelson-LeGall (1992) observed
that being female puts one at risk for gender discrimination, and belonging to a
racial minority group puts one at risk for racial discrimination. Race, gender,
cultural factors, and the school setting are among the complex, interacting
variables that affect the achievement of Afro-American females in American
schools.

Culture inﬂuences children’s learning and achievement. In America,
children whose cultural and cognitive styles differ greatly from those of the
dominant middle-class Euro-American culture Often experience signiﬁcant
difﬁculties in academic achievement. Today, more than 40% of the youngsters
in America are outside the cultural mainstream and are seriously underachieving
academically (Natriello et al., 1989).

Cohen (1969) found that basically two cognitive styles-~analytical and
relational—are present in all ethnic and cultural groups in America. The cognitive
style most frequently used by Afro-Americans is relational, whereas that used by
Euro-Americans is analytical. Cohen believed that the analytical approach to
learning is the one that is primarily used in schools today.

The strengths and values ofthe Afro-American family contrast sharply with
those of the Euro-American family. The Afro-American family serves as a

mediator for its Children; family members attempt to teach the children how to

50
survive and be successful In the dominant culture without losing their own
cultural identity.

Young Afro-American females face the cultural conﬂict of being outside
the mainstream of the cultural majority. They also encounter gender bias and
negative cultural messages, mythical and real, about being an Afro-American
female in a male-centric, Euro-American society.

It is not until the end of the ﬁrst year of school that Afro-American children
begin to underachieve (Ames, 1967; Entwisle 8 Alexander, 1989). They are
virtually equivalent to Euro-American children in terms of computational and
verbal skills at the beginning of ﬁrst grade (Entwisle, 1989). Blau (1981)
discovered that, when she considered a comprehensive range of variables, the
difference in IQ scores between the races was only 3.5 points. However, even
though Afro-American children are relatively equal to their Euro-American
Classmates when they start school, they begin to fall behind by the ﬁrst marking
period and continue to do so throughout their schooling. The literature reviewed
for this study indicated that underachievement of Afro-American Children stems
not from their lack of competence but rather from the failure of parents and
teachers to provide effective instruction, encouragement, and support as the
Children are required to master progressively more complex concepts and skills.

The experiences of Afro-American females in school seem to differ from
those of other racial/gender groups. Afro-American females seem to occupy a

"role" or "place" in the classroom that deemphasizes academic achievement.

51

They are frequently the target of racist and sexist remarks, which usually are
made by Euro-American males, and they often demonstrate lower self-esteem
than other identiﬁable groups in the desegregated school setting (Grant, 1984).
The key to improving the success of Afro-American children, particularly
Afro-American females, is to modify the means used to achieve the desired
outcomes (Gilbert 8 Gay, 1985). Much is now known about the cognitive style
and learning characteristics of many Afro-American children. It is anticipated that
the use Of effective instructional strategies, based on this knowledge and a better
understanding ofthe socialization process experienced by young Afro-American
females, could enhance the achievement Of all Afro-American students.
According to Piaget (1952), Children of all cultures progress through
certain developmental stages in their learning. Mathematical knowledge grows
as youngsters move through these stages. By using the nontraditional method
of mathematics instruction, which is based on known learning characteristics of
these stages and relevant cognitive learning theory, teachers might be able to
improve the mathematics achievement of many Children, including Afro-American
youngsters. This method focuses on active, informal learning and takes into
account the physical, mental, and emotional needs of youngsters. It is
compatible with instructional strategies that have been found to be effective with

young Afro-American learners.

CHAPTER III

METHODOLOGY

Intmdusﬂsm
The methodology used in carrying out the study is described in this
chapter. The problem and purposes of the study are stated ﬁrst, followed by a
discussion of the study population. The two instruments used to collect data for
the study are discussed next. The data-gathering procedures for both the
quantitative and qualitative portions of the study also are described, as are the

data-analysis techniques.

IbeEmblemjndEIuDQseseﬂbeﬁtudx

In 1991 , when the achievement test data for East Elementary School were
disaggregated, the disparity between the mathematical problem-solving and
applications scores of Afro-American and Euro-American females in grades 2,
3, and 4 became apparent. Whereas 49% of the Euro-American females scored
below the 50th percentile, 74% of the Afro-American females scored in that
range. The question arose, why was there such a great discrepancy in the
mathematics achievement of the Afro-American females? Thus, the problem to

be researched in this study became evident to the writer.

52

53

In the fall of 1992, the school district introduced an innovative,
nontraditional approach to mathematics instruction, which focused on a
conceptually based approach to learning mathematical concepts and
applications. This new method of mathematics instruction is based on the
educational theory developed by Piaget and others, that children must be the
architects oftheir own learning. In learning a particular concept, youngsters must
have many and varied opportunities to experiment with and/or "ac " on the
concept before they can attempt to use an abstract symbol, such as employing
this number concept in a paper-and-pencil activity. For example, youngsters
may work with manipulatives and counting objects in a variety of ways before the
concept of 3 + 1 = 4 is established in their minds. Eventually, as they arrange
objects, children build a relationship in their own minds about the congruity of
that number concept.

This "acting-upon" and "relationship-building" approach to learning is
enhanced by the insertion of mathematical thinking into real-life situations. For
school-age children, real-life situations occur at home, at school, at play, and at
work. Quantiﬁcation activities are employed at every opportunity in school, from
record keeping to game playing. Frequent problem solving using mathematical
and logical thinking occurs throughout the school day. Youngsters do not move
to the abstraction until they demonstrate that they understand the concept and

have had multiple opportunities to "act" upon it. The teacher seeks every

54
opportunity to use a real-life situation for quantifying and for mathematical
thinking and logic.

Educators in the district anticipated that the use of a nontraditional
approach to mathematics instruction would enhance the mathematics
achievement of both Afro-American and Euro-American females. Teachers also
anticipated that the nontraditional type of instruction would improve these
youngsters’ attitudes and perceptions about mathematics.

Implementation of this new method of mathematics instruction, based on
sound educational pedagogy, provided a baseline for comparing and contrasting
the effects of this treatment on a sample group of young learners. The Afro-
American females in this school population were not achieving at a satisfactory
level when teachers used the traditional approach to teaching mathematics.
Hence, one of the researcher’s purposes in conducting this study was to
determine whether achievement would improve as a result of the application of
a nontraditional method of mathematics instruction.

The researcher compared Afro-American and Euro-American females’
achievement in mathematics concepts and applications two years before and two
years after the application of a nontraditional, conceptually based method of
instruction. The subjects were third, fourth, and ﬁfth graders in one elementary
school

Responding to the literature stating that perceptions and attitudes about

mathematics may affect children’s achievement in mathematics, the researcher

55

expanded the study to also include interviews with 12 members of the sample
group. The purpose of this portion of the study was to investigate these subjects’

attitudes and perceptions toward mathematics.

IheﬁtudlLSamnle

The sample for this project comprised 28 Euro-American and 19 Afro-
American females in grades 3, 4, and 5 in one elementary school. East
Elementary School was chosen as the setting for the study primarily because of
the disparities in the achievement test scores of its students in 1991. These
disparities were discovered during the building’s school improvement process.
Before the implementation of the nontraditional mathematics instructional
program, 74% of the Afro-American females scored below the 50th percentile on
the mathematics concepts and applications portions of the CAT, whereas 49%
of the Euro-American females scored below the 50th percentile.

Each female in grades 3, 4, and 5 at East Elementary School was
assigned a student number, In order to protect her anonymity. NCE scores for
math applications and problem solving forthe years 1990. 1991, 1992, and 1993,
as well as recent mathematics computation and reading comprehension scores,
were gathered. The researcher used the school computer to select those
females in grades 3, 4, and 5 for whom complete test-score data were available
for the years 1990-91 through 1993-94. From a pool of approximately 70 Euro-
American females in the school, 28 were selected forthe sample. Likewise, from

a pool of approximately 35 Afro-American females, 19 subjects were selected.

56

It was not necessary to obtain parental permission for this portion of the study
because the identities of the youngsters being studied would not be revealed.

From the initial group of 47 sample members, 26 responded to the request
for interviews and returned permission notes from their parents. Of that number,
12 (six Afro-Americans and six Euro-Americans) were Chosen forthe interviews.
The researcher endeavored to select for the interviews those subjects who had
a signiﬁcant disparity between their total reading scores and their mathematics
concepts and application scores. The other primary characteristics shared by the
12 interviewees was their interest in the project, their willingness to be
interviewed, and their inclusion in the study sample.

The interviews took place two years after the inception of the
nontraditional mathematics instruction. Because the interviewees were not
randomly selected from the sample group, their perceptions and attitudes about
mathematics may not be generalizable to the total population, but they do
provide insight into the subjects’ perceptions and attitudes concerning

mathematics.

Instmmentation
Two instruments were used to gather the data for this study: the
California Achievement Test (1984 edition) and the interview guide developed by
the researcher. Students’ scores on the CAT, a series of norm-referenced,
objective-based tests for kindergarten through twelfth grade, were used for the

quantitative portion of the study. The test series was designed to measure

57

achievement in the basic skills commonly found in state and district curricula.
Because the tests combine the most useful characteristics of norm-referenced
and criterion-referenced tests, they provide information about the relative ranking
of students against a norm group, as well as speciﬁc information about the
instructional needs of students.

The subject areas measured by the CAT were reading and mathematics.
These scores were used to provide a measure Of test performance and to make
direct comparisons of achievement between the two subject groups. For the
purposes ofthis study, onlythe mathematics concepts and applications, reading,
and mathematics computation NCE scores were collected.

The structured interview was designed to explore the 12 interviewees’
perceptions and attitudes about mathematics. The ﬁrst ﬁve questions were
designed to put the student at ease and to gain some demographic data about
her. Questions 6 and 7 were designed to gain insight into how the student
perceived herself as a grown up. Questions 8 through 17 were intended to
gather information on the interviewee’s perceptions and attitudes about
mathematics. The ﬁnal section, which was adapted from a sentence-completion
questionnaire developed by Sandra Davis at the University of Minnesota, sought
additional information on the subjects’ attitudes and perceptions about
mathematics.

The interview was ﬁeld tested on a group of 12 second graders who were

not included in this study. One change was made in the instrument after the ﬁeld

58

study. Question 17, item 3, was changed from "When Ithink about math, I. . .
to "Doing math makes me feel. . . The revised instrument provided the
information sought for the qualitative portion of the study. A copy of the interview

questions may be found in Appendix B.

- t' - l i r

The superintendent of the school district involved in the study granted the
researcher permission to conduct this study. Both school-district and school-
building personnel were interested in determining whether the nontraditional
method of mathematics instruction would enhance the mathematics achievement
of all female students, and particularly Afro-American females, at East
Elementary School. The researcher also obtained approval from the Michigan
State University Committee on Research Involving Human Subjects (UCRIHS)

to carryout the study.

II E III E I' [ll SII

The quantitative data for this study were Obtained from the subjects’
permanent cumulative records (CA-60$). Subjects’ scores on the Math
Applications and Problem-Solving portion of the CAT for the years 1990, 1991,
1992, and 1993 were used in this study. Computation scores and total
mathematics scores were not used in the analysis of variance portion of the
study because mathematics concepts and applications was revealed in the test-

aggregation process as being the area of most signiﬁcant underachievement.

59

Normal curve equivalent (NCE) scores on the CAT of the sample
members were used in the quantitative portion of the study. The scores from two
years before and two years after the application of the nontraditional approach
to mathematics instruction were compared to determine whether the subjects’
scores had improved after the application of the nontraditional mathematics
instruction. The NCE score scale, ranging from 1 through 99, coincides with the
national percentile scale at 1, 50, and 99. NCE scores are similar to perCentile
ranks but also have an equal-interval scale. That is, the difference between two
successive scores on the scale has the same meaning throughout the scale. For
example, if a youngster in the study had an NCE score of 65, this means that this
youngster did better than 65% of the children in the country who took the test.
NCE scOres, unlike percentile or stanine scores, allow meaningful comparisons
to be made between different groups of students on the same test. Comparisons
can be made by averaging the scores for the groups.

In the quantitative portion of the study, the statistical analysis of the CAT
data was set up in a 2 x 4 format, the 2 representing the two groups of subjects
(Afro-American and Euro-American) and the 4 representing the four years of test
data being analyzed (two years before and two years during and after the
application of the nontraditional mathematics instruction). The mean
mathematics scores of the sample groups, presented in the 2 x 4 format, are

shown in Table 3.1.

60

Table 3.1: Mean mathematics scores of the total sample.

 

 

 

 

 

 

 

 

 

Group 1990 1991 1992 1993
Afro-Americans 37. 3 35.0 41.3 41.5
Euro-Americans 49.3 60.6 58.0 61.2

 

An F-test was used to measure variance within and between the two
groups on their mathematics concepts and applications achievement scores two
years before and two years after the application of the nontraditional method of
mathematics instruction. A p-value was computed to indicate the probability of
the outcome. The .05 alpha level was established as the criterion for statistical
signiﬁcance.

The following null hypotheses were formulated for this study:

Hypothesiﬂ: There will be no signiﬁcant difference in the mean

mathematics scores of the total sample before and after the application

of a nontraditional method of mathematics Instruction.

HypothesisZ: There will be no difference between the Afro-American and

Euro-American subjects in terms of their mean mathematics scores over

the four-year period of study.
TI 3 III E I' [II S! l

Forthe qualitative portion of the study, interviews were conducted with 12
females selected from the sample. The interviews were designed to explore
these subjects’ perceptions and attitudes about mathematics. The researcher

met with all of the female students in grades 3, 4, and 5 at East Elementary

School and explained the interview process. She told the youngsters that this

61

was a study about how they felt about mathematics. They were informed that the
research was being conducted by a student from Michigan State University with
the help Oftwo female college students--an Afro-American and a Euro-American.
They were not told that their principal was the person concluding the study, in
order to avoid any subtle coercion implied in the principal-student relationship.
The students had participated in various studies before, so this was not an
unusual situation for them.

Consent letters explaining the study were sent home with those students
who expressed an interest in participating in the interviews (see Appendix A).
Interviewees were selected from a list of students who returned permission slips
if there was a substantial discrepancy between their reading and mathematics
achievement scores.

The subjects’ responses to the interview questions were recorded on a
single data sheet for each question. The following codings were used to allow
the researcher to look for variables that might be associated with the question,
as well as the interaction between a subject and the interviewer.

lnterviewerllnterviewee

AA/AA Afro-American/Afro-American
AA/EA Afro-American/Euro-American
EA/AA Euro-American/Afro—American

EA/EA Euro-American/Afro-American

62
A sample page showing the coding of the responses may be found in Appendix
B.
D 'I' [II II t IT I? h
WW

Beginning in the fall of 1992, the school district changed to a
nontraditional, Conceptually based approach to mathematics instruction. Every
teacher at East Elementary School participated in the Michigan Mathematics
lnservice Project (MZIP). This staff-development project was the result of the
cooperative efforts of the Michigan Council of Teachers of Mathematics and the
Michigan Department of Education. The purpose of this project was to help
Michigan’s kindergarten through ninth-grade teachers Of mathematics to prepare
their students for the challenges of the twenty-ﬁrst century. Speciﬁc objectives
for the elementary school portion of the MZIP lnservice project were as follows:

1. To extend and update the mathematics background of Michigan’s
K-6 teachers.

2. To shift the emphasis of K6 teachers from teaching rote skills to
that of developing concepts and higher-level thinking skills.

3. To foster the development of problem-solving abilities.

4. To include probability and statistics, geometry, measurement, and
algebraic ideas in the K—6 curriculum.

5. To identify strategies for encouraging underrepresented and

underserved groups in the study of mathematics.

63

This training model consisted of approximately 40 hours of teacher
lnservice at the K-2 and 3—6 grade levels. Teachers were taught by project
leaders to be "teachers of teachers." The teachers for this project at East
Elementary School were from a nearby school district. The lnservice sessions
were held after school from 3:30 to 6:00 pm. once a week for approximately 16
weeks. Hands-on instructional materials such as attribute blocks, tangrams,
counting and sorting manipulatives, cubes, bloCks, geometric solids, fraction kits,
measurement devices, calculators, and other kinds of appropriate materials were
purchased, and the teachers were taught to use them in the classroom.

Mathematics instruction gradually was Changed from a traditional
approach using pencil, paper, and computation to an interactive, cooperative,
problem-solving approach. Instruction was provided in small groups, one to one,
and in large groups. Mathematics and problem solving were integrated
throughout the whole school day, across all subject areas. Problem solving and
graphing were used in such Classroom issues as who had hot lunch and who had
cold lunch, or who wore tennis shoes and who did not. Computers were used for
drill and practice and for higher-level thinking and problem solving. Students
continually were predicting, using mental math, graphing, sorting, and so on.

Mathematics learning evolved from a passive to an active, ongoing, and
integrated part of the school day. The interactive activities involved young
people in mathematics problem solving all day rather than in an isolated segment

of the school day. On a typical day, time spent on mathematics now totaled 90

64

minutes, spread throughout the day, rather than a single session of 35 minutes
a day. Children had opportunities to work in small, cooperative groups, as well
as to explore the mathematics materials on their own.

The focus of the nontraditional mathematics instruction was on active
learning, based on developmental theory. Teachers remained with the concrete
materials to build the theory or concepts and moved to the abstract (paper and
pencil) only when the Children had had many concrete experiences. A typical
day in a nontraditional classroom is described in the following section, followed
by a description of a typical day in a traditional classroom before the instructional

change.

T'Iin"nr"l"l

A typical day in a "nontraditional" second-grade Classroom at East
Elementary School begins with taking attendance and lunch count and noting the
calendar day, the number Of days left in the school year, and the number of
permission slips that have been returned. The teacher seizes every opportunity
to form quantiﬁcation activities out of the daily record keeping. As the teacher
moves into the morning’s language arts lesson, such activities as collecting and
distributing instructional materials, locating the correct page in the textbook, and
voting on favorite characters from the story are used to encourage mathematical
and logical thinking. During spelling, health, social studies, and reading,
opportunities for quantiﬁcation are sought. The teacher frequently asks

questions such as "Does everybody agree?" "How many do we need without

65

bringing extras?" and "Pass out just enough for our group." Such questions
encourage the development of relationships of numbers and problem solving to
actual daily living.

In the formal mathematics instructional period of the day, the teacher has
the children work in small, cooperative groups after a large-group demonstration
of a number concept such as set theory. Much group discussion takes place
during the small-group problem-solving process. The teacher asks questions
that encourage logical thinking and processing but avoid eliciting a "right"
answer.

In the afternoon, approximately 45 minutes are set aside for the
youngsters in this second-grade class to plan and do an activity of their choice.
The Class is divided into interest areas. The mathematics area draws two boys
and three girls as they construct all kinds of things from building blocks,
tangrams, uniﬁx cubes, fraction kits, measurement devices, calculators, and
various collections of objects to count and arrange in this center. SOme
youngsters select mathematics activities on the computers; others are busy in
the writing center or the art center. Four youngsters are playing games such as
Hi-Ho!_ Cherry-0 and Concentration, which provide ample opportunities for
quanﬁﬁcaﬂon.

The teacher moves about the room and converses with individual groups
of Children. She may ask the youngsters focusing questions but spends much

more time listening to what they want to tell her about their projects. She

66

occasionally takes notes if she observes that a particular youngster is
demonstrating an understanding of an important academic concept, such as the
logic of a particular measuring device to determine the relationship of size. As
she moves about, the teacher also notes evidence of students’ social growth,
such as cooperation in the group.

The ﬁnal part of the day involves writing and sharing a review of the
activity period. Once again, as the teacher passes out folders, messages, and
notes at the end of the day, she uses every possible opportunity to quantify the
tasks Of dismissal. Mathematics and logical thinking are integrated throughout

the curriculum and throughout the school day.

W

The day in this traditional second-grade classroom begins with greetings
and record keeping. Afterthe teacher and students greet each other, theteacher
takes attendance, counts the hands raised for hot lunch and cold lunch, and asks
the children to turn in their permission slips for an upcoming ﬁeld trip. Afterthese
initial activities are completed, the teacher gives the children directions for
ﬁnishing their morning seatwork, which consists of ﬁve sheets of paper that are
already on their desks. The teacher explains to the students that they have one
page of handwriting practice; one letter-sounds phonics drill page; one cut, paste,
and color mathematics "match the set" worksheet; one page of blank, lined paper

on which to write a story; and one sheet of 9" x 12" art paper on which to draw

67

a picture to accompany their story. The teacher reminds the Children that this is
their morning work, which they are to complete while she meets with each of the
four reading groups.

Each reading group comes to the teacher when she calls them. The
teacher asks that they bring their books, workbooks, and pencils with them. After
their reading group, the students return to their desks with an additional page of
reading-related seatwork. These reading groups take most of the morning. The
teacher might do some whole-group instruction on compound words or sentence
structure to complete the language arts block of instruction before lunch.

. After lunch, the teacher announces that it is time for mathematics.
Today’s task for the mathematics block of instruction is to learn the number-fact
family of 8 + 6 = 14. The teacher Chooses to use 14 red blocks at her teaching
station to demonstrate the number set of 8 + 6 = 14, 6 + 8 = 14, 14 - 6 = 8, and
14 - 8 = 6. After demonstrating the number set with the blocks. she has the
Children recite the equations written on the Chalkboard. Initially, the teacher has
the whole class recite as she points to the chalkboard. Eventually, she calls on
youngsters who seem hesitant to recite alone to check for mastery. As a game,
she now begins erasing parts of the equations and asks the children (ﬁrst as a
large group, then individually) to ﬁll in the blank spaces. She then assigns a
worksheet with this number family, which has a total of 14 teddy bears on it,
illustrating the number family of 6 + 8 = 14. Several "complete the blank"

equations are on this worksheet. An additional worksheet containing 15 drill-and-

68

practice problems, including the new number family students have just learned,
is assigned. The students are allowed to use the number line that is taped to
their desks if they need to do so.

Students take their papers to the teacher when they are ﬁnished, and the
teacher gives gold stars to students who have provided the correct answers on
their papers. After most of the children have completed this task, the teacher
announces that it is time for recess. The children who have not successfully
completed their worksheets must remain in the room so that they can ﬁnish.
After recess, the teacher may conduct a science or social studies activity if there

is time.

Chametﬁummam

The methodology used in conducting the study was explained in this
Chapter. The problem and purposes of the study were set forth. Next, the
selection Of the sample was described, followed by a discussion of the
instruments used in the study. Data-collection and data-analysis procedures
used in the quantitative and qualitative portions of the study were explained, as
was the nontraditional method of mathematics instruction with which the study
was concerned. Finally, typical days in both nontraditional and traditional
classrooms were presented. Results of the quantitative and qualitative data

analyses are presented in Chapter IV.

CHAPTER IV

RESULTS OF THE DATA ANALYSES

IniLQdumI'Qn

The researcher’s primary purpose in this study was to investigate and
compare the achievement in mathematics ofAfro-American and Euro-American
females in grades 3, 4, and 5 in a particular school setting before and after the
application of nontraditional mathematics instruction. A second purpose was to
investigate selected subjects’ attitudes and perceptions toward mathematics.

In this Chapter, the data gathered in an attempt to fulﬁll the study purposes
are presented. Results of the quantitative analysis are presented ﬁrst; included
are the outcomes of the hypothesis testing. Next, the qualitative data are
presented. Each interview question is stated, followed by the responses to that

question.

I3 I! [lh :1 III a I

The mathematics concepts and applications scores Of the sample
members two years before and two years after the application of the
nontraditional mathematics instruction were analyzed forthis portion of the study.

An F-test was used to compare the two subject groups’ (Afro-American and

69

70

Euro-American) mean achievement scores before and afterthe application ofthe
nontraditional instruction. An analysis was carried out to determine whether
there were signiﬁCant differences in the mean mathematics scores of the total
sample before and after the application of the nontraditional method of
mathematics instruction. Between-group variability in mean scores also was
analyzed using race as a factor. In this section, each null hypothesis is restated.

followed by the results for that hypothesis.

NulLtIvDothesisl
There will be no signiﬁcant difference in the mean mathematics scores of
the total sample before and after the application Of a nontraditional
method of mathematics instruction.
The nontraditional mathematics instruction at East Elementary School

began in 1992. The mean CAT mathematics scores of the sample over the four-

year time period covered in this study are shown in Table 4.1.

Table 4.1: Mean CAT mathematics scores of the total sample, 1990 through

 

 

 

 

 

1993 (N = 47).
Year Test Mean Std. Dev. Std. Error
1990 44.468 . 19.090 2.785
1991 50.234 26.604 3.489
1992 51.277 20.348 2.968
1993 53.255 16.857 2.459

 

 

 

 

 

 

A

mathem:

 

 

study (1%
overall n
F=278
instructl
sample
appeare
the nor
Dreced
Sgnﬁk

Fudhe

71

A signiﬁcant difference was found between the total sample's mean CAT
mathematics score for the ﬁrst year of the study (1990) and the last year of the
study (1993). When the overall mean of 44.468 in 1990 was compared with the
overall mean of 53.255 in 1993, the main effect was found to be signiﬁcant, with
F = 2.78 and p = .04. These data suggest that the nontraditional mathematics
instruction had a slight overall positive effect on the achievement level of the total
sample over the four-year period under investigation. The positive effect
appeared in the fourth year examined, or two years after the initial application of
the nontraditional method of mathematics instruction. On the basis of the
preceding data, Null Hypothesis 1 was not accepted. A slight but statistically
signiﬁcant difference was found between Year 1990 and Year 1993 ofthe study.

Further discussion of these data can be found in Chapter V.

C-

I” 0‘1 \I g! u?! C)

‘I (II 1_
'3- '5. I.

“I
O 0

CI ’3: t“ L:-

 

1990 1991 1992 I993

Figure 4.1: Mean CAT mathematics scores of the total sample,
1990 through 1993.

72 '
W

There will be no difference between the Afro-American and Euro-

American subjects in terms of their mean mathematics scores over the

four-year period of study.

The mean mathematics achievement of the two. racial groups was
compared for the four-year period under investigation. The mean CAT
mathematics score for the Afro-American females over the four-year period was
38.789, whereas the mean for Eu ro-American females over the same period was
57.286, as shown in Table 4.2. The difference between the two subject groups
over the four-year period was signiﬁcant, with F = 20.705 and p = .001. The very
low p-value reﬂects a Signiﬁcant main effect for race over the four-year period.
Afro-American females scored signiﬁcantly lower than Euro-American females

both before and after the application ofthe nontraditional method of mathematics

instruction (see Figure 4.2).

Table 4.2: Mean CAT mathematics scores of the Afro-American and Euro-
American subjects, 1990 through 1993 (N = 47).

 

Subject Group 1990 1991 1992 1993
Afro—American 37.316 35.000 41.316 41.526
Euro-American 49. 321 60.571 58.036 61.214

 

 

 

 

 

 

 

 

 

73

100

 

90

 

'CIO

 

7O

 

 

I Afro-American

E Euro—American

 

 

 

 

 

1990 1991 1992 1993

Figure 4.2: Mean CAT mathematics scores of the Afro-American and
Euro-American subjects, 1990 through 1993.

Further analysis was conducted to determine whether there was an
interaction between the two subject groups. A high p-value of .686 indicated that
there was little interaction between groups. That is, neither the Afro-Americans
nor the Euro-Americans experienced signiﬁcant gains as a result of the
application of the nontraditional method of mathematics instruction. The clarity
of the noninteraction and the ﬂatness of the growth of both groups from 1990
through 1993 are shown in Figure 4.3. There was little variance in the mean
CAT mathematics scores Of these two groups. The gap between the two groups

remained about the same over the four-year period.

74

100
90
80
7O
60
SO
40
30
20

IO

 

I Afro-Am
E Euro-Am

 

 

 

1991 1992 1993

Figure 4.3: Mean CAT mathematics scores of the two subject groups
over the four-year period, 1990 through 1993.

The scores of the Afro-American females dipped from a mean of 37.316
in 1990 to a mean of 35.000 in 1991. However, their mean score rose to 41.316
after the ﬁrst year of the nontraditional mathematics instruction. The second year
of that instruction, the Afro-Americans’ score remained about the same (41.526).

In contrast, the Euro-American females in this study started in 1990 with
a mean score of 49.321; their mean score rose to 60.571 in 1991 (both years
were before the application of the nontraditional mathematics instruction). In
1992, the mean of this group dipped slightly, to 58.036, with the inception of the
nontraditional mathematics instruction, and moved back up to 61.214 in 1993,

following the second year of that instruction. Again, little difference in

achi

Inst:

Asl

IIOI

ml

75

achievement was apparent as a result of the nontraditional mathematics
instruction.

On the basis of the preceding data, Null Hypothesis 2 was not accepted.
A signiﬁcant difference was found between the two racial groups with regard to
their mean scores on mathematics concepts and applications. However, no
signiﬁcant interaction was found between'the two groups with regard to
mathematics achievement. Based on the mean scores from this sample, the
nontraditional method of mathematics instruction did not appear to have had
much inﬂuence on the achievement of either group. Neither group seemed to be
signiﬁcantly inﬂuenced, either positively or negatively, by the nontraditional

mathematics instruction.

t f h ' tiv I

In the qualitative portion of this study, 12 subjects (six Afro-Americans and
six Euro-Americans) from the study sample were interviewed regarding their
attitudes and perceptions about mathematics. The following research questions
guided the development of the interview schedule:

1. What are the perceptions and attitudes of young Afro-American
females toward mathematics achievement?

2. What are the perceptions and attitudes of young Euro-American
females toward mathematics achievement?

3. I: What perceptions and attitudes toward mathematics achievement

do young Afro- and Euro-American females share?

Chit

char

their

1686

DOII

SCO

hat

SOT

bet

an

int

II

76

4. What additional variables, perceptions, and attitudes revealed in
the interviews affect, either positively or negatively, the mathematics
achievement of young Afro-American females?

The 12 subjects selected for the interviews shared the common
Characteristic of a substantial discrepancy between their total reading scores and
their mathematics concepts and application scores on the CAT. In the
researcher’s view, low-average to high-average reading scores indicated a
normal ability to learn. A discrepancy in a subject’s reading and mathematics
scores was an indication that the individual had a normal ability to learn but was
having less success with mathematics than with reading. It should be noted that
some of the Afro- and Euro-American females with the greatest discrepancies
between their reading and mathematics scores did not return permission slips
and seemed uninterested in participating in the interviews.

The mean reading score of the 12 interviewees was 47.583. The
interviewees’ mean score on the mathematics concepts and applications portion
of the CAT was 34.916. The mean reading Scores for the Afro-American and
Euro-American interviewees were 48.0 and 46.5, respectively. The mean
mathematics concepts and applications scores for the Afro-American and Euro-
American interviewees were 36.0 and 33.0, respectively (see Table 4.3).

The subjects’ responses to the interview questions are presented on the

following pages. In reporting the ﬁndings, the researcher made every effort to

refralr

young

Taole

 

IFI

00.

abs
nol

COI

77

refrain from interpreting the subjects’ responses. Rather, she Chose to let the

youngsters’ comments speak for themselves.

Table 4.3: Mean mathematics and reading scores of the 12 interviewees.

 

 

 

 

 

 

 

 

Mean Score
Subject GTOUP Mathematics Concepts Reading
and Applications
Afro-Americans ' 36.0 48.0
Euro-Americans 33.0 46.5
Total group 34.916 47.583

 

 

Do you have brothers and sisters?

All 12 of the interviewees had siblings. The average number of siblings
for all of the respondents was three. However, several of the Euro-American
females indicated their siblings lived away from home with grandparents, an
absent father, or a boyfriend. One Euro-American female indicated that she did
not know her two step-siblings because they lived in Texas with their dad. In
contrast, the Afro-American females who reported having siblings or step-siblings
mentioned no estrangements between siblings or relatives.

The most Obvious difference between the Afro-American and Euro-
American subjects wasthat the Afro-Americans had extended families who were
easily accessible, with little estrangement, whereas several of the Euro-

Americans reported unique and estranged family relationships. When the Euro-

78
American females spoke of the siblings and relatives from whom they were
separated, it was with anxiety.
Do you belong to any clubs such as Gil!
Scouts, Campﬁre Gids, church, etc. ?

All six of the Afro-American subjects indicated a range of activities, from
cheerleading to Choir to Girl Scouts. Most frequently, however, they mentioned
a strong church afﬁliation. Five of the six Afro-American interviewees reported
going to Church every Sunday with theirfamilies. In contrast, the Euro-American
females had signiﬁcantly fewer outside activities. Only one indicated
participating in Girl Scouts and baseball; just three reported any Church afﬁliation.

There was an obvious difference between the Afro-American and Euro-
American subjects in the use of support systems outside their families. The Afro-
Americans were involved in many more activities that placed them in extended
social support systems than were the Euro-Americans. Almost all of the Afro-
American interviewees reported a strong Church afﬁliation. The Euro-American

interviewees seemed to be socially isolated.

When you grow up, who would you like to be?

Responses to this question were typical of youngsters in this age group. ,
The Euro-American females reported that they would like to be a business
person, teacher, police ofﬁcer, singer, secretary, and preschool teacher. The
Afro-American females said they would like to be a baby nurse, teacher,

veterinarian, lawyer, artist, and Spanish teacher.

79

Some children’s responses to this question seemed to relate to the
present occupations or interests of their parents, such as "business person and
work where people come and get video tapes and tan like Morn does" or "a
police Ofﬁcer" because the subject’s grandparents frequently watched Rescue
911. One Euro-American female indicated an interest in being a bartender like
her mom.

The reasons the subjects gave for their chOsen occupations were typical
Of elementary schoolchildren. For example, one youngster wanted to be "a baby
nurse because I like kids and I like to be around little babies because they are
cute." The Afro-American female who wanted to be a veterinarian said that if she
could not be a veterinarian, she would be a doctor; if she could not be a doctor,
she would be a pediatrician. However, each subject was able to think of
something she would like to do when she grows up.

What do you think girls are supposed
to do when they grow up?

In responding to this question, all six of the Afro-American interviewees
reﬂected a practicality that seemed to be summed up in one subject’s answer:
"clean the house, grocery shop, shop for the kids-—can be nurses, lawyers, car
dealer, cook, or a waitress. That’s not a good job, but they can do it if they want
to." The EuroéAmerican females’ responses were similar but leSs professionally
oriented; they mentioned being cheerleaders, working in a day care center, and

being a waitress, cook, singer, or dancer. One Euro-American female could only

80

think of being a model. Almost all of the other respondents offered multiple
answers to this question. One Euro-American female’s surprising response to
what girls are supposed to do when they grow up was "be on the street."

The next portion of the questionnaire contained questions designed to

elicit the subjects’ perceptions and attitudes about mathematics.

What is your most favorite subject in ”school?

Seven of the 12 interviewees indicated that spelling was their favorite
subject in school. Not one of the Afro-American females reported that
mathematics was her favorite subject, whereas three of the six Euro-American
females indicated that mathematics was their favorite subject in school.

The fact that none of the Afro-American females indicated mathematics
was their favorite subject, whereas three Euro-American females did so, is
especially interesting when one considers that the average mathematics score
Of the Afro-American interviewees was 36 (based on a percentile rank of 100,
which means that 64% of the other females in the United States who took the
CAT did better on the mathematics test) and that of the Euro-American
interviewees was 33. Neither score is strong, but those females who indicated
they liked math best did not score any higher than those who did not mention
math as their favorite subject.

The interviewees’ failure to Cite mathematics as a favorite subject is

consistent with the overall average mathematics score of this group of females,

81
which was 34.5-barely above the lowest quartile. Not only did they not like
mathematics, they also did not achieve well in the subject.
When you have #99 time at school,
what do you droose to do?

At ﬁrst glance, both the Afro-American and Euro-American females’ use
of free time'at school was similar. Afro-American females’ responses to this
question were as follows: "go out for recess," "jump rope," "play games," "draw,"
"play kickball or baseball," "go to the library," "read to younger kids," and "spell
words and ﬁnd other words." Euro-American females’ answers included the
following: "play with my friends," "write stories and play at different centers in the
room like math and language," "run outside on the playground," "color," "draw,"
"read," "enjoy WM," and "play games on the computer."

Upon closer examination, it seems that the activities Chosen by the Afro-
American females were more active and interactive in nature than those Chosen
by the Euro-American females. Many of the activities mentioned by the Euro-
American females were introspective and ones they could do alone, whereas the
Afro-American females seemed to Choose more social activities. One Euro-
American female mentioned that she liked to play games on the computer
because she did not have many friends.

Only one of the 12 interviewees even mentioned mathematics as
something she chose to do in her free time. This Euro-American youngster

mentioned mathematics in the context of going to one of the activity centers such

82

as mathematics and language in the claSsroom. It is Clear that these females
were not interested In selecting mathematics activities during free time.
Language arts, communication, art activities, and active play were favorites of

these interviewees.

How do you feel about doing math?

Responses to this question revealed some major differences between the
Afro-American and Euro-American females. Only one of the six Afro-American
females indicated that she felt good while doing mathematics. Five of the six
Euro-American females stated positive feelings about doing mathematics, such
as "[it’s] kinda fun," "it’s easy," "I like graphing," "I like math," and "it’s fun." The
one Euro-American female who expressed negative feelings poignantly stated:
"Sometimes I don’t understand it, and they just make us do it by ourselves. One
day I had homework, and there was this question I didn’t understand. It was a
hard one, and my mom didn’t even know it. ldon’t like math because I get mixed
up (like the times table) and I do really bad-«I just don’t like math."

The ﬁve Afro-American females who expressed negative feelings about
doing mathematics used such phrases as "weird," "hard," "confusing," "it’s OK
but I sometimes don’t like it," "sometimes hard to understand," and "don’t like
plus, minus, and fractions." The prevailing negative feeling for the one Euro-
American female and the ﬁve Afro-American females was that math is "hard" or

"confusing."

83

The responses tothis question expressed much more feeling and intensity
than did those to the earlier question about the interviewees” favorite subjects or
even their choices of activities during free time. It would seem that the females
in this study had strong feelings about doing mathematics.

It is surprising that ﬁve of the six Afro-American females had negative
feelings about doing mathematics, even though their mean mathematics score
was slightly higher than that of the Euro-American females, who indicated
positive feelings about doing mathematics. Thus, it can be seen that the
interviewees’ feelings about mathematics did not necessarily relate to their

mathematics achievement.

Are you good at math?

The Afro-American females responded tentatively to this question,
expressing such qualiﬁed responses as: "I’m OK," "a little bit," "kind of, a little,"
"I don’t know," "sometimes, it depends on the material," and "l have trouble with
division and fractions." In contrast, ﬁve of the six Euro—American females gave
positive responses to this question: "yes," "a little,” "good because it is easy,"
"yes, get good grades," and "good in math."

However, whatever their perceptions, these young females were not
achieving well in mathematics. The mean mathematics score forthe total group
of interviewees was 34.9, which means that 65.1% of all the other females in

America who took the CAT math test achieved higher scores. The Afro-

 

84
American interviewees seemed to have a more realistic assessment than did the
Euro-Americans about their mathematics achievement.
Do you think mam is important?
Why or whynot?

All 12 of the interviewees stated that mathematics was important. Nine
of the 12 females related the importance of mathematics to counting money for
shopping or for working in a supermarket. Three of the 12 indicated that
mathematics is important for advancement in school and a good education. One
stated, "It’s hard to live without numbers." The responses to this question were
related to those noted earlier concerning what these youngsters thought girls are
supposed to do when they grow up. At this stage in their lives, the subjects
imagined themselves as money handlers In the supermarket, which is related to
their perception of themselves as mothers and caregivers when they grow up
and also reﬂects how they saw their mothers using mathematics in their lives.

The one subject’s observation that it is hard to live without numbers was
the most abstract response and probably indicated that her developmental
maturity was higher than that of the other interviewees. The other 11 females
still based their opinions on what they experienced and knew--women shop in

order to care for their families.

85

Does your mother think math is important?
Why or why not?

The responses to this question were all positive. Each interviewee
believed that her mother thought mathematics was important. Several of the
answers suggested that the subjects’ mothers had talked with them about being
better in mathematics and the importance of mathematics in getting a good
education. About half of the~ girls related their mothers’ attitudes about the
importance of mathematics to their jobs, such as working in a store, using
mathematics to count cards, and paying the family bills.

There was a strong sentiment among the interviewees that their mothers
valued mathematics and encouraged them to work hard to understand
mathematics. One youngster reported that she got grounded if she did not do
her mathematics homework.

Can you think of an y ways math could
be useﬁrl to you?

The responses to this question were consistent with those to earlier
questions. The interviewees saw the primary use of mathematics as aiding them
in shopping and paying bills. This response seems logical in that counting
money at the check-out counter was a common experience for all of the
youngsters. The association with money was a strong rationale for the
usefulness of mathematics. One interviewee did respond, however, that she
could not think of any ways that mathematics could be useful to her. It is clear

that these young females had a limited perception of how mathematics is used

86
in everyday situations, as well as potential vocations that might require
knowledge of mathematics forsuccess.
Which would you rather do, math worksheets
or count and build things?

Of the 12 interviewees, only two Afro-Americans reported that they would
rather count and build things than do worksheets. The other ten subjects
indicated that they would rather do worksheets. Several said that counting and
building things is for little kids and that "one should do stuff that’s right for your
own age." One respondent said that worksheets are better because they are
quicker.

The strong preference of these subjects fer the worksheets was probably
related to their comfort with "doing'the familiar” and not taking risks. It also
seemed to support current learning theories, which suggest that many females
learn in a linear fashion. That is, they memorize the model or conﬁguration and
then apply the model when it reappears. This process of learning works well as
long as they are doing computation, but when the need for abstract problem
solving arises, the linear model will not work for them because they have no
previous models or conﬁgurations at hand to use in solving a new problem.

This question was posed to elicit information from the subjects about their
learning modalities. Their responses seemed to support Tobias’s (1978) theory
that many female learners prefer to take no risks in learning mathematics.

Handling and manipulating the counters or building units was not familiar practice

87
to most ofthe subjects in this study. ltwould seem from their responses that this
group of females preferred to practice mathematics in the more traditional
manner.
The last group of items comprised open-ended statements about
fﬁ athematics, which were adapted from Davis’s questionnaire. These statements

We re designed to aid in analyzing people’s feelings toward mathematics.

Men I get bad grades in math. . . .

As expected, the responses to this open-ended statement were negative.
A“ of the participants indicated sadness, disappointment, upset feelings, anger,
and so on, when they received bad grades in mathematics. However, ﬁve of the
Six Afro-American females indicated that theyjust tried harder and did their work
over in order to pass. Two of the Euro-American females indicated that they

would try again and do their work over.
One respondent stated that she felt like crying when she received bad
grades in mathematics. All responses to this question were related to feelings
of lower self-esteem. All of the responses reﬂected a high intensity of negative

feeling.

Matt) is. . . .
Four of the Afro-American students responded positively to this statement
With such comments as "a little easy," "fun and exciting," "one of my favorite

subjects," "very fun," and "fun." One Afro-American interviewee simply said that

88

math was "hard," and the other responded similarly, saying that "math is very
hard, but I try to learn." The Euro-American females’ responses were more
varied. One indicated that "math is terrible," whereas three said that
mathematics is important because it will be needed in the future. Two Euro-

American interviewees indicated that "math is fun."

Doing math makes me feel. . . .

The responses to this open-ended statement were varied. The Euro-
American females said that doing mathematics made them feel "important,"
"confused," "happy," "good," "depends on the grade-happy if the grade is good,"
and "like I am helping someone in the future." There was no apparent
consistency in these responses. However, the one Euro-American female who
reported feeling confused about doing mathematics consistently communicated
these kinds of feelings throughout the interview. Although she had reported
confusion and difﬁculty with mathematics, she identiﬁed mathematics as her
favorite subject.

Two of the Afro-American females reported negative feelings such as
"dizzy, like . . . confused" and "kind of mad because I don’t understand." The
other four reported more positive feelings, such as "pretty happy," "real good
about myself" "like I am just getting higher and higher in math and my grades are

getting better," and "kind of important because I know I am succeeding."

89
When I drink about mam. . . .

All six Afro-American interviewees indicated an attitude of resignation in
their responses to this open-ended statement. One of them responded by stating
that she cringed when she thought about mathematics because she had to do
it and there might be something else she would rather be doing. Another Afro-
American stated that when she thought about mathematics she did not like it
because when she did not know how to do it, the teacher just said, "'Read the
directions yourself,’ and then doesn’t do Stuff with us." The other four Afro-
American interviewees Simply said something to the effect that they got the piece
of paper and pencil, started to work, and tried to understand. One said that she
thought she knew what she was doing in addition, subtraction, multiplication, and
division.

Five of the six Euro-American females responded with positive remarks
like "I feel good about myself," "I think about numbers and stuff like that," "I feel
happy because I am good at it," "I try my best," and "I am ready to do it." The
remaining Euro-American stated that she initially said to herself, "Aw, man," but
then she got her things out and the mathematics became fun.

The attitudes of the Afro-American and Euro-American females, as
reﬂected in their responses to this open-ended statement, were quite different.
The Afro-American females stated clearly thatthey did not like mathematics, that

they were not good in mathematics, but that they tried hard to understand

90
mathematics anddo the work. In contrast, this group of Euro-American females

seemed to have more positive thoughts about doing mathematics.

My most ﬁrn at math was when. . . .

Five of the 12 respondents indicated that their most fun at mathematics
occurred when the learning activity was group oriented and included
manipulatives such as the following: "When we did adding fractions last year
using M 8 M’s, and when we got one right, we ate an M 8 M." "When we did the
money thing using play money." "When we worked with big blocks and little ones
[uniﬁx cubes], and they taught us how those blocks could help on the worksheets
and we would know the answer easier." "When we did graphs and located the
points." "Last year when we got to measure stuff." The remaining seven
interviewees responded in a more traditional way to referring to worksheets,
learning times tables, adding and subtracting, and checking other people’s work.
The animation expressed in the descriptions of the ﬁve hands-on activities

indicated those ﬁve subjects’ high interest in interactive mathematics exercises.

:[II 12' Illl'

Although the interviews were tape recorded, the interviewers were asked
to write a brief description of the body language and perceived comfort level of
the young females who were being interviewed. Only one of the six Afro-
American subjects communicated some anxiety during the interview. She was

described as being nervous and ﬁdgeting throughout the interview. The other

' 91
ﬁve Afro-Americans, whether they were being interviewed by the Afro—American
orthe Euro-American student, were described as being excited, smiling, relaxed,
and having good eye contact.

Three Of the Euro-American females were described as smiling and as
having good eye contact, bodies erect, and speaking freely. The other three
seemed embarrassed, anxious, soft-spoken, and hesitant, whether the
interviewer was an Afro-American or a Euro-American. One Euro-American
female initially spoke with her hands over her mouth.

The two questions that seemed to be somewhat difﬁcult for several of the
respondents were:

1. What do you think girls are supposed to do when they grow up?

2. Can you think of any ways that math would be useful to you?
For these two questions, the interviewers used follow-up queries to encourage
the youngsters to think abOut and respond to the question. Apparently, the
abstract nature of the questions was confusing to these young subjects. A

summary of the interview responses is contained in Table 4.4.

t r mm
Results ofthe statistical analysis ofthe mathematics achievement scores
of the subjects before and after the application of a nontraditional mathematics
curriculum were presented in the ﬁrst part of this Chapter. The ﬁrst hypothesis
stated that there would be no signiﬁcant difference In the mean mathematics

scores of the total sample before and after the application of a nontraditional

92

Table 4.4: Summary of interview responses.

 

Question .

Afro—American Females

Euro-American Females

 

Do you have any
brothers and sisters?

Yes. We’fight a lot; Mom has to
break it up.

Yes; older. Get along with sister
that is 29.

Two step-brothers and two step-
sisters.

Yes.

Four. She's the Oldest and likes
being the oldest.

Three siblings; she's the oldest.
Has to clean up after others.

Yes, one brother. He’s a pain,
but they get along sometimes.

Two brothers. All live together
and get along.

One brother and one sister.
Doesn't know them; they live in
Texas with Dad. She lives with
grandparents.

Four older siblings. She is
youngest and is spoiled.

One brother and four sisters; gets
along with all of them.

Middle child. One is adopted;
two were by a boyfriend and now
live out of state with him.

 

Do you belong to any
clubs such as Girl
Scouts, Campfire
Girls, church, etc.?

 

 

Cheerleading; choir; goes to
church every Sunday.

Goes to church every Sunday.

Girl Scouts; cheerleading; goes to
church every Sunday with Mom
and Dad.

Pentecostal church; likes it; it's
fun.

Missionary outreach at Grace
Temple. Learns about the Bible
and sings in the choir.

 

Girl SCouts; played baseball in
summer; goes to church every
Sunday.

No.

Sings in church choir and goes to

' church every Sunday.

NO.

I No; has been to church once.

No.

 

 

93

Table 4.4: Continued.

 

Question

Afro-American Females

Euro-American Females

 

be?

When you grow up.
who would you like to

A baby nurse because I like kids
and I like to be around little babies
because they are cute and I like to
play with kids.

A teacher. I want to be like a
student teacher and go to college.

Vet. If that doesn’t work out, I
want to be a doctor. If that
doesn’t work out, then I’ll be a
pediatrician.

Lawyer; ﬁnds it fun; likes trials.
Artist.

Teacher of language; Spanish or
spelling.

A businessperson and work
where people come and get
videotapes and tan (like Mom).
A teacher.

A police Ofﬁcer because her
grandparents watch Bescue 91]
a lot.

Singer.

Secretary or teacher.

Preschool teacher.

 

up?

 

What do you think
girls are supposed to
do when they grow

 

Clean the house, grocery shop,
shop for the kids. Can be nurses,
lawyers, car dealer, cook, or
waitress. That’s not a good job,
but they can do it if they want to.

Be a maid.

I Whatever they want to do as long

as they believe In it. They can do
things that boys do. It doesn't
necessarily have to be girl things.

Go to college; become a nurse,
teacher, doctor.

Model, movie star, or just
anything.

Lawyer, teacher, policeman,
principal.

 

Cheerleading, play sports and
tennis. Women should have a
good job like... (couldn’t think of

anything).
Artist, doctor, teacher, coach.

I don't know...a model.

Work in a day care, waitress,
cook singer, dancer.

Anything--teacher, actress,
secretary, principal.

Teachers, moms, "be on the
street."

 

 

94

Table 4.4: Continued.

 

 

Spelling.

Spelling.

F
Question Afro-American Females Euro-American Females
What Is your most Spelling, language, and a little bit Math.
favorite subject in ' Of math.
school? Spelling.
Spelling.
‘ Math.
Science. E
' Math.
Reading, mysteries. ‘
' l Spelling.

Math and spelling.

 

What is your least
favorite subject in
school?

Social studies and science.
Math.

Spelling.

English and science-boring!
Science.

Math--but get lowest grades!

Language.
Math.
Language.
Social studies.
Reading.

Social studies and reading.

 

When you have free
time at school, what
do you choose to do?

 

 

Go out for recess, jump rope. In
classroom, I play games or draw.

Play games.

Play games such as kickball or
baseball.

I Draw; go to the library; read to

younger kids.

Play games (Life, 4 Corners, and
Up).

Likes to spell words and ﬁnd other

words.

 

 

Play with my friends.

Write stories and play at different
centers like math and language.

Run outside on the playground.
Color, draw, or read.

Draw, read World Book

, Anthglggy.

Play games on the computer (not
many friends).

 

 

95

Table 4.4: Continued.

 

Question

Afro—American Females

Euro-American Females

 

How do you feel about

I like most of it, like times tables--

It's kinda fun.

 

 

 

I don't know. I get A's and 8's on
tests.

Sometimes; it depends on the
material. I don’t really have to
work hard at it.

l have trouble with division and
fractions. I know my times table
beuen

 

doing math? that's the best in math. I know all
my times tables and some parts of Sometimes I don’t understand it,
dividing and rounding. Do not like and they just make us do it by
plus or minus or fractions. ourselves. One day I had
homework, and there was this
Weird, hard. question that I didn't understand.
It was a hard one--even my mom
l feel good. didn't know it. I don’t like math
because I get mixed up (like the
Sometimes it is confusing (times times table) and I do really bad. I
tables). just don’t like math.
It's OK, but I sometimes don't like It's easy.
it.
It's fun because it is easy.
Don't like it at all. Sometimes it’s
hard to understand. , I like graphing.
I like math but don't always do
good in it.
Are you good at I'm OK. Yes.
math?
A little bit. A little.
Kind of, a little. Yes.

Good, because it is easy.
Yes, I get good grades.

I feel like I am good at math.

 

 

96

Table 4.4: Continued.

 

Question

Afro-American Females

Euro-American Females

 

Do you think math is
important? Why or
why not?

Yes. math and reading are
important because when you get a
job, you have to know how to
read, spell, and add.

Yes.

Yes, because if you grow up and
work at a grocery store or
something like that, you have to
know how to give change to
people.

Yes. Shopping, understanding
how to spend money. Learn a lot
from math.

Yes; when you grow up. you are
going to need to know how to
count money.

Yes, because you need to know
how to count money.

Probably. Might help you get a
good education.

You need to know math so you
can advance to the next grade.
Sometimes you’ll have homework
and you'll need to know the
answer to a math question.

Yes. It is hard to live without
numbers. You need numbers to
count things.

Yes, because you have to know
how to count money.

Yes, because you need it for
shopping.

Yes; if you get a job at a store,
you need to know how to count
change.

 

Does your mother
think math is
important?

 

 

Yes; she tells me lots of times that
I should know my math...that math
is good.

Yes, because if I don't get my
work done, she grounds me and
makes me stay in the house.

Yes, because she is always telling
me to work on my times tables.

Yes. Mom adds well. My sister
thinks so, too. Morn helps with
homework.

She doesn't really know, except
‘ her mom pays the bills.

Yes, because she knows hers and

, she wants her children to know

theirs.

 

 

 

I don't know; she probably uses it
in herjob.

My mom didn't ever get her math
done. She knows how to do it
now, but she says I need to know
everything about math so I can
get a good education.

I don't know about mothers...but
grandmother does because she
counts cards and she has to have

l
numbers.

Yes, so her daughter can get a
good education.

Yes; needs to use it for bills.

Yes, because she works at a
store.

 

 

 

97

Table 4.4: Continued.

 

Question

Afro-American Females

Euro-American Females

 

Can you think of any
ways that math would
be useful to you?

l can use it in the future and now--
especially when you get a job.

I can’t think of anything.

If I become a vet, customers will
have to pay when they get their
dog, and I'll have to give them the

change back.

Shopping; helping children with
math.

Bankers need math to count
money.

Paying bills.

Probably in a job you can use it.
If you have a swimming class or
something, you can count the
people who swim at one time or
another.

You have to know math because
some teachers don't check their
books and when the class has to
do it, you'll need to know the
answer to make sure they get a
good grade.

; It helps me to know how to do

subtraction, addition, and
division.

To buy clothes; to count money;
cost of instrument.

, Add bills, shopping with money.

As a mother, helping children
with their homework.

l
l
l
l

 

Which would you
rather do—math
worksheets or count
and build things?

 

 

Math worksheets because
counting and building is for little
kids. You should do worksheets.
You should do stuff for your age.

Math worksheets.

Count and build things because
counting is quicker and you can
do it.

Math worksheets.

Count and build things.

Math worksheets.

 

 

Math worksheets because you
get done faster.

Worksheets because they have
easy directions, and it's like
things that we've already done on
a sheet of paper.

Worksheets because they help
me better than counting and

. building.

Math worksheets.

Math worksheets.

Math worksheets.

 

 

98

Table 4.4: Continued.

 

Question

Afro-American Females

Euro-American Females

 

When I get bad grades
in math, l....

I get disappointed.
I do it over.
I have to do my papers over.

I feel like crying; I get mad; I get in
trouble; I try harder.

I feel low, and I do it over to do
beuer

I do the papers over so I can learn
in order to pass.

I’m sort of upset.

l'll study at home.

i I feel bad.

I feel sad and I try again for
practice.

I feel bad; I do them over.

I get mad, and my mother doesn’t
say anything.

 

When I think about
math, l....

 

 

When I ﬁnish it, I know all of it, but
I have to ask some questions.

I don't like it because when you
ﬁrst get to it, you don’t know it,
and then the teacher says, "Read
the directions yourself,“ and she
doesn't do stuff with us.

I listen to the directions and do
what the teacher says in math.

i think I know what I am doing.
Addition, subtraction, times, and
division are easy.

I cringe because I have to do it,
and there could be something else
I would rather be doing.

I get a piece of paper and a pencil
and try to understand.

 

I start saying, “Aw, man," and
then I get my stuff out and it's
fun.

Try my best.

I'm ready to do it.

I feel happy because I'm good at
it.

I feel good about myself.

I think about numbers and stuff
like that.

 

 

99

Table 4.4: Continued.

 

Question

Afro-American Females

Euro-American Females

 

My most fun at math
was when....

I did my times tables.

l was doing the money thing...like
counting money.

When we did adding fractions last
year. We used pieces of M & M's,
and every time we got one right,
we got the chance to eat one.

We checked other people’s
papers. Lots of math problems to
do.

I learned my times tables.

I did times worksheets.

I do problems--like addition and
subtraction--where I think of the
answer and write it down.

We worked with big blocks and
little blocks; they taught us how
those blocks could help on the
worksheets. We‘ll know the
answer easier.

I do adding and subtracting and
division.

When I started dividing in ﬁrst
grade.

Doing graphs and locating the
points.

Last year, when we got to
measure stuff.

 

Math is....

 

 

A little easy.
Hard.
Fun and exciting.

One of my favorite subjects. Very
fun.

Fun.

Very hard, but I try to learn.

 

Fun and I like it.
Terrible.
Fun.

Important, because you need it in
the future.

Important, because you need it in
the future.

Important, because you have to
know how to count before you
can do math.

 

 

1 00
Table 4.4: Continued.

 

v

 

Question Afro-American Females Euro-American Females
Doing math makes me Pretty happy. Important.
feel....
Dizzy like...confused. Confused.
Real good about myself. Happy.
Like I am just getting higher and Good because when I help my
higher in math--and my grades mom, I get my share.

are getting better.
I am helping someone in the
Kind of important because I know future.

I am succeeding.
f Depends on the grade; I am
Kind of mad because I don't happy if it is good.
understand.

 

 

 

 

 

method of mathematics instruction. A signiﬁcant difference was found between
the subjects' mathematics achievement the ﬁrst year of the study and their
achievement the last year of the study, with means of 44.468 in 1990 and 53.255
in 1993. The data suggest that the nontraditional mathematics instruction had
a slight, overall positive effect on the mathematics achievement of the total
sample of females in this study over the four-year period. Thus, Null Hypothesis
1 was not accepted.

Null Hypothesis 2 stated that there would be no difference between the
Afro-American and Euro-American subjects in terms of their mean mathematics
scores over the four-year period of study. A signiﬁcant main effect for race was
found over the four-year period. The Afro-American females (mean = 38.789)

scored signiﬁcantly lower than the Euro-American females (mean = 57.286)

101

throughout the four years of the study. The high p-value (.686) strongly suggests
that there was no interaction between the two subject groups. Neither the Afro-
American females nor the Euro-American females demonstrated signiﬁcant gains
following the application of the nontraditional method of mathematics instruction.
The gap in mathematics achievement between the two subject groups remained
the same throughout the four years of the study. Null Hypothesis 2 was not
accepted.

The qualitative portion of the study consisted of interviews with six Afro-
American and six Euro-American females from the sample to discover their
attitudes and perceptions about mathematics. The participants were volunteers
from the sample who had a large disparity between their reading and
mathematics achievement scores. The mean mathematics score for the Afro-
American interviewees was 36; the mean mathematics score for the Euro-
American interviewees was 33.

The perceptions and attitudes of the 12 interviewees were generally ones
of math avoidance and limited perception of the importance of mathematics in
their lives and the world of work. Only three interviewees indicated that
mathematics was a favored subject. The subjects interpreted the importance of
mathematics in terms of money management and family shopping. Not only did
these subjects not like mathematics, they did not achieve well in mathematics.

They expressed feelings of confusion about and inadequacy in mathematics.

102

The primary difference in attitudes between the Afro-American and Euro-
American females was the Afro-Americans’ acceptance that mathematics was
hard for them and their declaration that it was not a favored subject. They
indicated a resigned willingness to do the required work but little else. In
contrast, the Euro-Americans seemed to deny their low achievement by
maintaining that they were good at mathematics.

The results of this study demonstrated that the young Afro-American
females did not like mathematics, found it difﬁcult, and tended to avoid it. The
method of mathematics instruction did not seem to inﬂuence their mathematics
achievement appreciably. Hence, factors other than the instructional approach
inﬂuenced the underachievement of the Afro-American females in this study.

A summary of the study, major results and discussion, reﬂections, and

recommendations are presented in Chapter V.

CHAPTER V

SUMMARY, MAJOR RESULTS AND DISCUSSION,

REFLECTIONS, AND RECOMMENDATIONS

Summary

The researcher’s primary purpose in this study was to investigate and
compare the achievement in mathematics of Afro-American and Euro-American
females in grades 3, 4, and 5 in a particular school setting before and after the
application of nontraditional mathematics instruction. A second purpose was to
investigate selected subjects' attitudes and perceptions toward mathematics.

Nineteen Afro-American females participated in the quantitative portion of
the study; they represented approximately one-half of the total Afro-American
female population in grades 3, 4, and 5 at East Elementary School. Twenty-eight
Euro-American females were selected from the same elementary school
population, representing approximately one-third of the total number of Euro-
American females in the identiﬁed grades. The researcher used student CA-60
ﬁles to gather achievement test data on all of the subjects in the areas of
mathematics problem solving and applications, mathematics computations, and

reading comprehension. Other information such as birth date, grade, and ethnic

103

104

identity also was gathered from the ﬁles. The computer was used to select
subjects for the sample for whom complete test data were available.

In the qualitative portion of the study, 12 subjects from the sample were
interviewed to discover their attitudes and perceptions about mathematics that
might affect their mathematics achievement. Six Afro-American females and six
Euro-American females who had a signiﬁcant discrepancy between their reading
comprehension and mathematics problem-solving scores were selected forthe
interviews.

In this chapter, the major results of the study are discussed. The writer’s

reﬂections and recommendations for further research also are presented.

M ' r R It n Di l n

QuantitativeAnalvses

The results of the statistical analyses of the mathematics achievement
scores of the 47 subjects who participated in this study are discussed in this
section. A signiﬁcant difference was found between the subjects’ mean
mathematics achievement scores for the first year of the study (1990) and the
last year of the study (1993). These data suggest that the nontraditional
approach to mathematics instruction had a slight overall positive effect on the
achievement of the total sample over the four-year study period. The p-value of
.044, with a .05 conﬁdence level, indicated slight growth between the ﬁrst and

fourth years of the study. The signiﬁcance of the difference in achievement

105

between the ﬁrst and fourth years of the study must be viewed with caution
because the p-value of .044 is very close to the .05 conﬁdence level and a rise
in total group mathematics scores in 1991, which was before the change in
instructional method. This unanticipated rise in scores, as reﬂected in the 1991
data, is attributed primarily to the Euro-American females, as seen in Figure 4.2
(p. 73). This explanation of the increase in 1991 for Euro-American females
cannot be attributed to any factors within the scope of this study.

The data also indicated a signiﬁcant main effect for race overthefour-year
period. The Afro-American females scored signiﬁcantly lower than the Euro-
American females both before and after the application of the nontraditional
method of mathematics instruction. The gap between the groups remained
about the same throughout the four years of the study.

Further analysis indicated that there was no interaction between the two
subject groups (p = .686). Neither the Afro-American females nor the Euro-
American females made signiﬁcant gains with the application of the
nontraditional method of mathematics instruction in the two years following Its
inception. A signiﬁcant gap remained between the two subject groups in their
mathematics CAT scores throughout the four years of the study.

Disgussign. The researcher anticipated that signiﬁcant increases in
mathematics achievement would take place with the nontraditional mathematics
instruction for both subject groups but, more particularly, for Afro-American

females. That is, it was anticipated that Afro-American females’ mathematics

106

achievement would be signiﬁcantly higher with the change in instructional
method. It was also anticipated that there would be signiﬁcant interaction
between the two groups of subjects with the application of this instruction. The
anticipated increases in mathematics achievement using the innovative
mathematics instruction did not occur. By itself, a methodological solution to a
systemic problem did not bring about a change in achievement. This ﬁnding was
disappointing to the researcher, who, like many of her teaching colleagues, was
seeking a single solution to a complex problem.

The study revealed larger gaps in mathematics concepts and applications
scores between the Afro-American and Euro-American subjects than originally
were anticipated. These gaps in achievement remained about the same
throughout the study period, regardless of the instructional strategy that was
used. Hence, the researcher concluded that a combination of other complex
factors contributed to the underachievement of many of the Afro-American
females in this study, such as the adequacy of the test instrument, the
relationship of reading ability to mathematical problem solving, the small sample
size, and teacher behaviors in classrooms.

Although the CAT is nationally normed and is used throughout the United
States to measure achievement, the researcher questions its ability to measure
the results of a nontraditional approach to mathematics instruction. As noted in
Chapter II, scores on achievement tests are of little or no value in formulating

instructional programs for children; neither can they be used to measure a

107

program. Rather, they can only be used to compare one child’s performance to
that of another child. The researcher’s original assumption was that the subjects’
low mathematics achievement scores were a result of traditional mathematics
instruction may well have been erroneous. The overall objective for using the
nontraditional instructional approach was to develop students’ mathematical and
logical thinking ability. The focus of this new approach to instruction was on
constructing mathematical concepts and relationships within students. The CAT
does not seek to measure "process" or what may be happening in students’
minds.

An additional problem with the use of the CAT is that test takers need
adequate reading skills to understand the directions and information speciﬁc to
each task measured on the test. The reading scores of both groups in the
sample were comparable to their mathematics achievement scores. The
relationship between reading development and mathematics problem solving on

the CAT was strong, as shown below:

Reading Test Mathematics Problem-
Mean Solving Mean
Afro-Americans 37.3 38.8
Euro-Americans 56.7 57.3

Another‘variable that the researcher believes signiﬁcantly inﬂuenced the
results of this study was the small sample size. In setting up the original project,
the researcher had in mind a 2 x 3 format that would include a sample of 30 Afro-

American females and 30 Euro-American females studied one year before and

108

two years after the application of the nontraditional mathematics instruction.
Instead, she chose a 2 x 4 format (two subject groups two years before and two
years after the application of the nontraditional mathematics instruction)'as a
stronger format. Complete four-year test data were available on only 28 Euro-
American females and 19 Afro-American females. This sample size was too
small for the ﬁndings to be generalizable to a large segment of the school
population. In addition, choosing a 2 x 4 format eliminated the youngest females
in the school from the study because they began ﬁrst grade with the
nontraditional method of instruction.

Another confounding variable that might have affected the
underachievement of these Afro-American females was teacher behavior.
During theteachers’ extensive inservice education in the nontraditional approach
to mathematics instruction, only one hour was devoted to equity and gender
issues. This hour-long session focused on encouraging underrepresented and
underserved groups in the study of mathematics. It is possible that certain
teacher behaviors, as described in the literature review, reinforced the Afro-
American females’ social place or role in the classroom. No effort was made to
increase teachers’ awareness of the complex social and cultural interactions that
often take place in integrated classrooms. These interactions frequently give
strong cultural messages to Afro-American females not to deviate from their

ascribed social positions in school.

109

Daily observation of these females in the school environment indicated
that more risk-taking and higher-level learning in mathematics were taking place
for these females than was reﬂected by the test data. It was common to see
males and females working and building in the mathematics area of the
classroom and for both males and females to be busily interacting and
developing patterns with geoboards and attribute blocks in their free time. The
reduction of mathematics anxiety and the integration of mathematics problem
solving into the learning process was apparent.

The signiﬁcant underachievement in mathematics of the Afro-American
females in this study was evident. The ﬁndings indicated the high probability that
an innovative curriculum, as a single factor, will not produce academic growth.
Consideration of and response to the complex social factors present in the
classroom is essential in order to realize the full potential of an innovative
curriculum. Thus, the need to search for and respond to the combination of
variables that negatively affect the achievement of Afro-American females in the

classroom was clearly supported by the results of this study.

Q l! l' E l

Twelve subjects from the sample were interviewed to discover their
perceptions and attitudes about mathematics. The interviews were conducted
in an attempt to discover the perceptions and attitudes of young Afro-American
and Euro-American females toward mathematics achievement, as well as the

perceptions and attitudes they share. In addition, the interviews were conducted

110
to discover any additional variables, perceptions, and attitudes that might affect
the mathematics achievement of young Afro-American females.

1 W. The young Afro-
American females who were interviewed in this study were resigned and practical
about mathematics and their mathematics achievement. These six subjects did
not appear to have mathematics anxiety or to fear risking failure if they tried
mathematics activities. They simply saw no immediate use for mathematics,
except as shoppers, and tended only to ”do math" as required by the teacher.
The six Afro-American interviewees did not necessarily like mathematics but
diligently went to work when they were given mathematics assignments. Not one
of them identiﬁed mathematics as their favorite subject in school. Although the
average total reading achievement score for this group was 48, the average
mathematics concepts and applications score was 36. Not only did they not like
mathematics, these subjects did not achieve well in it. Not one of the Afro-
American females indicated that she would choose a mathematics activity at free
time.

When asked how they felt about doing mathematics, ﬁve of the six Afro-
American females used such words as "weird," "hard," and ”confusing." None
of them indicated that they thought they were good in mathematics. The six Afro-
American subjects indicated a cautious and practical response concerning to
whether they were good at mathematics. They seemed to have an appropriate

self-assessment of their mathematics competence.

111

All six of the Afro—American females indicated that mathematics was
important but tended to think of its immediate value in terms of shopping and
taking care of the family. All of these subjects indicated that their mothers valued
mathematics and encouraged them to work hard at it. Yet none of the Afro-
American females saw much value to mathematics other than as shoppers and
managers of the family money.

Two’of the six Afro-American females indicated that they would rather
learn mathematics in an active way, such as counting and building things.
However, the other four Afro-American subjects seemed to prefer the more
traditional form of learning mathematics-doing worksheets. Five of the six Afro-
American females stated that, when they received bad grades in mathematics,
they just tried harder and did their work over. Four of these six youngsters said
that, when thinking about doing mathematics, theyjust got out their work and did
it. This response of acceptance of responsibility was typical of the Afro-Am erican
subjects in this study. Two of them indicated frustration with the unhelpfulness
of the teacher when it came to doing mathematics.

In general, the Afro-American interviewees seemed animated, relaxed,
and candid during the interviews, whether they were being interviewed by a
Euro-American or an Afro-American. All six Afro-American females were very
involved in activities such as Girl Scouts, sports, and church. They all had
siblings and were part of extended families that includedaunts and grandmoth-

ers; they had positive relationships with each other and with their families.

‘0 u (I. v.1 0‘ o r _ o-‘ " .. ‘ 1. ‘ . Eventhough
their average achievement score on the CAT test was only 33, the Euro-
American interviewees indicated that they liked mathematics. They shared such
positive feelings about doing mathematics as "[it’s] kinda fun," "it’s easy,“ and "it’s
fun.” However, even though they indicated that they liked mathematics, these
subjects did not choose mathematics activities at free time.

One Euro-American subject consistently expressed frustration with the
teacher, who "just makes us do the work by ourselves." Her responses tended
to reﬂect negative feelings about herself while doing mathematics. Interestingly,
she indicated that mathematics was her favorite subject and that during free time
she might choose an activity in the mathematics center.

All six Euro-American females thought that mathematics was important
and said that their mothers considered mathematics to be important as well.
They saw the importance of mathematics in their lives primarily as money
handlers for their families while shopping. All six of the Euro-American
interviewees indicated that they would rather do mathematics worksheets than
count and build things. That is, they preferred passive, abstract computation to
active, concrete manipulation of objects in their learning. All six Euro-American
females expressed feeling disappointed or sad when they received bad grades
in mathematics.

When completing the open-ended statement “Math is. . . ﬁve of the six

Euro-American females said mathematics was important. They said their

5Ul

Ale

Si.

113

success in mathematics determined how doing mathematics made them feel.
Also, these subjects said they felt happy if their grade was good and terrible if
their grade was bad.

Five of the six Euro-American females completed the open-ended
statement "When I think about math. . . with such positive responses as "I feel
good about myself." However, although these subjects gave positive responses,
their mathematics achievement was not strOng. Three of them cited a
mathematics activity such as graphing, working with uniﬁx cu bes, and measuring
“stuff" as being the most fun in mathematics. This recollection was in contrast
to their previously stated preference for mathematics worksheets.

These six Euro-American females seemed to be anxious youngsters
whose attitudes did not reﬂect their achievement in mathematics. All six of them
were experiencing estrangement from siblings and family and lacked extended
support systems to help them with their loneliness and frustration. They did not
have accurate perceptions of their ability to achieve in mathematics.

I. ‘0. -. '1‘ . 0. 9‘!’ ‘QtOI o n.- iri n .no -ofr-A 1" -_1
females. The six Euro-American and six Afro-American females who were
interviewed shared some perceptions and attitudes about mathematics. Seven
of the 12 interviewees stated that spelling was their favorite subject in school.
Only three said that mathematics was a favorite subject. Eleven of the 12 said
they would not choose a mathematics activity at free time, but they mentioned

a range of other activities they would choose. These included speaking,

114
listening, reading, writing, socializing, doing art activities, and playing. Only one
su bject suggested that a mathematics-center activity would be her choice.

All 12 interviewees acknowledged that mathematics was important to their

futures. Nine of the 12 related the importance of mathematics to shopping and
ta king care of the family’s budget. The remaining three indicated that
mathematics was important for such reasons as advancing in school and
r eceiving good grades.

. All 12 interviewees said their mothers thought math was important. Each
gaid her mother and/or grandmother encouraged her to work hard in
“1 athematics.

Most of the interviewees indicated that they preferred worksheets over
Counting and building activities when doing mathematics. This preference for
\worksheets is consistent with the research, which has suggested that females

. earn how to do mathematics much like they learn to read--by memorizing a
I'nodel or conﬁguration and then applying the familiar model when it reappears.
The interviewees felt about the same when they received bad grades in
mathematics. All 12 of them had negative feelings when they were not
successful in mathematics. Their responses suggested that they had lower self-
esteem when they did not do well in mathematics.
A'QOIo. ._ to ‘ .0‘ ‘9 0| v.10. l't 0.- I. - i - 2|.” l"
W. Several variables that

were unique to the Afro-American interviewees became apparent in the interview

115

process. All sixof these subjects seemed to be happy, productive, practical, and
so cial youngsters. Throughout the interviews, they appeared to be self-conﬁdent
a re d maintained a balanced attitude about themselves and others.

A prevailing attitude among the Afro-American females was one of
practicality. They identiﬁed the primary purpose of mathematics as handling
m <3ney and shopping for the family. They saw little additional practical value to
m athematics other than money management. They did not choose a

m athematics activity during free time but rather chose interactive, social activities
with their friends. None of the Afro-American females named mathematics as a
favored subject. They did not particularly like mathematics and did not choose

mathematics activities for fun. While they were doing mathematics, if they ran
3 nto problems they just tried harder.

The Afro-American females in this study had a mean reading score of 48,
as compared to a mean mathematics score of 36. Their reading scores placed
them within the average range of reading achievement. Yet these same females

scored signiﬁcantly lower in mathematics. It is possible that the Afro-American
interviewees’ negative feelings about mathematics and their failure to see its
importance and practicality in their lives and in their future work were factors in
their underachievement.

The young Afro-American females had a close and supportive relationship

with their Afro-American peers and families. All six of these youngsters’

responses to the questions, body language, and communication strategies were

116

similar. Their cultural compatibility may have been reinforcing negative cultural
m essages about mathematics achievement. At no time did the Afro-American
st udents venture beyond the notion that mathematics is for counting money and
s h opping. They did not like mathematics, saw no need for mathematics beyond
self-maintenance, and had the notion that mathematics was something they did

because they had to, but other subjects were more fun.

ti n m ' r r i
An important ﬁnding of this study is that the Afro—American females in the
sample were continuing to underachieve in mathematics in spite of this method
of mathematics instruction. They scored signiﬁcantly lower than the Euro-
American females both before and after the application of the nontraditional
mathematics instruction. It is apparent that a single innovative instructional
strategy, such as the nontraditional mathematics approach, cannot reverse the
complex problems of many of these young females. The underachievement of
‘the Afro-American females in the sample was systemic, not methodological.
The results of this study should be shared with the administrators,
teachers, and parents atthis school. These groups must work togetherto reduce
the effects of a social-economic system that may be reﬂected in some public
schools, which too often assigns children to certain roles and then keeps them
there.
School districts must begin addressing equity, gender, and racial issues

that are a part of the social/cultural make-up of the community. Teachers should

117

have opportunities to expand their own perceptions and attitudes about these

issues. In addition, teachers should have the opportunity to assess their

be haviors and attitudes in a nonthreatening manner that provides insights and
a It ernative behaviors.

The underachievement of the Afro-American females in this study is a
re ality. However, to know these youngsters as individuals and to see and enjoy
th «eir warmth and intelligence is the strongest reason of all to continue to seek
W ays to make it possible for all students to learn and be successful.

Gilbert and Gay (1985) stated that the key to improving the success of
Afro-American children is to modify the means used to achieve the outcomes.
‘I‘his modiﬁcation must extend beyond adopting innovative methods of

Mathematics instruction to establishing such achievable objectives as unbiased
testing, sensitized teacher expectations and behaviors, working partnerships
between schools and parents, and a realistic appraisal of social and cultural

factors in the classrooms that are blocking the progress of some children. The

primary implication of this study is clear: Educators and communities must

develop the resources of all children.

W
The following recommendations are made for additional research:
1. A longitudinal study should be conducted, using the same sample and
statistical format from the present study, to substantiate whether the slight

increase in mathematics achievement from Year 1 to Year 4 would continue, thus

118

su pporting the belief that the nontraditional method of mathematics instruction
is in closer alignment with good educational pedagogy than is the traditional
m ethod.

2. A study should be carried out, comparing the mathematics

8 Chievement of the sample from the present study with that of a similar sample
w I10 have not been exposed to the nontraditional mathematics instruction.

3. The mathematics achievement scores of the Afro-American and

E uro-American females who participated in this study should be compared to the

mathematics achievement scores of the Afro-American and Euro-American

males in the same population, to discover whether Afro-American males are

underachieving to the same degree as Afro-American females.

4. Further research should be conducted on the variables affecting
1he overall achievement of Afro-American females. A combination of complex
factors, thus far elusive, is negatively affecting the achievement of Afro-American

females in the schooling process. Racial, ethnic, and gender issues must be
identiﬁed, understood, and addressed so that Afro-American females can reach
their full potential throughout the schooling process.

5. Teachers’ attitudes and perceptions about mathematics
achievement must be studied and recommendations made for staff development
that focuses on the recognition and reduction of behaviors that negatively
inﬂuence females’ achievement in mathematics. Teachers must be part of the

solution to underachievement in mathematics, not part of the problem.

119

6. Teacher preparation and training for an innovative curricular
approach such as a "nontraditional" method of mathematics instruction must be
st udied. "Teachers teaching teachers" at the end of a long school day with no
re l ease time for planning and with little follow-up and support for the individual
te acher may signiﬁcantly inﬂuence the effectiveness of the implementation

:3 I" 00683.

APPENDICES

 

 

APPENDIX A

PARENTAL PERMISSION LETTER

 

120

September 1, 1993

D ear Parents,

Elementary School has been invited to participate in a study
examining the math achievement of young females before and after the application
of our "hands on" math curriculum. We are going to carefully analyze the math
8 cores of about 60 randomly selected females in grades 3, 4, and 5, of whom will be
Afro-American and 30 will be Euro-American. In addition, we will be seeking to
i r1 terview approximately 12 of these young females in order to explore for additional
8 md contributing factors that affect math achievement. Again, we will divide our
focus between six Afro-American females and six Euro-American females. We will
b e particularly interested in females who have a signiﬁcant discrepancy between
their reading scores and their math scores. We want to know why some young
females at School are not succeeding in math and whether our new
math curriculum is making a difference in their success.

 

These interviews will be audio-taped for the beneﬁt of the researcher, but the
student’s name will not be revealed on thetape. If selected, approximately 1 and 1/2
hours of interview time will be required for the in-depth interviews. The interviews
will take place during the school day at . All participating students
will be assigned student numbers, and the results will remain anonymous.

We are writing to request your permission to allow your daughter to participate in this
project. However, participation in this project is completely voluntary. You can be
assured that any wish to not participate is entirely respected, and no penalty for
nonparticipation will be directed toward your child. Your child may withdraw from
this study at any time without penalty.

The study is being administered by students from Michigan State University and
Albion College. However, if you have any questions or concerns about your

daughter’s participation in this project, please call Ms. Janys Roberson (629-2435)
at the school.

Your daughter has been informed about the project. If you wish to give permission
for your daughter to participate inthis study, please sign the form below, detach, and
return to the school. Thank you for your consideration of this request.

Sincerely, ‘

Ms. Janys A. Roberson

 

has my permission to participate in the study on
math achievement of elementary-age females. l indicate my voluntary agreement
to participate in this project by completing and returning this form.

 

(Parent) (Date)

 

APPENDIX B

INTERVIEW SCHEDULE

 

10.

11.

12.

13.

14.

15.

16.

121

STRUCTURED INTERVIEW

Student number

 

Age

 

Birth date

 

Do you have brothers and sisters? Tell me about them.

Do you belong to any clubs such as Girl Scouts, Campﬁre Girls,
church, etc.?

When you grow up, who would you like to be?

What do you think girls are supposed to do when they grow up?
What is your most favorite subject in school?

What is your least favorite subject in school?

When you have free time at school, what do you choose to do?
How do you feel about doing math?

Are you good at math?

Do you think math is important? Why or why not?

Does your mother think math is important? Why or why not?
Can you think of any ways that math would be useful to you?
Which would you rather do?

math worksheets
count and build things

122

17. Finish these sentences in a few words:*

When I get bad grades in math

 

Math is

 

Doing math makes me feel

 

When it comes to math, I

 

My most fun at math was when

 

*Adapted from Sentence Completion Questionnaire by Sandra L. Davis, University
of Minnesota.

123

STRUCTURED INTERVIEW

 

 

QUESTION:

RESPONSE:

Afro-American/Afro-American Afro-American/Euro-American
AA/AA AA/EA

Euro-American/Afro-American Euro-American/Euro-American

EA/AA EA/ EA

REFERENCES

REFERENCES

Akbar, N. (1978). CultuLalexpLesslonmLﬂleamcarlAmedcancDIIMA position
paper). (ERIC Document Reproduction Service No. ED 179 633)

Ames, L. B., &Ames, F. L. (1967, June). Search for children showing academic

promise in a predominantly Negro school. W,
11.0. 217-231.

Banks, C. McQuarty, G. &Hubbard, J. (1978). Toward areconceptualization
ofthe social-cognitive basis ofachievement orientations' In blacks. Review
oLEducationaLBeseaLch 48, 381 -.397

Baroody, A. J. (1987). Chi lrd rsns mathematical thinking; A dev vlnsgpme ntal

qll‘i' 9 9‘ 199 9.111 ._0_ 9‘ -.|‘9--.9l ‘ql‘ .New
York: Teachers” College Press.

Beady, C. H. ,& Hansel, S. (1981). Teacher race and expectations for student
achievement. AmedcanﬁducattonaLBeseamMoumalJﬁ, 191-206.

Binet.J- (1973). Ibedeveloomenmﬂnteiligenceimchﬂdren. New York: Arno
Press.

Blau, Z. S. (1981). W
W. NewYork: Macmillan.

' 1.. .. l‘ ‘Il‘l l‘ o u ._ .. 0. '1‘": al., 0 r . Monterey,
CA: CTB/MCGraW-Hill.

Children’s Defense Fund. (1974). ll t in
Washington, DC: Author.

Clements, D. H. (1989, December). Hands-on math: Vol. 1. Amnmejicleagher
32, 58.

Cohen, R. A. (1969). Conceptual styles, culture conﬂict, and nonverbal tests of
intelligence. W 11. 828-856.

124

125

Coleman-Bums, P. (1989). African-American women: Education for what?"

W63. 2.1. 145-160.
Comer, J. P. (1972). BMW. New York: Quadrangle Books.

Comer,J. P. (1980). West. New

York: Free Press.

Crain, R. L., & Mahard, R. (1978, Summer). Desegregation and black

achievement: A review of the research. Law—and_C_Qntemp_QLa£y
Emblems. 42. 17-56.

Dorsey, A. G. (1988, Winter). Art and creative development for young children.
Chlldhoodjducation.65.123.

Dweck, C. 8., Davidson, W., Nelson, T., 8. Enna, F. (1978). Sex differences in
learned helplessness. W1. 1_4.. 268-276.

Entwisle, D. R., 8 Alexander, K. L. (1988, May). Factors affecting achievement
test scores and marks of black and white ﬁrst graders. Etememtatyﬁeneel
.LQIImaI. pp. 449-471.

Entwisle, D. R., 8. Alexander, K. L. (1989, March). W
W (Report No. 34).
Baltimore, MD. Center for Research on Elementary and Middle Schools.

Gelman 8. Gallistell. (1986). the child’s understanding Qf number. Boston:

Harvard University Press.

Giddings, P. (1984). W
W. NewYork: William Morrow.

Gilbert II, S. E., 8. Gay, G. (1985, October). Improving the success in school of
poor black children. Wotan. pp. 133-137.

Ginsberg, H. P. (1989). Children’s anthmetie; IjQw they learn it and DSM yen
teaehjt. New York: Teachers’ College Press.

Grant, L. (1984). Black females’ "place" in desegregated classrooms. Seeinlegy
WI 51: 98'1 11.

Hale-Benson. J. E. (1986). W
styles. Baltimore, MD: Johns Hopkins University Press.

126

Hare, B. (1985). Reexamining the achievement central tendency: Sex
differences within race and race differences within sex. In H. McAdoo &
J. McAdoo (Eds), Blaekenjtanen. Beverly Hills, CA: Sage.

Havighurst, R. J. (1976, November). Human characteristics and school learning.

ElementatyﬁcllooLJoumal. 12. 101-109.

Heller, K. A., and others. (1982). hi r
W. Washington DC Ofﬁce
of Civil Rights.

Henry, R, Brazen, 8., 8. Henry, C. A. T. (19 ). Black student attitudes toward
standardized tests: Does gender make a difference? QoLegeﬁtneent
Joumal. 23. 346-354.

Hill, R. A. (1972). Ine_str_engtns_o_f_b_laek_famities. New York: Emerson Hall.

Hilliard, A. G. (1973). h ' II I f hil r

MW. New York: Institute of
African Research.

Hilliard, A. G. (1974, September). Restructuring teacher education for
multicultural imperatives (Research report). W.

Johnson, K. R., 8. Simons, H. D. (1972, January). Black children and reading:
What teachers need to know. EnLDettaLkannan, 53, 288-290.

Kahl, W. C. (1957, March). State aid for education. WW
Education. 89. 10-13.

Kamii, C. (1990). ' in r n' 'n
W. Chicago. University of ﬁlms

Kerckhoff, A. C. &Campbell, R. T. (1977, January). Black-white differences In
the educational attainment process. W, 50, 15—27.

Kleinfeld, J. S. (1973a, Summer). Intellectual strengths in culturally different
groups. EdugatigﬂaLﬁeseatcn. 4_3, 341 -.359

Kleinfeld, J. (1973b). UsingnonxemaLwarmttLtancreaseJeaMAcLoss:
ethuLaLexnenment. Fairbanks: Institute of Social, Economic, and
Government Research, Alaska University.

127

Ladner, J. A. (1971). Labeling black children. Social- -psychological implications.
JoumaLotAﬁeAmemaansues. 3. 43- 52.

Lewis, H. (1974). W. Atlanta, GA: Atlanta

University Press.

Lightfoot, S. L. (1976, December). Socialization and education of young black
girls in school. W. 18. 239-262.

Moore, E. G., 8. Smith, A. W. (1987). Sex and ethnic group differences in

mathematics achievement. MWLMathemMcs
Education. 1.8. 25—36.

Morgan, H. (1980, January-February). Howschools fail black children. Seeiat
EQlicy. J_Q, 49-54.

Munroe, R. H., Munroe, R. L., 8. Whiting, B. B. (1981). ﬁandnddkm
Wm. New York: Garland.

National Council of Teachers of Mathematics. (1989). Cunieulum and evaluatidn
standards tdr sehddl mathematics. Reston, VA: Author.

National Research Council. (1985). W669!
edm;atjen;_A_reseamh_agenda Washington, DC. National Institute of

Education, National Research Council.

Natriello, G. McDill, E. L. 8. Pallas, A. M (1990). Sc_o_o_linn_disad_vantaged
W. New York: Teachers College,

Columbia University.

Nelson- LeGall, S. (1989). Same sex and cross- sex help exchanges in the

classroom. MnalnLEdnchnalﬁsycndegx. 19. 67- 71

Nelson- LeGall, S. (1991). The condition of sex equity in education: Sex, race,

and ethnicity. SexequﬁxideduchnaLdoMnunitMJchiexementand
testing. Princeton, NJ: EducationalTesting Service.

Nelson-LeGall, S., 8 Glo-Scheib. (1985). Help seeking in elementary
classrooms: An observational study. m r E ti

Esxcbolngx. 10. 58-71.

Ogbu, J. U. (1981). Origins of human competence: A cultural-ecological

perspective. Qhﬂdmmgument. 52. 413-429.

128

Piaget, J. (1952). W. London: Routledge 8
Kegan Paul.

Powell, G. J. (1970). Sdbdojnesenmnandnandjeltndncem. South Hadley,
MA: Mt. Holyoke College.

Powell, G. J., Yamamoto, J., Romero, A. 8 Morales, A. (1983). The

nt ' ' r il . New York:
Brunner/Mazel.

Rashid, H. M. (1981, Fall). Early childhood education as a cultural transition for
African-American Children. 1' r h rt rl , 6, 55-63.

Renzulli, J. (1973). Talent potential in minority group students. Exceptidnel
Man. 39. 437-444.

Rodgers-Rose, L. F. (Ed). (1980). MW. Beverly Hills, CA: Sage.

Rohwer, W. D. (1971). Learning, race, and school success. Beam
EducatinnaLBeseaLcn. 41 191 -210.

Romberg, T. A. (1983). A common curriculum for mathematics. Yeamoek
(National Social Studies Education), 82(Part 1), 121-159.

Rosenberg N. (1972). W
W. Bethesda, MD: National Institutes of Health.

Rosser, P. L., 8 Oyemade, U. J. (1981). MW
' h vi l i ti

Rowntree, D. (1981). Staﬂstigsnuijlleutjeats. New York: Charles Scribner’s
Sons.

Rystrom, R., 8 Cowart, H. (1972, January). Black reading errors or white
teacher biases. Beadingddumat, 15, 273-276.

Sarason, I. G. (1973, June). Test anxiety and social inﬂuence. Bemenat
Jnnmal. 4.1. 261-271.

Scott-Jones, D., 8 Clark, M. L. (1986). The school experience of black girls. Ehj
Deltaﬁannan. 61. 520-526.

129

Simpson, T. (1962, March). Study, through biographies, of the history of the

elementary concepts of probability. senndLScieneeandMathematies 62,
172.

Slaven, R. E., Karweit, N. L., 8 Madden, N. A. (1989). W
students_at_[isk. Boston: Allyn 8 Bacon.

Spencer, M. B., Brokens, G., 8Allen, W. (Eds..) (1985). Beginningslbesmial
andaﬂectnedelenomentdtnackcmlie. Hillsdale. NJ: Erlbaum.

Starkey, P. (1981, March). Young children’s performance in number
conservation tasks: Evidence for a hierarchy of strategies. JdtnnaLdt

Genetldlisxchdlony. 198. 103-110.
Stodolsky, 8., 8. Lesser, G. (1967). Leammdoanenmmidsadxantaged.

New York: Ferkauf Graduate School of Humanities and Social Sciences,
Yeshiva University.

Tobias, S. (1978). Qxercomindmatllanxiety. Boston: Houghton-Mifﬂin.

Travers, K. T., 8 McKnight, C. C. (1985, February). Mathematics achievement
in US. schools: Preliminary ﬁndings from the second NAEP mathematics

study. W. on 407413-

Trent, W. (1984, May). Equity considerations in higher education. Race and sex
differences In degree attainment and majorI ﬁeld from 1976 through 1981.
AmedcamionmalnLEducatidn. 92. 280- 305.

Ward, 8. H. (1972, July). Self-esteem and racial preferences in black children.

AmMcandnumalnLanoosxcboiogx 42. 644-647.

Washington, V., 8LaPoint, V. (1988). BtacknnjtdtenandArnencarLInsmtmdns.
New York: Garland.

Weikart, D. P. (1980). 'I r t
pLeschLoanamnnmnanLougbagm Ypsilanti MI: High Scope
Press.

Williams, M. M. (1972, September). BacemlenmndeducanonaLachIexement
WW. Paper presented at the American Psychological
Association Annual ConventIon Honolulu Hawaii.

130

Witkin, H. A. (1979). Socialization, culture and ecology in the development of

group and sex differences in cognitive style. Humenmbmem. 22.
358-372.

Ysseldyke, J. E. (1983, April). Current practices in making psychoeducational

decisions about learning disabled children. Jeannaj_o_f_l_ea£mng
Disabilities. 16, 226-233.