Building math self-efficacy

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Title:
Building math self-efficacy a comparison of interventions designed to increase mathstatistics confidence in undergraduate students
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Forbes, Karen Joan, 1961-
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Mathematics -- Study and teaching -- Psychological aspects   ( lcsh )
Psychology thesis Ph. D
Dissertations, Academic -- Psychology -- UF
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Thesis (Ph. D.)--University of Florida, 1988.
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Includes bibliographical references.
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by Karen Joan Forbes.
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Vita.
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Typescript.

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BUILDING MATH SELF-EFFICACY: A COMPARISON OF
INTERVENTIONS DESIGNED TO INCREASE MATH/STATISTICS
CONFIDENCE IN UNDERGRADUATE STUDENTS






BY

KAREN JOAN FORBES


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1988 FLOA LMRA













ACKNOWLEDGEMENTS


I have been very fortunate to have a committee

composed of dedicated, caring, and gifted teachers and

scholars each of whom has made a unique contribution to

both my professional and personal development. Dr. Greg

Neimeyer has given me support and guidance from the

moment I began my graduate program. His patience and

consistently positive reinforcement as I struggled to

define myself as researcher, teacher, and therapist

were always greatly appreciated.

I would also like to thank Dr. Larry Severy who

first urged me to take on this project and who has been

a friend and supporter throughout my graduate program,

Dr. Helen Mamarchev who has been a wonderful role model

for my professional and personal life, and Dr. Paul

Schauble, who has helped me to set high goals for

myself as a professional psychologist and as a person.

Very special thanks go to Dr. Barbara Probert for her

wonderful friendship and her mentoring. Because she

always believed in me, it has been easier for me to

believe in myself.

This project would not have been possible without

the generous financial support of the College of

Liberal Arts and Sciences, the Department of

ii










Psychology, and the alumni fellowship committee of

Oberlin College. I also appreciate the efforts of the

Department of Statistics to support and to understand

the purposes of my research. Most importantly, many

thanks go to my committed, caring counselors and tutors

who gave their time and professional expertise so that

this project would be successful.

I would like to acknowledge all of the students

who participated in this project. I hope that this

experience helped them to feel more confident about

their mathematical ability. They were all intelligent,

thoughtful people, and I wish them much success in all

their endeavors.

Finally, there are several people who have given

me endless support and love throughout this process.

Dr. Kathleen O'Connor and Bill O'Connor have been

extraordinary friends and employers without whom I

literally could not have completed my dissertation. My

parents, Donald and Carole Forbes, my brother, Doug,

and my husband, Larry Gage, have never failed to let me

know how proud they are of my goals and my

achievements. Their love and constant support have

helped me to weather the bad times and to take joy in

the good times. I share all of my accomplishments with

them.


iii














TABLE OF CONTENTS

PAGE
ACKNOWLEDGEMENTS ...............................................ii

LIST OF TABLES ........ .. ..................... .. ..vi

ABSTRACT ........................................... viii

CHAPTER

ONE INTRODUCTION ................................ ... 1

Diminishing Numbers of Qualified
Mathematicians ....... .................. 1
Factors Affecting Mathematics Achievement..5
Mathematics Anxiety...................... 12
Self-Efficacy Theory and the Problem of
Math Avoidance ......................... 16

TWO REVIEW OF THE LITERATURE ................... 21

Images of Mathematicians .................. 24
The Effect of Teaching Strategies on
Mathematics Attitudes and Achievements..31
Mathematics Anxiety .......................41
Sex Differences in Mathematics
Achievement, Attitudes, and Anxiety ..... 51
The Effect of Self-Efficacy
Expectations on Human Behavior.......... 73
Interventions for Raising Mathematics
Participation and Achievement.......... 111
Hypotheses......................... ..... 128


THREE METHODS .... ... . ......... ............... 131

Subjects................................. 131
Instrumentation .......................... 134
Procedure........................ . . ... 141











RESULTS.................................... 144


Major Hypotheses: Math Self-Efficacy,
Math Anxiety, and Self-Estimate of Math
Ability................................ 144
Implications Grid........................ 149
Grades................................... 150
Correlational Analyses.................... 150
Post-hoc Analyses........................ 154

FIVE DISCUSSION................................. 162

Measures of Self-Efficacy: Self-Estimate
of Math Ability and Math Self-Efficacy
Scale.................................. 163
Math Anxiety.............................. 164
Implications Grid........................ 165
Final Course Grade....................... 167
Post-hoc Correlational Analyses .......... 168
Limitations of Confidence and Tutoring
Groups................................. 172
Conclusion............................... 175

APPENDICES

1 LETTER OF INTRODUCTION TO NEW MAJORS...... 177
2 MATHEMATICS SELF-EFFICACY SCALE ........... 179
3 FENNEMA-SHERMAN MATH ANXIETY SCALE........ 183
4 BEM SEX-ROLE INVENTORY .................... 184
5 BACKGROUND QUESTIONNAIRE .................. 185
6 IMPLICATIONS GRID ......................... 186
7 MATH CONFIDENCE GROUP OUTLINES............ 188


REFERENCES........................................... 194

BIOGRAPHICAL SKETCH .................................. 207


FOUR













LIST OF TABLES


TABLE PAGE

1 Mean Number of Sessions Attended by
Students for Whom Pre- and Posttest
Data is Available.......................... 133

2 Means, Standard Deviations, and t-test
Comparisons of Attenders versus
Non-attenders of Confidence Groups on MSES,
SE, MAS, and GPA .......................... 135

3 Means, Standard Deviations, and t-test
Comparisons of Attenders versus
Non-attenders of Tutoring Groups on MSES
SE, MAS, and GPA ................... ........ 136

4 Results of Pre-test ANOVA's Among
Confidence, Tutoring, and Control Groups
for MSES, SE, and MAS ............ .......... 145

5 Repeated Measures ANOVA's for MSES, SE, and
MAS..................................... .... ..146

6 Means and Standard Deviations of Pre- and
Posttest Data for Math Confidence,
Tutoring, and Control Groups for MSES, SE,
and MAS................. .. .......... ..... ....... ..147

7 F-Test Analysis of the Group X Time
Interaction Between Math Confidence
and Tutoring Grorups, Math Confidence
and Control Groups, and Tutoring and
Control Groups for SE ........... .......... 148

8 Repeated Measures ANOVA for Impgrid for
Math Confidence and Tutoring Groups........ 151

9 Means and Standard Deviations of Pre-
and Posttest Impgrids for Math Confidence
and Tutoring Groups................. o......151










TABLE


10 'Content Analysis of Confidence Group
Homework Assignment: "Describe a
Mathematician" ..... ... ............. ........ 152

11 One-way ANOVA for Grade for Math
Confidence, Tutoring, and Control Groups..153

12 Means and Standard Deviations for Final
Course Grade for Math Confidence, Tutoring,
and Control Groups ........................ 153

13 Matrix of Pearson Product-Moment
Correlations Among Dependent and Independent
Variables for Experimental Subjects....... 157

14 Matrix of Pearson-Product Moment
Correlations Among Dependent and Independent
Variables for Experimental and Non-
Experimental Subjects..................... 160

15 Stepwise Multiple Regression for
Dependent Variable: Final Course Grade....161


vii


PAGE










Abstract of Dissertation Presented to the Graduate
School of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of
Doctor of Philosophy

BUILDING MATH SELF-EFFICACY: A COMPARISON OF
INTERVENTIONS DESIGNED TO INCREASE MATH/STATISTICS
CONFIDENCE IN UNDERGRADUATE STUDENTS

By

Karen Joan Forbes

August 1987

Chairperson: Greg J. Neimeyer
Major Department: Psychology

Because many students lack confidence in their

ability to succeed in mathematics courses and careers,

the United States faces a serious shortage of people

with the skills necessary to sustain and to develop the

advanced technology on which our society depends. This

study was carried out to determine the comparative

effectiveness of two intervention strategies for

increasing mathematics self-efficacy in undergraduate

students and to assess the feasibility of implementing

a math/statistics confidence program within an academic

department at a major university.

Undergraduate students at a large southern

university were informed that the department of

psychology was implementing a pilot program to help

students deal with apprehension about required

statistics courses. One hundred and sixty-nine students

viii










completed a variety of pretest measures and 96

indicated an interest in being involved in such a

program. Students were randomly assigned either to one

of four counseling-based statistics confidence groups,

to one of three statistics tutoring groups, or to a

no-treatment control group. The experimental design was

a repeated measures paradigm where students were given

pre- and posttreatment assessments of math

self-efficacy, math anxiety, self-reported math

ability, and math self-concept.

Results showed that at posttest, math confidence

groups rated themselves as having significantly higher

math/statistics ability than either the tutoring or

control groups. Both experimental groups significantly

lowered their levels of math anxiety relative to the

control group. The differences between the two

experimental groups in the number of implications that

mathematical ability had on students' self-concepts

approached significance, with the math confidence

groups having fewer implications. There were no

significant differences among the three groups regarding

math self-efficacy or final statistics grade.

Correlational analyses performed on the experimental

data and the pretest data from the original sample of










169 students lend some support to earlier findings

regarding the relationships among math self-efficacy,

math anxiety, sex-role orientation, and academic

performance.















INTRODUCTION
CHAPTER ONE

Probably at no other time in our nation's history

is a knowledge of mathematics so essential. The

American people and their government representatives

are repeatedly being called upon to make decisions on

issues such as the Strategic Defense Initiative, the

federal deficit, the new tax laws, and the Space

Shuttle program. Although it is not necessary to be

intimately familiar with all of the details of these

programs, complete ignorance and the need for the

American people to rely on a few "experts" does not

bode well for the democratic decision-making process.


Diminishing Numbers of Qualified Mathematicians

Although a lack of understanding of scientific and

mathematical issues is a serious problem, an equally

serious threat is a dearth of trained mathematicians

and scientists. The 1986-87 Occupational Outlook

Handbook (Bureau of Labor Statistics, 1986) reports

that there is currently a shortage of Ph.D.

mathematicians which is expected to continue throughout

the decade. Individuals holding master's and bachelor's

degrees in mathematics will benefit from this shortage.

Many job opportunities in computer science and data













processing will become available, and by meeting

state certification requirements, many will find

openings in secondary education.

Mathematical work can be categorized as either

theoretical (pure) or applied. Although theoretical

mathematicians seek to develop new principles and

relationships without necessarily seeking practical

applications of their work, their findings are often

the basis of important applied technologies, as in the

case of Rieman's non-Euclidean geometry and the

creation of atomic power (Bureau of Labor Statistics,

1986). Applied mathematicians find employment in

business, government, engineering, natural and social

sciences; however, as the Bureau of Labor Statistics

points out, "the number of workers using mathematical

techniques is many times greater than the number

actually designated as mathematicians" (Bureau of Labor

Statistics, 1986, p.76).

Although an abundance of job vacancies is welcome

news for those with degrees in mathematically related

fields, it is troubling to think of shortages in areas

so vital to our nation's economy. These labor

statistics also raise questions about why so few of our

vocationally oriented college students are pursuing













math and science majors that would almost guarantee

them employment.

The United States Department of Education monitors

the number of degrees conferred by institutions of

higher education. In 1982-83, there was a 50% decline

in Bachelor's degrees granted in the discipline of

mathematics, 12,543 down from 24,801 in 1970-71. Of the

30 disciplines listed in the Digest of Education

Statistics (Grant & Snyder, 1986) only library science,

foreign languages, and letters had a similar or greater

decline (Grant & Snyder, 1986, p. 129). The number of

master's degrees in mathematics declined by 45Z from

1970-71 to 1982-83, and 42% fewer Ph.D.'s were granted

in 1982-83 than in the decade before. Again, only the

disciplines of library science, foreign languages, and

letters suffered a greater reduction in master's

degrees over the previous decade, but mathematics

emerged with the greatest losses in the numbers of

Ph.D.s granted (Grant & Snyder, 1986,

p. 129).

Education has been especially affected by the

diminishing numbers of math graduates because employers

in private industry and government offer much higher

salaries to people trained in mathematics, science, and













computer programming (Bureau of Labor Statistics, 1986,

p.129). During the 1981-82 academic year, 1,897

students received degrees in art education, 4,915 in

music education, and 17,391 in physical education. Only

529 degrees were granted in mathematics education and

558 in science education. In 1982, although 17,000, or

9Z of the nations's math and science teachers left

their jobs, only 700 recently graduated math and

science education majors began teaching.

In the case of elementary education, many teachers

do not enjoy mathematics, nor do they feel it is

important. Bulmahn and Young (1982) found that many of

the elementary educators they interviewed considered

math to be their worst subject, and as a result, felt

their career options had been limited. What was most

disturbing to the investigators was the feeling

expressed by many beginning education students that

"elementary teachers do not really have to be very good

at mathematics beyond the basic computations" (Bulmahn

& Young, 1982, p. 56). Furthermore, Kelly and Tomhave

(1985) found elementary education majors to be highly

anxious about performing mathematical tasks, a result

supported by Probert (1979).













Although it is debatable whether or not increasing

teacher salaries will entice students trained in math

and science to become elementary or secondary school

teachers, it would seem possible that the current

situation is forcing us into a vicious cycle. Because

our country is producing fewer qualified

mathematicians, there will be that many fewer in

classrooms motivating children to take advanced

coursework and to pursue careers in mathematics and

related fields.


Factors Affecting Mathematics Achievement

Along with the decline in the numbers of

mathematicians has come a corresponding decline in

standardized test scores. From 1961 to 1981, average

SAT scores fell from a total of 969 to 890, after which

a small increase occurred bringing the current average

to 906 (Bremelaw, 1986, p. 86). The Digest of Education

Statistics (Grant & Snyder, 1986) records the average

quantitative SAT score for females as 467 in 1967 and

449 in 1984. For males, the average quantitative score

declined to 495 in 1984 from 514 in 1967. What is

especially significant about these data is that

Americans are currently spending over $80 million and













countless hours on SAT preparation courses (McCabe,

1986).

The Digest of Education Statistics (Grant &

Snyder, 1986) lists the results of two international

studies of mathematics achievement which included

students from Belgium, Canada, England, Finland, Hong

Kong, Hungary, Israel, Japan, the Netherlands, New

Zealand, Nigeria, and Scotland. As compared to students

in Japan, our foremost technological rival, American

students performed quite poorly on mathematics tests.

On the average, Japanese students answered correctly

76% of algebra items, 69% of the calculus items, 76% of

the geometry items, and 72% of the probability and

statistics items. By comparison, American students

responded correctly to 43% of the algebra items, 29% of

the calculus items, 31% of the geometry items, and 40%

of the probability and statistics items. The median

percent correct for all 14 countries was 57% for

algebra, 44% for calculus, 42% for geometry, and 50%

for probability and statistics.


Mathematics Requirements in Secondary Education

Perhaps this wide gap in mathematical proficiency

can be attributed to the differential amount of time

spent on mathematics in American schools. Of the 14












countries in the above-mentioned international study,

three countries had a lower proportion of students of

appropriate age taking advanced mathematics classes

than students in the United States, five countries had

about the same proportion, and six had a higher

proportion.

In 1984, although only 10 states allowed fewer

than four years of high school English for graduation,

41 states did not require any more than two years of

mathematics (Grant & Snyder, 1986, p. 44). The average

number of Carnegie units in mathematics earned by high

school graduates in 1982 was 2.5, demonstrating that

students did not feel motivated to pursue mathematics

after they had completed their requirements. Students

in public schools took the fewest number of math

courses, an average of 2.5, while Catholic school

graduates earned an average of 3.3 Carnegie units in

math, and non-parochial private school students

averaged 3 math courses. It is interesting to note that

students of Asian descent took the greatest number of

math courses, an average of 3.1, while students of

Native American heritage had the lowest average number

of mathematics courses, 2.0. Likewise, 70% of

Asian-American students took the SAT or ACT as compared













to 30% of white students and 28% of all students

(Jacobson, 1986, p. 108).


Students' Perceptions of the Usefulness of Mathematics

There appears to be a direct relationship between

career aspirations of students and the number of

mathematics courses they complete in high school. In

the "High School and Beyond" survey done in 1982 by the

United States Department of Education, (cited in Grant

& Snyder, 1986) students were questioned about their

post-secondary plans. Those who had planned no further

education had taken an average of 1.9 math courses,

whereas those who planned to obtain a four year degree

had taken an average of 3.1 years of mathematics.

During the same year that the "High School and

Beyond" survey was conducted, Armstrong and Price

(1982) queried 1,788 high school seniors about the

factors that were most influential in their decision to

enroll in mathematics courses. The most significant

motivator was their perception of mathematics'

"usefulness" in their future lives. Evidently, the

average non-Asian American student does not perceive

mathematics to be important for his or her post-high

school life.












Students may not realize just how many college

majors and careers require extensive mathematics

preparation. In one of the first studies addressing the

problem of math avoidance, Lucy Sells (1973) dubbed

mathematics the "critical filter," because over 75% of

the college majors at the University of California at

Berkeley required advanced mathematics. Sells (1973)

was also one of the first investigators to point out

the differential enrollment of young men and women in

high school and college math courses. Since then a

great deal has been written about the reasons why men

are more likely to enroll in mathematics and science

courses and to pursue math-related careers. This is an

issue of considerable importance because "scientists

and engineers exert considerable influence on United

States society, any group that contributes few

scientists and engineers is at least partly

disenfranchised" (Goldman & Hewitt, 1976, p.50).

Results have been equivocal concerning the existence of

reliable differences in mathematical aptitude between

men and women, however a sex-role socialization

hypothesis has become one of the most popular

explanations given for math avoidance.














Confidence and Enrollment in Math Courses

The second most important variable that Armstrong

and Price (1982) found determined mathematics

enrollment was students' confidence in their ability to

do well in math courses. The "High School and Beyond"

study uncovered a supporting relationship between

actual test performance and enrollment in math courses.

Those who performed most poorly on an academic test

battery averaged the lowest number of math courses,

1.9, while those who scored highest took the greatest

number of mathematics courses, 3.3. This is in contrast

to enrollment in English courses where low scoring

students still took an average of 3.4 courses, as

compared to 3.8 courses taken by the highest scoring

students, and to social science where the difference

between low and high scoring students was 2.5 and 2.7,

respectively. In the case of English and social science

(usually History), students have little choice about

how many courses they take, regardless of how poorly

they perform in these classes. With mathematics,

students who are not successful may opt to deal with

their deficiencies by avoiding the subject altogether.













Strategies for Teaching Mathematics

The quality and type of math instruction have been

offered as a reason for students' poor performance and

subsequent avoidance of non-required mathematics

courses. Jay Greenwood, a mathematics educator in

Portland, Oregon, believes that students would be more

adept at and more interested in the subject if teachers

would abandon the "explain-practice-memorize" approach

to mathematics (Greenwood, 1984). Greenwood feels that

this teaching strategy "promotes and perpetuates that

all too common perception of mathematics as a subject

that appears easy and logical to a few 'brains' and

incomprehensible to most common folk" (Greenwood, 1984,

p.663).

There is little question that the data reported

above point to a current and increasing deficit of

Americans trained in mathematics and related fields.

What is not as clear are the factors responsible for

this trend. It is difficult to untangle the lines of

causality; should poor mathematics performance be

attributed to inadequate academic preparation or rather

are students choosing not to pursue additional math

courses because they have not performed well? Either

scenario produces even more questions. If the former is













true, why have educators decided that mathematics is

not important enough to require three to four years of

coursework; is this attitude promoting a feeling among

students that mathematics is not "useful?" If the

latter is the case, what is it about our educational

system that is not producing successful learners of

mathematics?


Mathematics Anxiety

Since the early 1970s the construct of "math

anxiety" has become a popular explanation for the

diminishing numbers of American students pursuing

advanced mathematics coursework and math-related

careers. Upon offering a behavioral therapy program

through the Colorado State University Counseling

Center, Frank Richardson and Richard Suinn found that a

third of the students responding indicated that their

problem related to anxiety about mathematics courses

(Richardson & Suinn, 1972). These two researchers and

therapists described mathematics anxiety as "involving

feelings of tension and anxiety that interfere with the

manipulation of numbers and the solving of mathematical

problems in a wide variety of ordinary life and

academic situations" (Richardson & Suinn, 1972).













In 1976, Sheila Tobias stated that math anxiety

was a promising construct for understanding avoidance

behavior in mathematics. Byrd (1982, cited in Reyes,

1984) found that students claimed math anxiety to be

the cause of avoiding not only math courses, but

certain jobs, science courses, some careers, tests,

balancing the checkbook, and colleges with heavy math

requirements. As early as 1954, Gough defined number

anxiety as "the presence of a syndrome of emotional

reactions to arithmetic and mathematics" (p. 344).

Fennema and Sherman's (1976) Math Anxiety Scale was

designed to "assess feelings of anxiety, dread,

nervousness, and associated bodily symptoms related to

doing mathematics" (p. 4).

Researchers such as Nancy Betz (1978) and Darwin

Hendel have produced data that suggest that large

numbers of college-aged students are experiencing math

anxiety (Hendel & Davis, 1978), while more recent

studies have failed to confirm these findings (Resnick,

Viehe, & Segal, 1982). However, each of these studies

reported that math anxiety was greater in the cases of

students with weak mathematical backgrounds. Hendel and

Davis (1978) believe that "one symptom of mathematics

anxiety is avoidance of mathematics" (p. 430).














Mathematics Anxiety as a Unique Construct

Reyes (1984) believes that even though there has

been a "flurry of activity" around math anxiety since

1977, much of the discussion has not been grounded in

research knowledge (p. 563). Many investigators do not

accept that 1) math anxiety is a separate construct

from test anxiety or that 2) anxiety, with all of its

attendant physical symptoms is an appropriate term.

Byrd (1982, cited in Reyes, 1984) defines test

anxiety aroused by evaluative situations. Wine (1971)

has divided test anxiety into two parts, worry, which

is the cognitive concern about one's performance, and

emotionality, the arousal of the autonomic nervous

system in evaluative situations. Dendato and Diener

(1986) describe Wine's (1971) Cognitive Attentional

Model as proposing that students suffering from test

anxiety are impaired by "worry, negative

self-evaluative statements, and task irrelevant

ruminations that compete for attentional capacity with

task relevant activity and interfere with the recall of

pertinent information" (p. 131).

Dew, Galassi, and Galassi (1984) investigated both

the uniqueness of the math anxiety construct and the

physiological correlates in their study entitled, "Math













Anxiety: Relation with Situational Test Anxiety,

Performance, Physiological Arousal, and Math Avoidance

Behavior." The heart rate, skin conductance, and skin

fluctuations of undergraduates were measured when they

were given math problems to solve under test equivalent

conditions. The students were also asked to complete a

variety of other assessments measuring test anxiety,

math anxiety, and math aptitude. Dew et al. (1984)

found that the measures of math anxiety were generally

correlated with test anxiety. Math anxiety was

inversely correlated with performance on math problems,

however it was not significantly related to any of the

physiological measures.

Proponents of the math anxiety construct believe

that individuals only experience the cognitive

attentional deficits described by Wine (1971) when the

situation involves mathematics. Rounds and Hendel

(1980) have suggested that "mathematics anxiety is less

a response to mathematics than a response to evaluation

of mathematics skills" (p. 146). Rounds and Hendel were

led to such a conclusion based on their analysis of the

Richardson and Suinn (1972) MARS. They concluded that

the MARS is not measuring a homogeneous factor called
"math anxiety," but rather two distinct factors which













they described as follows:1) Mathematics Test

Anxiety--apprehension about taking and receiving the

results of math tests, and 2) Numerical

Anxiety--everyday concrete situations requiring number

manipulation. (Rounds & Hendel, 1980).

Rounds and Hendel (1980) are concerned that the

Richardson and Suinn (1972) definition of math anxiety

is not broad enough to encompass the diversity of the

field of mathematics.


If the "solving of mathematical problems" is
considered within the context of mathematics
tests, the two MARS factors identified are a
good fit to that definition. However, the
fact that mathematics is a very broad field
makes this and other definitions of the
mathematics anxiety domian problematic.
(Rounds & Hendel, 1980, p. 145)


Self-Efficacy Theory and the Problem of Math Avoidance

Although the MARS describes an individual's

reactions to certain mathematically related situations,

it provides little or no explanatory value as to why

mathematics produces such anxiety. In order to

alleviate the distress that individuals are reporting,

counselors and educators need a theoretical framework

within which they can develop effective treatments.

Rounds and Hendel (1980) warn that the label, "math

anxious" is both limited and "linguistically ambiguous,













suggesting a pathological response to mathematics per

se" (p. 146).

Ultimately, math anxiety is only an important

construct insofar as it explains either poor

performance or avoidance of mathematics when to do so

is harmful to an individual. There are those who may

never experience nervousness because they can

successfully avoid all contact with the subject.

Avoidance is, in effect, a "cure" for math anxiety.

However, the true problem is that thousands of students

are avoiding the math courses that will prepare them

for a myriad of careers, that will allow them to be

knowledgeable about technological issues that face the

country, and at the very least, will enable them to

perform functions such as managing their personal

finances. Anxiety and avoidance are part of a cycle

whose origin is very difficult to pinpoint.

Recently, Nancy Betz and her colleagues have begun

to advance Bandura's Self-Efficacy Theory as a context

within which to explain individual's avoidance behavior

towards mathematics courses and careers. Bandura has

posited that a person's beliefs in regard to his or her

ability to perform a certain task are the major agents

of behavior and behavior change (Bandura, 1977). These













beliefs about one's abilities, or self-efficacy

expectations, are learned and modified by four sources

of information: 1) performance accomplishments,

2) vicarious learning or modeling, 3) verbal

persuasion, and 4) emotional arousal. Betz and Hackett

(1983) explain that Bandura believes anxiety to be a

co-effect of self-efficacy expectations; as

self-efficacy diminishes, anxiety increases.

Therefore, anxiety is not viewed as the cause of

avoidance behavior but as a byproduct of low

self-efficacy. And unlike Hendel and Davis (1978) who

see avoidance as a symptom of math anxiety,

self-efficacy theory would explain both anxiety and

avoidance as by-products of low self-efficacy

expectations.

Self-efficacy theory can offer a theoretical

framework for the results produced by studies proposing

teaching styles, sex-role stereotyping, and even

physiological deficits as underlying causes of math

avoidance behavior, since all of these factors can

affect one's self-statements and beliefs about one's

ability to perform. It also allows for people who may

have experienced similar mathematical instruction or

who have been exposed to similar sex-role stereotyping













to behave differently in situations calling for contact

with mathematics since their perceptions of the

situation and their abilities may lead to different

expectations.


Building Math Self-efficacy Expectations

Unlike explanations of math avoidance that rely

on nearly immutable factors such as gender role and

physiology, self-efficacy theory provides the mechanism

whereby a person's belief system can be modified,

leading to increased confidence and in many cases,

reduced anxiety. "Interventions focused on increasing

self-efficacy expectations via attention to the sources

of efficacy information should increase approach versus

avoidant behavior and concurrently, decrease anxiety in

relationship to the behavior" ( Betz & Hackett, 1983,

p. 331). As Bandura (1986) has recently noted, however,

no consistent relationships have been found between

changes in fear arousal and phobic behavior, so merely

modifying anxious responding will not guarantee an

increase in confidence about performing a certain task.

The following study attempts to show the

superiority of a multimodal intervention in increasing

self-efficacy expectations of undergraduate students

enrolled in an introductory statistics course as













compared to a traditional tutoring approach and a

control group. Unlike interventions that have focused

on reducing anxiety, the groups were based on Bandura's

notion that it is not fear or anxiety that produces

avoidant behavior, but expectations that one will fail

at a task (Bandura, 1986). Thus, the students were

involved in math confidence groups, rather than in

groups that focused exclusively on reducing math

anxiety.

In the subsequent chapter, a more detailed history

of the theorizing and research performed in the area of

math anxiety and avoidance will be discussed, including

the role of mathematics instruction, sex-role

socialization, and physiology. Bandura's theory of

self-efficacy and the studies which have attempted to

use this framework as an explanation for math avoidance

behavior will be explicated. Finally, a review of

previous counseling interventions used to alleviate

"math anxiety" will be discussed and contrasted with

the approach used in this study. Chapters Three, Four,

and Five will describe the measures and procedures

under which the study was executed, the results of the

various group by group comparisons and post-test

measures, and the conclusions suggested by the data.
















CHAPTER TWO
REVIEW OF THE LITERATURE


On January 26, 1987, the Public Broadcasting

System premiered a children's show entitled, "Square

One" whose intention it is to teach mathematics in a

way that is lively and entertaining. The following

remarks were culled from a review of the program by

Washington Post staff writer, Megan Rosenfeld.


As a certified math idiot, I approached
the new children's TV series on the
subject, "Square One Television" with
particular interest. . .It is geared
to children 8 to 12 years old-about the
age my math education ceased...Frankly,
I don't think there is much you can do
to make math interesting, and frankly,
this program doesn't do much to disprove
that idea. .1 am, alas, not the only
person suffering the heartbreak of math
illiteracy. In fact, I was able to find
someone worse at it than I am and marry
him. (1987, p. B8)


There are not many subjects in which adults will

publicly declare there inadequacies let alone state

that they are "idiots." However, mathematics seems to

be an area about which it is all too acceptable to be

ignorant. Ms. Rosenfeld continues by describing a

segment of the show in which she encountered particular

difficulty, a lesson about prime numbers, in which the













number fourteen was declared "not a prime number." "Now

admittedly, I had to take college remedial math twice

and never had 'new math,' but what the heck is a 'prime

number'--and why should we care?" (Rosenfeld, 1987,

p. B8). Imagine someone writing about something so

fundamental to our language as a verb and asking why

should we care what it is?

There are many tragedies embodied in this

light-hearted, sarcastic review. First is the idea that

mathematics is useless and that one can be a perfectly

successful adult without having further than a sixth

grade-level understanding of basic arithmetic

principles. Secondly, the author dismisses the

possibility that mathematics can be interesting,

leading the reader to infer that anyone who disagrees

is somewhat odd. Finally, Ms. Rosenfeld alludes to

having a child in the age group to which the show is

geared. One can expect that with two parents who

proudly announce themselves to be mathematical

illiterates, this child is not likely to grow up eager

to learn this subject.

There has been a great deal of research

documenting Americans' negative attitudes towards

mathematics. A somewhat smaller number of studies have













proposed either causes or solutions to this problem.

Many of these investigations have included, the term,

"math anxiety" in their titles. In 1978, Patricia

Casserly of the Educational Testing Service told the

National Council of Teachers of Mathematics that

mathematics anxiety was "often used to conveniently

lump together all sorts of phenomena associated with

learning mathematics" (p. 7, cited in Rounds & Hendel,

1980). It is as if when we can determine the "causes"

of math anxiety we could increase the academic

performance of students, raise the numbers of women and

minorities in mathematical professions, and produce a

more technologically sophisticated populace. In 1980,

Rounds and Hendel had already accumulated enough

evidence to dispute these claims, leading them to

suggest that both the construct of math anxiety and the

tests which purported to measure it were in need of

serious revision if they were to be useful in the

understanding of mathematics learning and performance.

The following review will examine the many studies

which have claimed to study "math anxiety," including

descriptions of negative attitudes toward mathematics,

the relationships between personality and demographic

characteristics and math attitudes and aptitude, and













the effects of teaching strategies on math anxiety and

performance. But even more importantly, an attempt will

be made to determine whether or not the whole concept

of math anxiety is useful in contributing to our

understanding of how a person is unsuccessful in his or

her efforts to become proficient in mathematics.


Images of Mathematicians

Try the following exercise. In your mind, picture

a mathematician. Attempt to visualize how the person is

dressed. Imagine yourself meeting this person for the

first time; how do you feel? Now create a mental

picture of yourself in a room filled with

mathematicians. Where do you sit in the room? Are you

comfortable in this environment. ("Picture A

Mathematician," Probert, 1983).

I have posed these questions on many occasions to

people of both sexes, of varying ethnic backgrounds,

age groups, and educational backgrounds. With few

exceptions, the images revealed a stereotype of a

mathematician as a white, male, with eyeglasses who is

uncomfortable socially and who dresses in a somewhat

outdated fashion.

Even more important than the rather negative

characterizations of mathematicians are the feelings of

intellectual inferiority reported by subjects. Many













persons have described the mathematician as being

gigantic in height while they shrink to tiny

proportions. One young woman even saw herself as a dot

rather than a human being. Needless to say, being in a

room filled with mathematicians was not a comforting

experience for these people!

Almost every stereotype has at its core a grain of

truth, but ironically, many of the students reported

knowing mathematicians who had very few, if any, of

these characteristics. Kogelman and Warren (1978), in

their book, Mind Over Math, relate an anecdote about a

party they attended. Even though they were both young,

attractive, casually dressed, and outgoing, when people

to whom they were introduced discovered that they were

mathematicians, the conversations ended abruptly.

Kogelman and Warren (1978) have developed a list

of "math myths," commonly held yet false beliefs about

mathematics and mathematicians. Many of these myths

fall broadly into two categories. Some of the myths are

authoritarian, that is they proclaim that mathematics

must be performed in a certain way, and that way only.

The second category is composed of myths having to do

with the types of people who can do mathematics; the

"math people" are male, smarter, more logical, have













better memories, and less creative than "language

people" (Kogelman & Warren, 1978).

A number of studies have tested empirically these

stereotypes about persons who are successful in

mathematics and related fields. McNarry and O'Farrell

(1971) found that students viewed scientists as

"helpful, wise, and important, but hard, old,

frightening, and colorless" (p. 1060). Lorelei Brush

(1980) expected that mathematicians would be viewed in

a similar way. She asked high school and college

students of both sexes to contrast their idea of a

"typical" mathematician with their idea of a "typical"

writer using a semantic differential. Finally, the

students' self-images were compared to each of the

above professions.

The semantic differential items were grouped into

factors which resulted from the principal components

factor analysis. Writers were described as creative,

individualistic, independent, and sensitive.

Mathematicians, however, were viewed as being rational,

wise, responsible, and cautious. There was almost

complete unanimity between high school and college

samples. Both sexes felt their own image of themselves

was more compatible with their stereotypes of a writer













than with a mathematician, even though the traits

assigned to mathematicians are considered exclusively

masculine (Bem, 1974).

A similar study was conducted by Naomi Rotter

(1982) in which she sampled students at engineering and

liberal arts colleges in New Jersey. Students were

asked to rate their peers who were engineering majors,

math/science majors, or liberal arts majors such as

sociology and psychology using 30 bipolar trait items

(i.e. ambitious-lazy, friendly-unfriendly,

attractive-unattractive, etc.). As compared to female

liberal arts majors, women majoring in engineering were

perceived as less friendly, less attractive, less

flexible, and as having a poorer sense of humor.

However they fared better in comparison to their male

engineering-major counterparts who were believed to

possess these traits to an even lesser degree.

These studies support the notion that in general,

students do not think of themselves as
"mathematicians." To call oneself a mathematician, is

to declare that one is purely rational, lacking in

creativity, and unattractive socially. It would appear

that the social stigma of being interested in and adept













at mathematics far outweighs the benefits of learning

the subject.

Brush (1980) proposes a number of explanations for

why young people view mathematics so negatively. She

feels that students are probably quite ignorant of the

daily lives of both mathematicians and writers.

Students ignore the tedious background research and

methodical plotting of storylines that writers must

perform and, furthermore, have a narrow conception of

the myriad careers writers pursue other than writing

novels. Likewise, mathematicians use many of the skills

the students' in Brush's and Rotter's studies

attributed only to liberal arts types of professions.

Brush (1980) writes:


A mathematician in her or his research
must be flexible in the ideas she or he is
juggling, prepared to reject others' frames
of reference and create a new image of a
problem. This flexibility and creativity
--this independence from givens--seems
antithetical to the common notion of rational
thought as a linear, clearly defined process
of arriving at a conclusion. (p. 234)


Mathematicians, themselves, may even foster these

stereotypes. Some have admitted that they did not want

those outside of the field to learn about what they

were doing. It was a tradition to share discoveries













only with those who intimately understood their work,

but unfortunately, this attitude prevented persons in a

position to support funding for mathematical research

from having the necessary information (McDonald, 1986).

An article in the Chronicle of Higher Education

points out that unlike other scientists who work with

concrete objects or substances, the realm of the

mathematician is largely abstract or imaginary. Often,

his or her work is solitary, done without the help of

technical staff or graduate and undergraduate

assistants. Irving Kaplansky, director of the

Mathematical Sciences Research Institute in Berkeley

has said, "It's the only profession I know of where you

can lie at home with your feet on the couch and tell

your wife that you're working" (McDonald, 1986, p. 5).

However, even mathematicians who thought their

work too abstract to ever be used for practical

purposes have been proved wrong. Increasingly,

scientists have found solutions to complex problems in

mathematical theories developed decades earlier. The

Chronicle reports that mathematicians often have

difficulty distinguishing their work from that done by

physicists, astronomers, and economists (McDonald,

1986, p. 5).













Ralph Slaught, chairperson of the Department of

Philosophy at Lafayette College, theorizes that most

colleges and universities hold the belief that
"aristocrats do not get their hands dirty" and thus

separate the "thinkers" from the "doers" (1987, p. 38).

He quotes from both James Adams, a Stanford engineer

who recalled how the Greeks provided formal education

only to the elite with foreigners, slaves, and

businessmen doing "work" and from John Dewey, who

proposed that "our distaste for work itself extends to

those who must do it" (Slaught, 1987, p. 38). Slaught

warns that we cannot merely take courses "about"

technology, but must be conversant with the methods of

science. "We cannot make intelligent decisions about

how to deal with technology without an adequate

understanding of the processes involved. Without

that we are forced to defer to the expert or

pseudo-expert and in doing so we surrender some of our

freedom" (Slaught, 1987, p. 39).

Somehow, we must impress upon our young people the

gravity of the consequences about being ignorant about

the principles underlying advanced technology. The

image of socially inadequate, isolated mathematicians

and scientists must be replaced by that of vital,














creative, and powerful contributors to society. The

classroom is the most obvious place to look for both

the birth of negative stereotypes of mathematicians and

the means to change these images.

The Effect of Teaching Strategies on Mathematics
Attitudes and Achievements

The preceding section focused on the effects of

myths about the personality characteristics of people

who are successful in math. Kogelman and Warren (1978)

have also uncovered myths about the field of

mathematics itself. Many believe that there is only one

right way to arrive at a correct answer and that

mathematicians divine these answers by having superior

powers of memory and logic and by knowing the "tricks"

inherent in mathematics.

Many educators and mathematicians have laid the

blame for these false beliefs at the feet of the

schools. Whereas teachers cannot be held responsible

for all of our nation's problems with mathematics, they

are often the first people to introduce mathematical

concepts to children and thus have great influence in

the development of attitudes toward the subject.

Greenwood (1984) describes the typical approach to

teaching mathematics as the "explain-practice-memorize"













teaching paradigm, of which he has numerous criticisms.

This method gives little or no attention to developing

children's logical thought processes or reasoning

abilities that are the basis for mathematical

principles (Greenwood, 1984). Rather it focuses on

computing "right" answers and instills in children the

feeling that there is a "trick" or magic solution to

math to which they do not have access but must accept

on faith. As Greenwood (1984) says, this

memorization-based approach perpetuates the notion that

mathematics is a subject that "appears easy and

logical to a few 'brains' and incomprehensible to most

common folk" (p. 663).

Another commonly voiced criticism is that basic

math skills are taught as distinct from higher order

problem solving. James Sandefur, a mathematics

professor at Georgetown University, asks us to imagine

that students took two years of grammar, one year of

spelling, and then spent two years studying authors'

use of symbolism before they finally read a novel

(Sandefur, 1987). Although this method of teaching

English seems ridiculous, Sandefur sees mathematics as

being taught in an equally absurd fashion.













Rarely do teachers of algebra teach formulas

within the context of solving a particular problem, and

because so few students perservere in math to take

applied courses, many never realize the importance of

math (Sandefur, 1987). Sandefur agrees with Slaught

(2/18/87) when he proposes that mathematicians enjoy

the "mystique" of their science and "don't want to

defile...[their] art by teaching applications" (1987,

p. 38). He criticizes his colleagues for failing to

develop new courses, as do professors in other

disciplines. Because non-mathematicians know so little

about the subject and are so easily intimidated by

those who do, it is the responsibility of the

mathematicians to develop new educational frontiers in

thier field (Sandefur, 1/21/87).

The International Association for the Evaluation

of Educational Achievement (1985) recently published

its Second Study of Mathematics Summary Report in which

it related the poor math performance of American

students to the manner in which the subject is taught.

In eighth grade, American students scored only slightly

above the international average in computational

arithmetic, whereas they were already significantly

below the average of other countries in problem solving














capabilities. However, by the twelfth grade, calculus

students in the United States ranked as low as the

bottom quartile among international students.

The study concludes by describing the American

eighth grade curriculum as "low intensity," dealing

only superficially with topics in the space of one or

two class periods. The authors of the study propose

that this approach prevents students from developing a

firm conceptual foundation on which to learn other

forms of mathematics. This method was most notably in

contrast with the more "intense" approach favored by

the Japanese. Furthermore, the study states that the

United States high school curriculum is highly

compartmentalized, teaching algebra I, geometry,

algebra II, and then trigonometry, analytic geometry,

or calculus in the fourth year. In other countries

mathematics is taught in a more integrated manner so

that students may see the relationship between

different areas of math. The authors of the study

recommend that the fragmented, low-intensity approach

to teaching mathematics in the United States be

forsaken in favor of a more integrated curriculum.

After graduating from high school in the United

States, students may perceive mathematics to be a














rather abstract discipline with few applications

outside of the classroom. If they pass their courses

using the "memorization" approach, they may graduate

from high school with little recall for the material

taught by their math teachers. These students may feel

they have missed the "trick" of doing math; they may

conclude that they do not have a "mathematical mind"

(Kogelman & Warren, 1978).

It is not difficult to see how these mathematical

myths about the proper way to do math and the proper

personality or temperament for mathematics came about.

In our system of math education, those students who

have a preference for logic, rote memorization, and

non-verbal approaches are more likely to succeed in a

classroom that emphasizes such a learning style. But

that is not to say that persons with alternative

learning preferences cannot enjoy or achieve in math.

A recent attempt at updating the strategies for

teaching math in California met with limited success.

Textbook manufacturers spent millions of dollars on

revisions designed to overcome negative attitudes and

outdated instruction methods only to be met with

resistance from teachers. Teachers continued to choose














traditional books because they looked and felt familiar

(All Things Considered, 1987).

Research on learning styles has already shown that

children learn more quickly and are able to master

subjects which they previously found impossible when

they are taught through their indiviudal learning

styles (Hodges, 1983). Unfortunately, those who become

teachers often learn only one method of teaching math

and they lack the knowledge to use alternative

approaches.


Teacher Attitudes Toward Mathematics

The problems with mathematics education are not

confined to the manner in which the concepts are

taught. Of equal concern are the attitudes about the

importance of mathematics that may be transmitted by

instructors. Bulmahn and Young (1982), a mathematician

and a psychologist, conducted a descriptive study to

investigate the mathematical experiences and attitudes

of elementary education majors in contrast to other

students.

Bulmahn and Young (1982) asked students to

complete a questionnaire which included questions

regrading demographic data, perceived difficulty of

academic disciplines, math courses taken, level of














anxiety about doing math, and its implication for

career choice. Not surprisingly, students favored

subjects in which they did well. There was a consistent

relationship between preferences for mathematics and

science and language and social studies, however

correlations between the two discipline dyads were

insignificant or negative.

In the second part of the study, students were

asked to compose essays on their mathematical

background. The investigators reported a large number

of students who revealed that math had been their worst

subject and who felt that their dislike for the subject

had limited their career opportunities. What was most

disturbing to Bulmahn and Young (1982) was the belief

held by many of these students that it was not really

necessary for them to know mathematics beyond the basic

computations. They concluded that for many elementary

education teachers, "mathematics is at best a necessary

evil" (Bulmahn & Young, 1982, p.55).


In general, the kind of person who is drawn
to elementary school teaching is not
necessarily the kind of person who enjoys
mathematics in the broad sense--from its
logical beauty to its real world
applications. As a matter of fact, these two
areas of preference, elementary school
teaching and mathematics, may have some













inconsistencies between them. (Bulmahn &
Young, 1982, p.55)


Bulmahn and Young (1982) express alarm at these

attitudes, yet their suggestions and conclusions

perpetuate some of the worst stereotypes about select

groups of people who can "do math." They suggest that

further research be done on whether being good at math

is incompatible with being a good elementary teacher

and whether people who are interested in math are

likely to be interested in teaching. They even suggest

that math specialists be placed in schools to

compensate for lack of teacher knowledge and expertise.

Although this is a possible short-term solution and one

the teachers might even welcome, it would seem more

important to change the negative attitudes of those

planning a career in elementary education, especially

when school systems are having difficulty paying

teachers' salaries, let alone the salaries of special

math experts.

To say that only certain female, verbal,

humanistic personality types can be teachers is equally

as bad as saying that only white, male, logical,

uncreative people can be mathematicians. Rather than

devising schemes to match personality types with our













stereotypes of what qualities certain professions

demand, educators should consider putting their

energies into disputing preconceived notions that so

many students have about careers.

Recently, there have been some attempts to help

teachers become aware of the way students can best

learn mathematics. One of the most inventive and

successful programs is at Indiana University, where

non-math/science professors are reliving their college

days by attending undergraduate physics classes. The

aim of the project is to have skilled, motivated

learners help physics professors devise better ways to

communicate their subject matter. Sheila Tobias, who

directs the project, explains:


Students who could critique the class-those
for whom the material doesn't make
sense-either fail or simply drop from the
rolls. Often the instructor never finds out
what is wrong. Faculty members have the
confidence, self-awareness, and ability to
analyze why they find something difficult
(McMillen, 1986, p. 18).


The professors take an introductory course

required for non-majors who plan to take science or

math courses. Over a quarter of the students withdraw

before the end of the semester, thus disqualifying

themselves from many majors. Many of these drop-outs













are women and minorities, according to the professor

who teaches the course.

The Department of Physics at Indiana University

has already made significant changes in the laboratory

sections of the class, emphasizing discussion and

writing, rather than a purely quantitative analysis of

the class material. The discussion leader employs the

Socratic Method, and the lab is called, ironically,

S.D.I., for Socratic Dialogue Inducing.

But even with these changes, the professors, those

proven-successful learners, become anxious. A

psychology professor found his "whole life flashing

before. .[him]. .it was the math that stumped me",

he said (McMillen, 1986, p. 18). A professor of English

said that he felt rushed and would think, "I don't

understand this. .1 have never understood this. The

problem starts when math is introduced to the course"

(p. 19). It is not difficult to imagine what

undergraduates feel when even experienced academicians

lose confidence in their ability to grasp scientific

and mathematical concepts.

It is somewhat understandable that non-math and

science professors feel anxious about material that

they have not studied for years, if ever. However when














people who are teaching math are anxious the

consequences are much more serious. Kelly and Tomhave

(1985) compared several different age groups who had

little or no math preparation, a group of students

participating in a workshop for math anxiety, and a

group of elementary education majors on their levels of

anxiety about mathematics. The elementary education

majors were the most anxious of any group except the

workshop participants; however the male elementary

education majors scored the lowest for math anxiety.

Because females make up the majority of elementary

school teachers, Kelly and Tomhave (1985) conclude that

elementary teachers may be passing on their own anxiety

about mathematics to the girls in their classroom. They

hypothesize that students surrounded by confident,

enthusiastic, and sensitive teachers, familiar with a

variety of strategies for teaching math, will be less

susceptible to negative feeling about math than

students whose teachers are anxious, negative, and

uncomfortable with their own ability to teach and to

learn mathematics (Kelly & Tomhave, 1985, p. 53).


Mathematics Anxiety

The majority of the research that has been done in

this area has come under the heading of mathematics














anxiety. In many cases, investigators have focused on

the "symptoms" of people who do not enjoy math or who

cannot succeed academically in the subject. Counselors

and educators have treated math anxiety as a unified

construct, yet more thorough investigations have shown

this not to be the case. There have been many studies

which have investigated the attitudinal and personality

variables which put one at risk for math anxiety. The

following section will discuss some of these findings.


Definitions of Math Anxiety

In the Elementary School Journal, Reyes writes,

"One of the difficulties with the mathematics anxiety

literature is in understanding what mathematics anxiety

is. It might be described as anxiety about mathematics,

but to understand the description it is necessary to

know what anxiety is" (1984, p.563). Webster's

dictionary describes anxiety as


An abnormal and overwhelming sense of
apprehension and fear often marked by
physiological signs (as sweating, tension,
and increased pulse), by doubt concerning the
reality and nature of the threat, and by
self-doubt about one's ability to cope with
it.


There have been many definitions of math anxiety,

some of which were discussed in Chapter One. The most













widely used is that of Richardson and Suinn (1972), in

which math anxiety is described as "involving feelings

of tension and anxiety that interfere with the

manipulation of numbers and the solving of mathematical

problems" (p.551). Fennema and Sherman's (1976)

definition is similar, if not more intense, "feeling of

anxiety, dread, nervousness, and associated bodily

symptoms related to doing mathematics" (p.4). Even

Gough's 1954 work in the area had number anxiety as

"the presence of a syndrome of emotional reactions to

arithmetic and mathematics" (p. 344).

All of these definitions concentrate on

physiological and emotional factors. However, as

Webster's definition implies, there are cognitive

factors involved as well. In fact, only the

dictionary's definition of anxiety gives a clue to why

the feeling exists--doubt about one's capacity to cope

with a threat. One can infer that if there were no

questions about an individual's ability to deal with a

threatening situation he or she would not have the

feeling of being overwhelmed.

One of the major problems with the literature on

math anxiety is that definitions are descriptive rather

than causal. When students avoid math, do poorly in the













subject, or even feel physically uneasy about the

prospect of facing problem solving, researchers have

called them "math anxious"; however this label gives us

few clues as to the causes of these behaviors.

Frary and Ling (1983) conducted a study

hypothesizing that mathematics attitudes might be the

result of some other stable personality measures.

Results of the factor analyses showed that the most

significant factor represented four of the five math

attitude scales as well as a moderate loading of test

anxiety. Because the math anxiety attitude scale had a

loading of .89, they designated this factor as

"mathematics anxiety." Finding little or no

relationship between the personality measures and math

anxiety, led them to conclude that "mathematics anxiety

is relatively superficial, perhaps responsive to simple

persuasion or desensitization" (Frary & Ling, 1983, p.

990).


Relationships between state, trait, and test anxiety

Reyes (1984) suggests that math anxiety be viewed

within the context of the literature on anxiety as a

general psychological construct. One of the most

significant theories of anxiety was put forth by

Spielberger (1972) who divided the construct into state














and trait anxiety. State anxiety is defined as an

unpleasant emotional state or condition which is

characterized by activation or arousal of the autonomic

nervous system (p. 482). This type of anxiety is linked

to specific times and situations and is evident when a

person feels threatened by a certain place or event.

Trait anxiety, however, is viewed by Spielberger as a

relatively permanent personality trait which the person

experiences across situations.

More recently, Byrd (1982, cited in Reyes, 1984)

presented a model of how a person reacts to an

anxiety-provoking situation. The individual meets with

a situation which she or he must experience as

threatening. The threat may produce physiological

reactions which are not easily controlled; however

behavioral reactions (nail biting, eating, etc.) are

more subject to conscious control. During a period of

cognitive reappraisal, the person decides how to cope

with the stressor. Some anxiety might actually improve

performance facilitativee anxiety) while more often the

anxiety hinders performance debilitativee anxiety).

In the case of mathematicics, any situation

involving mathematical computations might be perceived

as a threat and individuals may choose to cope with the













threat by avoiding the task or rationalizing that it is

not worthwhile. Of course, they could also choose to

study harder or to get assistance with the problem,

although this happens less frequently!

There have been many questions concerning the

differences between mathematics anxiety and test

anxiety. Wine (1971) has described the person suffering

from test anxiety as focusing on the self and

concomitant physiological reactions to the detriment of

the task performance. Test anxiety is almost always

debilitative because of the individual's lack of

attention to the task. Both Wine (1971) and Liebert and

Morris (1967) have identified two components of test

anxiety: worry and emotionality. Morris and Liebert

(1970) found that worry, the cognitive concern about

performance, was negatively correlated with test

performance. Emotionality, the arousal of the autonomic

nervous system, was not significantly related to

performance.

A number of studies have tried to uncover the

relationship between general anxiety, test anxiety, and

mathematics anxiety. Betz (1978) found a correlation of

-.28 between scores on the Fennema-Sherman Math Anxiety

Scale (MAS, Fennema & Sherman, 1976) and the A-trait














scale of the State-Trait Anxiety Inventory (STAI,

Spielberger, Gorsuch, & Lushene, 1970) and -.42 between

the MAS and the Test Anxiety Inventory (TAI,

Spielberger, Gorsuch, Taylor, Algaze, & Anton, 1978

cited in Reyes, 1984; high anxiety on the MAS produces

a low score; high anxiety on the STAI and TAI produces

a high score).

Hendel (1980) uncovered a correlation of .65 between

the MARS (Richardson & Suinn, 1972) and the Suinn Test

Anxiety Battery (Suinn, 1969) for 69 adult women who

had enrolled in a math anxiety reduction course. A

regression analysis indicated that for these women,

test anxiety was the most significant predictor of math

anxiety.

Dew, Galassi, and Galassi (1983) asked 769

students at the University of North Carolina at

Greensboro to complete a number of measures in order to

untangle the relationship of math anxiety to test and

general anxiety. Students filled out the MARS, the MAS,

the Sandman Anxiety Toward Mathematics Scale (ATMS,

Sandman, 1974, cited in Dew et al., 1983), and the

STAI. All three of the math anxiety measures were

moderately and more closely related to each other

(37.2%-62.4%) than to measures of test anxiety













(11.6%-36.%). Dew et al. (1983) suggest that Hendel's

findings may be due largely to his use of the MARS and

the STABS, both of which were constructed by Suinn and

which share items in common.

Finally, Dew et al. (1983) found that math anxiety

was equally related to both the emotionality and worry

component of test anxiety. Because worry is supposed to

be the more stable component of test anxiety, it was

expected that the correlation between worry and math

anxiety would be greater than for emotionality. Dew et

al. (1983) accepted this as further evidence that math

anxiety and test anxiety are not identical constructs.

In 1984, Dew, Galassi, and Galassi further

investigated the relationship between test, math, and

general anxiety. Sixty-three undergraduates completed

the measures listed in the description of their 1983

study, but in addition, students completed the

emotionality and worry components of the Deffenbacher

Post-Task Questionnaire and three mathematical problem

sets. Students also had their heart rate, skin

conductance, and skin fluctuations monitored.

As in the 1983 Dew et al. study, math anxiety

measures were more closely related to each other than

to test anxiety. The MARS appears to focus more on test














related math anxiety and situational worry than either

the MAS or the ATMS. None of the math anxiety nor test

anxiety measures accounted for variance in the

students' problem set performance above and beyond a

measure of math ability (SAT-M). However, the authors

caution that the SAT-M is not a "pure" measure of

innate math aptitude and could be influenced by years

of previous anxiety and negative attitudes about math.

Physiological measures bore little relation to

math anxiety; however this finding could have been the

result of the confounding of assessment

characteristics. This was also the case for avoidance.

Dew et al. (1984) measured avoidance by the number of

problems left uncompleted or completed out of order.

They hypothesized that after individuals are forced to

confront the anxiety-provoking situation, they

demonstrate few avoidance behaviors. The authors

recommended that avoidance of math problem solving

situations would be a more appropriate criterion

variable.

It would appear that although math anxiety and

test anxiety share some common variance, no study has

demonstrated a large enough correlation to dismiss math

anxiety as purely a manifestation of test anxiety in














mathematical evaluation situations. However, neither

construct is very useful unless it affects performance.

Reyes (1984) cites a number of studies that have shown

a relationship between high anxiety and low achievement

(Aiken, 1970; 1976; Betz, 1978; Callahan & Glennon,

1975; Crosswhite, 1972; Sarason, Davidson, Lighthall,

Waite, & Ruebush, 1960; Szetela, 1973). Frary and Ling

(1983) found that students with higher levels of math

anxiety tended to receive lower course grades, had

lower grade point averages, and took fewer math

courses.

Llabre and Suarez (1985) found that math anxiety

was unable to contribute significantly to the

prediction of algebra grades for college men and women

after controlling for math aptitude. They acknowledge

that the math component of the SAT is likely not a

"pure" measure of aptitude and could be confounded with

anxiety, a problem they say "plagues math anxiety

research" (Llabre & Suarez, 1985, p. 286). They also

found that math anxiety is less specific in men,

sharing 24Z of the variance with general anxiety, as

opposed to 4% for women. The authors saw implications

for treatment, recommending that women could more than














men interventions specifically designed for dealing

with math situations more than men.

Reyes (1984) pointed out that the relationships

between anxiety and performance have been correlational

rather than causal. Interventions that have been able

to reduce anxiety have not always been successful in

improving performance, producing further questions

about the power of anxiety to reduce performance

directly (Reyes, 1984).

Ultimately, identifying "math anxiety" as a

construct in its own right produces more questions than

answers. One of the areas about which there has been

the most controversy has been the differences in math

anxiety and performance between men and women. The

following section will discuss the research in this

area.

Sex Differences in Mathematics Achievement, Attitudes,
and Anxiety

In their much quoted book, The Psychology of Sex

Differences, Maccoby and Jacklin (1974) were among the

first to review the literature on the differences

between boys' and girls' quantitative abilities. They

noted that up until junior high there were few













differences, but that after age fourteen, boys'

performance in mathematics was superior to girls'.

In 1960, the Project TALENT studies reported no sex

differences in math achievement for ninth graders, but

by the senior year in high school, males did slightly

better (Flanagan, Davis, Dailey, Shaycroft, Ori,

Goldberg, & Neyman, 1964). Similar differences favoring

males have been found in international studies and

longitudinal studies done in the United States. Wilson

(1972) showed that males excelled at higher cognitive

tasks related to application and analysis but that

women were superior on lower level cognitive tasks. The

California Assessment Project (CAP, 1978) found no

overall differences in math performance between the

sexes in sixth grade, but found twelfth grade girls to

be weaker than their male peers in measurement

applications, geometry applications, and in probability

and statistics. Gifted males and females have been

found to perform differently on the quantitative

section of the SAT when the test is taken in junior

high. Fox and Cohn (1980) demonstrated that males

scored higher than females during each of the six years

they conducted their talent search.














Physiological Explanations of Sex Differences

There have been some attempts to explain these sex

differences as the result of innate, physiological

mechanisms. Some of these studies have involved the

investigation of differences in spatial ability since

it has been found to be related to math performance

(Fennema & Sherman, 1977; Mellone, cited in Maccoby &

Jacklin, 1974). Some of the hypotheses have included a

recessive gene for spatial ability located on the X

chromosome. The idea is that the chance of girls

inheriting the two X-linked, recessive genes necessary

for the trait to manifest itself is less than the the

chance for boys inheriting the one recessive gene

needed. Therefore, women tend to achieve less on tests

of math and spatial ability (Stafford, cited in Maccoby

& Jacklin, 1974). Studies by Bock and Kalikowski

(cited in Boles, 1980), DeFries et al. (cited in Boles,

1980), and Fennema and Sherman (1978) have disputed

this hypothesis.

Levy (1976) has advanced a hypothesis about

spatial performance and hemispheric dominance, in which

males are more similar to left handed males because

their brain hemispheres are less specialized for verbal

versus spatial-gestalt functioning. Levy tested














graduate and post-doctoral students using the WAIS

Performance scale and a test of verbal reasoning and

found that left-handed men did more poorly on the

performance scale than on the verbal. Their performance

IQ was significantly lower than the right handed males.

Levy hypothesized that optimal intellectual functioning

occurs when the right hemisphere is specialized for

gestalt, spatial functioning. She believes that left

handers have poorer spatial ability because they have a

greater frequency of verbal functioning in both

hemispheres (Levy, 1976).

Sherman (1977) found that although left-handedness

was relatively detrimental to male spatial scores and

beneficial to verbal performance, the reverse was true

for females. Spatial scores of left-handers were never

significantly lower than for right-handers within sex.

There were no significant differences between boys and

girls for math achievement in the ninth and tenth

grades; however both left and right handed boys scored

higher in math achievement in eleventh grade.


The Differential Coursework Hypothesis

Elizabeth Fennema and Julia Sherman challenged the

assumption that males always achieve more highly than

females. They hypothesized that sex differences were














caused by some third factor, namely the differing

numbers of math courses taken by girls and boys. In

addition, they suggested that courses involving spatial

relations and outside activities that included both

mathematics and spatial relations also contributed to

scores on standardized tests of mathematics.

In 1977, Fennema and Sherman tested male and

female students at four Wisconsin high schools. The

cognitive variables assessed were math achievement,

general/verbal ability, and spatial visualization. They

also measured the following affective variables:

attitude toward success in math, stereotyping math as a

male domain, perceived attitudes of mother, father, and

teacher toward the student as a learner of math, and

perceived usefulness of math. Other variables examined

were the number of courses with math prerequisites, out

of school math activities, and courses and activities

involving spatial skills.

Although females tended to score higher on

measures of verbal ability, the differences were not

significant. Males scored higher on the math

achievement and spatial ability tests; however the

differences were significant at only two schools. Boys

tended to engage in more math activities outside of














school, to take more math-related courses after the

ninth grade, and to take more courses requiring spatial

ability, differences that were significant at two

schools. At almost all of the schools, boys were more

confident about their ability and rated math as more of

a male domain.

At School 1, when scores of confidence in math

ability, perceived attitudes of parents, perceptions of

math as a male domain, and usefulness of math were used

as covariates, the sex differences in math achievement

became non-significant. At School 4, when both spatial

visualization and the six significantly different

affective variables were covaried out, achievement

effects also became non-significant. When the numbers

of math and spatial related courses were employed as

covarlates, differences in spatial visualization scores

at Schools 2 and 4 disappeared.

DeWolf (1981) conducted a similar test of the

differential coursework explanation of achievement

differences on students from Washington state high

schools. Students were juniors taking the Washington

Pre-College Test (WPC) which included the following:


1) Quantitative Skills A--ability to
determine whether enough information is
given to solve a problem














2) Quantitative Skills B--ability to
determine the relative size of the given
qualities

3) Applied Mathematics--applying
knowledge of arithmetic and elementary
algebra to solve practical problems

4) Mathematics Achievement--knowledge of
algebra and geometry

5) Spatial ability--ability to visualize
transformations in three dimensions

6) Mechanical Reasoning--ability to
understand physical principles as
applied to mechanical devices


Each student's transcript provided the grades and

the semester credits for total physics coursework and

for math coursework divided into general math, algebra,

geometry, and advanced mathematics. Mathematical

electives such as accounting and business math were not

included in the analysis.

Males scored significantly higher than females on

all six WPC tests and took a significantly greater

number of courses in algebra, geometry, advanced math,

and physics. The only difference favoring females was

in overall math grade point average. The multiple

regression analysis showed that sex accounted for a

significant amount of the variance in performance for

Quantitative Skills B, applied mathematics, and

mechanical reasoning.














When the amount of coursework in general math,

algebra, geometry, advanced math, and physics was

controlled, sex did not predict Quantitative Skills A

and math achievement. Coursework alone accounted for

the largest proportion of the variance of Mathematics

Achievement (51%) and 29% of the variance of

Quantitative Skills A. Spatial ability was uninfluenced

by sex once math coursework was controlled; however

Quantitative Skills B, Applied Mathematics, and

Mechanical Reasoning remained subject to prediction by

sex, regardless of the control exercised over

coursework preparation.

Although not all differences can be attributed to

differential patterns of coursework, DeWolf's findings

support Fennema and Sherman's hypothesis that sex

differences in math achievement are due in large part

to environmental differences, rather than to innate

physical factors. DeWolf hypothesizes that the sex

differences that remain in her study could be due to

bias in item content and inadequate control for other

coursework that may teach math and spatial skills.

Two researchers who have held firm to the idea

that boys have a greater tendency to excel in math than

girls are Camilla Benbow and Julian Stanley of Johns














Hopkins University. In their studies of gifted junior

high students who took the SAT they found that by age

thirteen, the gap between boys and girls begins to

widen significantly, especially in the higher end of

the score distribution where boys outnumbered girls

13:1 in scores over 700. Benbow and Stanley (1983)

stated that because boys and girls were matched for

age, intellectual ability, grade, and number of math

courses, differences could not be due to sex bias or

differential course taking.

Senk and Usiskin, in a less publicized article

(1983), offered an explanation for these differences.

They administered a group of geometry tests to students

in five states chosen to represent a cross section of

educational, socio-economic, and ethnic backgrounds.

Students were given a 25 minute test for their entering

knowledge of geometry at the beginning of the school

year (EG) and a geometry proof test and a multiple

choice geometry achievement test (CAP) at the

conclusion of the year.

The three forms of the proof test were not

equivalent in difficulty so the results are reported

separately. Raw mean total correct were higher for

males on two forms and for females on one form, but














none of these differences were significant. Mean scores

for girls on the test of entering geometry were

significantly lower than the scores for boys. When EG

scores were used as variables in computing proof test

total scores, girls scored higher than boys on each

form and significantly higher on form three. Therefore,

even though females begin their high school geometry

course with less knowledge than do males, at the end of

the year there are no consistent differences on the

solving of geometry proofs. Senk and Usiskin (1983)

find these results particularly striking because boys

scored significantly higher on the geometry achievement

test.

In response to Benbow and Stanley (1980), Senk and

Usiskin (1983) studied subsets of high performing

students. Of the 71 students who had perfect or nearly

perfect scores on any one of the three proof test

forms, 37 were female and 31 were male. The second

subset consisted of 12 girls and 7 boys who were in the

seventh and eighth grades, and had thus accelerated at

least two years. There were no sex differences on the

proof test, either adjusted or unadjusted. The third

subset consisted of students who scored in the top 3%

on the CAP, comparable to the Benbow and Stanley













requirement that students score in the top 3% on a

standardized measure of math achievement. In this case,

there were 58 males and 31 females, with significantly

more males than females scoring at the upper end of the

test. These results are similar to Benbow and Stanley

(1980;-1983) only on the multiple choice CAP test;

there were no sex differences for the proof test.

Senk and Usiskin (1983) conclude that "the more an

instrument directly measures a student's formal

educational experiences in mathematics, the less the

likelihood of sex differences" (p. 9). Because the

large sex differences in the Benbow and Stanley studies

occur only on tests in which the items are designed to

be unfamiliar to students, Senk and Usiskin suggest

that scores could easily be affected by experiences

outside of formal math instruction such as math

contests, computer use, and outside reading, all of

which claim more male participants (Senk & Usiskin,

1983). Because geometry proof writing is rarely

encountered outside the classroom, boys and girls are

more likely to have had equal exposure both inside and

outside of the classroom.














Sex Differences in Math Anxiety

Sex differences in math anxiety have also been

reported, often as a cause of sex differences in math

performance (Ernest, 1976; Sells, 1973; Tobias, 1976).

Dew et al. (1983) found that women had significantly

higher scores on the MARS and the MAS but not on the

ATMS. However, women scored higher on the other

measures of trait and test anxiety as well, suggesting

that women's tendency to report emotions more often

than men may be a factor here. Resnick et al. (1982)

found no significant sex differences on the MARS and

suggest that studies that have found differences may be

focusing on a unique group of women who have been away

from formal math instruction for a long time (Rounds &

Hendel, 1980).

Once again, math anxiety emerges as a construct

with little ability to predict behavior and academic

performance. When Tobias (1976) first discussed math

anxiety, she viewed it as an explanation for the

underrepresentation of women in math courses and math

careers. Although she does discuss the various ways in

which women are discouraged from pursuing math courses

and careers, it is not clear that these factors are

causing women to be anxious about math.















Negative Influences on Women's Participation in Math

At the same time that researchers find that the

gap in math achievement is widening for girls and boys,

males also begin to exhibit a greater interest in

pursuing mathematics coursework. Brush (1980) found

that boys show a significantly greater desire than do

girls to take high school math, and Fox (1980)

discovered that even when girls and boys do not differ

in their intentions to study higher level math, the

actual female enrollment in calculus is significantly

less than for males. This underenrollment takes its

toll on the numbers of women who pursue careers in

math.


Teacher influences

There are a number of environmental factors that

may conspire to dissuade women from enrolling in math

courses. As has been previously stated, the numbers of

female elementary teachers who dislike math are large.

These women may unintentionally encourage their

admiring female pupils to emulate them by avoiding math

(Kogelman & Warren, 1978). Teachers have also been

found to devote more attention to developing math

skills in boys as early as the second grade (Fox,

1980). Boys are more often called upon, spoken to, and














asked questions than are girls (Becker, 1981; Skolnick,

Langbort, & Day, 1982).

Teachers and counselors encourage young women to

pursue future math study and careers far less often

than they do boys (Ernest, 1976). Luchins and Luchins

(1980) interviewed women and men who had chosen careers

in math, and found that 24% of the former recalled

being discouraged by high school teachers as compared

with 2% of the latter. Frazier and Sadker (1973) found

that over 75% of teachers' criticisms of boys pertain

to improper conduct or problems with neatness, while

almost all of the praise they receive is for

intellectual and academic achievement. For girls,

however, the process is reversed; they are praised for

neatness and obeying the rules and criticized for their

lack of academic achievement.

A common message sent to boys is "you will succeed

if you just settle down and try" (Skolnick et al.,

1982). When boys fail, teachers attribute their

performance to lack of effort six times more often than

they do for girls (Dweck, Davidson, Nelson, and Enna,

1978). Boys are then encouraged to try again, while

girls are praised merely for trying at all. In this

way, girls are taught to be compliant and rigid and to














refrain from being creative, autonomous, and analytic,

skills essential to the study of mathematics (Frazier &

Sadker, 1973).


Family influences

Many of the same problems of lack of encouragement

and lowered expectations for girls' success in

mathematics arise in the home. Skolnick et al. (1982)

see the beginning of a cycle of math anxiety and

avoidance with mothers of daughters. Ernest (1976)

found that after grade six, both boys and girls tend to

seek help with their math homework from their fathers.

Both parents generally agree that the father is the
"math expert."

Armstrong (cited in Fox, 1980) found that fathers'

educational expectations for their daughters were the

most significant predictors of girls' math enrollment.

Unfortunately, most fathers do not expect their

daughters to perform as well as their sons in math

(Ernest, 1976; Fox, 1980; Hilton & Berglund, 1974). In

interviews with highly creative mathematicians, most

women were seen as being highly identified with their

fathers.

Women who have pursued careers in math are likely

to have no brothers or to be only children (Helson,














1971). Given that fathers are usually the parent to

whom children turn for expertise in math, this is not

surprising. If fathers expect their daughters and sons

to follow in the footsteps of the same sex parent, only

sons would be encouraged or expected to pursue math.

However, if there are no sons, fathers who believe math

to be important will have only their daughters to

encourage.


Role strain

Apart from the lack of encouragement and guidance

young women receive, they also face conflicts in terms

of sex role expectations from their peers. As was

mentioned above (Brush, 1980) the common

characteristics associated with success in mathematics

are viewed as masculine traits. For an adolescent

female hoping to be accepted by male peers, excelling

in math is not the key to success. Young women who are

particularly good in math have reported that their

peers consider them strange and that they have been

advised to follow more "traditional" female career

paths so that they do not alienate themselves from men

(Luchins & Luchins, 1980).

To avoid becoming social outcasts, young women

will often reduce this role strain by "playing dumb."














Sherman (1982) interviewed young women who had taken

four years of math in high school. Although only 17%

admitted to feigning ignorance in situations where they

could have provided expertise, 76% accused other young

women of "playing dumb!"

This phenomenon has been documented in girls as

young as ten-years-old. Steinkamp and Maehr (1984)

reported that when asked about their desire to take

courses in science, girls were less likely to show high

motivation when a male "visitor" was present, when

questions were administered orally, or in a small group

situation. The authors concluded that the girls were

reluctant to demonstrate interest because they were

aware of the masculine image of scientific study

(Steinkamp & Maehr, 1984). Luchins and Luchins (1980)

found that women mathematicians explain the paucity of

females in the field as the result of the pervasive

attitude that math is a male domain. Male

mathematicians explain the lack of women mathematicians

as due to women's lack of interest and inability to

think mathematically.

Hollinger (1985) has demonstrated the effects of

stereotypes on the career patterns of mathematically

talented women. Starting with research that has shown













the importance of self-perceptions of math ability on

success in math (Meece, Parsons, Kaczala, Goff, &

Futterman, 1982), Hollinger set out to uncover

differences in self-perceptions of math ability in

women aspiring to careers requiring differing levels of

mathematical and scientific knowledge.

Hollinger (1985) divided women into six

career-track categories: non-traditional math,

non-traditional science, neutral/traditional

math/science, nontraditional non-math, neutral

non-math, and traditional non-math. The three non-math

categories were combined since they did not differ

significantly. The analyses of variance indicated that

the four career aspiration groups differed on the

following self-perceptions of career relevant

abilities: mechanical ability, manual ability,

friendliness, math ability, artistic ability, and

science ability. These findings supported Hollinger's

(1985) hypothesis that the self-perceptions of multiple

abilities rather than just that of math ability

distinguishes non-traditional math career aspirants

from all other career groups.

The non-traditional math career group had

significantly higher self-perceptions of math ability














than the non-math career group. Interestingly, the

non-traditional math group also demonstrated

significantly lower self-perceptions of friendliness

and artistic ability than the other three groups. The

non-traditional science career aspirants reported

significantly higher estimates of math, mechanical, and

manual and science ability than the non-math groups but

not greater than the neutral/traditional math/science

groups. The only way in which the neutral/traditional

science career aspirants were different from the

non-math group was that the former reported higher

self-estimates of science ability. Hollinger (1985)

concludes that adolescents see their level of math

ability as a "threshold variable" and believe that the

level of competence in math required for neutral or

traditionally female math and science careers is lower

than for non-traditional math and science careers.

Hollinger's (1985) findings support previously

discussed results regarding stereotypes of

mathematicians. The low estimate of friendliness

reported only by the non-traditional math career group

"may reflect the influence of the stereotyped

perception of the mathematician as social isolate and

is consistent with findings indicating that women














mathematicians and women in non-traditional career

fields may be more aloof than other women" (Hollinger,

1985, p. 333). Another finding that supports the notion

of adolescent stereotypes is the low self-estimate of

creative ability demonstrated by the non-traditional

math group. This is a disturbing finding for those who

recognize the importance of creativity in mathematics.

Forbes (1985) sampled undergraduate women who had

declared either a non-traditional math/science major or

a traditional non-math/science major. When asked to

respond to questions about how their lives would be

affected as a result of pursuing a variety of

math/science and non-math science careers, both groups

experienced greater conflict when they imagined

themselves pursuing math/science careers as compared

with non-math/science careers. The greatest conflict

centered around the amount of time the women expected

that they would be able to spend with their family.

This effect was even stronger for the math/science

career aspirants. Sherman (1982) revealed a similar

finding. Young women who had taken four years of high

school math were more likely to worry about balancing a

family and career than were women who had taken fewer

years of math.














Summary

A great deal of the research on mathematics

achievement and avoidance has focused on sex

differences and the possible causes for women's

underrepresentation in math courses and careers.

Despite the fact that fewer young women than young men

are becoming interested in mathematics and the

sciences, the numbers of young men expressing interest

in these areas is still alarmingly small.

There are unique problems faced by young women,

however. Not only do they face the same problems of

curriculum and teaching strategies as do boys, but they

also experience outright discouragement from

significant adults and peers. One of the central themes

of this review has been the prevalence of the

stereotypes about mathematicians as socially inept,

uncreative, unemotional, logical, and isolated. While

these are not wholly pleasant qualities with which men

might wish to be associated, they run counter to the

traditional views of femininity to an even greater

degree.

Women seem to be under the impression that to

pursue a career in math and science will prevent them

from having a family life. It is not clear whether













women feel that they will literally not have time for

their family because of the demands of a math or

science career or that becoming a mathematician will

make them unattractive to potential partners. There is

indirect evidence that supports the latter. Rotter

(1982) showed that women engineers were seen as

uninterested in dating and lacking in social skills.

Hollinger's (1985) study demonstrated that the women

who aspired to math careers believed themselves to be

less friendly and Helson's (1971) women mathematicians

saw themselves as more aloof.

Boys are expected to do well in math by parents,

teachers, and counselors. They are given the

opportunity to experiment with mathematics and science

outside of the classroom. They are even supposed to

possess the personality and mental characteristics

believed to be necessary to be a successful

mathematician. Girls, however, have none of these

positive expectations and in fact, often experience

serious sex-role conflict if they are interested in and

talented in mathematics. There are very few, if any,

factors that can be relied upon to promote girls'

interest and participation in math and science.














But to focus exclusively on women and the

environmental factors which dissuade them from these

careers is to deny that boys are also seeking to avoid

math/science courses and careers. It is easy to make a

case for girls lack of participation, it is disturbing

that with all the positive expectations and training

they receive that boys still feel uncomfortable about

mathematics. An explanation must be proposed that

encompasses both the sex-role socialization literature

and account for the avoidance and anxiety demonstrated

by males.

The Effect of Self-Efficacy Expectations on Human
Behavior

The mechanism of self-efficacy has been developed

by Albert Bandura to account for a wide variety of

behavioral phenomena including coping behavior,

physiological stress reactions, reaction to failure

expectations, the development of intrinsic motivation,

career aspirations, and even math "anxiety" and

avoidance (Bandura, 1982). The following section will

describe Bandura's theory of self-efficacy in human

behavior, discuss the research that supports and

disputes the theory, and finally, posit it as an

explanatory mechanism for the widespread avoidance of













the avoidance of mathematics and the anxiety that

accompanies this avoidance.


Explication of the Theory


Knowledge, transformational operations, and
component skills are necessary but
insufficient for accomplished performances.
Indeed, people often do not behave optimally,
even though they know full well what to do.
(Bandura, 1982, p. 122)


Bandura's main interest is in how people come to

judge their own abilities and how these

self-perceptions affect behavior. "Perceived

self-efficacy is concerned with judgments of how well

one can execute courses of action required to deal with

prospective situations" (Bandura, 1982, p.122).

Individuals are constantly being called upon to make

decisions about their ability to cope with the world.

Self-efficacy beliefs are not trivial; attempting a

task without the requisite skills can be dangerous or

even fatal, in some cases.

Bandura (1977a) has shown that in general, people

avoid activities at which they know they cannot perform

well and engage in those about which they feel

confident. The degree of self-efficacy is also related

to the amount of effort expended and persistence in the














face of obstacles. People who have doubts about their

ability to cope will decrease their energy levels and

will often give up altogether, while those who firmly

believe in their ability will perservere and generally

perform at high levels.

People develop their judgments about self-efficacy

using four basic sources of information: 1) performance

attainments, 2) vicarious experiences of observing the

performances of others, 3) verbal persuasion and allied

types of social influences that one possesses certain

capabilities, and 4) physiological states from which

people partly judge their capability, strength, and

vulnerability (Bandura, 1982, p. 126).

Performance attainments, according to Bandura

(1982) are the most influential source of information.

Individuals can see firsthand whether they can or

cannot cope with a situation or task. Obviously,

successful experiences increase self-efficacy while

failure experiences lower it. The latter is the case

especially if the actor is expending adequate effort

and there are no negative environmental influences.

However, individuals cannot possibly "learn by

doing" for every possible circumstance. They must rely

on vicarious experiences as well. Brown and Inouye













(1978) showed that when a person sees someone whom he

or she believes to be of similar competence succeed, he

or she will raise his or her self-efficacy expectations

accordingly. However, when the model fails despite

trying hard, the observer lowers his or her efficacy

expectations. Besides giving social comparison

information, modeling can also provide facts about the

environment and its predictability and can teach

observers coping strategies for use when dealing with

difficult situations.

The third source of information for efficacy

expectations is verbal persuasion, used to influence

individuals to believe they have the requisite skills

to achieve their goals. Chambliss and Murray (1979)

have shown that verbal persuasion is especially

effective when an individual has at least a minimal

belief in her or his ability to successfully complete a

task. Bandura has stated that verbal persuasion boosts

self-efficacy enough that a person expends the effort

needed to succeed. It can also promote the development

of skills and a more permanent sense of personal

efficacy. Finally, individuals also look to their

physiological state for information about their

abilities. When people experience physical signs of













stress they will often interpret this as a sign of

weakness and vulnerability to failure.

However, Bandura posits that none of these four

sources of information is important on its own; they

become so only after coming under the influence of

cognitive appraisal.


The cognitive processing of efficacy
information concerns the types of cues people
have learned to use as indicators of personal
efficacy and the inference rules they employ
for integrating efficacy information from
different sources. The aim of a comprehensive
theory is to provide a unifying conceptual
framework that can encompass diverse modes of
influence known to alter behavior. In any
given activity, skills and self-beliefs that
ensure optimal use of capabilities are
required for successful functioning. If
self-efficacy is lacking, people tend to
behave ineffectually, even though they know
what to do. Social learning theory postulates
a common mechanism of behavioral change-
different modes of influence alter coping
behavior partly by creating and strengthening
self-percepts of efficacy. (Bandura, 1982,
p.127)


Bandura and his associates have conducted a number

of experiments designed to test this theory, many of

which have used snake phobia as the behavior to be

changed (Bandura & Adams, 1977; Bandura, Adams, &

Beyer, 1977; Bandura, Adams, Hardy, & Howells, 1980).

This disorder was chosen because of the minimal chance

that subjects would encounter snakes outside of the













experimental session, thus confounding the results of

the study. Persons are subjected to a number of

different treatment strategies. Those in the enactivee

mastery" group are gradually exposed to the fearful

situation with the assistance of "induction aids."

After these aids are withdrawn, subjects undergo

experiences where they confront their fears to verify

and then to generalize their efficacy.

In the vicarious treatment mode, subjects merely

observe a model performing increasingly more

threatening tasks without negative effects. The third

treatment modality involves the subject's imagination.

The assignments include the generation of images of

multiple models successfully coping with and mastering

the activity around which the phobia centers.

The results of these studies show that each mode

of influence can raise and strengthen self-efficacy

expectations. More importantly, behavior is closely

tied to the amount of change in the perception of

efficacy; the greater the level of perceived efficacy,

the greater the level of performance. Self-efficacy

also predicts perserverence, with high efficacy related

to greater persistence. Enactive mastery appears to













produce the greatest and most generalized increases in

coping, followed by vicarious and cognitive methods.

Self-efficacy is not merely a reflection of past

behavior. Bandura and his colleagues have shown that

some subjects do not increase self-efficacy until the

final mastery task even though they have coped

successfully with all of the tasks along the

progression. Some subjects increase their self-efficacy

expectations at much higher rates than their

performance would have predicted. Bandura (1982) states

that individuals are more influenced by their

perceptions of their successes and failures than by the

fact that they have succeeded or failed. Therefore,

self-efficacy is a better predictor of subsequent

behavior than previous behavior.

There have been a number of studies (Bandura &

Adams, 1977; DiClemente, 1981; Kendrick, Craig, Lawson,

& Davidson, cited in Bandura, 1982; McIntyre,

Mermelstein, & Lichtenstein, cited in Bandura, 1982)

which have shown that perceived self-efficacy is a

better predictor of future performance than past

performance. Bandura explains this as follows:

initially people increase their perception of their

abilities when their experience runs contrary to their













negative expectations or when they first gain new

skills to cope with the feared activity. Individuals

retain their weak self-perceptions while they continue

to test their new knowledge and skills. If they

encounter anything which shakes their confidence they

will register a decline in self-efficacy despite their

previous successes. After they become assured of their

ability to predict and to cope with the threat, they

become quite self-assured about their ability to manage

future challenges.

An even more exacting test of the causal

relationship between self-efficacy and action was

carried out using vicariously induced levels of

self-efficacy (Bandura, Reece, Adams, cited in Bandura,

1982). Subjects executed none of the coping strategies

but merely observed others. The models emphasized

predictability and controllability in their displays.

The former was demonstrated by repeatedly showing how

the feared objects were likely to behave across

situations. The latter was exemplified by the model's

use of highly effective techniques to manage the threat

in many circumstances. Subjects' self-efficacy

expectations were assessed at different points until

they reached predesignated low or medium levels. When













later asked to perform the threatening task, those with

higher levels of self-efficacy gained merely from

observing models produced higher performance

attainments.

In summary, self-efficacy expectations appear to

mediate behavior and behavior change. Just because the

individual is physically or mentally capable of

performing a task does not mean that he or she will in

fact be able to execute it successfully. Even

successful previous performances do not always affect

self-efficacy immediately. The degree to which persons

will raise or lower their beliefs about personal

efficacy depends upon such factors as task difficulty,

effort expended, outside help received, situational

circumstances, and the pattern or their successes and

failure over time (Bandura et al, 1980).


Physiological Arousal

Self-efficacy theory explains the relationship

between anxiety and cognitions and accounts for some of

the failures of behavioral theory and therapy to alter

anxiety reactions. According to behavioral conditioning

principles, formerly neutral stimuli become associated

with fearful experiences, thus generating anxiety long

after the association has occurred.













The social learning perspective holds that it is

an individual's perceived inefficacy in coping with

potentially threatening events that makes them

anxiety-provoking. When one can alter these events to

prevent, end, or ameliorate them, fear dissipates. When

coping efficacy is increased, anxiety should diminish.

For Bandura, the cognitive component of negative

arousal is more important than the actual physical

discomfort produced by the aversive stimulus. People

who believe themselves to be inefficacious focus on

their inability to cope and see many situations as

potentially threatening. They may increase the threat

disproportionately, and they tend to worry about

difficulties that are unlikely to arise. Where in some

cases anticipating threatening situations can lead to

the development of coping strategies, with these

individuals, their arousal can inhibit any cognitions

other than those of anxiety about the impending

aversive situation. Bandura (1982) cites Beck, Laude,

and Bohnert (1974) who found that fearful cognitions

occur just prior to the onset of anxiety attacks in

almost every case. The focus of these thoughts centers

around deep fears about the ability to cope.













A number of studies have shown the relationship

between fear arousal and self-efficacy (Bandura &

Adams, 1977; Bandura et al., 1977; Bandura et al.,

1980). In each of these studies, the methodology

involved subjects undergoing different forms of

treatment and then reporting the strength of their

self-efficacy expectations regarding the performance of

various tasks. Later, during behavioral testing,

subjects reported their level of fear before and during

the performance of the feared activity.

The results of these studies indicate that

individuals experience high anxiety before and during

the performance of tasks when they believe themselves

to be inefficacious. As their self-efficacy increases,

their fear arousal declines. When self-efficacy is at

its highest point, the previously threatening tasks are

completed with barely a trace of arousal.

Self-efficacy theory gives an alternative

explanation for the results achieved by systematic

desensitization. This approach is based on the

dual-process theory that anxiety promotes defensive

behavior which is in turn reinforced by the reduction

of the anxiety produced by the occurrence of the

conditioned aversive stimulus. The idea behind













systematic desensitization is to eliminate avoidance

behavior by eradicating the "underlying anxiety"

driving it. This is done by gradually increasing the

proximity of the stimulus while simultaneously relaxing

the subject. The association between the neutral

stimulus and fear is replaced by one between the

stimulus and relaxation.

That systematic desensitization produces

behavioral changes is not disputed, the assumption that

they come about because of the reduction of anxiety,

is. Bandura (1977b; 1986) and social learning theorists

view anxiety and avoidance behavior as coeffects rather

than as causes. "Aversive experiences, of either a

personal or vicarious sort, create expectations of

injurious consequences that can activate both fear and

defensive behavior. Being coeffects there is no fixed

relationship between autonomic arousal and actions"

(Bandura & Adams, 1977, p.289).

It is true that stressful situations foster

emotional arousal that may provide data that impinge on

personal commpetency. High levels of arousal are likely

to impede successful performance, so it is reasonable

for individuals to assume that they will perform less

well if they are experiencing fear arousal. Approaches













such as systematic desensitization which focus on

minute changes in physiological arousal reinforce this

association. From the social learning perspective,

reducing aversive arousal improves performance because

it is raising efficacy expectations; the theory

emphasizes the information gained from self-monitoring

of physical states rather than the physical energizing

properties. Because physiological arousal is only one

source of information about self-efficacy, it rarely

provides enough to eliminate avoidance behavior.

Bandura and Adams (1977) concede that when

subjects visualize feared situations in systematic

desensitization, there is bound to be some loss of

extinction effects when they actually perform the

feared behavior. However, they point out that since

anxiety arousal to the imagined stimuli is eliminated

in all subjects, dual-process theory cannot explain the

variability in the performance of subjects who have all

been equally desensitized.

To test the hypothesis that it is efficacy

expectations rather than anxiety reduction that are

changing behavior, Bandura and Adams (1977) administered

standard systematic desensitization treatments to

chronic snake phobics until their fear was completely














extinguished. The approach behavior and efficacy

expectations of the subjects were measured before and

after the treatment. Phobics have differing experiences

regarding the stimuli they fear and are also likely to

perceive their fear in differing ways. Bandura and

Adams (1977) hypothesized that by eliminating arousal,

self-efficacy would be enhanced but in differing

degrees across subjects. They also predicted that the

greater the efficacy expectations, the greater the

reductions in avoidance behavior.

The comparison of pre- and post treatment efficacy

expectations taken before the posttest confirmed that

systematic desensitization significantly raises

self-efficacy expectations for stimuli similar and

dissimilar to those used in treatment (in this case,

two different kinds of snakes), but to differing

degrees. The higher the level of perceived efficacy at

the end of treatment, the higher the level of approach

behavior. Microanalytic analyses where subjects'

abilities to predict their behavior on specific tasks

showed an 85% congruence for the similar snake and an

82% congruence for the dissimilar snake. Furthermore,

the higher the subject's level of self-efficacy

following treatment, the lower his or her anticipatory














arousal at the prospect of performing the previously

avoided task and the weaker his or her arousal during

the actual performance. Bandura and Adams (1977)

conclude that because thoroughly desensitized subjects

ranged from 10% to 100% in the completion of the

behavioral tasks, the knowledge of complete extinction

is of little value in computing behavioral change.

Measurements of self-efficacy appear to be much more

accurate predictors of performance.


Self-Efficacy Theory and Math Avoidance

Recently, researchers have been using

self-efficacy theory to understand academic and career

decision-making behaviors (Betz & Hackett, 1981; Betz &

Hackett, 1983; Hackett, 1985; Taylor & Betz, 1983). The

results of these studies hold promise for understanding

math avoidance behavior, most notably in women. Betz

and Hackett (1981) asked 124 female and 101 male

undergraduates to indicate how sure they were that they

could fulfill the educational requirements and job

capabilities for each of ten traditionally female and

ten traditionally male occupations. They were also

asked how interested and how seriously they would

consider each career. American College Test math and

English scores were also collected for each subject.













Overall, students believed the educational

requirements for physicians, engineers, and

mathematicians to be the most difficult, with 45%, 47%,

and 49% of the students believing themselves to be

capable of completing the academic prerequisites. The

careers perceived to have the easiest educational

requirements were elementary school teacher (88%),

social worker (87%), and travel agent (85%).

Sex related differences were found for 10 of 20

occupations. Males demonstrated higher self-efficacy

for accountant, drafter, engineer, highway patrol

officer and mathematician, whereas females felt more

capable than men of becoming dental hygienists,

elementary school teachers, home economists, physical

therapists, and secretaries. Males perceived the

educational requirements for becoming a physician as

most difficult, females believed engineering

requirements to be the most rigorous. The greatest

divergence of self-efficacy occurred for engineering

with 70% of the men believing they could complete the

requirements as opposed to 30% of the women.

Males and females also differed in their

self-efficacy expectations regarding the completion of

job duties. Males reported significantly higher













self-efficacy for the job duties of accountants,

drafters, engineers, highway patrol officers, and

mathematicians. Females had greater confidence in their

ability to perform the duties of a dental hygienist,

home economist, secretary, and social worker. Males

believed the job duties of an art teacher to be the

most difficult while females saw an engineer's duties

as the most demanding.

The data show that the sex differences were due

largely to females' low self-efficacy in regard to

non-traditional female careers. Males believed

themselves to be able to complete the educational

requirements of an average of 6.9 of the 10 traditional

(for females) careers and 6.9 of the 10 non-traditional

(for females) careers and the job requirements of 7 of

the 10 traditional occupations and 7.2 of the 10

non-traditional occupations. In contrast, females

believed themselves to be able to complete the

educational requirements of 8 out of 10 traditional

careers but only 5.7 of the 10 non-traditional careers,

and the job duties of 8 traditional careers and 6

non-traditional careers.

A stepwise multiple regression analysis was

performed for the total group of students and for males














and females to examine the relationships between

self-efficacy expectations, interests, and sex with

perceived range of career options. First of all,

females said they would consider more traditional

careers than males (6.3 vs. 2.4) whereas males reported

considering more non-traditional careers (4.4 vs. 2.6).

Thus, the sex differences in range of career options

closely resemble those for self-efficacy expectations.

The multiple regression further confirms these

findings.

Interests for both non-traditional and traditional

careers were the major predictors of the range of

traditional career options. The degree of interest in

traditional occupations was positively related to the

range of traditional options, but interest in

non-traditional careers was negatively related to

consideration of traditional options. Non-traditional

interests and non-traditional efficacy were positively

related to the range of non-traditional career options

considered by both females and males, and

self-efficacy, in terms of traditional occupations, was

negatively correlated with the range of non-traditional

career options considered by the students. Sex was a

significant predictor for the total group analysis,