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BUILDING MATH SELFEFFICACY: 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 SelfEfficacy 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 SelfEfficacy 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 SelfEfficacy, Math Anxiety, and SelfEstimate of Math Ability................................ 144 Implications Grid........................ 149 Grades................................... 150 Correlational Analyses.................... 150 Posthoc Analyses........................ 154 FIVE DISCUSSION................................. 162 Measures of SelfEfficacy: SelfEstimate of Math Ability and Math SelfEfficacy Scale.................................. 163 Math Anxiety.............................. 164 Implications Grid........................ 165 Final Course Grade....................... 167 Posthoc Correlational Analyses .......... 168 Limitations of Confidence and Tutoring Groups................................. 172 Conclusion............................... 175 APPENDICES 1 LETTER OF INTRODUCTION TO NEW MAJORS...... 177 2 MATHEMATICS SELFEFFICACY SCALE ........... 179 3 FENNEMASHERMAN MATH ANXIETY SCALE........ 183 4 BEM SEXROLE 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 ttest Comparisons of Attenders versus Nonattenders of Confidence Groups on MSES, SE, MAS, and GPA .......................... 135 3 Means, Standard Deviations, and ttest Comparisons of Attenders versus Nonattenders of Tutoring Groups on MSES SE, MAS, and GPA ................... ........ 136 4 Results of Pretest 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 FTest 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 Oneway 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 ProductMoment Correlations Among Dependent and Independent Variables for Experimental Subjects....... 157 14 Matrix of PearsonProduct 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 SELFEFFICACY: 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 selfefficacy 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 sixtynine 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 counselingbased statistics confidence groups, to one of three statistics tutoring groups, or to a notreatment control group. The experimental design was a repeated measures paradigm where students were given pre and posttreatment assessments of math selfefficacy, math anxiety, selfreported math ability, and math selfconcept. 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' selfconcepts approached significance, with the math confidence groups having fewer implications. There were no significant differences among the three groups regarding math selfefficacy 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 selfefficacy, math anxiety, sexrole 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 decisionmaking 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 198687 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 nonEuclidean 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 198283, there was a 50% decline in Bachelor's degrees granted in the discipline of mathematics, 12,543 down from 24,801 in 197071. 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 197071 to 198283, and 42% fewer Ph.D.'s were granted in 198283 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 198182 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 abovementioned 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 nonparochial 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 AsianAmerican 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 postsecondary 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 nonAsian American student does not perceive mathematics to be important for his or her posthigh 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 mathrelated 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 sexrole 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 nonrequired 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 "explainpracticememorize" 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 mathrelated 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 collegeaged 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 selfevaluative 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 Anxietyapprehension about taking and receiving the results of math tests, and 2) Numerical Anxietyeveryday 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) SelfEfficacy 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 SelfEfficacy 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 selfefficacy 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 coeffect of selfefficacy expectations; as selfefficacy diminishes, anxiety increases. Therefore, anxiety is not viewed as the cause of avoidance behavior but as a byproduct of low selfefficacy. And unlike Hendel and Davis (1978) who see avoidance as a symptom of math anxiety, selfefficacy theory would explain both anxiety and avoidance as byproducts of low selfefficacy expectations. Selfefficacy theory can offer a theoretical framework for the results produced by studies proposing teaching styles, sexrole stereotyping, and even physiological deficits as underlying causes of math avoidance behavior, since all of these factors can affect one's selfstatements 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 sexrole 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 Selfefficacy Expectations Unlike explanations of math avoidance that rely on nearly immutable factors such as gender role and physiology, selfefficacy 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 selfefficacy 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 selfefficacy 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, sexrole socialization, and physiology. Bandura's theory of selfefficacy 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 posttest 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 oldabout 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 lighthearted, 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 gradelevel 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' selfimages 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. ambitiouslazy, friendlyunfriendly, attractiveunattractive, 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 engineeringmajor 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 givensseems 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 pseudoexpert 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 "explainpracticememorize" 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 memorizationbased 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 nonmathematicians 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, lowintensity 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 nonverbal 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 sensefrom 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 shortterm 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 nonmath/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 classthose for whom the material doesn't make senseeither fail or simply drop from the rolls. Often the instructor never finds out what is wrong. Faculty members have the confidence, selfawareness, and ability to analyze why they find something difficult (McMillen, 1986, p. 18). The professors take an introductory course required for nonmajors 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 dropouts 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 provensuccessful 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 nonmath 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 selfdoubt 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 existsdoubt 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 anxietyprovoking 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 FennemaSherman Math Anxiety Scale (MAS, Fennema & Sherman, 1976) and the Atrait scale of the StateTrait 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. Sixtythree 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 PostTask 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 (SATM). However, the authors caution that the SATM 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 anxietyprovoking 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 Xlinked, 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 spatialgestalt functioning. Levy tested graduate and postdoctoral students using the WAIS Performance scale and a test of verbal reasoning and found that lefthanded 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 lefthandedness was relatively detrimental to male spatial scores and beneficial to verbal performance, the reverse was true for females. Spatial scores of lefthanders were never significantly lower than for righthanders 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 mathrelated 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 nonsignificant. At School 4, when both spatial visualization and the six significantly different affective variables were covaried out, achievement effects also became nonsignificant. 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 PreCollege Test (WPC) which included the following: 1) Quantitative Skills Aability to determine whether enough information is given to solve a problem 2) Quantitative Skills Bability to determine the relative size of the given qualities 3) Applied Mathematicsapplying knowledge of arithmetic and elementary algebra to solve practical problems 4) Mathematics Achievementknowledge of algebra and geometry 5) Spatial abilityability to visualize transformations in three dimensions 6) Mechanical Reasoningability 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, socioeconomic, 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 tenyearsold. 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 selfperceptions of math ability on success in math (Meece, Parsons, Kaczala, Goff, & Futterman, 1982), Hollinger set out to uncover differences in selfperceptions of math ability in women aspiring to careers requiring differing levels of mathematical and scientific knowledge. Hollinger (1985) divided women into six careertrack categories: nontraditional math, nontraditional science, neutral/traditional math/science, nontraditional nonmath, neutral nonmath, and traditional nonmath. The three nonmath categories were combined since they did not differ significantly. The analyses of variance indicated that the four career aspiration groups differed on the following selfperceptions 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 selfperceptions of multiple abilities rather than just that of math ability distinguishes nontraditional math career aspirants from all other career groups. The nontraditional math career group had significantly higher selfperceptions of math ability than the nonmath career group. Interestingly, the nontraditional math group also demonstrated significantly lower selfperceptions of friendliness and artistic ability than the other three groups. The nontraditional science career aspirants reported significantly higher estimates of math, mechanical, and manual and science ability than the nonmath 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 nonmath group was that the former reported higher selfestimates 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 nontraditional 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 nontraditional 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 nontraditional 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 selfestimate of creative ability demonstrated by the nontraditional 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 nontraditional math/science major or a traditional nonmath/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 nonmath science careers, both groups experienced greater conflict when they imagined themselves pursuing math/science careers as compared with nonmath/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 sexrole 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 sexrole socialization literature and account for the avoidance and anxiety demonstrated by males. The Effect of SelfEfficacy Expectations on Human Behavior The mechanism of selfefficacy 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 selfefficacy 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 selfperceptions affect behavior. "Perceived selfefficacy 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. Selfefficacy 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 selfefficacy 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 selfefficacy 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 selfefficacy 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 selfefficacy 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 selfefficacy 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 selfbeliefs that ensure optimal use of capabilities are required for successful functioning. If selfefficacy 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 selfpercepts 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 selfefficacy 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. Selfefficacy 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. Selfefficacy is not merely a reflection of past behavior. Bandura and his colleagues have shown that some subjects do not increase selfefficacy until the final mastery task even though they have coped successfully with all of the tasks along the progression. Some subjects increase their selfefficacy 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, selfefficacy 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 selfefficacy 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 selfperceptions while they continue to test their new knowledge and skills. If they encounter anything which shakes their confidence they will register a decline in selfefficacy despite their previous successes. After they become assured of their ability to predict and to cope with the threat, they become quite selfassured about their ability to manage future challenges. An even more exacting test of the causal relationship between selfefficacy and action was carried out using vicariously induced levels of selfefficacy (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' selfefficacy 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 selfefficacy gained merely from observing models produced higher performance attainments. In summary, selfefficacy 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 selfefficacy 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 Selfefficacy 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 anxietyprovoking. 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 selfefficacy (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 selfefficacy 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 selfefficacy increases, their fear arousal declines. When selfefficacy is at its highest point, the previously threatening tasks are completed with barely a trace of arousal. Selfefficacy theory gives an alternative explanation for the results achieved by systematic desensitization. This approach is based on the dualprocess 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 selfmonitoring of physical states rather than the physical energizing properties. Because physiological arousal is only one source of information about selfefficacy, 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, dualprocess 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, selfefficacy 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 selfefficacy 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 selfefficacy 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 selfefficacy appear to be much more accurate predictors of performance. SelfEfficacy Theory and Math Avoidance Recently, researchers have been using selfefficacy theory to understand academic and career decisionmaking 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 selfefficacy 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 selfefficacy 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 selfefficacy expectations regarding the completion of job duties. Males reported significantly higher selfefficacy 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 selfefficacy in regard to nontraditional 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 nontraditional (for females) careers and the job requirements of 7 of the 10 traditional occupations and 7.2 of the 10 nontraditional 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 nontraditional careers, and the job duties of 8 traditional careers and 6 nontraditional careers. A stepwise multiple regression analysis was performed for the total group of students and for males and females to examine the relationships between selfefficacy 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 nontraditional careers (4.4 vs. 2.6). Thus, the sex differences in range of career options closely resemble those for selfefficacy expectations. The multiple regression further confirms these findings. Interests for both nontraditional 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 nontraditional careers was negatively related to consideration of traditional options. Nontraditional interests and nontraditional efficacy were positively related to the range of nontraditional career options considered by both females and males, and selfefficacy, in terms of traditional occupations, was negatively correlated with the range of nontraditional career options considered by the students. Sex was a significant predictor for the total group analysis, 