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Rawley, Jo Ann K.
Factors perceived to contribute to mathematics avoidance or mathematics confidence in non-traditional age women attending a community college
h [electronic resource] /
by Jo Ann K. Rawley.
[Tampa, Fla.] :
b University of South Florida,
ABSTRACT: Over the past decade, the number of students entering postsecondary institutions immediately after high school has been decreasing, while the number of non-traditional aged students, defined as adults over 25, has substantially increased, with women making up the majority of this adult student group. Mathematics education is an area where non-traditional age women tend to have difficulty. Fifteen individual interviews were conducted with non-traditional age women enrolled in a community college, 10 identifying mathematics as the subject they would least enjoy and 5 identifying mathematics as the subject they would most enjoy. Data were analyzed by comparing the women's stories and drawing out common themes.Eight major themes and six sub-themes emerged: (1) Acquiring a college education is a personal goal; (2) Adequate study time is necessary to understand and to retain mathematical concepts; (3) Experiences with mathematics at an early age remain in one's memory, (3a) Poor experience with mathematics at an early age tended to make participants believe they could not learn mathematics, (3b) Positive experience with mathematics at an early age tended to provide participants a higher degree of self-efficacy in succeeding in mathematics courses; (4) Parental behavior and expectations play a role in children's self-perception, (4a) Absence of parental/family support tended to discourage participants from pursuing further education, (4b) Presence of parental/family support tended to encourage participants in pursuing further education; (5) Teacher behaviors and teaching methods matter, (5a) Negative teacher behaviors tended to cause some to develop poor mathematics self-concept, (5b) Positive teacher behaviors tended to encourage some to persevere in understanding mathematics; (6) Feelings of powerlessness may impede learning mathematics; (7) Self-esteem can survive in spite of past failure; (8) Motivation to understand mathematical concepts remained high.Seventeen implications for both faculty and students were drawn from the responses of the participants. Both metacognitive and affective factors present in learning mathematics were expressed and meanings attached to experiences were reported in participants' own words. Suggestions are offered explaining what teachers might do to reinforce positive metacognitions and reduce those that are negative. Recommendations for further research are provided and personal reflections are shared.
Dissertation (Ph.D.)--University of South Florida, 2007.
Includes bibliographical references.
Text (Electronic dissertation) in PDF format.
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Advisor: Arthur Shapiro, Ph.D.
x Higher Education
t USF Electronic Theses and Dissertations.
Factors Perceived to Contribute to Mathem atics Avoidance or Mathematics Confidence in Non-Traditional Age Women Attending a Community College by Jo Ann K. Rawley A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Education Department of Adult, Car eer and Higher Education College of Education University of South Florida Co-Major Professor: Arthur Shapiro, Ph.D. Co-Major Professor: Steve Permuth, Ed.D. Jan Ignash, Ph.D. Robert F. Dedrick, Ph.D. Denisse R. Thompson, Ph.D. Date of Approval: April 6, 2007 Keywords: anxiety, self-efficacy, teacher behaviors, parental support, motivation Copyright 2007, Jo Ann K. Rawley
Acknowledgements Completing this research study would not have been accomplished without the support of the members of my doctoral committ ee, who were patient with my progress, which seemed to be imperceptible at times, and who provided me with helpful advice and guidance through my move to Pennsylvania a nd the years that followed. Dr. Arthur Shapiro was there to inject hi s unfailing sense of humor even in the darkest times when I felt hopelessly overwhelmed. Dr. Jan Ignash wa s always available to help me locate the right paperwork and the ins and outs of maneuvering through the maze leading to graduation. She was my long-distance c onnection to the Tampa campus and I will always be grateful for her advice and enc ouragement. I would also like to thank Dr. Steve Permuth, Dr. Denisse Thompson and Dr Robert Dedrick, all of whom made important contributions to my journey through the doctoral program. I feel blessed to have had the opportunity to work with such talented individuals. I would also like to than k the administrators at Reading Area Community College for allowing me to conduct my research on the campus and for granting me the freedom to access student records and use their facil ities to interview participants. Dr. John DeVere and Dr. Fred Indenbaum welcomed me and my proposed research, and although I was a newcomer to the college, trusted and believed in me. Without their help, this research could never have been conducted.
Next, I would like to thank two special individuals who served as co-researchers with me in reading transcripts and offering th eir perspective on the stories the participants shared. Dr. Sandra Kern and Tomma Lee Fu rst were invaluable in their support and expertise. Without their unselfishness and dedication to providing the best possible educational experience for students, my wo rk on this study would have been far less productive. Thank you both for your time and encouragement. Thank you to my children for never givi ng up believing that their mother would finish this research, ev en though it took many years to accomplish. Thank you, Jon, Melodie, Dan, David, Norah, Justin, and Morgan. Finally, my thanks to the fifteen women who agreed to participate in this study and who granted me the honor and privilege of witnessing their memories and feelings. Through their stories, I became a bett er educator and a closer friend.
i Table of Contents List of Tables................................................................................................................. ...v List of Figures................................................................................................................ .vi Abstract....................................................................................................................... ...vii Chapter One: Introduction.............................................................................................. 1 Enrollment Trends................................................................................................1 The Non-Traditional Age Student........................................................................2 Challenges for the Community College..............................................................13 Purpose................................................................................................................19 Research Questions.............................................................................................20 Definition of Terms.............................................................................................21 Assumptions........................................................................................................25 Limitations and Delimitations.............................................................................26 Organization of the Study...................................................................................27 Chapter Two: Review of Literature...............................................................................29 Historical Background........................................................................................29 Women in the Labor Force.....................................................................31 Mathematics Course Selection and Affective Issues in High School.................32 Need for Information Age Skills in TodayÂ’s Labor Force..................................38 Understanding the Role of A cademic Self-Perception.......................................40 Mathematics Self-Concept and Gender..............................................................42 Social Conditioning and Stereotyping................................................................47 Parental Support......................................................................................47 Stereotyping............................................................................................48 Silence of Women...................................................................................49 Mathematics Anxiety..........................................................................................51 Teacher Influence on StudentsÂ’ Attitudes...........................................................61 Quality in Mathematics Education......................................................................66 Constructivist Approaches to Learning..............................................................70 Mathematics Education Initiatives......................................................................74 National Council of Teacher of Mathematics.........................................74 American Mathematical Asso ciation of Two-Year CollegesÂ— Crossroads in Mathematics...............................................................75 Womenwin at Mathematics Through Writing........................................77
ii Developmental Algebra: Re structuring to Effect Change.....................78 Mathematics for Elementary Teachers...................................................78 EQUALS.................................................................................................79 Project SEED..........................................................................................79 Phenomenological Research in Mathematics.....................................................80 Theoretical Background fo r Qualitative Research..............................................86 Summary.............................................................................................................90 Chapter Three: Method..................................................................................................92 Problem...............................................................................................................92 Purpose................................................................................................................93 Research Questions.............................................................................................94 Overview of Chapter...........................................................................................94 Value of Phenomenological Interviewing..........................................................94 Study Design: The Informal Conversational Interview.....................................97 Researcher.........................................................................................................100 Research Setting: Reading Area Community College.....................................102 Selection of Participants...................................................................................105 Procedure..........................................................................................................109 Data Analysis....................................................................................................113 Trustworthiness and Credibility........................................................................114 Summary...........................................................................................................117 Chapter Four: Findings................................................................................................119 Profiles of the Participants................................................................................122 Individual Participants..........................................................................122 Presentation of Themes and Sub-themes..........................................................131 Theme 1: Acquiring a college education is a personal goal.................131 Theme 2: Adequate study time is necessary to understand and to retain mathematical concepts.................................................133 Theme 3: Experiences with mathematics at an early age remain in oneÂ’s memory...................................................................136 Sub-Theme 3a: Poor experience with mathematics at an early age tended to make participants believe they could not learn mathematics.........................................137 Sub-Theme 3b: Positive experience with mathematics at an early age tende d to provide pa rticipants a higher degree of self -efficacy in succeeding in mathematics courses.............................................................139 Theme 4: Parental behavior a nd expectations play a role in childrenÂ’s self-perception..................................................................140 Sub-Theme 4a: Absenc e of parental/family support tended to discourage participants from pursuing further education...................................................................141
iii Sub-Theme 4b: Presence of parental/family support tended to encourage participants in pursuing further education...................................................................143 Theme 5: Teacher behaviors and teaching methods matter..................144 Sub-Theme 5a: Negative teacher behaviors tended to cause some to develop poor mathematical self-concepts.........................................................................144 Sub-Theme 5b: Positi ve teacher behaviors tended to encourage some to persevere in understanding mathematics...................................................147 Theme 6: Feelings of powerlessness may impede learning mathematics......................................................................................149 Theme 7: Self-esteem can su rvive in spite of past failure...................153 Theme 8: Motivation to understand mathematical concepts remained high....................................................................................156 Summary...........................................................................................................160 Chapter Five: Summary, Discussion and Implications................................................162 Problem ............................................................................................................162 Purpose..............................................................................................................163 Summary of the Findings..................................................................................164 Discussion ........................................................................................................166 Theme 1: Acquiring a college education is a personal goal.................166 Theme 2: Adequate study time is necessary to understand and to retain mathematical concepts.................................................166 Theme 3: Experiences with mathematics at an early age remain in oneÂ’s memory...................................................................167 Sub-Theme 3a: Poor experience with mathematics at an early age tended to make participants believe they could not learn mathematics.........................................168 Theme 4: Parental behavior a nd expectations play a role in childrenÂ’s self-perception..................................................................169 Sub-Theme 4a: Absenc e of parental/family support tended to discourage participants from pursuing further education...................................................................170 Theme 5: Teacher behaviors and teaching methods matter..................171 Theme 6: Feelings of powerlessness may impede learning mathematics .....................................................................................172 Theme 7: Self-esteem can su rvive in spite of past failure...................172 Theme 8: Motivation to unde rstand mathematical concepts remained high....................................................................................173 Implications for Practice...................................................................................178 Suggestions for Future Research......................................................................190 Personal Reflections..........................................................................................193
iv List of References.........................................................................................................195 Appendices....................................................................................................................2 28 Appendix A: Multiple Choice Questions for COMPASS...............................229 Appendix B: Adult Informed Consent.............................................................233 Appendix C: Interview Questions....................................................................236 Appendix D: Code Mapping: Constant Comparative Analysis......................237 Appendix E: Letter of Permission....................................................................238 About the Author................................................................................................End Page
v List of Tables Table 1 Total Fall Enrollment in Degree-Granting Institutions, by Attendance Status, Sex of Student and Control of Institution: 1947-2004.................................................................................................3 Table 2 Community College Fall Headcount Enrollment by Age and Gender: 1997..............................................................................6 Table 3 Total Fall Enrollment in Degree-Granting Institutions, by Attendance Status, Age, and Sex: 1980-2014.....................................7 Table 4 Percentage of 1999-2000 Unde rgraduates with Various Risk Factors, by Age.........................................................................................9 Table 5 Proportion of Women in Selected Occupations (1975 and 1995)......................................................................................33 Table 6 Relationship of Research Questions to Interview Questions................111 Table 7 Themes and Sub-Themes......................................................................121 Table 8 Constant Comparative Anal ysis of Participant Responses...................123
vi List of Figures Figure 1 Community College Enrollment by Age: 1997.......................................5
vii Factors Perceived to Contribute to Mathem atics Avoidance or Mathematics Confidence in Non-Traditional Age Women Attending a Community College Jo Ann K. Rawley ABSTRACT Over the past decade, the number of students entering postsecondary institutions immediately after high school has been decrea sing, while the number of non-traditional aged students, defined as adults over 25, ha s substantially increased, with women making up the majority of this adult student group. Mathematics education is an area where nontraditional age women tend to have difficulty. Fifteen individual interviews were conducted with non-traditional age women enrolled in a community college, 10 identifyi ng mathematics as the subject they would least enjoy and 5 identifying mathematics as the subject they woul d most enjoy. Data were analyzed by comparing the womenÂ’s stories and drawing out common themes. Eight major themes and six sub-themes emerge d: (1) Acquiring a college education is a personal goal; (2) Adequate study time is necessary to understand and to retain mathematical concepts; (3) Experiences with ma thematics at an early age remain in oneÂ’s memory, (3a) Poor experience with mathem atics at an early age tended to make participants believe they could not learn ma thematics, (3b) Positive experience with mathematics at an early age tended to provi de participants a hi gher degree of selfefficacy in succeeding in mathematics courses; (4) Parental behavior and expectations play a role in childrenÂ’s se lf-perception, (4a) Absence of parental/family support tended
viii to discourage participants from pursui ng further education, (4b) Presence of parental/family support tended to encourage pa rticipants in pursuing further education; (5) Teacher behaviors and teaching methods matter, (5a) Negative teacher behaviors tended to cause some to develop poor mathematics self-concept, (5b) Positive teacher behaviors tended to encourage some to pe rsevere in understanding mathematics; (6) Feelings of powerlessness may impede learni ng mathematics; (7) Self-esteem can survive in spite of past failure; (8 ) Motivation to understand mathem atical concepts remained high. Seventeen implications for both faculty and students were drawn from the responses of the participants. Both met acognitive and affective factors present in learning mathematics were expressed and meanings attached to experiences were reported in participantsÂ’ own words. Sugges tions are offered explaining what teachers might do to reinforce positive metacognitions and reduce those that are negative. Recommendations for further research are prov ided and personal reflections are shared.
1 Chapter One Introduction For women preparing to move ahead in the 21st century, education offers one of the best opportunities for advancement a nd for improving quality of life. Although education brings no guarantee of monetary or professional success, it can help open doors that are closed to the uneducated. Lack of education can be an overwhelming obstacle to women in AmericaÂ’s competitive technologica l society. Because of the increasing demand for technological skills, mathematic s education, in particular, has taken on heightened importance and is in the national spotlight. Successful completion of mathematics courses for women in undergraduat e education is critic al, not only to those individuals whose lives are a ffected, but to the economy of the nation as well. An awareness of factors perceived to contribute to mathematics avoidance or mathematics confidence in non-traditional age women atte nding a community college may be key to addressing issues of retention and persiste nce in AmericaÂ’s educational institutions. Enrollment Trends Over the past decade, the number of students entering postsecondary institutions immediately after high school has progressive ly decreased. Conversely, the registration of non-traditional age students, defined as the Â“mature,Â” Â“reentry,Â” or Â“adultÂ” learner over the age of 25, has substan tially increased (Klein, 1990; Krager, Wrenn, & Hirt, 1990; Padula, 1994; Scott, Burns, & Cooney, 1996). The majority of this group is female and is
2 becoming the fastest growing population now entering postsecondary education (Clayton & Smith, 1987; Donaldson & Graham, 1999). Continued success of postsecondary institutions may well depend on their ability Â“to understand and accommodate the unique dispositional, situational, and institutional needs of non-traditional female studentsÂ” (Carney-Crompton & Tan, 2002, p. 140). Statistics gathered by the U. S. Depart ment of Education, National Center for Education Statistics, show that women accounted for about 96% of the increase in college and university fall enrollment from 1976 to 1984. Recent data indicate this trend continuing with 66% of the increase in en rollment from 1994 to 2004 attributable to women. Between 1994 and 2004, the number of men enrolled rose 16% while the number of women increase d by 25% (see Table 1). The numbers speak for themselves. By 1996, women were a majority of both part-time and full-time enrollees (Bae, C hoy, Geddes, Sable, & Snyder, 2000). More than half of undergraduates were wome n in 1999-2000 (56%) (NCES, 2003). Data collected during 1999 and 2000, re ported that female stude nts comprised 61% of the student body on community college campuses in the United States (VanDerLinden, 2002). The Non-traditional Age Student Although female students of all ages who are seeking higher education predominate, those who are considered of non-traditional age are the focus of this research. Non-traditional age has generally been recognized as being over the age of 25 years (Chickering, 1981 ; Cohen & Brawer, 1996; Horn, MPR Associates & Carroll, 1996; Lyons & Pawlas, 1998; Winter & Harris, 1999). Some believe that this population,
3 Table 1. Total fall enrollment in degree-granting institu tions, by attendance status, sex of student, and control of institution: 1947 to 2004 Year Total Attendance status Sex of student Enrollment Full-time Part-time Men Women 1947 2,338,226 --1,659,249 678,977 1957 3,323,783 --2,170,765 1,153,018 1967 6,911,748 4,793,128 2,118,620 4,132,800 2,778,948 1976 11,012,137 6,717,058 4,295,079 5,810,828 5,201,309 1980 12,096,895 7,097,958 4,998,937 5,874,374 6,222,521 1984 12,241,940 7,098,388 5,143,552 5,863,574 6,378,366 1990 13,818,637 7,820,985 5,997,652 6,283,909 7,534,728 1994 14,278,790 8,137,776 6,141,014 6,371,898 7,906,892 1998 14,506,967 8,563,338 5,943,629 6,369,265 8,137,702 2000 15,312,289 9,009,600 6,302,689 6,721,769 8,590,520 2001 15,927,987 9,447,502 6,480,485 6,960,815 8,967,172 2002 16,611,711 9,946,359 6,665,352 7,202,116 9,409,595 2003 16,900,471 10,311,814 6,588,657 7,255,551 9,644,920 2004 17,272,044 10,610,177 6,661,867 7,387,262 9,884,782 Note. From Digest of Education Statistics: 2005, Table 170. Â“Fall Enrollment in Colleges and Universities.Â” U. S. Department of Educatio n, National Center for Education Statistics.
4 which includes disproportionate numbers of women and members of ethnic minorities, Â“are subjected to an interact ive web of entrenched values from long-standing elitist systemsÂ” (Richardson & King, 1998, p. 68). Although there are pressures to bring change and opportunities for wider access to hi gher education, many educators agree that this population has been overlooked (Ric hardson & King, 1998; Schatzkamer, 1986; Smith, 1994; Winefield, 1993). Non-traditional students repor t different concerns than traditional students (Bean & Metzner, 1985). They often share characte ristics such as working full-time while enrolled in classes part-time, having family re sponsibilities, desiring to upgrade skills and advance their careers, and havi ng been out of academic life for a period of time. Many have undergone a major life change, such as di vorce or loss of a job, and seek to develop self-confidence to cope with future uncertainty. Non-traditional age students, male and fe male, comprise nearly half the population in AmericaÂ’s community colleges. The National Center for Education Statistics reports, in National Profile of Community Colleges: Trends & Statistics that 46% of community college students are 25 years or older (see Figure 1). In reference to gender, NCES data show that more women than men pursue postsecondary education afte r the age of 30, perhaps co inciding with average postchildbearing years (Phillippe, 2000) (see Table 2). Table 3, showing statistics gathered by the U. S. Department of Education, National Center for Education Statistics, proj ects student enrollment figures in degreegranting institutions to 2014, separated according to age and gender. These data indicate that in 2000, women between th e ages of 25 and 34 years, wh o registered for a full-time
5 Figure 1. Community College Enrollment by Age: 1997 Note. From National Center for Education Statistics, 1999. American Association of Community Colleges. course load, represented 53% of fall enrollments in that age category; by 2014 that figure is projected to reach 61%. In the 35-year s-old-and-over category, women enrolled fulltime represented 57% of 2000 fall enrollments ; projections for 2014 rise to 67%. In addition to full-time attendees, part-time en rollments of non-tradi tional age women are growing as well. In 2000, women between th e ages of 25 and 34 years represented 60% of fall enrollments in that age group, with the 35-years-old-andover group of women reaching 65%. Projections for 2014 for part-time women enrollees show 60% for the 25to-34-years-old group and 63% for th e 35-years-old-andover category. Many non-traditional age women seeking higher education have special needs that go beyond those of the trad itional, 18-21-year-old female coming to college right out of high school, who is single and depending on he r parents for financial support. One of
6 Table 2. Community College Fall Headcount Enrollment by Age and Gender: 1997 Age Male Female Percent Female Under 18 86,209 129,523 60% 20 to 21 409,823 457,211 53% 22 to 24 333,067 391,630 54% 25 to 29 327,589 439,205 57% 30 to 34 200,582 319,889 61% 35 to 39 156,621 285,270 65% 40 to 49 198,156 381,813 66% 50 to 64 78,430 140,041 64% 65 or older 23,838 35,713 60% Age unknown 20,452 23,388 53% Note. From National Profile of Community College s, American Association of Community Colleges. the most potentially problematic of these needs is requiring childcare while attending college classes. Survey data collected during 1999 and 2000 reveal that 45% of single mothers who enrolled in community colleges because of a major life change indicated that the cost of childcare or dependent care was a problem while taking courses (VanDerLinden, 2002).
7 Table 3. Total fall enrollmen t in degree-granting institutions, by attendance status, age, and sex: 1980 through 2014 [in thousands] Sex and age 1980 1990 1995 2000 Projected 2014 Full-time 25 to 29 years 577 770 908 878 1509 30 to 34 years 251 387 430 422 584 35+ years 182 471 653 599 748 Men 25 to 29 years 360 401 454 415 596 30 to 34 years 124 156 183 195 213 35+ years 74 152 238 256 251 Women 25 to 29 years 217 369 455 463 913 30 to 34 years 127 231 247 227 371 35+ years 108 319 415 343 498 Part-time 25 to 29 years 1209 1213 1212 1083 1404 30 to 34 years 905 935 805 843 989 35+ years 1145 2012 2093 2150 2539
8 Sex and age 1980 1990 1995 2000 Projected 2014 Men 25 to 29 years 594 539 508 447 538 30 to 34 years 397 381 378 332 423 35+ years 382 672 748 757 935 Women 25 to 29 years 616 674 704 636 866 30 to 34 years 507 554 427 511 566 35+ years 762 1340 1345 1393 1604 Note. From Digest of Education Statistics: 2005, Table 172. Â“Projections of Education Statistics to 2014.Â” U. S. Department of Education, National Center for Education Statistics. In addition to the complexities of child care, many women have been away from a classroom atmosphere and the rigors of academic life for several years and often enroll as part-time students while holding a full-t ime job. The coexistence of family responsibilities and commitments with the demands of academia can create additional challenges and barriers to academic success for non-traditional age women (Padula, 1994). All of these responsibilities, with which non-traditional age women must cope, have been identified as risk factors and may result in lowe r rates of persistence. The number of risk factors is directly related to the like lihood of leaving postsecondary education without completing a program (B erkner, Horn, Clune, & Carroll, 2000). The data in Table 4 reflect numbers of students affected by the following risk factors:
9 Table 4. Percentage of 1999-2000 Undergraduates with Various Risk Factors, by Age Risk factors Age categories Delayed Part-time Have dependents Work full-time enrollment attendance or children while enrolled 18 or younger 9.6 26.3 5.6 16.1 19-23 years 31.2 34.0 11.1 24.2 24-29 years 62.5 61.6 35.4 52.1 30-39 years 72.9 73.1 61.0 60.8 40+ years 74.7 82.0 55.0 62.7 Note. From National Postsecondary Student Aid Study (NPSAS:2000). U. S. Department of Education, National Center for Education Statistics, 1999-2000. (a) delayed postsecondary enro llment; (b) part-time attenda nce; (c) responsibility for dependents; and (d) full-time workload while enrolled. One of the more serious risk factors is having been out of school for anywhere from five to twenty years. Although communica tion and verbal skills tend to operate as part of normal life activity, mathematical sk ills are often forgotten. Not surprisingly, a lack of confidence in succe ssfully performing calculations and solving problems in mathematics may develop (Ramus, 1997). Furt hermore, if these students had negative experiences in the mathematics courses they had growing up, this lack of confidence may escalate as they think about taking a mathematics course for the first time in several years.
10 Many women enrolled in postsecondary educ ation have three, four, or more of these risk factors present. These studen ts know their chances are limited and, as a consequence, they are more serious now about school than they were when they were in their teens or early twenties (Cox, 1993; Miglietti, 1994; Sc hatzkamer, 1986). According to Schatzkamer, older students do better work and have a different classroom attitude than most younger students. Consequently, th ey are actually more likely than younger students to exhibit a deep approach or a meaning orientation toward their academic studies, and they are conversely less likel y than younger students to adopt a surface approach or a reproducing orientation to th eir academic work (Richardson, 1995). They want to learn and to change their li ves (Cox, 1993; VanDerLinden, 2002). We cannot afford to overlook this important age group (Mullinix, 1995). Much of an adultÂ’s self-identity, whether male or female, is derived from his or her life experience. To ignore or reject such experience is to de value that individual (Crawford, 1980). Rarely are a dultsÂ’ prior experiences form ally integrated into the process of learning, and rarely are socialization issues addres sed in the curriculum. The focus tends to be on cognitive learning alone. Consequently, there is a neglect of the affective issues adults are facing as they return to the classroom. Feminist writers insist that learning should be regarded as a holis tic process (Belenky, Clinchy, Goldberger, & Tarule, 1986), and yet most of the mainstream literature on learning neither reflects the experiences of learners nor acknowledges that ideas cannot be separated from experience (Richardson & King, 1998). Because feelings an d personal interests play a vital role in learning, adult students are ofte n capable of more effective and elaborative learning than younger students precisely because they are likely to be more adept at examining and
11 exploiting their prior experien ce in order to make sense of new information and new situations (Merriam & Caffarella, 1991). Researchers studying adult development cha llenge mathematics teachers to create environments that are warm and accepting, c ooperative, adventuresome, and challenging (Bean & Metzner, 1985; Donohue & Wong, 1997; Fiore, 1999; Jackson & Leffingwell, 1999; Koelling, 1995). When adult learners re turn to school afte r being out of the academic world for a considerable period of time, they tend to be unsure of themselves and unclear about the expectations of the acad emic environment. They may be in the middle of major transitions in their personal or professional lives and are being forced to look in new directions. They may be asking fundamental questions about their identity and self-esteem (Grood, 1985; Ham, 1998; Lehmann, 1985; Menson, 1982; Miglietti, 1994). Many studies support an optimistic profile of the non-traditiona l age student as a very capable and resilient participant in the academic world, despite the number of challenges and stressors that differ from t hose experienced by more traditional age students. Carney-Crompton and Tan (2002) suggest a number of ways that postsecondary institutions can enhance the poten tial success of these students. First, because the transition to the ro le of student real istically entails a significant degree of adjustment to an al ready full repertoire of commitments and responsibilities, postsecondary institutions should o ffer orientation programs targeted specifically to these students and their life situations. Such programs should be offered prior to and in the in itial particularly st ressful stages of enrollment. These students should be info rmed about the real demands of higher
12 education, including course expectations a nd requirements, as well as the extent of time and personal energy required to succeed. Although most students are aware of the actual class tim e involved in a particular program, many are unaware of the amount of time that is frequently required for out-of-cl ass reading, written assignments, and other course activities. Non-traditional female students who have successfully completed some or all of their studies coul d provide realistic accounts of the demands as well as re wards of further education (p. 146). Study skills often must be developed or refreshed as the women assume their new role as students. Counselors and academic support staff may help he re to give advice, guidance, and support. Life experiences, although receiving no academic credit, should be recognized and respected. For non-traditi onal age students, emotions may have more influence on learning than intellectual abi lity (Fiore, 1999). Anxieties about academic performance and test taking, especially in mathematics, often surface (Mullinix, 1995; Parker, 1997; Richardson & King, 1998). This is sometimes due to a low mathematics self-concept (Zachai, 1999; Zopp, 1999), which is discussed in Chapter Two. Because of low self-concept and low conf idence, many non-traditional age students require more individual attention and support than may be necessary for the traditional, straight-fromhigh-school student (Lehmann, 1985; Rich ardson & King, 1998; Zachai, 1999). Although older students may be effective pr oblem solvers for many life demands, they may exhibit fewer skills for coping with an academic environment, particularly mathematics. From data gathered in Br itish universities, Woodley (1984) found that adult students tended to obtain better degr ees (i.e., first-cl ass honors) than younger students in the arts and the soci al sciences, but the reverse was true in science disciplines.
13 Woodley concluded that in the former subjects the extra life experien ce of adult students could be translated into greater academic success, whereas the break in their full-time education had resulted in a decl ine in their mathematical and sc ientific skills. These data add support to the researcherÂ’s hypothesis that mathematics courses create unique challenges to non-traditional age students. It appears that non-tr aditional age students will con tinue to make up a substantial percentage of the higher education student body. Because this is a relatively new phenomenon, it is necessary to develop a grea ter understanding of their unique goals and needs in an educational system that was orig inally established to facilitate the growth, training, and education of young adults (D onohue & Wong, 1997). Because women have been traditionally more open about expressi ng their feelings about mathematics, they provide an excellent resource fo r collection of data about the nature of their experience (Fiore, 1999). Challenges for the Community College Many people think of mathematics as one of the most logical, most impersonal branches of knowledge, yet it inspires mo re emotion than any other school subject (Marderness, 2000; Smith, 1994). Â“In the Unite d States most people would be ashamed to admit that they never could learn to read, ye t it is perfectly respectable to confess that one canÂ’t do mathÂ” (Zaslavsky, 1994, p. 5). The researcher has witnessed the emoti onal drama surrounding mathematics first hand, having worked for over a decade in a co mmunity college setting. It is a common occurrence to encounter a sizeable number of women who have been kept from achieving a degree due to an inability to pass the required mathematics. They may attempt the
14 mathematics early in their programs and find it to be a disappointing, failing experience. Instead of persevering, they bypass the mathem atics requirement until they finish all of the other required courses, then return to ma thematics at the end of their programs. By this time, mathematics has become, in their minds at least, a formidable and foreboding challenge, one that may add a year or more on to degree completion. Furthermore, it is not an uncommon occurrence to find women who are counseled to change their original degree choice and career aspirations in order to avoid the required ma thematics. Saddest of all are those who give up en tirely after investing their tim e and money in an education that proved to be a disappointment and unfulfilled dream. Community colleges, in accordance with their mission to democratize American higher education, hold out the promise of providing education in, among other areas, mathematics, to a wide range of students, many of whom had not previously been successful in this subject. Â“Ironically, the very fact that community colleges exist to give students a Â‘second chanceÂ’ may c ontribute indirectly to the co mplex factors that lead to the lack of success some students have ha d in mathematics...Â” (Seidman, 1985, p. 133). Knowing that community colleges will admit them regardless of prior experience with or previous success in mathematics, some may pe rsuade themselves that there is always the hope that if absolutely necessary, they will contend with mathematics and pass the course somehow. The problem is that mathematics, perhaps most of all subjects, is cumulatively organized. Progress in mathematics at one level rests on having achieved a solid foundation at earlier levels. The further one goes up the hierarchically arranged mathematics curriculum, the firmer the foundation of basic unde rstanding must be (Seidman, 1985).
15 All students are forced to deal with ma thematics in a way that affects their futures. The first hurdle, often a barrie r when entering postsecondary undergraduate education, is maximal performance on a standa rdized test. For acceptance to most fouryear universities, a minimum score on the SAT or ACT is required. For entrance into a community college, a college placement test is given to every student, regardless of previous high school grades. Performance on these standardized tests greatly affects student motivation and self-concept (Zachai 1999). Smith (1994) states that doing poorly on these tests makes most people feel dumb. Â“And I donÂ’t just mean dumb in mathematics, I mean dumb in everythingÂ” (p. 93). Most frequently, scores on the verbal portion of the test are adequa te; however, low performance on the mathematics portion of the test places students into remedial cour sesÂ— prealgebra or begi nning algebra. These courses are meant to be Â“refresherÂ” courses a nd move quickly over a lo t of material in a very short period of time. When students ha ve been away from using mathematics for years, it often is a formidable prospect to ma ster all of the mathematical concepts needed to establish a foundation for further learning. Often introductory mathematics courses are structured in the form of labs, where academic software is introduced and several levels of developmental mathematics are combined into one class. The instructor acts as a facilitator and tutor while the student is working in a self-paced fashion. In this scen ario, students are not onl y required to learn the mathematics but must also become familia r with software and quickly learn computer skills, causing additional frustration and a nxiety. With the way the curriculum is organized, coverage of material is essential; therefore, cove rage has taken priority over
16 comprehension. Students are not given the opportunity to understand what has taken place with any degree of depth (Berkman, 1995). The first few mathematics courses offe red in college should be carefully structured and taught by the schoolÂ’s best instructors (Arr iola, 1993; Schatzkamer, 1986), which might include full-time instructors tr ained in methodology for both mathematics and education. These courses s hould be entertaining, interest ing, and practical, with only the essentials that students are going to n eed to succeed in a te chnical career (Smith, 1994). Colleges would need to get help from businesses in order to decide what those essentials [italics added] might consist of and tie the material to some practical applications and interests of their students. It has been the researche rÂ’s experience that teachers assigned to introductory mathematics courses are generally part-time, retired, or persons coming from business or technical fields who want to try [italics added] their hand at teaching. Full-time faculty often prefer to teach the highe r-level mathematics courses wher e they can be assured of a class of students who do not have the uni que struggles and anxi eties that beginning mathematics students exhibit. As a result, teachers may lack a commitment to and understanding of the teaching and learning styl es required for underprepared or returning, non-traditional students to succeed. Good college remedial mathematics teachers are non-threatening and use a student-centered, active learning approach (Arriola, 1993; Cook, 1997; Mullinix, 1995). A typical introductory course in mathem atics may not be viewed as welcoming, meaningful and appealing (Ramus, 1997). An Exploration of the Nature and Quality of Undergraduate Education in Sc ience, Math, and Engineering dated January 1989,
17 reported that, Â“Introductory courses remain unapologetically competitive, selective, and intimidating, designed to winnow out all but the Â‘top tierÂ’Â” (cited in Tobias, 1990, p. 9). Students often see themselves as outsiders in a worl d that they cannot penetrate. Poor performance and repeated failure does not in still confidence and hope Motivations and interests are affected by pe rformance rather than the other way around (Parker, 1997; Scherer, 1990; Tobias, 1990). Tobias recommends extending comfort zones [italics added] for women who are struggling with ma thematical concepts (Tobias, 1994, p. 12). She believes that the failure of women and girls to conquer the world of mathematics represents not a failure of intellect, but rather a failure of nerve. Students enjoy learning and perform well within establis hed comfort zones; therefore, it is a valuable teacher who is committed to identifying ways to help stud ents get comfortable with mathematics. Interest in finding strategies for teac hing entry-level mathematics courses was demonstrated by a call for manuscrip ts in the October 2001 issue of Mathematics Teacher These manuscripts were sought for the purpose of providing material for articles on the following topics: examples of lessons that motivate beginning, hesitant, or uninterested students; ideas for making algebr a and geometry accessible to all students; uses of manipulatives, multimedia, and t echnology to make mathematics a hands-on, interactive, real-life experience for students; innovative ways to teach seemingly simple concepts; and classroom ideas that lead students to take mathematics courses beyond the minimum required for graduation. Although th is magazine is geared for high school mathematics educators, the principles and techniques are applicab le for postsecondary introductory mathematics courses as well.
18 My investigation has been inspired to a large degree by Marderness, whose research was conducted at a publ ic high school, involving ten 11th-grade girls enrolled in an on-level (not the most or least challe nging) mathematics class. Marderness (2000) investigated the thoughts a nd emotions experienced by young women engaged in various mathematical situations, especially those particular experiences where feelings of confidence were lowered. An existentia l phenomenological method was used which depends on acquiring articulate, expressive individuals who have experienced the phenomenon under investigation. Unlike other methods that are researcher-controlled, the existential phenomenological method depends on detailed descriptions of participants to provide the raw material [italics added] for the data analysis (Marderness, 2000). The study revealed common themes of experience, including five structural themes and two interactional themes within one fram ing theme. The framing theme was Perceptions of Self and Others Structural themes included Concern about Grades, Disappointment, Frustration, Giving Up, and Math Just IsnÂ’t Me. Pr essure from Self and Others and Influences of Teacher Behaviors were revealed as interactional themes (Marderness, 2000). MardernessÂ’ research underscores the im portance of teacher responsibility, not only to help students learn content, but also to prepare them for their roles as critical thinkers in todayÂ’s information age. When students, especially females, suffer from a lack of confidence to do mathematics and do not take any more than the minimum number of mathematics classes re quired, their future success is inhibited in a society that is increasingly dependent on mathematical li teracy (Steele & Art h, 1998). MardernessÂ’ research has contributed gr eatly to the body of literat ure on womenÂ’s mathematics
19 experience and this study builds on her findi ngs with a population of older women in a community college atmosphere. Purpose The first purpose of this study was to ex amine metacognitive and affective factors that are perceived to contribute to mathem atics avoidance or mathematics confidence in non-traditional age women atte nding a community college. The research setting is described in detail in Chapter Three. Stories of participants illuminate real ities for other non-traditional age women who are struggling with mathematics and also provide new insights and connections for women who are considering a retu rn to school and a career change It is hoped that this exploration and description of factors that are perceived to contribute to mathematics avoidance or mathematics confidence will add to the literature available to educators and prove to be beneficial in curriculum and pedagogical ini tiatives augmentation. The investigation was done using a phe nomenological approach to in-depth interviewing. The appropriateness of this research design has been eloquently proposed by Seidman (1998). Â”At the very heart of what it means to be human is the ability of people to symbolize their experience through languageÂ” (p. 2). Recent research studies using this approach, description, and th eoretical background of phenomenology are discussed in Chapter Two. Listening to and studying the stories and details of womenÂ’s lives are ways of knowing and understanding. By developing profiles of these women and making thematic connections among their st ories, this study focuses on factors in the educational experience that are perceived to contribute to mathematics avoidance or mathematics confidence.
20 The second purpose of this study was to e xplore and to describe the meaning nontraditional age women attach to their experi ence with mathematics. The question of meaning [italics added] addresses the intellectual and emotiona l connections between the participantsÂ’ feelings about mathematics and their lives. The third purpose of this study was to de termine the relationship, if any, between metacognitive and affective experience in lear ning mathematics. What is true in any given situation has rarely been investig ated (Bloom, Krathwohl, & Masia, 1964) and therefore was one of the unique pursuits of this study. The methodology of in-depth interviewing was chosen because it is a powerful way to gain insight into educational issues by describing and attempting to understand the experience of women whose lives may depe nd on and are enhanced through education (Seidman, 1998). Â“Older women returning to college are part of the continuing movement toward education for allÂ” (Schatzkamer, 1986, p. 322). This study investigated the follo wing three rese arch questions: 1. What metacognitive and affective factors are perc eived to contribute to mathematics avoidance or mathematics confidence in non-traditional age women attending a community college? 2. What meanings do participating non -traditional age women attending a community college attach to their experience with mathematics? 3. What is the relationship, if any, between metacognitive and affective experiences of participating non-traditional age wo men attending a community college in learning mathematics?
21 Definition of Terms For the purposes of this study, the following definitions are provided for clarification. Affect/affective The term affect has generally been used to refer to mental aspects of human nature that are diffe rentiated from reason. Those aspects include feelings or emotions like anger, fear, love, and others th at were thought to arise from stimuli without reasoned analysis (Beane, 1990). Modern research in psychology and philosophy, however, has demonstrated that thought a nd feeling occur simultaneously. Emotional responses are based on past experiences. Â“Feelings and emotions are not empty occurrences; rather they are feelings about or emotions tied to something. That Â‘somethingÂ’ is the content of pr oblematic situations that call for some sort of reaction or resolution. Â…no matter how irrational a react ion or proposed resolution may seem, it is based on some degree of belief that it is appr opriate in the given s ituationÂ” (Beane, 1990, p. 4). Consequently, affect refers to a br oad range of dimensions such as emotion, preference, choice, and feeling. These dime nsions are based on be liefs, aspirations, and attitudes regarding what is desirable in personal development and social relationships. Furthermore, these dimensions are connected to thinking or cogn ition because they are informed by what has been learned from past experiences (Beane, 1990) It is important to recognize the interrelationship between cognition and affect (Bloom et al., 1964; Goleman, 1995). Affective development is regarded as important by many institutions. This includes the development of such characteris tics as emotional maturity, tolerance, empathy, and leadership ability (Astin, 1985). Fo r the purpose of this paper, the affective
22 dimension refers to the operative emotional component when dealing with mathematics, at times consisting of positive [italics added] affective as pects such as confidence and exhilaration, and sometimes consisting of the negative [italics added] affective aspects such as anxiety, feelings of nervousne ss, tension, dread, fear, and unpleasant physiological reactions to stressful si tuations (Oxford, 1997; Sarason, 1986). Mathematics avoidance Closely related to the cons truct of mathematics anxiety, mathematics avoidance is behavior that seeks to evade confrontation with mathematics. Such behavior would include attempting to circumvent the mathematics sequence of courses, get around [italics added] the mathematics requirements, keep oneÂ’s distance from the instructor by sitting in the rear of the classroom, and staying detached from any discussion or contemplation that relates to ma thematics. Parents, peers, significant role models, and environmental, social, and psychol ogical factors contribute to mathematics avoidance (Campbell & Beaudry, 1998). Mathematics confidence Confidence in mathematics is a belief in oneÂ’s ability to complete the work successfully. It is the Â“I think I canÂ” mentalit y, a conviction that one can handle the task and handle it well. Partic ipation in mathematics at all levels beyond elementary school revolves around the conf idence factor (Armstrong & Price, 1982). Metacognition Metacognition, or awareness of the process of learning, is a critical ingredient to successful learning. Metacognition is an im portant concept in cognitive theory and consists of two basic processes occurring simultaneously. Individuals (1) monitor their progress as they learn and (2) make changes and adapt their strategies if it is perceive d they are not doing so well (Winn & Snyder, 1996). Blakey
23 and Spence (1990) define metacogniti on as thinking about thinking, knowing what we know and what we donÂ’t know [italics added] Â“Metacognative skills include taking conscious control of learning, planning and selecting strategies, monitoring the progress of learning, correcting errors, anal yzing the effectiveness of learning stra tegies, and changing learning be haviors and strategies when necessaryÂ” (Ridley, D. S., Schutz, P. A., Glanz, R. S. & Weinstein, C. E., 1992). Learning how to learn, devel oping a repertoire of thinki ng processes which can be applied to solve problems, is a major goal of education. Non-traditional age student According to the NCES, the term Â“non-traditional studentÂ” is not a precise one, although age and part-time status are common defining characteristics (Bean & Metzne r, 1985). Non-traditional students, described by Cohen and Brawer (1996), are individua ls who do not conform to the profile of the traditional 18-year-old student who enrolls full-time at a community college, completes the freshman and sophomore years, and transf ers to a four-year college to earn a baccalaureate degree. In searching through the vast amount of literature relating to college student populations, most authors, as we ll as the U. S. Department of Education, have concurred that a non-traditional student means he or she is 25 years or older and has any one or more of the following characterist ics: delays enrollment; attends part-time while working full-time (35 hours or more per week); has dependents other than a spouse (usually dependent children); r eenters a two-year college afte r being previously enrolled at a four-year institution; is considered financially independe nt for purposes of determining eligibility for fi nancial aid (Bean & Metzner, 1985; Horn et al., 1996; Klein, 1990; Krager et al., 1990; Padula, 1994; Scott et al., 1996).
24 Perception In psychology, perception is ment al organization and interpretation of sensory information or stimuli. An i ndividualÂ’s perception is influenced by the intensity of the stimulus, effects of previous stimulation, past expe rience, and his or her motivation and emotional state. Sometim es emotional disturbance can prevent perception completely (Smith, 2004). We r eceive information through our senses, then our mind selectively focuses on some of the in formation, ignoring what is considered less important. Evidence points to the conclusi on that early experience, learning, emotion, and motivation are important in defining what and how we perceive (Heil, 1983). Phenomenoloical method This model of in-depth, phenomenological interviewing (Seidman, 1998) allows the in terviewer and partic ipant to plumb the experience and place it in context. Phenom enology encourages a close relationship between the experiences of real life and th e ideas that guide our actions in practice (Boeree, 2007). This process of putting e xperience into language is a meaning-making process (Vygotsky, 1987). Self-efficacy Albert Bandura defined self-effica cy as the conviction that one can successfully execute the behavior required to produce the desired outcome (1977, p. 193). Expectations of what one can do determine how much effort will be expended and how long this effort will be sustained in the face of obstacles and aversive experiences. Perceptions of efficacy influence thought patt erns, actions, and emotions. What a person believes that he or she can do is derived from four sources: performance accomplishments, vicarious experiences, verbal persuasion, and physiologi cal states. Application of the concept of self-efficacy expectations to the realm of mathematics could help to understand and treat mathematics avoidan ce and anxiety (Bet z & Hackett, 1983).
25 Assumptions There are a number of assumptions that were made at the beginning of this study. Some of these follow: 1. Individuals construct knowledge in a social context, through individual cognitive processes and the social context and interactions in the home and in the classroom. 2. Participants would welcome the opportuni ty to talk about their experiences in their early education, recall teachers who they perceived as making a difference in their lives, and describe their feelings about lear ning mathematics currently in the community college atmosphere. 3. As the researcher, I would be able to listen intently, withhold judgment, and focus on what the participant is truly saying, rather than what I might think theyÂ’re saying, and to establish sufficient rapport a nd confidence to encourage participants to trust me and share their stories. 4. Participants would be able to r ecall and clearly share their memories and feelings about their educational experience. 5. Despite many differences, I am simila r enough to the participants that if I calmly, non-defensively listen to them, I would be able to clearly understand their lived experience and be able to re port their stories in a manner that accurately reflects that experience. 6. Knowing the factors pe rceived to contribute to mathematics avoidance or mathematics confidenceÂ—as shared by the participantsÂ—would be valuable in serving the needs of other non-trad itional age women enrolled in a community college.
26 Limitations and Delimitations Limitations Some of the limitations of this study include the following: 1. This study was limited to non-traditional age women attending a community college. 2. This qualitative research study is not meant to be generalizable since the findings are based on a small purposeful samp le; nevertheless, the study does provide an in-depth examination of the phenomenon. Co mmunity college teachers and students can determine whether or not the st udy applies to their situations. 3. Since the researchers are the main inst ruments in the collec tion and analysis of data in qualitative research, as the primary researcher, my beliefs and experiences teaching in a community college have informed this study. As an individual who has lived out many of the same experiences as the participants in this study, I have a particular viewpoint that has an effect on how I interpret the data. Therefore, the involvement of two co-researchers to read and develop themes from the data was instituted to assure me of the accuracy of my analysis of the participantsÂ’ stories and to increase the credibility of the findings. 4. ParticipantsÂ’ responses may have been limited by the capacity of their memories. In addition, the interview proce ss may have colored th eir recollections of people and past events. They may have di scovered new meanings in the process of retelling their stories. Delimitations No delimitations.
27 Organization of the Study Chapter One introduces and emphasizes th e need for exploring factors that are perceived to contribute to mathematics a voidance or mathematics confidence in nontraditional age women attending a community college. The importance of understanding the experience many non-traditi onal age women have in attempting to pass the required mathematics courses they need to move in to technological or sc ientific fields is presented. Current trends in postsecondary institutional enro llment of non-traditional age students are discussed along with the current challenges facing community colleges. The purpose of the study, research questions, defi nition of terms, assumptions, limitations, and delimitations of the study are presented. Chapter Two is dedicated to a review of the literature relevant to this study. Chapter Three examines the value of a phenomenological approach to in-depth interviewing and the study design, which was th e informal conversational interview. The research setting is described and a brief histor y of the researcher is reviewed. Selection of participants, procedures involved in decidi ng on interview questions and the process of transcribing, compiling and analyzing the data are explained in detail. The chapter concludes with a discussion of trus tworthiness and credibility issues. Chapter Four is comprised of the findings of this study. Eight major themes and six sub-themes emerged from a constant comp arative analysis of the transcribed data. Brief profiles of each participan t are presented as well as thei r stories, expressed in their own words.
28 Chapter Five includes a summary of the findings, a review and discussion of the major themes in connection with the literature, imp lications for practic e, suggestions for further research, and personal reflections.
29 Chapter Two Review of the Literature This chapter will address research findings from studies of academic self-concept, social conditioning and stereo typing of women, mathematics anxiety, teacher influence on studentsÂ’ attitudes about mathematics, constructivist approach es to learning, and mathematics education initiatives designed to improve mathematics performance. The first section of this chapter provides a hist orical backdrop including womenÂ’s role in the American labor force and the need for inform ation-age skills in our society. The final section will examine the fundamental prin ciples of phenomenological research approaches and explore the appropriateness of such methodological approaches for this study. Historical Background Concerns about mathematics performance by U. S. students came to the forefront of national concern following the successful Ru ssian launching of S putnik in October of 1957. AmericaÂ’s presumed scientific and t echnological superiority was called into question (Mayhew, Ford, & Hubbard, 1990). Th e immediate response was to concentrate on increasing the robustness of mathematic s and science education in the nationÂ’s schools. Along with an increased rigor in mathematics courses and mathematics testing, concern arose in the mid 1970s over mathematic s anxiety, which came to the attention of educators first as a feminist issue. Young women were not taking the high school
30 mathematics courses they needed for many co llege majors, and, as a consequence, were excluding themselves from promising a nd well-paying careers (Zaslavsky, 1994). Three decades later, President George H. Bush, in the 1990 State of the Union address, Â“proclaimed as one of his educational reform goals that American students, by the year 2000, should rank first in the world in math and scienceÂ” (Smith, 1994, p. 123). According to Reyes and Stanic (1988), know ledge of mathematics is essential for all members of our society. Â“To participate fu lly in our democratic processes and to be unrestricted in career choice and advancement, people must be able to understand and apply mathematical ideas. Unfortunate ly, certain groups are underrepresented in mathematics courses and do not achieve up to their potentialÂ” (p. 26). Women are among these groups (Reyes & Stanic, 1988). TodayÂ’ s women are less likely than men to earn mathematics degrees in postsecondary e ducation, earning a smaller and smaller proportion of these degrees at advancing levels of education, in other words, from the associate through doctoral degree levels (Fos ter, Squyres, & Jacobs, 1996). High school senior girls are more likely than boys to say that they did not take mathematics courses because they were advised that they did not need them (32% and 26%, respectively) or because they disliked the subject matter ( 35% and 22%, respectively). Females were more likely than their male peers to say poor performance in the subject kept them from taking additional mathematics classes (Foster et al., 1996). It was found that, initially, boys and girls are alike in their perceived mathematical capabilities, but gi rls begin to lose confidence in their mathematics ability and differ increasingly in this regard as they move into high school. Partly to blame are parentsÂ’ beliefs about their childrenÂ’s capabilities, whic h stem from the cultural
31 stereotype that girls are less talented in mathematics than boys (E ccles, 1989; Phillips & Zimmerman, 1990). Outside the home, studi es found gender bias alive and well in classrooms as well and, not surprisingly, al so in career guidance functions (Betz & Fitzgerald, 1987). More students are going on to college than ever before, but near ly a third of them find it necessary to take remedial courses in reading, writing, or mathematics (Ravitch, 2000). Presently, there are a significant num ber of non-traditional age women entering the nationÂ’s community colleges who are una ble to pass the mathematics requirement (i.e., 22% of non-traditional female students te sted into remedial mathematics courses at the researcherÂ’s commun ity college in the spri ng of 2003). This si tuation points to the important role of social-support systems for females at all levels of education. Women in the Labor Force Non-traditional age women enter or reenter the world of academia for a variety of reasons, many of which are job-related. Most of the women are working wives and mothers; they work because they need the money. A solid majority of women in the labor force in 1991 were either single ( 25%), divorced (12%), widowed (4%), separated (4%), or had husbands who earned less than $15,000 the previous year (13.5%) (Begun, 2000). A little more than a century ago, few women completed high school, and the fortunate few who did attended sex-segregated seminaries that emphasized social graces, not academics. Most women who went beyond high school could only prepare for teaching, social work, and nursi ng. Business schools prepared them for clerical duties, never for management (Begun, 2000). While many women still train for these traditionally female occupations, others, as a result of growing educational and
32 occupational opportunities, explor e other fields. According to the findings in Table 5, during the past 20 years, female participation in such traditionally male occupations as medicine, law, engineering, law enforcement, computer science, financial management, and college and university teachi ng has increased significantly. For women to continue to move into pr ofessional and technol ogical jobs, a strong background in mathematics is critical. The pr oblem lies in the fact that when these women were in high school, they may not have taken the mathematics necessary to allow them to succeed in college level courses. Many academically capable students prematurely restrict their educational and career options by discontinuing their mathematical training early in high school (Fotoples, 2000; Hall, Davis, & Bolen, 1999; Horn et al., 1996). Mathematics Course Selection and Affective Issues in High School The National Center for Education Stat isticsÂ’ primary assessment of what American elementary and secondary students know and can do in academic subjects is the National Assessment of Educational Progr ess. Often called the Â“NationÂ’s Report Card,Â” the NAEP has been taking academic sn apshots of AmericaÂ’s students since 1969. Funded by the federal government, the NAEP tests 4th, 8th, and 12th graders in several different subjects. Unlike the SAT exam, wh ich develops academic pictures of only the college-bound, the NAEP offers information on a sample of all studen ts. It is through this and other national tests that we learn how well girls begin their school careers and what happens with each yearÂ’s promotion to a new grade. In the 1980s, when the nontraditional age women in postsecondary education today were most likely junior or senior
33 Table 5. Proportion of Women in Selected Occupations Occupation Proportion female 1975 1995 Automobile mechanics 0 1 Cashiers 87 79 Carpenters 1 1 Computer systems analysts 15 30 Engineers 1 8 Financial managers 24 50 Lawyers 7 26 Physicians 13 24 Police and detectives 3 14 Registered nurses 97 93 Social workers 61 68 Teachers, college and university 31 45 Teachers, elementary 85 84 Waiters and waitresses 91 78 Note. From Â“Gender Differences in Occupational Employment,Â” by Barbara H. Wooten. Monthly Labor Review, 120 (4). Permission granted by the author.
34 high school students, data indi cate that only half of all high school graduates chose to enroll in mathematics courses beyond the 10th grade. These reports also indicate that fewer women than men enrolled in the more advanced courses in high school mathematics. Furthermore, women, in particul ar, did not appear to have attained a high level of mathematical competency, even if th ey completed four years of high school math (NAEP, 1988; NCES, 1984). Other research conducted dur ing the 1980s supports the findings of the national data. Eccles (Parsons), Adle r, and Meece (1984) found that self-concepts of mathematics ability were predictive of junior and senior high school studentsÂ’ course enrollment plans and performance in mathematics. They also document significant gender-related differences in junior high school studentsÂ’ mathematics related expe ctancies, values, and self-concepts of mathematic s ability. In 1990, Meece, Wigfield, and Eccles studied predictors of mathematics anxiety and its influence on young adolescentsÂ’ course enrollment intentions and performance in ma thematics. This study confirms predictions regarding the critical role (especially for gi rls) that perceptions regarding the value of mathematics play in determining studentsÂ’ inte ntions to enroll in advanced mathematics courses. NAEP 2000 assessment results i ndicate that, at the 8th and 12th grade levels, boys continue to outperform girls in mathema tics, although the gap is decreasing. This increase in mathematics achievement for girl s is reflected in National Science Foundation statistics which show that the percentage of bachelorÂ’s degr ees in science and engineering awarded to women has been steadily incr easing; in the mid 1980s women earned about 38% of bachelorÂ’s degrees in science and engineering; in 1995, they earned 46%, in
35 1998, the number reached 49%. (Begun, 2000). Th is is an encouraging report, however gender differences in college majors persist, w ith women still concentrated in fields like education and men more likely than women to earn degrees in engineering, physics, and computer science (Bae et al., 2000). According to a National Science Foundation study, Women, Minorities and Persons with Disabilities in Science and Engineering: 2002 one of the factors related to course taking and achievement in high school is a studentÂ’s attitude toward mathematics and science. Differing attitudes toward scienc e and mathematics and different perceptions about their performance in these subjects are evidenced by members of both sexesÂ….One factor in the differenc es between male and female students in science and mathematics achievement may be these differences in attitude. Females generally have less positive att itudes toward science and math than do males. In 2000, female 4th and 12th graders were less likely than their male counterparts to agree with the statement Â“I like mathematicsÂ” (an indicator of their attitudes about mathematics). In grades 4, 8, and 12, females were less likely than males to agree with the statement Â“I li ke science.Â” And among students in all three grades, females were less positive than males regarding their mathematics and science performance: sp ecifically, they were less likely than males to agree with the statements Â“I am good at mathem aticsÂ” and Â“I am good at scienceÂ” (NSF, 2002). When reading these results, it is important to keep in mind that there is no simple, causal relationship between membership in a subgroup, such as female, and mathematics
36 achievement. A complex mix of educational and socioeconomic factors may interact to affect student performance (NAEP, 2000). Teachers can help students appreciate the value of mathem atics by explicitly relating mathematics to studentsÂ’ everyday lives and by counseli ng students about the importance of mathematics for various caree rs. Â“Unfortunately, in over 400 hours of classroom observation, Eccles and her collea gues observed fewer than a dozen instances of these instructional behavi orsÂ” (Meece et al., 1990, p. 69). In addition to the critical role that studentsÂ’ perceptions of the value of mathematics and their own mathematics ability play, evidence further s uggests that affective factors may also play an important role in achievement patterns, especially in females (Meyer & Fennema, 1986; Rounds & Hendel, 1980) and have strong relationships with course and career selections (Armstrong & Price, 1982; Ethington & Wolf e, 1988; Tocci & Engelhard, Jr., 1991). Affect permeates the en tire school and the expe riences of young people who attend. Â“A theory of learning or schooling that ignores or deni es affect is incomplete and inhumanÂ” (Beane, 1990, p. 7). When young people come to school, they bring with them their whole selves, including th e affective aspects. One of these aspects is a belief system of attitudes about themselves. Each day adds new experiences that may confirm, change, refine, or otherwise al ter their existing belief system s, preferences, or attitudes toward themselves and others. Because th ese aspects are part of the makeup of the individual, they cannot be left at the school door or set aside. The student who is afraid or lacking in confidence is not likely to barge into new expe riences as if those feelings did not exist (Beane, 1990).
37 Beliefs shape behavior and can create a ch illy climate in the classroom, according to Sadker and Sadker (1994). TeachersÂ’ beliefs that boys are smarter in mathematics and science begin in the earliest school years. Many adults think that boys possess innate mathematical and scientific ability. Girls can also achieve, they believe, but they have to try harder. Sometimes it is counselors who harm when they mean to help. Feeling sorry for girls who find their mathematics and scienc e courses difficult, they excuse them, a dismissal less likely to be offered to male students (Sadker & Sadker, 1994). When girls select out of mathematics, they are making decisions that will affect the rest of their lives. Without the right high school courses, college courses are out of re ach. Without college courses, females are filtered out of careers that remain overwhelmingly and solidly male (NSF, 1990). Scenes experienced by a student in the classroom, whether positive or negative, have a permanent effect on the quality of his or her life (Tomkins, 1991), depending on how many times the scene is reh earsed. Â“The effect of any se t of scenes is indeterminate until the future either further magnifies or attenuates such experienceÂ…The consequence of any experience is not singular but plural. There is no single effect, but rather there are many effectsÂ…Â”(Tomkins, 1991, p. 87). Any scen e which is sufficiently similar to bring about the same type of respons e is thereby made more simila r and increases the degree of connectedness of the whole family of scenes (Tomkins, 1991). Consider a scene in which a student is bombarded with demands for rapid responses at the same time being threatened with humiliation. Such threats increase the perceived difficulty of the demanded performa nce sufficiently to slow the student down
38 while simultaneously increasing the demand for more competence and more speed (Tomkins, 1991). When students internalize success and extern alize failure, an approach that Sadker and Sadker refer to as, Â“the male approach,Â” they are able to tack le new and challenging tasks and persevere in the face of difficulty. Students who attribute success to effort and failure to lack of ability (the female appr oach) exhibit Â“learned helplessness.Â” When confronted with difficult academic material, th ey do not persist but rather say, Â“I think I canÂ’t,Â” and give up (Sadker & Sadker, 1994). Need for Information Age Skills in TodayÂ’s Labor Force How large a role do women play in the labor force today and what is to be expected in the next decade? Acco rding to the November, 2001, issue of Monthly Labor Review the rate of growth for women in the labor force between 2001 and 2010 will increase at a faster rate than that of men (Fullerton & Toossi, 2001). In 2008, women will make up about 48% of the labor force and men 52%. In 1988, the respective shares were 45% and 55% (Monthly Labor Review, February 14, 2000). The workforce of the future will be shaped by several emerging trends: an aging workforce; increased numbers of women, minorities, and immigrants ; a declining pool of youth from whom to draw; a continuing movement toward the service-producing sect or; and a demand for highly skilled workers (Begun, 2000). In 2010, women will account for 48% of the labor force compared to 47% in 2000 This 1% may seem small, however it represents an increase of 9,884,000 women (Fullerton & Toossi, 2001). In addition to the aging of the future wo rkforce, there is a continuing demand for workers who have acquired computational ski lls. Our world is becoming increasingly
39 more quantitative due to technological adva nces through the rise of computers and a more global job market. Therefore, occ upations in the United States are becoming increasingly dependent on knowledge of com puters and quantitative expertise. Â“As we enter the 21st century, three critical challenges face the nation: competing in a global economy, reversing the growth of a perman ent and disenfranchised underclass, and developing a workforce with Information Age skills. Without doubt, meeting these challenges will depend on the achievement of our educational systemÂ” (McCabe, 1999, p. 23). Dr. Susan Sclafani, advisor to United Stat es Secretary of Education Rod Paige, in a speech delivered at the 2002 annual meeting of the National Council of Teachers of Mathematics (NCTM), stressed the importance of a highly sk illed workforce. In 1950, the job mix in the United States was 20% pr ofessional, 20% skilled, and 60% unskilled. Comparing those figures to the year 2000: the job mix was 20% professional, 65% skilled and only 15% unskilled. Sclafani projected that by the year 2020, there would exist 15 million more jobs requiring education and skills than people to fill them. Saying, Â“I was never good in math,Â” wonÂ’t work in this new economy (Sclafani, 2002). Quantitative literacy is important through a variety of job fields, not just in mathematics, as many people previously believed. As evidence of this fact, NCTM offers, in its 2001-2002 catalog, the book, Why Numbers CountÂ—Quantitative Literacy for TomorrowÂ’s America which shows ways in which mathem atics plays a critical role in everyday life. Women who desire to move into new fields of opportunity are finding that undergraduate degrees that prep are them for such fields re quire a significant number of mathematics courses. Careers in nursi ng, veterinary techno logy, engineering,
40 architecture, and criminal justi ce, just to name a few, are ex amples of careers that require college-level mathematics. Understanding the Role of Academic Self-Perception Perception, the way people experience, pro cess, define, and interpret the world around them (Lewis, Goodman, & Fandt, 2001) influences the way individuals communicate and become aware of sensations and stimuli that exist around them. Acting as a filter, perception helps indi viduals take in or see only cert ain elements in a particular situation (Lewis et al., 2001). There is not only a constant interaction with oneÂ’s world but also a continuous contact with oneÂ’s inner reality; it is a personal choice of a point of view (Bach, 1965). The way that we perceive and react to an event is largely responsible for the ultimate effect of that event upon us If we can understand and make sense out of an event and draw some objective, meaningful conclusion, the impact of that event will be less dreadful. Meaninglessness, on the ot her hand, can be very disturbing (Ghadirian, 1983). An individual becomes slowly both the bene ficiary and the victim of the world he or she has most often experienced and rememb ered. One is the beneficiary when stored scenes have been positive and suffers as the victim when the scenes have been negative. One feels free [italics added] when good scenes ar e retrieved and feel s victimized when bad scenes are remembered (Tomkins, 1992). Many elements may affect a personÂ’s per ception in any given situation. These include an individualÂ’s age, sociocultural or ientation, personal sk ill and capability in problem-solving and decision-making processe s, memory capacity, quality of vision and hearing, energy level, or amount of stress at a given time (Badger, 1996). In addition,
41 personal beliefs and expectations play a major role in a personÂ’s perception of his or her capacity to learn and achieve (Ghadirian, 1983) Several studies have shown that the beliefs that learners hold about their capab ilities have a strong influence on task engagement and achievement (Bandura 1982; Richardson & King, 1998; Schunk, 1989). Â“It is studentsÂ’ interpretati ons of their achievement outcomes and not the outcomes themselves that have the strongest eff ects on studentsÂ’ affective reactions of achievementÂ” (Meece et al., 1990, p. 68). E ach student may perceive the college experience differently. Studies have shown that these per ceptions have both negative and positive effects on student attrition and persistence (Hatcher, Kryter, Prus & Fitzgerald, 1992; Klein, 1990; Lamport, 1993). A vivid illustration of how sensory percep tion may lead to mi sinterpretation is dramatized in the poem entitled, Â“Blind Men and An Elephant.Â” The poem, written by American poet John Godfrey Saxe (1816-1887), is based on a fable told in India many years ago, and describes the experience of six blind men who went to see the Elephant. Each Â“sawÂ” the Elephant as something differe nt: a wall (touching the ElephantÂ’s side), a spear (touching the ElephantÂ’s tusk), a snak e (touching the ElephantÂ’s trunk), a tree (touching the ElephantÂ’s knee), a fan (touching the ElephantÂ’s ear), and a rope (touching the ElephantÂ’s tail). Â“Each was partly in the right, but all we re in the wrong.Â” Likewise, on a college campus or in th e classroom, studentsÂ’ perceptions of a remark, a gesture, or a situation, may not be alike. For example, a competitive classroom atmosphere may be perceived as exciting a nd motivating to one student yet may appear threatening and intimidating to another. Re search indicates that, while competition is motivating for many traditional age students, for non-traditional students, a cooperative
42 orientation rather than a competitive orientati on enhances the studentÂ’s success in college (Donohue & Wong, 1997). Perceptions of the am ount of effort expended to achieve a goal may differ. One student may view extensive hours of studying as a sign of dedication and tenacity, where another student may perceive the effort as a sign of low ability (Harju & Eppler, 1997). Of particul ar interest to this researcher is an understanding of the perceptions students may have of their own ability, teacher behaviors, and the college expe rience that affect mathematics confidence either positively or negatively. Mathematics Self-Concept and Gender Â“The self is not something with which individuals are born but something they create out of their experien ces and interpersonal relati onshipsÂ” (Hamachek, 2000, p. 230). Each person has a self (a sense of personal existence), a self-concep t (an idea of personal identity), and a certain level of self-esteem (f eelings of personal worth). The self grows, develops, and understands itself in a social cont ext. The interactions students experience and the feedback they receive are each important components in the development of a sense of self-concept (Hamachek, 2000). More specifically, academic self-concept re fers to studentsÂ’ perceptions of their academic abilities (House, 1992). A positive se lf-concept is an important mediating variable that may promote academic achievement and other valuable educational outcomes (Marsh & Yeung, 1996). Marsh, Craven, and Debus (1991) show that enhancement of self-concept can improve acad emic performance and is strongly related to subsequent course sel ection (Marsh & Yeung, 1997).
43 Self-concept has been shown to be intim ately involved in the cultivation of cognitive competencies (Bandur a, 1997). Furthermore, percei ved beliefs of oneÂ’s ability (self-efficacy) contribute inde pendently to intellectual performance rather than simply reflecting cognitive skills. Â“Perceived efficacy exerts a more substantial impact on academic performance, both directly by a ffecting quality of thinking, and good use of acquired cognitive skills and indirectly by heightening persistence in the search for solutionsÂ” (Bandura, 1997, p. 216). Individuals of high efficacy pers ist while those of low efficacy are more apt to quit (Bandur a, 1982; Bandura & Schunk, 1981). The higher the studentsÂ’ efficacy beliefs, the higher the academic challenges they set for themselves (Pintrich & DeGroot, 1990). Students gain knowledge about themselves by comparing how they measure up with those around them. When students are uncer tain about any aspect of the self that seems important to themÂ—abilities, compet ency, personal worthÂ—they do not compare themselves with just anyone, but only with similar others (Gilbert, Gieser, & Morris, 1995). Comparison of self with others is usef ul in two ways: it allows individuals to gain knowledge about themselves (cognitive info rmation) and to assess how they feel about themselves (affective information) (Hamachek, 2000). According to Donohue and Wong (1997), se lf perceptions of traditional and nontraditional students may differ significantly, sugg esting that age is a variable affecting oneÂ’s self-perception. In addition to age, ge nder may also be a defining factor in selfperception. Mathematics self-concept in women is believed to be related to issues of power, justice, and oppression, not aptitude. Psychosocial theorists propose that factors such as gender-role orientation and/or att itude toward mathematics play critical and
44 complementary roles. Many such theorists be lieve that the traditional feminine genderrole orientation and associated gender stereo typing interfere with womenÂ’s successful mathematics achievement and discourage involvement in mathematics-related careers (Landerman, 1987; Swindell, 1988). Women may attribute good performance in the classroom to effort rather than ability; they can achieve when they put forth great effort but, at heart, they may not believe that they have a true ability for mathematics (Eccles (Parsons) et al., 1984; Richardson & King, 1998; Seegers & Boekaerts, 1996; Smith, 1994; Stipek, 1984). Studies conducted in 1977 and 1978 by Fennema and Sherman show that males consistently exhibit higher levels of confid ence in their ability to learn mathematics than females. These gender differences were also found when there were no differences in achievement (Meyer & Koehle r, 1990). Successful performances do not necessarily enhance efficacy-related perceptions ; the impact of this information depends on how it is cognitively appraised and in terpreted (Schunk, 1984). Although a selfderogating bias in attributions has been found in skill areas other than mathematics, gender differences are most consistent in the mathematics domain (Ryckman & Peckham, 1987). Does the same self-concept seem to hol d true in women when the subject under discussion is not mathematics? Mythology a bout mathematics includes the idea that if one is good [italics added] at language arts, one is, inevitably, not good [italics added] at mathematics (Tobias, 1991, p. 92). In this rese archerÂ’s experience, it is not uncommon to be informed by a female student that she is getting AÂ’s in Eng lish but is Â“no good at math.Â” Marsh (1990) studied how mathematics and English self-concepts are formed and what may influence them in their formation. The study found that mathematics self-
45 concept was dependent on whether a student had an external or internal frame of reference. An external frame of reference compares oneÂ’s self-perception of skill to the perceived skills of other students, whereas an internal frame of reference will make a comparison between oneÂ’s ability in mathema tics as compared to oneÂ’s performance in English. The Big Fish-Little Pond Effect wa s also described where an average-ability student is placed in a high-ability group of classmates, causing self-concept to suffer. The implication is that averag e ability students may see them selves as doing poorly when placed in a group of high-ability students. Li kewise, when studentsÂ’ English skills are high, this may concurrently affect their per ceptions of their mathematics skills as being low. The results of the study imply that if a studentÂ’s frame of reference can be determined, such an insight may help to e xplain self-concept. MarshÂ’s findings also indicated that when mathematics achievement was strong there was an accompanying positive relationship to math ematics self-concept (1990). Unger and Crawford (1996) have linked a low self-concept to feelings of not belonging in the mathematical arena. Students, especially girls, of ten begin elementary school with positive feelings about their abil ity to perform well in mathematics. By adolescence, however, many girls have conc luded that they are Â“not good at mathÂ” (Silver & Kenney, 2000). Sax (1992) conducte d a study of college-age women who, in spite of performing slightly better than men on tests of mathematics ability, scored lower than men on tests of mathematics self-conf idence. Using data from 192 colleges and universities, representing a total of over 15,000 students, Sax discovered that women rated themselves significantly lower on mathematics ability than men.
46 In-depth interviews have the potential to explore and clarify this complex phenomenon of lower mathematics confidence in women of equal or greater ability than men. The discrepancy may be due to the fact that females are often reluctant to express confidence in an area that has historically been labeled a male domain (Sherman & Fennema, 1977; Tocci & Engelh ard, Jr., 1991). As a result, women are more likely to attribute success in mathematics to great e ffort or luck. Although over the last two decades achievement levels of females has in creased, the myth of male superiority and dominance in the field of mathematics rema ins virtually unchanged (Caporrimo, 1990). For many women, the belief that they are outsiders to the world of mathematics becomes ingrained in their belief system during adolescent years and remains intact throughout their adult lives (Crawford, 1980; Marderness, 2000). Teacher attitudes and encouragement (or lack of) have a signifi cant influence on a studentÂ’s choice of major/career fields. StudentsÂ’ attitudes about the subjects they study are often tied to performance. Such attitudes can affect enth usiasm for learning a subject and the effort devoted to studying it (Eccles 1994). WomenÂ’s attitudi nal differences toward mathematics begin early in elementary school, and are reinforced by subject area encouragement from teachers. In fact, resear ch suggests teachers, more than family or peers, influence womenÂ’s decisions to c ontinue/discontinue high school mathematics (Scherer, 1990). A negative mathematics self-concept involves perceptions of mathematical incompetence and low mathem atics self-esteem which may result in a perpetual lack of mathematical succe ss (Sax, 1994; Wadlington, 1992). There is evidence that a lack of encouragement to study mathematics goes hand in hand with the experience of individuals who lack pow er in our society (Drew, 1996).
47 Social Conditioning and Stereotyping Social conditioning cannot be overlooke d when discussing self-concept and selfconfidence of women. Studies have shown th at women speak freque ntly of problems and gaps in their learning and so often doubt th eir intellectual competence. For many women, the real [italics added] and valued lessons did not necessarily grow out of their academic work but in relationships with friends and teachers, life crises, and community involvements (Belinky et al., 1986). Boys and girls differ with respect to attitudes, selfconfidence, values, career aspirations, and expectations regarding mathematics performance. These differences are l earned behaviors (Fox, Tobin, & Brody, 1979). Socialization agents are parents, teachers, c ounselors, peers, the school environment, the media, and books (Fox et al., 1979). Parental Support Studies have shown parents, to a greater extent than teachers, hold gender-differentiated beliefs about their sonsÂ’ and daughtersÂ’ mathematics achievement (Eccles-Parsons, Adler, & Kaczala, 1982; Tocci, 1991). Furthermore, childrenÂ’s self-concept of ability and their c onfidence in mathematics are more directly related to their parentsÂ’ beliefs about thei r mathematics aptitude and potential than to their own past achievement in mathematics (E ccles-Parsons et al., 1982 ). Parents in the Eccles-Parsons et al. (1982) study thought that mathematics was more difficult for their daughters, that their daughters had to work harder in order to do well in mathematics, and that enrollment in advanced level mathem atics courses was less important for daughters than for sons. To the extent that parents c onvey the expectations inherent in these beliefs to their children, parents may he lp socialize the gender differences in studentsÂ’ attitudes toward mathematics (Dickens, 1990; Yee & Eccles, 1988).
48 Yee and Eccles (1988) found a consiste nt pattern emerging from their data indicating that although girls and boys were doing equally well in mathematics, both mothers and fathers credited boys with tale nt and girls with effort. Parents may inadvertently encourage boys to develop highe r estimates of their mathematical ability while undermining both their own and their daughtersÂ’ estimates of the daughtersÂ’ mathematical talent. When parents do th is, it may cause boys to develop greater confidence in their continued future success, while girls begin to doubt their continued success in an activity that they presume ge ts increasingly difficult (Eccles, Adler, Futterman, Goff, Kaczala, Meece, & Midgley, 1983). Tocci and Engelhard, Jr. (1991) conducte d a study using nationally representative samples of 13-year-old students in the Unite d States and Thailand. The study indicated that, for students in both countries, there was a positive relationship between parental support and student attitudes toward mathema tics. AdolescentsÂ’ pe rceptions of their parentsÂ’ reactions to mathematics, along with the amount of encouragement they receive to study and do well at it, may affect the st udentsÂ’ attitudes towa rd mathematics. Researchers of adolescent be havior frequently underestimate the power of the home environment (Tocci & Engelhard, Jr., 1991), especially in the area of mathematics achievement, which appears to be particularly susceptible to the influence of parental beliefs (Chipman, Brush, & Wilson, 1985). Stereotyping Women generally have lower stat us than men, as is evidenced by findings that stereotypical femi nine traits are evaluated less favorably than stereotypical masculine traits (Broverman, Vogel, Broverman, Clarkson, & Rosenkrantz, 1972). Consequently, in mixed-gender discussions of gender neutral topics, men display a
49 greater amount of verbal and nonverbal power This suggests that in mixed-gender groups, women are given fewer opportunities to make contributions receive less support for their contributions, and are less influential than men (Carli, 1990). Stereotypes influence perceptions and pe rformance in school and are often cited as contributing to girlsÂ’ shortcomings in schools (Fennema, Peterson, Carpenter, & Lubinski, 1990). In an analysis of th e role of teacher beliefs on mathematics performance, Fennema et al. (1990) found th at teachers attributed boysÂ’ mathematical success to ability 58% of the time, and attribut ed girlsÂ’ mathematical success to ability only 33% of the time. Female successes were du e to effort 37% of the time, while malesÂ’ successes were due to effort only 12% of th e time (p. 178). In the same study, teachers attributed characteristics such as volunteer ing answers, enjoyment of mathematics, and independence to males. Silence of women Gilligan (1982) asserted that women who have been taught to accept the judgments of the men in their lives rather than make decisions for themselves, lack the opportunity to develop confiden ce-building experiences. As a result, many women fall silent. This pattern of silence is evidenced by the results of a study conducted by Norman (1997) that indicated that girls do not ask questions in class and therefore tend to have higher levels of mathematics a nxiety. Belenky et al. (1986) learned, through intensive interviews with women, that, in ge neral, they often felt alienated in academic settings and experienced formal education as either peripheral or irrelevant to their central interests and development. Differing socialization processes of male s and females are primarily responsible for the fact that many women speak Â“in a different voiceÂ” (Carli, 1990; Gilligan, 1982;
50 Tannen, 1990). In WomenÂ’s Ways of Knowing (Belenky et al., 1986), a five-category model served as a framework for the diffe rent ways women expressed themselves: silence, received knowledge subjective knowledge, procedural knowledge, and constructed knowledge. The 135 participan ts in this study, although representing different ages and backgrounds, shared comm on perspectives. These perspectives emerged from the telling of their lived experiences. If participating in classroom discussi on is a necessary part of successful performance, then males have an edge (Sa ndler, Silverberg & Hall, 1996; Tannen, 1990). Speaking in a classroom is more in harmony with boysÂ’ language experience than girlsÂ’ since it involves putting oneself forward in front of a large group of people, some of whom are strangers and at leas t one of whom (the teacher) will be judging the speakersÂ’ knowledge and intelligence by their verbal eloquence (Tannen, 1990). Debate-like formats as a learning tool tend to be mo re preferred by men than by most women. Adversativeness, the act of expressing oppositi on, is fundamental to the way most males approach any activity, however, this is contra ry to the way most females learn and like to interact (Kenschaft, 1991; Ong, 1981). Â“In the interactions between teacher and students, girls are silenced; they become spectators, wallpaper flower s, listeners of the boys who, given more time and attention, form the domin ant valued core and command the action of the classroomÂ” (Martel & Peterat, 1994, p. 159). Marderness (2000) cites several relate d studies conducted with women and specifically their mathematical experiences. One such study by Erch ick (1996) dealt with capable, female elementary teachers who genera lly did not assume the role of the silent
51 learner in other areas of their lives but reve aled feelings of being unable to speak and unable to hear when faced with mathematical situations: The women spoke of how they Â“just never got it,Â” no matter the particular content or instructor or level. One woman de scribed her feeling that mathematics was like building blocks, and she had missed th e very first one. These women asked no questions, and few asked the teacher for help. They felt voiceless, fell silent, and stayed silent regarding math ematics well into adulthood (p. 112). From the findings of Erchick and othe rs it would appear that many women believe that their voices are silent and have no weight when decisions regarding mathematics curriculum, scheduling, course content, and instructor attitudes are concerned. Mathematics Anxiety In the mid 1970s the problem of mathematics anxiety came to the fore as a feminist issue. Young women were not taki ng the high school mathem atics courses they needed for many college majors, and, as a cons equence, they were excluding themselves from promising and well-paying careers (Z aslavsky, 1994). Interv iews were conducted with such women who were now adults. They felt they could not cope with mathematics. Their experiences were often inspiring, but more frequently devastating. Zaslavsky suggests that pinpointing the prob lem is a first step in overcom ing negative feelings about mathematics. Research on memories, attit udes and beliefs (Martin, 1994) has helped identify individuals who repor t feelings of mathematics a nxiety. There are factors in societyÂ— inadequate schools, poor teachi ng, inappropriate mathematics programs,
52 stereotypes about who can and cannot do math ematicsÂ—that indicate the problem of mathematics anxiety is a complex construct. Mathematics anxiety does not appear to have a single cause. Research has shown that it is the result of different factors, such as an inability to handle frustration, excessive school absences, poor self-concept (Ha ll et al., 1999; Norwood, 1994; Wadlington, 1992), parental attitudes toward mathema tics (Dahmer, 2001; Eccles, 1994), teacher attitudes toward mathematics, and empha sis on learning mathematics through drill without understanding (Norwood, 1994). In addition, mathematics anxiety has been shown to be related to weak sk ills in mathematics. Lasher (1981) believed that reducing mathematics anxiety and building skills in ma thematics were processes that must be done simultaneously. Although there are different theories about the causes of and cures for mathematics anxiety, there is little disagreemen t in the literature a bout the fact that it exists. Indeed, noteworthy research on mathematics anxiety has shown the phenomenon to be real, widespread, and seriously to interfere with learning, severely impacting peopleÂ’s perceptions of themselves and of their career choices (Norman, 1997; Probert, 1983; Smith, 1994). The prevalence of mathematics anxiety in the college populati on is discussed by Betz (1978), who reports that 68 % of the students enrolled in college mathematics classes experience high levels of mathematics anxiet y. Women suffer from mathematics anxiety to a greater degree than men (Atkinson, 1988; Douglas, 2000; Fischer, 2000; Furner, 1996; Ho, Senturk & Lam, 2000; Hopko, 2000; Rabalais, 1998; Skiba, 1990; Spanias, 1996; Tobias, 1991). Furthermore, a relations hip between age and mathematics anxiety
53 exists such that the higher the studentÂ’s ag e, the higher the studentÂ’s mathematics anxiety (Springer, 1994). Greenwood (1984) argues that mathematic s anxiety is a problem whose solution lies almost entirely within th e domain of mathematics educat ion. Steele and Arth (1998) support this view and assert that the majo r source of mathematics anxiety lies in the impersonal teaching approach characterized by the explain-practice-memorize paradigm. When mathematics is taught without m eaning, students are forced to memorize unconnected bits of information. As a result mathematics doesnÂ’t make sense to them. Because it doesnÂ’t make sense, students rely on memorization. When students become convinced they will never understand mathematics and can learn only through memorization, Â“they may begin to fear math and may even become mathematics avoidersÂ” (Steele & Arth, 1998, p. 19). Tobias (1991) expresses her belief that mathematics anxiety is a political issue with millions of adults Â“blocked from professional and technical job opportunities because they fear or perform poorly in math ematicsÂ” (p. 91). Hers is a political conviction that mathematics is being used to channel students into j ob classifications and to lower their achievement and career goals. She feels that there is a cure but it will involve a change in both studentsÂ’ and teacher sÂ’ attitudes. Instead of teaching students the steps to take when they remember [italics added] how to solve problems, teach them how to cope when they forget [italics added]. Coping skills such as taking problems apart, relaxation techniques to combat panic, and positive self-talk are vital in the teaching of mathematics (Peskoff, 1997), yet feelings, attitudes, and anxieties about mathematics are not discussed in many mathematics classrooms (Furner, 1996).
54 Mathematics anxiety feeds on helplessness, the burden of being out of control and alone (Fischer, 1993; Tobias, 1991). It is de fined in terms of three components (Ohman, 1993): a subjective experience consisting of a feeling of foreboding; perceptions of bodily responses, such as sweating, palpitations shortness of breath; behaviors associated with escape and avoidance. Fu rthermore, inability to influence events that significantly affect oneÂ’s life may also give rise to f eelings of futility and despondency as well as anxiety (Bandura, 1982). The emotion is unpleasa nt, is directed toward the future, and is out of all proportion to the threat (Hembree, 1990). Students magnify the formidableness of the task and their personal weaknesses, think over their pa st failures, worry about the consequences of failing, imagine disconcerting scenarios of things to come, and otherwise think themselves into emoti onal distress and inadequate performance (Sarrason, 1975; Wine, 1982). Mathematics anxiety has been identified as a factor limiting educational and career choices of college students, particul arly women (Betz, 1978; Zettle & Raines, 2000). Why such emotional responses to mathem atics? It is spoken of as Â“the worst curricular villainÂ” (Smith, 1994, p. 135), associated with terrib le anxiety, bad grades and test scores, avoidance, power, mysteriousne ss, hostility, and pressure. For many, it is a devastating experience (Tobias & Wei ssbrod, 1980; Garcia, 1998; Ma, 1999). Some individuals consider learning mathematics to be useless in the context of their lives and work; yet there is a sense that knowing mathematics gives one a powerful feeling of being more knowledgeable than othe r people. There is a mystique surrounding mathematics that is hard to overcome (Seg eler, 1986). People who can solve problems
55 we canÂ’t or who can solve problems more qui ckly than we might, seem to be more intelligent and swifter of mind (Smith, 1994). Fascination and curiosity in studying mathematics anxiety can be demonstrated by the considerable number of studies conducted and the resulting articles and books that have been written on the subject (Hanson, 1988; Mill er, 1999; Norwood, 1994; Voit, 1983). Some of these materials are self-hel p guides that provide hints and creative suggestions for people who find th ey are in trouble with mathematics. Examples include, Math, A Four Letter Word! (Sembera & Hovis, 1996), Mastering Mathematics: How to Be a Great Math Student (Smith, 1998), Winning at Math, Your Guide to Learning Mathematics through Successful Study Skills (Nolting, 1997), Math Study Skills Workbook: Your Guide to Reducing Test Anxiety and Improving Study Strategies (Nolting, 2000), Succeed With Math: Every StudentÂ’s Guide to Conquering Math Anxiety (Tobias, 1987), Overcoming Math Anxiety (Tobias, 1995), Conquering Math Anxiety: A Self-help Workbook (Arem, 2003), and Math: Facing an American Phobia (Burns, 1998), just to name a few. These publica tions begin by validating the existence of mathematics anxiety and assure the reader th at he or she is among a vast number of others who feel the same way. Next are id eas that may help to alleviate nervousness, test-taking hints, study suggestions and the like. Most materials of this type focus solely on what the student can do to overc ome the anxiety. A variety of cures [italics added] are suggested ranging from mental imaging to hypnosis. Although past research has revealed many interesting facets of mathematics anxi ety, there is no current study underway, of which the researcher is aware, that focuse s solely on women of non-traditional age who are enrolled in postsecondary institutions Research is needed that uncovers and
56 describes feelings and stories coming from the learner. These stories may serve to unveil new strategies and stimulate understanding for educators to use in developing educational programs and processes which will speak to the problem from a pedagogical, curricular, and institutional perspective (Marderness, 2000). Indeed, mathematics anxiety is an affective issue that must be addressed not only by students but also by educators (Koelling, 1995). McLeod (1992) contends that affective issu es play a central role in mathematics learning and instruction. Mo reover, the National Council of Teachers of Mathematics expresses the importance of affective issues by including two goals related to affect in their 1989 Curriculum and Evaluation Standa rds for School Mathematics: (1) students should learn to value mathematics; and (2) students should become confident in their ability to do mathematics (NCTM, 1989). Likewise, the Principles and Standards for School Mathematics released in 2000, builds on a nd consolidates messages from previous NCTM documents by stating that knowing mathematics can be personally satisfying and empowering. Â“Â…those who unde rstand and can do mathematics will have significantly enhanced opportuni ties and options for shaping their futures.Â” In this document, standards for grades preK th rough 12 include (a) providing environments where thinking is encouraged, uniqueness is valu ed, and exploration is supported (p. 73); (b) providing instruction that is active and inte llectually stimulating (p. 142); (c) seeing to it that mistakes are seen not as dead ends but rather as potential avenues for learning (p. 145); (d) providing rewards for sustained effo rt and progress, not the number of problems completed (p. 145); (e) recognizing sensitivity of individuals to p eer-group perceptions (p. 210); and (f) encouraging confidence in each studentÂ’s own mental and physical
57 capacities (p. 287). The Principles emphasize that, when mathematics is made interesting and relevant, Â“many apparently uninterested student s can become quite engagedÂ” (p. 371). Tobias (1990) calls for more individual attention and support, more meaningful and appealing introductory mathematics course s, and greater guarantees of welcome and success by colleges and universities. The intimidating, competitive and selective characteristic of most mathematics courses is seemingly designed to winnow out all but the top tier [italics added] of students. Tobias calls this an outsider-insider problem where an individual can only join the insider group by demonstrating mathematical prowess, something perceived to be unatta inable by many non-traditional age students. Educators have the power to affect mathematics anxiety (Fiore, 1999; Forbes, 1988; Marable, 1994). A change must take place and it has to start with the mathematicians and the teachers of ma thematics (Smith, 1994; Stodolsky, 1988). Anxiety in any subject area hinges on a studentÂ’s belief in his or her ability to master the concepts. On the average, students like math ematics in the elementary grades but when the content and level of abstra ctness increases in junior hi gh and high school, they lose interest and perceive mathematics as a negative entity. As their dislike increases, so does their judgment of the subjectÂ’s difficulty. A concomitant belief that ability plays a critical role in learning mathematics is also a part of this picture (Stodolsky, 1988). Studies have shown, however, that aptitude, i. e., intelligence, is hindered by emotional factors. Intrusive worries about mathematics temporarily disrupt mental processes needed for doing arithmetic and drag down mathematics competence. Students find it difficult to hold new information in their mi nds while simultaneously manipulating it.
58 This capacity, known as working memory, is crucial for dealing with numbers (Bower, 2001). The theory that there is a relations hip among working memory, mathematics anxiety, and performance is supported by recen t studies conducted by Ashcraft and Kirk (2001) on college students. In this study, 66 participants were administered mathematics anxiety and working memory test s. It was found that there is a reduction in the available working-memory capacity of high-mathematicsanxiety individuals wh en their anxiety is aroused. This reduction should depress levels of performance in any math or math-related task that relies substantially on worki ng memory, including no t only addition with carrying, but presumably any counting-based task. It specific ally includes math in which procedural knowledge is essent ial, for example, situations requiring carrying, borrowing, or sequencing a nd keeping track in a multistep problemÂ….The anxiety reaction is one of a ttention to or even preoccupation with intrusive thoughts and worry. Because such thoughts and worry are attended, they therefore consume a portion of the limited resources of working memory. This reduces the available pool of res ources to be deployed for task-relevant processingÂ….Math anxiety, when arouse d, functions exactly like a dual-task procedure; that is, performance to the primary task is degraded because the secondary task, the anxiety reaction, compromises the capacity of working memory. The draining of resources imp lies continued, inappropriate (and selfdefeating) attention to th e cognitive components of th e math-anxiety reaction and to intrusive thoughts, worry, preoccupati on with performance evaluation, and the
59 likeÂ….The effect may be the result of an inability to inhibit attention to intrusive thoughts or distracting inform ation or, perhaps equivale ntly, a failure to focus attention and effort on the task at hand. In either case, the available processing capacity of working memory is compromised, with transitory but important effects on cognitive performance (Ashcraft & Kirk, 2001, p. 236). Students who have a low sense of efficacy to manage academic demands are especially vulnerable to achieve ment anxiety (Bandura, 1997). Academic activities are often infused w ith perturbing elements. Many parents impose on their children stringent academic demands that are difficult to fulfill. Accomplishments that fall short of those standards are devalued and lead to unpleasantness at home. A similar dram a is played out in schools, where academic deficiencies displease teachers and lower status and evaluation by oneÂ’s peers. To add further to the stress, students who adopt stri ngent standards for themselves, as indeed many do, must cont end with self-censu ring reactions to their own substandard performances as we ll as with the reactions of others.Â…The stakes become considerably higher at upper levels of schooling where performance grades determine entry to future pursuits that affect life coursesÂ….Academic deficiencies foreclos e many life paths and erect barriers to others that are difficult to surmount (Bandura, 1997, p. 235). The influence of efficacy beliefs on anxi ety over scholastic ac tivities has been examined most closely in relation to mathem atics, where a low sense of mathematical efficacy is accompanied by high mathema tics anxiety (Betz & Hackett, 1983).
60 Individuals experience high anti cipatory and performance distress on tasks in which they are inadequate, but as their self-efficacy incr eases, their fear declines (Bandura, 1982). A final perspective offered for mathema tics anxiety is the idea that much of mathematics anxiety is due to test anxi ety (Bandura, 1997; Furner, 1996). There is evidence that males and females differ in th eir responses to evaluative pressure and performance outcomes. When confronted with actual or potential failure in important achievement situations, boys have been f ound to engage in behavior indicative of increased achievement, while girls ofte n display the opposite tendency (Dweck & Gilliard, 1975). Hembree (1990) did a meta-analysis on th e nature, effects, and relief of mathematics anxiety, integrating the results of 151 studies. Under the general heading of anxiety, two subconstructs specifically re lating to academics were identified: mathematics [italics added] anxiety and test [italics added] anxi ety. Test anxiety, according to research conducted by Lieber t and Morris (1967), consists of two components: emotionality [italics added], behavioral in nature, and conscious worry [italics added] or concern, a cognitive el ement. This theory conceptualizes an Â“interference model of test anxiety, in whic h test anxiety disturbs the recall of prior learning, thereby degrading performanceÂ” (Hembree, 1990, p. 34). An alternative deficits [italics added] model was proposed by Tobias (1985) that points to poor study habits and test-taking skills as the culp rits causing low test scores. Within this model, test anxiet y does not cause poor performan ce; the reverse is true. An awareness of doing poorly in the pa st causes test anxiety.
61 Hembree came to the following conclusions after doing the meta-analysis: (a) mathematics anxiety depresses performance; (b ) mathematics anxiety is related directly to debilitating test anxiety a nd inversely to the anxiety driv e that facilitates performance during testing; (c) females displayed higher le vels of mathematics anxiety than males, especially in college (the highest levels occurred for students preparing to teach in elementary school); (d) higher achievement consistently accompanies reduction in mathematics anxiety; and (e) treatment can restore the performance of formerly highanxious students to the performance level associated with low mathematics anxiety (Hembree, 1990). Teacher Influence on StudentsÂ’ Attitudes One cannot go very far in reviewing th e literature on math ematics education before being confronted with issues regard ing teacher behaviors that produce anxiety responses in students. Teachers are consider ed to be a major force in contributing to student achievement, more important than e ither the method or curriculum (Greenwood, 1984; Swetman, Munday, & Windham, 1993). Certai n personality traits of teachers are related to mathematics anxiety in thei r students (Norwood, 1989; Spanias, 1996). Jackson and Leffingwell (1999) investigated t ypes of teacher behaviors that created or exacerbated anxiety and found that anxiety-prod ucing problems occurred in three primary clusters of grade levels: elementary level, especially grades 3 a nd 4; high school level, especially grades 9-11, and college level, es pecially freshman year. Teaching behaviors that caused students to have anxiety in mathem atics classes were classified as either overt or covert. Although the beha viors at all three levels were fascinating and thoughtprovoking, concentration will be directed to responses at the college level.
62 In the Jackson and Leffingwell (1999) study, 27% of 146 students indicated that their freshman year was the starting point of mathematics-related stress. Students could not understand some teachers because of th eir poor pronunciation. In some cases, English was not the teacherÂ’s first language. The speed at which lect ures were delivered was too rapid for some students. Students were told to leave class if they did not understand the material. Teachers belittled students for not having the prerequisite knowledge. When seeking assistance, students were often told that the teacher did not have enough time to help them. One teacher said, Â“If you donÂ’t like math, get out.Â” Students were told to go to the mathematics lab if they were that Â“dumbÂ” (Jackson & Leffingwell, 1999, p. 3). Teachers gave poor explanations or rushed through explanations. Relying on assumed prerequi site knowledge, teacher s told students that they should know the material. If they did not, then the teacher did not have time to waste on them. Teachers did not explain materi al sequentially or at an instructional pace that was understandable. One college teacher wrote equations with one hand and erased them with the other hand as he proceeded, w ithout concern for studentsÂ’ needs. Students saw long and complex tests as punishment and as a vindictive form of retaliation against students who asked questions. Some teacher s were offended at having to teach entrylevel mathematics classes and vented their frus trations on students. Teachers told female students that girls should not take mathema tics classes. Teachers used a condescending and demeaning manner to tell female student s that because they did not understand the lesson in class, the teacher would explai n it after the lecture. Teachers showed insensitivity to students who were older th an the traditional 18-to-22-year-old bracket when these students expressed anxiety about returning to school after many years.
63 Overt, or observable, behaviors can be e ither verbal or nonverbal; for example, a teacher might scowl, use belittling humor, inte rrupt or allow peers to interrupt, attribute success to luck, beauty, or relationships ra ther than talent, or make a derogatory comment. Covert behaviors, although veiled, can have the same detrimental effects as the overt behaviors. These include failure to respond to a studentÂ’s question, relying heavily on worksheets without explaining co ntent, asking easier questions, making seemingly helpful [italics added] comments, doubting accomplishments, expecting less, giving less feedback, less criticism, less he lp, less praise, avoi ding eye contact with students or sighing in a demeaning manne r (Jackson & Leffingwell, 1999). Because mathematics requires sequential-thinking skills, any stress in the mathematics classroom will have even more adverse effects because of the nature of the subject (Zaslavsky, 1994). The results of Jackson and LeffingwellÂ’s study have significant implications for mathematics instructors. Students tend to in ternalize their teachersÂ’ interest in, and enthusiasm for, teaching mathematics. Conve rsely, if students think that the teacher is not happy teaching and does not enjoy being with them in the classroom, they will be less motivated to learn. The survey of student responses showed that the negative memories were so profound that mathematics anxiety coul d persist for 20 or more years (Ernest, 1976; Hanson & Gentry, 2001; Jackson, 1999). Teachers can help enhance studentsÂ’ valuing of mathematics in several ways, in cluding explicitly relating the value of mathematics to studentsÂ’ everyday lives, ma king mathematics personally meaningful, and counseling students about the importance of mathematics for various careers (Eccles,
64 1994). Female mathematicians frequently c ite inspirational teach ers as being a major factor in their choice of career and sign ificant role models (Leroux & Ho, 1994). In a qualitative study of female mathema tics majors at a very competitive college, Gavin (1996) found that almost ha lf attributed their decision to major in mathematics to the influence of a high school teacher. They needed someone outside of their family to tell them that they had mathematical talent and should continue to pursue it. Students mentioned female mathematics professors w ho served as mentors and role models in college. In addition to encouraging students, their styles of teaching also provided a nurturing environment where students fe lt free to ask questions (Gavin, 1996). Because teacher attitudes toward mathema tics, positive or negative, can affect students in a similar manner, it is important to note the extent of literature on the anxiety of mathematics teachers at the elementary level (Allen, 2001; Richardson, 1980; Sachs, 1994; Swetman et al., 1993; Trice & O gden, 1987; Williams, 1988). Allen (2001) concluded that although mathematics anxiety is prevalent among citizens of our country, nowhere is it more prominent than in Am ericaÂ’s elementary-level classrooms. By examining 43 mathematics autobiographies of female elementary school pre-service teachers, AllenÂ’s results identified experiences with mathematical content as a significant predictor of mathematics anxiety for both ma thematics specialization participants and those who selected a specialization other than mathematics. In addition, the study showed that mathematics anxiety and particip antsÂ’ experiences with mathematics content were significant predictors of mathema tics avoidance behaviors. Sachs (1994) investigated the degree of mathematics a nxiety in a group of elementary education students (pre-service teachers) and found that two-thirds of the group had moderate to
65 high anxiety, blamed their instructors for it, were low in confidence, and rarely took part in class discussions. It appears that correction needs to be made in the training and preparation of elementary education teachers; however, ch anging the long-standing attitude of someone who has maintained and fostered a negative att itude for years is a difficult task. As far back as the 1960s, research directed toward improving attitudes toward mathematics in elementary school teachers was conducted (Dutton, 1962) with discouraging results, suggesting that attitudes deve loped early and maintained over many years can be difficult to change. Research conducted in the 1980s (Ke lly & Tomhave, 1985; Larson, 1983; Martinez, 1987; Trice & O gden, 1987; Widmer & Chavez, 1982) reveals mathematics anxiety was Â“rampant among elementary t eachersÂ” (Swetman et al., 1993, p. 422). Current research on mathematics anxiety indica tes that females suffer more than males, and because many elementary teachers are fema le, it is not surprising that a high degree of mathematics anxiety is found among elemen tary teachers (Swetman et al., 1993). Even more alarming is the fact that mathematics anxiety usually results in mathematics avoidance. Â“The tragedy of this situation is that prospective teachers who avoid taking mathematics courses are not as we ll-prepared to teach it as they could beÂ” (Swetman et al., 1993, p. 422). Trice and Ogden (1987) found that teachers who were most intimidated by mathematics planned less time for mathematics instruction and were observed teaching non-mathematics content during the time allocated for mathematics. The researchers also discovered that this av oidance of mathematic s conveys a negative attitude toward mathematics to the students. The scenario becomes a vicious circle:
66 mathematics-anxious teachers communicate a negative attitude toward mathematics to the students, who in turn take fewer mathema tics courses and then proceed to become illprepared mathematics-anxious teachers who create more students who are mathematicsanxious (Swetman et al, 1993). Most educators recognize the importance of affective variables in the learning process; there is an emotional side to learning as well as cognitive (Hodges, 1983). Teachers who provide supportive, encouraging e nvironments attempt to increase student confidence levels and improve achievement and interest in the subject matter. Providing the right [italics added] environment, however can be complicated, because students respond to stimuli in varying ways. Competition may provide motivation for some students, whereas for others, competition may appear personally threatening. Individual attention, while considered a plus to many st udents, may cause others to be embarrassed and feel Â“on the spotÂ” (Marderness, 2000). Quality in Mathematics Education Mayhew et al. (1990) share a perplexing dilemma that c oncerns the importance of intellectual dexterity with numb ers and other abstract symbols. They correctly recognize that in a society dominated by science and t echnology and likely to be highly dependent in the future on computers, the need for ma thematical ability is obvious. They also correctly identify a pervading fear and dread of mathematics, not only in high school dropouts, but also in th e general population: It has been urged that al l students should have four years of mathematics in secondary school followed by more mathematics in college. Unfortunately, such
67 a recommendation runs counter to the rather deep-seated fears and feelings of antagonism regarding mathematics that exist in the larger society (p. 86). Quality in mathematics education may be taking a new turn, as evidenced by articles such as the one by Berkman (1995), suggesting that the mathematics curriculum be centered on cooperation rather than comp etition. Competition implies that learning mathematics is being able to come to a solu tion quickly. This error in thinking denies learners a deeper understanding of the subject. Berkman says too often instructors are driven by coverage of material rather than comprehension/mastery. If mathematics is boring, monotonous, routine, and uncreative, by all means, get it over with as fast as possible. But, if mathematics is interest ing, useful, entertaining, and pertinent to our lives, then teach it at a pace that allows students to enjoy the process and achieve confidence and mastery. Most curriculums p ack too much information into too little time, at a significant cost to the learner (Brooks, 1999). Stud ents must be given problems to explore rather than tasks to complete, which produces understandi ng of content rather than mere memorization (Schroeder, 1998). TeachersÂ’ beliefs shape the way in which they teach mathematics (Carter & Norwood, 1997; Reyes, 1980). Teachers communi cate expectations in their interactions with students during classroom instruction, th rough their comments on studentsÂ’ papers, when assigning students to instructional groups, through th e presence or absence of consistent support, and in their contacts with significant adults in a studentÂ’s life. These actions also influence studentsÂ’ beliefs about their own abilities to succeed in mathematics (NCTM, 2000). Students learn mathematics through th e experiences that
68 teachers provide. The teaching they encount er in school shapes their understanding of mathematics, their disposition toward mathematic s, and their confidence in mathematics. NCTMÂ’s vision for mathematics education, strongly influenced by constructivist views (Oxford, 1997) and described in Principles and Standards for School Mathematics (2000), is quite ambitious and brings with it a commitment to both equity and excellence. NCTM shares with students, sc hool leaders, and parents the re sponsibility to ensure that all students receive a high-quality mathema tics education, in environments that are equitable, challenging, supportiv e, and technologically equi pped for the twenty-first century (NCTM, 2000). The Principles and Standards (PSSM) describes the mathematical content and processes that stud ents should learn. The six principles for school mathematics address the following th emes: (a) equity; (b) curriculum; (c) teaching; (d) learning; (e) assessment; and (f) technology. PSSMÂ’s equity principle states that all st udents, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to learn mathematics, and receive the support necessary to achieve success. This principle challenges a pervasive societal belief in North America that only some students are capable of learning mathematics. This be lief leads to low expectations for many students, including females (NCTM, 2000). To accommodate students effectively and sensitively, teachers need to understand and confront their own beliefs and biases. PSSMÂ’s assessment principle focuses on more than just using tests to certify studentsÂ’ attainment. Assessment should be an integral part of instruction that informs and guides teachers as they make instructi onal decisions. Assessment should not merely be done to [italics added] students; ra ther, it should also be done for [italics added] and
69 with [italics added] students to guide and enhance their learning (NCTM, 2000). Assessment and instruction must be integrated so that assessment b ecomes a routine part of the ongoing classroom activity rather than an interruption. Such integration provides information teachers need to make appropria te instructional decisions (NCTM, 2000). Teachers should be gathering information continually about studentsÂ’ progress by asking questions during the course of a lesson; in fact, a type of evaluation might include dialogues between teacher and student in orde r to assess thinking. Research indicates traditional testing may not always provide a true picture of stude nt learning. Teachers can use a variety of testing techniques: asking oral questions, observing student demonstrations, having students keep a mathema tics portfolio or jour nal. Assessment is more than a test that is given at the end of a chapter (Ste ele & Arth, 1998). As the NCTM Principles and Standards are implemented across the United States, the way in which they are accepted and put into practice will depend largely on the beliefs that teachers have about mathem atics and the teaching of mathematics. For example, the view of learning mathematics from a constructivist approach, that is, having children c onstruct their own knowledge about mathematics may precipitate a large gap between teachers Â’ beliefs and their cognitive understanding of the recommendations of the Standards (Carter & Norwood, 1997). Although an instructor may be in concu rrence with the goals of a new way of teaching, he or she may not be willing to cha nge the strategies that have been used for years, and may have no problem ignoring met hodologies listed in the reform material. Routine operations become highly ritualized; a lesson plan is in place that is repetitive, predictable, and hence highly re sistant to change. Faculty members value their autonomy
70 highly and guard vigilantly against attempts to limit it. Sometimes this attitude takes the form of blind resistance to almost any ch ange (Astin, 1985). At the root of much resistance are issues of power, control, and vulnerability (Lucas, 2000). Furthermore, one could argue that the hier archical organization of Am erican higher education is resistant to change. Â“Intellectuals who have achieved power and influence in the larger society have a stake in accepting and perpet uating the status quo, no matter how much in need of reform or renewal it may beÂ” (Asti n, 1985, p. 192). Most colleges still maintain considerable control over their most vital f unctions, two of which are course content and teaching techniques (Astin, 1985). Cohen and Ball (1990) agree that st rategies, philosophies and content recommendations in the NCTM Principles and Standards are not simply excised and inserted into the classroom experience. So metimes the changes made may be superficial or incomplete. For example, some of th e pedagogical ideas ma y be enacted without sufficient attention to studentsÂ’ understandi ng of mathematics content (NCTM, 2000). Teachers continue to behave and teach in accordance with their existing beliefs built by their own personal experiences unless ther e is a powerful reason to change. The recognition for the need to change is a mandatory first step (Carter & Norwood, 1997). Constructivist Approaches to Learning Constructivism is an educational theory which argues that students are motivated to learn only if they are active learners, constructing their own knowledge through their own discoveries. All learning is digested by the learner and unde rstood in relation to what the learner already knows (Oxfor d, 1997; Ravitch, 2000). What students know
71 [italics added] consists of internally cons tructed understandings of how their worlds function. New information either transforms their old beliefs or fails to do so. Belenky et al. (1986) emphasize the potency of connections in learning. Linking topics to studentsÂ’ lives is a prerequisite for the most effici ent learning of abstractions. Mathematics becomes exciting when it is not divorced from other subject areas and when it is linked to the real world. This type of curriculum honors the learning style of females (Karp, Brown, Allen, & Allen, 1998). Constructivists would agree that the dynamic nature of learning makes it difficult to capture on assessment instruments that limit the boundaries of knowledge and expression (Brooks, 1999). Emphasis on perfor mance usually results in little recall of concepts over time, while emphasis on lear ning generates long-te rm understanding. Some formal knowledge is also involved. For instance, in mathematics, students must be engaged in active problem solving which requires that they master the basic skills of adding, subtracting, multiplying, and divi ding. In other subject areas, such as the social sciences, helping students search fo r personal understanding and valuing different, often contrasting, points of view is reasona ble (Brooks, 1999). However in mathematics, in many cases, there is only one correct an swer, with multiple avenues leading to its discovery. Acknowledging that there exists only one correct answer is what makes following constructivist approaches in mathema tics classrooms a challenge. It is up to the teacher to ensure that the truths [italics added] arrived at in the classroom are consistent with disciplinary knowledge (Prawa t, 1992). Boeree (2007) believes that rote learning, i.e. the memorization of facts, will always be with us. However, it is not
72 entirely devoid of meaning. Â“The trick is to encourage students by making the necessity of the rote learning meaningful.Â” There is a difference between talking at a nd talking with students, both in Â“giving students opportunities to invent mathematics and in encouraging positive beliefs about learning mathematics. Students should see their job not as finishing assigned tasks but as making sense of, and communicating about, mathematicsÂ” (Clements, 1997, p. 200). The classroom environment should be perceived as one in which students are free to explore ideas, ask questions, and make mistakes (Cobb, 1988). Yager (1991) asserts that constructivism requires a dramatic change in beliefs about education. Putting the studentsÂ’ effo rts to understand ahead of the traditional telling-listening relationship is paramount. Most agree that this involves a decisive change in the focus of teaching and the role of the teacher (Prawat, 1992). The traditional telling-listening relationship between teacher a nd student is replaced by one that is more complex and interactive. Â“Teachers who take this path must work harder, concentrate more, and embrace larger pedagogical respons ibilities than if they only assigned text chapters and seatworkÂ” (Cohen, 1988, p. 255). As previously stated, in order for teachers to accept such significant change, their beliefs about teaching and learning must change dramatically. First, there is the tendency to think of both learning and content as fixed entities, rather than dynamic and continually undergoing change and revisi on (Prawat, 1992). S econd, there is the tendency to equate activity with learning instead of seeing that student engagement is not the best measure of educational value. Third, there is the popular view of curriculum as a fixed agenda with certain material to cover at all costs. Constr uctivists favor a more
73 interactive and dynamic approach to curriculum, believing that it should be viewed more as a matrix of ideas to be explored over a period of time than as a road map (Prawat, 1992). A constructivist approach enabling students to understand may be accomplished by (a) seeking out and using student ques tions to guide lessons, (b) accepting and encouraging student initiation of ideas, (c) promoting student self -regulation and action, (d) using studentsÂ’ experiences and interests to drive lessons, (e) encouraging uses of alternative sources of info rmation, (f) using open-ended questions and encouraging student elaboration when possibl e, (g) encouraging students to suggest causes for events and situations and to pred ict consequences, (h) seeki ng out student ideas before presenting ideas from the text, (i) allowing tim e for reflection and analysis, (j) facilitating reformulation of ideas in light of new e xperiences and evidence, and (k) encouraging social interaction (Cleme nts, 1997; Yager, 1991). Some would caution that cons tructivism is not an instru ctional approach; it is a theory about how learners come to know (Airasian & Walsh, 1997; Clements, 1997). Since knowledge is constructed by the learner, constructivism makes the assumption that all students can and will learn. The vision of the constructivis t student is one of activity, involvement, creativity, and the building of personal knowledge and understanding (Airasian & Walsh, 1997). Â“In th e constructivist approach, we look not for what students can repeat, but for what they can generate demonstrate, and exhibitÂ” (Brooks, 1999, p. 16).
74 Mathematics Education Initiatives Failure in mathematics can be the limiting factor in an undergraduateÂ’s choice of major (Berenson, Carter, & Norwood, 1992). Because many people perceive mathematics as a barrier that needs to be hurdled in order to gain entrance to their selected careers, programs have been deve loped and some are just beginning to help show the relevance, utility, and criticality of mathematics. These truths must be demonstrated to students, parents, adviso rs, faculty in other disciplines besides mathematics, and to administrators. Sensitiv ity to gender and age are imperative (Hovis, Kimball, & Peterson, 2003). The following are current mathematics education initiatives, some still in experimental stages, that are tackling long-standing perceptions about mathematics as a barrier. National Council of Teachers of Mathematics. With the release of Curriculum and Evaluation Standards for School Mathematics in 1989, the National Council of Teachers of Mathematics (NCT M) moved to the forefront of efforts to improve mathematics education in the Un ited States and Canada. The Standards were the result of three years of plannin g, writing, and consensus-buildi ng among the membership of NCTM and the broader mathematics, scienc e, engineering, and education communities, the business community, parents, and school administrators (NCTM, 2000). In 1991, NCTM published the Professional Standards for Te aching Mathematics (PSTM), which was produced as a companion to the Curriculum and Evaluation Standards and whose goal was to provide guidance to those involved in cha nging mathematics teaching. The Assessment Standards for School Mathematics, released in 1995, es tablished objectives against which assessment pr actices could be measured These documents are based on
75 the assumption that all students are capabl e of learning mathematics. In 2000, NCTM released Principles and Standards fo r School Mathematics, which is a compilation of the lessons learned and experiences gained since th e first document was written. It delineates six Principles that should guide school ma thematics programs and ten Standards that propose content and process goals. Th e newest innovation from NCTM are the EStandards, an electronic edition of Principles and Standards with interactive applets, short videos, and links to ot her resources. NCTMÂ’s effort s in addressing reform in mathematics education have been making a di fference, influencing state standards and curriculum frameworks, teacher educ ation and classroom practice. Principles and Standards has provided a catalyst for the con tinued improvement of mathematics education. American Mathematical Association of Two-Year CollegesÂ—Crossroads in Mathematics. In 1995, seeking to bridge a gap in mathematics programs between high school mathematics and college calculus, the American Mathematical Association of Two-Year Colleges devel oped standards known as Crossroads in Mathematics Educators became aware of the fact that e ach year greater numbers of students were entering college mathematics starting below the level of calculus, yet very few were persisting to higher levels. This situation is still occurring an d the failure of many of these students to persist in mathematics not only prevents them from pursuing their chosen careers, but it also has a nega tive impact on our nationÂ’s eco nomy as fewer members of the workforce are prepared for jobs in tech nical fields. A second standards document, Beyond Crossroads, was developed in 2006 to renew and extend the goals of the first document which were to address the special circumstances of students who are enrolled
76 in introductory college mathematics. The part icular type of student s addressed in this document may be described as non-traditional, meaning they Â“are older, work a full-time or part-time job while attending college, are re turning to college after an interruption in their education of several y ears, need formal developmen tal work in a variety of disciplines and in study skillsÂ…Â” American Mathematical Association of Two-Year Colleges [AMATYC], 2007). The major goals of Beyond Crossroads include improving mathematics education in two-year colleges and encouraging more stud ents to study more mathematics. One of the principles of Beyond Crossroads is that there would be in creased participation by all students, specifically referring to women and others who have traditionally been underrepresented in the mathematics discipline. Standards are presented in the document for intellectual development, content, and pedagogy. Mathematics education has traditionally focused on content knowledge, knowing certain pieces of subject matter. This document takes the position that knowi ng mathematics means being able to do mathematics and that problem solving is the heart of doing mathematics. Students should understand mathematics as opposed to thoughtlessl y grinding out answers. The standards for pedagogy in the Beyond Crossroads are compatible with the constructivist point of view, which is based on the premise that knowledge cannot be Â“givenÂ” to students. Instead, it is something that they must constr uct for themselves. Recommended is the use of instructional strategies that provide for student activity and student-constructed knowledge. Furthermore, the standards ar e in agreement with the instructional recommendations contained in Principles and Standards for School Mathematics Faculty who teach introductory college ma thematics must increase the mathematical
77 power of their students by actively involving them in meaningful ma thematics problems that build upon their experiences, focus on broad mathematical themes, and build connections within branches of mathem atics and between mathematics and other disciplines. Guidelines for pedagogy include: active involvement of st udents rather than passive listening, use of technology rather than paper-and-pencil drill, multistep problems rather than one-step single-answer problems, mathematical reasoning rather than memorization of facts and procedures, con ceptual understanding rather than rote manipulation, realistic problems encountered by adults rather than contrived exercises, open-ended problems rather than problems w ith only one possible answer, and a variety of teaching strategies rather than lecturi ng. These standards provide a new vision for introductory college mathematics focused on the needs of non-traditional-age students who have come into their college experience without the necessary mathematical skill to begin at the college level. Programs reflecting the original Crossroads standards continue to show remarkable results. One such program is the Â“WomenwinÂ” program at Miami-Dade Community College, in Miami, Florida. Womenwin at Mathem atics Through Writing The Womenwin at Mathematics Through Writing project is designed to help women build on their verbal skills to improve their knowledge of mathematics. Research indicates that writing about mathematics may lead to a better understanding of the subject. Preliminary results show a trend that students in th e writing groups are more li kely to pass from collegepreparatory mathematics to college-level ma thematics in one semester; whereas their non-writing counterparts are more likely to repeat college-pre paratory course work. The
78 investigators feel that the Â“emphasis has move d from Â‘calculatingÂ’ to Â‘understandingÂ’ and great things are happening to the students both academically and emotionallyÂ” (Austin & Ballester, 1999, p. 20). Developmental Algebra: Rest ructuring to Effect Change. Another innovative Crossroads program took place at William Rainey Ha rper College, in Palatine, Illinois. Objectives of the program included the deve lopment of curriculum materials for teachers who want to make changes in the way th ey teach algebra, particularly at the developmental level. A specific goal was to determine the effect a studentÂ’s confidence has on learning. Key to this project was the sh ift in instructor towa rd facilita tor and the shift in student from passive to active involvement in the classroom. Findings indicate that, although not all faculty a nd students responded positively to the new structure, those who did expressed empowerment. A field study at several othe r community colleges around the United States indicated Â“significant shifts in studen tsÂ’ self evaluation of their abilities to do mathematics following the use of the project materialsÂ” (McGowen & DeMarois, 1999, p. 26). Mathematics for Elementary Teachers Providing meaningful mathematics for preservice teachers majoring in elementary e ducation was the princi pal goal of a twocourse sequence develope d jointly by faculty at the University of Michigan Flint Campus and Mott Community College. Students worked in groups in an active laboratory setting with discovery-based, handson activities. A locally deve loped instrument called the Mathematics Concept Scale (MCS) was admi nistered to students completing the sequence and also to a group who were still ta king the old curriculum. Findings indicated that attitudes toward mathematics of stude nts exposed to the new courses became more
79 like those of experienced mathematicians, whil e the attitudes of th e students still taking the old curriculum did not change (Sharp & Sutton, 1999). EQUALS. EQUALS programs, based and coor dinated at the Lawrence Hall of Science, University of California, provide workshops and curriculum materials in mathematics for teachers, parents, and co mmunity members. The program includes creative innovations such as Family Math I and II, Math Matters BEAM program, and algebra institutes for mathematics instructor s. An important focus of EQUALS is to provide ideas on multiple assessment: portfolios, teacher logs, performance tasks, openended questions, and student journals. Their concern is that standardized tests are not always understood and are biased against minorit ies, including females. Their program is ever expanding and is presented on an interactive website. Project SEED Project SEED is a growing progr am whose purpose is to increase the number of minority and educationally di sadvantaged youth majoring in mathematics and related fields. The goals of the program are to increase academic self-confidence, develop problem-solving and critical thinki ng skills, and raise mathematics achievement levels. The program is four-pronged: cla ssroom instruction, staff development, parent workshops, and curriculum development. Th e parent workshops, covering topics ranging from supporting a childÂ’s mathematics educatio n at home to algebra curriculum designed to help parents increase thei r own mathematical knowledge, ha ve been enthusiastically appreciated by parents. The Project SEED specialist works to make the class the arbiters of knowledge so that they feel a sense of ownership in the mate rial. This is in oppositi on to the traditional focus of teacher as expert. The use of sile nt feedback hand signals allows students to
80 constantly interact and provide feedback to one another in a non-disruptive, polite and yet very clear fashion. The constant feedback also allows all students to participate without risk. This risk-free environment then makes it easier for students to take academic risks without fear of embarrassment or humiliation. This provides a setting for social growth along with the development of critical thinki ng abilities. Various learning and teaching techniques that Project SEED has used to he lp students develop mathematics confidence rather than mathematics avoidance may support the findings of the proposed study. What all of these initiatives have in comm on is their desired outcome: to take the process of Â“doingÂ” mathematics out of the domain of only a small percentage of the college-prepared, college-ready elite, and place it in the realm of something possible for everyone to achieve, even the at-risk, ma th-anxious, educationally disadvantaged individual. The programs star t with the assumption that studying mathematics can be an enjoyable, rewarding activity, one that is both meaningful and relevant. They demonstrate how mathematics can be integrated with other discipline s. Seeing the need for making mathematics understandable, comfor table, and even fun is a beginning point from which may emerge new ways of teaching mathematics, improving the mathematics curriculum, and fostering an appreciation of ma stering mathematical concepts rather than just memorizing how to manipulate numbers. Phenomenological Research in Mathematics The phenomenological method is useful in cases where a par ticular type of experience is the subject of scrutiny (Mar derness, 2000). Success with this method requires articulate, expressive individual s who have experienced the phenomenon under investigation and who are willi ng to share, in detail, their stories. Unlike other methods
81 that are controlled by the researcher the phenomenological method depends on participants to provide the material for data analysis. In cases wh ere participants cannot contribute detailed descripti on, additional contributors need to be sought. Researchers must be willing to continue the interview process until no new themes surface. In the last two decades, phenomenological research ha s contributed to the discovery of valuable information useful to educators. Grood (1985) studied four women over 30 years old and how they each responded to the demands of being college freshmen. The investigation focused on their academically-related behaviors, needs, concer ns, and learning styles, as well as their reactions to and interactions with college in structors and younger cla ssmates. Qualitative research methods were employed including: participant observation in the classroom, structured essays, in-depth interviews, and visits to pa rticipantsÂ’ homes. Results indicated similarities among participants with respect to motives for college enrollment, orientation towards learning, obstacles to sc hooling, adaptation strate gies, interactions with students and teachers, physical concerns and attributions for academic success and failure. Grood offered implications to educ ators related to modifi cations in studentsÂ’ behavior: creating and maintaining open lin es of communication with instructors and classmates, and learning about and availing themselves of th e various services offered on campus. GroodÂ’s findings were further augmen ted by Schatzkamer (1986). Through a qualitative analysis of in-depth interv iews, Schatzkamer attempted to understand, describe, and explain, from a feminist pers pective, the educational experience of returning women students in community college s. She chose a qualitative design because
82 a quantitative approach Â“could not tell [her] what [s he] wanted to know: the realities of the experience of another human being and how that person thinks and feels about her experienceÂ” (p. 30). Three 90-minute in-depth phenomenological interviews were held with each of 18 returning women students at nine community colleges in four states. Participants were 25 to 70 years of age and ha d returned to traditional schooling after an absence of four to fifty years. Findings we re reported through pr ofiles, in the womenÂ’s own words (Schatzkamer, 1986). Although womenÂ’s studies are pertinent to understanding self -concept and gender issues in education, there are ot her areas of study that relate to both sexes. For example, Hartman-Abramson (1990) did a study of the phenomenology of mathematics anxiety and found that it is not uncommon. While sh e found that earlier studies had been done using quantitative analysis, she utilized a qualitative approach and focused upon adultsÂ’ fear of numbers through the eyes of those who experienced this phobia. Ten mathematics-anxious persons were selected for the study and repres ented a cross-section of any major American metropolitan area: five adults from the United States and five from other countries (Brazil, Jordan, Lebanon, and the Soviet Union). The information related by these interviewees was transcribed and distilled into descriptive data. The material was then analyzed according to th e standard qualitative research design of phenomenological reduction. This technique enabled the research ers to detect nuances of the experiences of those with numerical phobia. Additionally, this work points to the sources of mathematics anxiety, reveal s aspects of its nature and provides recommendations for dealing with a fear of figures (Hartman-Abramson, 1990).
83 Building on Hartman-AbramsonÂ’s findings another qualitativ e-design study on mathematics anxiety was conducted by Bisse (1994) utilizing focus group interviews with 14 students, and individual interviews w ith 10 students. The Math Test Anxiety Survey (MTAS) was used to identify mathem atics-anxious students at the beginning of the research period and also at the end of the rese arch period to find out if any exposure to developmental mathematics over the length of a summer session had any influence on studentsÂ’ mathematics test anxiety. Th e focus group interview and the individual interviews helped to establish categori es and sub-categories, which permitted the placement of studentsÂ’ responses into tabl es and matrices. Conclusions reflected studentsÂ’ perceptions in eight categories that were responsib le for mathematics anxiety. These conclusions were used to create a pr ofile of a mathematic s-anxious student, in terms of worries and concerns. Recommenda tions addressed necessary improvements of the curriculum, assessment, placement, teaching methodologies, and classroom environment (Bisse, 1994). Research by Parker (1997) went be yond describing mathematics anxiety and focused on the ways adults overcome it. Th e purpose of the study was to understand the nature of the transition that adults make as they move from being mathematics-anxious to being more comfortable with mathematics. Parker chose 12 formerly mathematicsanxious adults and conducted a series of semi -structured in-depth interviews and personal documents. Three areas of inquiry were ex amined: (a) the participantÂ’s mathematical history, (b) how the particip ant overcame mathematics anxi ety, and (c) the impact of overcoming mathematics anxiety. Analysis of the transcripts using the constant comparative method resulted in inductivel y derived categories descriptive of the
84 experiences of overcoming mathematics anxiety during adulthood and how those experiences affected the participantsÂ’ liv es. A six-stage process of overcoming mathematics anxiety was uncovered. First, ad ults perceived a need to become more comfortable with mathematics. Recognition of the need was followed by making a commitment to address the problem. Third, th e mathematics-anxious adults took specific actions to become more comfortable with math ematics. Learning how to get the most out of mathematics, they refined their study techniques, used le arning tools, attended tutoring sessions, and applied relaxation techniques. These time-consuming actions required them to make sacrifices. Fourth, the adults rec ognized that they had reached a turning point and were no longer mathematics anxious. The adultsÂ’ mathematical perspectives changed. Finally, the adults became part of the support syst em for others seeking help with mathematics, just as others had help ed them overcome their mathematics anxiety. In support of further research on math ematics anxiety, a qu alitative study was conducted by Zopp (1999) at McHenry County College, a small community college in northern Illinois, with eight students selected to participate. Criteria for selection included being non-traditional-ag ed (25 and older) and havi ng a high score on the Math Anxiety Rating Scale (MARS). Case study methodology was used whereby three interviews were conducted with each of the pa rticipants. Data coll ected were categorized according to three research questions. Qu estion 1 investigated how adult students describe the causes and nature of math ematics anxiety. Question 2 addressed participantsÂ’ perceptions of useful strategies in over coming mathematics anxiety. Question 3 asked how adult students describe the impact of a program designed to reduce mathematics anxiety. Their responses incl uded feelings about mathematics anxiety,
85 confidence about mathematics achievement, w illingness to ask for help, changing career goals, and connections of mathematics to work and life. The findings in the study are of importance to educators, curriculum specialis ts, teacher trainers, and all individuals having a role in the educati on of adult students, especially at the community college level. Although mathematics anxiety appears to be a common problem for adults, it is not the only problem adult stud ents face when returning to higher education. Ham (1998) investigated the lives and real ities of adult basic education learners who had experienced a gap in their schooling. The selected partic ipants were a diverse group, but through their stories, patterns emerged concerning their chil dhood, their teen years and their adulthood. The study revealed that some of the more difficu lt adults to reach are returning to school. Caring and competent adult educators must he lp connect the two worlds. The challenge presented to educators is to make sure that the system does not fail the adult learners in their quest of becoming the idealÂ—a lifelong learner. Lang (1999) used a combination of qualita tive and quantitative research designs to investigate the various facets of developm ental education in community colleges. The study compared the academic performance of students who successfully completed developmental courses with the performance of students who entere d the college ready for college-level courses. The qualitative port ion of the study consisted of collecting data through a focus group and telephone interviews Findings indicated that students who were ready for college-level classes were mo re likely to be white and male than their counterparts who participated in developm ental courses. Other factors of age,
86 socioeconomic level, and en trance to the college by high school diploma/equivalency tests were not significantly diffe rent between the two groups. Theoretical Background for Qualitative Research Although relatively new in the field of research on teaching, a qualitative, interpretive, phenomenological approach emer ged as significant in the 1970s in the United States (Erickson, 1986). The empirical phenomenological approach involves a return to lived experience in order to obtai n comprehensive descrip tions that provide the basis for a reflective structural analysis that uncovers the essences of the experience. The human science researcher dete rmines the underlying structures of a given experience by interpreting the originally gi ven descriptions of the situ ation in which the experience occurs. The aim is to determine what an experience means for the persons who have had the experience and are able to provide a co mprehensive description of it. Although the interviewer can try to avoid making assumpti ons and inferences, and will seek to make the meaning revealed by the interview as true to the participantsÂ’ intent and reflection as possible, the interviewer must recognize that the meaning is, to some degree, affected by the interaction between the two of them. From the individual descriptions, genera l or universal means are derived. The understanding of meaningful c oncrete relations implicit in th e original description of experience in the context of a particular situation is the primary target of phenomenological knowledge (Moustakas, 1994). Thus, the primary concern of interpretive-phenomenological rese arch is particularizability, rather than generalizability (Erickson, 1986). Interpretive re search and its guiding theory developed out of interest in the lives and perspectives of people in society who had little or no voice.
87 The specifics of action and of meaningperspectives of actors in interpretive research are often those that are overlooke d in other approaches to research. There are three major reasons for this. One is that the people who hold and share the meaning-perspectives that are of interest are those who are themselves overlooked, as relatively powerless member s of society. [A second reason that] these meaning-perspectives are not represen ted is that they are often held outside conscious awareness by those who hold them, and thus are not explicitly articulated. A third reason is that it is precisely the meaning-perspectives of actors in social life that are viewed theo retically in more usual approaches to educational research as either peripheral to the center of research interest, or as essentially irrelevant, part of the Â‘sub jectivityÂ’ that must be eliminated if systematic, Â‘objectiveÂ’ inquiry is to be done (p. 124-5). Developing the interview as a research method involves a challenge to broaden, and enrich the conception of knowledge and rese arch in the social sciences. The research interview does not only yield qua litative texts rather than quantitative data, but reflects alternative ways of thinking about the subj ect matter of the social sciences. Many apparently methodological problems do not st em from the relative newness of the interview method or from insufficiently deve loped techniques, but are the consequences of unclarified theoretical assumptions. Th e mode of understanding implied by qualitative research involves alternative conceptions of social knowledge, of meaning, reality, and truth in social science researc h. The basic subject matter is no longer objective data to be quantified, but meaningful relations to be interpreted (Kvale, 1996).
88 There is a move away from obtaining knowledge primarily through external observation and experimental manipulat ion of human subjects, toward an understanding by means of conversations with the human beings to be understood. The participants not only an swer questions prepared by an expert, but themselves formulate in a dialogue th eir own conceptions of their lived world. The sensitivity of the interview and its cl oseness to the subjectsÂ’ lived world can lead to knowledge that can be used to enhance the human condition (p. 11). Qualitative data, in the form of words rath er than numbers, have always been the staple of certain social science disciplin es, notably anthropology, history, sociology, and political science. However, more and more researchers in fields with a traditional quantitative emphasisÂ—educational studies, for exampleÂ—have shifted to a more qualitative paradigm (Miles & Huberman, 1994) The strategy of qualitative methods is derived from a variety of philosophical, episte mological, and methodol ogical traditions. Qualitative methods are derived most direct ly from the ethnographic and field study traditions in anthropology (Pelto & Pe lto, 1978) and sociology (Bruyn, 1966). More generally, the holistic-inductive pa radigm of naturalistic inquir y is based on perspectives developed in phenomenology (Bussis & Chit tenden, 1976; Carini, 1975), symbolic interactionism and naturalistic be haviorism (Denzin, 1978), ethnomethodology (Garfinkel, 1967), and ecologica l psychology (Barker, 1968) (cited in Patton, 1980). An integrating theme running through these traditio ns is the fundamental notion or doctrine of verstehen The verstehen approach assumes that the social sciences need methods that differ from those used in agricultural experime ntation and natural science because human
89 beings are different from plants and nuclear particles. The verstehen tradition stresses understanding that focuses on the meaning of human behavior, the context of social interaction, an empathetic understanding based on subjective experien ce. The tradition of verstehen or understanding places emphasis on the human capacity to know and understand others through sympathetic intr ospection and reflection from detailed description and observa tion (Patton, 1980). The verstehen tradition is rooted in phenomenology and existential philosophy more generally. Phenomenology is the sense of understanding social phenomena from the actorsÂ’ own perspectives, describing the world as experienced by the participants, with the assumption that the important reality is what people perceive it to be (Kvale, 1996; Moustakas, 1977). Mishler (1986) suggests that stories are a way to knowledge and understanding and move the discussion of in terviewing beyond the boun daries set by the traditional approach. Words, especially when they are organized into stories, have Â“a concrete, vivid, meaningful flavor that ofte n proves far more convincing to a readerÂ— another researcher, a policymaker, a practit ionerÂ—than pages of numbersÂ” (Miles & Huberman, 1994, p. 15). Yet, graduate programs in education have, in the past, been almost totally committed to building knowle dge in education through experimentation (Seidman, 1998). Although this approach is important and valuable, some researchers do not agree that scientific investigation in this form is the only valid design for inquiry. VanKaam (1966) believes that a preconcei ved, experimental design imposed on the Â“subjectsÂ” of an experiment, and statistical me thods, Â“may distort rath er than disclose a given behavior through an imposition of restri cted theoretical constructs on the full meaning and richness of human behaviorÂ” (cited in Moustakas, 1994, p. 14). Mishler
90 (1986) agrees with this thinking and states that in standard interviewing practice, Â“respondentsÂ’ stories are suppr essed in that their responses are limited to Â‘relevantÂ’ answers to narrowly specified questionsÂ” (p. 68). The essence of the phenomenological method is this: Examine experiences carefully, without theoretical pr ejudice; discover the essentia ls of those e xperiences; and communicate what you discover to others for verification (Boeree, 2007). A phenomenological approach suggests suspe nding our own theories, expectations, categories, and measurements and going to the source. The aim of this qualitative study was to remain open to the participantsÂ’ comm unications of meaning and at least approach an understanding of their understanding. Summary In conducting the review of literature and considering the current environment of questioning of our nationÂ’s educational goa ls and how we will reach those goals, it appears that this study was very timely. It is clear that our country needs as many talented students as possible to pursue math ematics and science at advanced levels; yet it is clear that fewer girls and women are pursuing majors and car eers in these fields. They remain, at least in the minds of many wo men, a male domain. A low academic selfconcept in many women, low self-efficacy, st ereotyping, and socia lization factors are evident and tend to preserve the status quo. Debilitating mathematics anxiety paralyzes many students and frustrates their efforts to succeed. Teachers play a critical role in influencing female students either positively or negatively. Mathematics initiatives have increased awareness of inequities and pr omoted new ways of teaching and new approaches to learning, including constructiv ist methods of classroom interaction.
91 Due to the valuable contributions phenome nological research has made in the area of mathematics education, it a ppears that a qualitative resear ch design was a valid process which has allowed the researcher to share, to a degree, in the actua l experience of the participants. The strengths of qualitative resear ch lie in its inductive approach, its focus on specific people describing their experience as they see it, and its emphasis on words rather than numbers (Maxwell, 1996). Since this study is concerne d with discovering and understanding the phenomenon of mathematics avoidance or mathematics confidence, a topic that is often sensit ive and ambiguous, the phenomenological approach seemed especially appropriate. An open-ended quest ioning of this phenomenon, as opposed to a more traditional survey and analysis of quantitative data, has provided a greater opportunity for the researcher to understand participantsÂ’ perceptions of their experience, unravel some of the factors contributing to it and translate this understanding into greater sensitivity to studentsÂ’ needs.
92 Chapter Three Method Problem It is clear that our countr y needs as many talented st udents as possible to pursue mathematics and science at advanced levels in high school, college, a nd graduate school. Yet, in the last few decades, it has been clear that fewer girls and women are pursuing majors and careers in either mathematics or science (Reis & Pa rk, 2001). Women are more educated, more employed, and employed at higher levels today than ever before, but they are still largely pigeonholed in pink-collar [italics added] jobs according to the American Association of University Wome n (AAUW) Educational Foundation report, Women at Work (2003) [italics added]. The report goes on to say that the new high-tech economy is leaving women behind because they donÂ’t have the keys to open the door to this high-tech sector of the work force. National census data show that the highest proportions of women with a college educati on are still in traditionally female careers: teaching and nursing (AAUW, 2003). A report by the National Research Council, entitled Everybody Counts: A Report to the Nation on the Future of Mathematics Education emphasizes that undergraduate mathematicsÂ—the mathematics of the college experienceÂ—is vitally important, perhaps even more than elementary or secondary school mathematics. Â“More than any other subject, mathematics filters students out of programs leading to scientific and
93 professional careers. From high school through graduate school, the half-life of students in the mathematics pipeline is about one yearÂ…Â” (Smith, 1994, p. 135). Undergraduate mathematics is the linc hpin for revitalization of mathematics education. Not only do all the scie nces depend on strong undergraduate mathematics, but also all students who prepare to teach mathematics acquire attitudes about mathematics, styles of teaching, and knowledge of content from their undergraduate experience. No reform of mathematics ed ucation is possible unless it begins with revitalization of undergraduate mathematics in both curriculum and teaching style (c ited in Smith, 1994, p. 135). For a myriad of reasons, mathematics inspires more emotion than any other school subject (Segeler, 1986; Zaslavsky, 1994) Furthermore, anxiety and fear of mathematics may keep a woman from comple ting her degree program and succeeding in her academic goals. It has been found that women are more likely than men to suffer from debilitating mathematics anxiety (Cle well, Anderson, & Thorpe, 1992; Stewart, 1990; Tobias, 1990). Purpose The first purpose of this study was to ex amine metacognitive and affective factors that are perceived to contribute to mathem atics avoidance or mathematics confidence in non-traditional age women atte nding a community college. The second purpose of the study was to explore and describe the meani ng participating, non-tr aditional age women attach to their experience with mathematics. The third purpose of the study was to determine the relationship, if any, between me tacognitive and affective experience in the learning of mathematics.
94 Research Questions 1. What metacognitive and aff ective factors are perceived to contribute to mathematics avoidance or mathematics confidence in non-traditional age women attending a community college? 2. What meanings do participating non-tradi tional age women attending a community college attach to their experience with mathematics? 3. What is the relationship, if any, between metacognitive and affective experience of participating non-traditional age women attending a community college in learning mathematics? Overview of Chapter The method for conducting this study is pr esented in the following eight sections of this chapter. The first section is an introduction explaining the value of a phenomenological approach to in-depth inte rviewing. Section two describes the study design: the informal conversational intervie w. In section three, a personal testimony regarding my own experience in learning mathematics is disclosed, followed by a description of the research se tting in section four and the selection of participants in section five. The procedure involved in impl ementing the study is reviewed in section six, including a briefing on the results of a pilot study. Section seven clarifies the way the data were compiled and analyzed. The fi nal section concludes with a discussion of trustworthiness and credibility issues. Value of Phenomenological Interviewing Phenomenology is the attempt to describe, rather than explain or analyze, an experience as directly as possible, without any thought about the origin, or cause, or
95 nature of an experience (Kvale, 1996). F eelings are best described and understood through phenomenological research Â—in-depth interviewing that investigates, delves, digs beneath the surface to discover and describe what lies beneath a shallow, superficial surface view of a subject. The strength of in-d epth interviewing lies in the reality that we can come to understand the details of othe rsÂ’ experience from th eir point of view (Seidman, 1998). [Interviewing] Â“has led me to a deeper understanding and appreciation of the amazing intricacies and, ye t, coherence of peopleÂ’s experi ences. It has also led me to a more conscious awareness of the power of the social and organizational context of peopleÂ’s experience. It has al so given me a fuller apprecia tion of the complexities and difficulties of changeÂ” (Seidman, 1998, p. 112). My use of the term phenomenology [italics added] primarily refers to the interviewing process based largely on Se idmanÂ’s (1998) work. In-depth, intensive interviewing is the major way a qualitati ve researcher seeks to understand the perceptions, feelings, and knowledge of people (Patton, 1980). The task is to provide a framework within which people can respond in a way that represents accurately their points of view about their world. The challe nge is to get close enough to the people and situation studied to be able to understand the depth of what is happening. The aim is to capture what actually takes place, what the pa rticipants actually say. The interview data consist of a great deal of pur e description--direct quotations from those being interviewed (Patton, 1980). The commitment to get close, to be factua l, descriptive and quotive, constitutes a significant commitment to represent the participants in their own terms. This does not mean that one becomes an apol ogist for them, but rather that one
96 faithfully depicts what goes on in their live s and what life is like for them, in such a way that oneÂ’s audience is at least part ially able to project themselves into the point of view of the people depict ed (quoted in Patton, 1980, p. 36). Educators need this curiosity in order to understand, and thereby to act, to enable students under their guardianship to succeed a nd thrive in the particular setting, whether that setting is elementary, secondary, or postsecondary education. Understanding and humility are not bad stances from which to try to effect improvement in education (Seidman, 1998). For investigation of the problem of th is research, I found that a quantitative, experimental study could not tell me the real ities of the experience of another human being and how that person thinks and feels about his or her experience. Quantitative measures are succinct and easily collected for analysis, whereas qualitative measures are longer, more detailed and irregular in cont ent. Quantitative data are systematic, standardized and presented in tables and char ts. In contrast, quali tative data may be difficult to analyze because responses are neither systematic nor standardized. Nevertheless, where it may be possible to disc ount quantitative data as being biased or rigged (as in what may be perceived as a loaded [italics added] questionnaire), one cannot so easily dismiss the emotions and feelings revealed in participantsÂ’ own reflections (Patton, 1980). Consequently, I be lieve that a qualitativ e study of in-depth interviews has uncovered what questionnaires or rating scales c ould not. Prior to conducting this research, I was inspired and guided by comments and concerns of undergraduate and graduate students, facu lty members, university professors and
97 administrators who suggested that a qualitati ve study of attitudes regarding mathematics would be of interest and value to educators. This study sought to provide an understa nding of factors in the educational experience that are perceived to contribute to mathematics avoidance or mathematics confidence in non-traditional age women attending a community college, and the meanings the participants attach to their experience with mathematics. An understanding of these factors is needed in educational research, first because they inhabit the Â“invisibility of everyday lifeÂ” (Erickson, 1986). Everyday life is largely invisible to us because of its familiarity and because of its contradictions, which people may not want to face. We do not realize the patterns in our actions as we perform them (Geertz, 1984). Second, there is a need for specific insight through documentation of concrete details if one is attempting to understand the points of view of the actors involved. Third, we need to consider the lo cal meanings that experiences have for the people involved in them (such as getting an Â“FÂ” on a test or being called to the board) because events that seem ostensibly the same may have distinctly differing local meanings to different people. It may be found that the differen ces in participantsÂ’ perceptions of their experiences are quite small and that a small adjustment made by educators may lead to a big di fference in student learning. Study Design: The Informal Conversational Interview The informal conversationa l interview is the pheno menological approach to interviewing. The interviewer uses it in orde r to maintain maximum flexibility to be able to pursue information in whatever directi on appears to be appropriate, depending on the information that emerges from talking to one or more individuals in th at setting. Most of
98 the questions flow from the immediate cont ext. Although the inte rviewer may compose a standard list of open-ended que stions to begin the interview, no complete predetermined set of questions is possible under such circumstances, because the interviewer does not know beforehand what is going to happen and wh at it will be important to ask questions about (Patton, 1980). The questi ons may change in response to the distinctive nature of carrying out the intervie ws or in response to changes in the researcherÂ’s perceptions and understandings as the interv iews progress (Erickson, 1986). The data gathered from informal conversational interviews are diffe rent for each person interviewed. According to Patton, the phenomenologica l interviewer must Â“go with the flowÂ” (p. 199). The interviewer is also part of the pro cess and must be disciplined and dedicated to keeping each interview true to what the participant is saying. On the other hand, the interviewer must be aware that the meani ng is, to some degree, a function of the participantÂ’s interact ion with the intervie wer (Seidman, 1998). The challenge is to minimize any distortion of meaning created by interviewer bias. When studying data, researchers may use br acketing, which is a mental exercise in which the researcher identifies, then sets aside taken-for-granted assumptions. It allows the researcher to complete his or her own reconstruction of the experience under investigation in orde r to become aware of personal biases and preset opinions. Bracketing reveals what Â“everyone knows,Â” what people assume but rarely say (Erickson, 1986). For example, one might say, Â“YouÂ’re going the wrong way Â” [italics added]. Who is to say that an individual is going the wrong way? If one does not know where that individual came from or where he or she is heading, who can judge which way is the wrong way? The [wrong way] then is bracketed. It is held aside from judgment in order
99 to create a path to a clearer view of something. By doing this, a researcher can understand something on its own terms without interference of his or her own frame of reference. Another example of bracketing is one that may appear humorous, yet it frequently occurs in households. One might say, Â“The toilet tissue is on backwards Â” [italics added]. Who is able to decide the direction of b ackwards? To one individual the direction appears to be backwards; to another it is certainly not. [Backwards] is bracketed, temporarily suspended from judg ment until an investigation has taken place. Â“Presuming that critical analysis provides ne w or different data to assess, the release [italics added] of the brackets provides the opport unity for a clash of the old and the newÂ…resulting in yet another new [italics added]Â…which ma y or may not culminate in a change of values or positionÂ” (Kessel, 2007). The researcher has made a conscious effort to bracket all presuppositions, standards, and prior commitments, not in orde r to deny their existence or importance, but simply as a methodological move to see clearl y, without bias, what the participants are saying. To accomplish this, I underwent a bracke ting interview prior to the beginning of the participant-interviewing pha se of the study. This was done with the help of another faculty member at the community college wh ere I am employed. This faculty member holds an Ed.S. degree from the University of Florida and is Director of the collegeÂ’s Center for Counseling and Academic Developmen t. He is a published author and has 20 years experience in counseling and supervis ion of student personnel services. The bracketing interview was transcribed and reviewed by me and provided valuable considerations and insights for the study.
100 Researcher (Jo Ann Rawley) I am particularly intrigued by a descri ption by Kvale, presenting the interviewing process using a traveler metaphor (1996). The in terviewer is pictured as a traveler on a journey that leads to a tale to be told upon returning ho me. The interviewer-traveler wanders through the environment and enters in to conversations with the people. The traveler explores the country, as unknown terri tory or with maps, roaming freely around the territory. The traveler may also deliberately seek specific sites or topics by following a method of questioning. The interviewer wande rs along with the local inhabitants, asks questions that lead the participants to tell their own stories of their lived world, and converses with them. What th e traveler hears and sees is described qualitatively and is retold as stories to be shared with the people in the interviewerÂ’s own country, and possibly also to those with whom the inte rviewer wandered. The potential meanings in the stories are differentiated and unfolded thr ough the travelerÂ’s interpretations and are proven trustworthy and credible by their imp act upon the listeners. The journey may not only lead to new knowledge; the traveler might change as well. The journey might instigate a process of reflection that lead s the interviewer to new ways of selfunderstanding, as well as uncovering previously taken-for-granted values and customs in the travelerÂ’s own country. Through conversatio ns, the traveler can also lead others to new understanding and insight. I have drawn upon many years of experience in tutoring mathematics at different levelsÂ—middle school, high school and college. I have worked with students in middle and secondary school, recent high school graduates, adults retu rning to college, aspiring teachers, and graduate students studying to pass standardized exams. I am fortunate to be
101 currently working in a community college where students from all ethnic, socioeconomic, and age groups are represented. I have w itnessed many eyes light up as understanding took the place of confusion and fear. Most importantly, I understand how debilitating mathematics anxiety can be from personal experience. One experience, or defining moment rela ting to mathematics, occurred in my senior year of high school. N eeding another mathematics credit to qualify for entrance as a freshman at Shippensburg State Teachers College, I was placed into a tenth-grade algebra I class. Being accustomed to earni ng AÂ’s in all of my other courses, I was devastated by the failing grades I was receiving on my algebra tests. I stayed after school to receive additional help from the teacher, only to be more confused after the session was over. Taking tests was a nightmare acco mpanied by severe anxiety. Even more humiliating was the fact that the students in the 10th grade class seemed to have no problem whatsoever with the material and rare ly asked questions. A final grade of a C (a gift) in the algebra class caused a drop in my GPA and placed me four th in the graduating class. It was not until many years later, in a community college, that a mathematics teacher was able to undo the psychological damage of that high school experience and give me the confidence to pursue mathematics as a career. The teacher thoroughly explained each new mathematical concept, connecting it to previously known and understood concepts. Contextual examples of using this new concept were given on the board and students were given the opportunity in class to practic e a problem or two on their own. Each class began by allowing tim e for questions before new material was presented, thus clearing up misunderstandings and encouraging student feedback. Tests were challenging. However, credit was given fo r evidence of critical thinking even if the
102 answer was incorrect. The teacher was availa ble outside of class time to talk. She was interested in studentsÂ’ goal s and encouraged risk-taking and perseverance. In this mentoring role, she remained in contact wi th me after the class ended and suggested participation in several confidence-building on-campus activities, one of which was helping in the mathematics tutoring center. It is experiences such as the one just described that I hoped would emerge from the participantsÂ’ stories and, in doing so, shed light where there was confusion and bri ng insight, understanding, and motivation. I identified my interest and my desire to channel it properly in order to minimize any distortion that might have resulted in the way the interviews were carried out. A conscious effort was made to ensure that I was not Â“reading intoÂ” or Â“interpretingÂ” participantsÂ’ comments for them. Research Setting: Reading Area Community College The study took place at Reading Area Co mmunity College in the state of Pennsylvania. The college is approved by the Department of Education of the Commonwealth of Pennsylvania as an institut ion of higher education, and is authorized to award the Associate in Arts Degree, th e Associate in Applied Science Degree, the Associate in General Studies Degree, and the Certificate of Specialization, as well as appropriate diplomas and cer tificates. The Commission on Higher Education of the Middle States Association of Colleges has granted the college full accreditation. The Associate in Arts (A.A.) degr ee is designed for students who ar e planning to transfer to a four-year college or university and carries a requirement of at least 60 credit hours of study, with not less than 15 cred it hours earned at the loca l college. For students not desiring to pursue a bachelorÂ’s degree, th e college offers the Associate in Applied
103 Science (A.A.S.) degree, which broadl y prepares students for employment upon graduation and is referred to as a Career Program. The Associat e in General Studies (A.G.S.) degree is an individualized curricu lum which allows students to design their own degree programs for professi onal development or transfer and has a minimum course requirement of 60 credit hours, with not less than 15 hours earned at the local college. Certificates of Specializati on are designed to give stude nts an opportunity to gain specialized knowledge to advan ce in their jobs, learn new sk ills, update the skills they have, or to help them change careers. Many candidates elect to enroll in the certificate program first and then, after completion, cont inue in the Associate in Applied Science degree. Diplomas offer college credit and pr ovide students with specific technical job skills for workforce entry or promotion. The college is a publicly-supported, co mprehensive community college and serves a student body of approximately 4500 studen ts. It is located in a busy, downtown section of a city, on the banks of a river, which gives it both a vibrant, community feeling, yet one of peace and re flection. Classes at the coll ege are small, averaging 18 students, allowing teachers to get to know each student by name. The atmosphere is friendly and supportive; teachers act as mentor s and guides in addition to their teaching responsibilities in the classroom. At the community college, students are welcomed from all academic backgroundsÂ—with competitive SAT scores or no SAT scores at all. Some are bound for advanced degrees or seeking to refresh thei r skills, while others may have doubted they were college material [italics added]. For those who wi sh to continue their education, credits can be easily transferred to other colleges and universities, due to the Academic
104 Passport program, which guarantees students with associate degrees from the college admission to any of 14 universities in th e Pennsylvania State System of Higher Education. All students are required to take placeme nt tests before registering for credit courses at the college. Based on the scores they receive, students will be advised concerning the appropriate courses to take as they begin their college careers. Such advice is based on test scores and followup interviews. In some cases, students may move directly into freshman level English or mathematics courses, in others, they may be advised to consider noncredit developmental courses. One of the collegeÂ’s institutional goals is to provide students with effective de velopmental services that link into college level course work and remedial programs that allow them to reach their potential. To achieve this goal, free tutoring and academi c counseling are available to everyone on a walk-in basis. Tutoring is offered in a tuto rial center where tutors are available during posted hours. In addition to the walk-in tutorial center, the college participates in three federally funded programs that offer individual tutori ng: The Student Support Services Program (SSSP), the ACT 101 EMPHASIS Program, and the Carl Perkins Program. Students eligible for the Student Support Services Pr ogram receive personal counseling, tutoring and college success strategies courses free. The ACT 101 EMPHASIS Program provides supportive services for students who have good potential to succeed in college but who need to overcome academic and financial barriers. The EMPHASIS Program is funded by the U. S. Department of Education through ACT 101, the Pennsylvania Higher Educational Opportunity Act of 1971. The Ca rl Perkins Program provides academic and
105 counseling support services to academically and financially challenged students who are pursuing technological degrees. Although there are some support services in place for struggling students, there are no special programs operating expressly fo r women. Many women do not qualify for individual help and find the hour s the tutorial center is open, and the tutorial assistance available, insufficient to meet their needs. Especially in the mathematics area, the supply of tutors cannot accommodate the demand by substantial numbers of students who require one-on-one help. Furthermore, the majo rity of student tutors may not be sensitive to the unique needs of non-tradit ional students, which are descri bed in Chapter One. It is expected that the results of this study ma y provide suggestions to benefit the student support programs at the college. Selection of Participants At RACC, students are drawn from a wi de surrounding local area as well as internationally. Potential part icipants for this research we re selected from a pool of nontraditional age women who applied to the college and enrolled in a mathematics course. The researcher was not invol ved in any way with the in struction in any of the mathematics courses. Placement in the part icipant pool depended on answers to several questions on the college entrance/placement pr ofile. This 28-questi on profile is called COMPASS and is designed to give information about incomi ng students, which is then used to anticipate student n eeds (see Appendix A). The CO MPASS data are entered via computer. The researcher used 13 of the 28 questions to determine qualification for the participant pool. Answers to these 13 questions provided necessary demographic informationÂ—gender, ethnicity, birth date, parentsÂ’ education, size of householdÂ—and
106 uncovered those students who had strong posit ive or strong negative feelings about mathematics. The researcher selected only those participants who chose mathematics as the subject they would most enjoy or least enjoy (que stions 22 and 23). I wondered if a sufficient number of partic ipants could be obtained from this type of selection process. Th erefore, a year before this study was conducted a trial was performed during a six-month period with a sample of over 1200 in-coming college enrollees. One hundred eighty-tw o students chose mathematics as the subject they would least enjoy; 77% (140) were fe male and 68 of these were of non-traditional age. Fortyseven students chose mathematics as the subject they would most enjoy; 43% (20) were female and 5 of these were of non-traditional ag e. Using this trial as a forecaster, I was confident there would be a sufficient number of sample participants from which to draw. Thirty potential par ticipants were sent a letter through the mail describing the research and were invited to respond by c ontacting me either by email or telephone. Response to this letter was very low. On ly one person contacted me and volunteered to be interviewed. Consequently, telephone calls were made to as many potential participants as could be reach ed. Three of the participants could not be reached due to phone numbers that had been disconnected. Of the original 30 women, I was able to reach 27. I found that this met hod of contact proved to be much more successful; all of the persons to whom I spoke agreed to at le ast stop by my office to discuss the research study. Of the 27 women, 18 agreed to be in terviewed. Due to sc heduling conflicts and lack of interest on their part, 3 of the 18 were never included in this study. In gaining access to potential participan ts and making contact with them, I made every effort to achieve as much equity in the interviewing rela tionship as possible by
107 assuring them that the outcomes of this resear ch would in no way affect their grades, that participation was voluntary, and that the result s from the research would be kept strictly confidential. I wanted to minimize differe nces in perceptions of power and authority between the participants and my self. I believe that when th e participants saw a genuine interest in getting to know their backgrounds and experience with mathematics, they felt important and flattered that they had been chosen to pa rticipate in this research. Following the initial selection and positive response of each participant, further essential criteria for selection included the following: the participant was willing to engage in an in-depth interview and granted the researcher the right to tape-record and publish the data in a disserta tion and other publications. No attempt was made to choose participants based on socioeconom ic background or ethnic group. As soon as prospective participants we re identified, a contact visit was set up before the actual interview process began. In these meetings, I explained who I was, what my study was about, how I intended to use the data collected, and the amount of time and the nature of the commitment necessary from participants. The goal was to be explicit about the work and tr y to create a situation in wh ich the potential participants made an active choice about whether to par ticipate in the study. Once that choice was made, a date and time for the interview was proposed. After having made contact with each participant, secured her agreement to participate, and set up the time and place for th e interview, one final step remained as a bridge between the contact process and the interview process. At the appointment for the interview, but before it actually started, the participant was given a consent form (IRB Adult Informed Consent Form, see Appendix B.). This form served to clarify further the
108 purpose and nature of the research. Pa rticipation in the study was voluntary and participants were free to withdraw from the study at any time. Using a list of 10 interview questions (s ee Appendix C) as a starting point, the conversation was spontaneous according to the flow of ideas expressed by each participant. If an interesting comment was voiced, further unplanned questions were posed in order to follow the participantÂ’ s line of thought. The interviews were interactive, as the interviewer responded to the participantÂ’s answ ers, giving feedback when necessary in order to encourage the part icipant to continue sharing her thoughts and feelings. Feedback might have included comm ents such as, Â“How did that [Â…] make you feel?Â” Â“Did that [Â…] happen again?Â” Â“Thank you for sharing that with me.Â” There was no set limit on the length of an interview; length depended on each participant reaching a point at which she ha d depleted all her de scriptions and thoughts regarding the interview ques tions. A quiet, comfortabl e environment, free from distractions, was chosen to put participants at ease and facilitate the interview process. Interviews were tape d and later transcribed by me, which provided another opportunity to reflect on particip antsÂ’ stories. Recording assu res an accurate and detailed account, including pauses, sighs, la ughter, and so forth. After transcribing th e interview, I read it, underlining important passages a nd assigning a code to each, while comparing the story in front of me to those of prev ious participants. Aspects of individual experience which were considered important included conflict, both between people and within a person, hopes expressed, language i ndicating beginnings, middles, and endings, frustrations and resolutions, and indications of isolation as well as indications of community. Gender, class, or et hnic issues that may have ar isen were of importance, as
109 were indications of hierarchy and power affec ting the participantsÂ’ experience. I strove to include every aspect of the interview that may have proven to be enlightening to those interested in learning about the results of this study. Interviews were conducted beginning in the fall term, 2005, and c ontinued through the fall term, 2006. I created a profile of each participantÂ’s experience, thus opening up the material to analysis and interpretation. The narrative fo rm of such a profile, written in the first person, allowed me to transform this learning into telling a story. Crafting profiles is a way to find coherence in the events of a pa rticipantÂ’s experience and to link such an experience to the social and organizational context within which she operates. In addition to these profiles speaking for themse lves I was able to explore and comment on salient issues within indi vidual profiles and point out connections among profiles. After compiling the data, I shared with each participant the specific material from her interview and removed all identifying data, such as the participantÂ’s name. Participants were requested to examine careful ly the description of the narrative derived from their responses, verify accuracy and make any additions and corrections necessary. I did not use any part of a prof ile that was not verified as accurate by the participant. This verification by participants provided trustworthiness of the data. Based on the visible emotional responses of some of the participants upon reading the tran scription of their interview, I was encouraged in belie ving there was indeed trustworthiness and credibility in the data that had been gathered. Procedure The primary source of data for this study was the in-depth interview. Interview questions (see Appendix C) provi ded a structure for the invest igation and were designed
110 to encourage participants to think about their past life expe riences with mathematics and the meanings they made from these experi ences. Anfara, Brown, and Mangione (2002) suggest providing readers of a research study wi th a cornerstone for the analysis of data. In following their recommendation, Table 6 serves to demonstrate how interview questions were related to the three majo r research questions in this study. The interview aimed at establishing th e context of each womanÂ’s experienceÂ— how she came to seek a higher education degree the reasons she chose to enroll at the college, and what preparation in mathematic s she had previously acquired. Questions were asked about her memories, in elementa ry grades through high school, up until the time she became a college student, going as far back as possible. Some questions were aimed at eliciting descriptions of experiences behaviors, and actions. Other questions were aimed at understanding the metacogniti ve and interpretive processes of the participant, delving into what was per ceived. These questions focused on goals, intentions, desires, Â“How could that experi ence have been made more constructive?Â” Another set of questions were aimed at unde rstanding the emotional responses of people to their experiences. Â“How did you f eel whenÂ…?Â” Â“How would you describe your reaction toÂ…?Â” Each participant was asked to recons truct her present experience in her mathematics class by relating stories about it in detail. Some questions were aimed at eliciting descriptions of expe riences, behaviors, and actions Â“What are your feelings now regarding mathematics?Â” Â“Do you e xpect to be successful in passing the mathematics course you are currently taking?Â”
111Table 6 R elationship of Research Questions to Interview Questions Research Question Interview Question 1. What metacognitive and affective factors are perceived to contribute to mathematics avoidance or mathematics confidence in non-traditional age women attending a community college? 2. What meanings do participating nontraditional age women attending a community college attach to their experience with mathematics? 3. What is the relationship, if any, between cognitive and affective experience of participating non-traditional age women attending a community college in learning mathematics? Q2, Q4, Q5, Q6, Q8 Q1, Q3, Q4, Q5, Q6, Q7, Q8, Q9, Q10 Q2, Q4, Q7, Q9, Q10
112 The interview encouraged the participant to reflect on the meaning her experience held for herÂ—the effect her mathematics performance had on her self-esteem, selfconfidence, and motivation. The participant wa s encouraged to look at how the factors in her life interacted to bring he r to this present situation. Qu estions were asked about selfconcept, such as, Â“Would you define yourself as being smart?Â” Â“How do you compare your skills in other areas, such as English, with your skill in ma thematics?Â” Â“How will your achievement in mathematics affect your car eer goals?Â” The data consisted of a great deal of description of people, activities, and interactions as well as direct quotations. A pilot study with one college stude nt was completed for the purpose of becoming familiar with the interview process and the time required in transcribing the data. I used different ques tions to ascertain which one s would bring out the most stimulating responses and interactions. Th rough the pilot study, I learned some of the practical aspects of making contact, setti ng up the interview, c onducting the interview, and using a tape recorder succe ssfully. I spent time to refl ect on the experience and make adjustments and revisions based on what wa s discovered. The pilot study interview lasted for 30 minutes, even though an hour was set aside, and the participant was given time to respond thoroughly to every questi on. Transcription of the interview took approximately one hour. The location in which I conducted the pilot study was rather dist racting with other students too close by; therefore, I chose a private office to meet with the other participants in my study. I also revise d the order of the questions by moving one question to an earlier point in the interv iew to achieve a better flow of thought.
113 The process of data collection contin ued until saturation of information was reached, a point in the study at which the inte rviewer began to hear the same information reported. In-depth, phenomenological interv iewing, applied to participants who all experience similar structural a nd social conditions, gives grea t power to the stories of a relatively few participants (Seidman, 1998). A total of 15 participan ts were interviewed, followed by the important process of readi ng, modifying and verifying their profiles. Data Analysis Merriam (1998) recommends that data analysis occur simultaneously with collection. Transcripts of th e interviews were analyzed and coded by looking for key themes, underlining statements I felt were significant and making notes in the margins to compare with other participantsÂ’ stories. The words that the participants used to describe their feelings were compared and I noted the similarities and contrasts among them. Codes were assigned to responses given by pa rticipants; as new or different responses occurred, new codes were added. When all of the interviews were completed, responses were compared and common patterns, or themes, became evident. By paying close attention to the words of the participants, connections and similarities were gleaned. Themes were identified that appeared to be shared by common structural and social forces. See Appendix D for detailed coding elements and patterns. This technique, utilized in this study, is refe rred to as constant comparativ e analysis (Glaser & Strauss, 1967), and helped me to assign codes to the data, recognize patterns, and develop themes. The participantsÂ’ stories were written in the form of profiles, or vignettes, short narratives that covered an aspect of a partic ipantÂ’s experience. These profiles present the
114 participantsÂ’ stories in contex t, using their own words, in first person, to reflect their feelings and realities. Trustworthiness and Credibility Ensuring trustworthiness and credibility in qualitative research involves conducting the investigation in an ethical manner (Merriam, 1998). Because human beings are the primary instrument of data co llection and analysis in qualitative research, interpretations come directly from their rich descriptions. Therefor e, the trustworthiness of this study was determined, in part, by the voiced experiences of the participants themselves. Each participant read the transc ription of her interview and then read her profile containing specific quotes from the in terview. Each participant was asked to verify accuracy and make any additions and correc tions. In the litera ture, this process is sometimes referred to as conducting member checks. Creditability was also sought by involvi ng two of my colleagues who read all 15 interview transcripts and independently wrote what each recognized as common meanings and themes expressed by the particip ants. Their findings were compared with my own and we reached consensus on the final list of themes. According to Merriam (1998) Â“rigor in qualitative research derives from the researcherÂ’s presence, the nature of the inte raction between researcher and participants, the triangulation of data, the in terpretation of perceptions, and rich, thick descriptionÂ” (p. 151). I recognize the possi bility of human error in listeni ng to and interpreting the words of the participants. However, being aware of the dangers of bias in interpretation, I did everything in my power to minimize any distor tion due to my role in the interview.
115 Further trustworthiness of participantsÂ’ st ories is the impact they have upon those who listen to them or read them (Kvale, 1996) As previously stated, I was surprised by the emotional reaction most of the participants evidenced when they read their profiles and the findings produced by this study. Se veral of the women were overcome with emotion and shared how meaningful this study had been to them personally and how appreciative they were to have had a chance to participate. There is trustworthiness to the description of the data due to the audio recordings of the interviews and the verbatim transcripti on of those recordings. As the researcher, I strove not to impose my own framework of beliefs, or my own meanings onto the data; rather I tried to understand th e participants whom I studied and the meanings they attached to their words. My questions, for the most part, were not closed, short-answer questions, but were structured to give part icipants the opportunity to reveal their own perspective. Credibility of the data analysis is comp licated by questions such as, Â“How do we know that what the participant is saying is true?Â” Â“If it is true for this participant, is it true for anyone else?Â” Â“If another person were doing the interview, would we get a different meaning?Â” Â“If we were to do the in terview at a different time of year, would the participant reconstruct her experience diffe rently?Â” Â“If we had picked different participants to interview, would we get an entirely dissimilar and perhaps contradictory sense of this issue?Â” (Seidman, 1998 p. 17). Merriam (1998) contends, Because what is being studied in educati on is assumed to be in flux, multifaceted, and highly contextual, because information gathered is a function of who gives it and how skilled the researcher is at gett ing it, and because the emergent design of
116 a qualitative case study precludes a priori controls, achieving reliability in the traditional sense is not only fanciful but impossible (p. 206). Instead of using the term Â“r eliabilityÂ” in the traditiona l sense, Lincoln and Guba (1985) suggest that, with qualitative resear ch, one should judge the Â“dependabilityÂ” or Â“consistencyÂ” of the results obtained from the data. Â“R ather than demanding that outsiders get the same results, a researcher wish es outsiders to concur that, given the data collected, the results make senseÂ—they are consistent and dependa bleÂ” (Merriam, 1998). In attempting to make sense of the credibility dilemma, comparing interviews was a safeguard. Comparison placed participan tsÂ’ comments in context and allowed for checking the comments of one participant agains t those of others. Furthermore, reading and verifying the accuracy of the transcribed interview encouraged participants to think about what they had said and correct or clar ify anything that may have been ambiguous. Comparison of participantsÂ’ interviews, the passage of time over which the interviews occurred (one year), the internal consiste ncy and possible external consistency of the words of the interviews, the syntax, diction, and even nonverbal aspe cts of the interview all provide a measure of confidence in the tr ustworthiness and credibility of these data. Triangulation using multiple investigators and soliciting feedback from them was a strategy used to increase the credibility of this study and identify researcher biases and assumptions. Two of my colleagues studied the transcripts of the interviews and interpreted them independently, giving their ow n perspective to the findings. The first colleague is a learning specialis t employed by the college. This individual holds a BA in Psychology and a BS in Education and is certifie d to teach in the state of Pennsylvania. She has worked in the field of education fo r 20 years and is currently responsible for
117 oversight of the Americans with Disab ilities Act and Federa l 504 guidelines for accommodations for students with disabil ities at a community college. She has previously been employed as a remedial math ematics instructor fo r adult learners and was responsible for adapting mathematics curr iculum to meet individual strengths and needs and for completing skill assessments for students. The second colleague is a professor in the Business Di vision at RACC as well as coordinator of the Business Management program. In these roles, she works closely with students in both an instructional and advisory capacity. She ha s had 30 years of experience teaching higher education. She earned a bachelorÂ’s degree in Business Education from Bloomsburg State University, a masterÂ’s degree in Counseling at the University of Delaware, and holds a doctoral degree in Adult Education from the Pe nnsylvania State University. The three of us collaborated and reached consensus on the final list of themes. Summary This is a qualitative rese arch study and is not meant to be generalizable since the findings are based on a small, purposeful samp le selected precisely because my aim was to understand the particular experience in dept h, not to find out what is generally true of the many. The extent to which the findings in th is study apply to other situations is up to the people in those situations. A phenomenol ogical approach to in-depth interviewing was used to examine factors perceived to contribute to mathematics avoidance or mathematics confidence in non-traditional ag e women attending a community college. This approach allowed me to gather desc riptive information a bout the phenomenon of mathematics avoidance or mathematics confidence by interviewing those participants who have experienced one or the other, or possibly both. By usi ng information provided
118 by the participants, I discovered themes that recurred and were shared. Participants were selected from a pool of students who, afte r completion of a COMPASS profile, chose mathematics as either the subject they most en joy or least enjoy. After participating in an in-depth interview, data collected was transcribed, checked for accuracy by each participant, and read by the researcher a nd two colleagues. Consensus on themes was sought. Comparisons were made among the par ticipantsÂ’ stories and the data were coded, analyzed, and crafted into prof iles. By sharing common themes from the stories of the participants, factors that contribute to math ematics avoidance or mathematics confidence have emerged from the study and may lead to a better understanding of the phenomena. Identifying such factors adds to or supports the existing literature as well as provides information for educators to more effectivel y meet the needs of this growing group of students.
119 Chapter Four Findings The first purpose of this study was to examine, by means of in-depth, phenomenological interviewing, the metacogn itive and affective factors that are perceived to contribute to mathematics a voidance or mathematics confidence in nontraditional age women attending a community co llege. The second purpose of this study was to explore and describe the meaning partic ipants attached to their experience with mathematics. The third purpose of this st udy was to determine the relationship, if any, between metacognitive and affective experien ce in the learning of mathematics. This chapter will present profiles of 15 women and findings, using constant comparative analysis, of 15 in-depth intervie ws. Details regardi ng procedure and data analysis can be found in Chapter 3. All pa rticipants were non-traditional age women enrolled at Reading Area Community Colle ge. Interviews were audio taped and transcribed by me. Transcript s were typed with first names only, and in the final form pseudonyms were used. Interviews were conducted, be ginning in the fall of 2005 and ending in the winter of 2006. During this period of time, appr oximately 4500 students enrolled at RACC for the first time and took the COMPASS placement test. Non-traditional age females who chose mathematics as the subject they woul d most enjoy or least enjoy were chosen randomly and contacted first by letter, then by telephone. Time did not allow taking note
120 of those who were not interest ed in being participants. It could not be expected that randomly chosen adults would cooperate with a researcher unknown to them. It became clear that some degree of personal acquainta nce with me was a prior condition to their willingness to participate. Telephone conversations, for the most part, were very productive and, by making voice contact with th e women, an initial measure of trust was built. A preliminary finding of this study was that when a student recognizes an instructorÂ’s interest in her or his college experience and when that instructor takes time to make a personal phone call, a powerful impact is made on a studentÂ’s self-image and selfefficacy. The findings are presented in two form s: profiles and themes/sub-themes. Profiles provide a brief description of why each participant was seeking an education and the challenges that each woman perceived as fo rmidable in the path to completion of her academic goals. Themes and sub-themes provide a framework from which to compare and contrast the stories of the participants. Rich description from the participantsÂ’ own words allow in-depth insight into the meaning th e participants made of their experiences. The themes and subthemes presented represen t only the thoughts and meanings that the participants in my study shared and are not meant to be generalizable to other populations. Nevertheless, these themes a nd sub-themes are a means for those in mathematics education to continue the conve rsation on the meanings non-traditional age women give to their mathematics experience. There were eight common major themes and six sub-themes as listed in Table 7. The profiles will be presented first (in no particular order), and a discussion of the themes and sub-themes that emerged will follow.
121 Table 7. Themes and Sub-themes (1) Acquiring a college education is a personal goal. (2) Adequate study time is necessary to unders tand and to retain mathematical concepts. (3) Experiences with mathematics at an early age remain in oneÂ’s memory. (a) Poor experience with mathematics at an early age tended to make participants believe they could not learn mathematics. (b) Positive experience with mathematic s at an early age tended to provide participants a higher degree of self-efficacy in succeeding in mathematics courses. (4) Parental behavior and e xpectations play a role in ch ildrenÂ’s self-perception. (a) Absence of parental/family support tended to discourage participants from pursuing further education. (b) Presence of parental/family support te nded to encourage participants in pursuing further education. (5) Teacher behaviors and teaching methods matter. (a) Negative teacher behaviors tended to cause some to develop poor mathematics self-concepts. (b) Positive teacher behaviors tended to encourage some to persevere in understanding mathematics. (6) Feelings of powerlessness may impede learning mathematics. (7) Self-esteem can survive in spite of past failure. (8) Motivation to understand mathematical concepts remained high.
122 Profiles of the Participants The 15 participants who were intervie wed were all non-traditional age women enrolled at Reading Area Community College. These individual par ticipants included six White, six Hispanic, and three African-Ameri can students. There were no questions relating to ethnic background or race, neither was any notice taken or importance placed on these factors for the purposes of this study. Five of the participants had positive feelings about mathematics and ten describe d their feelings in varying degrees of negativity. Table 8 is an overv iew of the responses of the 15 participants interviewed in this study. The following section begins by identifying the voices in the stories that were shared and includes a brief description of each of the participants including an account of why they were seeking a college education and what challenges they were facing in order to provide perspective to the data reported in this chapter. Pseudonyms are used to protect identities and maintain confidentiality. Individual Participants Alice is a vibrant, enthusiastic student who had previously worked in several service sector jobs including retail and waitr essing. Â“I love working with peopleÂ…thatÂ’s what IÂ’ve always done.Â” She said she is pur suing a college educa tion because Â“whatever I do, I give my 100%. And I decided that, si nce I do give my 100%, I might as well get paid for my 100%; so thatÂ’s why I decided to get a degree. I left my job in order to be able to go to school. So IÂ’m looking at my school as my job. So I want to be able to give myself at leastÂ—anywhere from five to eight hours a day of studying, because thatÂ’s what I would be doing if I was working.Â” Alice is currently enrolled in the LPN program at
123 Table 8. Constant Comparative Analysis of Participant Responses Most/ Least Pseudonym Ethnicity #1* #2 #3 #4 #5 #6 #7 #8 *Key #1 College a personal goal? (Y/N) #2 High importance of study? (Y/N) #4 Parental support/encouragement? (Y/N) #3 Experience in past mathematics classes #5 Teacher behaviors important? (Y/N) P Â– positive memory #6 Feeling powerless? (Y/N) N Â– negative memory #7 Positive self-perception (efficacy)? (Y/N) Memory from: #8 Level of self-confidence/motivation E Â– elementary Y high level J Â– junior high N lacking high level of confidence H Â– high school L ALICE Hispanic Y Y N (E 4) N Y Y N Y L SHEILA White Y Y N (J) N N Y Y N L LISA Af-Amer Y Y N (E) N Y N Y Y L DOROTHY Af-Amer N N N Y Y N Y N L BARBARA White Y Y N (E 4) N Y N Y Y L SUE White N Y N (E 1) N Y Y Y Y L HEATHER Af-Amer Y Y N (H) Y Y Y Y Y L LAUREN White Y Y N (H 10) N Y Y Y Y L MARY Hispanic Y Y N (J 7) N Y Y N Y L CONNIE Hispanic N N N (E) N Y Y Y Y M ROSE Hispanic Y Y P N Y N Y Y M BRITTAN Hispanic N Y P Y Y N Y Y M COURTNEY White N Y N (E) N Y N Y Y M JENNIE Hispanic Y Y P Y Y N Y Y M DIANNE White Y Y P Y Y N Y Y
124 RACC and notes her main challenge to attain ing that degree is passing mathematics. Sheila is a motivated woman who has a va riety of work experience and is Â“very diversified,Â” as she describes herself. Â“IÂ’ve always been very successful in moving up the corporate ladder, so to speak, to some degree. And I get as farÂ…I believeÂ…as far as I can go without a degree.Â…I can never get past the next real leve l. ItÂ’s like somebody doesnÂ’t really take you seriously. So I just made the decision that, after my children had all grown up, that I was going back to school a nd pursue a degree and see if that does actually make a difference to where I want to go.Â…When you look at my resume, I have a really good variety of work experience, but ye t I canÂ’t seem to break into another salary level no matter how much experi ence I take into a position.Â” Sheila continues, Â“I spend a lot of time [studying] on weekends. IÂ’m tired. IÂ’m at the age where I want to get to work and put school behind me. ItÂ’s hard to juggle [family, work and school]Â…even though my children arenÂ’t little, itÂ’s st ill hard.Â” Sheila is currently enrolled in a business manage ment program at RACC and is afraid that mathematics will be difficult for her to master, therefore she has put off taking those courses until the end of her course schedule. Lisa is a woman of color, born in the Virg in Islands, who always wanted to be the first one in her family to get a college edu cation. After starting on coursework toward a business administration degree, she was for ced to quit due to personal problems. Â“Actually I wanted to have my own busine ss ever since I was in high school, but the business administration part got too hard fo r me.Â…One of my problems is taking notes [in mathematics class]. I us ed single sheets [of notebook pa per] and I would write down so many notes on it, but then I would forget to put them in the right order and they would
125 end up being mixed up in other material. I saw others had a spiral notebook and they were keeping their notes in or der. I plan on getting a spir al notebook for my math next time.Â” Despite being out of school for many y ears, including a period of time serving in the United States Army, Lisa is now enrolled at RACC in the culin ary arts program and failed in her first attempt to pass the mathematics requirement. Courtney is an LPN who feels that her options in the nursing profession are very limited and therefore is back in college to pur sue an RN degree. She is 50 years old and was overwhelmed with the amount of prerequi sites she needed when first beginning her program. Â“I determined that I need to study every dayÂ—and chunks of time, not cramming the night before [a test]. I realized I had to do itÂ—just c ontinually feed myself the information. For the basics of math course I just drilled myself daily. I learn by the visual and hearing, the two toge ther just work for me.Â” For Courtney, mathematics and chemistry are her main challenges in progressing toward graduation. Dianne was a factory worker for years be fore getting laid off. She is a single mother of three and had been out of school for eleven years when she took her GED and passed. She has positive feelings about mathematics and is pursuing an accounting degree at RACC. Â“I always wanted to get a degree in something, but itÂ’s hard when you have three kids. At first I wanted a degree in social services and then I looked at their incomes and things and I changed my mind. I decided to go into accounting. IÂ’m not a good note taker but I try to be. I try to write down the important things that I need to know. Eleven years is a real long time span of not going to school at all to going back to school to learn how to do notes and everything li ke that. ItÂ’s like be ing taught to walk all over again. ItÂ’s rough.Â”
126 Sue never thought that she would need a college education; however, after losing her manufacturing job, she decided to try her hand at early childhood education. She is a mother of two and a grandmother who t ook a test through Car eerLink at RACC and tested high in the area of working with ch ildren. Mathematics is an enigma to Sue because she experiences failure in trying to retain the information she learns in the classroom and tutoring sessions. She describe d her experience by sta ting, Â“I had one-onone tutoring and I could do the stuff in th e tutoring lab and then it just went somewhereÂ…I just couldnÂ’t get it for the test There would be problems put on the board and I would just go blank. I would just lose it. When asked about her study habits, Sue replied, Â“IÂ’m really conscious of getting my assignments done just like when I was in school, and I think IÂ’m really even more adaman t about it now then when I was in school. Things just have to be so. IÂ’m always second guessing myself and I do ask a lot of questions if I donÂ’t understand anything. Now IÂ’m finding out, after all my terms, that professors are approachable. Before I was re ally scaredÂ—my first two terms I was really scaredÂ—but now IÂ’m finding out theyÂ’re human and they do the same things and make the same mistakesÂ…and a lot of them sh are, so it makes you more comfortable.Â” Heather, an African-American student, cam e to RACC after ea rning a bachelorÂ’s degree in religious studies at another college. Unfortunate ly, she found that the degree she had did not help her in fi nding employment. She is curr ently finishing her two-year degree in general studies a nd has struggled with the ma thematics requirement. She describes herself as Â“a slow studier.Â” Â“Studying comes hard for me when I study by myself. If I study with someone itÂ’s easier, but if I study by myse lf, itÂ’s hard. I donÂ’t have that attentiveness. Something is l acking.Â…If we were sitt ing in a group, I wouldÂ—
127 nine times out of tenÂ—lead the group. But if I had to go and study that same information [by myself], I would have a difficult time w ith that.Â…Test taking has always been hard for me. I have always earned my grades. Everybody earns their grade, but I mean I spend a lot of time in order to earn my grades So thatÂ’s something thatÂ’s lacking in my brain or something. I donÂ’t know. IÂ’ve taken all kinds of study tips and clues and sat down with A students and done everything I can to try to enhance my techniques and itÂ’s still my problem.Â” Lauren spent many years doing factory wor k, raised a family, and says that now Â“itÂ’s my turn.Â” She works two part-time jobs a nd takes care of her grandchildren. She is at RACC to earn an RN degree and remembers th at her weakness in high school was always in mathematics. When this interview occu rred, Lauren had finally passed algebra I after her third attempt. Â“My study habits are probabl y not the greatest. But I put time aside for studying because I know that, for every hour of class, I should have two hours of studying at home. And I do work two part-time jobs. I am very glad that the tutoring center is at the college, because IÂ’ve used the math tutoring ve ry much. IÂ’ve felt like it was my second home at times.Â” Mary is a single mother with teenager s at home who never thought that she was college material. Â“I actually did not think about it (going to colle ge) in high school. Eventually I wanted to finish school, get a job, and just work. I was always scared to do more studying as far as school was concerned because I wasnÂ’t quite a smart person or someone who was able to think up the informa tion and put it into play like it was being taught.Â…When I study, I study during my lunc h hour and I also study when I go home after I make dinner. I try to stay focused because, when I first came back to school, I was
128 not that excited at first. But then, once I got in [to college] and I was learning, and teachers here took their time to actually work with me, I was able to understand what they were teaching me. So it became very im portant to me.Â” Mary is enrolled at RACC in order Â“to get a better jobÂ…to enhance my clerical skills, communication skill s, and that nature. I always wanted to do it (go to colleg e). I just didnÂ’t get around to do it. Every time I started school, I had to stop because I had a troubled teenager at home. But once that was all over with, then I finally enrolled and started coming to school.Â” Mary describes her in securities with mathem atics and school in general. She is at RACC in the office technologies program and ha s not yet been able to pass the GED test because of the mathematics section. She is hoping that the developmental mathematics courses at RACC will help her understand the ma terial well enough to pass it. She has put off taking those courses and says, Â“IÂ’m worried now because IÂ’m almost done with my degree and I fear that starting at the botto m of math and having to take three or four math classes scares me because, what if I donÂ’t pass the first one. I canÂ’t go on to the second one. You knowÂ…and then I thinkÂ…well, now if I have to go through all this, come June I probably wonÂ’t be able to gra duate. And that has me more scared than anything else. IÂ’ve done pretty good with all my other classes here at RACC, but math scares me to death.Â” Connie works full-time in a hospital as a nursing assistant, seven days on and seven days off. She leaves her classroom and goes straight to work. She says, Â“IÂ’ve been a nursing assistant for fifteen years, so IÂ’m tired of doing the di rty work. I want something different. I want to work in an office and just do something different.Â…I study however I can, because my life is very, very full. I do realize that time
129 management plays a very important role in school. Sometimes I have to study at work because I work seven days on and seven days off. When IÂ’m working, I leave right from school and go to work. So when I have dow n time at work, I have to study. When I come home, sometimes I have to study.Â” Connie is now working on a business management degree at RACC. She is plagued by past memories of school and describes them as Â“disheartening andÂ…never, never pl easantÂ…I never did math homework because I was very much ashamed because I didnÂ’t und erstand it and it seemed like I just could never get it.Â” Rose is a Hispanic woman who, as a young child, was taken away from her family and grew up as a ward of the state of Pennsylvania, shuffled from one foster home to another until getting married and starti ng her own family. Her dynamic spirit is evident when she says, Â“I want to have my children know that they have an educated mother. And I want to let them know that minorities can advance in lifeÂ…if you have the willpower, you can do whatever you want.Â” Ro se shared an uncomfortable situation in her mathematics class, Â“The students who ha d graduated from Reading High had already taken algebra there, but they were taking it again. They had more knowledge of it than I did in certain chapters of the book. I felt a little intimidatedÂ…and th en especially being older too.Â” Rose works as a bartender part -time and is at RACC in the general studies program. Despite her difficult upbringing, she ha s a great love of mathematics and tries to encourage others when th ey become frustrated. Brittan is a vibrant Hispanic student w ho worked in a factory for 17 years until it closed. Her children are grow n and her company gave her the opportunity to go back to school for retraining. She has chosen account ing as her major at RACC since she has
130 always liked mathematics. Â“IÂ’ve always want ed to be an accountant. But I had kids, and they were first, and I just continued to wor k.Â” Brittan dropped out of school in the ninth grade but never lost her desire to get an education. She is working on completing enough credits to get her high school diploma a nd is Â“whizzing through the mathÂ…I had no problemÂ…I picked up everything that I di dnÂ’t know before. So far I have a 97.5% average in my class.Â” Dorothy is a high school dropout who went back to get a GED later in life. It took her three years to pass the GED test beca use of the mathematics section and she says she has Â“blocked outÂ” all of her high school memo ries and is very afraid of mathematics. Â“I really want to learn it now. IÂ’m scared, but I want to.Â…Math, thatÂ’s my only main concern. I can read. I can wr ite. I can do a lot of things. But math, itÂ’s justÂ…Â” A busy mother of five girls, Dorothy is enrolled at RACC to earn a degree in counseling and aspires to be qualified to work with troubled teens someday. Jennie is finishing a degree at RACC af ter many years of Â“on and off,Â” taking classes and working full-time while raising a family. She enjoys mathematics and the challenge of earning good grades. Â“Yes, I en joy the challenge of passing my tests. Sometimes, when my class is in the morning, usually I get up earlier and I try to study one hour before I come to classes (because at that time my children are sleeping and they let me study). Sometimes when I finish my class and I have some timeÂ—half an hour, one hourÂ—between classes, I take a little ch ance there and I study a little bit. Finally, nighttime when my husband comes home at el even oÂ’clock and my children are in bed again, thatÂ’s when I do my homework.Â” She hopes to earn a high enough GPA to graduate with honors.
131 Barbara grew up feeling she was not a good student in high school and therefore would never be a good student in college. Af ter working in an accounting department for years, at the urging of her supervisor, sh e enrolled at RACC and is now pursuing an accounting degree. Â“I was encouraged by my s upervisor to do it (enroll in college). I think that he saw something in me that I didnÂ’t. I always had the thoughtÂ…both my sisters went to college and I always t hought about itÂ—or wished I hadÂ—but thought I couldnÂ’tÂ…because I wasnÂ’t a good student in high school, I just didnÂ’t think IÂ’d be a good student here.Â” Barbara is finding that he r experience is far di fferent than she had imagined and that she is succeeding in all of her courses, including mathematics. Â“I make sure I do my homework. I usually do that at my first oppor tunity. I think IÂ’m pretty good with my study habits. I thi nk I really apply myself well to study.Â” The following section presents eight major themes and six sub-themes that were evident in comparing and contrastin g the 15 participants in this study. Presentation of Themes and Sub-themes The themes and sub-themes presented are not meant to be representative of the thoughts and feelings of all non-traditional ag e women enrolled in a community college. What I purposefully set out to do was to be tr ue to the participants I interviewed. The constant comparative analysis (technique described in Chapter Three) of interviews with 15 participants revealed eight major th emes and six sub-themes (see Table 7). Theme 1: Acquiring a college education is a personal goal. Ten of the 15 participants described thei r motivation to enroll in the community college as a personal goal (see Table 8), not just to earn more money. Most of them wanted more rewarding jobs, offering secu rity for themselves and their children.
132 Alice described herself as the type of pers on who is not interest ed in doing things half-heartedly when she said, Â“I would have to say the reason I am pursuing a college education is because I feel like, whatever I do, I give my 100%Â…and I might as well get paid for [it].Â…I love helping people.Â” Sheila, Lauren, and Mary all waited for thei r turns to go to college. Sheila stated, Â“I just made the decision that, after my children had all grown up, I was going to go back to school and pursue a degree.Â” Lauren was emphatic when she said, Â“My children all went through college and I thought it was my turn. Even though my youngest thought it was a little too late to start now, I said, Â‘No, I donÂ’t think so!Â’ IÂ’m giving it a shot. And, IÂ’m amazed at how well my grades areÂ…because theyÂ’re better than when I was in high school.Â” Mary recalled, Â“Every time I st arted school, I had to stop because I had a troubled teenager at home. But once all that was over with, then I finally enrolled and started coming to school.Â” LisaÂ’s hope as a young woman wa s that she would be the fi rst in her family to get a college education. That hope was never realized, due to several personal set-backs, however, now she is renewing her dream and is moving through her program one course at a time. She works a part-time shift at a large, downtown hotel, wa lking to work in the dark at 5:30 a.m. each morning, then back hom e again, covering the ten-block distance to her modest apartment. Barbara watched as both her sisters went off to college while she did not. Â“I always had the thought [to go to college]Â…I wished I had [gone to college]Â…but thought I couldnÂ’t.Â…I wasnÂ’t a good student in high school, so I didnÂ’t think IÂ’d be a good
133 student here....I thought IÂ’d be the one always behind and not doing well. Kind of like high school all over again. But thatÂ’s not been the case.Â” Heather has made her choice to be a life long learner. She already had one degree when she enrolled in college for a second. Sh e shares, Â“I wanted to be more qualified for the challenge that I was facing.Â” Heather faced down an inst ructor who told her she was never going to be able to pass his mathematic s class and, thanks to her determination and many hours spent in the mathematics tutoring lab, she proved him wrong. Rose felt it was her duty, as a mother and member of a minority, to get a college education. Â“I want to have a better future. I want to have my children know that they have an educated motherÂ…and I want to le t them know that minorities can advance in lifeÂ…and thatÂ’s why IÂ’m doing it.Â” Jennie is in college to make her fath er, her husband, and her daughter proud of her. Â“I owe it to my fatherÂ…I want to give him that gift. Secondly, my husband, ...and I want to have my diploma hanging there to show to my daughter.Â” Five of the participants had other reas ons for coming to the community college, including needing better jobs or being fo rced out of their wo rkplaces due to global competition, requiring them to seek training in alternate careers. Theme 2: Adequate study time is necessary to understand and to retain mathematical concepts. When asked for a description of their study habits, 13 of the 15 participants recognized the need for time management skills in order to schedule adequate study time (see Table 8, p. 123).
134 Alice said she was Â“Â…looking at my school as my job. So I want to be able to give myself, at least, anywhere from five to eight hours a day of studying, because thatÂ’s what I would be doing if I was working. IÂ’m us ually either at home or IÂ’m here at the [math tutoring] lab. So I usually get about th ree hours a day in the la b and then the rest of the time I get at home studying.Â” Five to eight hours of studying seemed like quite a lot to me, until I heard HeatherÂ’s response to my question. Â“I study all day, all night, all the time (laughs). I like to study immediately after my courses. While itÂ’s still fresh in my mind I like to work on it. For me, the library is my study place. Wh at I do is rewrite my notes and questions. And I review the work periodically so that IÂ’m not cramming for exams. I think I take pretty good notes. I would say, pe r class, I study at least thr ee to four hours. They say you need one or two hours [for every hour in cl ass]. I need much more than what they say that you need.Â…IÂ’d need at least two times more.Â” It was clear that Sheila had given stu dying a lot of thought when she said she had Â“tried several different [study] techniques, a nd what works for me is to go over it all and then I stop. And I come back the same day and go over it a little bit again.Â…I absorb more if I continually go back and go over itÂ…the more I see it, the more it registers for meÂ…the more times I see it, itÂ’s almost photographic in my mind.Â” Barbara emphasized, Â“I make sure I do my homework. I usually do that at my first opportunity. I think IÂ’m pretty good w ith my study habits. When I get to school early, I get it out. I look over it again. And before a test IÂ’ll probably go through all the chapters and do those reviews. So I think I really apply myself well to study.Â…like over
135 the weekend, maybe IÂ’ll sit there for an hour, and then IÂ’ll go outside and walk aroundÂ…play with the dog or someth ingÂ…then IÂ’ll come back to it.Â” Sue said she was Â“really conscious of ge tting my assignments done just like when I was in school. And I think IÂ’m really even more adamant about it now than when I was in school. I ask a lot of questions if I donÂ’t understand anything.Â” When asked specifically about mathematics material, she shook her head, Â“With math it was justÂ…constantly I had to be at it. I did th e CDs. I was in the tutoring lab. Anytime I wasnÂ’t in class, I was in the tutoring lab.Â” CourtneyÂ’s response was much like SueÂ’s. Â“Quickly I determined that I need to study every dayÂ…and chunks of time, not like cr amming the night before. I realized I had to do itÂ…just continually feed myse lf the informationÂ…with the CDs on the computer or reading, or whatev er. Just everyday I would do a little bit.Â…Those CDs that come with the books reinforce the learning. I learn by the visual and hearingÂ…the two together just work for me.Â” Lauren, although working two part-time j obs, found time to study. Â“I put time aside for studying because I know that for ever y hour of class, I should have two hours of studying at home.Â…I try to put an asterisk on my notes where a teacher emphasizes something and IÂ’ll jot that down maybe at the top of my paper when I rewrite my notes so that it stands out for me. I canÂ’t just hear it and then have it I have to re ad it a couple of times. I do take notesÂ…I have to. And thatÂ’s very important.Â” Mary took her homework along to work. Â“When I study, I study during my lunch hour and I also study when I go home after I make dinner.Â”
136 Rose wanted a Â“silent placeÂ” to study. She studies Â“at night, because itÂ’s always quiet. [I study in] small bits, because I ge t a little wound after st udying long periods of time, so I break everything down into certain times.Â” In contrast to Rose, Jennie said she Â“needs noise.Â” Â“I either need the TV on or the radio.Â” She steals whatever amount of tim e she has to study. Â“Sometimes, when the class is in the morning, I get up earlier a nd I try to study one hour before I come to classÂ…because at that time my children are sleeping and they let me study. Sometimes when I finish my class and I have some timeÂ…half an hour, one hour between classesÂ…I take a little chance there a nd I study a little bit. Fina lly, nighttime when my husband comes home at eleven oÂ’clock and my child ren are in bed again, thatÂ’s when I do my homework.Â” Brittan surprised me when she said, Â“We ll, to be honest with you, I think I study too muchÂ…only because I havenÂ’t been in school for so long and there is nobody home.Â…I have a computer room and I sit in there and IÂ’m consta ntly, constantly doing homework. I always pushed my kids, Â‘You ha ve to do good.Â’ So now itÂ’s my turn and now itÂ’s like, Â‘I have to do good.Â’Â” Two of the participants re sponded to my study habits questions a bit defensively. Connie reminded me that Â“I study when I can, because my life is very, very full.Â” Likewise, Dorothy talked about her busy, noisy household and said sh e liked studying but Â“after while I get bored.Â” Theme 3: Experiences with mathematics at an early age remain in oneÂ’s memory. Eccles (1989) argues that initially, boys and girls ar e alike in their perceived mathematical capabilities, but gi rls begin to lose confidence in their mathematics ability
137 as they move through the lowe r grades into high school. Wh y should this be the case? Poor experiences with mathematics at an ea rly age undermined confidence and increased mathematics anxiety for several participants. Sub-theme (3a): Poor experience with ma thematics at an early age tended to make participants believe they coul d not learn mathematics. Of the 15 participants interviewed for this study, 10 participants had memories of poor experiences dating back to their years prior to community college enrollment (see Table 8, p. 123). Alice said her earliest memo ry of mathematics was in four th grade. Â“I donÂ’t think it was a good memory. I didnÂ’t gr asp it. And, I wasnÂ’t made to learn it. They just said, Â‘OK, just move her to the next grade.Â’Â” Sheila remembers, Â“Math was never a good subject for me. I really didnÂ’t struggle until the eleventh grade. I went to vo-techÂ…half a day we went to vo-tech and the other half we went to the high school. But, because I was majoring in chemistry, I had math at vo-tech. But I decided, when I wasnÂ’t doing wellÂ…and the chemistry teacher said to me, Â‘You know what; youÂ’re doing grea t in theory, but I donÂ’ t think youÂ’re gonna pass this because you canÂ’t get the math.Â’ So I figured IÂ’d go back to Reading High and just do general courses to just meet the math requirement.Â” (Intervi ewer note: At that defining moment, Sheila gave up her desire to pursue the field of chemistry.) Lisa remembers feeling Â“left behind in my math somewhere in elementary school.Â…In high school I had a teacherÂ…and he taught algebra. He taught OK, but, to me, the book was hard. When he taught, itÂ’s li ke you have to learn that math in the same day he taught itÂ…because youÂ’d do the homework, and heÂ’d teach you something else the
138 next day. That was hardÂ…it was hard on meÂ…a nd IÂ’m not even sure I really got it. I could have gone on to trigonometry, but I c hose not to go on in mathÂ…I took a vo-tech course instead of taking math. I was just tired of mathÂ…it was so difficult. I just needed more time with it and somebody who really cared about me.Â” Courtney, when recalling her first me mories of mathematics, said, Â“Yes, I remember having to count the haystacksÂ…or bundles of tensÂ…those confused me so much. And itÂ…forget itÂ…plus I was a year youn ger than the rest of my classmates. I donÂ’t know if that was why, but I just couldnÂ’t get the concept of those bundles. I felt a lot of anxietyÂ…performance anxietyÂ…big time.Â” Sue remembers Â“being in first grade. I can remember my teacherÂ…she was great with the ruler. And if you didnÂ’t add right, you got smacked on your hands with the ruler. And then every teacher along the wa y would try and sit and help meÂ…and itÂ’s likeÂ…the foundation wasnÂ’t there. Then, when I was in fourth grade, we moved from one school to the other and they were a litt le farther ahead, and I just felt like I never caught up with thatÂ…I always seemed like I was behind.Â” Lauren shares an experience in the tenth grade. Â“I had a bad experience in tenth grade. I got the answer that he (the teach er) wanted, but I did it my wayÂ…but he wanted it in algebraic formÂ…and I just could not understand how to break it down and put it in algebraic form. He called on me and made me go up to the front of the class to the blackboard. He had a problem on the board and I put down the answer and I turned around and sat down. And he made me come ba ck up in front of the class and asked me why I did that. And the diffe rent questions he asked me Â…the kids were laughing at meÂ…and I said thatÂ’s the only way that I understood how to get the answer. And he
139 literally made an ass out of me. A nd heÂ…I donÂ’t know what you want to call itÂ…badgered me, belittled me, to the point wher e I stood there and cried. And I just had a problem with algebra from there onÂ…nothing was clicking for me.Â” Barbara recalls a ninth grade algebra cla ss, Â“I just failed miserably. I donÂ’t know what my grade was, but it was horrible. I remember the teacherÂ…and I donÂ’t know if she didnÂ’t want to bother to help meÂ…I donÂ’t know. I couldnÂ’t get her to help me. I remember her being up at the board talking but I probably just tuned out because I didnÂ’t understand what she was saying.Â” Five of the 15 participants either lack ed any negative memories from their past schooling or had positive memories to report. Dorothy, for example, was not able to recall any memories relating to mathematics. Even after attempting to prod her to remember either a classroom experience, a par ticular instructor, or any experience at all, she only replied, Â“I donÂ’t remember, I mean, maybe I block it out (laughs), but, I donÂ’t knowÂ…IÂ’m scared of math.Â” Sub-theme (3b): Positive experience with math ematics at an early age tended to provide participants a higher degree of self-efficacy in succeeding in mathematics courses. Four of the 15 participants had positive me mories of learning mathematics. Rose remembered her love of fractions in elemen tary school. Â“I love fractions. I donÂ’t know why, but I love fractionsÂ…division, multipli cation, addition, subtractionÂ…algebra?Â…soso, but if I work hard enough, I can advance in it as well.Â” Brittan said she Â“always liked math. Al ways, always. But I donÂ’t remember any of my teachersÂ’ names. But I was always ge tting good grades in math. I know that there was a teacher there but thatÂ’s all I know (laughs).Â…IÂ’ve always liked learning math.
140 When the kids were in schoolÂ…when they n eeded help with their homework, [I helped them]. I also do a lot of budgeting with different [friends of mine]. [If] they canÂ’t make it with the money that theyÂ’re making, I will go to their house, and fix it up for them. I helped out quite a few people.Â…It gives you a good feeling. Â…And I started thinking, Â‘Now, wait a minute! I should be getting paid for something like this (laughs)!Â” Some of JennieÂ’s memories of mathema tics were related to her father, who sat down with her and was Â“the first one who show ed me the math.Â” She also remembered a Chinese teacher, Â“I loved the way he taught. And IÂ’ll always remember him.Â” Earlier in her life, she recalled, Â“When I was back in my countryÂ…me and another guyÂ…we were the smartest in the classroom. She (our te acher) invited us to go to her house and she would teach us what to do. So he was the sm artest one of the guys and I was the smartest one of the girls. So the teach er used to teach us, and then we used to go back to the classroomÂ…we used to tell them in our [italics added] way.Â…I felt proud.Â” Dianne remembered that she Â“always enjoyed math. I remember one day we were learning numerators and denominators and that just came so easy to me. Those were one of my favorites and I remember my teacher teaching that. I always had good grades in math. But you put me in English or soci al studies and I wasnÂ’t good with that.Â” Theme 4: Parental behavior and expectations play a role in ch ildrenÂ’s self-perception. It was noteworthy that only five of the 15 participants gave positive responses to my question, Â“How does (or did) your pare nts feel about your math performance in school?Â” (see Table 8, p. 123)
141 Sub-theme (4a): Absence of parental/family support tended to discourage participants from pursuing further education. Alice recalls that her parents were no he lp in the academic area. Â“I donÂ’t think that our education was an inte rest of my momÂ’sÂ…not so much that it wasnÂ’t an interest but just that she didnÂ’t have time. She ran her own business at the time to keep a roof over our heads and she just didnÂ’t have the time to help us. So there was no one there to help.Â” Sheila felt her parents ju st didnÂ’t care. Â“Honestly, a nd I think it reflected [in my attitude]Â…my parents didnÂ’t care one way or the other. My mo m only had a seventh grade education and didnÂ’t know to get involved. My fath er, on the other hand (I was the youngest out of four brothers), Â…I think I cam e along at a time in my familyÂ’s life when, for my father, it wasÂ—I donÂ’t want to say it was a choreÂ—but it was just, Â‘Quick, sheÂ’s the last one, letÂ’s get her on her way and if sh e just gets throughÂ…Â’ I can recall that my father worked as a draftsman so I automa tically assumed that he was very good with numbers, and he probably was, but just a poor teacher. And he could not teach me math. He used to get extremely frustrated and ther e would be many times when he would just slam the book closed and say, Â‘I canÂ’t do it!Â’ You know, he would get frustrated. And I think subconsciously thatÂ—to th is dayÂ—that has led me to belie ve that I donÂ’t like math. Psychologically I feel like IÂ’m never goi ng to be good at math. If you donÂ’t get encouragementÂ—you donÂ’t need a tickerta pe paradeÂ—but you need some kind of reinforcement.Â” Lisa, Mary, Connie, and Rose also ca me from families that lacked formal education. Lisa stated, Â“My mother only had a fifth grade education. She tried to go back
142 to school, but, for some reason, she stopped. She never really attempted to teach me anything.Â” Mary was fortunate enough to ha ve an older sister to help her get her homework done. Â“My parents, they dropped out of high school real early. They had to go to work back in Puerto Rico so they didnÂ’ t get a whole lot of education. They would ask my sister to help me.Â” Connie said he r parents Â“didnÂ’t even complete high school. So thatÂ’s a motivation for me to keep going on. I want my children to go to college. I donÂ’t want them to have to struggle. I donÂ’t want them to have to go through some of the things that I had to go thr ough. I was never encouraged at home. Â…They (parents) couldnÂ’t give me what they didnÂ’t have.Â” Ro se couldnÂ’t look to her parents for help. Â“When I studied, I studied by myself. My parents werenÂ’t very knowledgeable with school. I was an honor student in school. I was actually showing my parents how to do this stuff.Â” When asked about her parentsÂ’ behavior, Courtney regrettably shared that, Â“They kind of just told me I was stupid. Â‘You know you can do better.Â’ But I didnÂ’t really have the tools to do better. I didnÂ’ t know how to get better with what I was doing in school.Â” Sue recounted her memory of her parents by saying, Â“M y mom had mental problems and my stepfather cared for her most of the time. I just felt education was just never a big pointÂ…it was, Â‘YouÂ’re going to work in a factory like your mother did and youÂ’re gonna get married and have babiesÂ…and, what do you need an education for?Â’ And they had no patience. My stepfather taught me to drive because my mother said she wouldnÂ’t have any patience. To the day she died, the woman never drove in the car with me. So thatÂ’s the way things were. Nobody ev er said, Â‘Have you considered education?Â’ It was like, Â‘OK, youÂ’re going to get good gr ades, but yet weÂ’re not going to give you
143 time to do your homework and get your homework done right, because youÂ’re going to have to take care of the kids so that I can ta ke my medicine and go to bed.Â’ So I really didnÂ’t have the time that I could have put in to be a really good st udent when I was going to school.Â” Lauren said her parents Â“had no clue at all what algebra wa s all about. And I didnÂ’t know how to explainÂ…and ma ybe I got some feedback from them as to what they thought about that kind of math, which kind of stuck with me alsoÂ…their negative view of it. Â‘YouÂ’re not going to need that in a factory.Â’Â” Barbara said that her mother Â“just th rew her hands up and didnÂ’t know how to help me. And I just fell by the wayside all the way around when it came to math. I canÂ’t remember my dad ever doing any kind of homework with us.Â” Sub-theme (4b): Presence of parental/family support tended to encourage participants in pursuing further education. In contrast to the 10 participants who shared there was no parental help given, JennieÂ’s experience was positive. Â“My father used to sit down with me. He used to show me the Roman numbers. My father showed me that until I could do itÂ…division and all that stuff. My father was the first one who showed me the math. My father educated me from the beginning, since I was real sma ll. He is very proud of me now.Â” Dorothy remembers that her parents Â“were constantly always after me. Both my mom and dad said, Â‘You need your education to get ahead.Â’ Also my older brother and sister would help me [with math]. I might have gone to them for math help, but I canÂ’t remember much of it.Â”
144 Heather recalled that she di d not feel any pressure to perform in school. She only spoke about her mother, who Â“thought she was good at math (laughs) but she wasnÂ’t all that hot (laughs). She tried to help me with long divisionÂ…and she stayed up all nightÂ…and it was the wrong answerÂ…we s till laugh about it to this day.Â” Brittan and Dianne both said that alt hough their parents were proud of them as students in school, it was thei r brothers and sisters who provided encouragement when it came to education in general. Theme 5: Teacher behaviors and teaching methods matter. Responses of the participants in the st udy revealed a common belief that a teacher can make a difference, that good teachers can build confidence in mathematics ability and help relieve anxiety, while poor teachers can have a devast ating effect on students. Sub-theme (5a): Negative teacher behaviors tended to cause some to develop poor mathematics self-concepts. Connie stated, Â“I know that the instruct or plays a major role in the learning process for me. A lot of instructors figure you ge t it or you donÂ’t. ItÂ’s so frustrating to a student who doesnÂ’t get it to have that look. Body langua ge plays a major role in teaching. I mean, if you look at someone lik e theyÂ’re crazy because they donÂ’t get it, how to you think that makes that person feel if they already have no self-esteem in that area of their life? I donÂ’t have a problem with instructors be ing on top of me. Of course, thatÂ’s not their job, but sometimes it helps. Just because weÂ’re adults, doesnÂ’t mean that we sometimes donÂ’t need that extra push.Â” Sheila recalls her junior hi gh teachers, Â“Believe me when I tell you, when I got to junior high, the teachers, IÂ’d ha ve to say, werenÂ’t great. Th ey were nothing to hoot and
145 holler about, believe me. And I didnÂ’t just th ink that then. As IÂ’ve gotten older and was able to go back and look and thinkÂ—are some people just like that? Are some teachers just like, Â‘Oh, go away,Â’ and tune you out, so to speak? If you wa nted to go into the classroom and put your head down for the en tire period, you could put your head down. It didnÂ’t phase them.Â” Upon entering the community college, She ila encountered a challenge relating to her mathematics instructorÂ’s teaching method. Â“I felt like we were kind of all over the placeÂ—chaotic and not methodical. We were in this chapter and then we were in the back of this chapter, but then we would come to the front of this chapter. And, in my mind, things go like this (partici pant gestures), so when you go to the back of the chapter, IÂ’m already out of sync. I want to know, Â‘W hy are we back here?Â’ To me, if that was part of the chapter, shouldnÂ’t it have been at the beginning? I got ve ry disorganized, very out of sync. I would ask questions and I would only know how to ask them how I was processing them in my head. They may not ha ve made sense to that instructor, but it was the best way that I could communicate where I was in the problem. And itÂ’s hard when youÂ’re trying to communicate like that a nd the person youÂ’re communicating to isnÂ’t getting you. So now it becomes an even more frustrating situation.Â” Lisa spoke about a teacher sh e had in ninth grade. Â“She could have simplified the learning. All she did was follow th e bookÂ…whatever the book taught.Â” Barbara recalled a teacher in a ninth gr ade algebra class. Â“Oh I just failed miserably. I donÂ’t know what my grade was, but it was horrible. I remember the teacher, her name was Mrs. P. And I donÂ’t know if she didnÂ’t want to bother to help meÂ…I donÂ’t know. I couldnÂ’t get her to help me.Â” When I asked Barbara if she had actually gone to
146 the teacher to ask for help, she replied, Â“Yes, and I just rememb er getting an answer that I didnÂ’t really understand, so I just gave up. I remember her being up at the board talking, but I probably just tuned out because I didnÂ’t understand what she was saying.Â” Mary blamed a teacher she had in tenth grade for insisting that students come up to the board and do the math. She failed to get the correct answer Â“so he had someone else come up and do it.Â” Mary felt her le arning experience coul d have been more constructive if the teacher, Â“knowing that I didnÂ’t know how to do it, maybe working with me and showing me, instead of having so meone else do it.Â” After that experience, Mary never asked for help because Â“as far as math was concerned, I just felt that I couldnÂ’t do it at all.Â” Sue related a conversati on she had with her mathematics instructor at the community college. Â…Â”It doe snÂ’t look like youÂ’re going to get anywhere, Sue. If you want to keep coming to class, fine. But the way you are in class, IÂ’m really worried about you. I can see when you come in [to class], you get so nervous. ItÂ’s not that youÂ’re stupid; some people just cannot gr asp it.Â” Sue shared that she wa s Â“really afraid of her. I donÂ’t know why. I didnÂ’t really know anything about her. And it seemed for each math teacher that was the way it was.Â” Earlier in her life, Sue had a teacher tell her, Â“YouÂ’re never going to be good in math, no matter how hard you try, youÂ’re never go ing to be good in math.Â” SueÂ’s earliest memory of math was in the first grade with Miss F. She remembered that her teacher Â“was great with the ruler. And if you didnÂ’t add ri ght, you got smacked on your hands with the ruler.Â”
147 Heather recalled an instructor who Â“didnÂ’ t seem too approachable. He actually discouraged meÂ—or tried to. He made a st atement in the class, Â‘Those of you that havenÂ’t gotten it this far, you might as well fo rget it, because youÂ’re going to fail.Â’ That was a motivating factor for me. I thought, Â‘How can you stand up there and say that? The class is not over. Hey, itÂ’s not over until itÂ’s over.Â’ I didnÂ’t think he had the right and I was determined not to be one of those people that he had crossed off.Â” Lauren described an instructor she enc ountered at the commun ity college. Â“IÂ’ve had prealgebra with a teacher and she was very demanding. She said, Â‘Well, if you canÂ’t get it here in class, then go down to the tutoring center, that what itÂ’s for.Â’ I knew how long she had been teaching and I thought maybe she needed a break. I donÂ’t think thatÂ’s the way youÂ’re supposed to talk to your st udents to make them feel good about their math. Yes, the tutoring center is there to he lp us, but I feel that the teacher should have taken a little more pride in the students tryi ng to get it while theyÂ’re in her class.Â” Sub-theme (5b): Positive teacher behaviors te nded to encourage some to persevere in understanding mathematics. Alice remembers a teacher who Â“would encourage me and tell me, Â‘Oh Alice, youÂ’ve got a mind like a trap. YouÂ’re just really good at me morizing things.Â’ And that just encouraged me to want to do it.Â” Connie remembered an instructor who Â“was just so calm. He wasnÂ’t all over the place. He was just so calm and explained things in a way that I understood it. Sometimes I have to go step by step even with simple thingsÂ…it has to be step by step.Â” Rose spoke of a teacher who Â“always t ook the time to show you how to do math andÂ…especially with algebra. That was my hardest subject, but she showed me how to
148 understand negatives and positives and all that She was a real good teacher. She was very patient and she explained everything st ep by stepÂ…and she made everything (long pause) clear.Â” Brittan shared that she was very plea sed with her mathematics instructor at RACC. Â“There was never one day that I went to the math class that I wasnÂ’t happy with the way she taught me. I understood everyt hing. She made it so much easier to understand.Â…And because I sit up front, she was quick to point to me to answer questions, which was nice. Sometimes IÂ’d be scared, thinking, Â‘Oh, IÂ’m not sure.Â’ But, for the majority of them, I answered them right.Â” Courtney remembered a teacher whom she especially liked. Â“He was really cool. He was a great teacher. He was just relaxed in the way he taught. He was not, Â‘You got to get thisÂ…got to get this.Â’ He didnÂ’t pick and choose whom he thought was better.Â” Sue described a teacher, Mr. M. who had a positive effect on her. Â“I sat in the back of the class the first couple weeks and, at the start of the one class, he said, Â‘You will sit up front, and you will sit up front th e rest of the year, and you will do your homework, you will pay attention, and you will k eep your mouth shut so you can work.Â’ I did good that year and he would help me.Â” There was another teacher that stood out in SueÂ’s memory. Â“I had one teacher who would spend at least five minutes with each child. Â‘Did anything happen in the home that youÂ’re upset about? Did you do all your homework?Â’ Then, as you get older, that Â’s lost and you donÂ’t feel like youÂ’re an individual. And there are so many other issues with teachers that they donÂ’t have time [to spend with each student individua lly]. ItÂ’s not their fault.Â”
149 Lauren struggled to pass her college in troductory mathematics course and, after her third try, finally succeeded. She attributes her success to her instructor. Â“Mr. F. was very, very good. I canÂ’t say that IÂ’ve ever ha d a math teacher that was as patient as he was. And, if you couldnÂ’t get it one way, a nd you had a puzzled look, heÂ’d say, Â‘Let me try this another way.Â’ And he would show the problem another way and IÂ’d think, Â‘Now that way I can get it.Â’ And that was wonderf ul. He took the time. I respect him and I like him. He had homework assignments that we could do and he always said if there were any problems with homework, weÂ’ll go over it in class. He was very relaxed, very positive, very open. He would talk to us, not down to us, not at us. He had a very soft voice. I donÂ’t think he had an anger spot in him. He was just very, very kind, and he understood where some of us were coming fr om that were strugg ling. But he also understood the other ones that were catching it an d were right with him in class. He was just very understanding. He told me, Â‘I enjoyed having you in class and just give yourself time, this will comeÂ…you will get it.Â’ I think he really helped me even out the negative experience I had in tenth grade.Â” Dianne liked when her mathematics teacher in elementary school would have contests in the classroom with multiplication flash cards. Prizes were given out to the winners. Dianne recalls Â“I wa s fast at that. I know I won a couple of times. I enjoyed the competition.Â” Theme 6: Feelings of powerlessnes s may impede learning mathematics. Although only seven of the 15 participants spoke of feelings of powerlessness regarding their lack of unders tanding of mathema tics (see Table 8), it is included as a
150 common theme since the particular stories of participants who described such feelings were extremely compelling. AliceÂ’s interview revealed three seriou s issues which render ed her powerless. First, she regarded asking for help with he r mathematics as a sign of weakness. As a result, she neglected going to the math lab fo r help until it was too late. She recalls thinking, Â“TheyÂ’re going to think youÂ’re an idiot because you donÂ’t know it.Â” In addition to her reluctance to ask for help, Alice felt her classroom was too crowded. Â“And that made it very stressful for me, because I just fe lt like there just was no room in the class. You just felt likeÂ…you were justÂ…just so overcrowded. I donÂ’t know, it just made it hard for me to take in the information because the room was so crowded. I felt overwhelmed.Â” Third, Alice f ound the presentation of mathema tics material very rushed. Â“I think it was very, very rushed. Â…Like I ne ver had the algebra before. It was all new to me. That made it more stressful because you didnÂ’t know what to expect.Â” Because Alice was hindered by her fear to ask for help and because she did not have any control over the size of the class or the speed with which the materi al was presented, she was, as she saw it, powerless to change her circumstances. Whenever a test was scheduled, Sheila w ould rehearse a traumatic scene that had occurred years before, where her father, in frustration, would slam the math book closed because she didnÂ’t understand what he was tryi ng to teach her. Â“I think subconsciously thatÂ…to this dayÂ…[that incident] has led me to believe that Â…IÂ’m never going to be good [at mathematics]. So many things go through my head as IÂ’m studyingÂ…you know, the mental block.Â” SheilaÂ’s debi litating memory of that scene with her father caused her to
151 experience a mental block, particularly when studying for a test, and she felt powerless to move beyond it. Sue allowed incidents with fellow students to affect her self-confidence and selfefficacy. She explained, Â“There was a girl in my class th at sat behind meÂ…and the class hadnÂ’t really started and [my instructor] was standing in fr ont of my desk and she was asking how I was doing. I said I didnÂ’t th ink I was doing very well.Â…I donÂ’t know how the girl behind me got involved, but she sai d, Â‘Well, I surely w ouldnÂ’t want an F on my [italics added] transcript.Â’ And I turned around and said, Â‘Well, do you think I [italics added] do? IÂ’m sorry if IÂ’m not as smart as you are.Â’ After class a nd the girl had left, I was trying to go out of the room quick and [my instructor] said to me, Â‘IÂ’m really sorry. You know, not everybody can get math.Â’Â…And th en I was a little ups et because one of my bosses was taking algebra with meÂ…and he was getting it and I wasnÂ’tÂ…and I thought, Â‘Uh-oh.Â’Â” These incidents, on top of other previous occurrences where she was up at the board and Â“got an answer wrong a nd a couple of kids laughed,Â” caused Sue to lower her expectations that she could understand the mathematics. When I asked Sue how she might have turned these negative experiences into something positive, she replied, Â“I donÂ’t think ther e was anything positive to come out of any of my experiencesÂ…there was no way I could have tu rned it around. I didnÂ’t know the answer to the math problemÂ…I had multiplied wrongÂ…and the girl behind meÂ…it was just the way she was so out front with itÂ…and nobody wa nts to have an F on their transcriptÂ…and she really wasnÂ’t involved. She was just, more or less, listening, and I think thatÂ’s what really bothered me.Â” When one is convinced th at there is nothing to be done to change a situation, that individual, for a ll practical purposes, is powerless.
152 When instructors use their power to embarrass a student, the result can be devastating to the student involved. Laur en experienced extreme embarrassment when, in tenth grade, an instructor used his power to humiliate her in front of the entire class. Â“Instead of making me feel like a total jerk in front of the class, he could have said, Â‘If you donÂ’t understand it, come over to my desk af ter class and weÂ’ll di scuss it.Â’ I guess I could have been more aggressive and just s it down and not stand up there and take it, but that wasnÂ’t the way I was raised. So I stood there and I was trying to answer his questions, and the more I tried to answer Â…I was just over my head.Â” LaurenÂ’s powerlessness to deal with that mathematics teacher still invades her current feelings about mathematics. Maria was another particip ant who was embarrassed by an instructor in tenth grade. Her mathematics instructor used the strategy of calling students to the board. Â“I had an embarrassing experience going up to the boardÂ…being told to do a math problem that I didnÂ’t know how to do. That was some thing the teacher did all the time. He wanted the kids to come up and do the math on the board. I was embarrassed and went back to my seat. So he had someone el se come up and do it and show me how it was done.Â” When I asked Maria how that experien ce might have been turned into a positive one, she replied that she felt the instructor wa s unapproachable. When a student sees the instructor as unapproachable, a wall is erecte d between that student and the instructor. How much learning can take place in such a situation? Connie expressed the confusion she expe rienced when the steps to finding a solution to a mathematics problem were expl ained in more than one way. Â“I knew the math. It just gets so confusing in my head when I take a test. And I know what I was
153 doing, but I allowedÂ…if I know how to do someth ing already, itÂ’s not a good idea for me to allow somebody to show me a different wa y. And I know I tried to perform something on the test [in a way] that I wasnÂ’t really familiar with and it cost me points, which caused me to get kicked out [of the LPN progr am]. Connie realized that, Â“The teacher plays a role, but the initial part I have to play. I have to perform. I have to give it back to you. If IÂ’m not understanding it, I canÂ’t give it back to you.Â” Connie saw herself as powerless to keep new, unfamiliar ways of solving the mathematics from confusing her previously learned method. Theme 7: Self-esteem can survi ve in spite of past failure. In attempting to identify a level of self-esteem which was characteristic of the 15 participants interviewed for this stud y, the question was asked, Â“Would you define yourself as being smart?Â” Twelve gave a positive affirmation to that question. Only three felt that others were much smarter th an they. Overwhelmingly the findings seem to uncover the phenomenon that, ir regardless of academic perf ormance, the participants deemed themselves to be smart. In virtually all cases, smartness [italics added] was attributed to various accomplishments, ot her than academic, that these women had experienced. Sheila asked me to, Â“Define smart. There are smart people in the world that are very booksmart and areÂ—I donÂ’t want to say stupidÂ—but lack common sense, so I donÂ’t consider them smart. IÂ’d like to think of myselfÂ—and I donÂ’t want to sound egotisticalÂ— but I am smart. I have some common se nse and I have some book knowledge. IÂ’m not a bookworm; IÂ’m not an Einstein, but IÂ’ d like to think that IÂ’m smart.Â”
154 Courtney proudly answered, Â“Smart? Yes, smart enough to get the grade IÂ’m getting here at RACCÂ…you got it! I learnedÂ—discoveredÂ—my learning style. As soon as I started college here, I discovered how I c ould learn better. Before that, I didnÂ’t know if I was smart enough to do it. But I thought that this was just the ri ght time and the right place...the right time in my life to try.Â” Dianne qualified her affirmative answer by saying, Â“IÂ’m smart in some ways; not smart in other ways. IÂ’m smart as to coming in to my job and with my kids. IÂ’m not stupid or I wouldnÂ’t be where I am right now.Â” Sue also said she was smart Â“to a point. I seem to be better since I came to college because I didnÂ’t know I ha d a lot of this in me. And I did make the deanÂ’s list! I didnÂ’t apply myself while I was in high sc hool because of family issuesÂ…so I didnÂ’t really give it my all. But I now realize that I knew a lot of things. IÂ’m finding out now that a lot of young people that are just comi ng out of high school really donÂ’t know very much. So I donÂ’t feel that IÂ’m really, really smart, but si nce IÂ’m coming to college, IÂ’m more aware of knowledge and how to get it. There are two ways of being smart. You can be actual smart that God gave you or you can be smart where you can learn yourself. And, in order to be smart, you have to know how to get that intelligence. One way is to ask questions and read.Â” Heather felt she was Â“pretty knowledgeable and pretty resourceful. IÂ’m a good learner although IÂ’m slow at comprehending. So if you give me an opportunity, given the time, I can understand a lot of things. I not onl y retain the material, but I think I apply it too. And that, to me, is the most important th ing. But a lot of times that doesnÂ’t occur so it hinders me in life, due to the fact th at the time and opportun ity is not there.Â”
155 Lauren said, Â“I like to think that IÂ’m above average. It doesnÂ’t come easy for me. I do have to work at it. When you say, Â‘smart,Â’ I donÂ’t know exactly which way you mean itÂ—if itÂ’s booksmart or streetsmart. IÂ’m an older student coming back to school and IÂ’ve already raised a familyÂ—I have gr andchildrenÂ…so I think in some instances I may be smarter than some of the younger ones in college. If you ask me if IÂ’m smartÂ— booksmartÂ—IÂ’d have to say that I am above average, yes.Â” Connie responded, Â“Yes, I believe that I am very smart. I have problem-solving skillsÂ—maybe not for mathÂ—but I have them fo r life. Like, if this doesnÂ’t work, you donÂ’t stay stuck in the closet, you move on to the solution. IÂ’m very smart; my husband told me so.Â” Rose found that question humorous and repl ied, Â“I am smart, yes. I have a head on my shoulders. I already have a year in co llege so, if I wasnÂ’t smart, I wouldnÂ’t even have completed a year. I have a lot of know ledge in a lot of areas, and I never thought I had thatÂ…and I doÂ…thanks to college.Â” Dorothy and Brittan had similar responses They both said they were smart Â“in particular areas.Â” Dorothy went on to say, Â“I guess throughout life you learn a lot, especially when you have five girls. You go tta be smartÂ…you gotta be on top of things. So I guess in that way I consider myself smart.Â” Jennie shared, Â“I want other people to tell me IÂ’m smart, not me saying it. I donÂ’t want to show off. Oh, IÂ’m smart. I understand a lot.Â” Barbara also answered affirmatively, but went on to say, Â“I never used to [think I was smart], but yes, I am. I get good grades. I donÂ’t do foolish things in my personal life or financially or anything lik e that. I think IÂ’m pretty heads up and common sense.
156 When I was in school, I didnÂ’t feel that way at all. I have two younger sisters and theyÂ’ve both gone to college. My one sister actually is a PhD and she has two masterÂ’s degrees. I was always the Â‘not smart one.Â’ I was always better at baking pies a nd things like that. And so I thought that was my nicheÂ…not being smart.Â” The three participants who responded negatively to my question gave their reasons. Alice felt, Â“No. It takes a lot to get into my headÂ…to absorb the information. IÂ’ve got to go over it and over it and over it Â…it takes me longer than it does the average person, I think. I would define a smart person as one that, when the instructor goes over the information, theyÂ’re graspi ng it right away. Me? IÂ’m lik e, Â‘What are they talking about?Â’ So then, I mean I just donÂ’t get it that easy, as easy as most people would get it.Â” Lisa said she defines herself as Â“used to be ing smart. IÂ’m more laid back this time when I study. IÂ’m trying to say to myself, Â‘I donÂ’t necessarily have to have an A this time.Â’ All I want is at least a C to pass the co urse. Before I think I used to be very hard on myself. But IÂ’ve had some problems in the pastÂ…I had two breakdownsÂ…and it will take a while to get used to going back to math. But since IÂ’ve been in college, my environment has proved to me that I can [italics added] read and think.Â” Maria was quick to answer, Â“No, IÂ’m not smar t. Sometimes, just like the smallest things, I can turn it into a big task when itÂ’s really only a small task. If I ask help from someone else and they do it, IÂ’m like, Â‘O my gosh, IÂ’m dumb! I couldnÂ’t think of that. Why couldnÂ’t I think of that?Â’ So I donÂ’t think IÂ’m smart at all.Â” Theme 8: Motivation to understand math ematical concepts remained high. One of the most fascinating discoveries made from the interviews of the 15 participants in this study was that, in sp ite of the sometimes disheartening experiences
157 throughout their past schooling, in spite of a lack of parental/family support, and in spite of less than desirable behavior s on the part of teachers, and in spite of feeling powerless at times, these women still exhibited a self -confidence and expressed the motivation to and belief that they would succeed. Alice stated, Â“Whatever I do, I give my 100%. And I decided that, since I do give my 100%, I might as well get paid for my 100%, and so thatÂ’s why I decided to get a degree.Â” When Alice was asked if she e xpected to be succes sful in passing her mathematics courses at RACC, she responde d, Â“Oh, absolutely, ab solutelyÂ…whatever it takes. I know I have help here, so thatÂ’s why I know that as long as I search to get that help, IÂ’m going to get th e help that I need.Â” Lisa said her feelings about mathematics remained positive. Â“ThereÂ’s no reason a woman shouldnÂ’t know as much math as a man, because they go to the same classes, the same schools. They should be able to understand it just as well. ThatÂ’s why I say, Â‘IÂ’ve got to finish my education!Â’ IÂ’m used to being in the smart gr oup, and then, all of a sudden, all of these things in life get the be tter of meÂ…and I know fully well I am up to this level. Yes, IÂ’m going to do everything I can to pass it (referring to her mathematics class)! Courtney said her feelings about mathem atics currently are, Â“Awesome. I love math because thereÂ’s an answerÂ…thereÂ’s go t to be an answerÂ…one answer. Actually what happened is that I fell in love with l earning. ItÂ’s true. I liked learning about subjects I was interested in, but to learn about a subj ect that I wasnÂ’t in terest in, and then discover I liked it, was just an eye-opener for meÂ…too much fun. I can do it. I know what I have to do to get it done.Â”
158 Dianne stated, Â“To me, math is easy to learn. ItÂ’s always growingÂ…thereÂ’s always more and more to learn about math. And you use math every day of your life. I look forward to studying math. I wouldnÂ’t be scared or I wouldnÂ’ t back down. I would be looking forward to that (referring to ta king mathematics courses). IÂ’m not saying I wouldnÂ’t struggle, but I know that I would be able to do it.Â” Sue described mathematics, Â“ItÂ’s a cha llenge but IÂ’m not going to give up because I need that to graduate. I want to graduate with some kind of a degree so I can better myself. Nobody can do it but me and IÂ’ve decide d that IÂ’ve just gotta get through it. I gotta put a lot of time in it and IÂ’m gonna do itÂ…even if I have to take it again. Just like the computer. I didnÂ’t let it defeat me when I started taking word processing. My teacher, Ms. F. says, Â“You can do it, you just gotta put your mind to it.Â’ Down the road, IÂ’d really like to transfer [t o a four-year university]. All these other people are saying about furthering their careers a nd getting a degree. It might take me a course at a time, but you build a house one brick at a time. So that Â’s how I look at it. IÂ’ve got to do this. IÂ’ve got to get through that math to get me somewhere else.Â” Heather credited her mother for some of her determination. Â“My mother thought she was good at math but she wasnÂ’t all that ho t. She tried to help me with long division (she only went to eighth grad e) and she stayed upÂ—talk a bout dedication and sticking to itÂ—she stayed up all night. I remember being asleep and I would wake up and she was still trying to do this problem. And IÂ’d go in to school and I was so proud that my mother did it. And it was the wrong answer! We s till laugh about it to this day. It was the wrong answer and she was up all night working on that problem. So maybe thatÂ’s where some of my determination to hang in there comes from.Â”
159 Lauren was facing a chemistry course wh en her interview took place. She said, Â“IÂ’ll do the best I can. IÂ’ll go to the tuto ring center for help. IÂ’ll keep taking them (referring to her mathematics courses) unt il I passÂ…yes, IÂ’ll do the best I can.Â” Mary had a positive outlook. Â“IÂ’m confident. IÂ’m going to seek help. I know that help is available. IÂ’m hopi ng that, because I go to school in the evening, I hope that help will be available to meÂ—someone to help me in the evenings, because during the day I work. But I am going to look for extra help.Â” Connie was emphatic about succeeding. Â“I intend to stand up to my adversity. I have a friend who has struggled for three year s straight with the math. She gave me so much hopeÂ…she did it. I will never forget that, because thatÂ’s what it takes. On the days when I feel like pulling the cover up over my hea d, I think of her. Who am I to cop out? I donÂ’t have the right to do that.Â” Rose expressed her passionate desire fo r Â“a better future. I want to have my children know that they have an educated mother. I want them to know that minorities can advance in lifeÂ…if you have the will power, you can do whatever you want. And thatÂ’s why IÂ’m doing it (referring to being in college). I came from a backgroundÂ—I was taken away from my family and everythingÂ—I was in foster homes all my lifeÂ—I was a ward of the state. It just ma de me stronger. It made me r ealize that without an education, youÂ’re not going to succeed in life.Â…I think math is a challenge; itÂ’s food for the head. I think, out of everything, math is the numb er one thing. You can succeed by knowing about numbers in life, because everything is about numbers.Â…IÂ’m a very determined person. [IÂ’ll be successful in my mathematics cour ses] even if I have to go to four or five days of tutoring just to advance in it, yes.Â”
160 To the question of being successful in passing all of her mathematics courses required for her degree, Brittan answered, Â“D efinitely. I love challenges. ThereÂ’s no reason why I canÂ’t pursue something and pursu e it with a good grad eÂ…not just passing by the skin of my teeth. IÂ’m hoping I can ca rry this excitement all the way through [my college education].Â” Barbara said she gained confidence afte r doing well in an algebra class. Â“My teacher said that [understanding] math give s you confidence. And yes, I knew what she was talking about there because I finished up th at class and, yes, I did. I have that whole new confidence. I can do anything now.Â” Rose had a word of advice to others who are intimidated with mathematics. Â“Just keep sticking in there. DonÂ’t ever give up because, if you give up, you are saying youÂ’re a failure. You donÂ’t want to ever give up. Y ouÂ’ve got to strive if you want something in life. ThatÂ’s the way I look at it. I strive d all my life and itÂ’s with mathÂ…so math will never put me down. It will just make me want to strive harder to succeed in that area.Â” Summary Fifteen individual interviews of non-tr aditional age women enrolled in Reading Area Community College were conducted. Te n chose mathematics as the subject they would least enjoy and five chose mathematic s as the subject they would most enjoy. Data were analyzed by comparing profiles of each of the particip ants and drawing out common themes that emerged. Eight major th emes and six sub-themes emerged: (1) Acquiring a college education is a personal goa l; (2) Adequate study time is necessary to understand and to retain mathem atical concepts; (3) Experien ces with mathematics at an early age remain in oneÂ’s memory, (3a) Poor experience with mathematics at an early age
161 tended to make participants believe they could not learn mathematics, (3b) Positive experience with mathematics at an early age tended to provide participants a higher degree of self-efficacy in succeeding in mathem atics courses; (4) Parental behavior and expectations play a role in childrenÂ’s self-perception, (4a) Absence of parental/family support tended to discourage participants from pursuing further education, (4b) Presence of parental/family support tended to encourage participants in pursuing further education; (5) Teacher behaviors and teaching methods matter, (5a) Negative teacher behaviors tended to cause some to develop poor mathematics self-concepts, (5b) Positive teacher behaviors tended to encourage some to pe rsevere in understanding mathematics; (6) Feelings of powerlessness may impede learni ng mathematics; (7) Self-esteem can survive in spite of past failure; (8 ) Motivation to understand mathem atical concepts remained high. The findings of this study are important in order to emphasize the attention that needs to be devoted to this female populati on, particularly in the area of mathematics education, and the valuable contributions nontraditional age women can make if colleges can attract and retain them in college classrooms.
162 Chapter Five Summary, Discussion, and Implications This chapter will accomplish the following purposes. First, the problem and purpose of the study will be reviewed. A su mmary of the findings of the study will be then be provided. Next, major themes that emerged from the study will be reviewed and discussed in connection with the literature. Implications for practice will be addressed along with suggestions for further researc h. Finally, I will conclude with my own reflections. Problem It is clear that our countr y needs as many talented st udents as possible to pursue mathematics and science at advanced levels in high school, college, a nd graduate school. Yet, in the last few decades, it has been clear that fewer girls and women are pursuing majors and careers in either mathematics or science (Reis & Pa rk, 2001). Women are more educated, more employed, and employed at higher levels today than ever before, but they are still largely pigeonholed in pink-collar [italics added] jobs according to the American Association of University Wome n (AAUW) Educational Foundation report, Women at Work (2003) [italics added]. The report goes on to say that the new high-tech economy is leaving women behind because they donÂ’t have the keys to open the door to this high-tech sector of the work force. National census data show that the highest
163 proportions of women with a college educati on are still in traditionally female careers: teaching and nursing (AAUW, 2003). A report by the National Research Council, entitled Everybody Counts: A Report to the Nation on the Future of Mathematics Education emphasizes that undergraduate mathematicsÂ—the mathematics of the college experienceÂ—is vitally important, perhaps even more than elementary or secondary school mathematics. Â“More than any other subject, mathematics filters students out of programs leading to scientific and professional careers. From high school through graduate school, the half-life of students in the mathematics pipeline is about one yearÂ…Â” (Smith, 1994, p. 135). Undergraduate mathematics is the linc hpin for revitalization of mathematics education. Not only do all the scie nces depend on strong undergraduate mathematics, but also all students who prepare to teach mathematics acquire attitudes about mathematics, styles of teaching, and knowledge of content from their undergraduate experience. No reform of mathematics ed ucation is possible unless it begins with revitalization of undergraduate mathematics in both curriculum and teaching style (c ited in Smith, 1994, p. 135). Purpose The first purpose of this study was to examine, by means of in-depth, phenomenological interviewing, the metacogn itive and affective factors that are perceived to contribute to mathematics a voidance or mathematics confidence in nontraditional age women attending a community college. The second purpose of the study was to explore and describe the meaning pa rticipating non-traditional age women attach to their experience with mathematics. Th e third purpose of the study was to determine
164 the relationship, if any, between metacognitive and affective experience in the learning of mathematics. Summary of the Findings Findings from this study may be applicable to a universal popul ation. That is, the identified themes may not be specific only to non-traditional age women. Nevertheless, this qualitative study supports and extends the existing literature by providing depth through interviews of partic ipants, discovering the meani ngs they make of their experiences in their own words. The study fo llowed the interpretive form of educational research described by Carr and Kemmis (1986), where education is considered to be a process and school a lived experience. To unde rstand the meaning of that process is to gain knowledge by an inductive, theory-genera ting mode of inquiry. Multiple realities are constructed socially by i ndividuals. Schwandt (1998) argues that individuals construct knowledge in an effort to make sense of their experience and that this knowledge is continually tested and modifi ed in light of new experiences. I was interested in understa nding the experience of learning ma thematics from the perspective of the women themselves, regardless of whether that experience was a positive or negative one. Also of interest to me was discovering factors, if any, that served to differentiate those women who had not succeed ed thus far from those who may have had a struggle but who, nevertheless, were su ccessful in completing their mathematics courses. Findings are grouped into eight major them es and six sub-themes: (1) Acquiring a college education is a persona l goal; (2) Adequate study time is necessary to understand and to retain mathematical concepts; (3) E xperiences with mathematics at an early age
165 remain in oneÂ’s memory, (3a) Poor experien ce with mathematics at an early age tended to make participants believe they could not learn mathematics, (3b) Positive experience with mathematics at an early age tended to provide participants a higher de gree of self-efficacy in succeeding in mathematics courses; (4) Parent al behavior and expectations play a role in childrenÂ’s self-perceptio n, (4a) Absence of parental/fam ily support tended to discourage participants from pursuing further educati on, (4b) Presence of parental/family support tended to encourage part icipants in pursuing further educa tion; (5) Teacher behavior and teaching methods matter, (5a) Negative teacher behaviors tended to cause some to develop poor mathematics self-concepts, (5b) Positive teacher behaviors tended to encourage some to persevere in understanding mathematics; (6) Feelings of powerlessness may impede learning mathematics; (7) Self-esteem survived in spite of past failure; (8) Motivation to understand mathematical concepts remained high. The trustworthiness of this study was determined, in part, by the voiced experiences of the participants themselves. E ach participant read th e transcription of her interview and then read her pr ofile containing specific quotes fr om the interview. Each participant was asked to verify accuracy and make any additions and corrections. Creditability was also sought by involvi ng two of my colleagues who read all 15 interview transcripts and independently wrote what each recognized as common meanings and themes expressed by the particip ants. Their findings were compared with my own and we reached consensus on the final list of themes.
166 Discussion Theme 1: Acquiring a college education is a personal goal. Acquiring a college education was not simply one in a list of many other priorities for the participants. It was expressed as a lifelong desire and at the top of the list of personal goals. Many of these particip ants had raised children, supported their households by working in factories or se rvice occupations, an d postponed their own dreams until it was finally their turn [italics added]. They were serious about doing well academically. They were serious about doing well in mathematics. Some of them had succeeded in passing the required mathematics courses, some had not. Those who had not were unwilling to accept defeat. Some were discouraged momentarily but were determined to try again. A serious attitude about school, evidenced by the participants in this study, supports the research (noted in Chapter One) of Cox (1993), Miglietti (1994), and Richardson (1995), who found that non-traditional age stude nts often exhibit a deep approach or a meaning orientation toward their academic studies. Theme 2: Adequate study time is ne cessary to understand and to retain mathematical concepts. The question of study habits, or lack thereo f, was an area I, as the researcher, was curious to explore. All 15 women had their own unique study strate gies, favorite places to study, best times of day to study, and various approaches to note-ta king. Some needed quiet; others had to have noise. Most pref erred studying in small chunks of time rather than cramming for hours. The common thread among all of the pa rticipants, however, was the fact that studying was very importa ntÂ—important enough to take their books to
167 work, spend hours in the library, get up early before the rest of the family, or stay up late after others had gone to sleep. In other words, they realized the effort required to learn, understand, and retain information. With ma thematics, however, some of them were beginning to question if learning the material was beyond their ability, regardless of how much time they spent studying. As reviewed in Chapter One, Carney -Crompton and Tan (2002) found that study skills often must be develope d or refreshed as non-tradit ional age women assume their role as students. Counselors and academic support staff may help he re to give advice, guidance, and encouragement. Theme 3: Experiences with mathematics at an early age remain in oneÂ’s memory. The third major theme that was evident fr om listening to the participantsÂ’ stories was that they remembered experiences with ma thematics from their early school years. If the experience being reported was a positive one, the memory was often expressed in rather vague terms. However, if the expe rience was unfavorable, or negative in some fashion, participants were able to reca ll the words spoken to them, body language, and accompanying emotions in vivid detail. Five of the 15 participants shared positive past experiences and good memories of learning ma thematics. Less than positive memories were more common, however, and this pa rticular phenomenon emerged as a shared experience for 10 of the women, representing two-th irds of the participan ts in this study. Six of these participants shared experiences from elementary school (one remembered an experience as early as first grade), two f ound junior high school to be where they remembered falling behind in mathematics, and two experienced difficulty beginning in the tenth grade.
168 Sub-theme (3a): Poor experience with mathematics at an early age tended to make participants believe they co uld not learn mathematics. Having a poor experience with mathematic s at an early age was perceived by the participants as having a signi ficant effect on their self-effi cacy when performing in the college mathematics classroom. This findi ng supports research studies discussed in Chapter Two. Hamachek (2000), for example, asse rts that Â“the self is not something with which individuals are born but something they create out of their experiences and interpersonal relationshipsÂ” (p. 230). Each of the participants had a self-concept and a certain level of self-esteem that, as they grew, was influenced by their surroundings. As they moved through elementary school, j unior high school, and high school, the interactions they experienced and the feedback they received from parents, teachers, and peers, were important compone nts in the development of th eir self-concept. Tomkins (1992) refers to individuals as victims [italics added] when the world he or she has experienced contains negative scenes. Belenky et al. (1986) learned, through inte nsive interviews with women, that, in general, they often felt alienated in academic se ttings. If feelings of alienation are carried through oneÂ’s school experiences it is no mystery that cognitive ability to understand mathematical concepts is hindered. Bandura (1997) argues that se lf-concept has been shown to be intimately involved in the cult ivation of cognitive competencies. Stodolsky (1988) agrees that an individua lÂ’s belief in his or her abil ity plays a critical role in learning mathematics. Sue explained, Â“I can remember my teacher [in first grade]Â…she was great with the ruler. And if you didnÂ’t add right, you got smacked on your hands with the ruler.Â”
169 Connie could not remember when her problems with mathematics began. However she did say, Â“I donÂ’t remember my grade school ex perience vividly, but I do remember spurts of it and it was never pleasantÂ…never pleasant. Now when I get the right answer [to a mathematics problem] IÂ’m excited about it because of past experiences which were so disheartening.Â” Since, from elementary grades on up, some of the participants were told that they lacked the ability to succeed in mathema tics, they accepted that as fact. Sheila remembered the chemistry teacher telling he r, Â“You canÂ’t get the math.Â” This study supports the findings in both CrawfordÂ’s re search study (1980) and MardernessÂ’ research study (2000), that a belief that one is an out sider to the world of mathematics becomes ingrained in oneÂ’s belief syst em during oneÂ’s early years a nd this belief remains intact throughout oneÂ’s adult life. Theme 4: Parental behavior and expectat ions play a role in childrenÂ’s selfperception. Parental support and encouragement, or parental indifference toward academic achievement, seemed to play an important role in the participantsÂ’ self-perception of how well they might fare in what they perceive d as more difficult subjects, such as higher levels of mathematics or college level courses. Parental support wa s lacking for 10 of the participants, representing two-thirds of the women interviewed. In some cases, the lack of support was due to the parentsÂ’ own lack of schooling. In other cases, parents and relatives did not see the value of getti ng an education, particularly mathematics education, for girls. Rather, parents encourag ed the participants to concentrate on finding a job, getting married, raising a family, and expected them to be satisfied with those
170 options. Five of the participants said they had parents who cared about their education and encouraged them to do well in mathematics. Four of these five participants evidenced a positive attitude toward mathema tics and were achieving success in their mathematics courses. ChildrenÂ’s self-concept of ability and thei r confidence in mathematics are more directly related to their pare ntsÂ’ beliefs about their mathem atics aptitude and potential than to their own past achi evement in mathematics (Eccles-Parsons et al., 1982). As noted in Chapter Two, Tocci and Engelhard, Jr. (1991) found a positive relationship between parental support and stud ent attitude toward mathematic s. They also argued that perceptions of parentsÂ’ reactions to mathematics, along with the amount of encouragement children receive to study and do well at it, may affect the childÂ’s attitude toward mathematics. The power of the home environment should not be underestimated. Students need to see indications from others, including their parents, siblings, and peers, that mathematics is im portant (Fennell, 2007). Sub-theme (4a): Absence of parental/fa mily support tended to discourage participants from pursui ng further education. Several of the participants came from homes where one or both of their parents lacked a formal high school education. This be ing the case, the participants assumed that their parents Â“just didnÂ’t care.Â” Sheila rema rked, Â“Â…my parents didnÂ’t care one way or the other.Â” In several cases the participants were told that mathematics was not important for girls to learn. Sue was told by her father Â“YouÂ’re going to work in a factory like your mother did and youÂ’re gonna get married a nd have babiesÂ…and, what do you need an education for?Â” LaurenÂ’s parents convinced her, Â“YouÂ’re never going to need that in a
171 factory.Â” Several participantsÂ’ responses revealed that their parents felt frustrated because they couldnÂ’t help their child with mathematics. Barbara remembered that her mother Â“just threw her hands up and didnÂ’t know how to help me.Â” Other participants shared that their parents were just too busy to be concerned with their edu cation. AliceÂ’s mother ran a business and Â“just didnÂ’t have the ti me to help us.Â” Sue had to take care of her younger siblings and wasnÂ’t given the time or opportunity to Â“be a really good student when I was going to school.Â” In c ontrast to those part icipants who did not receive parental support, the findings of this study show that in each case where the participant remembered receiving parental support, there was a corresponding motivation to be successful in the academic arena, particularly in mathematics courses. Theme 5: Teacher behaviors and teaching methods matter. All of the participants in this study shared the belief that a teacher makes a difference, not only affectively in a studentÂ’s belief in himself or herself, but in the cognitive learning of the mathematics material itself. I found it very surprising how vivid the memories of the participants became when they described issues with past teachers, whether the incident occurred in their early schooling or in th eir current experience in the community college. Their memories were cl ear, even to the point of recalling exact words that were spoken by the teacher, the t eacherÂ’s body language they perceived to be either negative or positive, punishments inflic ted when an incorrect answer was put forth, and, in some instances, the numerical grad es they received for their mathematics performance. Those who received encouragem ent, praise, attention, and mentoring from a teacher, whether in their past or curr ently at the community college, responded by persevering, working harder, and in many cas es, mastering the mathematics material.
172 Greenwood (1984) and Swetman et al. (1993) state that teac hers are considered to be a major force in contributing to student achievement, more importa nt than either the method or curriculum. Hodges (1983) emphasizes the importance of affective variables in the learning process, that there is an emo tional side to learning as well as a cognitive side. Teachers who provide supportive, encouraging environments increase student confidence levels and improve achievement and interest in the subject matter. Theme 6: Feelings of powerlessn ess may impede le arning mathematics. The sixth theme that surfaced from the participantsÂ’ stories was never explicitly spoken. However it was implicit in some of thei r rich description and in the emotion that was evidenced in their voices. Feeling powerle ss to move away, to take oneself out of a situation, to change the circum stance, to speak back, to say stop [italics added], to take control, or to regain oneÂ’s self-respect can be not only physically and mentally painful but also restricts and inhibits learning (Erchick, 1996). Severa l of the participants shared events that occurred in mathematics cla sses in which they found themselves to be powerless. These events served to hinder th eir learning and erect a mental barrier to understanding the mathematics material. Theme 7: Self-esteem can survi ve in spite of past failure. A surprising, yet welcome, finding from this study was that 13 of the 15 participants thought of themselves as smart [italics added]. Interestingly, 7 of those 13 were struggling with mathematics, having fa iled to pass their mathematics courses in previous attempts. The participants revealed what I call survival techniques [italics added] in order to maintain a level of self-e steem that would enable them to continue to persevere in order to move thr ough their mathematics requirements.
173 In most cases, the participants attributed their smartness [italics added] to having common sense, displaying wisdom as wives a nd mothers, proving an ability to juggle family and work responsibilities with their co llege classes, or achieving a level of success in their college courses thus far. As noted in Chapter Two, Marsh and Yeung (1996) assert that a positive self-concept is an important mediating variable that may promote academic achievement and other valuable educational outcomes. Marsh, Craven, and Debus (1991) show that enhancement of se lf-concept can improve academic performance and is strongly related to s ubsequent course selection. Perceived beliefs of oneÂ’s ability (sel f-efficacy) contribute independently to intellectual performance. In fact, Pintrich and DeGroot ( 1990) found that the higher the studentsÂ’ efficacy beliefs, the higher the academic challenges they set for themselves. Individuals of high self-efficacy persist while those of low self-efficacy are more apt to quit (Bandura, 1982). A positive self-concept, evidenced in the findings of this study, lends credibility to the final theme which ha s to do with self-confidence and motivation. Theme 8: Motivation to understand ma thematical concepts remained high. Collins (1982) conducted studies giving children mathematical problems to solve, and found that efficacy beliefs predicted in terest in, and positive attitudes toward, mathematics, whereas actual mathematical abil ity did not. Efficacy beliefs, therefore, are foundational to learning and persistence. The participants in this study expressed a high sense of efficacy. Thirteen of the 15 participants expressed self -confidence that they would u ltimately perform successfully in their mathematics courses. The two who were unsure said they hoped to pass and would do their best. This self-confidence pr oduced a high level of motivation that was
174 evident in all of the participants. Schunk (1989) found that perceived self-efficacy was a better predictor of intellectual performance than skills alone and that self-efficacious individuals will intensify their efforts and, if necessary, change the environment, i.e. seek tutoring, restructure work schedules, form study groups, read additional mathematics material, watch publisher-produced teaching videos, or temporarily set aside other interests, in order to succee d. Participants in this study exhibited a leve l of motivation necessary to complete th eir educational goals. ParticipantsÂ’ self-efficacy appeared to emerge as a result of raising families and working to provide financial s upport as single mothers or as h ousewives. Also critical to participantsÂ’ self-efficacy was having access to the student support that the community college offers in the areas of tutoring, a dvising, and simple, yet extremely significant, caring about the welfar e of students. Have the purposes of the research been attained? Do we have a description of factors these participants perc eived to result in mathematics avoidance or mathematics confidence? The responses of the participants in their own wordsÂ—their expressed desire to achieve a college degree, their diligence in applying themselves to study, their past experience in mathematics classes, the ex istence and degree of parental support and encouragement they received or did not rece ive, teacher behaviors and manner in which the mathematics was taught, f eelings of powerlessness to im prove their situations, their self-esteem and motivationÂ—contain comm on factors which may contribute to mathematics confidence or mathematic avoidance. Do we have a sense of the meaning that the participants attached to their mathematics experience? Perhaps the most pr evalent meaning the participants expressed
175 was their belief that they were smart [italics added]. Despite pa st struggles, and in some cases, repeated failure in mathematics courses, they continued to di splay a level of selfesteem and were motivated and self-confident that they would succeedÂ—if they continued to striveÂ—in thei r mathematics courses. Can we determine any relationship between metacognitive and affective experience in learning mathematics? Rich desc ription from participantsÂ’ stories in their own words revealed both positive and negative metacognitive factors. For example, positive metacognitions included (1) recognizing th at help was available, (2) that prior learned mathematics concepts were benefi cial in connecting to new mathematics concepts, (3) that problems could be worked out in more than just one way, (4) that the instructor was helpful and cared, and (5) th at being Â“stuckÂ” was only temporary. Along with these positive metacognitions were positiv e affective factors such as (1) enjoying the competition, (2) having a favorite grade, (3) feeling satisfied at earning a grade, (4) feeling the caring of the teacher, (5) respec ting the teacher, and (6) feeling excitement about oneÂ’s grade. Along with positive metacognitive awareness, participants expressed some negative thoughts. Negative metacognitions included (1) recognizing the fast-paced presentation of mathematics material, (2) the overcrowded classroom, (3) the sometimes hostile behavior of peers (being referred to as an Â“idiotÂ” or Â“stupidÂ” ), (4) the fact that oneÂ’s powers of concentration are limited by stress, (5) the fact that high school mathematics was not adequate preparation for college, (6) the observation that the teacher was not approachable, and (7) the recognition that oneÂ’s capacity to absorb information had been reached and there was nothing to be done to expand that capacity. Along with
176 these negative metacognitive factors, the following negative affective factors were expressed in the participantsÂ’ own words: (1) feeling very stressed, (2) overwhelmed, (3) anxious, (4) humiliated, (5) lacking in self-confidence, (6) discouraged, (7) blown away, (8) lost, (9) flustered, (10) frus trated, (11) feeling like a tota l jerk in front of the whole class, (12) embarrassed, (13) disheartened, (14) hopeless, (15) intimidated, (16) disappointed, and (17) upset with oneself. When feelings of alienation (affective information) are carried through oneÂ’s school experiences, it is no mystery that c ognitive ability to unde rstand mathematical concepts is hindered. Nevertheless, metac ognitive information, such as knowing which teaching style works best, which learning style makes learning easier, or what type of study regimen results in the best perfor mance on a test, can result in affective information, such as how one feels about hims elf or herself, how mu ch anxiety one feels when it is time for a test, or how much one feels cared for and respected by those he or she interacts with daily. Stories from the par ticipants in this study re vealed that it is not so much what mathematical concepts are l earned, but how they ar e learned, which will determine the affective objectives that will be attained at the same time as the cognitive objectives. A relationship between metac ognitive and affective experience was not apparent in all participants. However there was a correlation in most of the stories as may be seen in Table 8, in Chapter Four (p. 123). Qualitative research is based upon Â“the view that reality is constructed by individuals interacting with their social worldsÂ” (Merri am, 1998, p. 6). I, as the researcher, am interested in understandi ng the meanings non-traditional age women attending a community college have construc ted relating to learning mathematics, and
177 how they make sense of their world and the lived experiences they have had. It is assumed that meaning is embedded in the participantsÂ’ experiences and that this meaning is mediated through my own per ceptions. Patton (1985) explains: [Qualitative research] is an effort to understand situations in their uniqueness as part of a particular context and the intera ctions there. This understanding is an end in itself, so that it is not attempting to predict what may happen in the future necessarily, but to understa nd the nature of that settingÂ—what it means for participants to be in that setting, what their lives are like, whatÂ’s going on for them, what their meanings are, what th e world looks like in that particular settingÂ—and in the analysis to be able to communicate that faithfully to others who are interested in that setting (p. 1). According to Merriam (1998) qualitative research is not a linear, step-by-step process. Collecting the dataÂ—meeting with each participant and audiotaping her storyÂ— and analyzing the dataÂ—listening intently to each participan t and observing tone of voice and body languageÂ—occurred simultaneously to a la rge degree. My analysis truly began with the first interview, which happened to be my own. By allowing a colleague to conduct the interview and by taki ng on the role of a participan t, I was made aware of my own feelings and responses to the questions I had compose d. It was from this very personal perspective that I then set out to conduct interviews of my own and strove to bracket my own experiences, and the meaning I made from them, in order to more fully understand the stories told by th e participants. During the pe riod of time over which my interviews were conducted, there were in sights that emerged, impressions that I
178 questioned, and tentative hypotheses which led to a final refinement of shared themes and believable findings. Implications for Practice The findings in this study of non-traditio nal age women can inform the practice of teachers in higher and adult education in a number of ways. Primarily, there must be a willingness to understand the importance of e ducation to these womenÂ’s futures and the degree of weight that is placed on their shou lders to learn mathematics. Wolfgang and Dowling (1981) found that non-traditional stude nts tend to focus on internal motivations instead of external motivations when attend ing postsecondary school These students are going to college because they want to learn. Following are 17 implications for both faculty and students drawn from the metacognative responses of the participants Suggestions are o ffered explaining what teachers might do to reinforce the positive metacognitions and reduce those that were negative. (1) According to the partic ipants in this study, a positive impact is made on their learning when they perceive their teachers care and are willi ng to listen to and answer their questions. Particip antsÂ’ positive metacognition, recognizing that help in mathematics was available, can be reinfo rced by teachers scheduling their office hours around the times of their mathem atics classes. I know of faculty who schedule office hours at 7:30 a.m. for the convenience of being able to leave when their teaching load is finished later in the day. There are others who schedule office hours only on a Tuesday or Thursday, which is of little help to students who are on a Monday/Wednesday/Friday schedule. Telling students they should go to the tutoring center for help is acceptable;
179 however, students often feel Â“brushed offÂ” unless the teacher has already taken some personal time to see where they are having pr oblems. Often the tu toring center is not able to provide one-on-one help for an extended period of time. (2) Recognizing that new mathematics conc epts seem more easily digested when connected to previously learned concepts was a second metacognition. Teachers could be careful to make connections between a nd among mathematical concepts as each is introduced. A worthwhile practice is to encourage students to look at the problem carefully, describe it fully, and seek the esse nce of it. Point out how the new concept takes the previous one a step further or help s to make the material useful in another discipline, such as engineer ing, architecture, or medicine Help students to see the problem as legitimate [italics added], meaning it is enga ging, relevant or entertaining. It must not be routine, repe titive, rhetorical, or c ontrived (Boeree, 2007). (3) Sometimes realizing that a problem can be solved using alternative strategies is exciting to students; but, it can also be in timidating. It is impor tant for teachers to listen to their students. What is their unders tanding, not only of math ematics, but of the world? Some students will have more expe rience with methods of problem-solving or creativity or have more general background fr om which to draw. We cannot take these things for granted. For adult women, partic ularly, with busy schedules and many details to manage, settling on one problem-solving stra tegyÂ—the one they perceive is the best oneÂ—is sometimes preferred over learning multiple paths to a solution. (4) Kern (2006) cited caring on the part of the instructor as an important concern for adult students, one that could not be ove remphasized. KernÂ’s findings were supported
180 by this study. ParticipantsÂ’ metacognition th at the teacher was he lpful and cared, was a critical element in their persistence and performance in mathematics. Teachers can show they care in any nu mber of ways, including acknowledging the reality that students have lives outside of the classroom. As discussed in Chapter One, adult women are balancing many obligations and college is just one of them. Some flexibility may be helpful in turning in assi gnments or taking tests. I have heard women tell of their frustration in trying to complete a mathematics course, where they were told that, since they were unable to attend the scheduled test (due to a mandatory work requirement or caring for a sick child), they were given a grade of zero, with no other options. Even more troubling are the storie s women have shared of their being dropped from the roll in a mathematics course, due to circumstances beyond their control. This, in my view, is comparable to being punished for a crime you did not commit. While there are pedagogical standards that must be met, flexibility in meeting such standards is appreciated by adult students. In addition to flexibility, demonstrati ng empathy with students forms a bond that encourages learning. Empathy means that on e tries to feel what it would be like to walk in the other personÂ’s shoes [italics added], showing sensitivity, understanding, and responsiveness. While in other areas of their lives, adult students may be the local expert and amaze their friends, in school matters, the teacher is always better than they. Comparison with an authority takes away some of the potential for pride (Boeree, 2007). Teachers can help adult students build a sense of pride by sharing their own struggles, challenges they ar e facing, or simply by listeni ng without passing judgment or giving advice. They should then bolster st udentsÂ’ hopes by directing them to focus on
181 their goal and the pride that will accompany ha ving reached it. In other words, teachers should try to connect with their students by being personable, being available, being enthusiastic, and encouraging them to persev ere. Not surprisingly, teacher characteristics that participants noted as be ing especially meaningful to them were patience, calmness, friendliness, respect, and understanding. (5) Occasionally students reach an impasse in their learning mathematics, where they feel bewildered and are at a loss to know how to proceed. Participants in this study described this as being stuck [italics added]. However, a participant metacognition revealed an awareness that this condition, although frustr ating, was only temporary. A fitting metaphor may be seeing a light at the en d of the tunnel or ge tting a glimpse of the sun behind heavy cloud cover on a dreary da y. One can imagine getting past the momentary distress. When stude nts are in this perplexing stat e, here is where a teacherÂ’s encouragement is most needed. (6) Metacognitions of a negative nature were more numerous than positive ones. The first was recognizing the fast pace of the mathematics material. Prawat (1992) questions the popular view of curriculum as Â“a fi xed agenda, a daily course to be run that consists of present means (i.e., certain materi al to cover) and predet ermined ends (i.e., a discrete set of skills or competencies)Â” (p. 358) He favors a more interactive approach to curriculum where it is viewed more as Â“a matrix of ideas to be expl ored over a period of time than as a road map. One would enter this matrix at various points, depending on where students are in their current understa ndingÂ” (p. 358). Viewing mathematics as dynamic, rather than as a static body of knowledge, would emphasize the importance of student reasoning instead of following a teacher-
182 centered approach where students learn by carefully attending to the teacherÂ’s demonstrations and explanations and respondi ng to his or her ques tions. The studentcentered approach asks students to expre ss their own ideas, which not only supports studentsÂ’ efforts to make sense of the conten t, but also allows the teacher to understand what they are thinking. How else can a teacher know what the studentsÂ’ needs and difficulties are? One implication from the part icipants in this study is that many times the fast pace of the mathematics material is overwhelming. Teachers should try to capitalize on remarks from students and incorporate them in to the mainstream of the lesson or shift the discussion to clarify the studentsÂ’ difficult ies. Â“Attentiveness to student cognition is one of the defining features of constructiv ist teachingÂ” (Pravat, 1992, p. 367). Simply moving through the mathematics content, chap ter after chapter, doe s not assure that learning, i.e., meaningful learning, has taken place. (7) One participant in this study was aw are that she felt nervous and distracted due to the overcrowding in the mathematics classroom. Overcrowding is a problem that faces many educational institutions, partic ularly in urban areas where the student population is growing faster th an facilities can be built to provide adequate space. Research has consistently shown that ove rcrowding negatively affects both classroom activities and instructional techniques. Cr owded classroom conditions make it difficult for students to concentrate on their lessons. The implication is that smaller class size significantly increases the amount of learning that takes place. (8) Comments from fellow classmates, although not intentionally meant to be offensive, may be hurtful or di scouraging to a student. Severa l participants in this study shared that demeaning comments from classmatesÂ—or even anticipating such comments
183 from classmatesÂ—kept them from participati ng in class discussion or asking questions in class. Sandler (1996) found that hostile behavi ors, such as rolling of the eyes, or hostile remarks, such as calling another student stupid [italics added], often leave women feeling angry, demeaned, and uncomfortable. Their class participation may drop considerably and some may drop out of class. Teachers, when taking notice of this kind of hostile overt behavior, should not ignore it, but activ ely discourage and expr ess disapproval of it by indicating that such behavior is not acceptable. (9) A repeated theme of participants in this study was that stress affected their power of concentrationÂ—in the classroom, wh en the teacher was explaining a problem on the board, and when studying for a test, or during the process of taking a test. As noted in the literature review relating to mathematic s anxiety in Chapter Two, research has shown that stress produces a decrement in mathematics performance (Ma, 1999; Norman, 1997; Rabalais, 1998). College mathematics teac hers can find ways to reduce the anxiety women may carry with them into the classr oom by displaying an understanding attitude. They could invite women to express their opinions since women, more frequently than men, hesitate to speak up and show asser tiveness (Karp, et al., 1998). Many women have been socialized to be silent, especially in formal mixed groups, such as a college classroom (Sandler, 1996). Stude nts, particularly women, must be encouraged to discuss, openly talk about their discove ries, and feel comfortable entering into debate, when appropriate. Teacher behaviors establish the context for a classroom environment that is either hospitable to non-tradi tional age women or intimidating to them (Sandler, 1996). One such technique is to bring the idea of stress out into the open. Help students to explore their anxieties about mathematic s through discussion. Sharing my own
184 anxiety about mathematics and how I was able to gain control over it has been successful to some degree in encouraging other students. (10) Inadequate preparation in high sc hool has placed approximately one-third of enrolling freshmen into developmental clas ses in community colleges and universities (Ravitch, 2000). Being placed into developm ental mathematics comes as a shock to many students if they had taken several years of algebra or calculus in high school. To non-traditional age students, taking developmental courses is expected since many of these students have been out of school for years. They accept the fact that their mathematical skills need to be refreshed. However, many are surprised at how little mathematics they had been taught when they were in high school (Fotoples, 2000; Hall et.al, 1999; Horn et.al, 1996; Zaslavs ky, 1994), and how little they had retained (Karsenty, 2002). In the 1970s and 1980s, when the participants in this study were in high school, many girls were not encouraged to take higher level mathematics which would prepare them for college. (This trend is changing as recent statistics show that more girls than boys are now en rolling in degrees in scien ce and engineering fields.) Teachers should be aware that non-traditional age students are learning mathematics that they may be seeing for the first time, as opposed to younger students who have had a lot of the material in high school. Participan ts in this study appr eciated a teacher who demonstrated patience with them and unde rstood that they were grappling with mathematical concepts, however basic [italics added], that were new to them. (11) A non-threatening classroom atmos phere can only been developed when the teacher is non-threatening. Several of the part icipants in this study shared that, when the mathematics course first began, they were af raid to volunteer answers, afraid to ask
185 questions, or afraid to talk to the teacher f ace to face. In some cases, throughout the days and weeks that followed, a rapport was built between teacher and student, and the teacher was now described as approachable [italics added]. Teacher personality traits may be a factor that promote or hinder learning in the mathematics classroom (Norwood, 1989; Spanias, 1996; Jackson & Leffingwell, 1999). Injecting humor and sharing personal experiences that relate to the mathematical problems being solved makes the class fun and interesting. Studies have shown that a teacher who is approachable will smile, show interest in getting to know each student as an individual, learn studentsÂ’ names, interact with students outside of the classroom, i.e., in the hallway, in the parking lot, in the cafeteria, etc., and demonstrate a genuine inte rest in seeing students do well (Kern, 2006). (12) Time management becomes crucia l to non-traditional age women who are working and raising children in addition to ti me spent in the college classroom and often experience tremendous stress in trying to ha ndle the multiple roles of students, mother, spouse, and family breadwinner (Cross, 1981). For this reason, they need additional sources of support. The participants in the st udy stated that they were thankful for the help that was available at the community colle ge in the form of tu toring. One participant expressed her hope that there would be tuto ring services available during the evening hours for her to use before her evening cla ss. Mathematics tutoring centers should be receiving support, both financially and otherwise, from the institutions they serve. They should be staffed with friendly, knowledgeable personnel who are trained to work with students who are struggling with mathematical concepts. Centers should have copies of all the textbooks being used by the teachers pl us creative materials such as manipulatives for students to construct knowledge based on th eir own learning style. The hours of the
186 mathematics tutoring center should reflect th e hours that classes are being offered and take into account studentsÂ’ sche dules. Having this service av ailable serves as a powerful motivator for students to persevere and achieve. (13) A challenge remains of attempting to reverse and counteract harmful, even humiliating experiences that students, partic ularly women, carry with them from past experiences with mathematics in elementary school, junior high sc hool, or high school. Francis (Skip) Fennell, current president of the National Counc il of Teachers of Mathematics states, A studentÂ’s view of what it means to know and do mathematics is shaped in elementary school; yet in the United States elementary teachers are, for the most part, generalists. Their preservice teacher education typically includes two or three courses in mathematics content and one course in the teaching of mathematics. A mathematics speciali st is needed because the preservice background and general teaching responsibilit ies of elementary teachers do not typically furnish the continuous developm ent of specialized knowledge required for teaching mathematics today (November, 2006). Perhaps, since participants recounted negative experien ces from their elementary school years, renewed attention should be give n to the preservice trai ning of elementary teachers to better prepare them for the ta sk of making mathematics meaningful and understandable to their elementary-age students. I suspect that there are similar needs at the middle and high school levels. Middle and high school mathematics teachers need ongoing content and pedagogical assistance as well.
187 Prior to coming to college, many women ne ver found their inner voice so they did not ask for more teacher support. Rather they became convinced that they could never be good at mathematics. Because of the ways wo men are socialized, one area they tend to find difficult is higher-level pr oblem solving. While one of their strengths is following rules, a weakness is in taking chances a nd risks (Karp et al., 1998). Teachers can encourage women to participate in the mathem atics classroom by being attentive to them and involving them in class discussion. They can make an effort to make their classes interesting and teach in an enthusiastic manner. (14) As already stated, participants rega rded patience as an important quality for teaching mathematics. So often speed is pr ized over understanding. Karp et al., (1998) argue that Â“it is not surprising that many well-educated women who can successfully solve mathematics problems are still uncomfort able with mathematics because they are unsure of how formulas Â‘work.Â’Â” Not enough time has been given to really understand and critically think through the mathematical concepts. Williams (1993) suggested that one goal of instruction is to make students co mfortable with doing mathematics as part of their everyday getting on the world. Wale n and Williams (2002) proposed that teachers attempt to engage their students with mathema tics in ways that are personally meaningful and relevant to the context of their concerns thus helping them to become comfortable enough doing mathematics so that it beco mes a tool for solving real problems. (15) Teaching mathematics in an organi zed, step-by-step, process was appreciated by the participants in this study. This traditional [italics added] approach to learning mathematics, which encourages rote learni ng of procedures, i.e. rules, algorithms, symbols, has been studied to ascertain whethe r procedural knowledge eventually leads to
188 concept development. The notion that there are stages of development in mathematics and learners typically go through a procedur al orientated phase before they can effectively use their conceptual knowledge is studied in Davis, Gray, Simpson, Tall, and Thomas (2000). The women in the current study felt that as they were able to comprehend the steps to solve a mathematical problem, they we re also able to understand more clearly the reasoning behind the process. These findings support the work of Rittle-Johnson, Kalchman, Czarnocha, and Baker (2002), who studied the relationship between procedural and conceptual knowledge in the mathematical classroom and hypothesized that throughout development, conceptual and procedural knowledge influence one another in mutually suppor tive and integrated ways. Also supported are Baker, Czarnocha, and Prabhu (2004), who suggest that an optimal environmental for learning clearly involves coordination of pr ocedural and conceptual knowledge. An implication arising from this stud y is that procedural knowledge and conceptual knowledge influence one another continually throughout the learning process. Sketching visual pictures on the board, e xplaining problems in different ways, and purposefully tying new concepts to previously learned concepts all help women grasp the mathematical concepts. (16) Teachers should be learners too. Unfortunately, not all classrooms have teachers who are learners. Berkas and Pattison (2006) report, In our work with groups of teachers, we have found two distinct cultures, a culture of passivity and a cu lture of collaborative lear ning. The passivity culture is characterized by teachers who are se t in their ways and believe that low-
189 performing students are responsible for th eir own lack of success. We hear statements like, Â“kids these days just donÂ’t care,Â” and, Â“kids are lazyÂ—they have no work ethic.Â” By contrast, teachers who belong to a collaborative learning culture do not accept perception data about their students but strive to learn from students about students. They work to understand not only their studentsÂ’ learning styles and gifts but also the personal backgr ound that influences their learning. They plan lessons collaborativ ely and use a variety of instructional methods to engage and motivate their student s. In short, teachers in this culture are always learning and fue ling their own inner fires. (17) Although many community college s have orientation programs for new students, a special program specifically fo cused on the needs of adult women could emphasize their unique issues and concer ns (Carney-Crompton & Tan, 2002; Johnson, Schwartz & Bower, 2000). It would state cl early that the institution was committed to their success and recognized their spec ial needs. Emphasis should be placed on promoting the benefits of learning mathematics and create incentives for women to pursue courses in mathematics that would prepar e them for fields in science, engineering, or information technology, which are the bette r-paying, higher-status, and faster-growing occupations in our society. All of these implications have the potential to offer a window into the world of the non-traditional age woman who enrolls in a community college and sits in an introductory mathematics classroom. Teachers at all levels of education need to be supported in their efforts to make mathem atics education relevant, meaningful, and motivating for their students. Fiore (1999) sugge sts that Â“instructors can teach students
190 who have had painfully negative experience s in mathematics through encouragement, positive talk, and accommodationÂ” (p. 405). If students believe that they can learn mathematics and that the instructor cares abou t their learning, they wi ll push themselves harder. We should not underestimate the power of encouragement and positive talk in the mathematics classroom, nor should we Â“ underestimate the damage that negative talk from a teacher or parent can have on self -esteem and performance, regardless of how long ago it occurredÂ” (p. 405). Suggestions for Future Research Because the stories of the participants in this study included memories of mathematics classes that took place in the el ementary grades, and because it is in the elementary grades that teachers make sense of the manipulation of numbers and symbols, more research would be helpful in studying el ementary school teachers and their level of mathematics anxiety and their attitudes towards mathematics. Allen (2001) contends that these attitudes, whether positive or negative, perpetuate themselves in students. Participants in this study overwhelmingl y agreed that their college mathematics teachers played a crucial role in their percepti ons of themselves and their ability to learn mathematical constructs. Research conducte d by Arriola (1993) s howed that, for adult students in college developmental mathem atics classes, teachers should be nonthreatening and use a student-centered, active learning approach which stresses understanding over memorization and rote comput ations. More resear ch is needed to determine the ways in which differences, other than cognitive, can help or hinder mathematics learning.
191 As already mentioned, the participants in this study evidenced a noticeable degree of emotion when sharing their past and pr esent experiences with mathematics. The effects of emotion on cognitive functioning can be substantial (Walen et al., 2002). As educators, we are all very interested in helping our students learn, and more than that, helping to make learning meaningful in their lives. Acco rding to Boeree (2007) Â“an examination of meaningful learning makes it clear that the amount of the affect is an intrinsic measure of the meani ngfulness of the experience, i.e. of how import ant it is to the person.Â” Understanding ma thematical concepts was very important to the women in this study. If teachers could present mathemati cal concepts as challenging without being overwhelming, and could demonstrate the findin g of the solution as a real possibility, students would be motivated to learn. We cannot allow distress and anxiety to discourage our student s, to drive them into avoidance. Mandler (1989) points out th at Â“stress tends to d ecrease attention to peripheral events and to focus attentional conscious capacity on those aspects of the situation that the individual considers im portantÂ” (p. 9-10). Further research on the effects of emotion when sitting in a mathem atics classroom, note taking, peer behavior, taking mathematics tests, the timed aspect of test-taking, and the various formats, lengths, and rapidity of mathematic s tests would be useful. Self-motivation and aspirations were evident in the stories of the participants in this study. These elements will determine, in large part, what is made of the educational opportunities presented to them. A belief in one Â’s self-efficacy (Â“I know what to do and I know how to do it.Â”) is the ingredient necessa ry to continue to persevere as long as it takes to accomplish the task. Bandura (1997) posits that Â“the major goal of formal
192 education should be to equip students with the intellectual tools, efficacy beliefs, and intrinsic interests need ed to educate themselves in a variety of pursuits throughout their lifetimeÂ” (p. 214). Further research is need ed to study how psychos ocial processes are involved in the cultivation of cognitive competencies and contribute to their development. In addition, research studying how self-efficacy expectations impact the domain of mathematics could have consid erable utility for the understanding and treatment of mathematics avoidance. This study reinforces the recommendati ons of Carney-Crompton (2002) in that future research could clarify the role that age, intrinsic motivators, and child-rearing responsibilities play in the decision of non-traditional age wo men to return to school and in their strategies to survive and thri ve in a demanding academic environment. Postsecondary institutions might re-examine their commitment to providing opportunities for many types of students, including adult wo men, and ensure that financial resources as well as social support are available and accessible. Finally, further research is needed to study factors such as cl ass size, length of semesters, and length of class time. What class size is most conducive to mathematics learning? What length of se mester is the best overall for students to grasp the mathematical concepts in enough depth to bui ld self-confidence? What length of class time is best for teachers to present mathema tics material in such a way that engages studentsÂ’ varying learning styles? Non-traditional age women continue to face unique problems when it comes to learning mathematics. Although community college campuses have welcomed these women and acknowledge their presence, barrie rs to completing a degree continue to
193 exist. Administrator, faculty, and staff all mu st work together to create a user-friendly environment, especially in the area of math ematics education. We can all agree that, Â“adults and mathematicsÂ” is a complex subj ect where Â“emotional f actors are just as important as cognitive ones in the psyc hological learning processÂ” (Wedege, 1999, p. 206). Findings from this study may be applicable to a universal popul ation. That is, the identified themes may not be specific only to non-traditional age wo men. It is possible that interviewing a group of non-traditional age men would produce the same type of responses. A qualitative study interviewing male participants and searching for meanings they made from their mathematical experience may answer this question. Personal Reflections Several of the women in this study found strength and inspiration from their simple participation in this research. When they read their own transcripts and then saw, in the research findings, how the other partic ipants shared many of their own feelings, they commented about how that connection between themselves and the other participants had impacted their thinking about themselves. This metacognition, in fact, seemed to be an immediate boost to their self-esteem. Conducting this qualitative research study wa s an arduous task since it forced me to step into the shoes of 15 women whose lived experiences, although different in many ways, were much alike in that the learni ng and understanding of mathematics was a critical part of their self-c oncept. I could see myself and my own experience in many of their stories. In my work as a mathematics instructor in a community college, it is an ongoing rewarding experience to be a part of list ening to the stories of other non-tr aditional
194 age women attending this college. Hearing th eir descriptions of th eir lives and feelings about themselves relating to mathematics is inspiring and helpful. Without the perspective gained from this particular resear ch, I would not have the insightÂ—neither would I have a compelling desire to use that insightÂ—to stimulate a desire in my students to learn. My calling and privilege is to be an enabler, particularly to non-traditional age women, as they pursue the fulfillment of their academic goals.
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229 Appendix A Multiple Choice Questions for COMPASS 1. Learn about RACC What ONE source of information had the MOST influence on your attending RACC? A. Website B. Television ads C. Newspaper ads D. Friends E. Parents F. High school teacher or counselor G. College fair H. Presentation/interview with a college staff member I. Publication mailed to your home J. Billboards 2. Influence to attend What influenced you the MOST to attend RACC? A. Low cost B. Location/close to home C. Program of study D. Academic reputation E. Support services F. Campus climate G. Small class size H. Friends are attending RACC I. Parents 3. RACC 1st choice Was RACC your first choice when choosing a college? A. Yes B. No 4. ParentÂ’s 4 yr degree Do either of your parents have a 4-year college degree? A. Yes B. No 5. Size of Household What is the total number of people living in your household, including yourself? A. 1 B. 2 C. 3 D. 4 E. 5 or more 6. Library use How often do you use a library? A. Daily B. Weekly C. Monthly D. Rarely E. Never use it
230 Appendix A (Continued) 7. Library Resources What type of library resources do you use the most? A. Primarily books B. Primarily magazines & newspapers C. Primarily video & DVDÂ’s D. Primarily internet computers E. Many library resources F. None 8. Internet use How often do you use the internet? A. Daily B. Weekly C. Monthly D. Rarely E. Never use it 9. Internet used for What do you use the intern et for most of the time? A. Primarily email B. Primarily chat rooms C. Primarily shopping or banking D. Primarily research E. Primarily news & current events F. Primarily games G. Primarily downloading software H. Many purposes I. Never use it 10. Can you type on PC Can you type on a computer keyboard? A. Yes B. No 11. Internet at home Do you have internet service at home? A. Yes B. No 12. PC at home Do you have a computer at home? A. Yes B. No 13. Software use Do you use any of the following software? A. Primarily word processing (WORD) B. Primarily data base management (ACCESS) C. Primarily spreadsheet (EXCEL) D. Primarily presentation (PowerPoint) E. Primarily WORD and ACCESS F. Primarily WORD and EXCEL
231 Appendix A (Continued) G. Primarily WORD and PowerPoint H. Primarily ACCESS and EXCEL I. Primarily EXCEL and PowerPoint J. None of these 14. On-line courses Would you be interested in taking courses on-line? A. Yes B. No 15. Work exp. in health Do you have any work experience in the health services field? A. Yes B. No 16. HS math level What was the highest level of high school math you completed with a Â“CÂ” or better? A. General Math B. Pre-algebra C. Algebra I D. Geometry E. Algebra II F. Trigonometry G. Pre-calculus H. Calculus I. Unsure 17. HS chemistry level What was the highest level of high school chemistry you completed wit h a Â“CÂ“or better? A. No chemistry B. Chemistry C. Advanced chemistry D. Unsure 18. HS biology level What was the highest level of high school biology you completed with a Â“CÂ” or better? A. No biology B. Biology C. Advanced biology D. Unsure 19. 2nd language Â– speak Do you speak a language other than English at home? A. No B. Yes, but not often C. Yes, regularly 20. 2nd language Â– read Do you read a language other than English? A. No B. Yes, but not often C. Yes, regularly
232 Appendix A (Continued) 21. Type of English Which of the five major types of English have you learned? A. American Â– USA B. British Â– England C. African/Middle Eastern D. South Asian Â– India/Pakistan area E. Caribbean 22. Most enjoy Â– sub What subject will you MOST enjoy? A. Business B. English C. Health care D. Humanities/fine arts E. Math F. Natural science (biology, chemistry, environment) G. Social science (history, psychology, sociology) H. None 23. Least enjoy Â– sub What subject will you LEAST enjoy? A. Business B. English C. Health care D. Humanities/fine arts E. Math F. Natural science (biology, chemistry, environment) G. Social science (history, psychology, sociology) H. None ESL Testing Questions Only 24. ESL-EngLng How would you BEST describe your feelings about learning English? A. Enjoyable B. Frustrating C. Challenging D. Boring E. Necessary 25. ESL-EngCulture Which answer BEST describes your adjustment to life in America? A. Very difficult B. Difficult C. Not bad D. Easy E. Very Easy
233 Appendix B Adult Informed Consent
234 Appendix B (Continued)
235 Appendix B (Continued)
236 Appendix C Interview Questions 1. Why are you pursuing a college education? 2. Describe your study habits. (time, place, etc.) 3. Would you define yourself as being smart? Why or why not? 4. What is your earliest memory of math? 5. Describe a female or male teacher who had a positive impact on the way you feel about mathematics. 6. Describe your most stressful mathematics classroom experience from kindergarten through college. 7. How could that experience (from question 6) have been made more constructive? 8. How do (or did) your parents feel about your math performance in school? 9. What are your feelings regarding mathematics? 10. Do you expect to be successful in passing the math courses in your RACC program of study? Time necessary to transcribe ________________________
237 Appendix D Code Mapping: Constant Comparative Analysis Pattern Variables 1. Strong evidence of personal goals 5. Influence of teacher apparent 2. High level of study importa nce 6. Classroom stress common, either currently and/or in the past 3. Good self-perception/Good selfefficacy 7. Lack of family contribution by either father or mother 4. Memories were vague/ memories were clear/memory of doing poorly and feeling left behind 8. Strong feelings of being successful and having a plan to succeed. Initial Codes 1A. Financial Goal 5A Influence-Mentor 1B. ** Personal Goal 5B. **Influence-Teacher 1C. Work Related 5C. Influence-Peer 2A. Importance of study-Low 6A. **Classroom experience stress-yes 2B. Importance of study-Moderate 6B Classroom experience stress-no 2C. **Importance of study-High 7A. Family member support 3A. **Self-Perception-G ood 7B. Father support 3B. Self-Perception-Poor 7C. Mother support 3C. **Self-Efficacy-Yes 8A. **Feelings currently-positive 3D. Self-Efficacy-No 8B. Feelings currently-negative 4A. **Memories vague 8C. **Succeed-yes 4B. **Memories clear 8D. Succeed-no 4C. **Remember doing poorly 8E. **Succeed-plan 4D. **Feeling left behind Surface Content 1A. Financial Goal (7) 5A. Mentor (5) 1B. Personal Goal (11) 5B. Teacher (10) 1C. Work related (6) 5C. Peer (0) 2A. Study importance-high (11) 6A Classroom experience-stress (7) 2B. Study importance-moderate (6) 6B. Teacher (4) 2C. Study importance-low (2) 6C. Peer (2) 2D. Regular times (3) 2E. Regular place (0) 7A. Family member (4) 2F. Short periods (5) 7B. Father (2) 2G. Long periods (5) 7C. Mother (6) 3A. Self-perception Good (12) 8A. Power to change things (1) 3B. Self-perception-Poor (7) 8B. Powerless to change things (11) 3C. Self-efficacy-yes (10) 8C. Giving up (7) 3E. Self-efficacy-no (7) 3F. Comparison to others (4 ) 9A. Feelings positive (14) 9B. Feelings negative (4) 4A. Memory clear (12) 4B. Memory vague (10) 10A. Succeed-yes (15) 4C. Remember doing poorly (15) 10B. Succeed-no (0) 4D. Left Behind (13) 10C. Plan (11) 4E. Pride (5) 4F. Embarrassment (6) ** highest reported response rate from participants
238 Appendix E Letter of Permission
About the Author Jo Ann Rawley was a non-traditional ag e student when she entered the world of academia. She began her postsecondary education at St. Petersburg College, in Tarpon Springs, Florida, completed her Bachelor of Science and Master of Science Degrees at National Louis UniversityÂ—Tampa Center, and completed her Doctor of Education Degree at the University of S outh Florida. Her interest in mathematics education grew while working as director of the Math La b in the Learning Support Center on the Tarpon Springs Campus of St. Petersburg College. Sh e recognized the need to give attention to non-traditional age students, pa rticularly women, who struggled with mathematics and often experienced mathematics as an obstacl e in reaching their educational goals. She initiated research into womenÂ’s career choi ces, mathematics anxiety, and other feminist issues. Currently she is employed as an inst ructor in the Business Division at Reading Area Community College in Berks County, Pennsylvania.