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MATHEMATICS, CULTURE, AND QUESTIONS: A STUDY IN THE CULTURE OF THE MATHEMATICS CLASSROOM THROUGH A RANDOM QUESTIONING EXPERIMENT By PAIGE ALLISON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Paige Allison This work is dedicated to my children, Julie and Tommy, who have been with me every step of the way; to my Mother, Selma; to those who motivated and inspired me to do this work; and to all the students and teachers who have touched my life and made this work possible. ACKNOWLEDGMENTS I am most grateful to all those who have been a part of this process. My first contact with the University of Florida Department of Anthropology was with Allan Burns, through Pat Waterman, who encouraged me to do the work that I wanted to do. Dr. Burns said to me during our first conversation, "Don't ever let anyone tell you that you can't do it." I took his advice to heart and it has served me well. As my committee chair, he fostered me as an anthropologist and directed me adroitly, carefully nudging me in the right direction, even when I wasn't sure that I wanted to go there. He stuck with me through it all, and I am truly thankful for all of his help, support, and advice over the years. Without Allan's initial and long term support and encouragement I might never have embarked on this project. The many kindnesses I have received from Russ Bernard over the years have been invaluable. He has always been there for me. I will never forget the thrill of learning some clever way of counting or measuring things and the way his enthusiasm for his work was such an integral part of his teaching. Not only did the subject matter appeal to me; Dr. Bernard is very simply a fine teacher who is inspired by his work and passes that inspiration on to others. I am most appreciative of Sue Boinski, who has been my constant navigator and reassurance that all will be well. Sue has been a mentor and helped me negotiate my way through the maze of graduate school in such a way that she seems invisible to all but me. As professional mothers, we have shared support systems and advice when either of us was in need. To sum it up, Sue has always had my back. Several years ago, I had the pleasure of having dinner in San Francisco with Catherine Emihovich. I came home from that meeting remembering her phrase, "Life is long." Everyone else says, "Life is too short." I added her saying to my collection and used it often. What a gift it was to find her moving to UF. I asked her to meet me for lunch upon her arrival and explained to her that I was an "at risk" student. Before lunch was over, she had me pointed in the right direction again. I appreciate her support and her continued advice and assistance. I extend many thanks to Thomasenia Lott Adams, who most graciously came to my rescue when I was in need. From the day she defended her dissertation until the present, she has always been a positive role model who conducts herself with great grace and dignity. It is a special honor to have her, a student of Mary Grace Kantowski, as a member of my committee. I also want to extend my sincere appreciation to all those people who are a part of my life who have in one way or another supported me or contributed to me making it to this point. To those whom I have lived, loved, and laughed with, I thank them all. Many thanks go to my extended family of dancers, drummers, musicians, and kindred spirits of many varieties who have been my refuge and strength. Special thanks go to Julie and Tommy who began this degree with me as little children who would come and sleep on the couch near where I worked on late nights and who have given me their love and support through thick and thin. To hear my daughter say to me, "You know I'm proud of you, don't you?" brings a big smile and tears to my eyes. I wish also to thank my fellow students, friends, and neighbors who have wondered just how long this was going to go on and who have patiently encouraged me and never given up; and my students, who have taught me so much over the years and without whom this project would never have come to fruition. I wish to thank Robert, my friend of the heart, for everything! Sincere gratitude goes to Heather McIlvaineNewsad, who continued to pursue this project even when I had given it up. Heather's persistent crusade for funding and the generosity of Western Illinois University were critical in the implementation of this study. Additionally, I appreciate the firm but kind words of my friend Sheila Jeffers, who said, "Just do it, Paige!" Spending a weekend with these two women made me realize that I simply had to do this project. The invaluable, support, friendship, data analysis, and editing contributions of Hank Green facilitated the seamless integration of qualitative and quantitative data and the completion of this project. I extend many thanks to my dissertation group, and to the SOR group who have always been helpful, kind, and encouraging. An especially exuberant thank you goes to Stefany Burrowes who adroitly talked me through the entire dissertation process and the peripheral life processes in such a wonderfully kind, enlightening, supportive and juicy manner. Stefany's faithful support facilitated me completing this project sanely, and with grace and style. A long overdue thank you is extended to the faculty, staff, and students of the UF Department of Anthropology and the faculty, staff, and students of OGHS and the OCSB. Without the help, encouragement, facilitation, and support of all these wonderful folks this project would not have been possible. I would also like to thank James "Skippy" Albury for his tireless service and devotion to the Empire, Butch and Arnold for always curling up next to me and just being there for me as I worked, and Ernie Lado for being a really fine father to our children. There are far too many names to remember or list, so to all those who have been a part of this project in any way, I offer my sincerest gratitude and appreciation. TABLE OF CONTENTS A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TA BLE S ............... .............. ................ ......... .............. .. xii LIST OF FIGURES ......... ....................... .......... ..................... xiv A B S T R A C T .......................................... ..................................................x v 1 IN T R O D U C T IO N .................................................................................. 1 B background .......................... ........ ................................... ...............1 Research Questions and Objectives ..... .....................................................2 Applied and Theoretical Significance of the Research ..............................................6 A applied Significance .................................................. .. ... T theoretical Significance ................................................................... ............... R research Site and P population ........................................ ......... ............... .. ....8. The Cultural Context of Oak Grove High School: A Sense of Place ..........................9 D ata C collected ....................................................... 13 F field D ata ................................................................. ...... .. . ........ 13 A n a ly tic D ata ................................................................................................. 1 3 Organization of Chapters ................................. .......................... .........15 2 LITERATURE REVIEW .............................................................. ...............16 T theoretical F ram ew ork ...................................................................................30 C critical R ace T h eory ...................................................................................... 3 0 C critical P ed ag ogy ....................................................................................32 Research Objectives, Expectations, and Hypotheses ..................................... 39 3 RESEARCH DESIGN AND METHODOLOGY ................................................42 Introduction .................................................................. ..... .. ............... 42 Quantitative Research Design and Methodology: The Random Questioning Strategy E xperim ent............................................... ...............42 Perform ance M measures ......................................................... 45 P articip action ...................................................................................... 4 6 A attitudinal M measures .................................. ......................................... 46 T h e scales .............................. ........ ..... ................................................4 6 Implementation and administration of the scales .................................47 Teacher Perceptions and Feed Back ........................ ...................................... 48 Teacher C om pensation .............................................. .............................. 48 V alidity and Confounds......................................................... .............. 48 Lim stations .................................... ............................................49 Qualitative Research Design and Methodology: The Focus Groups..........................50 In tro d u ctio n ................................................................................................... 5 0 Methodology ........... ............. ......... .......... 51 E thnographic M ethods.............................................................. .....................53 V ariables .............................................. 56 Limitations ............... ........ ......... ......... 56 4 RESULTS FROM QUANTITATIVE PORTION OF THE STUDY: THE RANDOM QUESTIONING STRATEGY EXPERIMENT .............................. 58 In tro d u ctio n ............................................... .. .................. ................ 5 8 D ata Classification and V ariables......................................... .......................... 60 A cadem ic Perform ance M measures ........................................ ......................... 62 National Proficiency SurveyMathematics............... ......... ..............62 Student Grades....................................................... ... ............ 62 Discussion of Academic Performance Measures ............................................62 P participatory M measures ....................................................................... ..................63 Attitudinal and Belief M measures ...... .............................................. .. ............... 64 Evaluation of Survey D ata........................................................ ............... 64 Anxiety ............... ......... ...................... ....... ....... 64 M ath V alue .......................................... ......... .................. ......... ..... ..........64 SelfC oncept......... ......... ............................................ ............... 66 E njoym ent ..................................................................... 67 M otiv action .....................................................................67 B e lie fs .................................................. ................ 6 8 A ttitu d e s .................................................... ................ 6 9 D discussion of Survey D ata ............................................................................ 69 Betw eenGroup Com prisons ...................................................................... 70 Mathematical performanceNPSS mathematics......................................71 M ath anx iety ............................................. ................ 72 V alue of m them atics.............................................................. ............... 74 SelfConcept in m them atics ............................................ ............... 75 Enjoym ent of m them atics.................................... .................................... 76 M them atical m otivation......................................... .......... ............... 77 B eliefs about m them atics ........................................ ....................... 78 Attitudes toward mathematics ........................................... ..............79 Discussion of BetweenGroup Comparisons.....................................................79 Sum m ary ...................................... ................. ................. .......... 86 K ey F in d in g s.................................................... ................ 8 8 R Q S ................... ....... ................................ ................ 8 8 Tl and T2 BetweenGroup Comparisons.........................................................88 5 RESULTS FROM QUALITATIVE PORTION OF THE STUDY ........................... 90 T h e F o cu s G rou p s............. .................................................................. ........ .. ...... .. 9 0 Objectives and Questions ........................................ .............................90 D ata A n a ly sis ................................................................................................. 9 1 T h e F ocu s G rou p s........... ..... ......................................................... .... .... .... .. 92 M ajor Them es and SubThem es................................... .................................... 93 W hat Teachers D o ............................................... ........ .. ............ 95 Teaching and instruction ........................................ ......................... 95 Ethnicity and gender teacher............ ............................. 104 W hat Students D o................................... .. ......... .. .. .... .. ........ .... 109 Student effort................................ .......... ........ ... .... .. ........ .... 110 The value of education ........................................................ 112 Future plans .................................... ......... .. ...............................113 Student feelings and stress ......... ................ ............... 114 Student behavior ..... ........... .. ......... ........................ .... ........ ..... 116 M ath rating ....................... ......... ...............................121 Textbooks ................................. .... ............... 124 Best Practices ...................................... ......... 124 Ethnicity and gender student ................ ............ ........... ....... 128 Classroom Questions and the Random Questioning Strategy (RQS) .............133 C classroom questions........................................................ ............... 133 Focus groups R Q S............................ ....................... ...... ....136 Teacher perceptions and feedback surveys and comments RQS ............138 Students' perceptions, attitudes, and feedbacksurveys and comments R Q S ........................................................................... 14 1 F ollow U p R Q S Q questions .................................................... ............... ....14 1 What Administrators Do: The School Environment .....................................144 Class size .................................. .......................... ... ........ 145 G ood and bad classes ........................................................................... 146 V iew s From the Field ........................... ........ .. ...... ...............148 Outside Participant Observations ........................................... ............... 148 A L ifetim e in the F ield ............................................... ............ ............... 149 Leadership for Social Justice....................................... .......................... 156 T teaching assignm ents ........................................ .......................... 158 C u rricu lu m .............................................................16 3 Suggested im provem ents.................................... ..................................... 169 Practicing leadership for social justice......... ......................................... 172 D discussion of Qualitative R esults................................... ............................. ....... 173 K ey F in ding s ................................................................................177 Focus Groups and Participant Observation...................................................... 177 T each ers .....................................................................177 Students ................................... .............................. ........ 177 Adm inistrators .................. ............................ ...... ................. 178 RQS Feedback ............ .................................... 178 Focus G roup D ata Table......................................................... ............... 179 x 6 CONCLUSIONS AND CONTRIBUTION...................... ..... ............... 181 Data Summary and Conclusions....................... ...... ...............181 Quantitative Data SummaryThe Random Questioning Strategy E xperim ent ................................... ............................181 Qualitative Data SummaryFocus Groups, Surveys, and Observations .........183 Conclusions and Interpretations ................ ................................ .................. 186 Random Questioning Strategy Experim ent...................................................... 187 B etw eenG roup Com prisons ........................................ ........ ............... 187 C contributions .......................................... .......... .. .... ................. 189 Personal ContributionsUnderstanding the Research Process ........................189 Practical ContributionsProblems with Educational Research ....................190 Methodological ContributionsBlending Qualitative and Quantitative Research ............. ...... ....... .................... .... .. ........ 192 Theoretical Contributions ................... ......... ..... ............194 Implications for Educational Policy ............................... ............ ............. 197 Im plications for Adm inistrators ...................................... ............... 197 Im plications for Teachers.......................................... ......... ... ............... 198 Future Research ............. ........... .... ...... ......... ....... .. ..... ........... .. 199 SAMPLE FOCUS GROUP QUESTIONS ................................ ............................... 201 Focus G group Q uestions.................... .... .......................................... ............... 201 Introduction, General Business and Grand Tour....................................201 School Envirom nent ................. ... ............... .. ................... ...............202 Su gg estion s ...................... ................................... .................................2 02 G ender and E thnicity ...................... .. .. ......... .. ..................... ............... 202 E th n ic ity : .......................................................................................2 0 2 G en d e r ...............................................................2 0 3 Questioning ........................................................................ ......... ................. 203 G getting H elp/Student Effort ....................................................... 204 Class Atmosphere ............................... ............... 204 F future P lan s ...................... ................................... .................................205 Best Practices ............... ......... .............. ...................205 T e stin g /S tre ss .............................................................................................. 2 0 5 R Q S ...............................................................2 0 6 C lo su re ...................... .. ............. .....................................................2 0 6 LIST OF REFERENCES ..................................................... .. .......207 BIOGRAPHICAL SKETCH ................................................ ............... 222 LIST OF TABLES Table page 1 Performance and Attitudinal Measures Random Questioning Strategy ExperimentDistribution of Students by Treatment, Gender, and Ethnicity .........44 2 Question Answering Behavior Random Questioning Strategy Experiment Distribution of Students by Treatment, Gender, and Ethnicity.............................44 3 Student Focus Groups: Perceptions and Experiences In the Math Classroom Distribution of Students by Group, Gender, and Ethnicity .................. ........... 54 4 Performance and Attitudinal Measures Random Questioning Strategy Experiment Distribution of Students by Treatment, Gender, and Ethnicity ........59 5 Question Answering Behavior Random Questioning Strategy Experiment Distribution of Students by Treatment, Gender, and Ethnicity............................59 6 Focus Group Interviews: Major themes and subthemes......................................93 7 Focus Groups: Positive StudentTeacher Relationship...............................96 8 Focus Groups: Explains W ell ........... .. ............ ..................... ... 98 9 Focus Groups: Extra Help ............. ... ...................... ........................ 100 10 Focus Groups: Repetition/Works Problems......................................100 11 Focus Groups: Explains Poorly or Not at All .................................... ........101 12 Focus Groups: U nfair Treatm ent...................................... ......................... 103 13 Focus G groups: E ethnicity ........... .................................................. ............... 105 14 Focus G roups: G ender................................................. .............................. 105 15 Focus Groups: Classroom Participation.............................. ..... ........ .......... 110 16 Focus Groups: OffTask Activity ........ ........................... ..................... 111 17 Focus Groups: OutofSchool WorkStudy................................... 112 18 Focus Groups: Failure to Engage in OutofSchool WorkStudy ........................112 19 Focus Groups: The Value of Education ..... ................................................113 20 Focus Groups: Student Stress/Frustration .............................. ........ ...............114 21 Focus Groups: Student Behavior .............................. 117 22 Focus Groups: Math RatingPositive ...................................... ............... 122 23 Focus Groups: Math RatingNegative.....................................................122 24 Focus G roups: A sk Q questions ........................................................................ 134 25 Focus Groups: Answer Questions ..................................................................... 134 26 F ocu s G roups: B ad C lasses .................................................................................. 147 27 Focus Group Data: Major Themes and SubThemes................. ............. .....179 LIST OF FIGURES Figure page 1 ANOVAThe Value of Mathematics Scale By Ethnicity................................65 2 ANOVAThe Selfconcept in Mathematics Scale By Ethnicity .........................66 3 ANOVAMotivation in Mathematics Scale Treatment vs. Control....................68 4 ANOVABeliefs About Mathematics Scale by Ethnicity ...................................69 5 Between Groups Pre and PostTest Rankings National Proficiency Survey M ath em atic s ....................................................... ................ 7 2 6 Between Groups Pre and PostTest Rankings Mathematics Anxiety Scale...........73 7 Between Groups Pre and PostTest Rankings Mathematical Value Scale .............74 8 Between Groups Pre and PostTest Rankings SelfConcept in Mathematics S c a le ............................................................................. 7 5 9 Between Groups Pre and PostTest Rankings Enjoyment of Mathematics Scale ..76 10 Between Groups Pre and PostTest Rankings Mathematical Motivation Scale.....77 11 Between Groups Pre and PostTest Rankings Beliefs About Mathematics Scale..78 12 Between Groups Pre and PostTest Rankings Attitudes About Mathematics S c a le ............................................................................. 7 9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MATHEMATICS, CULTURE, AND QUESTIONS: A STUDY IN THE CULTURE OF THE MATHEMATICS CLASSROOM THROUGH A RANDOM QUESTIONING EXPERIMENT By Paige Allison December 2005 Chair: Allan F. Bums Major Department: Anthropology This dissertation reports the results of an experiment to test whether removing control of whom to call on from the teacher in highschool mathematics classes, affects students' academic performance in, attitudes and beliefs about, mathematics and levels of classroom participation. In experimental classrooms, students were called on randomly to answer the teacher's questions. In the control classrooms, the teacher decided whom to call on for each question. Students in both intervention and control groups were given standardized academic tests in their subject area and questionnaires about their attitudes toward and beliefs about mathematics. Results indicate that calling on students at random does not affect change in student performance or performance rates, and does not affect their attitudes or beliefs about mathematics. However, ethnicity is a significant factor in accounting for differences in academic performance, in attitudes and in beliefs. White students perform at the highest levels followed by black and Hispanic students. Despite this, white male students have consistently higher anxiety about mathematics than do their female counterparts. Black students have the highest enjoyment of mathematics and highest motivation scores and Hispanic students have the highest scores on the value of mathematics. Despite this, both groups' academic performance scores are the lowest. Within ethnicity, gender is significant with black and white female students academically outperforming their male counterparts, and the reverse being true for Hispanic students. Focus group data show that competent teachers who develop a positive working relationship with their students, provide good quality and quantity instruction, and treat students fairly are the most effective; large class size and disruptive student behavior are detrimental to the learning process. Being prepared for and paying attention in class, doing routine homework and study, taking notes, asking questions, and seeking out extra help are the behaviors exhibited by highly successful students, along with a positive attitude. Curriculum offerings are often not broad enough to adequately address the varying needs of the entire student body. CHAPTER 1 INTRODUCTION Background As a high school mathematics teacher in the public schools of north central Florida for the past 20 years, I have had a firsthand opportunity to observe some of the inequities in math education. In the early 1990s I became aware of research by Myra and David Sadker (1994, 1985) and others on gender inequity in mathematics classrooms (AAUW 1992). As I became more familiar with this research, I began my own informal experiments in my classroom. I became more aware of which students I called on to answer questions. I was clearly calling on more male than female students. As I continued to try to remedy this situation, I found that in order to maintain some semblance of classroom order I had to call on more male students to keep them from acting out and to maintain their attention. Later I became aware of further research (Weisbeck 1992) showing that teachers respond to pressure in the classroom. This was clearly what I was experiencing; I was being pressured by male students to give them more of my attention either by answering their questions or by calling on them to keep them in line. I also became aware of research on women's ways of knowing and the tendency for female and minority students to be what are called connected learners, which means that they learn by access to others' experiences and by having a connection with the material, while male students tended to be separated learners who learn individually with little need for connection to the material (American Association of University Women AAUW 1992; Belenky et al. 1986; Perry 1970). I began to consider applications of all this work within my own classroom. I noticed that female students tended to respond to activities that were relevant or connected to their current interests and experience and that they responded less well to applications that were separated from their areas of interest. Given the opportunity to learn within context and alongside other students, the female students seemed to be more involved in the classroom. Additionally I observed that unless heavily coaxed, female students rarely contributed to the questionandanswer portion of the classroom routine. Time and again, female students expressed that they would rather ask me privately for help or that they were too shy or embarrassed to ask in class. Seeing the direct applications of these theories in the mathematics classroom led me to the present area of research. Research Questions and Objectives There is real, although subtle, intimidation that takes place in the classroom reinforcing the idea that women and minority students cannot do math as well as white male students can (AAUW 1992). Myra and David Sadker (1994) have shown that teachers tend to call on white male students more frequently than other students, respond to their questions and requests for help differently, and provide them with entirely different experiences in the classroom. This practice is routine across the curriculum but is especially evident in the area of mathematics. It is well documented that women and minorities on average tend to perform to a lesser standard than white men on portions of standardized math tests (AAUW 1998; Croom 1997; Fennema 1996). In addition, the traditional math curriculum is presented at the time that is most developmentally appropriate for male students, putting female students at a distinct educational disadvantage (AAUW 1992). Built into these practices is the expectation that women and minority students cannot perform as well in the classroom as white male students. This resulting intimidation of female students, for the most part, is not intentional. Nevertheless, like racism and sexism in general, it is a product of history and is part of contemporary American life (AAUW 1998; Leder 1990). John Ogbu (1994) states that lack of opportunity and also attitudes toward minority students and minority students' responses to these attitudes contribute to the poor educational performance of minority students. Even people who make an effort to rid themselves of sexist and racist baggage find themselves falling into the trap of behaving in ways that perpetuate this problem. Research has shown that even when teachers are aware of these behaviors and genuinely try to change them within their own classroom, these behaviors persist (Roth 1996). The research reported here was an attempt to test the effect, if any, of behaviors among high school math teachers that inhibit response by young women. The idea was to take the choice out of the teachers' hands of which student to call on by forcing the teachers to call on students at random. Central to the idea that all students have the same chance of being called upon each time is the expectation that all students have the potential to answer each question. Since Rosenthal and Jacobson's early work on expectancy (1968), research has continued to show that students live up or down to their teachers' expectations. While other factors also affect students' performance, teacher expectations have been consistently demonstrated to both positively and negatively affect student performance (Brophy 1983; Good and Brophy 1987; Goldenberg 1992, 1989a, 1989b; Wentzel 2002). Central to this issue as well is the idea of equity vs. equality. It is imperative not only that all students receive the same amount of attention in the classroom but also that they receive the kind of attention that is most appropriate for them. The prevailing view in education is that since all students can go to school, they have the same opportunities for success. We know, however, that women and minority students do not pursue mathematicsrelated careers to the extent that men do. There is a distinction between equity and equality in education. An equitable educational experience implies that students receive the education appropriate for them to achieve a shared standard of excellence as that of their counterparts (Scott 1988). By explaining and documenting students' participatory strategies, perceptions, and behavior in this setting, we will gain a better understanding of the mathematics classroom dynamic, especially as it relates to the questionandanswer component of mathematics education. This knowledge in turn will enable us to implement institutional reforms that will deliver to students an equitable educational experience and equal opportunity to achieve a standard of excellence. This project has four objectives: to (1) examine how teachers inadvertently communicate the message to students that female and minority students cannot do math; (2) gain a clearer understanding of the mathematics classroom dynamic in general; (3) determine the strategies used by male and female students to ask and answer questions in the classroom environment; and (4) determine the best practices, behaviors, and attitudes of women and minority students who perform at a high level in mathematics. The research will document students' perceptions of the mathematics classroom, document their strategies for participation or avoidance of participation, and test whether the removal of inadvertent bias in choosing which students to answer questions improves students' attitudes toward mathematics and their academic performance. A key question is how teachers communicate to students the cultural idea that women and minority students are destined to perform at a lower level than their white male classmates. Another key element of this research is student behavior. Parents and society in generalnot only teacherscommunicate to children a socially acceptable mode of behavior. Girls are conditioned to "act like a lady," to quietly wait their turn, and to value their relationships with other people. Girls are often the most well behaved students in the classroom. However, this "good behavior" may disadvantage them with respect to learning. Female students are rarely aggressive in the classroom. They sit quietly and wait their turn. In my experience, seldom does a female student exhibit forceful behavior in order to participate in class. By contrast, male students often exert pressure on the teacher by acting out in the class, waving their hands, calling out, interrupting, and generally drawing attention to themselves. In an effort to maintain order in the classroom, teachers respond to this pressure. I hypothesize that there are significant differences among male and female students in their general perception of the mathematics classroom and in their participation strategies. In my position as a mathematics educator in the public schools of Oak County, Florida, I was in a position to implement a research program in mathematics classrooms to test my hypotheses. I was able to investigate student perceptions, behaviors, and participation strategies using direct classroom observations and ethnographic interviews. These data yield insight into students' perceptions of the mathematics environment. From the experimental data and the focus groups, I was able to examine closely the culture of the mathematics classroom and the factors that contribute to some groups of mathematics students becoming marginalized. In this research project I draw on theory and methods from cultural anthropology, educational psychology, and women's studies to examine the effect of teachers' unintentional communication of negative expectations on female and minority students. Additionally, I investigate students' perceptions of the mathematical environment in general and document student participation strategies and behavior in the mathematics classroom. * Objective 1: To elicit and document students' perceptions and experiences as students of mathematics in order to gain a clearer understanding of students' participatory behavior in the mathematics classroom. * Objective 2: To elicit and document the best practices of highly successful women and minority mathematics students and to identify and document the attitudes, behaviors, perceptions, and circumstances that contribute to the mathematical success of these students. * Objective 3: To measure the effectiveness of a random questioning strategy for increasing academic achievement and improving attitudes toward mathematics for female and minority students in a secondary setting. * Objective 4: To determine the extent to which gender is the primary factor with respect to ethnicity, class, and age for success in the mathematics curriculum. * Objective 5: To document the effect that use of the random questioning strategy intervention has on teachers' and students' classroom activities and behaviors. Applied and Theoretical Significance of the Research Applied Significance Mathematics is known to be the gatekeeping course for many college preparatory courses in high school and to highpaying scientific and technical career tracks in college. Mathematics even serves as a gatekeeper for athletic clearinghouse scholarship eligibility. According to Sells (1980, 1975), 75 percent of college majors are out of reach if a student does not have a full set of high school mathematics skills. This affects men of color and women in particular, who tend to engage in high school mathematics at a lower level than white men. Theoretical Significance Gender Gaps (AAUW 1998) documented that teachers receive little or no training (less than 2 hours per semester) in gender equity issues from schools of education. Even less time is spent on issues of cultural and ethnic diversity. Established teachers who were trained prior to the recognition of the need for gender and multicultural teacher education have had no such training unless their individual school districts have provided it. Research over the last decade has shown that males, females, and minorities have different classroom experiences, which in turn influence their future career choices (Schwartz and Hanson 1992). Research by Ma and Willms (1999), Simpson and Oliver (1990), Lipps (1995), Finley, Lawrenz and Heller (1992), and Kahle and Meece (1994) indicates females and minorities express more negative attitudes toward math and science than do males, regardless of their ability. Not surprisingly, classroom experiences influence students' future choices of science, technology, engineering, and mathematics (STEM) related careers. Attitudes present in the classroom are typically a reflection of cultural attitudes in general. Education can roughly be described as the means by which a culture acquires, transmits, and produces knowledge for its people to function within that culture and adapt to the given environment in which that culture survives. Education is the process by which culture continually adapts and perpetuates its own existence (Levinson 2000). Culturally, the purpose of school is to impart norms such that young people are taught to live within cultural expectations. Schools teach children to think and act within the expectations of their culture, since thinking out of the box and beyond cultural boundaries is a threat to cultural preservation. This is done within the school setting via what Jules Henry (1963) described as "noise." Noise consists of the lessons a student learns in school that are not directly related to the subject area at hand; that is, children learn many social rituals to help them in a consumption based culture to hate, compete, benefit from the failure of others, find the best angle, and so forth. School is a place of mass enculturation that serves to maintain the status quo (Henry 1963). The present research will address the behaviors that contribute to the notion that women and minority students cannot do math and will document students' cumulative lived experience in mathematics class and best practices of successful women and minority students. By addressing these issues, this research will contribute further understanding to the persistently unanswered question of why women and minority students tend to perform to a lesser standard than that of white males in mathematics. Research Site and Population This research took place in Oak Grove High School (OGHS) in Oak County, Florida (OGHS and Oak County are pseudonyms). I am a member of the mathematics department and teach at OGHS. This was convenient in terms of carrying out the research, and it also met the requirements of the study in terms of ethnic composition, class, and gender for the research population. Oak County is a highly heterogeneous community in terms of both ethnicity and socioeconomic status (SES). OGHS is one of the county's larger secondary schools and is the most central of the city high schools in Oak Grove. OGHS is an ESOL (English for Speakers of Other Languages) Center with a very heterogeneous multicultural population. Thus, OGHS was an excellent environment for a series of focus groups with students about what goes on in the mathematics classroom. It was also a place where I could field a quasiexperiment examining the effects of randomly choosing students to answer questions in the classroom on academic performance and attitudes toward mathematics. Anthropological research requires a different approach than many of the studies cited in this paper, which have utilized surveys as their primary data collection method. This study is a multimethod approach that includes both qualitative and quantitative methodological and theoretical approaches to examine issues of gender and ethnicity in the mathematics classroom. This multimethod approach provides an interesting and thoughtprovoking platform from which to carry out this research project. Anthropology mandates that the researcher be personally engaged with the research community (Bernard 2002). In other words, anthropologists use participant observation and spend as much time as possible within the research community. Thus, in addition to survey data, experiment data, and student performance data, this study includes extensive ethnographic, or first hand, interviews, focus groups, and researcher observations. The data collected in this study are examined across gender and ethnic lines for comparative analysis on several levels. The Cultural Context of Oak Grove High School: A Sense of Place Oak Grove is the cultural center of its region and is described as one of the most desirable places to live in Florida, with a population of approximately 117,000. OGHS is one of three city high schools in its district. The larger district (county) has six high schools in total, including three rural schools. Of the three city high schools, I would describe Oak Grove as the median high school, situated in between the more affluent Buckhead and the lower income area school, Wilston. All three high schools have magnet programs attempting to lure choice students to their schools. Oak Grove's magnet programs are the Institute of Health Professions where students are enrolled in a specialized curriculum geared toward the allied health professions such as nursing and other medical technical positions. Wilston hosts an International Baccalaureate (IB) program, while Buckhead hosts an entrepreneurial center. Oak Grove is literally the middle of the road school in this area, but OGHS is in the process of bringing a program similar to the IB in an effort to increase funding and bring back to OGHS the advanced students it has typically lost over the years to the upper level program at Wilston. Oak Grove has many of the characteristics of an inner city school despite the fact that Oak Grove is a small city. The students at OGHS range from poor urban minority students living below the poverty level (including some who are actually homeless), to very affluent highSES students and everything in between these two extremes. As the ESOL center for the region, any students who arrive and cannot pass an English exam are enrolled in the ESOL center until they learn enough English to successfully function in their zoned school. With a high influx of academic, research, and medical personnel in the area, many students of different ethnicities attend OGHS. Given the full range of SES as well as a wide range of ethnicities at OGHS, there is a multicultural atmosphere with a high degree of inclusion and tolerance for others. It is not unusual to see mixed ethnicity couples together in the hallways and at school social events. The gay and lesbian students are not as open as are the mixedethnicity couples. There is an unofficial antidiscrimination club that informs students about different religions, ethnicities, and lifestyles. It encourages open mindedness and embraces diversity. The student population at OGHS is quite diverse and seemingly accepting of each other. There are occasional fights, but for the most part they are not based on race or ethnicity. Over the years there have been what would I term gang scares resulting in dress code changes and the banning of all hats and other headgear from campus to avoid tagging with hat brims or bandanas. However, in my experience at OGHS over the past 14 years, I have noticed little obvious gangrelated activity. I have had students who I know have shot someone, and I have had the children of state politicians in my classroom. This heterogeneous mix of students gives OGHS what feels to me a fairly balanced and authentic atmosphere. There is a small but visible contingent of punkgrungemetal kids who reject any labels but are usually quite intelligent, thoughtful, pleasant students who express themselves uniquely. OGHS has a large and active athletic program with football, basketball, volleyball, and tennis teams that have won state championships. There are two cheerleading squads, a large and awardwinning band, and a chorus program as well. OGHS is a wellrounded school with a little bit of everything, socially, ethnically, politically, and in terms of curriculum. I have observed a strong sense of social justice among the student body as well as many voices in the extremes, but all in all it feels like a wellbalanced group of students. Schools are a reflection of and a mechanism for perpetuating the culture in which they are situated. The social imbalances in the community of Oak Grove are reflected in the population of OGHS. OGHS is the oldest and first high school in its district and, while not in its original building, OGHS has existed as a public high school for over 100 years. The other two high schools were built in the 1970s when the black high school, Washington High, was closed and all the city's high school students were rezoned to these three schools. I have only heard stories, but apparently there were race riots or close to that when OGHS was first integrated. A young principal, William Daniels (pseudonym), shepherded OGHS through this time and was its principal for nearly 30 years. I firmly believe that the quiet, comradely leadership style of Dr. Daniels had a significant influence on the character of OGHS that still exists today, though he has not been principal there for nearly a decade. He began his school administrative career at OGHS wearing corduroy suits and hair covering his ears in the 1970s and eventually became the district superintendent. The faculty at OGHS is fiercely independent and headstrong, but it is a fine faculty. Dr. Daniels was an adroit leader and had the unique ability to bring together the collective identity of a diverse faculty for the good of the students and the school. The academic programs at OGHS range from remedial arithmetic and reading to advanced placement courses in all areas of the curriculum. Like all schools, there are the leaders, the troublemakers, the complainers, the competent, and the not so competent on this faculty. From year to year each of these components ebbs and flows, but in general, the faculty, like the student body, is representative of the wider community ethnically, ideologically, and socially. There is a great love for OGHS from those who have known it for a while. Some people detest it and cannot wait to get another job, but for the most part, people are happy to work here and have a sense of loyalty to OGHS and its students. While OGHS is a pretty good place for most, the social, economic, and educational inequities in the larger culture are reflected and to some degree perpetuated at OGHS. To the casual observer, the social and economic hierarchy is not readily visible, and in general it is a good place to learn and work, which I think contributes to the equalizing effect at OGHS. OGHS is a place where students who apply themselves have a good chance of school success. However, OGHS does find the lower SES and ethnic minority students in higher numbers in the lower level classrooms with the upper SES and mostly white middleclass students in the more advanced and prestigious academic environments. Hence, while the opportunity for school success is there for most students, it is not without the cultural and systemic barriers found in most United States educational institutions today. Data Collected Field Data Student perceptions and participation strategies in the mathematics classroom were explored via a series of classroom observations and focus group interviews. These data were analyzed for underlying themes and subthemes. Several of the focus groups consisted of students identified as highachieving female and minority students. Data collected in these groups were analyzed and assembled into a list of best practices that contribute to the success of students who often are expected to do less well as a result of their gender or ethnic classifications. Analytic Data Three teachers were given a handheld computer (HHC) with randomly generated lists of the names for each class (with replacement). Whenever the teacher had a question, instead of calling on the students whose hands were raised or choosing the students themselves, the teacher called upon the next student on the random list. Thus, each student had the same opportunity to be called on for each question asked in class. When students were called on, they could either try to answer the question or they could just say "pass" with no penalty. The teacher recorded on the HHC whether the student passed or attempted to answer the question, and, if the latter, answered it correctly. Similar data were collected by classroom observation in control classrooms. Each class, both experiment and control, was given a pre and postbattery of previously validated mathematics attitudinal and belief survey items as well as pre and postacademic tests. Student grades were also used as a measure of academic achievement. In this experiment, there are two types of data being analyzed: attitudinal and performance. These data are meant to test whether the intervention affects students' academic performance, beliefs, and attitudes toward mathematics. The data on participation (pass, attempt, correctincorrect) are meant to test whether the random questioning strategy (RQS) contributes over time to students participating at different levels. Key measures include whether students choose to pass less over time, whether they attempt more over time to answer, and whether they answer correctly more as time goes on. The additional postproject questions addressed students' experiences with and perceptions of the RQS. They comprised a series of Likerttype items, with a scale from 1 to 5. Students were also free to give written comments about the RQS. In combination, the qualitative and the quantitative data contribute more than either would alone. It is through grounding the experimental portion of this study in the actual lived experience of the classroom that the RQS results may be interpreted in context (Burns 1979). By interpreting the results of this study within the context in which it takes place, the quantitative data take on a more personal characteristic and the qualitative data are supported by numerical measures. Organization of Chapters The balance of this dissertation is organized as follows: Chapter 2 presents a focused review of the literature and theoretical framework leading to the objectives and hypotheses. Chapter 3 outlines both the quantitative and qualitative research design and methods for this project. Chapter 4 reports the quantitative data from the RQS experiment. Chapter 5 reports the qualitative data from focus groups, surveys, interviews, and participant observation. Chapter 6 brings the quantitative and qualitative data together in a fashion that illustrates how they inform and enrich each other and provides a deeper and richer understanding of the culture of the mathematics classroom. CHAPTER 2 LITERATURE REVIEW Throughout the centuries, women and minority scholars of mathematics have been denied access to knowledge, the privilege of publishing their work as their own, and membership in scientific and mathematical societies; and in general they have been omitted from the historical record, thus denying them credit for their achievements. Despite such obstacles some of these scholars have still risen to the forefront in both mathematics and science. Those recognized are but a few of the many who have excelled and not been acknowledged (Adler 1972; Campbell and CampbellWright 1995; Fennema 1990; Fordham 1996). Mathematics and science are like works of art: Without context it is often difficult to understand the artist's intention. Supplying that background often makes it easier to understand the work. Similarly, the context of mathematical and scientific achievement gives that knowledge meaning. The absence of women and minority scholars from the historical record leaves a void for students who are often disenfranchised when it comes to the pursuit of mathematical and scientific knowledge. The lack of meaningful role models in mathematics and science leaves these students feeling little or no connection to the discipline and few role models for them to emulate (Fordham 1996; Swetz 1997). This lack of women and minority mathematicians in the historical record has reinforced the societallevel assumption that women and minorities are not adept at math. If they were adept, the reasoning goes, they would be better known. The underrepresentation of female and minority role models in mathematics and science leaves these students feeling no connection to the discipline and with few role models for them to emulate (Swetz 1997). By providing educators with a system that removes gender and cultural bias from questioning interactions in the classroom, this research study is designed to test whether a more equitable educational experience can be achieved. There continues to be a real, albeit subtle and often unconscious intimidation that takes place in the classroom which reinforces the cultural perceptions that women and minority students are unable to do math at the same level as their white male counterparts (AAUW 1992; Fordham 1996; Spender 1997). Physicist Sally Ride, the first U.S. woman in space, says that many young women and minorities continue to face "subtle obstacles" that block their paths to math and science careers. These roadblocks include educators who place more credence in answers from young men, and school counselors who discourage young women and minority students from taking advanced mathematical courses (Mervis 2001). Myra and David Sadker (1994) have shown that educators tend to enact these behaviors: (1) they call on white male students more frequently than other students, (2) they respond to their questions and requests for help differently, and (3) they provide them with entirely different experiences in the classroom. This practice is routine across the curriculum but is especially evident in the area of mathematics. The fact that women and minorities on average perform less well than white males on certain math portions of standardized tests is a well documented phenomenon (AAUW 1998; Fordham 1996, Croom 1997; Fennema 1996; National Center for Educational Statistics (NCES) 1998; Young and Fisler 2000). One consequence is that the retention of women and minorities in math and sciencerelated majors declines as the educational level increases (Gray 1996; Rosser 1997; Boland 1995; Leahey and Guo 2001). Roth (1996) has shown that even when teachers, who pride themselves on being free of sexist and racist baggage, are aware of these behaviors and genuinely try to remedy them within their own classroom, they still find themselves exhibiting behaviors that perpetuate this problem. Weisbeck (1992) indicated that teachers respond to pressure in their classrooms. By focusing on the concept of teachers responding to pressure in the classroom, rather harsh accusations of teachers being blatantly sexist are redirected. This intimidation, for the most part, is not intentional, but a byproduct of specific cultural presuppositions linked to both the historic and contemporary contexts in which we live. By taking the choice of which student to call on out of the educator's hands by implementing a computergenerated random questioning strategy, the chance that one group of students will be called upon disproportionately will decrease. Central to the idea that all students have the same chance of being called upon each time is the expectation that all students have both the opportunity and the potential to answer each question. The selffulfilling prophecy first examined over 30 years ago (Rist 1970; Rosenthal and Jacobson 1968) has become somewhat of a cultural icon (Weinburg 1987). The fundamental premise that students tend to live up to their teachers' positive expectations and down to negative expectancies has been supported in the literature, with the caveat of a strong likelihood of possible spurious correlation(s) between teacher expectations, teacher activity, and other possible underlying factors (Brophy 1983; Good and Brophy 1987; Goldenberg 1992, 1989a, 1989b; Wentzel 2002). More effective schools tend to have faculties that hold higher expectations for their students than faculties in less effective schools, and expectancy effects are often predictable for those teachers who have firm expectations for their students and whose actions in the classroom are guided by those expectations (Brophy 1985). In his case study of teacher expectations, Claude Goldenberg (1992) concluded that it is not necessarily what teachers expect; it is more what teachers do. Hewett (1984) holds that a more fundamental issue is how a teacher defines a situation because individuals define situations according to their perception and then act according to that definition. Additionally, teachers tend to provide more and better learning opportunities to highexpectancy students, often giving less attention to lowexpectancy students (Mitman and Snow 1985). Another way to think about this is via the results of the 1999 TIMSS Video Study (U.S. Department of Education (USDE) 2002). This research suggests that in the United States, we spend more time reviewing previously covered material than five of the six other nations in the study (Australia, Czech Republic, Hong Kong SAR, Japan, Netherlands, Switzerland), and of those countries, the United States has the second lowest score. Results from this study indicate that in U.S. classrooms in the 8th grade, teachers spend over half of their instructional time reviewing previously covered material. If teachers feel it necessary to review over half the time, it stands to reason that the expectations for students is fairly low. Given these data, it appears that there may exist a lower expectation in this country for mathematical excellence than in the other highperforming countries in this study. The highestscoring countries spend approximately onequarter of their time in review, with others spending approximately onethird of their instructional time in review. By spending a majority of their class time in review, teachers communicate the expectation that students cannot move on because they don't know their mathematics well enough, thus limiting their opportunity to excel at a more advanced rate due to the amount of time spent covering review rather than new material. When we look at the content preparation for teachers of mathematics in the high poverty, lowachieving high schools, it becomes apparent that many teachers in these schools are simply not prepared mathematically to take their students further. Mathematics teachers in highpoverty, lowachieving high schools are much more likely to be underqualified or out of field than in lowpoverty, highachieving high schools (Nation's Report Card 2004, USDE 2004). This lack of subject area preparation for mathematics teachers offers a plausible explanation for the extensive review in U.S. mathematics classes, as many teachers simply are not prepared to teach higher level material and thus cannot take their students further in the mathematics curriculum. Given that women and minorities are often assumed to have a lowered standard of mathematical excellence in the U.S., this research addresses not only the idea of expectancy, but of what it is that teachers do in the classroom. By giving each student the same chance to participate in class, will indicators of academic success, participation levels, attitudes, and academic performance be affected? Gender differences seem to be the result of social factors most consistently. Many of these differences are the result of different socialization experiences across gender lines (Harway and Moss 1983). Males are encouraged to exhibit aggressive, independent and dominant behavior from birth. This behavior reveals itself in the classroom as they talk out, raise their hands repeatedly, wave at the teacher, and generally display assertive behavior. Such behavior clearly pressures teachers to direct their attention to the male student. Females, however, have been enculturated to be cooperative, follow the rules, and value the relationship they have with their teacher. Because of this, girls are less likely to pressure their teacher for attention. They are more inclined to raise their hands and wait quietly for attention. If they do not get that attention after several tries, they are not likely to raise their hands again or be aggressive in their quest for knowledge. By following the rules for "appropriate" female behavior, not pressuring the teacher or displaying assertive behavior, the female student works to preserve her relationship with the teacher by "doing the right thing." Girls' good behavior works against them, while boys' bad behavior works in their favor (AAUW 1998; Goldenberg 1992; Guinier 1997). Teachers, having been societally conditioned in the same manner (Walden and Walkerdine 1985), often hold male and female students to different standards of behavior. Behavior that is often excused for male students, with the rationale that they are just boys, or boys will be boys, would not be tolerated under normal classroom conditions from girls. Girls out of their seats, yelling across the room, interrupting the teacher or other students would most likely be corrected or told to wait their turn quietly in a classroom setting (AAUW 1998). Our society conditions people to believe that mathematics is a male domain and that women cannot do math as well as men (AAUW 1998; Campbell and Storo 1996; Leder 1990). A substantial body of literature documents the influence of parental beliefs on children's achievement attitudes and academic performance. Parental beliefs and expectations have been related to the child's performance history (Jacobs 1991; Parsons, Adleter, and Kaczala 1982; Entwisle and Baker 1983; and Entwisle and Hayduke 1978, 1981; and selfperceptions of academic ability and achievement expectations (Hess, Holloway, Dickson, and Price 1984; Parsons et al. 1982; Stevenson and Newman 1986). Others studies suggest that parents and other adultsespecially educatorshold culturally based beliefs about the appropriateness of certain behaviors and proper roles for males and females (Connor and Serbin, 1977; Fagot, 1973, 1974; Jacobs and Eccles 1985; Perloff 1977; Boland 1995). In general, parents and teachers have educational expectations that tend to be lower and often do not include the STEM fields for females, and parents tend to attribute a daughter's success to hard work while they attribute a son's to talent (Boland 1995, Grayson and Martin 1988; Rosser 1997; Sadiker and Sadiker 1994; Stage et al. 1985; Weisbeck 1992 ). These assumptions about suitable gender roles and stereotypes are likely to influence parental and educator judgments about students' academic abilities (Boland 1995; Eccles 1984; Eccles et al., 1983; Jacobs and Eccles 1985). Girls more than boys tend to respond to parent expectations and aspirations for them (Boland 1995), while boys tend to have higher selfconcepts with respect to mathematics, and girls tend to have higher selfconcepts in regard to reading (Crain and Bracken 1994). StantonSalizar (2001) and Lopez and StantonSalizar (2001) indicate that while the parents of lowstatus Mexican American students tend to have high goals and expectations for their children, they often lack the resources, skills, social networks, or ability to facilitate their children in achieving these goals. The parents of highachieving Mexican American children have been shown to be less likely to accept lower grades from their children and to model skills related to school success than the parents of lower achieving Mexican American students. While the parents of the lower achieving students held high expectations for their children, they appeared not to understand that Cs and D's indicated that their children were not performing well in school (Okagaki et al. 1995). Some research suggests that Hispanic students have lower selfconcepts that African American and white children (Wasserman et al. 1990), while earlier research suggests that there is no difference (Healey 1969). Frisby and Tucker (1993) found that in African American students' concept of self esteem, academics was not a component by which they evaluated themselves. Comparisons of African American and white American students' selfesteem shows that African American students' selfesteem is either higher than that of white students or that there is no difference (Drury 1980; Tashakkori and Thompson 1991; Wright 1985). African American mothers tend to have higher educational expectations for their daughters than for their sons (Oyserman et al. 1995), which may be the result of differences in gender socialization among African American children. Women and minority students are often encouraged to choose career tracks that are less rigorous and require fewer science and math courses. Furthermore, female and minority students often take only the mathematics courses that are required of them for their high school diploma or for college entrance (AAUW 1998; Feagin, Vera and Imani. 1996; Feagin and Sikes 1994; Leder 1990; Mervis 2001). This societal perception that girls can't do math and that math is a male domain results in girls electing not to further their mathematical careers because they perceive there is no use for them or they will require too great a sacrifice in terms of femininity and family (Gray 1996). Mathematics is a critical filter for most academic success. Lowincome and minority students who take algebra and geometry attend college at rates similar to higher SES whites. However, only about half as many lowincome and minority students take these courses. In 2003, over half of the black and Hispanic students in the U.S. attended highpoverty schools compared to only 5 percent for their white peers; and 40 percent of the black and Hispanic student population attended schools where the enrollment was at least 90 percent minority. In addition, these same highpoverty, highminority schools' students are more often taught English, math, and science by underqualified outoffield instructors than their peers in lowpoverty and lowminority schools (The Nation's Report Card 2004). When girls are exposed to successful women models, they tend to have higher expectations of success and spend more time on school related tasks (Campbell 1984). Unfortunately, many minority students often have few meaningful successful role models to emulate (Fordham 1996) and have little motivation to pursue a career requiring more than the minimal math and science requirements (Fordham 1996; Fordham and Ogbu 1986). According to lack of opportunity and teacher attitudes toward minority students and minority students' response to these attitudes contribute to the poor educational performance of minority students. The 1995 Trends in International Mathematics and Science Study indicates that gender differences in math and science are virtually nonexistent at the 4th grade level. At the 8thgrade level, differences in mathematics were still minimal but become apparent in science. By the 12th grade, males had significantly higher achievement in mathematics and science than females. Results from the 1999 TIMSS indicate that there were no changes in math and science achievement for boys and girls; however, there were no comparative high school data between the 1995 TIMSS and the 1999 TIMSS (Ansell and Doerr 2000, USDE 1999). Coinciding with this drop in math and science achievement from 8th to 12th grade, researchers have found that girls experience a drop in selfesteem in their teenage years (Gilligan 1982) and that gender differences in selfconcept include higher selfconcept in physical abilities and mathematics for boys with girls having higher selfconcepts in reading and English (Harter 1982; Marsh 1987; Marsh and Jackson 1986; Byrne and Shavelson 1987; Marsh et al. 1984; Marsh et al. 1985). However, some research suggests that African American girls do not experience the same drop in selfesteem as white females, but that African American girls as a population have higher selfesteem, more positive body image, and greater social assertiveness than white girls. African American girls also demonstrate higher academic performance than their black male counterparts (AAUW 1991). This difference suggests that while gender is a contributing factor in math and science success, gender differences are not universal across ethnic strata. Over the last two decades the gender gap between male and female math and science achievement scores has narrowed at the elementary, middle, and high school levels (AAUW 2000, 1998; NSF 1999). While an increasing number of women and minorities are pursuing careers in STEM, this progress has been uneven. Girls now appear to meet the achievement and course requirements necessary to enter STEM majors and fields in numbers equivalent to boys. While enrollment in STEMrelated majors for women is increasing, young women who do major in the STEM fields drop out of STEM majors at higher rates than young men with the same grades, thus selfselecting out of the system (Boland 1995; Pattatucci 1998; Rosser 1997). However, the same cannot be said about minority students with the exception of Asian/Pacific Islanders (Campbell and Hoey 2002). Less than 50 percent of AfricanAmerican and Hispanic students demonstrate the basic knowledge of science and math skills based upon National Assessment of Educational Progress (NAEP) testing (NSF 1999). Mathematics is known to be the gatekeeping course for many college preparatory courses in high school and to highpaying scientific and technical career tracks in college. Sells (1980, 1975) illuminated the role that mathematics plays as a filter for entering the highest ranks of mathematics and math oriented majors. Seventyfive percent of college majors are out of reach if a student does not have a full complement of high school mathematics. Since mathematics serves as a bridge to highpaying, secure career choices, the avoidance of mathematics and science careers by women and men of color not only has a negative effect on their career choices but also deprives society of a substantial population of creative contributing mathematicians, scientists, and engineers. There is no obvious biological evidence for cognitive differences in women and men. In terms of general intelligence, there appear to be no differences. In almost all areas of verbal ability, women have the advantage, but at the highest end of the quantitative spectrum, men hold the advantage. It is important to remember that on the whole, males and females are overwhelmingly alike. In the areas where cognitive differences have been isolated, these conclusions are based on aggregate data and cannot be generalized to any one individual due to withinsex variance (Halpern 2000; Sperling 1999). Given the absence of any conclusive biological evidence for the gap in women's and men's achievement in mathematics, we must turn to other possible sources of the problem. All cognitive data come from testing instruments of one sort or another. All tests are to some extent biased, reflecting the cultural training and life experiences of those who develop and administer them (Kottak and Kozaitis 1999). Most aspects of our schools, curricula, courses of study, teaching methods, and the subject matter of mathematics were developed primarily by men and reflect the life experiences and goals of men (Damarin 1990). Furthermore, timed tests and mixed testing situations, which are the primary indicators of gender difference in mathematical achievement, have been shown to be biased against girls and women (AAUW 1998; Inzlicht and BenZeev 2000). In 1995, women accounted for 55 percent of all the bachelor's degrees awarded, but only 35 percent of the degrees in math and computer science. In 2001, women accounted for 32 percent of the degrees in math, computer science, and engineering. Women account for 22 percent of the science and engineering work force but constitute 46 percent of the total work force. This fact is further evidenced by the continued male\female gap in GRE and SAT scores even after they are adjusted for the number and type of math courses taken. The median income for women in 1998 was 76 percent of that for men. These trends are also reflected in the salaries commanded by women in STEM careers (Halpem 2000; Ansell and Doerr 2000; Margolis 2000; Rosser 1997; Spertus 2004). Despite claims from some, the evidence for the gender gap in mathematics achievement is clear. It still exists, but why? Work by Belenky et al. (1997, 1986) describes five categories of knowing or how people come to know things. Silent knowers accept what they know without stating it; received knowers attain knowledge from authority figures through listening; subjective knowers listen to their own internal voices. Procedural knowers fall into two categories: separate knowers who learn individually, and connected knowers who learn by access to others' experiences and connection with the material. Finally, constructed knowers judge evidence within context; this is an integration of both the separated and connected approaches to knowing. Women and students of color are predisposed to being connected knowers (Belenky et al. 1997, 1986; AAUW 1992), while men are more likely to be separate knowers (Perry 1970). Traditionally, mathematics has been taught and approached from a perspective that is congruent with separate knowing, stressing concepts like deductive proof, absolute truth and certainty, and the use of algorithms; and emphasizing abstraction, logic, and rigor. Textbooks have predominantly contained problems focused on traditionally male activities and often lack any mention of minority and female role models. Some publishers have sought to resolve this issue by inserting pictures of women and people of color in the photographs, inserting short bios of successful women and minority scholars, and using multiethnic and gendered names in the problems. However, while a step in the right direction, this is just a band aid approach for correcting a much deeper problem. For the most part, the achievements of black intellectuals are absent from academic texts (Fordham 1996). Simply pasting in a few instances of the achievements of women and people of color does not change the fundamental context of schooling or the existing cultural presuppositions about women and minorities and mathematics in the U.S. today. Because of this separated andro and Eurocentric focus, mathematics texts often seem foreign, contrived, and incomprehensible to female and minority students (Rosser 1995). Because of their lack of connection to their textbooks, the primary method of communication in the mathematics classroom (USDE 1999), many students are put at a disadvantage when it comes to mathematics education. Educational equity implies quality education and equal opportunities for all students. Factors such as sex, class, socioeconomic status, race, and ethnicity often contribute to unequal educational outcomes. According to Lee, educational equity is "a concern for unequal educational outcomes by social background" (1998:41). The concept of educational equity leads to these questions: (1) Do students receive the most appropriate education to achieve a shared standard of excellence? (2) Do students have the same opportunities on graduation as a result of their educational experience? (3) Do students need the same educational experiences to achieve those outcomes? and (4) Is the same educational experience for all students an equal educational experience? In U.S. culture, the concept of equity vs. equality is difficult since most of the focus is on equality meaning sameness (Scott 1988). However, if the outcomes of an equal education are not equal, then we have failed in providing an equitable educational experience to our students. Failure to take into consideration the varied characteristics of students in a heterogeneous society is failure to address the educational needs of the populace as a whole. By explaining and documenting students' participatory strategies, perceptions, and behavior in this setting, this research attempts to gain a better understanding of the mathematics classroom dynamic as it relates to questioning. The goal is for this knowledge to enable institutional reforms that will result in a more equitable educational experience for all students. Mathematics pedagogy has undergone many changes in the last two decades, including an increase in the kind of collaborative learning strategies that benefit females and ethnic minorities (Pearson and West 1991). When parents and educators understand that females and minorities can succeed in math, their attitudes may change. If educators understand and respect different learning styles and interests, new curriculum models can be developed that reflect the interests of girls and minorities and emphasize practical, reallife applications. Other changes may include an emphasis on more cooperative learning strategies which focus on the process of solving the problem rather than the answer itself, raising teachers' awareness of culturally and gender biased educational material during textbook development and selection, and convincing females and minorities that they can learn and use advanced mathematics professionally outside of the classroom (Schwartz and Hanson 1992). Theoretical Framework Critical Race Theory Throughout history, schools have been some of the most effective instruments for the enculturation and continued oppression of black people in this country (Allen 1991). During the time of legal segregation (18801964), even in segregated schools where black teachers taught black students, they were often not under the control of black educators. In a society where segregation and separatism was the norm, students in segregated schools received the message that white was good and black was bad. In order to be educated, an African American had to be enculturated to the ways of the white American, and at the same time, understand that while they were to take on the ideals, ways and practices of the white American, they could never be white and attain the same status as white Americans (Ogbu 1993; Feagin et al. 1996). Despite the implementation of civil rights laws and legislation against hate crimes, five years into the 21st century, symbols of racial intolerance persist in our schools and in society at large. Along with facing this intolerance, black children are often punished for reacting to such symbolism. Black students tend to be overpoliced and are more likely to be punished for breaking school rules than white students (Skiba 2002; Schwartz 2001; Gregory 1995; McFadden 1992; Shaw and Braden 1990; Wiley 1989). African American children are pressured to become, as one student described it, "AfroSaxon" or to "pass" for white in terms of their clothing, hairstyles, behavior, and language. They are pressured through the imposition of societal norms to abandon their own identity and culture for that of the white American in order to be successful in the educational system, all the while knowing that they will remain in second class status despite giving up their identity for that of the predominant white middle class (Feagin and Sikes 1994; Fordham 2000; Ogbu 1993). In a similar vein, Fordham (2000) also focuses on students' response to systemic and institutionalized racism, but she focuses on the identity politics of the situation where she characterizes black women as doubly oppressed. In an educational or business setting, a system of passing exists whereby women and minorities must pass as white men by adapting their behavior, mannerisms, cultural practices, and ways of functioning in the system in order to be successful and accepted by those in power, be it colleagues or teachers and professors. For white women to be successful in a man's world, they must pass for white men in a sense. Similarly, black men must also pass for white men by taking on their dress, language practices, mannerisms, and so forth. However, within this system of passing, black women are doubly oppressed, as they have to pass as white women passing as white men. Within the educational system, this results in a doubly oppressive situation where black girls' identity is threatened and therefore more highly rated among young black women. By refusing to conform to the cultural and institutional norm, "those loud black girls," as Fordham has termed them, are maintaining their identity within a system of multiple oppressions (Fordham 2000). Critical Pedagogy In The Pedagogy of the Oppressed, Freire (1968) conceptualizes oppression in an analysis of the mechanisms of oppression within the context of colonialized Latin American society. The effects of unequal relationships between those with power and the powerless are deeply rooted. Freire describes this effect as cultural invasion. As a result of cultural invasion, the powerless lose their culture and identity. The invaders are perceived as superior and the oppressed group perceives themselves as inferior. As the oppressed take on the culture of their oppressors, the oppressors do the thinking for them. As a result of the oppressed being overwhelmed with the cultural norms and ideology of the oppressors, they tend to become silenced. One of the tools of cultural invasion that facilitates this process is the cultural myth. Freire defines the cultural myth as a lie promoted by those in power to facilitate their position of authority. The needs, knowledge, wisdom, and experience of the oppressed are not considered important and are therefore ignored, devalued, and considered inferior. Such cultural myths are internalized by the oppressed, and they often come to believe that they are ignorant and dependent on those in power (Freire 1998, 1968; Freire and Macedo 1998; Gadotti 1994; Smith 1997). One mechanism by which the cultural myth is perpetuated in education is through the selffulfilling prophecy and lowered expectations for the oppressed group. When the overriding cultural myth is exposed for the lie it is, the way is opened for informed action on the part of the oppressed group or individual. It is by rejecting these cultural myths that the oppressed come to a sense of consciousness. Rather than taking the cultural myth as truth, they come to see the oppressive cultural myth as a lie. Once this lie is exposed and seen by those it has held back, an enlightenment takes place so that the victims of the cultural myth are now able to see the myth and its power, which enables them to understand it and form meaningful strategies to combat its effects. Freire (1968) uses the term praxis to indicate informed action as the integration of reflection and action, practice and theory, thinking and doing. Taking action for oneself is taking freedom. In the Freireian sense, freedom means becoming more human and taking on the position of subject rather than object. Freedom is gained through struggle within the individual by transcending the boundaries, of self, and through struggle with others by transcending boundaries set for them by the oppressor and their myths (Freire 1998, 1968; Freire and Macedo 1998; Gadotti 1994; Smith 1997). One method by which the oppressed take on the position of subject rather than object is by practicing everyday forms of resistance. James Scott (1986) refers to everyday forms of resistance as the constant struggle between relatively powerless groups and those who seek to extract something from them. This type of struggle is rarely characterized by outright defiance but more often by passive forms of resistance such as slowcompliance or falsecompliance, pilfering, sabotage, contrived ignorance, and so forth. This type of everyday resistance is often demonstrated in classrooms where students choose not to participate, vandalize school property, cheat, and generally resist doing what is expected of them in the school setting. Since what is expected of them in many ways is to relinquish their ideals, ways of operating in this world, and their cultural identity, refusing to cooperate or slow cooperation within the school system is their way of gaining freedom. While Freire's work focuses on the mechanisms of oppression, the work of Ogbu illuminates the reactions of oppressed people to their situation. Ogbu characterizes African Americans, Native American Indians, Mexican Americans (early Mexican Americans who were conquered in the Southwest), and Native Hawaiians as involuntary minoritiespeople who were brought to the United States (or any other nation) permanently against their will as a result of slavery, conquest, or colonization. Once a group becomes an involuntary minority, membership is more or less permanent and is passed down to future generations. Involuntary minorities differ from voluntary minorities in several ways; mainly, voluntary minorities come to the United States or any other country of their own volition for more and better opportunities than those available in their homeland. As a result of this difference, the dual frames of reference for these two groups differ significantly. The voluntary minority has their homeland to look to for comparison to the present situation. They view discrimination as an obstacle to overcome on the path to a better life. The involuntary minority's referent other is that of the white middle class. They view discrimination and secondclass status not as obstacles to overcome but as a permanent insurmountable situation. The involuntary minority perceives the ways of the dominant group as a threat to their cultural identity and selfhood within their group. The dual frame of reference of the involuntary minority is oppositional. To take on the behavior and ideals of the dominant group threatens their membership and status within their own group. This oppositional frame of reference tends to be applied in areas where the criteria of performance, competence, and reward are established and evaluated by the dominant group or their minority representatives. Clearly, school is one of these areas. The effect of school being situated within the oppositional frame of reference is that schooling is perceived as a process by which the involuntary minority, in this case African Americans, is acculturated and that their own culture is displaced. For this reason, some involuntary minority students tend to regard school as tantamount with to the enemy. To adopt the attitudes and behaviors that contribute to academic success can be perceived as rejecting the culture of their group and taking on the culture of the oppressor. Furthermore, there are many examples of involuntary minority students who have adopted the necessary academic attitudes and behaviors and achieved academic success, yet due to discrimination have been denied access to positions for which they are qualified. So, for some involuntary minority students, to do what is necessary to be successful in school is to reject the attitudes and behaviors of their own cultural group and risk rejection by their own group as well as the dominant group (Ogbu 1993; StantonSalazar 2001). The work of Freire and Ogbu offers additional insight into the relationship between gender and mathematics. Freire's ideas on education come from and reflect his perspective on oppression. According to Freire (1968), there are two views of humankind: the objective, where humans are moldable and adaptable parts of the world; and the subjective, where humans are independent and able to transcend and change their world. Freire chooses the latter. Humans can think, reflect for themselves, and disassociate themselves from the world. Consciousness, according to Freire (1968) has three distinct levels: magical, naive, and critical. Magical consciousness is characterized by people accepting the will of a superior force, which means accepting life as it is and injustice as a fact of life. People operating with magical consciousness are silent and docile. People with naive consciousness have gained some insight into their own problems, but this insight remains personal rather than part of a bigger picture. Those who operate within critical consciousness have made connections with the world outside the individual looking at society and its injustices and have reconfigured these injustices as facts, rather than as myths, which can be changed rather than accepted. Freire (1968) frames his analysis of oppression with respect to the mechanisms of oppression rather than focusing on class. One of the tools of cultural invasion that facilitate this process is the cultural "myth." As an example of a cultural myth, consider the idea that women are irrational and are not as suited as men to rational pursuits. Historically, reflecting this general societal assumption, activities such as mathematics, science, and philosophy, which supposedly require the use of a "rational mind," were considered inappropriate for women. This myth was and continues to be underwritten by religion, education, and other forces of socialization such as various media. Women, of course, were not instrumental in the development of this myth, but many women support it and it continues to keep women oppressed, silent, and excluded from highstatus areas of intellectual activity. The obverse of the irrational woman myth is that men are expert in areas requiring rationality, which supports the fact of men having power in most matters of consequence. Men continue to dominate the system of socialization that supports the continuation of the irrational woman myth and the exclusion of women from positions of power. Many women in American society believe that they cannot participate at the same level in male oriented activities due to their irrationality despite no real supporting evidence for this claim and evidence that in some cases, men have been shown to be less rational than women (Penman 2003). Freire (1968) holds that oppressed groups must initiate "cultural action for freedom" in order to escape the culture of silence. Traditionally the content of culture is defined by the bourgeois class, which holds the definition up as the standard by which all other culture is measured. That is, according to Freire, bourgeois culture is defined as the only real culture; all other cultures are deemed inferior. However, according to Freire, culture is a human construct initiated and created by humans through praxis, or informed action. To transcend boundaries one must address boundary situations imposed either by oneself or society. Both boundaries in my opinion are the result of cultural myths, but the difference is who is the imposing force. For example, a subsidiary of the irrational woman myth is the myth that "girls can't do math." A young female student may simply accept that she can't do math the first time mathematics becomes at all difficult for her and stop trying. This is an example of a selfimposed boundary. An example of a boundary situation set by others would be a young female student who is very gifted in mathematics and has done well throughout her high school career. She has decided that she wants to be a civil engineer and to design public places that are more people friendly and environmentally kind than those she currently experiences. She goes to her guidance counselor and says that she wants to be an engineer. While not directly telling her that this is a man's profession, the counselor repeatedly suggests other majors that she could choose that would help people or have an environmental focus like psychology, nursing, teaching, botany, and resource recovery. From Freire's (1968) standpoint, freedom would be achieved by both of these students were they to recognize the boundary situation they are experiencing, engage in thoughtful reflection about the boundary, and take action to transcend it. After action has been taken, further engagement in thoughtful reflection and further action would take place. However to achieve freedom, both the internal and the external boundaries must be overcome. To transcend the external boundaries, however, the internal ones must be addressed (Freire 1998, 1968; Freire and Macedo 1998; Gadotti 1994; Smith 1997). Again, consider the young woman who wants to be an engineer. She leaves the guidance office unsatisfied with the results of her visit but not quite sure why. She thinks about this herself and discusses it with some of her peers and a mentor. She realizes that she is dissatisfied because she has not been heard. Her wants, desires, and interests have been disregarded. She wants to be a civil engineer, but that issue was not addressed. She makes another appointment with the guidance counselor to discuss the academic requirements, financial options, and schools for her to attend in order to become a civil engineer. The counselor again brings up the alternative, "more suitable" careers, but this time the student politely focuses on civil engineering and is firm about her desires and interests and insists on pursuing the career of her choice. The counselor is not at all sure that this is the right choice for the student but does provide her with the information requested. After breaking the stereotypical boundary set for her by the guidance counselor and leaving with what she came there for, the student reflects on the situation and on obstacles she may encounter in the future. This young woman is now looking at her life critically and seeing beyond the myth that "girls can't do math" and its parent myth of "the irrational woman." She has exposed the myth for the falsehood that it is. Research Objectives, Expectations, and Hypotheses I have embarked on a research agenda that focuses on the culture of the mathematics classroom and on factors that contribute to the persistent idea that women and minorities can't or don't do math. Through ethnographic interviews and observations, these factors can be identified and then tested for their contribution to math performance and attitudes. By grounding my research questions in the material reality of school, I have an opportunity to identify and examine the factors contributing to the inequitable educational outcomes in mathematics for women and minority students. The following hypotheses have grown out of the literature and my experience as an observing participant. Objec Objec Objec ;tive 1: To elicit and document students' perceptions and experiences as students of mathematics in order to gain a clearer understanding of students' participatory behavior in the mathematics classroom. Hi: There will be differences in students' participatory strategies in the mathematics classroom across gender and cultural lines. H2: There will be differences in students' attitudes toward and beliefs about mathematics across gender and cultural lines. H3: There will be differences in the attitudes and practices of highly successful women and minority students compared to those of their group that do not succeed at the same level. H4: The experiences and perceptions of students in the research population will vary along gender, ethnic, and socioeconomic lines. ;tive 2: To measure the effectiveness of a random questioning strategy for increasing academic achievement and improving attitudes toward mathematics for female and minority students in a secondary setting. H,: Implementation of a random questioning strategy will result in increased academic performance and improved attitudes toward mathematics for female and minority students. ;tive 3: To determine the extent to which gender is the primary factor with respect to ethnicity, class, and age for success in the mathematics curriculum. H6: Gender will be a more significant variable than ethnicity, class, or age with respect to academic performance and attitudes toward mathematics in a secondary setting. H,: Implementation of the random questioning strategy will have the same directional effect for minority students and women. Objective 4: To document the effect of the random questioning strategy on teachers' and students' classroom activities and behaviors. Hg: Teachers' questioning behavior will change: The types of questions asked and encouraging behaviors within their questioning interactions will change as a result of the random questioning strategy. H9: Students will be more attentive in class. Hio: Students in RQS treatment classes' levels of participation will increase over time. Hi : Students in RQS treatment classes will have more correct answers over time. H12: Gender will be a more significant variable than ethnicity, class, or age with respect to students' participatory and question answering behaviors in the RQS experiment. Objective 5: To document best practices data from highachieving female and minority students. H13: Attitudes of highachieving students will be more positive toward mathematics than those of the general study population. H14: Classroom practices and behaviors of highachieving students will be different than those of the general study population. H15: The attitudes and practices of highly successful women and minority students will be closer to those expressed by white males in the research population. AAUW (1998) suggests that future researchers in this area should analyze educational data by gender, ethnicity, and socioeconomic status (SES) in order to provide a more detailed picture of the STEM experience of all students. This research answers this challenge by addressing seven main objectives: (1) to examine the messages that educators inadvertently communicate to students that female and minority students cannot succeed in math; (2) to gain a clearer understanding of the dynamics of the mathematics classroom; (3) to determine the participatory behavior of students when answering questions in the mathematics classroom environment, (4) to identify the best practices, behaviors, and attitudes of female and minority students who are high mathematical achievers; (5) to document students' perceptions of the mathematics classroom; (6) to analyze student participation rates; and (7) to test whether the removal of inadvertent gender and cultural biases from educators' questioning practices improves students' attitudes and academic performance in mathematics. In the following chapter I will examine the quantitative and qualitative research methods employed to address these objectives in this study and the supporting methodology behind them. CHAPTER 3 RESEARCH DESIGN AND METHODOLOGY Introduction This is a multimethod study based on both quantitative and qualitative data. The quantitative data are the result of a quasiexperiment to test the effectiveness of a random questioning strategy (RQS) that takes the decision of whom to call on out of the hands of the teacher in the mathematics classroom. This results in each student having the same chance of being called on for each question in the classroom. The qualitative data come from a series of focus groups with students about the mathematics classroom. The focus groups identified students' experiences, needs, desires, behaviors, and suggestions with respect to mathematics and mathematics education. These two kinds of data inform each other and produce a more complete picture of the mathematics classroom than would be produced with either kind of data alone. Quantitative Research Design and Methodology: The Random Questioning Strategy Experiment The central question in this project was: Does removing inadvertent bias in teacher questioning practices affect student participation, attitudes, and academic performance in the mathematics classroom? To answer this question, I designed and implemented the random questioning strategy experiment (RQS). In this experiment, the choice of whom to call on during regular classroom activities was taken out of the teacher's hands and determined by a randomly generated list on a handheld computer (HHC). Each time the teacher asked a question, rather than choosing the student to call upon, the teacher chose the next student on the list, removing the possibility of bias in calling on students. The RQS was implemented at OGHS by three teachers, in 25 classes, over 10 weeks during the spring and fall semesters of 2003. The subject areas for these classes were prealgebra, Algebra I, Algebra II, and Geometry. Each teacher in the experimental group implemented the RQS strategy in all of their classes. One teacher taught five sections of geometry, another taught two geometry honors and three geometry, and the third intervention teacher taught one algebra I honors and three algebra I. Classes taught by two other teachers were used as controls; one teacher had three sections of algebra IA/prealgebra and three sections of algebra II; the other had three prealgebra and two geometry. Data on math performance and attitudes were collected from 453 students in 25 classrooms. There were 292 students in the 14 treatment classes and 161 students in 11 control classrooms. Table 1 shows the distribution of students by gender, ethnicity, and treatment for performance and attitude measures. Student participation data were also gathered through direct observation of the students' mathematics classes. As students were called on in class, I recorded whether they chose to pass or answer, and if they answered, whether their responses were correct or incorrect. Participation data were gathered on 497 students in 20 classes: 307 students in 14 treatment classes and 190 students in 6 control classes. Table 2 shows the distribution of students by treatment, gender, and ethnicity for the participation data. Students with incomplete data or in ethnic groups other than the three major categories examined in this experiment were not included in the final analysis. University of Florida Institutional Review Board Approval # 2003U72. Table 1. Performance and Attitudinal Measures Random Questioning Strategy ExperimentDistribution of Students by Treatment, Gender, and Ethnicity TREATMENT CONTROL White Black Hispanic White Black Hispanic Male Female Male Female Male Female Male Female Male Female Male Female Total 73 84 44 71 8 12 27 37 32 52 4 9 453 Table 2. Question Answering Behavior Random Questioning Strategy ExperimentDistribution of Students by Treatment, Gender, and Ethnicity TREATMENT CONTROL White Black Hispanic White Black Hispanic Male Female Male Female Male Female Male Female Male Female Male Female Total 74 81 48 76 9 12 58 53 26 24 10 10 481 The experimental design was a classic twogroup comparison, without randomization. Each of two groups is compared pre and postintervention, with control groups receiving no treatment. The National Proficiency Survey in the appropriate subject area and course grades served as measures of knowledge, while the attitudes toward math were measured using two previously validated scales, the Innovative Math Assessment Project (IMAP) (Telese 1993; Dossey et al. 1986) and the Mathematics Attitude Inventory (MAI) (Resnick 1985; Schmid 1985; Sandman 1979; Tapia 1996). In the classes selected to receive the RQS, teachers were provided with a pre programmed HHC that had a random list (with replacement) of names for each class for each day. In each class, teachers were asked to carry on their lessons and classroom activities as usual but to select students to call on by consulting the HHC. That is, each time a teacher asked a question, instead of calling on students who raised their hands, or allowing students to call out answers, teachers worked their way through the random name list. For this experiment, students were told that they could choose to pass on any question, with no penalty. Data were collected from the HHCs weekly. Teachers in control classes were asked to change nothing about their standard classroom method except for giving the pre and posttest performance and attitude instruments and collecting informed consent forms. Given the resources available for this research, a choice had to be made between having uncontrolled teacher effects and uncontrolled treatment effects. Since the focus of this experiment was to test the effect of the RQS, it was more important to control for treatment effects than teacher. I assigned different teachers to treatment and control classes to avoid diffusion of treatments. However, without further experiments, I cannot assess the effects of this choice. In the following sections I examine the three different types of data gathered in this experiment: (1) students' academic performance measures as determined by the National Proficiency Survey in Mathematics and by the grades they earned in their respective math classes, (2) level of students' participation as determined by their responses to the RQS and classroom questions, and (3) attitudes and beliefs about mathematics from survey items on the IMAP and MAI. Performance Measures * National Proficiency Survey Series * Student Grades Student performance was assessed both pre and postintervention with the National Proficiency Survey Series (NPSS) in mathematics. Each class was administered the form of the NPSS (Lehmann 1992; Riverside 1989) most appropriate for that course. Students in prealgebra took the general math test while students in algebra I and II and geometry took the test specifically designed for those respective courses. Students' first and second semester grades, as reported by their teachers, were also compared and evaluated for change as a pre and postexperimental evaluation measure of student academic performance. Participation Participation was measured for each question event during the RQS by the teacher noting whether or not a student answered a question correctly, attempted to answer the question at all, or chose to pass on a hand held computer. The same data was tallied for control classes by observation using a modified focal individual group scan method (Fragaszy, Boinski et.al. 1992). These data were examined to determine if students were more likely over time to attempt to answer a question rather than to pass. These data were also examined for differences along gender and ethnic lines. Attitudinal Measures Attitudinal measures were taken pre and postintervention to assess changes in students' attitudes toward and beliefs about mathematics and whether these changed as a result of the RQS intervention. Student feedback regarding the RQS experiment was also collected and evaluated. The postRQS survey had an additional ten questions pertaining to students' perceptions about the use of the handheld computer and the RQS. The scales Seven scales from two attitudinal and beliefs assessment instruments were used for this portion of the study. Five scales from MAI and two from the IMAP were used to assess student beliefs and attitudes toward mathematics. Mathematics attitudes are conceived as a multidimensional phenomenon; thus, differing scales measuring specific aspects of mathematics attitudes are required (Sandman 1980). The Anxiety Toward Mathematics Scale measures the uneasiness a student feels in situations involving mathematics. The Value of Mathematics in Society Scale measures a student's view regarding the usefulness of mathematical knowledge. The SelfConcept in Mathematics Scale measures students' perceptions of their own competence in mathematics. The Enjoyment of Mathematics Scale measures the pleasure a student derives from engaging in mathematical activities. The Motivation in Mathematics Scale measures students' desire to increase their knowledge and understanding of mathematics beyond classroom requirements. The Beliefs Scale measures students' beliefs and ideas about mathematics. The Attitudes Scale measures students' attitudes toward mathematics. The MAI was developed in connection with a largescale evaluation project supported by the National Science Foundation to measure the attitudes toward mathematics and changes in these attitudes for secondary school students grades 7 through 12. The construct validity of these scales has been supported by extensive field testing. The Cronbach's Alpha coefficients for these scales range from a minimum of .76 to a maximum score of .86, thus establishing factorial validity (Sandman 1980; Schmid 1985). To insure validity of all scales from the IMAP, all items paralleled questions on the National Assessment of Educational Progress instrument dealing with students' perceptions of mathematics (Telese 1993; Dossey et al. 1988). All forms were coded with the pertinent demographic and group information, scanned, and scored. Implementation and administration of the scales Items from the IMAP and the MAI were combined into one pre and postsurvey instrument. All questions were scored on a five point Likerttype scale, with 1 being strongly agree and 5 being strongly disagree. All items were adjusted to a common format with appropriate items being reversescored. Pretest surveys were administered to treatment and control groups before any implementation of the intervention. Posttest surveys were administered approximately nine weeks later, at the conclusion of the experiment, for all groups. Teacher Perceptions and Feed Back Teachers implementing the RQS were asked to share their experiences with the RQS, their thoughts, feelings, and opinions about the study. How teachers perceive this intervention is critical as they determine the educational methods practiced in their classroom. Teacher Compensation Participating teachers were paid time and a half for their participation, up to $700 for teachers in the RQS and $400 for teachers in the control classes. Funds were disbursed only to those who remained with the project until its completion and all data and equipment had been turned in and accounted for. The teachers recruited for this project were colleagues known to me who indicated a willingness and desire to participate in this study. Validity and Confounds There are potential confounds in the experiment caused by a number of factors. Intentional and unintentional verbal and nonverbal behaviors by the teacher who used the RQS could affect the study. Clearly, certain inflections of voice or body language could communicate teachers' attitudes, cultural biases, and expected responses. Verbal cues and nonverbal communication are a fact of life in experiments involving human subjects. Certainly, both intentional and unintentional nonverbal and verbal cues occur in such a setting. To avoid a diffusion of treatments confound, the decision was made that teachers would have either treatment or control classes, but not both. Since all students in the study did not have the same teacher, there is the possibility of an instrumentation (teacher) confound. However, teacher effects may offset each other. The maturation confound is another possibility. Over the period of the experiment, students surely came to understand the experiment better. They also learned more of their subject than they knew at the beginning of the experimenta simple function of schooling. This can be accounted for with control classrooms. Mortality was also an issue in this study, as there were both student and teacher participants who chose not to remain a part of the study to completion. Every effort was made to isolate these effects in the data analysis. The focus of this study was on the one thing that was feasible to control: which student gets called on to answer a question. The hypotheses are sufficiently robust and the data and analysis rigorous enough to overcome these classroom subtleties. Limitations Field research is complex and messy, and performing such research in a public high school, with its many bureaucratic impediments, can be a challenge. Although consent and approval of the school district was obtained for this research project, it involved working with minors, which requires permission slips signed by parents to use each child's data in the analysis. This was much more difficult to achieve than anticipated. The original project design was for approximately 750 students. However, signed informed consent forms were obtained for only about 500 students. This is a sufficiently large sample for data analysis, but there is no way to determine if there is any systematic bias in nonresponders. The original timetable for this study called for initial implementation in late January. However, actual implementation did not get under way until early March. This pushed the experiment out to the end of the school yeara time when students may not have taken the assessment instruments as seriously as they would have earlier in the year. Also, the original two teachers with control classes failed to complete the study, making it necessary to collect the control data in the fall of the following year. The original design called for using free and reduced lunch data as a proxy for SES. However, the SES and ethnicity data were too similar to make a distinction between them. The accuracy of data collection was not as consistent as it needed to be to provide unequivocal answers to the research questions. Some teachers were much more diligent than others in collecting the demographic information and informed consent forms and in administering the pre and postevaluation instruments. In replications of this research, I would have a research assistant go into each classroom to give uniform directions on filling out the consent forms and demographic information forms and to maintain a uniform testing environment for all instruments. Finally, this research took place in one Florida high school with 5 teachers and approximately 750 students over a period of ten weeks. Generalization beyond this context is inappropriate. However, the results reported in the following chapters indicate that further studies of this nature are clearly in order. Qualitative Research Design and Methodology: The Focus Groups Introduction High schools are complex organizations, where the students, teachers, administrators, and staff often have conflicting needs and demands (Burs 1979). The results of the RQS experiment must be interpreted within the complex reality of the setting in which it took place. One of the great contributions of anthropology to the social sciences is the insistence that researchers be engaged with the research community through participant observation and cultural immersion. As a researcher who is also a member of the culture being studied, I bring certain insights to the analysis of the RQS experiment that would not necessarily be apparent to an outside researcher. On the other hand, being a member of the culture has its liabilities: It is easy to miss important behavioral cues because they seem so ordinary, and it is easy for colleagues in the school to take the experiment less seriously than they might if the researcher were a stranger. Anthropologists often work directly with people as both advocate and researcher (Burs 1993). In the ongoing dialogue over equity in public education, the voices of many stakeholders are heard. However, perhaps the most important voice, that of the student, is often over looked. Both Burns (1989, 1993) and Emihovich (1999) address the importance of giving voice to the stakeholders in a given situation. Competing interests and differing perspectives require that as many voices as possible be heard. Transformation, collaboration, and policy reform occur through testing groups' different meanings against each other until consensus is achieved. Through these focus groups, not only have I attempted to illuminate the culture of the mathematics classroom from the students' perspective, but I have attempted to give an often unheard populationthe population mostly affected by educational policya voice. To gather as much information from as many students as possible with the least disruption to students, teachers, and classroom schedules, formal focus groups were implemented for this part of the study. Methodology It has been my experience, as an educational anthropologist who has done many ethnographic interviews with students, that while they are usually willing to answer my questions, they often don't feel totally free to express themselves. Students were with their peers in the focus groups conducted for this project, and they appeared to feel much more at ease expressing themselves and volunteering information that they thought would help me understand the mathematics classroom than student informants with whom I had previously conducted individual interviews. Focus groups are interviews with a small group of people covering a specific topic. They usually consist of six to eight relatively homogenous people who are asked to speak to questions asked by the interviewer over a period of time ranging from one to two hours. Focus group participants not only answer questions with their own individual responses; they are also exposed to others' responses and can make additional comments based on what others say, often enriching their initial response or that of another group member. The focus group interview technique was initially developed for market research in the 1950s with the understanding that consumer decisions are often made in a social context (Merton 1990). Similarly, the act of schooling takes place in a social context, making the focus group a fitting and highly efficient qualitative data collection technique appropriate for gathering the desired data. Focus groups also provide some quality control in the data collection process because they come together on the most important topics and issues at hand, weeding out inaccurate and extreme views. Furthermore, focus groups tend to be fun and enjoyable for the participants. Focus groups have many advantages, but they also have some weaknesses. The number of questions that can be asked is limited due the large number of responses. Differing personality types require that the facilitator be skilled in managing the interview so that it is not dominated by a few people and so that less verbal participants get a chance to speak. There may also be personality conflicts and diversions as well as threats to confidentiality if people know each other in the group. However, a skilled interviewer can usually redirect the conversation back to topic and remind participants that all comments are to remain confidential. If someone is uncomfortable with the possible lack of confidentiality, that person may choose not to participate. When focus groups are implemented carefully and in an appropriate and respectful manner, they are efficient and provide an interview setting that lies within the social context of schooling, allowing for the gathering of perceptions, outcomes, and the impact of those outcomes as students experience them in the mathematics classroom (Patton 1990). Ethnographic Methods There were 13 focus groups about students' experiences in the classroom with a total of 89 students participating. Students ranged in age from 14 through 19 and were enrolled in the 9th through 12th grades at OGHS. The focus groups were held on three topics: general experiences in the mathematics classroom (G), best practices (BP), and white males (WM). There were three BP focus groups with a total of 22 students who were identified by their teachers as highachieving minority and white female students. There were seven G focus groups, with a total of 54 minority and white female students. These students came from both the general and honors curriculum and had indicated that they would be willing to participate in a focus group about mathematics. There were two (WM) focus groups with a total of 13 students from the general and honors curriculum who had indicated their willingness to participate in a mathematics focus group. There were a total of 27 black females, 13 black males, 6 Hispanic females, 1 Hispanic male, 29 white females, and 13 white males. All racial/ethnic categorizations are reported as the students' self identified race/ethnicity or are from school records when selfidentification data were not available. The BP comprised students who traditionally have lower math scores than their white male counterparts: white females, and black and Hispanic females and males. This group was specifically selected to identify the behaviors, practices, and attitudes that contribute to the success of these groups of students that are generally expected to do less well. The general group (G) consists of the same groups of students as the BP group; however, these students are not high achievers in math. This group was selected in order to get better insight into their experiences in the mathematics classroom. The white male (WM) group is typically the most academically successful group and the group to which all less academically successful groups are compared and to which white females, and black and Hispanic females and males are considered "others." Table 3 indicates the make up of each of the focus groups. Table 3. Student Focus Groups: Perceptions and Experiences In the Math Classroom Distribution of Students by Group, Gender, and Ethnicity Date Category BF BM HF HM WF WM Total 4/30 BP 1 1 1 2 5 5/1 BP 4 4 8 5/2 G 1 1 3 2 7 5/6 G 1 3 3 7 5/7 G 4 1 1 3 9 5/8 G 2 2 4 8 5/13 BP 3 2 4 9 5/14 G 5 1 1 2 9 5/15 G 2 2 3 7 5/16 G 4 1 2 7 5/19 WM 4 4 5/21 WM 8 8 5/22 WM 1 1 Totals 27 13 6 1 29 13 89 B = Black H = Hispanic W = White F = Female M = Male BP = Best Practices G = General WM = White Male Focus groups involve from 5 to 15 participants from a preidentified group of people. Under the leadership of a facilitator, focus group members are asked to respond to a predetermined set of questions on a specific topic during approximately 90 minutes. Participants are usually given an incentive of some sort to encourage participation and to compensate them for their time and contributions (Schensul 1999). Students who were identified as either highachieving minority or female students and students from the general student population who were willing to participate in focus groups on math were given a slip of paper by their mathematics teacher inviting them to a free pizza and soda luncheon that would take place during an extended lunch period. All students responding favorably and returning informed consent forms were given a free lunch of pizza and soda as they participated in the focus group process. All focus groups were conducted by trained anthropologists with specific objectives, procedures and practices for conducting and evaluating the focus groups in a uniform manner. I directed most of the focus groups. Four focus groups that contained students in my classes were directed by a fellow anthropologist and coresearcher who is trained in ethnographic field methods. The idea was to create an environment in which all students would feel free to express themselves honestly. All focus groups were initiated with the leaders identifying themselves and asking participants to identify themselves and give verbal assent to participate in the focus groups. This was followed by a brief description of the purpose of this research project and how the focus groups would work. Students were assured of their confidentiality and anonymity. After all the students had settled in with their lunches, the interview began with a grand tour question (Spradley 1979) on what it is like to be in their math classes. They were asked to think not just about the classes they were currently enrolled in but also about their experiences with mathematics over the years they had been in school. Students were asked to talk about practices and procedures that they liked and didn't like, things that helped them to learn and things that made it more difficult to learn, and anything they felt pertinent to helping me understand the culture of the mathematics classroom from their perspective. The list of questions for the focus groups is shown in the Appendix. Students were initially encouraged to begin the conversation after the grand tour question with a prompt such as, "What do you like about your math classes?" and letting the conversation go on from there with the group leader facilitating the flow of information and allowing the participants to speak freely as well as including the specific questions for the interviews. This process allowed for the gathering of specific data of interest as well as letting the students bring up their own topics, interests, concerns, and complaints, giving a more indepth look into the mathematics classroom from their perspective. Variables The focus groups were not intended to provide data for quantitative analysis. Nevertheless, the same three major variables are important for both the focus groups and the RQS: gender, ethnicity, and success in the mathematics classroom. By comparing data from across the three focus groups, we can illuminate the practices of those minority and female students who perform at the highest levels in mathematics. Limitations Implementation of the focus groups went very smoothly except for the day that only one student showed up, and that focus group became an individual interview. Most of the focus groups had students from different classes so that few students in a group had the same teacher. In a few of the focus groups, most of the students did have the same teacher for the same subject. There were times in these interviews where it turned into a freeforall against their teacher. The students in these groups who did not have the same teacher were shouted over as the others used the safe atmosphere of the focus group to vent about their teacher. The data gathered in these groups were valuable for the study, but these groups were more difficult to handle and the tapes were more difficult to transcribe because it was often difficult to tell who was speaking. Having either a second group facilitator take notes or having the facilitator insist that all informants give their names or some identifying information before speaking each and every time would help to avoid this difficulty. Doing this, however, could easily interfere with the easy conversational flow of the focus group. An indepth examination and analysis of the RQS and focus group data gathered in this project appears in the following two chapters. CHAPTER 4 RESULTS FROM QUANTITATIVE PORTION OF THE STUDY: THE RANDOM QUESTIONING STRATEGY EXPERIMENT Introduction This chapter addresses the analysis of the quantitative data generated in the RQS experiment. Here I will look at the academic, attitudinal, and participation data from the pre and postacademic tests, student grades, attitudinal scales, and question answering data from RQS experiment. These data will directly address the following hypotheses: H2: There will be differences in students' attitudes and beliefs toward mathematics across gender and cultural lines. H4: The experiences and perceptions of students in the research population will vary along gender, ethnic, and socioeconomic lines. H,: Implementation of a random questioning strategy will result in increased academic performance and improved attitudes toward mathematics for female and minority students. H6: Gender will be a more significant variable than ethnicity, class, or age with respect to academic performance and attitudes toward mathematics in a secondary setting. H7: Implementation of the random questioning strategy will have the same directional effect for minority students and women. Hio: Students in RQS treatment classes' levels of participation will increase over time. Hi,: Students in RQS treatment classes will have more correct answers over time. H12: Gender will be a more significant variable than ethnicity, class, or age with respect to students' participatory and question answering behaviors in RQS treatment classes. Academic performance and attitudinal data were analyzed in a series of ANOVAs and ttests to determine the effect of the RQS on student achievement and attitudes toward mathematics. Participatory data were analyzed using odds ratios and backwards stepwise regression to determine the effect of the RQS on student questioning behavior. As mentioned in the previous chapter, performance and attitudinal data were gathered from 453 students in 25 classes, including 292 in the 14 treatment classes and 161 in the 11 control classes. Table 4 shows the distribution of students by treatment, gender, and ethnicity of performance and attitudinal measures in the RQS experiment. Participation data was based on student participation in the classroom to determine whether students' participation patterns changed as a result of the RQS. Participatory question answering behavior data consisted of 497 students in 20 classes, including 307 in the 14 treatment classes and 190 students in the 6 control classes. Table 5 shows the distribution of students by treatment, gender, and ethnicity for participatory behavior data. Table 4. Performance and Attitudinal Measures Random Questioning Strategy Experiment Distribution of Students by Treatment, Gender, and Ethnicity TREATMENT CONTROL White Black Hispanic White Black Hispanic Male Female Male Female Male Female Male Female Male Female Male Female 73 84 44 71 8 12 27 37 32 52 4 9 Table 5. Question Answering Behavior Random Questioning Strategy Experiment Distribution of Students by Treatment, Gender, and Ethnicity TREATMENT CONTROL White Black Hispanic White Black Hispanic Male Female Male Female Male Female Male Female Male Female Male Female 74 81 48 76 9 12 58 53 26 24 10 10 Data Classification and Variables Three types of data were gathered in the RQS experiment: academic performance, attitudes toward math, and response by each student (pass, answer correctly, or answer incorrectly) when called upon in class. The classification variables are treatment, teacher, course, ethnicity, gender, age, grade, and SES. The variable treatment is whether or not a student received the RQS. Teacher is the instructor each student was assigned. Course was the math course that the student was enrolled in at the time of the experiment. Gender, ethnicity, age, and grade were all determined by student reports unless this information was missing or unclear, in which case these data were gathered from school records. Free and reduced lunch status was used as a proxy for SES. Students on free and reduced lunch were categorized as lower SES than those on full pay lunch. Age, grade and course are all highly intercorrelated, so I used course to represent this grouping. Course may comprise mixed grades (an advanced 9th grader can be in a class that has mostly 11th and 12th graders in it, while a 9th grade class can have lower level 12th graders in it trying to make up credit deficits), but since math knowledge is cumulative, I chose course as my measure of development as it is a less variable way of classifying students than age or grade. Outcome variables are repeated measures Time 1 (T1) and Time 2 (T2), and pre and posttest, measures for academic performance (NPSS), class grades, and seven attitudinal and belief scales with respect to mathematics. The scales measured anxiety toward mathematics, the value of math in society, selfconcept in mathematics, enjoyment of mathematics, motivation in mathematics, beliefs about mathematics, and attitudes toward mathematics. The most appropriate statistical tests for these data are paired sample ttests for bivariate analyses and analysis of variance (ANOVA). Both are part of a family of tests that seek to determine how much of a measurement for individuals can be attributed to the particular subgroups they are in. These tests compare mean scores across groups and determine if they are statistically different when group variances are taken into consideration. Analysis of RQS participatory data was accomplished using odds ratios and multivariate logistic regression in an effort to develop a main effects model. All analysis here takes pvalues less than or equal to .05 as significant. Analysis was done with SPSS 10.0.5 (1999) and SYSTAT 10.2.1 (2002). Inherent in the design of this experiment are three basic confounds. First is the interaction between teacher and treatment. I chose not to assign the same teacher both treatment and control classes to avoid a diffusionoftreatments confound. Unfortunately, it is difficult to separate teacher effect from treatment/control effect. To minimize this confound as much as possible, several assumptions about the data and research design were made for this part of the analysis. While we know that the attitude and belief scales are probably intercorrelated since they were administered on one instrument, all analysis is done under the assumption that the scales are independent of each other. These assumptions affect the Type I experimentwide error rate, the likelihood of incorrectly rejecting the null hypothesis, but these assumptions will allow a clearer picture of this situation and guide future research. The first step in this analysis was to determine if there is a significant difference in scores for treatment and control groups. Academic Performance Measures National Proficiency SurveyMathematics Using paired sample ttests, assuming different group variances, students' scores on the National Proficiency Survey in Mathematics were significantly higher from T to T2 for both the treatment and control groups. This change is to be expected, as learning is the point of schooling. For the treatment group, t = 3.387 and p = 0.001 with 234 degrees of freedom. For the control group, t = 3.203 and p = 0.002 with 130 degrees of freedom. Comparing the data on test score change between treatment and controls, however, reveals that there is no significant difference in performance scores. (The tstatistic comparing treatment and control groups is 0.36, p = 0.718 with 309 degrees of freedom.) Variation as a result of treatment, ethnicity, and gender may have been obscured in the ttests, so all of these classification variables were entered into an analysis of variance. The ANOVA results confirm that none of the factors (ethnicity, treatment, gender) accounts for change in test score from T to T2. Student Grades ANOVA also confirms that treatment, SES, gender, and ethnicity had no effect on change in semester grade, and there are no significant interaction effects on grade changes. Discussion of Academic Performance Measures In summary, none of the predictor variables (treatment, SES, ethnicity, and gender) either alone or in any combination, was a significant predictor of academic outcomes as measured by the NPSS Mathematics or in student grades. Participatory Measures Data on individuals' participation in RQS and control classrooms were analyzed using odds ratios with logistic regression to investigate two questions: Are students in the treatment group more likely to participate than those in control groups? and Are students who participate more likely to answer correctly over time? The logistic regression approach is the most appropriate and powerful statistical method for answering these questions. It also allows us to determine what the key predictors of participation and answering correctly are and it eliminates the need for simpler univariate analyses. Results of this multivariate model indicate that none of the key predictors gender, ethnicity, class, grade, treatment, or controlindividually or in any combinationwere significant in predicting the outcome participation variables. Based on the data collected in these classrooms and using this statistical approach, it is impossible to determine if students in RQS classrooms perform any better than students in control classrooms. The main effects model did not fit the data well enough to make any determinations. At this point, these data do not suggest any particular characteristics that may help to predict classroom participation. In other words, there is no clear evidence that the RQS increases participation. There is also no evidence that the RQS increases the number of students' correct answers. Discussion of RQS Data: The results of the RQS data indicate that we cannot determine what individual characteristics may contribute to participation in these classrooms. In other words, there is no evidence that a particular gender, ethnicity, or grade level participates more or less in either the RQS classrooms or the control classrooms. This suggests that the bias suspected in control classrooms may not be present. Teachers may have some means of moderating possible biases with pedagogical approaches and techniques that are not readily apparent. Attitudinal and Belief Measures Each class was administered a pre and postattitudinal survey with seven scales measuring students' attitudes, beliefs, and ideas with respect to mathematics. The scales were (1) anxiety toward mathematics, (2) the value of math in society, (3) selfconcept in mathematics, (4) enjoyment of mathematics, (5) motivation in mathematics, (6) beliefs toward mathematics, and (7) attitudes toward mathematics. The post survey had an additional ten questions pertaining to students perceptions of the handheld computer random questioning strategy that was administered only to treatment groups to gather additional information on how the students felt about the intervention. These data are addressed in the next chapter. Evaluation of Survey Data All analysis for survey data was performed in the same manner as the performance measures using ANOVA in order to determine which variables or combination of variables had a significant effect on attitudinal scale scores between T1 and T2. Anxiety The Anxiety Toward Mathematics Scale measures the uneasiness a student feels in situations involving mathematics. Neither ethnicity, treatment, nor gender significantly account for change in anxiety; nor were there any significant two or threeway interactions. Math Value The Value of Mathematics in Society Scale measures a student's view regarding the usefulness of mathematical knowledge. Variation in math value change is accounted for by ethnicity (F= 4.59 and p= 0.011 with two degrees of freedom). The Fstatistic reveals that ethnicity has a significant effect on the change in math value scale scores between pre and post surveys. The average change for white students' scores is approximately 1.5 scale points, with black students' mean scale change at or near 0, and Hispanic students' mean scale change was approximately 1 scale points. Treatment/control and gender were not significant factors for this scale; nor were there any two or threeway significant interactions. In the graph shown in Figure 1, ethnicity is on the horizontal axis with 1 for White, 2 for Black, and 3 Hispanic. On the vertical axis is change in math value scores. Examining this graph, we see that white students' math values changed positively, black students' math values hardly changed at all, and Hispanic students' math value scores changed negatively. Least Squares Means 3 C 1 3 I I I 1 2 3 ETHNICITY 1White 2Black 3Hispanic Mean Change in The Value of Mathematics Scale White Black Hispanic 1.5 +0 1 Figure 1. ANOVAThe Value of Mathematics Scale By Ethnicity SelfConcept The SelfConcept in Mathematics Scale measures students' perception of their own competence in mathematics. Variation in selfconcept is accounted for by ethnicity. F = 3.288, and p = 0.039 with two degrees of freedom. The average change for white students' scores is approximately 1.5 scale points, with black students' mean scale change positive, but at or near 0, and Hispanic students' mean scale change was approximately 1 scale points. Treatment/control and gender were not significant factors for this scale, nor were there any two or threeway significant interactions. In Figure 2, ethnicity is on the horizontal axis of the graph, with 1 for white, 2 for black, and 3 Hispanic. On the vertical axis is change in selfconcept scores. Examining this graph one can see that white students' math values changed positively, black students' math values had a positive change, but near 0, and Hispanic students' math value scores changed negatively. Least Squares Means 1 2 3 2 I I 1 ETHNICITY 1White 2Black 3Hispanic Mean Change in The SelfConcept in Mathematics Scale White Black Hispanic 1.5 +0 1 Figure 2. ANOVAThe Selfconcept in Mathematics Scale By Ethnicity Enjoyment The Enjoyment of Mathematics Scale measures the pleasure a student derives from engaging in mathematical activities. Neither ethnicity, treatment/control, nor gender significantly account for change in enjoyment; nor were there any significant two or three way interactions Motivation The Motivation in Mathematics Scale measures students' desire to increase their knowledge and understanding of mathematics. Three of the four questions on this scale that factor load the highest have to do with the desire to do some mathematical work beyond classroom requirements (Welch and Gullickson 1973). Hence, the motivational construct of this scale must be interpreted within this limited meaning. Variation in motivational change is accounted for by treatment. F = 4.50 and p = 0.035 with one degree of freedom. Ethnicity and gender were not significant factors for this scale. The average change for treatment students' scores was near zero scale points, with control students having a mean scale change of approximately 2.3 scale points. There were no two or threeway significant interactions. In the graph shown in Figure 3, motivational change is on the vertical axis and treatment/control on the horizontal axis, with treatment being 1 and control 0. Note that for motivation change, the treatment group has almost no change, while the control group's motivation increased. These results appear to indicate that the RQS had a negative motivational effect on students. This could also be due to the possibly confounding effects of teacher or timing as mentioned previously. Least Squares Means 3 0 2 U I 0 0 1 0 1 TREATMENT 1 = Treatment 0 = Control Mean Change in Motivational Scale Scores Treatment Control +0 2.3 Figure 3. ANOVAMotivation in Mathematics Scale Treatment vs. Control Beliefs The Beliefs Scale measures a student's beliefs and ideas about mathematics. Variation in change in beliefs is accounted for by ethnicity. F = 3.397, and p = 0.035 with two degrees of freedom. The average change for white students' scores is approximately 1.25 scale points, with black students' mean scale change being positive, but at or near 0, and Hispanic students' mean scale score negative, but also at or near 0. Treatment/control and gender were not significant factors for this scale; nor were there any two or threeway significant interactions. In Figure 4, we see change in beliefs with respect to ethnicity. Beliefs is on the vertical axis while ethnicity is on the horizontal axis with 1White, 2Black, 3Hispanic. For white students, the change is positive, for black and Hispanic students the change is zero or negative but nearly zero. Least Squares Means 1 o 0 2 1  2 II I 1 2 3 ETHNICITY 1White 2Black 3Hispanic Mean Change in The Beliefs About Mathematics Scale White Black Hispanic 1.25 +0 +0 Figure 4. ANOVABeliefs About Mathematics Scale by Ethnicity Attitudes The Attitudes Scale measures students' attitudes toward mathematics. Neither ethnicity, treatment, nor gender significantly account for change in attitudes toward mathematics; nor were there any significant two or threeway interactions. Discussion of Survey Data Gender did not significantly account for change, as expected, on any of the scales. However, in all scales that did show significant change, ethnicity was a significant factor, accounting for that change in all but one. These were the math value, selfconcept, and beliefs scales. Treatment accounted for significant change on only one scale: motivation. Ethnicity accounted for significant change on the Math Value, SelfConcept, and Beliefs scales. The changes between T1 and T2 shown on these scales were positive for white students, near 0 or negative for blacks, and in each case negative for Hispanics. The aim of this portion of the study was to test whether or not an intervention such as the RQS, which gives all students equal access to participate in the question asking and answering process in the classroom, would increase performance and improve attitudes of women and minority students. These results indicate that this is not the case. Gender did not emerge as a significant factor associated with changes in beliefs and attitudes about mathematics, as expected. These results show that change between T1 and T2 are very similar for females and males, indicating that no genderrelated differences are apparent in these data. However, ethnicity was a significant factor for several scales, as was expected, and this indicates that changes in T1 and T2 scores are very likely related to ethnicity for those scales. Change on the Motivation Scale is accounted for by treatment. Motivation for the treatment group has very little change, while motivational change for the control group increased. Regardless of these circumstances, motivational change is accounted for by treatment, indicating that either the RQS had no effect on motivational scores or those scores were a function of timing, teacher, or any number of other factors. However, one can conclude that the RQS did not have a significant effect on motivational change. In future research, implementing controls for the timing and teacher confounds will give us a clearer picture of this situation. BetweenGroup Comparisons After analyzing the pre and posttest and survey data for change as a result of the intervention, it became clear that knowing the differences between gender and ethnic groups at T1 and T2 would be informative to this analysis. Pre and postcomparisons between all groups, white, black, and Hispanic by gender were compared to get a better picture of how each of the groups compared with each other at T and T2. In this section I will give a brief report of the results with interpretation to follow. A series ofttests shows that on both T1 and T2 data for all groups, females had significantly higher academic performance than males. On the attitudinal and beliefs items, females scored higher than males on the selfconcept scale on the presurvey. However, there was no significant difference in scores by gender on the postsurvey for selfconcept. On the enjoyment, motivation, beliefs, and attitudes scale, there were no significant differences related to gender. The ttests were supported by a series of ANOVAs. All analysis is by ethnicity and gender, comparing mean rankings for each group. Looking at the betweengroup comparisons, the change in ranking and comparison of certain rankings leave some interesting questions to be addressed. On most of the items it appears that white and black students' score rankings showed little change from Tl to T2. while Hispanic males' score rankings tended to increase and Hispanic females scores tending to decrease over time. Mathematical performanceNPSS mathematics On the subject area tests, as illustrated in Figure 5 below, white students performed the best both preand posttest, with black students having very little change in their standing. However, from T1 to T2, Hispanic females' rankings decreased dramatically and Hispanic males' rankings improved. Looking at these scores strictly by ethnicity, they are as expected, but when we examine the effect of gender within ethnicity, the dimension of gender emerges. While one could speculate as to what is affecting Hispanic females' scores from T1 to T2, something is contributing to their decreased performance that is different from white and black students and Hispanic males whose rankings remained relatively the same or improved. Examining the graph below, note that on the pretest for all ethnic groups, female scores were higher than male scores, but on the posttest, one can see that that while all scores increased, the increase in Hispanic males' scores was much greater than that of Hispanic females. PrePost Subject Area Test Ranking Pre Post 1 WF WF 2 WM WM 3 HF HM 4 BF BF 5 HM BM 6 BM HF Key: B=Black W = White H= Hispanic M = Male F = Female Estimated Marginal Means of PREPERCE Estimated Marginal Means of POSTPERC 70 70 0) 60 S \ GENDER GENDER C male f male E E S50 female 40 female white black/mixed hispanic white black/mixed hispanic ETHNICIT ETHNICIT Figure 5. Between Groups Pre and PostTest Rankings National Proficiency SurveyMathematics Math anxiety As shown in Figure 6 below, this trend continues on the anxiety scale, with very little change in the ranking of black and white students, but Hispanic females' anxiety ranking went from the lowest score on the presurvey to the highest on the postsurvey. PrePost Anxiety Scale Ranking Pre Post 1 WM HF 2 BF BF 3 WF WM 4 BM WF 5 HM BM 6 HF HM Key: B=Black W = White H= Hispanic M = Male F = Female Estimated Marginal Means of PREANX Estimated Marginal Means of ANSPOST black/mixed GENDER S male female hispanic white black/mixed ETHNICIT Figure 6. Between Groups Pre and PostTest Rankings Mathematics Anxiety Scale Examination of the graphs in Figure 6 illustrates that on the preanxiety scale, white males had the highest anxiety rankings and Hispanic females the lowest, but on the postanxiety scale, white males' anxiety dropped, with white females' anxiety increasing. However, the most drastic change is the change in Hispanic females' anxiety rating, which went from the lowest on the presurvey to the highest anxiety ranking on the post survey. 170 165 160 155 150 S145 140 S135 L 130 whl te ETHNICITY GENDER male female hispanic , 74 Value of mathematics For math value, illustrated in Figure 7, Hispanic students ranked the highest with no change pre to post survey, with very little change in the other groups with the exception of black females' math value ranking, which fell from third to sixth in the rankings. PrePost Mathematical Value Ranking Pre Post 1 HM HM 2 HF HF 3 BM BM 4 BF WF 5 WF WM 6 WM BF Key: B=Black W= M = Male Estimated Marginal Means of MVPRE White H = Hispanic F = Female Estimated Marginal Means of MVPOST white ETHNICITY 256 254 252 250 24 8 246 GENDER  244 1 male m 242 S female W 240 black/mixed hispanic whlte black/mixed ETHNICITY Figure 7. Between Groups Pre and PostTest Rankings Mathematical Value Scale In the graphs in Figure 7, note that on both pre and postsurveys, not only were Hispanic students' math value scores the highest, but the Hispanic females' scores got closer to those of their male counterparts over time. Also, note the drop in black females' scores, illustrating their drop from fourth to sixth place in the rankings from T1 to T2. 270  265 260 255 250 S245 2 240 E w 235 I SelfConcept in mathematics The sizeable drop for Hispanic females was again apparent on the selfconcept scale (Figure 8), with white females ranking the highest both pre and post and white males having among the lowest math selfconcepts. Black students' rankings had very little change. PrePost SelfConcept Ranking Pre Post 1 WF WF 2 HF HM 3 HM BF 4 BF WM 5 BM BM 6 WM HF Key: B=Black W = White H= Hispanic M = Male F = Female Estimated Marginal Means of SCPRE Estimated Marginal Means of SCPOST 215 210 21 0 205 20 5 S200 4 200 1 195 195 o195 190 GENDER GENDER S190  18 o O male 3 male L 180_ female w 185 female whtte blacklmixed hispanic whtte black/mixed hispanic ETHNICIT ETHNICIT Figure 8. Between Groups Pre and PostTest Rankings SelfConcept in Mathematics Scale Examining the graphs in Figure 8, note that at T1, Hispanic students' scores were among the highest of all the groups, but at T2 Hispanic females' scores dropped to the lowest of all groups and the gap between black students' selfconcept scores in mathematics increased over time. Also note that white males' selfconcept in mathematics scores is among the lowest at both T1 and T2. 76 Enjoyment of mathematics Black students scored the highest on the math enjoyment scale both pre and post with white students' rankings among the lowest on both implementations with Hispanic students showing very little change from T1 to T2, as evidenced in Figure 9. Between Groups Pre and PostRankings Enjoyment of Mathematics Scale PrePost Enjoyment Ranking Pre Post 1 BF BF 2 BM BM 3 HM HM 4 WF HF 5 HF WM 6 WM WF Key: B=Black W = White H= Hispanic M = Male F = Female Estimated Marginal Means of ENJPRE Estimated Marginal Means of ENJPOST 155 155 150 150 145 145 S140 a S1 140 135 2 2 135 130 GENDER GENDER 125 1 male 13 male S120 female 1251 female white black/mixed hispanic white black/mlxed hispanic ETHNICIT ETHNICIT Figure 9. Between Groups Pre and PostTest Rankings Enjoyment of Mathematics Scale Examining the graphs in Figure 9, note that their shapes have very little change from T1 to T2 except for white females' scores dropping below those of white males and Hispanic females' scores rising closer to their male counterparts. Also note that from T1 to T2, black students' enjoyment scores remained consistently higher than all other groups. Mathematical motivation Mathematical motivation scores shown in Figure 10 indicate that white students had the lowest mathematical motivation both pre and post, with black females' motivational ranking remaining among the highest. Hispanic students' motivational rankings had very little change. PrePost Mathematical Motivation Ranking Pre Post 1 BM BF 2 BF HM 3 HM BM 4 HF HF 5 WF WM 6 WM WF Key: B=Black W = White H= Hispanic M = Male F = Female Estimated Marginal Means of MOTPRE Estimated Marginal Means of MOTPOST 145 150 140 145 135 140 a 130 D 135 125  ) 2 130 S120 GENDER GENDER ~ 15 ] / D125  S115 male 1 3 male S110__ female 120 female white black/mixed hispanic white black/mixed hispanic ETHNICIT ETHNICIT Figure 10. Between Groups Pre and PostTest Rankings Mathematical Motivation Scale Examining the mathematical motivation scales in Figure 10, note that the white males' scores increased to the level of white females from T1 to T2. Nevertheless, white students' scores remained the lowest of all groups. Black students' scores also remained relatively the same, with black females surpassing males at T2. Hispanic students' scores also remained relatively the same from T1 to T2. 78 Beliefs about mathematics There was a substantial drop in black males' beliefs about mathematics from the highest most positive in first place to fifth place from T1 to T2. Hispanic females had the most negative beliefs about mathematics both pre and post, with Hispanic males' beliefs about mathematics increasing from fifth place to second (Figure 11). PrePost Beliefs Ranking Pre Post 1 BM WF 2 WF HM 3 WM BF 4 BF WM 5 HM BM 6 HF HF Key: B=Black W = White H= Hispanic M = Male F = Female Estimated Marginal Means of BIPRE Estimated Marginal Means of BLFPOST 178 180 177 178 Sm17 a 176 i i 174 175 S172 S174 21 GENDER 170GENDER Female 168 male L 1721_ female V 1661 female wh te black/mixed hispanic white black/mlxed hispanic ETHNICIT ETHNICIT Figure 11. Between Groups Pre and PostTest Rankings Beliefs About Mathematics Scale Note the relative change in position of black females and males from T1 to T2, with black males having the most positive beliefs about mathematics at T1 to very low at T2 with Hispanic females having the lowest beliefs scores consistently. Also notice the increase in Hispanic males' scores from T1 to T2. Attitudes toward mathematics For attitudes toward mathematics (Figure 12), white students again were among the lowest ranks in both pre and postsurveys, with Hispanic males and black females consistently ranking in the top positions. PrePost Attitudes Ranking Pre Post 1 HM HM 2 BF BF 3 BM HF 4 WF BM 5 HF WM 6 WM WF Key: B=Black W = White H= Hispanic M = Male F = Female Estimated Marginal Means of ATTPRE Estimated Marginal Means of ATTPOST 145 145 140140 140 135 S135 / r 130 125 1 I GENDER I GENDER 125 S120 / male 1 male w 115 female L 120 female white black/mixed hispanic white black/mixed hispanic ETHNICIT ETHNICIT Figure 12. Between Groups Pre and PostTest Rankings Attitudes About Mathematics Scale Examining these graphs, we can clearly see that white students' scores remain among the lowest from T1 to T2, and black females' and Hispanic males' attitudes toward mathematics are consistently among the highest. Discussion of BetweenGroup Comparisons These betweengroup T and T2 rankings yield some very interesting and surprising findings, many in contrast to what one might expect: Academically, white students are performing at the highest levels with black males consistently ranking near the bottom. The considerable drop in Hispanic females' scores accompanied by a considerable increase in Hispanic males' scores points to some factor or factors affecting Hispanic student's academic scores, beliefs, and attitudes differently than white and black students' scores, as illustrated in Figure 5. This trend continues when we examine the anxiety scores of Hispanic students. From T1 to T2, Hispanic females' math anxiety increased from the lowest at T1 to the highest at T2. Black females have the second highest math anxiety ranking at both T1 and T2, with very little change for white students and black males (Figure 6). Contrasting with the performance and anxiety scores (Figures 5 and 6), Hispanic students' had the highest math value score rankings (Figure 7) for both T1 and T2. Black females' math value ranking dropped to the lowest at T2, yet black females have some of the highest scores on enjoyment, beliefs, and attitudes toward mathematics (Figures 9, 11, and 12). This trend of dropping scores for Hispanic females along with rising scores for Hispanic males continues to be apparent on the self concept scale (Figure 8). Interestingly here, white males' selfconcept in mathematics ranking is among the lowest, while white females' selfconcept in mathematics is the highest both T1 and T2 with black students' rankings having very little change over time. On the enjoyment scale (Figure 9), black students have the highestranking scores, with Hispanic males ranking 3rd for both T1 and T2. White students' math enjoyment scales are consistently ranked among the lowest, with white females ranking the lowest of all at T2. In terms of mathematical motivation (Figure 10), black females consistently rank near the top with white students consistently ranking at the bottom and Hispanic males ranking just slightly higher than Hispanic females for both T1 and T2. Continuing this trend in Hispanic females' scores, Hispanic women exhibit the most negative beliefs about mathematics, while Hispanic males' beliefs scores substantially increase from fifth to second place from T1 to T2. Beliefs about mathematics (Figure 11) are consistently the highest for white females. While there was little change in black females' and white males' beliefs about mathematics, black males' beliefs scores dropped from first to fifth place over time. Similar to their motivational scores (Figure 10), white students' attitudes toward mathematics scores ranked among the lowest for both T and T2. Hispanic males and black females consistently ranked the highest scores for attitudes toward mathematics, yet Hispanic females and black males have among the lowest post test scores on the beliefs (Figure 11), selfconcept (Figure 8), and academic scales (Figure 5). Black female and Hispanic male students' scores are consistently similar for performance, enjoyment, motivation, attitudes, beliefs, and math value (Figures 5, 7, 9, 10, 11, 12). Hispanic females display some of the highest levels of anxiety about mathematics (Figure 6). These trends indicate that within ethnic groups, gender plays an important role in student attitudes, beliefs, and academic performance. Comparing academic performance with attitudes and beliefs over time yields an interesting picture, and often it is not what one would expect. While white students tend to have the highest academic scores (Figure 5), their math enjoyment (Figure 9), mathematical motivation (Figure 10), and attitudes toward mathematics (Figure 12) are among the lowest of all six groups. Gender comes into play for white students, with the sizeable differences in female and male for selfconcept in mathematics (Figure 8) and beliefs about mathematics (Figure 11), with white females' consistently scoring higher than white males. Academically (Figure 5), white female and male students are performing at the highest levels, but their high academic scores are situated in the lowest levels of enjoyment (Figure 9), motivation (Figure 10), and attitudes (Figure 12) toward mathematics with respect to their peers. This indicates that white students, females particularly, are successfully struggling for mathematical success in an environment that is highly anxious for them; yet, they perceive the value (Figure 7) of what they are doing as fairly low. However, despite those obstacles, they are continuing to perform. Gender is a significant variable among white students, with white males having higher anxiety (Figure 6), lower selfconcept (Figure 8), and less positive beliefs about mathematics (Figure 11) than those of their white female counterparts. Similarly, gender differences are apparent in black students' rankings. Academically, black males' scores are ranked among the lowest, with black females consistently ranking higher than their male counterparts near the middle of all groups (Figure 5). These academic standings are situated in a student population where the females have the second highest math anxiety (Figure 6) rating and their perception of the value of mathematics (Figure 7) is among the lowest. In contrast, black students consistently have the highest scores for mathematical enjoyment (Figure 9), with female scores higher than male. Black students' mathematical motivation scores (Figure 10) are steadily among the highest of all groups. While there was very little difference in belief scores (Figure 11) for black females, black males' beliefs about mathematics scores dropped drastically from the highest at T1 to fifth place at T2. Examining these rankings indicates that while black students are academically among the lowest, black females are consistently performing at higher levels than their black male counterparts, but with a very high level of anxiety and with a low perception of the value of mathematics. Despite the high anxiety and low perception of the value of mathematics, black students report that they are enjoying mathematics more than their white and Hispanic peers, with black females reporting higher enjoyment levels than black males. Another surprising contrast here is the contrast between performance and motivation. Black students have some of the highest motivational scores and some of the lowest academic scores in contrast to white students, who have the highest academic scores but the lowest motivational scores. Gender appears to be a significant factor in both academic performance and attitudes and beliefs toward mathematics for black students. Black females are in a similar situation to that of white females; they are performing at higher levels than their male counterparts but with very high levels of anxiety and a low perception of the value of mathematics. Similarly, there is a clear differential between mathematical motivation and academic performance for black students, which indicates a discontinuity between attitudes and performance that is related to both gender and ethnicity. As evidenced for both black and white students, gender appears to be a considerable factor within ethnic/racial groups. Gender also appears to have a sizeable effect on attitudes, beliefs, and academic performance for Hispanic students. However, within the Hispanic group of students there is an interesting gender dynamic that does not appear to be the case for the black and white groups. Academically, a considerable drop in Hispanic females' scores is accompanied by a similarly considerable increase in Hispanic male's scores from T1 to T2. Likewise, on the selfconcept scale, Hispanic females' scores dropped drastically while the Hispanic males' scores rose over time. Hispanic females had the most negative beliefs about mathematics while Hispanic males' beliefs scores rose substantially from near the lowest at T1 to near the highest at T2. Hispanic females' math anxiety increased similarly, from the lowest anxiety levels to the highest anxiety levels of all groups from T1 to T2. Hispanic males' math enjoyment and motivation scores are consistently higher than those of their Hispanic female counterparts. Hispanic males along with black females had the highest attitude toward mathematics rankings for both T1 and T2. Thus, for the Hispanic students in this study, gender is evident as a significant factor in terms of academic performance, attitudes, and beliefs about mathematics. Within the black and white groups, females seem to be outperforming their male counterparts despite the fact that this superior academic performance is often accompanied by attitudes and beliefs that would not seem to support such performance. However, within the Hispanic group, the male students are outperforming their female counterparts and tend to have more positive attitudes and beliefs associated with this better academic performance. Hispanic females are consistently ranking lower from T1 to T2 both academically and in terms of attitudes and beliefs toward mathematics. For Hispanic students, males seem to be out performing females, and this better performance is associated with more positive beliefs and attitudes about mathematics. Not only are Hispanic males outperforming their female counterparts, but, Hispanic females' academic, attitudinal, and beliefs scores in general have a notable decrease over time which is different from others in both their same gender and ethnic groups. These findings would support the idea that black and white females realize the necessity of success in the mathematics classroom, and despite this discomfort or dislike, they are performing to a higher standard than their male counterparts. In the Euro 