NORTHWESTERN UNIVERSITY
Geometries of Inequality: Teaching and Researching Critical Mathematics in a Low- Income Urban High School
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Education and Social Policy, Department of the Learning Sciences
By
Andrew Morgan Brantlinger
EVANSTON, ILLINOIS
June 2007
2 ABSTRACT
Geometries of Inequality: Teaching and Researching Critical Mathematics in a Low-
Income Urban High School
Andrew Morgan Brantlinger
The purpose of this dissertation study was to build on scholarly work in the area of critical mathematics by such scholars as Gutstein (2003) and Gutierrez (2002). Critical pedagogy generally is designed to foster political agency and critical consciousness in students (Freire, 1971). That is the aim of critical mathematics in addition to the general goal of enhancing mathematical power that is outlined in the standards of the National
Council of Teachers of Mathematics (NCTM, 2000). Proponents of critical mathematics argue that a Freirean approach to mathematics instruction has the potential to be more empowering and equitable than dominant versions of mathematics instruction because it allows subordinated students to use mathematics to develop their understandings of sociopolitical matters that affect themselves and their communities.
This dissertation relied on practitioner research; I used qualitative methods to examine my own critical mathematics instruction and curriculum design. It was conducted in a remedial geometry course in a low-income urban high school. Analyses were done on: (1) my approaches to critical curriculum design and my own teacher
3 beliefs, (2) the critical component of my mathematics curriculum, and (3) videotapes of classroom discourse and student participation in standards-based and critical activities.
The first analysis was conducted primarily on an instructional diary that I kept as I engaged in curriculum design and instruction; it documented evolutions in my approach to critical curriculum development and in my beliefs about secondary mathematics and the potential it offers for political and academic student empowerment. The curriculum analysis was based on a sociological framework developed by Dowling (1998) for the analysis of school mathematics texts. Using his approach, I found that the attention my critical mathematics placed on critical themes and contexts tended to displace the development of mathematical power. In the discourse analysis I examined the ways students engaged in, or resisted, standards-based and critical mathematics activities and also raised issues of power, status, and equity. There were a number of interrelated findings regarding student participation, including: marked differences in student participation in critical and standards-based activities and a positive shift in student stances towards the expression of personal agency, student-centered (subjectified) control of the classroom activities, and their ability to verbally reason about mathematical functions and take risks in classroom participation.
My dissertation findings raise important issues for those interested in issues of equity and social justice in mathematics education to consider. In particular, I found my critical mathematics program provided my low-income students of color with potential political use-values while not providing them the fullest exchange-value possible.
Because, as a teacher and curriculum designer, I had not set out to trade in exchange- value for use-value, I dissatisfied with some of my dissertation findings about critical
4 secondary mathematics. At the same time, my findings were not entirely negative for critical mathematics education; some of my students appeared to find critical activities to be more engaging and meaningful than standards-based activities. The balance between teacher and student responsibility for learning tipped toward the students in a constructive way in some critical mathematics activities. In addition, my discourse analyses, showed that, over time, there were higher rates of student subjectivization and increased classroom participation – especially regarding student willingness to verbalize their reasoning behind finding mathematical solutions in standards-based activities. Further research is recommended to: (1) enhance the mathematics power of critical mathematics and standards-based texts and activities; (2) explore the ways that the personal and political emphasis of critical mathematics might contribute positively to students participation and risk-taking in the mathematics classroom; and (3) understand how the negative aspects of interpersonal classroom power relations (e.g., when teacher is presented as authority and student as follower, feelings and actions based on the status of being a higher- or lower-achieving students) can be reduced through critical mathematics programs.
5 ACKNOWLEDGEMENTS
First and foremost, I dedicate my dissertation to the students at Guevara (the focus of my dissertation) and Park Vista. Their honesty, humor, insights, agency, and resistance, taught me more than I could have taught them. I am still learning from reflecting on our time together. I am also grateful to the Guevara and Park Vista administrations for allowing me to research my mathematics teaching.
The assistance and words of encouragement I received from my dissertation co- chairs, Professor Miriam Sherin and Professor Carol D. Lee, were invaluable. This dissertation simply would not have been possible without their guidance and support throughout my time as a graduate student at Northwestern University. I am also very grateful to Professor James P. Spillane for his mentoring and the help he gave me in framing my study and dissertation defense. As will be obvious to the reader, the practitioner research of Professor Eric Gutstein on the teaching of mathematics for social justice was the inspiration for my dissertation. I was fortunate to have him as a role model. I am also honored that each of these scholars were on my dissertation committee.
There were many others who helped support and shape my dissertation. First and foremost, was my mother Ellen Brantlinger. She was the first person to give me feedback on my draft chapters. I am very lucky to have a parent who could help me with my writing and help me think through important issues. More importantly, her thought on educational matters has had a profound affect on my dissertation and my career path.
Professor Norman Weston was a great listener and encouraged me get through some of my most challenging times. My mother and Norman helped me see that, despite my struggles with critical mathematics, my dissertation was valuable. Finally, I should
6 acknowledge my former colleague and good friend, Marc Siciliano, for our ongoing conversations about urban mathematics and science education. These conversations helped me clarify and expand on many of the ideas contained in this dissertation. While he does not yet have a doctorate, there are few educational thinkers I respect more than
Marc.
Many others provided support and inspiration: my family (Leroy Robinson Jr.,
Jayla Robinson, Susan Robinson, Jeremy Brantlinger, Patrick Brantlinger, the Italian
Andersons, the Minnesota Obeys, and the Canadian Smiths), friends and education scholars (i.e., Sergei Abramovich, Laura Adamo-Martinez, Alice Alston, Nieves Angulo,
Wanda Blanchett, Sukey Blanc, Leslie Bloom, Alan Collins, Laurel Cooley, Yak Epstein,
Mary Foote, Lynda Ginsburg, Serigne Gningue, Jerry Goldin, Louis Gomez, Rochelle
Gutierrez, Whitney Johnson, David Kanter, Eva Lam, Pauline Lipman, Danny Bernard
Martin, Chris Massey, Michael Meagher, Nick Michelli, Hector Morales, Marjorie
Orellana, Janine Remillard, Alexander Russo, Roberta Schorr, Beverly Smith, Sunita
Vatuk), my undergraduate research assistants (i.e., Jodi Anderson, Sean Morales, Keene
Roadman), teacher-friends-and-colleagues (i.e., Liza Bearman, Bill Bigelow, Yalitza
Brambila, Carolyn Brown, Patty Buenrostro, Gersome Carrera, Maritza Castillo, Haiwen
Chu, Shannen Coleman, Mirah Collins, Maggie Comparini, Eileen Donoghue, Liz
Dozier, John Ehresman, Suze Ferrier, Paula Fleshman, Laura Gellert, Gabi Gonzalez,
Efrain Gonzalez, Lidia Gonzalez, Manuel Gonzalez, Doug Goodwin, Jill Goodwin, Laura
Grandau, Catherine Hanna, Mark Harvey, Shana Henry, Darby Hollinrake, Anne Horn,
Lori Huebner, Rich Kaplan, Kevin Korschgen, Shane Kulman, Michael Lach, Jenna
Leitner, James Lynn, Adelric McCain, Danny Newmann, Peter O’Neil, Ana Perez,
7
Daniela Rebic, Ernest Richter, Leroy Robinson Sr., Orlando Rodríguez, Taylor Rogers,
Ed Rokos, Osvaldo Román, Angela Sangha, Uzma Siddiqui, Margaret Small, Steve Starr,
Amy Torres, Janet Walkoe, Kam Woodard, Carol Woodburn), Northwestern classmates
(i.e., Lauren Banks Amos, Megan Bang, Matthew Berland, Malayna Bernstein, Lawrence
Brenninkmeyer, Adam Colestock, Robert Daland, Nancy Deutsch, Lisa Dorner, Julia
Eksner, Katie Gucer, Erica Halverson, Sarah Hart, Heather Hill, Claudia Hindo, Anneeth
Hundle, Heather Johnson, Moshe Krakowski, Leema Kuhn, Victor Lee, Melissa Luna,
Anh Ly, Kristi Madda, Katie McHugh, Alyssa Montgomery, Nattalia Paterson, Lesley
Perry, Virginia Pitts, Enid Rosario, Ken Rose, Ben Shapiro, Tiffany Simons, Amalia
Smith, Anika Spradley, Jennifer Stephan, Su Swarat, Janae Townsend, Carrie Tzou, Ellen
Wang, Katie Weitz White, Michelle Wilkerson, Suyun Woo, Beth van Es, Le Zhong,
Anita Zuberi), other non-educator friends (i.e., David Aronoff, Ana Cholo, Estelle Clare,
Joe Dapier, John Ferrier, Steve Frenkel, Christine Gans, Norman Hogeveen, Juan
Antonio Jackson, Shannon Kelly, Agnes Kildisaite, Tomi Latimer, Carol Lahoun, Russell
Martin, Fluvia Morales, Lucee Nañaz Portillo, Jonny Polonsky, Leyli Radjy, Sanam
Radjy, Marlene Ranjitsingh, Mickey Rebic, Osvaldo Roman, Emel Sherzad, Solomon
Snyder, Meera Subramanian, Balázs Szekfű, Melanie Torres, Matt Walker), and all of my former secondary and post-secondary students who allowed me to grow as a professional and person.
8
TABLE OF CONTENTS
1. Introduction to the Dissertation Study/17
1.1. The Big Picture: Concerns about the Quality of Mathematics Education/17
1.2 Three Types of Mathematics Instruction/19
1.3 Goals and Rationale for the Study/23
1.4 Overview of my Dissertation Study/26
1.4.1 Undertakings of the Dissertation/26
1.4.2 Research Questions/29
1.4.3 Potential Contributions of this Study/29
1.4.4 Overview of Chapters/31
2. Literature Review/33
2.1 Concerns about Social Class and Racial Bias in Mathematics Education/33
2.1.1 Class-based Critiques in Mathematics Education/34
2.1.2 Cultural Relevance Concerns and Critical Race Theory/38
2.1.3 Culturally Relevant Mathematics/43
2.2 The Nature and Origins of Critical Pedagogy/44
2.2.1 Hegemony and Hegemonic Instruction/47
2.3 The Nature of Discourse and Uses of Discourse Analysis/50
2.3.1 The Importance of Foucault/51
2.3.2 Dearth of Discourse Analysis Studies in Critical Education/54
2.3.3 Hegemonic and Counter-Hegemonic Discourse Structures/55
9 2.3.4 Subjectivity and Objectivity: Teacher Positioning of Students/57
2.3.5 Student Stances – Agency, Resistance, and Conformity/57
2.3.6 Gee’s Social Language and Discourse Model as Tools of Inquiry/59
2.4 Empowerment from Standards-Based Reform and Critical Perspectives/61
2.4.1 Traditional and Reform Mathematics Instruction/62
2.4.2 Critiques of Traditional Mathematics Instruction/63
2.4.3 Standards-Based Reform Mathematics Instruction/65
2.4.4 Critical Mathematics Perspective/69
2.4.5 Discourse and Reform Mathematics Education/73
2.4.6 Dowling’s Vygotskian Critique of Reform Mathematics/74
2.5 Critical Mathematics/81
2.5.1 Examples of Critical Mathematics in the Education Literature/83
2.5.2 The Relationship Between Critical and Reform Instruction/85
2.6 Conclusion/88
3. Methods for Data Collection and Analysis/89
3.1 Research Purpose/89
3.2 Research Questions/90
3.3 Why a Qualitative Design was Selected/91
3.4 The Value of Practitioner Research/91
3.4.1 The Need for Instructional Design/95
3.4.2 Congruency Between Instructional Design and Qualitative Research
Processes/97
3.5 Design Specifics/98
10 3.5.1 The Setting and Circumstances of My Dissertation Study/98
3.5.2 Chicago: Dual Economy and Two-Tiered School System/101
3.5.3 The Night School Students/105
3.5.4 Discipline at the Guevara Night School Setting/107
3.5.5 The Guevara School Facility/110
3.5.6 Teaching and Learning in the Night School Setting/113
3.5.7 The Night School Routine/115
3.5.8 Gaining Access to the Guevara Night School Program/116
3.6 Sources of Data & Data Collection Procedures/117
3.6.1 Transcriptions of Videotaped Instruction/117
3.6.2 Participant Observation Fieldnotes/119
3.6.3 Instructional Diary/120
3.6.4 Formal Beliefs-Related Documents/121
3.6.5 Inequalities & Area Unit Text/122
3.6.6 The Bees and Rhoad Comparison Texts/122
3.6.7 Student Interviews/125
3.6.8 Students’ Written Coursework/126
3.6.9 Pre- and Post-Assessments/127
3.6.10 Post-Activity Data Management/127
3.7 Data Analysis: Analytical Frames and Techniques/128
3.8 Self-Analysis of My Critical Instructional Design Process and Related Beliefs as a
Teacher-Designer/128
3.8.1 Situating the Self-Analysis of Curriculum Design and Beliefs/129
11
3.8.2 Iterative, Inductive Coding of My Curriculum Diary and Proposal
Material/130
3.9 The Comparative Textual Analysis/134
3.9.1 Dowling’s Sociological Language of Description/134
3.9.2 Dowling’s Structural Level/135
3.9.3 Dowling’s Textual and Textual Resource Levels/141
3.9.4 My Methodological Approach to the Comparative Textual Analysis/144
3.10 Analysis of Classroom Discourse and Participation/146
3.10.1 My Method for Discourse Transcription and Analysis/146
3.11 Analyses of Ethnographic Fieldnotes and Student Interviews/150
3.12 General Methodological Issues/151
3.12.1 Researcher Role and Positionality/151
3.12.2 Dealing with Practitioner-Researcher Bias/152
3.12.3 Consent, & Ethical Concerns/153
3.12.4 Credibility Considerations Related to Self-Study/154
3.13 Conclusion/155
4. Teacher Reflections on the Design and Implementation of Critical Mathematics/156
4.1. Overview of Chapter: Model of Design Approach and Beliefs/157
4.2 Approaches to the Design of Critical Mathematics Materials/158
12 4.2.1. Initial Design Approach: Critical Recontextualization of Geometry/158
4.2.2 Latter Design Approach: Interweaving Separate Critical and Mathematical
Activities/168
4.3 Beliefs about the Real World Utility of Secondary Mathematics/172
4.3.1 Initial Utilitarianism or Conceptions of Secondary Mathematics as a Tool
Subject/173
4.3.2 My Shift to a Semi-Utilitarian Stance/176
4.4 Beliefs about Critical and Mathematical Empowerment/184
4.4.1. Initial Additive Version of Critical-Mathematical Empowerment/184
4.4.2 Shift Away from the Position of Additive Critical Mathematical
Empowerment/187
4.5 Analysis Phase and Current Belief Status/191
4.5.1. Shift to a More Profound Skepticism of Mathematical Utility/192
4.5.2 Viewing Critical and Mathematical Empowerment as Generally
Oppositional/193
4.6 Discussion/196
4.6.1 Academic Mathematics as a Non-Transformative, Self-Referential
System/197
4.6.2 Rhetoric of Utility and Targeting Non-Elite Students with Utilitarian
Mathematics/202
4.6.3 The Current State of the Field and Utilitarianism/204
4.6.4 The Disciplinary Cost of Critical Mathematics/208
4.7 Conclusion/211
13 5. Textual Analysis of a Critical Mathematics Unit/212
5.1 The Three Curricular Texts/212
5.2 Distribution of Domain Messages/216
5.2.1 Findings about Distribution of Domain Messages/216
5.3 Distribution of Mathematical Discourse/223
5.3.1 Findings about the Distribution of Mathematical Discourse/224
5.4 Distribution of Mathematical Content/229
5.4.1 Findings about the Distribution of Content/229
5.5 Discussion/232
5.5.1 Responding to Ravitch and Clarifying my Views/237
5.6 Conclusion/240
6. Discourse and Participation in Reform Activities/242
6.1 Overview of Four Findings Discussed in This Chapter/244
6.2 Initial Student Reactions to Mathematical Subjectification/249
6.2.1 From Beginning of First Hour of Week 2: Analysis of Excerpt 6.1/252
6.2.2 Students’ Initial Traditional Model of School Mathematics: Analysis of
Excerpt 6.2/261
6.3 Evidence for a Valuable Shift in Students’ Discourse Models and Discursive
Participation in Reform Activities/265
6.3.1 Changes in Students’ Discourse Models about School Mathematics/266
6.3.2 Changes and Continuity in Students’ Discursive Participation/269
6.4 Reform Discourse and Participation in Week 7/272
14
6.4.1 Small Group Work in Week 7: Analysis of Excerpt 6.3/275
6.4.2 Whole Class Discussion in Week 7: Analysis of Excerpt 6.4/281
6.5 Discussion/286
7. Discourse and Participation in Critical Mathematics Activities/290
7.1 Issues at Play in Critical Mathematics Lessons/290
7.1.1 Student Participation in Critical Classroom Discourse/291
7.1.2 Strength and Resilience of Hegemonic Discourse Model for Social
Inequality/291
7.1.3 Prominence of Vernacular Social Language/292
7.1.4 Tension between Critical and the Mathematical Instructional Goals/292
7.1.5 Unpredictability of Student Response to Critical Activities/293
7.1.6 My Inexperience with Critical Pedagogy/294
7.2 Charting New Critical Territory in Week 2/295
7.3 The Race and Recess Activity in Week 2/297
7.3.1 Newfound Agency and Student Subjectification: Analysis of Excerpt
7.1/298
7.3.2 Lived Experience & Deno’s Hegemony: Analysis of Excerpt 7.2/302
7.3.3 Providing Closure Through a Traditional Teaching Script: Analysis of
Excerpt 7.3/309
7.3.4 Reflecting on the Race and Recess Activity/313
7.4 The Inequality and Area Project in Week 7/314
15 7.4.1 Overview of the First Day of the Inequality & Area Lesson/315
7.4.2 Overview of Inequalities & Area Lesson as Enacted/316
7.4.3 Resisting the Critical I&A Writing Prompt: Analysis of Excerpt 7.4/317
7.4.4 Critical Ignition in a Small Group Discussion: Analysis of Excerpt 7.5/321
7.4.5 Whole Class Income Distribution Activity: Analysis of Excerpt 7.6/324
7.4.6 Elaborate Critical Student Engagement: Analysis of Excerpt 7.7/327
7.5 Discussion/333
7.6 Conclusion: Knowledge-Base Necessary for Critical Mathematics Pedagogy/338
8. Conclusion/343
8.1 Overview of Results /343
8.2 The Value of the Research Design/346
8.2.1 Importance of Insider Research/347
8.2.2 The Benefits of Field-Based Research/350
8.2.3 Grounding the Analysis in Theory/351
8.2.4 Acknowledging Teacher Wisdom/352
8.3 Connections to Mathematics Education Discourses/353
8.3.1 Attempts to Improve Mathematics Instruction for Urban Youth/353
8.3.2 The Need for Mathematics Skills and the Impact of Specializing
Instruction/355
8.3.3 The Efficacy of Traditional, Reform, and Critical Mathematics Texts/357
8.3.4 Issues of Empowerment, Status, and Agency/358
8.4 Recommendations for Teaching and Future Research/359
8.5 Rethinking Critical Mathematics: Where I Stand Now/365
16
References/371
Appendices/393
Appendix A: Inequalities and Area Unit/393
Appendix B: Race and Recess/432
Appendix C: Overview of Lessons/435
Appendix D: Thirteen Student Profiles/439
Appendix E: Pre- and Post-Interview Protocols/458
Appendix F: Pre- and Post-Assessment Results and Assessment/466
17 CHAPTER ONE: INTRODUCTION TO THE DISSERTATION STUDY
My study was designed to advance the understanding of the teaching of what has alternately been called “critical mathematics” (Frankenstein, 1989, 1991, 1995;
Gutierrez, 2002; Skovsmose, 1994) and “teaching math for social justice” (Gutstein,
2003, 2005). I use the simpler critical mathematics (CM). At the broadest level, my research focuses on issues of mathematics education, urban education, and educational equity. My interest in these areas reflects, in part, my experience as a secondary mathematics teacher in the Chicago Public School system. While a secondary mathematics teacher, I taught in neighborhood schools that served mostly low-income students of color. These schools were similar to the two urban schools that were included in my study.
1.1 The Big Picture: Concerns about the Quality of Mathematics Education
Since at least as far back as the turn of the twentieth century, the following issues have been of interest in mathematics education: the purposes of school mathematics, what topics should be taught and in what order, how mathematics should be taught (e.g., didactic versus problem-solving approaches), and what versions of mathematics (e.g., business math, Advanced Placement statistics) should be taught and to whom (NCTM,
1970). Opinions about these issues are often contentious (Schoenfeld, 2004). In the late nineteen fifties there was an attempt to accomplish widescale reforms in mathematics education with the New Math movement and the reaction to this reform effort was equally strong in calling for a return to “the basics” or traditional instruction (Kline,
1973; NCTM, 1970; Schoenfeld, 2006). Debates about what pedagogy and curriculum to
18 use and whether or not instruction should be the same for all students or be tailored to meet the needs of certain groups of students continue today.
At the time this study was done, discussions of improving mathematics and science education in U. S. schools had resurfaced on the national agenda. The recent
National Academy Report (2005), for example, expresses concern about the U. S. maintaining its competitive edge in science, technology, and mathematics. Indeed, U. S. students do underperform on international exams when compared with students from other developed countries (McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, &
Cooney, 1987; Stigler & Hiebert, 1999). It is also known that students from lower SES backgrounds and students of color do not do as well on mathematics exams as their white, middle class counterparts; that is, there is an achievement gap that corresponds to students’ race and class (NCTM, 2000; Tate, 1997). In sum, then, educators and the general public are concerned with both excellence and equity in mathematics education and most believe there is considerable room for improvement.
There are a number of ways that policymakers and educators have attempted to address the current perceived shortcomings in U. S. mathematics education. Three of these are particularly visible. First, federal legislation (i.e., No Child Left Behind) that holds all schools accountable by raising mathematics standards and providing tough sanctions to compel schools to meet standardized outcomes. Second, but not mutually exclusive, many mathematics educators attempt to improve mathematics instruction by getting them to attend more to their students’ mathematical thinking (Sherin, 2002). A third, but not necessarily unrelated, approach suggests that mathematics curriculum be made more relevant to lived experiences of students, particularly, to those students whose
19 experiences and cultural ways of reasoning and communicating have historically been marginalized in educational settings (D’Ambrosio, 1997; Gutierrez, 2002; Gutstein,
2003, 2005; Moses & Cobb, 2001; Moses, Kamii, Swap, & Howard, 1989).
1.2 Three Types of Mathematics Instruction
Critical mathematics essentially means including politicized themes and contextualized mathematics problems into the school mathematics curriculum – at least, this is what empirical examples suggest (Frankenstein, 1989, Gutstein, 2003, 2005; Tate,
1995). CM, as outlined by Gutstein (2003, 2005) and Gutierrez (2002), can be seen as a reaction to the two dominant forms of mathematics instruction typically found in U. S. schools, namely, traditional and (standards-based) reform instruction. I briefly describe these before outlining the focus of my study, critical mathematics, in more detail.
Traditional instruction is the default mode of schooling in which the teacher typically instructs didactically from the front of the room. The students’ role is either to listen to the teacher or to work individually at their desks. As Schoenfeld (2006, 2004) notes, traditional instruction has been around for more than a half century in U. S. schools and has deeper historic roots (Cohen, 1988). At its best, traditional mathematics teaching is organized by clearly defined learning objectives and results in students developing fluidity with mathematical procedures. However, there is a considerable body of theoretical and empirical work that shows that traditional instruction leads to many undesirable effects (Boaler, 1997; Schoenfeld, 1988; Skemp, 1976). Students in traditional mathematics classrooms often fail to understand the conceptual connections between the big ideas of mathematics and, while many develop procedural fluency, they
20 often fail to realize how these procedures might be applied in unfamiliar problem contexts. Students in traditional mathematics classrooms also develop narrow beliefs about mathematics (Boaler, 1997; Schoenfeld, 1988). For example, many students in traditional classrooms come to believe that if they cannot solve a problem within five minutes then something is wrong with the problem or, perhaps, their own mathematical capabilities.
To address these concerns, U. S. mathematics educators have collaborated to recommend what are referred to as standards-based reforms in mathematics education.
These reforms are encapsulated in the National Council of Teachers of Mathematics
(NCTM) Principles and Standards documents (1989, 1991, 2000). Reform instruction as outlined by the NCTM adheres to constructivist theories of learning, which means that instruction is structured to be responsive to and build on students’ current mathematical understandings. In theory, then, in reform classrooms there is a balance between conceptual and procedural ways of knowing mathematics. Students in reform classrooms should be positioned more authoritatively with respect to school mathematics and would be encouraged to explore, create, explain, and justify to a greater extent than students in traditional classes.
Concerns about equity are also at the heart of current reforms in mathematics education. The NCTM (2000) encourages mathematics teachers to hold high expectations for all of their students. This organization also takes a strong stand against tracking.
Nevertheless, many scholars believe the reform movement does not go far enough to make powerful mathematics accessible to the working-class and minority students who attend under-resourced urban schools (Apple, 1992; Gutierrez, 2002; Gutstein, 2003,
21 2005; Secada & Berman, 1999). Some of these scholars note that reform instruction appears to be based on middle class discourses and middle class norms of reasoning.
They argue that unless such norms are made explicit to students from non-dominant backgrounds, equitable outcomes are unlikely. Moreover, they point to a tacit acceptance of the societal status quo embedded in the reform agenda (Apple, 1992; Gutstein, 2005).
They claim that dominant mathematics instruction centers around a narrow set of goals; that is, personal gain (future employment) or national economic competitiveness, rather than learning mathematics for such socially democratic ends as economic justice or solving environmental problems. In fact, the standards-based reform curriculum that I used when I taught in Chicago Public Schools had “prepare students to use secondary mathematics in the world of work” as a stated goal (Fendel, Resek, Alper, & Fraser,
2000). As the critical scholar Rochelle Gutierrez (2002) notes: “I have seen very few reform materials that ask students to think critically about society or its major institutions” (p. 150).
Critical mathematics is designed to address what are seen as shortcomings in reform mathematics education reform with regard to excellence and equity. In the theoretical CM classroom, the mathematics curriculum would include “real world” problems that touch on issues of power that influence the lives of low-income urban students of color such as unequal resource distribution and school segregation. The belief is that engagement with critical problems will introduce students to the ideas that (i) mathematics, as part of critical literacy more generally, can be used as a tool to understand and fight oppressive conditions; (ii) working class students and students of color have valuable knowledge and ways of knowing that have traditionally been ignored
22 by schools and that this knowledge impacts school learning; and (iii) such students have a critical role in making America a more democratic society.
Mathematics educators who work in this area, such as Gutstein (2003, 2005) and
Gutierrez (2002), argue that equitable mathematics education requires explicit engagement with social justice themes. In fact, Gutstein maintains that, to be truly empowering, school mathematics should be reconceived as a critical literacy rather than a functional literacy. In other words, school mathematics should be taught as a real world tool that allows critical reflection on sociopolitical issues and powerful institutions that shape their lives. Gutstein claims that without attention to issues of justice, students become “domesticated” by a functional version of the mathematics curriculum that serves to reinforce unequal status quo organization of school and society.
As conceptualized by Gutstein (2003, 2005) and Gutierrez (2002), CM builds on standards-based reforms. Gutstein (2005) argues that students in CM classrooms should develop “mathematical power,” citing the following statement by the NCTM (2000) as his definition of as a definition mathematical power:
Students confidently engage in complex mathematical tasks. … draw on
knowledge from a wide variety of mathematical topics, sometimes approaching
the same problem from different mathematical perspectives or representing the
mathematics in different ways until they find methods that enable them to make
progress. … are flexible and resourceful problem solvers. … work productively
and reflectively … communicate their ideas and results effectively. … value
mathematics and engage actively in learning it. (p. 3)
23 Gutstein notes that the NCTM Standards move mathematics away from what traditional mathematics instruction, or what the critical educator Paulo Freire calls the “banking model of education” (i.e., focus on memorization), towards what the NCTM and Gutstein consider to be more empowering school mathematics. Students in CM classrooms are positioned authoritatively (or subjectively) with respect to the discipline of mathematics just as they are in reform classrooms. Again, the difference between critical and reform mathematics is that the former has an overt set of political goals that go well beyond mathematical power. In addition to mathematical power, CM is designed to help students develop: (1) critical consciousness – an understanding of the forces and institutions that shape their lives; (2) critical agency – a sense that they can fight for justice and make a positive difference in the world; and (3) an understanding of the cultural relevance of school mathematics to their lived experiences and cultural backgrounds.
1.3 Goals and Rationale for the Study
The purpose of my study was to understand what CM looks like in a secondary classroom with students of color from low SES families. At this point in time, critical education, inclusive of CM, remains primarily theoretical. Scholars describe its benefits, but few empirical studies of critical pedagogy have been conducted in public schools. In the case of CM, this likely is due to the limited number of mathematics educators doing work in the CM area. There are, in fact, only a handful of reports of studies in classrooms where CM is used (for examples, see Gutstein & Peterson, 2005). This lack of interest appears to be changing, as teaching for social justice has become the focus of increased attention in recent years (Ayers, Hunt, & Quinn, 1998; Darder, 2002; Darling-Hammond,
24 French, & Garcia-Lopez, 2002). While interest in critical pedagogy is less apparent in the areas of science and mathematics than in the humanities or social sciences, there are mathematics and science educators doing work in this area (Calabrese Barton, 2003;
Gutstein & Peterson, 2005). At the secondary level, however, there is nothing in the area of CM with the possible exception of the Algebra Project (Moses & Cobb, 2001; Moses,
Kamii, Swap, & Howard, 1989). Given that the details of the Algebra Project curriculum have not been made available to the research community (Hall, 2002), it is not clear if it is in line with CM as theorized by Gutstein (2003, 2005) and Gutierrez (2002) or if it is in line with reform mathematics as outlined by the NCTM (2000). However, while his problems and problem contexts may not be as explicitly political as those outlined by
Gutstein (2003), Moses’ stated (non-curricular) message appears consistent with CM at the broadest level.
When I began to design my study, I was influenced by the CM work of Gutierrez
(2002) and Gutstein (2003). Following these scholars, I saw CM as a conceivable next step to address what I saw as the major shortcomings of the standards-based reform movement in regards to equity, student engagement, and the goal of raising students’ consciousness about sociopolitical matters that affect them. As mentioned earlier, I used standards-based Interactive Mathematics Program (IMP) (Fendel, Resek, Alper, &
Fraser, 2000) when I was a full time teacher in the Chicago Public School system. In large part, because of my use of this curriculum, students in my classrooms experienced a different version of school mathematics than many of their peers. My students generally seemed to appreciate the standards-based reform approach; in my last year of teaching, for example, I was voted “the teacher I learned the most from” in the year senior class
25 poll. However, while I saw reform instruction as superior to traditional instruction, I no longer saw it as effective as I thought it could be when I first read work by some of its main proponents (Ball, 1993; Lampert, 1986; Schoenfeld, 1988). While I conducted no rigorous research during my more than four years I taught in Chicago Public Schools, general observation informed me that reform mathematics did not significantly alter students’ socio-political perspectives or, relatedly, their life chances. In addition, I worried about how students who planned to attend college would fare in post-secondary mathematics courses. Despite developing a good understanding of the advanced mathematics that they were learning, many lacked the level of procedural fluency required by university mathematics departments. While the IMP curriculum is superior to traditional curricula in many respects (Webb, 2003), it does not give students an opportunity to rehearse or apply the procedural skills they encounter and often fail to introduce students to more formal expressions of mathematics content (Wu, 1997).
Moreover, despite its thick real world contextualization, I felt that my students did not see themselves or their community knowledge represented in the IMP curriculum.
Hence, when I came across Gutierrez’ (2002) article on CM, I began to hope that mathematics education could go further than the reform movement in terms of equity and curricular relevance. I was particularly interested in CM’s implications for engagement, learning, and empowerment with the population of students who historically have been filtered out of courses and schools where they would have access to college preparatory mathematics (Tate, 1997). And because, as far as I understood, there were no mathematics teachers who taught from a critical perspective in the urban area where I lived, I decided to study my own CM teaching.
26
1.4 Overview of my Dissertation Study
I was the primary teacher and researcher for the study. I collected data on CM in two sites: a night school course at Guevara High School and a summer school course at
Park Vista High School (pseudonyms, as are all names for people and places mentioned in this report). For reasons that I detail in Chapter 3, the night course at Guevara High
School eventually became the focal point for my study. The night course at Guevara ran two hours a night, four days a week, for nine weeks. The course was equivalent to one semester’s worth of instruction – the students earned one semester’s worth of geometry credit towards graduation upon completion of the course. The students were high school juniors and seniors from predominantly low- to moderate-income Hispanic and African-
American communities. Many took this course to make up for a past failure in geometry because they needed the credit in order to graduate. Despite their past failure, the night school students had a variety of experiences with school mathematics (see Chapter 3 and
Appendix D). Mathematics was an obstacle for many, but certainly not all, of the enrolled students (see Appendix F for results of pre- and post-examinations). Many students had more generalized academic problems. Of the 33 who initially enrolled, 27 students successfully completed the course.
1.4.1 Undertakings of the Dissertation
Because no CM curriculum was available at the high school level, major undertakings in my study were the design and field-testing of CM curriculum and pedagogy (Magidson, 2002; McKernan, 1991). Curriculum design was a necessary
27 component of my study because of the dearth of viable CM materials at the secondary level. From my secondary teacher perspective, the key issues for CM design were that:
(1) students from subordinated communities would see it as believable, engaging and relevant to their lived experiences; (2) lessons would be politically and mathematically accurate (i.e., realistic applications of mathematics to politicized problem contexts); and
(3) lessons would provide the mathematics skills and knowledge necessary for college. I understood from the onset that turning secondary mathematics into an area of critical literacy would be empowering or equitable only if all three criteria were satisfied. My impression from reading the literature on critical pedagogy that this was difficult yet possible.
I set various goals for the study of my own teaching. Broadly speaking, I wanted to understand what implementing CM meant from the perspective of an experienced secondary mathematics teacher as well as from the perspective of urban students. I was particularly interested in four themes: (1) curriculum design issues, (2) relationship of
CM materials to available traditional and reform materials, (3) the nature of CM instruction, and (4) the outcomes of instruction, including students’ reactions to CM.
First, I paid careful attention to issues I faced as I undertook curriculum design. I documented these in an instructional diary (Lampert, 2000; Magidson, 2002; McKernan,
1991). I used the instructional diary as a space to detail my thinking and concerns during the design process. For instance, I took notes on the sources of materials I drew from and how I modified or further developed them to have them fit a CM mode. I shared my problems of finding problem examples relevant to my particular students’ lives.
28 Second, I was interested in how the CM curricular materials I developed compared with reform and traditional materials used in similar secondary mathematics courses. I wanted to understand the impact of reconceptualizing secondary mathematics as a critical literacy. To do this part of the study, I drew on a sociological framework developed by the British scholar Paul Dowling (1998). Dowling’s theoretical ideas allowed me to conduct a careful textual analysis, and compare my CM curricular content with two published secondary mathematics texts.
Third, I was interested in my own instruction and my students’ responses to it at the micro-level of spoken language. Hence, I had my courses videotaped and I used transcriptions of discourse from the videotapes to examine the classroom interactions captured on this medium. Given my focus on equity and social justice, I was particularly interested in the power relationships that developed in my classroom and how they played out between students and between the students and me. I was interested in the implementation of reform materials from the IMP curriculum (this foundational curriculum comprised approximately 80% of the curriculum) and my own CM materials.
In both reform and CM activities, I examined how students took up – or failed to take up, even rejected – the opportunities provided to discuss their mathematical ideas.
Fourth, and finally, I examined the outcomes of my instruction, including the mathematics students learned and their changed beliefs about secondary mathematics and the sociopolitical world. In order to get at mathematics learning, I examined students’ in- class work as well as a set of pre- and post-exams. In terms beliefs, I kept track of students’ emotional and cognitive reactions to the critical components of the curriculum, and noted how ideas in pre-interviews compared to what they said in the post-interviews.
29 There was no control group for this study. While I would have liked to have had a control classroom, the opportunity did not present itself. Furthermore, given that I had not yet gotten CM pedagogy “right,” the use of a control group to measure its effectiveness would have been largely premature. At the same time, I was able to reflect back on my four years of teaching secondary mathematics to equivalent groups of students in previous classes and compare those memories with the CM teaching in this study.
1.4.2 Research Questions
Corresponding to the four purposes of my study, listed above, I generated the following research questions:
1. What issues do secondary math educators face in developing and implementing
CM curriculum that needs to meet local math standards?
2. How does the incorporation of justice-related themes into the conventional secondary math curriculum transform this curriculum?
3. What is the nature of classroom discourse and student participation in a CM classroom and how does this evolve over time?
4. How does engagement in CM activities influence the mathematics that students learn as well as students' epistemological beliefs about mathematics, attitudes about themselves as learners of mathematics, and actual learning of mathematics?
1.4.3 Potential Contributions of this Study
My intention in designing this study was to make a number of contributions to the literature in mathematics and critical education. I hoped to collect sound empirical
30 evidence of CM. With the possible exception of the Algebra Project (Moses & Cobb,
2001), this is the first study to focus on the development and implementation of an explicitly political mathematics curriculum in an urban high school. While similar to
Gutstein’s (2003, 2005) research on his own teaching, my students were older, came from a more diverse set of cultural backgrounds, and were less privileged by the school system than most of the students in his study (i.e., they were not honors students). To be clear, my secondary students included adolescents from African American, Mexican American, and Puerto Rican-American backgrounds. Some of the Latino students were from undocumented families. All had failed geometry and were in the course to make up for past failure.
Second, the study adds to the understanding of instructional design in secondary mathematics (Magidson, 2002). Design issues and curriculum trade-offs that I faced as I infused critical themes and goals into the required secondary curriculum are documented.
These issues have wide implications because they are similar to what other educators who wish to design more culturally and sociopolitically relevant materials encounter.
Third, and relatedly, the study furthers the understanding of the use of real world and sociopolitically relevant mathematic problem contexts in school. Discussions about the need for real world and cultural relevance in the mathematics education are prevalent in the literature (Boaler, 1993, 1997; Chazan, 1996; Dowling, 1998; NCTM, 1970, 2000;
Schoenfeld, 2006). These often link to issues of student engagement, equity, and social justice. A frequent assertion in the literature is that a culturally relevant mathematics curriculum could increase engagement and understanding among African American and
Latino students (Gonzalez, Andrade, Civil, & Moll, 2001; Gutstein, Lipman, Hernández,
31 & de los Reyes, 1997; Ladson-Billings, 1995; Moses & Cobb, 2001; Tate, 1995).
Unfortunately, this assertion has not undergone sufficient empirical testing.
Fourth, I bring two relatively new methodological approaches to the investigation of curriculum and instruction in secondary mathematics classrooms. Using a sociological framework for curriculum analysis developed by Dowling (1998), I conduct rigorous analyses of the critical and utilitarian components of my curriculum. Using analytical ideas from Pruyn (1999) and Gee (2005), I conduct discourse analyses of the micro-level interactions between my students and myself in reform and critical activities.
Fifth, and finally, as practitioner-research, my study speaks to teachers in a way that much research in mathematics education does not. I document and discuss problems from the perspective of an experienced secondary teacher who attempts to create an empowering mathematics curriculum for his students from communities that have historically been disempowered by schools and school mathematics (Ladson-Billings,
1995; NRC, 1989; Sells, 1978). While I focus on CM in particular, many issues I faced were quite general and would be found in most mathematics classrooms (e.g., issues of engagement, deciding when to let students struggle and work out solutions on their own and when to intervene).
1.4.4 Overview of Chapters
In Chapter 2, I provide an overview of discussions of equity-related issues in the mathematics education literature as well as general arguments made by critical educators relative to equity. In Chapter 3, I give detailed accounts of the study setting and my qualitative research methodology. In Chapter 4, I present findings from the analysis of
32 my approaches to instructional design and teacher beliefs as stated in my instructional diary. In Chapter 5, I address the results of a curriculum analysis in which I compared one of my CM unit texts with a reform text and a traditional text. In Chapter 6, I discuss the results of my discourse analysis of reform activity from Weeks 2 and Weeks 7 of the nine-week night school course. In Chapter 7, I include evidence from my discourse analysis and student participation in CM activities in Weeks 2 and 7 of the course. I conclude with a discussion of the implications and contributions of my dissertation study in Chapter 8.
I expect the results of my study to be useful to other teachers, curriculum designers, researchers, teacher educators, and policymakers. I hope that others considering CM pedagogy will engage with some of the issues I raise here – issues that are not always discussed in the literature on CM and that I had to deal with for the first time in face-to-face interactions with students. While I anticipate that not all readers will agree with all of my conclusions, the results should foster a conversation about what the most empowering mathematics curriculum is for secondary students who are tracked into inferior “non-specialist” (lower track, remedial) mathematics courses or attend racially and economically segregated schools.
33 CHAPTER TWO: LITERATURE REVIEW
I situate my study in current discourses about mathematics education, critical education, and educational equity, so provide an overview of literature related to these themes. I start by reviewing the perspectives about schooling held by critical theorists and advocates of culturally relevant pedagogies. Then, I discuss Antonio Gramsci’s ideas about hegemony and Paulo Freire’s suggestions for student consciousness raising and action through critical literacy. I next address James Paul Gee’s theories about the nature of discourse in some depth. I also elaborate on Marc Pruyn’s theoretical framework and findings because, like me, he studied discourse in an educational program inspired by the ideas of Freire. I then focus specifically on mathematics education, outlining traditional mathematics instructions, reform mathematics, and critical mathematics (CM) educational reform. I theorize about the interrelationships among these three instructional approaches. Finally, I introduce Paul Dowling’s sociological appraisal of mathematics education. His model of analysis of mathematics texts provided a model, as well as essential tools, for my analysis of textual materials.
2.1 Concerns about Social Class and Race Bias in Mathematics Education
Progressives and critical theorists contend that much is needed in and out of schools if a democratic, egalitarian society is to be created and maintained (Apple, 1992;
Darder, 2002; Gutierrez, 2002; Secada, 1996). Although including the specific and detailed critiques of a wide range of critical educators is beyond the scope of this literature review, there are two general positions that stand out as especially relevant to reform and critical mathematics pedagogies: (1) economic and class-based critiques and
34 (2) critiques based on cultural difference and systemic racism. Although considerable overlap exists in the commitments of proponents of each perspective (Darder, 2002; hooks, 1994; McLaren, 1998), the distinction lies in what is seen as the primary explanatory factor for the lack of equity in education; that is, the economy/social class or racism/ethnicity. At the risk of oversimplifying, the first set of critical theorists – who are mostly White and often male – point to the pressures of global capitalism and class distinctions as the primary explanation for the lack of equity in education (Apple, 1992;
Anyon, 1980; E. Brantlinger, 2003; Bourdieu, 1977, 1984; Darder, 2002; Freire, 1971,
1994; Giroux, 1988; Lipman, 2004; McLaren, 1998). The second category, in contrast, involves critical race theorists and scholars of color – many of whom are African
American women – who claim educators do not fully take into account the ways in which cultural differences and racism influence teaching and learning (Delpit, 1988; Ladson-
Billings, 1995,1997; Lee &Slaughter-Defoe, 1995; Moses & Cobb, 2001; Perry, Steele,
& Hilliard, 1998). I first turn to the class-based critiques and then consider cultural differences and racism. As far as this study is concerned, both perspectives are powerful; hence, whether race or class factors dominate depends on the educational and community context being examined.
2.1.1 Class-based Critiques in Mathematics Education
Many critical educators fault education, including recent educational reforms, for not adequately taking into account how global capitalism and dominant political ideologies and agendas impinge on classroom learning (Apple, 1992; E. Brantlinger,
2003; Darder, 2002; Gutierrez, 2002; Gutstein, 2003, 2005; Lipman, 2004). They point
35 out that while the goal of achieving equity through providing all students an excellent education is laudable, they also claim that thinking this can be accomplished on the short run in a capitalist society is shortsighted. By basing arguments about the need for educational reforms in discourses of economic competitiveness (i.e., America will lose its competitive edge if students fail to learn enough, especially in science and mathematics), certain reformers tacitly accept an economic system that thrives on inequality and scarcity (Apple, 1992; Freire, 1971; Marx, 1963). Pressures for educational competitiveness fail to address the fact that big business and government agencies need a relatively small core of workers with competencies gained, or credentials earned, through advanced education. Apple (1992) claims that the limited mobility facilitated by academic push makes the price of creating a truly equitable educational system. He implies that talk about equity by reformers in mathematics education is little more than self-serving talk.
Another contention made is that while educational reform may reduce some inequalities by filling a limited amount of middle class jobs with more women and minorities, individual gains are unlikely to reduce the fundamental and widening gap between the economic haves and have-nots in U.S. society (Plotnick, Smolensky,
Evenhouse, & Reilly, 1998). Apple (2000) and Lipman (2004) note that in an essentially unregulated free market economy, a growing body of workers must fill the low paying service-sector jobs that global capitalism increasingly produces. Apple claims that it is logical to assume that there will always be a need for working class labor that does not necessarily require an advanced education. Conley (1999) argues discrepancies in wages and family wealth are the real barrier to achieving educational equity. Critical educators
36 contend that, along with a reduction in racism and sexism, broad and systemic changes need to be made to ameliorate economic and structural inequalities in U. S. society
(Anyon, 2005; Berliner, 2006; Darder, 2002; Freire, 1971; McLaren, 1998).
While educators, generally, are prone to espouse the value of education for social mobility, low-income students and their families are aware of their own circumstances and tend not to be convinced by the rhetoric of opportunity (E. Brantlinger, 2003;
Demerath, 2003; MacLeod, 1987; Willis, 1972). Hence, it is conjectured that many working class youth reject education, including school mathematics, as irrelevant precisely because they realize that they are unlikely to use it in future academic settings; that is, mathematics is distant from their long-term goals (Chazan, 1996, 2000; Powell,
Farrar, & Cohen, 1985). I do not suggest that working class students’ orientations to school mathematics cause gaps in achievement outcomes. Socioeconomic disparities figure as a major factor in achievement gaps (Kozol, 1991). Middle class youth have access to better teachers as measured by their academic preparation and instructional experience (Oakes, 1990, 2005; Tate, 1997a). Furthermore, the instruction working class students receive is very different from that of privileged youth (Anyon, 1981, 1997; E.
Brantlinger, 2003; Gamoran, 2000; Lipman, 2004). Haberman (1991, 2000) refers to the instruction – or lack thereof – that urban students receive as a “pedagogy of poverty.” He argues that in low-income urban areas, schools tend to function more as either day camps or custodial centers rather than as a place to learn (Haberman, 2000).
In the context of British mathematics education, Dowling (1998) shows a marked difference in the nature of textbooks used with elite students and those from working classes (as evident in the British text series SMP 11-16). Dowling’s empirical evidence
37 reveals that texts for so-called “lower ability” students focus on real world relevance to the exclusion of principled mathematical preparation, while texts to prepare “higher ability” students for further academic mathematics generally ignore real world relevance.
Such class-based educational differences exacerbate, even cause, inequitable outcomes as well as students’ perspectives about learning.
The work of structuralist sociologist, Pierre Bourdieu (1977, 1984), on social class reproduction theory is relevant to understanding class distinctions in behaviors and outlooks regarding school mathematics. Bourdieu offers the concept of habitus, which includes the idea that people’s subjective understandings relate to their objective realities.
People brought up in distinctive social (class) circumstances, who are treated differently by powerful institutions such as schools, develop class-based understandings that, in turn, affect their orientation to school. Parents employed in professional and entrepreneurial jobs can provide a distinctive residential milieu and different experiences for their children than can be offered by working class parents (Lareau, 2003). Working class parent are likely to have had lesser status in K-12 schools, which results in distinctive experiences and outcomes. Most would not have attended college.
Due to these unique experiences at home as well as markedly different treatment in schools, students from working class families develop different orientations towards school than do middle class students (E. Brantlinger, 1993, 2003; Lareau, 2003). This orientation, combined with general lack of opportunity, means working class youth may not persist in such subjects as school mathematics (Chazan, 2000), although there certainly are exceptions (Boaler, 1997, 2004). While students’ orientation are important, the school system is complicit in reproducing the social order by not making advanced
38 courses available and not providing sufficient academic support for working class students and students of color (Oakes, 2005; Tate, 1997a). In contrast, middle class students persist in school mathematics, in part, because they believe mathematics knowledge is necessary for advancement within academic settings. To be clear, middle class students do not necessarily believe school mathematics is more relevant to their lives outside of school than their working class counterparts, however, they do think they stand to gain from the institutional practices that position them for success. It is not surprising, then, that school outcomes vary by social class and, since class status correlates with racial status, educational outcomes also vary by race/ethnicity.
2.1.2 Cultural Relevance Concerns and Critical Race Theory
A second set of scholars, predominantly persons of color, focus on ethnic and racial differences and differential treatment according to race. Some identify themselves as proponents of critical race theory, which is considered a branch of critical theory because of their grounding interest in power relations (Ladson-Billings, 1997; Tate,
1997b). Some believe that educational achievement discrepancies are due to the lack of cultural relevance of curriculum for Black students (Delpit, 1988; Perry, Steele, &
Hilliard, 2003). Their critiques center on the cultural mismatch between the dominant forms of education and the cultures of communities of color. Theories of cultural difference or mismatch explain, at least in part, the problems that many students of color, and perhaps also working class white youth, face in both traditional and reform classrooms. These scholars make the argument that middle class, Eurocentric norms of individualism, competition, efficiency, and (the Western European version of) rationality
39 are problematically woven into the school curriculum and day-to-day instruction (Boykin
& Toms, 1985; Murrell, 1997; Walkerdine, 1988). Although Boykin and Toms believe that African Americans are rational, efficient, and even individualistic, they also contend that African American communities tend to emphasize other cultural values and forms that schools typically do not embrace. For example, such African American cultural forms as communalism and spirituality are rarely the norms of U.S. classrooms (see also
Tate, 1995). In fields such as literacy instruction, considerable progress has been made in building on the cultural forms of students of color (Lee, 1995). This is less true in the area of mathematics.
Another strand of critical race theory focuses on racism and stereotyping among school personnel. It is well established that students who talk and behave in ways appropriate to their home cultures are frequently defined as “deviant” in school (Delpit &
Perry, 2003; Walkerdine, 1988, 1999). These scholars also address the issue of
(European-American) culturally exclusive content in textbooks and tests. It is problematic when the people responsible for developing and implementing curricular decisions see subjects like mathematics and science as culture-free. They complain that school personal rarely recognize, or admit, that the general historic, economic, and political roots of American schools and society have a western European orientation
(bias) (Ladson-Billings, 1995; Powell & Frankenstein, 1997; Sleeter, 1997).
By focusing on cognitive processes and individual student development as generalized phenomena, European American educators, including some who see themselves as progressive, have staked out a position of universality and political/cultural neutrality. They believe that their theories and innovations apply equally to all students at
40 all times and in all settings. They also assume that their educational strategies affect all students in the same way so are fair and equitable (see critiques by Apple, 1992;
Haberman, 1991; Lipman, 2004; Martin, 2003; Popkewitz, 2004). Murrell (1997) levels a common criticism encountered in the writings of scholars of color:
Contemporary educational thought and policy neither take account of the cultural
motifs of African-Americans nor do they address the conditions of their
subjugation in American society. More serious is that the value system
traditionally promulgated in public schools is frequently at odds with many of the
cultural values of African and African-American culture regarding educational
achievement. (p. 26)
It is likely that the authors of such documents as the National Council of Teachers of
Mathematics (NCTM) Standards (2000) fail to truly deal with the fact that cultural differences and privilege affect learning in mathematics classrooms (Martin, 2003).
Although I believe the criticism of cultural biases are valid, I hesitate to be too critical of such national movements as NCTM because I believe its authors to be well intentioned.
Furthermore, there may be good reasons for them to remain relatively silent on issues of social justice and culturally relevant mathematics in today's conservative and punitive political climate. Nevertheless, following the advice of Delpit (1988), Dowling (1998), and Walkerdine (1988), I suggest that progressive reform pedagogies that require students to discover, invent, and explore, may rely on non-generalized cultural forms that need to be uncovered and unpacked if they are to be effective for students from distinctive cultural milieus (see also Bernstein, 1990). If the cultural forms embedded in educational reforms do inhibit access to the culture of power, then they need to be
41 explicitly taught to students who have not been brought up in a privileged, middle class communities (Delpit, 1988).
Theories of cultural distinctions in learning styles are unlikely to fully explain racialized school outcomes. Scholars point to various levels of historical and institutionalized racism and attribute differences in school outcomes by class and race to the inferior education that poor children and children of color receive compared to that provided to middle class, European American children (Lee & Slaughter-Defoe, 1995;
Martin, 2003; Noguera, 2003a; Perry, Steele, & Hilliard, 2003). Despite significant school desegregation in the mid-nineteen-sixties that resulted from Brown vs. Board of
Education (1954), schools in U. S. cities have become increasingly (re)segregated since
Nixon took office (Kozol, 2005; Orfield, Eaton & the Harvard Project on Desegregation,
1996). Low-income students of color are isolated in schools that receive markedly inferior resources (Anyon, 1997; Darling-Hammond, 2004; Kozol, 2005, 1991; Lankford,
Loeb, & Wyckoff, 2002; Lipman, 2004; Berliner, 2006). Even when students of color and
White students attend the same schools, the former are segregated into lower level tracks, including special education placements, at extremely high rates (Oakes, 2005). Teachers often hold harmful views of students of color and their communities (Ladson-Billings,
1997; Lipman, 1998; Paley, 1979; Zevenbergen, 2003).
Ogbu (1978) takes the discussion in a different direction by arguing that neither cultural mismatch nor racial bias in schools fully accounts for unequal racialized school outcomes. Ogbu points to the success that some immigrant groups have in U. S. schools despite their cultures not being well matched to the U. S. school culture. Ogbu theorizes that, in contrast to students from some voluntary immigrant groups, African American
42 students fail to succeed academically at high levels because they feel alienated from schools as a result of their status as involuntary immigrants. Due to a history of enslavement and subsequent segregation and subjugation, African Americans, Native
Americans, and Hispanic Americans do not trust societal institutions and have adversarial relations with school personnel from primarily European American backgrounds. Ogbu claims that African American youth may equate school effort with “trying to act white,” therefore resist schooling in ways whites and students from voluntary immigrant groups do not.
As Perry, Steele, and Hilliard (2003) argue that theories about cultural mismatch and Ogbu’s theories are not mutually exclusive. These authors content that both have explanatory power, yet neither fully explain the range of experiences of African
American students. While many acknowledge that Ogbu’s theory of the impact of involuntary immigrant status on school achievement is powerful, they criticize him for ignoring the historic struggle of African Americans to build schools and become literate despite opposition (Lee & Slaughter-Defoe, 1995; Noguera, 2003a; Perry, Steele, &
Hilliard, 2003). They point out that Ogbu focuses almost entirely on flaws in the learner
(i.e., their culture) rather than on the racist, oppressive, and inequitable nature of the school system. Ogbu’s critics also worry about the intractability of his primarily historic and political perspective because it appears to rule out school- and classroom-level changes that could improve the learning and educational status of African American students. Haymes (2002) and Murrell (1997) agree that a distinct African American epistemology, resulting from the oppression that African Americans face in the U. S., does exist. They add that this epistemology, in and of itself, cannot be reduced to an
43 oppositional stance to schools on the part of African American youth. In contrast to
Ogbu, these scholars believe schools should acknowledge and build on the lived experiences, cultures, and language of African Americans to be more effective rather than cast cultural diversity aside as problematic (Haymes, 2002; Lee, 1995; Murrell, 1997;
Perry & Delpit, 1998).
2.1.3 Culturally Relevant Mathematics
Although there is considerable criticism of traditional and reform mathematics for not being relevant to the lived experiences and cultures of poor children and children of color, the fact remains that there are few empirical examples in which culturally relevant materials have been constructively infused into mathematics curriculum (Gutstein,
Lipman, Hernández, & de los Reyes, 1997; Gutstein, 2005; Gutstein & Peterson, 2005).
Part of this may be due to the abstract nature of academic mathematics or, perhaps, the perception by teachers and curriculum writers that mathematics is universal (acultural), and therefore it is impossible to make it more culturally relevant (Sleeter, 1997).
Mathematics has a unique epistemology (D’Ambrosio, 1985) that makes drawing parallels between it, other disciplines, and everyday experiences difficult if not impossible (Dowling, 1998; Steen, 2004).
It might be noted that because modern science and mathematics (and other subjects) are conglomerates of knowledge gleaned from various cultures, the result is that they are not culturally coherent with any one race or ethnic group. Furthermore, these subjects are tied to disperse scientific communities that are not situated in any one location or affiliated with a particular nation or race. Hence, mathematics and science
44 may approach the status of being universals. At the current time, however, mathematics and science are practiced, created, and regulated by middle class scholars who tend to be located in, or relocate to, wealthy nations. In the U. S. context, the practices and discourses of mathematics and science resonate with, and are regulated by, middle class
White patriarchal culture (see Joseph, 1997). The control that elite classes held on the development of (academic) mathematics can be traced back at least as far as the Greeks
(D’Ambrosio, 1985).
Whether it is the nature of mathematics or people’s perceptions of mathematics that is key, making academic mathematics more relevant to diverse, often subordinated, groups is a difficult enterprise. As I argued above, students’ intrinsic interest in school mathematics may be tied primarily to how they perceive their futures and their academic needs to realize their futures rather than with how mathematics is taught per se. If and when low-income students and students of color have equal access to advanced educational opportunities, one would expect the current achievement gaps to be reduced
(Tate, 1997a).
2.2 The Nature and Origins of Critical Pedagogy
Among the most cited works by critical educators is Paulo Freire’s classic book
Pedagogy of the Oppressed (1971). In this book, Freire lays out the theoretical and pedagogical foundations for what has come to be called critical, transformative, Freirean, or emancipatory, education. As Freire taught his impoverished and oppressed Brazilian children and adults to read and write, these students were simultaneously expected to reflect on their oppressive conditions and be inspired to work, from the bottom up, to
45 improve their situations. Freire argues that educators should be concerned with the elements of students’ home cultures – and oppressive conditions they face – so they become literate in order to understand and improve their worlds. Freire’s consciousness- raising educational techniques have been used with oppressed children and adults throughout Central and South America and in many African countries as well. Critical educators and practitioner-researchers such as Darder (2002), Christensen (2000),
Gutstein (2003, 2005), hooks (1994), McLaren (1998), and Shor (1996) have reinterpreted Freire’s critical literacy work for the U.S. context. In so doing, they have had to worry about the effects of mass schooling and bureaucratic and market forces that affect them. Although Freire did not discuss literacy as being inclusive of mathematics instruction, critical mathematics educators such as Frankenstein (1989, 1991, 1995),
Gutierrez (2002), Gutstein (2003, 2005), and Skovsmose (1994) read him as if he did.
Following Freire, critical educators begin with the assumption that teaching is inherently political, which in the broad sense means that it concerns the "organization of social relations" and the "problem of ethical action" (Connell, 1993, p 13). Critical educators hold that a central purpose of education should be to prepare students for active participation in public life to create a more just and democratic society (Freire, 1974;
Giroux, 1988; McLaren, 1998). They see themselves as serving the particular political interests of the students they teach and the communities these students come from as opposed to serving such powers as the state, the business community, and even the local school administration. A Marxist, Paulo Freire began to write in the late 1960s at a time when revolutions were taking place in many Latin American and African nations.
46 Currently, most critical educators in the U.S. are left wing and identify as socialists or neo-Marxists.
Critical educators critique traditional, didactic instruction based on Freire’s (1971) claims that teachers objectify students by using a banking method of instruction in which students are expected to accumulate and store knowledge just as banks store money.
Freire (1971) outlines the traditional banking methodology as follows:
The teacher teaches and the students are taught; the teacher knows everything and
the student knows nothing; the teacher thinks and the student are thought about;
the teacher talks and the students listen – meekly; the teacher disciplines and the
students are disciplined; the teacher chooses and enforces his choice, and the
student comply; the teacher acts and the students have the illusion of acting
through the action of the teacher; the teacher chooses the program content, and the
students (who were not consulted) adapt to it; the teacher confuses the authority
of knowledge with his or her own professional authority, which she and he sets in
opposition to the freedom of the students; and the teacher is the Subject of the
learning process, while the pupils are mere objects. (p. 73)
Regarding mathematics education, Freire’s critique of didactic instruction resonates with accounts of the overall harm traditional mathematics instruction does to students (Boaler,
1997; Skemp, 1978; NCTM, 2000; Schoenfeld, 1988). These scholars note how the emphasis on expert knowledge and universal truths in traditional classrooms silences students and leaves little space to position students subjectively or authoritatively with respect to mathematics (e.g., to explore and explain mathematics for themselves). Boaler
(1997) shows that a negative outcome of traditional instruction is that students see
47 mathematics as inert and boring and see themselves as passive and often inadequate learners. Boaler’s findings resonate with Freire’s ideas about the debilitating impact of traditional instruction.
2.2.1 Hegemony and Hegemonic Instruction
A key concept in critical theory is hegemony, defined as the political, economic, and ideological control of subaltern groups by dominant groups in society. Marx (1963) viewed hegemony as resulting primarily from historical material-economic circumstances. The threat of violent reaction to the proletariat’s revolutionary aims was the primary force used to keep them in chains. In the mid twentieth century, Antonio
Gramsci (1971/1929-1935) argued that hegemony was not determined entirely by material power relations, and overt threats of violence, but also by cultural practices.
Pruyn (1999) describes Gramsci’s reinterpretation of hegemony this way:
[Gramsci’s] Prison Notebooks (1971) marked a major reinterpretation of
Marxism, especially in regard to cultural institutions and revolutionary
organization. Gramsci expanded Marxism past its economic roots to include
political, social and cultural theoretical elements (Gramsci 1971, Holub 1992).
While economics were still key, Gramsci brought new attention to the role culture
plays in maintaining the economic order and its revolutionary potential for
changing it (Gramsci 1971). Gramsci’s primary contribution to the theory of
agency was through this elaboration of “hegemony” and “counter-hegemony.”
According to Gramsci, it is through hegemony that a society’s cultural
institutions reproduce and reinforce the economic system – building on Marx’s
48 notion of a cultural superstructure (the state, family, law, media, religion, schools,
et cetera) that supports the economic infrastructure of capitalism (Marx 1963;
Marx & Engels 1972). Morrow and Torres (1995) describe Gramsci’s idea of
hegemony as the “ideological predominance of bourgeois values and norms over
the subordinate social classes … by moral and intellectual persuasion” (253).
Schools, for example, as one of the many cultural tools of the bourgeoisie, can
serve to cement the existing economic/social order in place by presenting these
cultural practices as “natural” and “normal.” According to Gramsci, hegemony –
social and ideological control and domination – operates at the level of “consent”
in “civil societies” (1971). Coercive force is brought to bear by the bourgeois state
only as a last resort, when the dominant ideology enters a crisis. (pp. 17 – 18,
emphasis Pruyn’s)
As Pruyn and other critical scholars note, schools play an important role in the maintenance of the social order through hegemonic practices that are invisible, or generally appear neutral, to most students and teachers. Pruyn defines hegemonic instruction as the educational practices that “reinforce, rather than challenge, the dominant cultural, political, economic and educational order” (p. 39). Typically teachers, whether consciously or unconsciously, further entrench dominant class power by using traditional, or progressive for that matter, methods that privilege middle class children.
As Pruyn (1999) clarifies:
Within these hegemonic pedagogical practices, the classroom is a teacher-
centered place where information is “transmitted” from the all-knowing teacher to
the “empty” students. Classroom practices such as these support the hegemonic
49 order by keeping teachers in positions of authority and power in classrooms and
keeping students subordinated to that authority. (p. 76)
Traditional instruction is hegemonic in that it reinforces existing unequal class relations through empowering those on top of the social order and disempowering those on the bottom (Anyon, 1981; Bowles & Gintis, 1976; Lipman, 2004). In traditional classrooms there is no forum for challenging status quo practices and ideas. Schools maintain the social order through such hegemonic practices as within and between school segregation and differentiated instruction. The views of Gramsci and Pruyn generally are consistent with class correspondence or reproduction theory in which structuralist sociologists portray a deterministic set of social relations in which the nature of society inevitably shapes the structure of schools and ultimately the nature of the study body
(Bowles & Gintis, 1976; Bourdieu, 1977, 1984).
Although still based on Marx’s idea of the material basis of social outlooks,
Gramsci and Freire describe hegemony as more dynamic and conflicted. Both argue that individual agents have the power to resist and confront oppressive social structures.
While it is understood that dominant groups try to maintain their power and keep others subordinate, in contrast to the determinism of Marx and structural sociology, Freire and
Gramsci were optimistic that oppressed groups could come to understand their social circumstances and confront dominant class hegemony.
Gramsci’s cultural reconceptualization of hegemony is important to critical educators because it opens up a larger space than did classical Marxism for critical actors
(e.g., teachers, students, proletariats) to re-educate themselves and confront oppressive cultural powers. Freire saw critical consciousness raising and informed resistance to
50 oppression as the means to transform social and economic systems. Pruyn (1999) notes that teachers can instruct in a manner that either supports or confronts dominant class hegemony. Critical pedagogy is counter-hegemonic because it is explicitly designed to challenge hegemonic ideologies and to educate subordinated students so they will be empowered to overturn oppressive sociopolitical structures that limit their lives. As Pruyn
(1999) shows, however, educators can claim to be teaching from a critical perspective while essentially reinforcing hegemonic classroom relations via didactic instruction.
2.3 The Nature of Discourse and Uses of Discourse Analysis
Pruyn’s (1999) study show the important role that discourse plays in critical pedagogy, however, a simple, clear definition of discourse is difficult to find. Murfin and
Ray (1997) write:
Discourse refers to the language in which a subject or area of knowledge is
discussed of a certain kind of business is transacted. Human knowledge is
collected and structured in discourses. ... society is made up of a number of
different discourses or discourse communities, one or more of which may be
dominant or serve the dominant ideology. Each discourse has its own vocabulary,
concepts, and rules–knowledge of which constitutes power.... Some
poststructuralists have used discourse in lieu of text to refer to any verbal
structure. (p. 89-90, emphasis theirs)
Discourses are not simply descriptive nor are they a transparent pathway from experiences to people’s meaning and thinking. Verbal discourses are not of secondary importance to nonverbal activity or the structure of the material world. Neither are
51 discourses a straightforward reflection of reality. In such institutional settings as schools, for example, discourses constitute (create and establish) institutional practices, position people within these practices, and, therefore, shape individual identities.
Several theorists informed my thinking about discourse and discourse analysis, including Cazden (2001), Gee (2005), Luke (1995), Mehan (1979), and Pruyn (1999).
Gee (2005) and Cazden (2001) do not identify themselves as critical theorists, yet it might be argued that they fall into the camp because of their focus on inequitable power relations, social class, and race.
2.3.1 The Importance of Foucault
Many of the discourse analysts cite Foucault as a primary influence in their conceptions of discourse (Fairclough, 1995, 2003; Gee, 2005; Luke, 1995; Moje, 1997;
Pruyn, 1999). All point to Foucault’s ideas about the constructing character of discourse as Luke (1995) clarifies:
Foucault described the constructing character of discourse, that is, how both in
broader social formations (i.e., epistemes) and in local sites and uses discourse
actually defines, constructs, and positions human subjects. According to Foucault
(1972, p. 49), discourses “systematically form the objects about which they
speak,” shaping grids and hierarchies for the institutional categorization and
treatment of people. These knowledge-power relationships are achieved,
according to Foucault, by the construction of “truths” about the social and natural
world, truths that become the taken-for-granted definitions and categories by
52 which governments rule and monitor their populations and by which members of
communities define themselves and others. (p. 8-9)
In other words, discourse is political in constituting that to which it refers. Following
Foucault, Gee (2005) views language, or what he calls “language-in-use,” as constructive. According to Gee, language-in-use functions to “support the performance of social activities and social identities and to support human affiliation within cultures, social groups, and institutions” (p. 1). Hence, because language is always part of larger activities, discourse analysis involves the study of the structure of spoken or written language and it relations to the social structure of the material world. Following Foucault,
Gee uses a capital “D” to write Discourse because:
[T]o study language-in-use we need to study more than language alone, we need
to study Discourses. Discourses are ways with words, deeds and interactions,
thoughts and feelings, objects and tools, times and places that allow us to enact
and recognize different socially situated identities. (p. 35)
The discourse analysts that I draw from describe how language is used at local levels and how these local level uses of language are tied to larger macro-level
Discourses, practices, and sociopolitical formations. Therefore, in addition to mathematics instruction, I focus on Discourses about such topics as mathematical competence, tracking, educational policy, and capitalism. This is different from traditional approaches to the study of discourse in education as Luke (1995) points out:
More traditional (scientifically-based) research on classroom discourse have as a
principal focus the study of language development and use per se rather than the
relationship between discourse and larger social formations. Furthermore, these
53 models tend to analyze language as a way of explaining the psychological intents,
motivations, skills, and competencies of individuals rather than explaining how
discourse systematically constructs versions of the social and natural worlds and
positions subjects in relations of power. (p. 8)
It is these local language linkages with larger social formations that allow macro- level conclusions to be made about observations. Luke (1995) and Fairclough (1995,
2003) state that critical approaches to discourse analysis that link micro-level activity to macro-level relations are important in establishing how school Discourses have differing material effects for students from different backgrounds. As Luke explains:
An approach to critical discourse analysis can tell us a great deal about how
schools and classrooms build “success” and “failure” and about how teachers’ and
students’ spoken and written texts shape and construct policies and rules,
knowledge, and, indeed, “versions” of successful and failing students. (p. 11)
While my focus was local in the sense that it centered on mathematics activity and classroom learning, I linked my discourse and textual analyses to larger sociological or sociopolitical structures and Discourses described by Fairclough (1995, 2003), Luke
(1995), and Gee (2005).
The most prominent frameworks I used in the analysis of discourse from my classroom transcripts came from two particular scholars: Mark Pruyn (1999) and James
Paul Gee (2005). Because Pruyn’s research approach and analyses were so similar to mine, his ideas were especially important. Pruyn focused on Spanish literacy instruction in a Freirean program for adult Latino immigrants in Los Angeles. He looked specifically at how three different teachers in the program used language to position their students in
54 relation to literacy learning and at the discursive stands (expression of agency, resistance) students took in response.
2.3.2 Dearth of Discourse Analysis Studies in Critical Education
While there is growing interest in the role of language in mathematics classrooms, few mathematics education researchers have taken a close look at how discourse occurs in traditional or reform instruction and particularly not on the nature of discourse in urban schools that serve students of color or the working poor (Khisty, 1995; Valero, 2004).
Based on a careful literature search, I found no research that examines the workings of
CM instruction at the level of spoken language. While Gutstein (2003, 2005) outlines a set of activities he used and includes written student responses, he does not describe what students said and did during critical lessons.
This dearth of empirical evidence about micro-level happenings in spoken discourse is true of research on critical pedagogy more broadly. Pruyn claims that previous studies of Freirean-inspired pedagogy “often failed to elaborate on what kinds of face-to-face talk were being used in these classrooms; that is, on the specific discursive classroom practices that contributed to the success or failure of such programs” (p. 4). He went on to note that while student agency was a primary concern of previous studies of critical pedagogy, “it was difficult to discern from the case studies how this was specifically fostered and developed, and, maybe even more importantly, how it came into existence in the course of everyday classroom practices” (p. 4). The studies reported in
Pruyn’s (1999) Discourse Wars in Gotham West begin to address the deficiency of micro-level analyses of critical pedagogy in the area of literacy instruction.
55
2.3.3 Hegemonic and Counter-Hegemonic Discourse Structures
Different pedagogical orientations of hegemonic and counter-hegemonic instruction lead to different classroom practices and discourses. Analyses of didactic pedagogies have shown that traditional teachers tightly control classroom discourse and activity (Cazden, 2001; Mehan, 1979). The Initiation-Response-Evaluation (IRE) pattern presented in Figure 2.1 represents the defining characteristics of traditional pedagogy.
Consider this example cited by Cazden (2001):
Initiation Teacher: Where were you born, Prenda?
Response Prenda: San Diego.
Feedback Teacher: You were born in San Diego, all right. Can you come up and find
Initiation San Diego on the map?
Response Prenda: (goes to the board and points)
Feedback Teacher: Right there okay
Figure 2.1. Example of a Traditional Initiation-Response-Feedback Pattern
Traditional teaching scripts, such as indicated in the above format, are essentially closed to student input and initiative and are unresponsive to their ideas. The teacher initiates a lesson with a question, students respond briefly within defined parameters, and the teacher either explicitly evaluates the students’ response or repeats the cycle with new initiations and implicit evaluations. By using these traditional techniques, they communicate to students that their own ideas do not count and reinforce hegemony. In the case of traditional mathematics courses, the effect is that students come to understand mathematics as pre-ordained and mostly irrelevant to their own interests and futures
56
(Boaler, 1997). Boaler describes the effect of traditional instruction on student perceptions of mathematics this way:
There were many negative consequences of the students’ belief in the rule-bound
nature of mathematics. One of these was that their desire to remember different
rules meant that they did not try and interpret and understand what they were
doing… A second negative consequence was that when students encountered
questions that did not require an obvious and simplistic use of a rule of formula,
many did not know what to do. In these situations they would give up on
questions or ask the teacher for help. A third problem was provided for the
students who thought that mathematics should be about understanding and sense-
making (Lampert 1986). These students experienced a conflict at Amber Hill
because they wanted to gain meaning and understanding but felt that this was
incompatible with a procedural approach. (p. 37)
Traditional classrooms that emphasize externally generated knowledge and universal truths silence students and leave little space for student subjectivity. In contrast to traditional top-down instructional practices, both reform and CM instruction position students as active, rather than passive, social subjects. Correspondingly, reform and critical teaching scripts are more open to student initiative and responsive to student ideas
(Cazden, 2001; Pruyn, 1999). These student-centered or responsive techniques are an indication of symmetrical power relations between students and teachers (Cazden, 2001;
Gutierrez & Larson, 1994; Pruyn, 1999).
57
2.3.4 Subjectivity and Objectivity: Teacher Positioning of Students
Closely related to hegemonic and counter-hegemonic discourses are the notions of student objectification and subjectification. Pruyn (1999) defines these as follows:
Student Objectification – Teacher discursively positions students as the “objects”
of instruction.
Student Subjectification – Teacher positions students as active social subjects
(discursively placing students in roles in which they can co-construct classroom
knowledge with the teacher and their classmates). (p. 77)
Again, from both CM and reform perspectives, traditional instruction tends to objectify students by positioning them as passive learners, while constructivist and CM instruction tends to subjectify students with respect to mathematics by positioning them as knowledgeable and creative (Gutstein, 2003; NCTM, 1991, 2000).
The fact that I discuss subjectification with respect to school mathematics as well as with critical education complicates matters. As discussed previously, critical education involves student empowerment, in part, through consciousness-raising regarding socioeconomic inequalities and the expression of critical sociopolitical agency in society generally, whereas subjectification in reform mathematics instruction involves positioning students authoritatively with respect to school mathematics. The two meanings of subjectification are not theoretically exclusive, however, I kept this distinction in mind when I conducted my analyses.
2.3.5 Student Stances – Agency, Resistance, and Conformity
58
In addition to clarifying the various ways that teachers are positioned in classroom power relations, Pruyn (1999) describes three stances that students take in classroom activity: critical student agency, resistance, and conformity. Each are viewed as reactions to the subjectification and objectification involved in instruction. Pruyn defines student agency or adopting a critical agentive stance as follows:
Critical Student Agentive Stance – The student assuming an “agentive stance”
takes action that attempts to alter existing hegemonic pedagogical practices. The
student actively engages problematic issues that arise in the curriculum, or in
classroom relations with the teacher and/or classmates, and takes some form of
action to change them. This student is active, a subject acting on the world in
order to change it. One of the main goals of critical pedagogy is to move students
from conformism and/or resistance to agency (Freire 1970, 1985, 1997; Freire &
Horton 1990; Freire & Macedo 1987, 1998). (p. 79). Agency replaces passive
resistance as counter-hegemonic attitudes develop (Gramsci 1971). A student
assuming an agentive stance elaborates or acts out a preferable set of counter-
hegemonic pedagogical practices. Further, this student acts in a concerted way
(individually or with other students) to make these alternative and liberating
practices central to the practice of the classroom. (pp. 79 – 80)
Recall that CM and reform instruction both subjectify students, albeit in different ways.
Students exhibit critical agency when they accept and further their subjective positioning.
However, I prefer to reserve the term critical agency for student participation in critical activities, and student engagement as students’ supported participation in reform
59 mathematics. Additionally, engagement is a term more often used in the mathematics education literature.
Pruyn discusses conformity and resistance as possible student stances.
Conformity, or adopting a conformist stance, involves compliance with the traditional teacher’s rituals and commands regardless of how these benefit or match the needs and goals of students. Conformity to traditional forms of instruction might also be said to occur when students explicitly challenge critical or standards-based reform pedagogy.
Resistance, or adopting a resistant stance, can include acting out, refusal to comply with instruction or classroom rules, and passive aggression (e.g., not submitting work, not cooperating in group activities, tuning out). While resistance may accomplish students’ short-term goals (e.g., lessening their work-load, reducing their sense of powerlessness), resistant actions tend to worsen their school and life situations (see Willis, 1977).
2.3.6 Gee’s Social Language and Discourse Model as Tools of Inquiry
While not meant to apply to critical pedagogy per se, Gee’s (2005) approach to discourse analysis provides sound ideas about how to link micro-level classroom discourse to macro-level cultural and political phenomena. Specifically, Gee’s concepts of social languages and Discourse models allowed me to go beyond Pruyn’s (1999) framework for the discourse analysis in my CM instruction. While Pruyn’s framework links to Gramsci’s macro-level idea of hegemony, Gee’s allowed me to link to larger macro-level institutional practices and societal-level conversations.
Gee’s (2005) construct of a Discourse model was particularly important to my analyses. According to Gee, a Discourse model is essentially a cultural model (see
60
D’Andrade, 1987), however, he considers the expression cultural model a poor one because Discourse models are link to “specific socially and culturally distinctive identities people can take on in society” (p. 61). In other words, African American siblings might grow up in the same cultural milieu but, at the same time, might develop different Discourse models by participating in unique pursuits as adolescents and distinctive professions as adults. Gee (2005) gives this definition:
Discourse models [are] the largely unconscious theories we hold that help us
make sense of texts and the world. Discourse models are simplified, often
unconscious and taken-for-granted, theories about how the world works that we
use to get on efficiently with our daily lives. We learn them from experiences we
have had, but, crucially, as these experiences are shaped and normed by the social
and cultural groups to which we belong. From such experiences we infer what is
“normal” or “typical” (e.g., what a “normal” man or child or policeman looks and
acts like) and tend to act on these assumptions unless something clearly tells us
that we are facing an exception. (p. 71)
The Discourse model construct allows simple identifications to be made, such as between hegemonic and counter-hegemonic beliefs. A collection of “theories” or “storylines” that students and teachers bring to bear in classroom discourse and activities can be identified.
Those that support the status quo would be classified as hegemonic, whereas those that challenge status quo power relations are counter-hegemonic, and still others are partial and inconsistent.
61 Gee’s construct of social language was also important to my analysis of classroom discourse and participation. The idea of a social language is related to
Bakhtin’s (1981) notion of speech genres. Gee outlines social language(s) as follows:
People use different styles or varieties of language for different purposes. They
use different varieties of language to enact and recognize different identities in
different settings; they also use different varieties of language to engage in all the
other building tasks … I will call each such variety a “social language.” For
example, a student studying hornworms might say in everyday language, a variety
of language often referred to as “vernacular language,” something like
“hornworms sure vary a lot in how big they get,” while the same student might
use a more technical variety of language to say or write something like
“hornworm growth exhibits a significant amount of a variation.” The vernacular
version is one social language and the technical version is another. Investigating
how different social languages are used and mixed is one tool of inquiry for
engaging in discourse analysis. (p. 20)
The distinction between vernacular (everyday) and technical (school-like) social discourses that Gee makes turned out to be particularly important for the analysis of the critical activities I implemented in the CM teaching component of my study.
2.4 Empowerment from Standards-Based Reform and Critical Perspectives
Both reform constructivist reformers (e.g., NCTM, 2000) and critical educators have student empowerment as a common goal, however, what each means by empowerment is different. From the standards-based reform position, student
62 empowerment is with respect to the discipline of mathematics. Reform teachers position students to do such things as construct, create, explain, justify, and challenge mathematical ideas. Again, traditional teachers do not position students subjectively with respect to mathematical ideas, instead they tend to objectify students. Because the authority held solely by the teacher is shifted onto students, reform instruction is a covert challenge to the hegemony implicit in traditional instruction. However, reform instruction generally does not position students to consider macro-political phenomena in the classroom as is necessary for critical education (Gutierrez, 2002; Gutstein, 2003, 2005).
While disciplinary mathematical empowerment is important to CM educators, within-subject-matter empowerment or acquiring mathematical knowledge as cultural capital would not be enough. As Gutstein (2003, 2005) and Gutierrez (2002) note, reform instruction fails to position students to question oppressive institutional and societal structures that impact their education and life chances. In contrast, CM instruction goes beyond reform instruction’s relatively covert challenge to mathematics or school-as- usual. Critical education overtly positions students to consider issues of power in the classroom and in society. Correspondingly, CM, in theory at least, deliberately constructs the mathematics curriculum around sociopolitical themes that are believed to be relevant to students’ lived experiences (Gutierrez, 2002; Gutstein, 2003, 2005).
2.4.1 Traditional and Reform Mathematics Instruction
There is a considerable body of theoretical and empirical work that focuses on dominant forms of mathematics instruction in U.S. schools; that is, reform instruction as outlined by the NCTM (1989, 1991, 2000) and the didactic mathematics pedagogy that is
63 traditional in American classrooms (see NCTM, 1970; Schoenfeld, 1988, 2004). Scholars point to harmful effects of traditional pedagogy while arguing for the superiority of reform approaches (Boaler, 1997; Schoenfeld, 1988, 2004). In this section, I briefly define traditional instruction and critique it from the standards-based reform perspective
– a perspective I mainly share.
2.4.2 Critiques of Traditional Mathematics Instruction
Traditional instruction is teacher-centered, didactic, and top-down (Bransford,
Brown, & Cocking, 1999). According to Schoenfeld, 2006), it has been around in name for more than fifty years, but the didactic instructional form has a long history in
European, U. S., and colonial schooling. Good teaching in the modern traditional mathematics classroom is characterized by clearly organized lectures with well-defined objectives and frequent evaluation of student knowledge (Gagne, 1985; Skemp, 1976).
Traditional teachers generally ascribe to the view that basic skills should be mastered before students attempt applications to practical situations or engage in independent thinking and problem solving (Bloom, 1956; Gagne, 1985; Hirsch, 1987). At best, traditional mathematics results in individual mastery of generalizable mathematical skills, the development of fluidity with mathematical algorithms, and the ability to use conventional mathematical logic or formal reasoning about mathematics functions. The assumption is that formal reasoning and procedural skills learned in traditional mathematics classrooms will provide the basis for learning increasingly advanced levels of mathematics and understanding mathematics functions in related fields. Traditional
64 educators tend to assume that skills and algorithms learned in traditional mathematics classrooms would transfer usefully to settings outside of school (NCTM, 1970).
Traditional approaches to teaching mathematics have increasingly been called into question by mathematics education researchers (Boaler, 1997; NCTM, 2000;
Schoenfeld, 1988, 2004; Skemp, 1976). Scholars focus on such negative results of didactic mathematics instruction as students' lack of understanding, lack of interest, and lack of mathematical empowerment. Boaler (1997, 1998) finds that many students in the
United Kingdom and the United States believe that traditional mathematics is irrelevant to their everyday lives, and see little connection between school mathematics and the everyday problem-solving situations they encounter in out-of-school contexts.
Boaler’s (1997) study of two British mathematics classrooms, one traditional and one project-based (i.e., progressive, reform-oriented) supports the claim that traditional math pedagogy actually inhibits intuitive and flexible mathematical problem solving (see also Schoenfeld, 1985, 1988; Skemp, 1976). Boaler draws on situated learning theory
(Lave & Wenger, 1991) to illustrate how students who spend their time doing highly abstract and decontextualized traditional mathematics exercises are socialized into thinking of mathematics as highly regimented and disconnected from everyday applications. Boaler (1997) notes that students in traditional mathematics classrooms come to sense that they and their ideas do not count. Boaler describes the effect of traditional instruction on students’ perceptions of mathematics this way:
There were many negative consequences of the students’ belief in the rule-bound
nature of mathematics. One of these was that their desire to remember different
rules meant that they did not try and interpret and understand what they were
65 doing… A second negative consequence was that when students encountered
questions that did not require an obvious and simplistic use of a rule or formula,
many did not know what to do. In these situations they would give up on
questions or ask the teacher for help. A third problem was provided for the
students who thought that mathematics should be about understanding and sense-
making (Lampert 1986). These students experienced a conflict at Amber Hill [a
traditional school] because they wanted to gain meaning and understanding but
felt that this was incompatible with a procedural approach. (p. 37)
Boaler claims that because students in traditional mathematics classrooms are accustomed to receiving highly contrived problems in highly scaffolded ways, when they encounter a slight variation in the problems they do not have the confidence or flexibility of knowledge to proceed (see also Schoenfeld, 1985, 1988). In contrast, the students in the progressive, project-based setting she studied exhibited more mathematical confidence and ingenuity when given the same unfamiliar problems. Criticism of didactic forms of instruction is not new. At the turn of the last century, Dewey (1916) criticized schools for not starting with students’ interests and Whitehead (1929) pointed out that students who learned by rote acquired an “inert” form of knowledge that did not transfer to out of school settings.
2.4.3 Standards-Based Reform Mathematics Instruction
To address concerns about traditional instruction, U. S. scholars have collaborated to design and disseminate standards-based reforms in mathematics education (NCTM,
1989, 1991, 2000; Senk & Thomson, 2003). Reform instruction adheres to constructivist
66 learning principles; it is structured to be responsive to and build on students’ current mathematical understandings in ways that traditional pedagogy does not. Students in reform classrooms are positioned more authoritatively with respect to mathematics. They are asked to explore novel mathematical tasks and communicate about their own ideas about the tasks with each other and the teacher. Ideally, students in reform classrooms would be thinking and doing mathematics much like academic mathematicians (Lesh &
Doerr, 2003; Stein, Smith, Henningsen, & Silver, 2000). At the beginning of Chapter 6, I include the NCTM’s (2000) vision of a reform mathematics classroom that complements this discussion. I will not repeat it here for the sake of brevity.
Reform curricula are likely to set mathematics problems in “real world” contexts
(Fendel, Resek, Alper, & Fraser, 2000; Schoenfeld, 2006). This real world problem contextualization in reform texts is generally more elaborate than that found in traditional texts – at least those designed for elite students (Dowling, 1998). Setting mathematics in elaborate contexts is done for at least two reasons: (1) to help students to see the relevance of what they are learning, and (2) to address Whitehead’s (1916) inert knowledge problem or the lack of transfer of knowledge learned in mathematics classes to contexts outside of school (Boaler, 1997; NCTM, 2000). Advocates of reform mathematics assume that complex, contextualized problem solving in the mathematics classroom facilitates the transfer of math knowledge to real world settings outside the classroom (Brown, Collins, & Dugout, 1989; Fendel, Resek, Alper, & Fraser, 2000; Lesh
& Doerr, 2003; NCTM, 2000; Schoenfeld, 2006).
It should be noted here that the NCTM standards should not be equated with the standardization, accountability, and high-stakes testing agendas that are currently
67 sweeping the nation (Apple, 2001; Kohn, 2004; Lipman, 2004). A teach-to-the-test rationale is not the intention of national supporters of standards-based reform mathematics (NCTM, 2000; Schoenfeld, 2006). Indeed, reform instruction defines itself in opposition precisely to the type of teacher-centered, teach-to-the-test instruction that standardization promotes (NCTM, 1989, 1991, 2000). However, how standards-based reforms play out in actual mathematics classrooms when teachers feel opposing pressures to teach from both teacher- and student-centered perspectives, remains an open question
(see Spillane, 2004).
Since the publication of the first NCTM Principles and Standards (1989, 1991), many mathematics teachers have moved towards the version of reform instruction outlined in these NCTM documents and the newer Standards 2000 documents. Regarding the extensiveness of the adoption of reform mathematics, Civil (1995) claims "at a time in which many schools are to a [greater] or [lesser] extent trying to adopt (or adapt) some of the messages in the reform documents … talking about traditional school mathematics may begin to be inappropriate" (p. 2). Perhaps this is because elementary educators are aware that recent empirical evidence suggests that a committed implementation of reform materials improves students’ conceptual understandings and problem solving ability in mathematics (Schoenfeld, 2002; Senk & Thompson, 2003; Stein, Grover, & Henningsen,
1996). Perhaps just as encouraging is the finding that the gap in mathematical achievement between European American and African American working class elementary students apparently is reduced when teachers faithfully implement reform materials and pedagogy at the elementary level. Schoenfeld (2002) points out that the ratio of White students to Black students who met the concepts and problem-solving
68 standards dropped from more than four to one to about three to two after all elementary schools in Pittsburgh adopted reform materials. There is similar evidence at the secondary level for the reform mathematics initiative called Interactive Mathematics
Program (IMP) (Webb, 2003). Nevertheless, the Third International Mathematics and
Science Study (TIMSS) suggests that standards-based reforms have not had as much of an impact in changing instructional practices of U. S. teachers as was desired or expected
(Stigler & Hebert, 1999).
The results of improved scores of minority students, and the reduction in achievement gaps between Black and White students, are encouraging for those of us who value equity and endorse the constructivist philosophies of learning integral to reform instruction. In fact, establishing educational equity is among the NCTM’s (2000) core principles. This equity focus is of primary interest because it is recognized that mathematics has traditionally been a "critical filter" that keeps subordinated populations out of academia and high-paying jobs. In a 1989 report from the NRC (cited by
Schoenfeld, 2002), "Everybody Counts," the following case was made:
More than any other subject, mathematics filters students out of programs leading
to scientific and professional careers. From high school through graduate school,
the half-life of students in the mathematics pipeline is about one year; on average,
we lose half the students from mathematics each year, although various
requirements hold some students in class temporarily for an extra term or a year.
Mathematics is the worst curricular villain in driving students to failure in school.
When mathematics acts as a filter, it not only filters students out of careers, but
frequently out of school itself. (p. 7)
69 Such blocks to social mobility contribute to the maintenance of societal divisions that weaken democratic ideals in the United States (Ladson-Billings, 1995; Moses & Cobb,
2001; NRC, 1989; Sells, 1978; Schoenfeld, 2002).
To address equity concerns, higher mathematics achievement for all students has become a major goal of the reform movement. Evidence of this sense of urgency about elevating mathematics gains for all students infuses the NCTM Standards 2000 documents. Reforming pedagogical practice to increase access to mathematics is seen as key to U.S. students’ participation in a post-industrial economy that increasingly requires technological expertise and academic achievement (NCTM, 1989, 2000; Schoenfeld,
2002). These goals and incentives are widely shared. Bob Moses, the civil rights activist turned math educator, claims that improving math education for African Americans is an issue that currently is as urgent for Blacks as voter registration was in the 1960s; that is,
"economic access and full citizenship depend crucially on math and science literacy"
(Moses & Cobb, 2001, p. 5). At the same time, as Moses would likely note, standards- based reforms designed primarily by middle class White academics and tested with middle class White children may not be the best instructional program for all students, and for students of color in particular (Martin, 2003; Valero, 2004).
2.4.4 Critical Mathematics Perspective
In terms of social class, the key question that critical mathematics educators ask is: What effect does reform instruction have on working class students and students from non-dominant cultural backgrounds? Contrary to the dominant belief in math education circles, and the positive findings in places like Pittsburg (Schoenfeld, 2002), the verdict is
70 still out on the equity impact of reform mathematics (Schoenfeld, 2006). Critical educators and some scholars of color (e.g., Gutierrez, 2002; Martin, 2003) argue that reform mathematics likely operates in complex ways with populations of students who have traditionally received an inferior mathematics education. While standards-based reforms may improve some aspects of learning, they may weaken others (Apple, 1992;
Walkerdine, 1988). Like progressive pedagogies more broadly, standards-based reforms in mathematics education appear to be based on middle class norms of reasoning.
Progressive pedagogies (e.g., reform mathematics) may exacerbate problems students from non-dominant groups have in school precisely because progressive pedagogies fail to make these norms explicit (Delpit, 1988).
Politically progressive critics of the mathematics reform movement worry that the
NCTM Principles and Standards do not go far enough in outlining how to achieve equity in mathematics education by actually showing what a democratic, egalitarian, society might look like (Apple, 1992; Gutierrez, 2002; Gutstein, Lipman, Hernandez, & de los
Reyes, 1997; Secada, 1996). Critical scholars voice concern about instruction that is solely in line with the NCTM Standards, pointing out that, outside of a critique of tracking, their recommendations for equitable practice lack detail. For example, the authors of the NCTM Principles and Standards make such non-specific statements as
"excellence in mathematics education requires equity” and “equity means high expectations and strong support for all students," but offer little of substance to help math teachers achieve these goals.
Boaler (2002) and Lubienski (2000) discuss the uncertainty about whether reform instruction is wholly beneficial for working class students. Lubienski (2000) suggests that
71 students from lower SES backgrounds have more difficulty engaging in the kind of open- ended tasks advocated by the reform movement than do students from higher SES backgrounds. She found that lower SES students generally disliked experiences with reform instruction and preferred more didactic approaches, while their higher SES counterparts appeared to feel that they did better the types of open-ended tasks advocated by standards-based reformers. In contrast, in Great Britain, Boaler (1997) found the opposite: working class students understood open-ended project-based mathematics to be superior to traditional mathematics instruction, particularly after they had a chance to get used to it. Of course, differences between British and American working class students may help explain the contradictory findings. The students in the class that Lubienski studied came from economically diverse communities while Boaler’s participants were mostly working class.
The apparent success of reform instruction in Pittsburgh referenced earlier supports Boaler’s argument that standards-based reforms will benefit working class students and students of color. Boaler’s (2004) study of Railside High School shows that, in addition to outperforming higher SES students in neighboring schools, students of color from low SES communities can come to see reform instruction as superior to traditional instruction. This finding resonates with my experiences teaching reform mathematics to students of color in low SES high schools in Chicago. While there may be initial resistance to instructional innovations, like Boaler, I believe working class students come to see the logic behind reform practices (see Chapter 6). That said, I realize that this may be the result of a power imbalance between teacher and student rather than a simple matter of the superiority of reform over traditional instruction.
72 Walkerdine (1997, 1998) offers an argument that resonates with Lubienski’s findings, but does not necessarily contradict Boaler’s (2002) position. She shows that at least some of the conceptual and abstract mathematical reasoning currently advocated by mathematics education reformers in Britain and the United States is based in middle class norms of behavior, discourse, and class (material) privilege. Walkerdine argues that working class students are required to calculate, or be calculating, in their everyday lives, while middle-class kids have time to do mathematics as a leisure activity in their homes.
Walkerdine further points out that her experiences growing up working class meant that numbers became associated with her family’s limited finances. As a result, school mathematics was emotionally loaded for her. Walkerdine believes that middle class kids worry less about the limitations that money places on their lives, so learning school mathematics is less emotionally problematic. The implication is that the version of conceptual reasoning integral to reform activities are detached from the everyday experiences of working class students or, at least, the curriculum fails to validate the experiences, languages, and reasoning of non-mainstream students. Hence, even reform mathematics instruction may alienate working class students whereas middle class students find such reasoning less of a stretch.
Making the claim that working class children do not conceptualize mathematics or reason about it the same way (i.e., as effectively) as middle class children is risky.
Clearly, working class people and people of color do think abstractly, conceptually, and in a sophisticated manner about mathematics and other subjects (Labov, 1972;
Walkerdine, 1988). However, Walkerdine argues that what is called abstract, or labeled an abstraction, in mathematics education is a set of discursive practices that detach the
73 doer from their mathematics. It is this distance that makes mathematics seem abstract, especially to working class students. Clearly more research needs to be done in this area.
2.4.5 Discourse and Reform Mathematics Education
The current standards-based reform movement highlights the importance of student-centered discourse and teacher’s facilitation of sophisticated mathematics discourse (NCTM, 1989, 1991, 2000). It is argued that reform instruction is superior to traditional instruction, in large part, because it shifts discursive authority away from the teacher and text and to students (Cazden, 2001; NCTM, 1991). Students’ ideas are elevated from the minimal, even absent, status they have in traditional classrooms to the central focus of instruction in reform classrooms. In traditional classrooms, the flow of language is unidirectional from teacher to student. In contrast, spoken and written texts have a different function and flow in reform classrooms – they are more multidirectional from student to student and from students to the teacher as well as from the teacher to students. The reform vision highlights the importance of student-centered mathematical conversations and student-initiated explorations to arrive at mathematical understandings and truths.
While there is growing interest in mathematics discourse and the role of language in mathematics classrooms, few researchers have looked closely at how discourse occurs with reform or traditional mathematics instruction (Cazden, 2001; Knuth & Peressini,
2001; Walkerdine, 1988), and particularly have not studied the nature of discourse in low-income schools (Valero, 2004). Studies of language use by bilingual learners in mathematics classrooms by Khisty (1995), Morales (2004) and Moschkovitch (2002) are
74 exceptions. In an attempt to address these shortcomings, my study involved careful and informed analyses of classroom discourse and student participation during mathematics instruction.
2.4.6 Dowling’s Vygotskian Critique of Reform Mathematics
An interesting and important critique of progressive reforms in mathematics education comes from British sociologist, Paul Dowling (1998). Dowling calls the dominant constructivist position held by U.S. and British mathematics reformers
“pedagogical constructivism.” He calls scholars who articulate this position “pedagogical constructivists.” Dowling’s position differs from that of pedagogical constructivists in several ways. First, he emphasizes the fact that mathematics is a sociocultural arbitrary with specialized discourses and practices. He claims that pedagogical constructivists generally fail to recognize this point and instead imply that mathematical knowledge has its primary basis in the material, rather than the social, world. In other words, such mathematics educators hold the Platonist or naïve realist view that mathematics is a universal that is discovered primarily through interactions (e.g., problem solving, modeling) with the physical environment, rather than seeing mathematics as something that is intersubjectively created and regulated by a powerful elite. Dowling rejects the mathematics-as-a-universal assumption as well as the progressive idea that mathematics teachers can be seen first and foremost as facilitators:
That which regulates mathematical knowledge is fundamentally discursive and
not physical. Whilst the physical world may provide starting points, the
mathematical interpretation of these starting points must be made explicit by the
75 only person in the classroom who is able to make them explicit: the teacher.
Mathematics comprises principled knowledge. Whether or not we accept that
there is a fundamental structure to the ways in which we develop cognitively, the
detail of mathematical knowledge is essentially a sociocultural arbitrary. Sooner
or later, if someone is going to learn mathematics, someone else is going to have
to tell them about it. (p. 44)
Second, and relatedly, Dowling discusses what he calls the “myth of reference,” or the idea that “the mathematician [can cast] a knowing gaze upon the non-mathematical world and [describe] it in mathematical terms” (p. 6). He notes that many pedagogical constructivists devote considerable scholarly and curricular attention to such things as real world relevance of mathematics, real world problem solving, and real world modeling (see Lesh & Doerr, 2003). Dowling argues that instead of referencing the real world, mathematics is primarily self-referential. He elaborates:
[T]he principles which regulate mathematics and those which regulate shopping
constitute distinct systems. One may recruit elements of the other: a shopper may
use a memory of a multiplication table; a mathematics textbook may incorporate a
domestic setting. But precisely what is recruited is regulated by the recruiting
rather than the recruited practice. Mathematics is not about shopping because the
shopping settings which appear in mathematical texts are not motivated by
shopping practices. Mathematics is no more about shopping than Picasso’s Les
Demoiselles d’Avignon is about prostitution. In its pure, ideal-typical form, the
myth of reference consumes the non-mathematical setting within a mathematical
play leaving only a trace to remind us that there is something outside of
76
mathematics. This is why the settings in the extracts from SMP 11-16 Book Y1 [a
secondary book used with elite students in Britain] are so pruned of non-
mathematical specificity. The merest residue is sufficient to the purpose of the
myth which is to claim an external motivation for mathematics. Behind the myth
lies mathematics as a self-referential system. Its utterances are not references, but
simulacra. (p. 16)
Dowling contends that the myth of reference has its basis in stratified social class relations; the mathematician, after all, occupies a position of high status in society and academia. This elevated status, rather than the power of mathematics per se, allows the mathematician to look down on the everyday practices of non-mathematicians and express them, or recontextualized versions of them, in mathematical terms. However, as
Dowling points out, while the mathematician might be able to describe (a subset of) real world phenomena in mathematical terms, it does not mean that mathematics is broadly useful or transformative in non-mathematical affairs. This does not prevent mathematicians (e.g., Paulos, 1988) and policymakers from pathologizing non- mathematicians for failing to use mathematics in their everyday lives.
In fact, the idea that mathematics is useful, outside of disciplinary mathematics and closely related fields, points to a related myth that Dowling refers to as the “myth of participation.” This myth refers to the idea that mathematics participates powerfully in people’s everyday affairs or, in the case of critical mathematics, the sociopolitical
(Skovsmose, 1994). Indeed, Dowling claims he wrote The Sociology of Mathematics
Education (1998), in part, “to challenge the core myth that practices which are organized
77 and developed within one activity [mathematics] can be effective, even transformative, within another” (p. 24)
Dowling observes that pedagogical constructivists buy into the myths of reference and participation. Hence, his assertion that mathematics is regulated primarily by the highly discursive and relatively arbitrary practices of mathematicians is in conflict with dominant, pedagogical constructivist views. Dowling’s position differs, for example, from the stated position of the NCTM (1989, 1991, 2000). In the NCTM Standards documents, the authors claim or imply, that school mathematics: (1) is useful in the non- academic world (utilitarian position); (2) is regulated primarily by the material world
(myth of reference position) rather than by expert academic discourses (self-referential view); and, therefore, (3) is learned primarily through a set of Piagetian-like experiences in the material world (myth of participation view).
I discuss the NCTM’s current utilitarian position and its impact on my study extensively at the end of Chapter 5, so here focus on Dowling’s latter two claims. The authors of the NCTM Professional Standards (1991) adopt a pedagogical constructivist stance, for example, when they state, “the teacher of mathematics should orchestrate discourse” by using the following seven principles:
(1) posing questions and tasks that elicit, engage, and challenge each student's
thinking;
(2) listening carefully to students' ideas;
(3) asking students to clarify and justify their ideas orally and in writing;
(4) deciding what to pursue in depth from among the ideas that students bring up
during a discussion;
78 (5) deciding when and how to attach mathematical notation and language to
students' ideas; and
(6) deciding when to provide information, when to clarify an issue, when to
model, when to lead, and when to let a student struggle with a difficulty;
(7) monitoring students' participation in discussions and deciding when and how
to encourage each student to participate. (p. 35)
The NCTM (1991) positions mathematics teachers to be primarily concerned about the facilitation and promotion of students’ mathematical ideas in discourse. The teacher poses the right questions, listens carefully, and recognizes mathematically important ideas when they come forth from students. The assumption is that, while they may lack the conventional notation and language, students can (re)discover many of the ideas that regulate school mathematics with minimal teacher guidance. While the reform mathematics teacher can take a more didactic stance on occasion (principle 6), this is apparently only a secondary role. The NCTM document (1991) does state, “[b]eyond asking clarifying or provocative questions, teachers should also, at times, provide information and lead students” (p. 36). How much telling, and when, is never clarified nor is a distinction drawn between elementary and secondary mathematics.
As a Vygotskian, Dowling (1998) understands that student-centered discourse can be mathematically empowering, however, he asserts that pedagogical constructivists overstate the discovery role of students in mathematics. Chazan and Ball (1995) comment on this point, stating:
Teachers' considerations are complex, their moves subtle. Yet, [current]
conceptions of the teacher's role often seem focused on what teachers should not
79
do. They should not tell students things; they should not be the source of
knowledge. If teachers stay out of the way, the argument goes, students will
construct new understandings. (p. 16)
Going farther, it is problematic to assume that the concepts and principles, particularly of secondary mathematics, can largely be intuited as the NCTM (1989, 1991, 2000) documents imply. The student-centered pedagogical constructivist approach breaks down at some point in the mathematical apprenticeship and does so increasingly at the secondary level. Repeating Dowling’s statement, “sooner or later, if someone is going to learn mathematics, someone else is going to have to tell them about it” (p. 44)
According to Dowling, pedagogical constructivism is based on the misguided application of Piaget’s developmental theory of learning to the learning of school mathematics. He claims that pedagogical constructivists build an entire program of school learning around the following three assumptions:
(1) children need concrete experiences if they are to acquire sound mathematical
concepts; (2) like adults, children learn best when they investigate and make
discoveries for themselves; (3) children refine their understanding and develop
conceptual structures by talking about their own thinking and what they have
done. (p. 36)
Indeed, the above assumptions are consistent with the NCTM’s (1991) position on discourse quoted earlier. Dowling contends that pedagogical constructivists achieve this child-centered platform via a condensation of important concepts from Piaget’s developmental theory that “conceals a shift in the meaning of the terms as they move
80 from Piagetian psychology” to a child-centered theory of learning school mathematics (p.
36). As Dowling clarifies:
Specifically, there has been a shift from ‘mathematico-logical experience’ and
‘schemata’, in Piaget, to concrete, which is to say practical experiences and
understanding. This is partially achieved by the association of ‘experience’ with
‘concrete’, in the above extract. The latter term is another reinterpreted term, the
meaning of which has shifted from a position within the developmental sequence
concrete operations/formal operations to having a meaning associated with
practical or physical. … The reinterpretation of ‘conceptual structures’ is
achieved in the third assumption in the above extract: ‘children refine their
understanding and develop their own conceptual structures’.
Refining/development are achieved ‘by talking about their own thinking and what
they have done’, so that there is an equivalence established between
understanding and the development of conceptual structures. (p. 36, British
punctuation style)
Dowling contends that Piaget would be dissatisfied with the pedagogical constructivist application of his developmental theory in such a field as secondary mathematics. That uneasy fit may lead to and understanding of why Piaget avoided making specific suggestions for instruction in schools. Piaget would likely disagree, for example, with the assumption that concrete experiences structure higher ordered mathematical thought in the absence of explicit pedagogical mediation.
Dowling (1998) argues that instead of basing instruction on Piaget’s ideas, that educator should look to Vygotsky to develop a theory of learning that takes into account
81 the socially arbitrary, self-referential, and highly discursive practices of mathematics.
Dowling claims that the difference between a Vygotskian and a pedagogical constructivist position is that, “Vygotsky’s construction, but not Piaget’s, refers to linguistic instability and thus allows for the development of an epistemology which constitutes rationality as socially constructed” (p. 44). Dowling states, “as far as pedagogic action is concerned, the Vygotskian approach demands that instruction be explicit” (p. 45). This means that, at some point, the principles and discourses that regulate academic mathematics have to be stated and illustrated for students in order for them to progress. In a Vygotskian mode, mathematics cannot be reconstructed solely, or even primarily, through student-centered problem solving, discovery, and student-to- student communication. Dowling’s theories and methods of analysis of mathematics and mathematics instruction are important and I draw heavily from them in my own analyses.
2.5 Critical Mathematics
Critical mathematics can be seen as an extension of Freire’s (1971, 1985) ideas about critical literacy (reading and writing) instruction to mathematics instruction
(Frankenstein, 1989, 1991, 1995; Gutierrez, 2002; Gutstein, 2003, 2005). Following
Freire’s lead, students in CM classrooms are asked to “read and write the world with mathematics” (Gutstein, 2005, book title). Students are to use mathematics to make connections between their lived experiences and macro political phenomena. CM educators talk about the need for students, particularly ones from subordinated cultures, to “mathematize,” or build mathematical models of, the political/economic conditions in which they find themselves (D’Ambrosio, 1999, Gutierrez, 2002; Gutstein, 2003, 2005).
82 CM educators attempt to foster individual and social agency by giving students the opportunity to use CM to reflect and act on the problems they face in their lives (see
Tate, 1995; Turner & Strawhun, 2005). For this to happen, students learn mathematics in a critical context.
From the CM perspective, achieving equity in mathematics education does not mean teaching the required curriculum more efficiently, but rather altering the dominant instructional methodologies and goals. Regarding the implementation of a social justice math curriculum in an urban school setting in Chicago, Gutstein (2003) outlined several long-term goals for CM:
1) Students see what they are learning as connected to their experiences.
2) Students develop a sense of agency, both as mathematics students and
sociopolitical beings.
3) Students develop positive social and cultural identities.
4) Students begin to see mathematics as socially constructed and not neutral
politically.
5) Students should do as well as students in standards-based reform classrooms
are doing in mathematics.
It should be noted that the long-term outcomes for CM are similar to the goals of theorists who write about culturally relevant mathematics instruction (e.g., Gutierrez, 2002;
Ladson-Billings, 1994, 1995; Moses & Cobb, 2001; Tate, 1995). There is considerable interaction between, and overlap in thinking among, scholars with critical and scholars with multicultural perspectives on mathematics teaching. Scholars in both areas assert that curriculum and instruction should be built around students’ home languages and
83 cultures. They condemn traditional assimilationist teaching and maintain that teachers should value multiple ethnically-related worldviews and non-standard dialects. Finally, culturally relevant and critical pedagogies both require teachers to have an implicit and explicit political commitment to students and their communities.
2.5.1 Examples of Critical Mathematics in the Education Literature
As Gutstein et al (1997) and Gutierrez (2002) note, there has been little research in the areas of multicultural and critical education that focuses on mathematics. I add that much of the existing CM literature is theoretical and descriptions do not go beyond surface-level appearances or individual lessons. To varying degrees, Frankenstein (1989,
1990, 1991, 1995, 1997), Williams and Joseph (1993), Skovsmose (1994), Tate (1995),
Gutstein (2003, 2005) and contributors to the book Rethinking Mathematics edited by
Gutstein and Peterson (2005), have described curriculum, instruction, and learning in a
CM mathematics classroom.
Frankenstein (1989, 1990, 1991, 1995, 1997) taught basic arithmetic to working class adult students in a community college setting. To my knowledge, Frankenstein’s
(1989) critical arithmetic textbook Relearning Mathematics was the first attempt at developing a comprehensive CM curriculum. This text mixed together three types of activities: self-reflection on such topics as math phobia, statistical analysis of data from politicized situations, and learning basic arithmetic skills. With the exception of this textbook, Frankenstein’s other work is primarily theoretical. In addition to extending
Freire’s (1971) ideas to mathematics, Frankenstein’s goal in these articles appears to be more about motivating the need for CM rather than detailing its inner workings in her
84 classroom. That said, however, Frankenstein (1991) does document some student resistance to her politicized curriculum, but does not do so in detail and she does not describe how she, as a CM teacher, responds to student resistance.
At the elementary school level, Tate (1995) describes an Afro-centric mathematics project in which a teacher has her African American students use mathematics to understand and ultimately take-on the distribution and overabundance of liquor stores in the neighborhood that surrounds their school. Turner and Strawhun
(2005) describe a similar mathematics project in which a group of middle school students use mathematics to measure overcrowding in their school and use mathematical data on which to base their arguments in their fight for improved educational conditions. In both of these examples, students used CM to understand and attempt to change their own immediate worlds for the better.
The Danish scholar, Skovsmose, cites three CM projects in Towards a Philosophy of Critical Mathematics Education (1994). The examples of CM he cites include computer programming in LOGO, designing a park, and figuring out how to spend money on a local Danish recreation center. While these projects extended over long periods of time and touched on the social/economic conditions of students’ lives, only the recreation center project touched on issues of power. As such, Skovsmose’s activities might be characterized as quasi-critical. Interestingly, Skovsmose notes that: (1) the projects he investigated headed in unpredictable directions, (2) students he observed did not take up a critical stance when left to their own devices, and (3) students did not necessarily recognize the open-ended projects as relating to school mathematics.
Apparently, then, CM teachers need to provide explicit scaffolding if they want students
85 develop a critical stance and recognize the mathematical reasoning they use in open- ended projects.
In the U.S. context, Gutstein (2003, 2005) drew on generative themes (i.e., neighborhood gentrification, racial profiling) that affect the communities of his Mexican-
American students. Activities in Gutstein’s CM classroom included simulating the distribution of wealth in the world, looking at expected values and probability of arrest through racial profiling, understanding Eurocentric biases in the Mercator Map Project
(Gutstein et al, 1997; Gutstein, 2003, 2005). In his recently published book, Reading and
Writing the World with Mathematics (2005), Gutstein is candid about problems he faced as a White male educator attempting to teach mathematics from a critical perspective in a predominantly Mexican-American school. With the exception of an activity at the beginning of the book and excerpts from student work, however, Gutstein does not generally detail what CM looks like in the daily instructional flow. Pruyn’s (1999) study shows that it is necessary to get beyond surface level appearances (e.g., summaries or general descriptions of projects) to the level of discourse in order to understand whether or not critical pedagogy is actually subjectifying.
2.5.2 The Relationship Between Critical and Reform Instruction
While some critical scholars express doubts about the use of progressive/reform curricular materials in non-mainstream settings, many also consider themselves to be constructivists who generally support the type of reform instruction outlined by the
NCTM (1989, 1991, 2000). The two perspectives are not necessarily at odds. Both constructivist reformers and criticalists view traditional instruction as problematic.
86 Reform mathematics and CM share the goal of student empowerment. Both attempt to position students authoritatively with respect to mathematics. Both have students engage in making/testing hypotheses and explore mathematical patterns to develop
“mathematical power” (Gutstein, 2005; NCTM, 2000). Critical educators realize that they cannot simply abandon the dominant curriculum and outcomes. As Gutierrez (2002) points out, CM students should not only develop critical stances on “the relationship between mathematics and society” but they should also develop proficiency in
“dominant” (both traditional and reform) mathematics (p. 151). CM and reform teachers and texts ask students to create, explain, justify, challenge, and accept mathematical ideas. From a critical perspective, such a subjective positioning is a covert challenge to the hegemony implicit in traditional mathematics instruction because the traditional authority held solely by the mathematics teacher is shifted onto students. At the same time, Gutierrez (2002) notes that “there may be a tension between critical and dominant versions of school mathematics” (p. 151).
While this student-centered subjective positioning and constructivist reasoning are essential aspects reform mathematics, CM advocates believe it does not go far enough.
Criticalists understand that subordinated students must be overtly taught to recognize and challenge hegemony (Pruyn, 1999). Another philosophical and epistemological difference from reform educators is that critical educators see all learning as value-laden, hence no curriculum or pedagogy is neutral. Standards-based reformers are not as attuned as their critical counterparts to the importance of the history of oppression and marginalization of poor children and children of color in society and schools. Hence, they are less likely to recognize that particular mathematics content is class- or race-biased.
87 Noting that students in reform mathematics classrooms are encouraged to question each other and their teachers about conceptual mathematics, Gutstein (2003) and Gutierrez
(2002) see themselves as extending or refining many of the ideas of the standards-based reform movement to make mathematics education more culturally and politically relevant to the lives of students of color. Gutstein (2005) discusses using the reform curriculum
Math in Context with his middle school students when they were not engaged in critical activities. He maintains that this reform curriculum provided his students with
“mathematical power,” but felt that the materials do not go far enough in terms of cultural and political empowerment. None of the NCTM Standards explicitly state that students are to ask questions or think critically in the Freirean sense of questioning life circumstance. Indeed, as Gutierrez (2002) observes: "in its current state, I do not see that reform mathematics necessarily positions students to consider issues of power in society, something that (for me) is at the core of equity" (p. 150). In response to these lacks, CM builds on the subjective positioning of reform mathematics by also asking students to consider issues of power in the mathematics curriculum and, by doing so, poses a more explicit challenge to hegemony. CM educators construct the mathematics curriculum around sociopolitical themes believed to be relevant to (subordinated) students’ lived experiences in ways reform instruction does not (Gutierrez, 2002; Gutstein, 2003).
Although CM educators share many of the views and goals of reform educators, they see mathematics as a tool that students can learn to use to (1) analyze and reflect on their situation in the world and (2) ultimately transform the world. Along these lines, critical educators attempt to give students the opportunity to see how mathematics, for example the public use of statistics, is constructed and used to obscure economic,
88 political, and social issues (Gutierrez, 2002; Noss, 1994; Skovsmose, 1994; Williams &
Joseph, 1993).
2.6 Conclusion
In this chapter, I provided a brief overview of the critical theory and cultural relevance perspectives that provide the grounding for CM instruction. Gramsci’s notions about hegemony and Freire’s ideas about how to teach poor children and students of color to combat hegemonic relations were covered. I introduced Gee’s theories about the nature of discourse and rationale behind discourse analysis. I outlined Dowling’s critique of the dominant form of constructivist pedagogy (i.e., reform pedagogy) as not coming to grips with the socially constructed nature of mathematics in instruction. I delineated various critiques of traditional mathematics instruction and gave a summary of the major themes important to reform mathematics. Finally, I clarified how CM theorists rely on reform mathematics concepts and instructional approaches, but also the ways they differ.
These various reviewed literatures offer essential background for my research questions, research methods, and analysis of my study’s results.
89
CHAPTER THREE: METHODS FOR DATA COLLECTION AND ANALYSIS
As discussed in the first two chapters, my study was designed to advance the understanding of critical mathematics (CM) for culturally diverse youth in secondary urban classrooms. CM involves developing mathematics problems that touch on issues of power (e.g., resource distribution, racial or social class segregation) likely to affect the lives and education of students from subordinated working class and poor communities.
To this point in time, scholarship on CM instruction has been largely theoretical. Given
CM curriculum’s potential for increasing cultural relevance and educational equity, studies of how it actually transpires in practice are needed.
In conducting this study, I was the primary teacher, instructional designer of the
CM component of the curriculum used, and the researcher. While I was a full-time teacher in secondary urban schools for four and a half years prior to enrolling in graduate school, I had never taught mathematics from a critical perspective. By documenting my own development of CM curriculum and instruction, I focus on dilemmas that teachers who wish to begin teaching school mathematics from a critical perspective might encounter. While I also collected data in a six-week summer school course, for various reasons that will be addressed in this chapter, this report focuses primarily on the nine- week night school course I taught at Guevara High School (a pseudonym). All thirty- three students who initially enrolled in the course needed geometry credit to make up for past failure and graduate from high school.
3.1 Research Purpose
90 Mathematics educators who write about CM argue that it has the potential, especially for marginalized students, to be more empowering in a political and mathematical sense than traditional instructional approaches that prevail in schools
(Gutierrez, 2002; Gutstein, 2003, 2005). However, because little empirical evidence is not available to validate this claim, whether or not CM is more empowering in practice remains an open question. In my research, I attempted to examine what CM looks like in my secondary classroom. I focused data collection and analyses on the following four points: (i) how critical curriculum and instruction might be organized and what an experienced secondary mathematics teacher (myself in this case) faced in this organization, (ii) how the incorporation of justice-related themes relate to and differ from the dominant secondary mathematics curriculum, (iii) what is the nature of the classroom discourse and student participation in CM activities, and (iv) how does participation in the CM course influence student learning of geometry as well as their attitudes and beliefs about mathematics, themselves as mathematics students, and as political beings.
3.2 Research Questions
The following research questions correspond to the purposes listed above:
1. What issues do secondary math educators face in the development and implementation of CM curriculum that needs to meet local math standards?
2. How does the incorporation of justice-related themes into the conventional secondary math curriculum transform this curriculum?
3. What is the nature of classroom discourse and student participation in a CM classroom and how does this evolve over time?
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4. How does engagement in CM activities influence the mathematics students learn, the students' epistemological beliefs about mathematics, their attitudes about themselves as learners of mathematics, and their feelings about learning mathematics?
3.3 Why a Qualitative Design was Selected
Because my research was designed to be a descriptive, in-depth look at CM in practice, qualitative design seemed to be the most appropriate research approach. As
Piantanida and Garman (1999) note, qualitative studies in education “aim to generate deeper understandings and insights into complex educational phenomena as they occur within particular contexts” (p. 132). Various other reasons justify my use of qualitative methods, including that my study was designed to be exploratory and descriptive. It was intended to include, and give voice to, students from populations that historically have received an inferior mathematics education (Ladson-Billings, 1995; Oakes, 2005; Sells,
1978; Tate, 1997), and whose perspectives have not been considered (Valero, 2004).
There has been little previous site-based research on CM, so its advocates rely largely on theoretical rationale for its use. Based on my literature review, I believe I am the first researcher to document a critical approach to mathematics instruction at the high school level. The work of the Algebra Project (Moses & Cobb, 2001) at the secondary level is relevant, but, to my knowledge, these researchers have not yet shared most of their curricular materials with the research community (Hall, 2002).
3.4 The Value of Practitioner Research
92 Given that my beliefs as a mathematics teacher and my instruction were a central focus of my research, it can be called practitioner-research (Anderson, Herr, & Nihlen,
1994; Hobson, 2001). Practitioner research is an umbrella term that stands for a number of closely related ideas. As Anderson, Herr, and Nihlen, (1994) note, there are a dozen or so terms for practitioner research in the educational literature, including “teacher research,” “teacher inquiry,” and “action research” (Carr & Kemmis, 1985; McKernan,
1991). Whatever the name, researchers share a number of assumptions that McKernan
(1991) says “rest on three pillars”:
first, that naturalistic settings are best studied and researched by those participants
experiencing the problem; second, that behavior is highly influenced by the
naturalistic surroundings in which it occurs; and third, that qualitative
methodologies are perhaps best suited for researching naturalistic settings. (p. 5)
Anderson, Herr, and Nihlen (1994) claim:
[P]ractitioner research is “insider” research done by practitioners (those working
in educational settings) using their own site (class, institution, school district,
community) as the focus of their study. It is a reflective process, but is different
from isolated, spontaneous reflection in that it is deliberately and systematically
undertaken, and generally requires that some form of evidence be presented to
support assertions. What constitutes “evidence” or, in more traditional terms,
“data,” is still being debated.
Most practitioner research is oriented to some action or cycle of actions
that practitioners take to address a particular situation. … Like all forms of
inquiry, practitioner research is value laden. Although most practitioners hope that
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practitioner research will improve their practice, what constitutes “improvement”
is not self-evident. It is particularly problematic in a field such as education,
where there is no consensus on basic educational aims. Practitioner research takes
place in educational settings that reflect a society characterized by conflicting
values and an unequal distribution of resources and power. (p. 2 – 3)
The practitioner research approach I took is similar to work in mathematics education by Ball (1993), Chazan (2000), Gutstein (2003, 2005), Heaton (1994, 2000),
Lampert (1986, 2001), Magidson (2002), and Romagnano (1994). To varying degrees, these scholars have all researched their own mathematics instruction using qualitative methods such as participant observation, analysis of their own teacher diaries, document inspection (e.g., of student work), and analysis of videotaped teaching episodes.
The overriding reason that I chose to conduct practitioner research was that it provided me with more flexibility to experiment with critical pedagogy than other research designs. There were additional reasons for choosing to do practitioner research.
First, the approach allowed me to document my internal conversation on such issues as cultural relevance, racial and social class privilege, school segregation, and problematic pedagogy. Second, of the abundance of research done on teachers’ instruction, some celebrate their teaching, however, much of it is negative and ungenerous to those who agree to participate in the research. If I were to have conducted research on someone else’s teaching, I would have risked projecting myself as an omniscient researcher who felt qualified to critique another teacher’s attempt to teach CM, when I had never taught it myself.
94
Third, the insider perspective that I took in my study provides an outlook on mathematics education that is unavailable through the use of other methodologies.
Tracking one’s own reflections provides valuable insights into philosophical, ethical, methodological, and practical issues that mathematics teachers and their students face.
Others seem to agree that this approach is important; the self-study work of several well- known mathematics educators (e.g., Ball, 1993; Lampert, 2001; Schoenfeld, 1994) is often cited, referred to in teacher education courses, and current policy in mathematics education is informed by it. That said, however, practitioner-research is “still somewhat unusual in mathematics education” (Magidson, 2000, p. 17) and may not yet be seen as valid or productive in some circles.
Fourth, and relatedly, there was a dearth of information about White teachers being self-critical regarding issues of diversity and equity in the mathematics education literature and I wanted to help fill this gap. The autobiographical writing of teachers in the broader education genre (e.g., Esme Codell, 1999; Sylvia Ashton-Warner, 1963) sometimes ends up being a "celebration of self." One exception in the literature is Vivian
Paley (1979), a White teacher who struggles deeply with racial and cultural issues that arise in her study. Except for the miracle stories that have been made into films about particular superstar teachers’ success in the “urban jungle,” there is little at the secondary level that involves teachers developing relevant curricula for urban or lower SES students while reflecting critically on the process. It should be noted that work by Chazan (2000),
Gutstein (2005), and Romagnano (1994) begin to address this deficiency in the area of mathematics education.
95 Fifth, I wanted to examine CM materials and instruction as well as the difficulty or ease of developing such curriculum. I feared that I would have a hard time recruiting a teacher to use my CM materials and to remain engaged in the project from start to finish or, indeed, even to buy into my ideas about CM. In other words, I wanted to control the nature of the CM curriculum development and to implement it in the way I felt was appropriate. As noted earlier, I also wanted to study the thinking that a teacher goes through in initiating an innovative way of teaching and it seemed most efficient and effective to keep track of my own thoughts in such a project rather than the thoughts of what was likely to be a less committed and open teacher.
Finally, there were broad theoretical and political reasons for self-study.
Specifically, it was a means to collapse some of the dualities found in academic research
– dualities between theory and practice, and between researchers and those researched.
Because this study was grounded in the realities of the classroom and the school community, my research challenges ideas about whether expertise lies with scholars in academia or teachers on the frontline in schools. It raised the questions about who should create knowledge about students and schools (Anderson, Herr, & Nihlen, 1994; Carr &
Kemmis, 1985; Cochran-Smith & Lytle, 1993; Lampert, 2001; Magidson, 2002;
McKernan, 1991).
3.4.1 The Need for Instructional Design
While my study clearly was practitioner-research, my attention to curriculum design and textual analysis meant that it could also be described as “curriculum action research” (Carr & Kemmis, 1985; McKernan, 1991) or “instructional-design-research”
96 (Magidson, 2002). Because there were no social justice math curricula available at the secondary level, a key component of my study was developing curriculum. Magidson
(2002) uses the term instructional design rather than curriculum design, arguing that “the word ‘curriculum’ generally refers to a large body of material – one or more year’s worth of materials for teaching a particular subject area such as mathematics” (p. 11). From
Magidson’s perspective, instructional design indicates a narrower focus as was the case with the materials I produced for my night course teaching. Like Magidson (2002), I document my approach to instructional design from the perspective of a teacher. In addition, I provide a subjective firsthand account of my instructional design process.
While the idea of a teacher as a curriculum designer or curriculum researcher might seem unusual, it is in-line with the research that Dewey (1916, 1929, 1990) and his colleagues engaged in at the University of Chicago Laboratory School at the turn of the twentieth century (Lagemann, 2002). Indeed, Lagemann argues that if Dewey’s vision for educational science had prevailed, rather than the technical vision of such positivists as the psychologist E. L. Thorndike, teachers as instructional designers and researchers would not be atypical now. Nevertheless, according to Lagemann, Thorndike’s vision won out, resulting in the highly psychologized, techno-rational approach that has kept teachers outside of the instructional design process. Worse still, rather than being seen as designers, teachers are often been seen as a design problem; various university-based curriculum designers have talked about designing “teacher-proof” materials so that there would be no “lethal mutations” to the intended curriculum (Cohen, 1989). Presently there is increased interest in including teachers on design teams or giving teachers a larger role in curriculum decisions (Ball & Cohen, 1996). Nevertheless, curriculum design continues
97 to be monopolized by teams of experts in academia or in large educational publishing firms (McKernan, 1991). In other words, general curriculum design tends to be removed from local school contexts. It also is aimed at a universal or generalized classroom, due in no small part to profit and funding incentives (Kohn, 2004). The outsider, top-down approach results in instruction becoming rigid and unresponsive to particular classroom contexts and students.
3.4.2 Congruency Between Instructional Design and Qualitative Research Processes
As Magidson (2002) and McKernan (1991) note, the design process is cyclical or iterative. The same assertion has been made about the nature of qualitative research
(Emerson, Fretz, & Shaw, 1995; Merriam, 1998). My dissertation study can be seen as a set of smaller repeating cycles of design, implementation, and reflection. Given the observational and reflective energy I put into each CM activity, each one can be seen as forming its own small design cycle (see McKernan, 1991).
I took the idea of reflection on the instructional design process seriously. Indeed, after I finished teaching the summer course at Park Vista, I began to take a closer look at my CM curricular materials. However, at the time, having just designed them, I felt that I was too close to my CM materials to be truly critical. After taking some time away from my dissertation study, I searched for an analytical framework that would allow me to distance myself from my curricular materials during analysis. I discovered the book, The
Sociology of Mathematics Education (1998), by the British scholar Paul Dowling. This
98 book provided a set of tools that I could use to analyze my CM materials from a systematic, and somewhat personally distanced, perspective.
3.5 Design Specifics
The primary analytical focus of my research was the CM component of my curriculum and instruction, although the reform approach used in the nine-week course is addressed in Chapter 6. Critical activities made up approximately 20% of the curriculum in the night course; the remaining 80% were largely reform activities that were not explicitly political. Given that ratio, it might seem that I was mostly focused on reform mathematics, however, I conducted three separate analyses in order to get at different aspects of CM. First, a textual (curriculum) analysis was done in order to understand the mathematical effects of including critical themes in the traditional and reform curriculum.
The unit analyzed was my critical I&A text (see Appendix A). Second, I analyzed my approach used to the design and implementation CM materials and my beliefs related to my role as a teacher-designer. Here my curriculum diary and various drafts of my study proposal were the artifacts analyzed. Third, I conducted a discourse analysis of transcribed classroom verbal interactions between students and between students and myself while CM was being taught.
3.5.1 The Setting and Circumstances of My Study
I collected data for my dissertation study at two sites: (1) a night school program at Guevara High School and (2) a summer school program at Park Vista High School
99 (again, both pseudonyms). Both programs were offered to allow students who had failed geometry previously to make up for past failure.
I initially intended my study to be the story of progression from a novice CM educator at the Guevara night school program to a more effective CM instructor in the summer program at Park Vista. With this aim in mind, I collected data on my CM implementation in both settings. While I did generally improve as a critical instructor, the expectation that I would essentially “get critical math pedagogy right” the second time around so I could report on the successes did not turn out as expected. This was one reason I decided to focus my report solely on the Guevara setting.
There were two other reasons that I narrowed my focus to data collected at
Guevara. First, the night school course extended over a nine-week period rather than the two shorter (three-week) summer school sessions at Park Vista. In addition, there was considerable student turnover in the summer program; some students attended the first semester, others the second, and still others attended both. The extended time in the
Guevara night school setting with one group of students meant I had more time to develop respectful relationships with students and to implement my critical curriculum.
Second, enrollment in the night school course was more similar to real classroom circumstances because it included 28 students who received credit for the course compared to approximately 15 students per each summer school session at Park Vista.
While there were drawbacks to studying an untried pedagogical approach with a larger class, there were also advantages; for instance, the larger class size meant more students were present to spark whole class discussions. The wider range of students allowed me to observe how achievement hierarchies evolved, and dissipated, over the course. The
100 summer courses at Park Vista were interesting and different from the course taught at
Guevara. I plan to analyze the collected data, particularly the interview responses and student work samples, at a later date. However, the main reason for focusing only on
Guevara was the sheer volume of data collected and time constraints of dissertation work.
Realistically, I could only analyze part of the data from one setting.
The Guevara night school geometry course was a good study site for several reasons. First, I was familiar with the teaching situation because I taught secondary mathematics in demographically similar Chicago Public Schools (CPS) for more than four years. I say “more than four” because I worked for a year on a funded math and science project at a CPS high school in another neighborhood district, did my student teaching in CPS, taught secondary mathematics summer school for seven straight summers, spent the first six months after completing my Masters in Education teaching math at a high school with predominantly Spanish-speaking students, and then transferred to another CPS high school where I taught three years. A further reason for focusing on the night course was that it ran for two hours a night (four days a week), which meant that I was not constrained by the usual 45-minute periods of the regular CPS school day.
With this extended time, there was more space open for critical and mathematical conversations. The time-intensive nature of the course also made it easier to arrange for the course to be filmed by undergraduate assistants. On the other hand, planning for and teaching an innovative course that ran two hours a day, four days a week presented a fairly intensive schedule burden, especially because I also had graduate school obligations.
101 3.5.2 Chicago: Dual Economy and Two-Tiered School System
Chicago is segregated in terms of race, ethnicity, and social class. Over the past several decades, long-standing social and geographic divisions in the region have been exacerbated as the U. S. economy shifted from manufacturing to service-oriented jobs.
While the wealthy generally have benefited from societal and economic changes, corporate disinvestment in their communities has hit the working class and many people of color hard (Anyon, 1997; Castells, 1998; Lipman, 2004).
Lipman (2004) frames post-industrial Chicago as a city that is divided into two social classes, a predominantly white professional class (e.g. corporate managers, lawyers) and a larger servant class, primarily of color, who work dead-end jobs with low pay and few benefits. The polarizing effects of the dual economy are widespread, as
Lipman describes:
[T]he face of today’s downtown Chicago and the ring around it is marked by
high-end lofts and shops carved out of converted manufacturing space, new condo
developments, and a central core corporate and convention center with expensive
retail outlets, cultural venues, and parks. The not-so-public face is a city of
deindustrialization, displacement of settled working-class and low-income
neighborhoods, and socially isolated, deeply impoverished communities
(Betancur & Gills, 2000a). As are other major international cities, Chicago is a
dual city spatially as well as socially and economically. (p. 28)
This economic dichotomy and geographic isolation is reflected in a two-tiered public school system. There is a small set of elite schools (and within-school programs) designed for middle class students and a larger number of public schools designed for
102 working class students most of whom are of color. In the first type of school, instruction caters to individual student needs, while in the second type, instruction focuses heavily on preparing students for high stakes examinations.
Beginning in 1995, the Chicago school system, headed by Paul Vallas, undertook a highly publicized reform movement that held schools, teachers, and students to increasingly high achievement accountability standards (Lipman, 2004). There were sanctions for schools that failed to live up to relatively rigorous academic standards-- underperforming teachers would be released and underperforming schools would be closed. Despite the fanfare that accompanied these reforms, they have done little to improve education for the less privileged or make the stratified school system fairer
(Kozol, 2005; Lipman, 2004; Martin, 2003). Lipman argues that accountability measures may never have been designed for improving equity, but rather are part of a broad set of municipal policies that pave the way for gentrification and globalization. Lipman describes it this way:
In short, when we examine the distribution of new programs against CPS’s claim
to expand educational opportunity, we find that although high-profile, allegedly
academically challenging programs and schools are scattered throughout the city,
almost all whole-school college-prep programs initiated since 1995 are clustered
in middle-class, white, or gentrifying areas. This has been a huge investment by
the city. According to the Catalyst (Schaeffer, 2000, p. 13) the new regional
magnets got almost half of CPS construction and renovation money between 1996
and 1999. Meanwhile most academically challenging programs in low-income
communities involve only a portion of the students, and vocational, military, and
103 DI schools – most involving all students in the school – are clustered in low-
income African American and Latino/a areas. Thus, programs created and
expanded under the 1995 policy agenda have reinforced the inequitable
distribution of challenging academic programs, and they support gentrification,
displacement, and spatial dualization. (p. 55)
In addition to serving economic interests of large corporations and middle class home buyers, the reformist educational policies implemented in 1995 imposed “high stakes” for schools serving low-income students and students of color. Reforms that allegedly held all schools accountable, selectively targeted schools attended by low- income students of color while essentially leaving schools for economically privileged students alone. CPS leaders rationalized the differential imposition of sanctions on the grounds that low-income students had low test scores. Yet, according to Lipman (2004), the measures that resulted from the accountability movement worsened conditions for those who were already oppressed. As Lipman observes:
To the extent that the new policies prompt schools to overemphasize standardized
tests and basic skills (as opposed to an intensive effort to develop more
thoughtful, intellectually challenging pedagogies), they widen educational
inequalities by institutionalizing a narrowed curriculum and less intellectually
challenging work. … Smith, Lee, and Newmann (2001) … found that “didactic
instruction, “ in which “students’ time is typically spent: (1) listening to the
teacher, (2) reciting answers to questions, or (3) practicing skills and information
retrieval by completing worksheets or exercises,” was more common in
classrooms with low achievement, in “problem classrooms” (read: discipline
104 issues), in classrooms with irregular attendance, in schools with more low-income
students, and in schools with low achievement levels. (p. 43)
Perhaps even more problematic, as Lipman points out, the skills and dispositions learned in didactic, test-centered classrooms link directly to those needed for service industry jobs (e.g., obedience, mindless drill). So, the accountability movement essentially enforced educational practices that prepared low-income students for futures in low-skill service sector jobs, thus reinforcing social class distinctions.
Like many urban high schools that serve low-income students, Guevara High has been on probation for low scores on standardized tests since the accountability movement began in 1995. Teachers at schools such as Guevara feel pressed to raise test scores. One of my colleagues in the night school program used a set of test-prep booklets as her primary text for algebra. She believed this to be the most effective way to prepare her students for the mathematics they would encounter on standardized exams. However, despite accountability policy promises, fewer than ten-percent of Guevara students currently test at, or above, grade-level in mathematics. There also is a chronically high dropout rate, with about forty-percent of entering freshmen failing to graduate within five years. The drop rate increased after 1995.
Related to the earlier discussion of preparing students for service sectors jobs, more than a third of the regular day school students, and at least as high a percentage of night school students, participate in vocational Education-To-Career (ETC) programs.
According to Guevara’s school website, the ETC program includes preparation for
“culinary arts,” “fashion design,” and “information technology” (e.g., introduction to
Microsoft Office). Vocational education can be a rational solution to the deep-seated
105 problems rooted in socioeconomic inequalities. On the one hand, vocational education may keep students in school who would not otherwise stay, which translates into higher graduation rates and greater opportunities for graduates. On the other hand, vocational programs are problematic considering that students already may have received “teach to the test” instruction that increases their likelihood of ending up in service-sector jobs. In the case of being relegated to such specialized programs based on prior (low) achievement, what is billed as free choice is not a choice at all. There are no vocational options at elite magnet high schools in Chicago. From my observations, it is apparent that
CPS vocational programs do not have the high tech equipment to provide students with viable skills needed for the occupations for which they supposedly prepare students.
3.5.3 The Night School Students
Thirty-three students attended my Guevara night course at some point in the first week. Three of these students did not continue to come after the first week. As policy demanded, two more students were dropped from the course after their fourth absence.
Of the enrolled students who completed the course: (1) ten were full time night school students, (2) five attended both the regular day and night school programs at Guevara, (3) ten attended regular day school elsewhere before coming to night school at Guevara, and
(4) three worked full-time outside of school and were attending night school to complete one or two courses in order to graduate. The ten full time night school students were not allowed in the regular day program for a variety of “reasons” (pregnancy, gang problems,
106 fights, attendance, in-class behavior) and some were upset about being blocked from day school (see Appendix D for more detail on individual students).
Of the twenty-eight students who passed the nine-week course, 13 were of
Mexican descent, 7 were of Puerto Rican descent, 6 were African American, 1 was
Honduran, and one student was of Pakistani descent. Many were either immigrants (1st generation in the U.S.) or the children of immigrants (2nd generation immigrants).
Because of a history of academic failure, several students were already nineteen. Most were eighteen. While I did not collect data on the socioeconomic status of their families, over 90 percent of the students at Guevara received free or reduced lunch. Many night school students also had full- and part-time jobs outside of school. While some students discussed having married parents, the majority came from single parent households, lived with a divorced mother, or lived on their own or with older siblings. One student lived with his pregnant wife. Four young women already had children and three were pregnant.
It is also important to note that most students were about to graduate and had a wide range of post-secondary plans. Four of the young men had already enlisted in the military or did so when they were in my course. Others wanted to be engineers, nurses, teachers, businesswomen, and lawyers. Unlike their military bound classmates, these students planned to attend college after high school. Those who wanted to be auto mechanics, beauticians, and nurses’ aides, planned to attend a technical or post-secondary vocational school.
During the course, I learned that the enrolled students experienced a host of personal and social problems. Guevara had a well-known gang problem. Several of my students, young women as well as young men, were either in gangs or had been in them
107 in the past. Two young women were involved in fights at school during the nine-week course. The police caught one of my male students when he reportedly was about to shoot at rival gang members. Two young men readily admitted to me that they occasionally dealt drugs and another admitted he had done so previously. While many students claimed to abstain from “partying,” others admitted to smoking pot, drinking, and doing hard drugs. They confessed to “partying” on a regular (nightly) basis. Several students came drunk and high to class during the first two weeks. One night I came across unfamiliar students dealing cocaine in the school bathroom. I although I did not realize it at first, one student was drunk enough that he threatened to fight me after I asked him to get to work. I provide more details about individual students from the night course in
Appendix D.
3.5.4 Discipline at the Guevara Night School Setting
When I began teaching in the Guevara night school program, there were two features that stood out to me, namely, the tight security in and around the school, and the relatively poor physical state of the school facility. First, in part because I had recently visited two of the more elite college prep high schools in Chicago, the tight level of security at Guevara High stood out as excessive. The following fieldnote excerpt illustrates my feelings about this:
When I arrive at [the night school facility], I go through the security check. It is
not unlike a security checkpoint at the airport. It’s run by three Latino [male]
officers in light blue, police-like uniforms with metallic badges with their last
names on them. … They’re standing by an old metal detector. It has an x-ray
108 machine with a conveyor-belt that’s used to search backpacks and coats for
weapons, apparently. Besides some sort of an escalator, the security checkpoint is
the only thing located on the first floor lobby – a huge space, (probably 200 x 100
feet). One entire floor devoted to security.
It is about 3 o’clock in the afternoon and about a dozen, mostly female
Latino/a and Black, students are standing around in the lobby, some with their
coats and backpacks half on, draped to the floor. I overhear the oldest officer tell a
female student to “be careful” on her way home because there has just been
reports of “gunfire” in the neighborhood to the south of the school.
I notice that the guards are all wearing guns on their belts so that they are
visible. I note to myself that this security seems much tighter than at most of the
schools I’ve been in. It’s as if these students are one step away from jail. When it
was my turn to go through the metal check, the youngest and largest of the guards
– my height (6 feet) but probably 100 pounds heavier – asks me what I was doing
at the school and who I wanted to see. I explained. He looked at me and asked,
“You want to teach night school?” I responded, “yeah.” He then asked, checking
me out with a grin, “Do you have a vest and a helmet?” I laughed in response. He
lowered his voice and put a hand on my shoulder stating, “No. I’m serious. Bring
a vest and a helmet. This is night school. Only the sta-ars come out at night.” He
chuckled some more. I just smiled back at him not knowing what to say.
(Fieldnotes, 11/04/03)
Guevara students were also policed – or protected depending on your perspective – outside the school by the Chicago police force. Often a police car was parked in front of
109
Guevara when I arrived at 5:30 p.m. and two or three police cars were parked in front of the exit when the students and I left the building at the end of the evening.
From what I observed while teaching in the system, Guevara’s tight security was not unusual for neighborhood (i.e., non-magnet) CPS secondary schools. These same conditions are typical of low-income schools elsewhere (Casella, 2001; Devine, 1996), in fact, schools that serve low-income students of color may prioritize security/discipline over learning (Casella, 2001; Devine, 1996; Noguera, 2003a, 2003b). In response to a night school class assignment, one of my students, Yasmin, wrote:
School to me is run like a jail. At least my school is. We have to wear uniforms.
We can’t even bring our own (see-through) bookbags. We are given one and we
all have ID numbers which is how we are known. Some school like mine run it
like a jail because its for our “protection.” I think they go overboard. In my school
we wear uniforms because too many people were wearing gang colors. (Student
Work, November 19, 2003)
While Yasmin and I agree that low-income urban students of color are dehumanized by being identified by ID numbers and treated as potential offenders by CPS, other night school students had a more dominant and hegemonic position. Students indicated that they believed that draconian security measures were meant to protect them from their more dangerous peers and, perhaps, themselves. Yasmin’s classmate, Stephie, responded this way:
Without rules students would do whatever they want. People wouldn’t listen to
the teacher. Students would come and go when they please! Students would bring
110 weapons to school. Students would talk on their cell phones when every they
want. Students wouldn’t pass their classes. (Student Work, November 19, 2003)
Indeed, as in many urban areas, the threat of violence in and around the school was real
(Noguera, 2003a). It is not surprising, then, that violence turned out to be a frequently discussed topic among students during the breaks and in the “third spaces” (Gutierrez,
Rymes, & Larson, 1995) of my night school course.
3.5.5 The Guevara School Facility
In addition to the tight security, the semi-neglected state of the school facility made an unsettling impression on me. The facility was not necessarily in disrepair but was worn-looking and was constructed of inferior material, especially when compared with the college preparatory school facilities I had recently visited. Guevara was not built as a school; the big, forty-year old, steel and glass, building, had been converted from an office building. Because it was nine stories high, escalators went from floor to floor, however, they were regularly out of service. The night school programs were held on the fourth and fifth floors. Some night school students hung out on the sixth floor before class and during breaks, apparently to avoid the security that was centered around the classrooms.
My 30' by 30' classroom was big enough to accommodate my 27 students. Like the school facility generally, the room was worn looking– the dark red carpet covering the floor had years of spills etched into it. The glaring florescent lights flittered enough to be noticeable. There were two cabinets with useless locks dangling from them because
111 the doors had been pried open enough to fit a human arm through. The outside windows had rubbery plastic curtains drawn unevenly over them.
According to the night school administrator, the regular day teacher whose room we occupied “was miffed” that his room was being used for night classes. Perhaps to retaliate, he left little space for us. The blackboard and surrounding walls were plastered with posters of U. S. presidents, so that, among other things, there was no room to write on the chalkboard. The large patriotic posters sent a blunt conservative message. They included large pictures of George W. Bush, Nixon, Ford, and Truman. Two posters were of the first President Bush. The picture of Lincoln seemed out-of-place among the others.
On a side wall, there were smaller posters of Presidents Kennedy and Carter, and Martin
Luther King. President Clinton was conspicuously absent from the display, as were women and Latino/as.
Of more immediate importance to my plans for collaborative student group work,
I was told that the desks, which were in seven neat rows, could not be rearranged. At the same time, the night school administrator did not discourage me from allowing students to work collaboratively and actively. In fact, he told me in no uncertain terms that lecturing would not work with “these” students. My access to technology was limited to an overhead projector that I had to carry back and forth to the classroom every evening from an office down the hall. There was, however, a television mounted to the ceiling in one corner of the room. Below it was an enormous, outdated IBM computer. While the televisions in other night school classrooms were often in use, the computers in my room were “off limits.” I initially was unaware of this rule. One day, when the night school administrator saw a female student of mine checking her email on this computer before
112 class, he reprimanded her explicitly, and me implicitly. It had never crossed my mind that students’ access to technology that was purchased for them would be off limits.
Concerning technology, I wrote the following in my diary, a venue that was more introspective than my field notes:
I think about the “digital divide.” Please. I have no chalkboard space and no
[working] overhead projector [initially]. … When I asked about getting a
classroom set of calculators I was told that the night school students are expected
to buy [outdated] solar calculators for $9 a piece. About a third of my students do
own cheap calculators of this sort, but I have no access to graphing calculators or
computers despite having [a computer] in the room. I also had to buy a set of
protractors and rulers for the class. Osvaldo told me that he was impressed by the
fact that I had bought them new things. (November 25, 2003)
Osvaldo’s comment shows the extent to which students in underperforming schools are conditioned to think that they do not deserve the same resources they see
(perhaps on television) privileged students have in their schools. It seems that these students are so socialized by inadequate material and social conditions, and are so isolated from students of other social classes, that they do not necessarily see these lacks as different or unfair. In Chapter 7, I discuss how Osvaldo and other classmates have internalized negative dominant messages about themselves and their peers. They frequently make comments that reveal they perceive their own communities as violent, intellectually inferior, and less deserving. Such extremely unequal treatment of students in terms of resources and tight security routines sends a powerful message to students and their teachers about their comparative worth. Again, this is one of the human effects of
113 the highly stratified school system in terms of resource distribution. Osvaldo’s statement reveals the micro-level influences of macro-level practices.
3.5.6 Teaching and Learning in the Night School Setting
Lipman (2004) describes CPS as a system that demoralizes low-income students and the teachers in their schools. The following entry from my teacher diary illustrates how the differences between the elite schools and Guevara affected me:
Teachers at [elite schools] are treated like professionals. They have “faculty”
offices and bathrooms while we [teachers in neighborhood schools] are lucky to
have our own classrooms or a “teachers’ lounge.” Elite schools are not
overcrowded like [Guevara.] I don’t feel like a professional at [the Guevara night
school]. Night school seems like a step above – or a step away from – a factory or
prison with the kids all in uniform and ever-present security. I’m also being
“surveilled” by the night school administrator and security – almost as much as
my students. They stop by my room with random regularity. I try to keep a smile
on my face when they visit because it makes me seem like a “team player” and
moves them along quicker. (Diary – 11/24/03)
In such an environment, it was not surprising that a handful of night school students seemed more concerned with getting through my course for accreditation than learning mathematics for its own sake. When, in an interview conducted before my course began, I told Jayla that there were 33 students enrolled in the course, the following exchange followed:
114 Jayla: Yeah, but like, by the middle, you’ll probably have, like, twelve students.
Yeah – cause we only have two days of absence. And, three tardies equals one
absence. So not that many people…
Me: Is that what happened last semester in [night school] math?
Jayla: Ye-ah. Like we didn’t have enough chairs to sit in and everything was all
packed up. But at the end of it – I mean – we had more than enough chairs to sit
in, cause everybody dropped out. (Pre-Interview, November 7, 2003)
Indeed, while five students dropped or were dropped from the course due to poor attendance, of the 33 students who came to class at some point during the first week, 28 made it through the course. This relatively high retention rate was apparently unusual for the Guevara night school.
Academically speaking, it appeared that not much was expected of students in the night school program. In the first few nights, the students indicated that they should get by without doing much thinking. Indeed, at the outset of my course, several students made it clear (albeit to varying degrees) that they would do the minimum amount of work necessary to pass the course. In the same pre-interview, Jayla summed up her experiences as a student in the night school program as follows: “we don’t really have to do anything in night school, but just be here, and don’t talk, and just keep quiet.” In fact, many of my students seemed surprised that I expected them to work diligently and collaboratively in order to pass my course. The following is an entry in my instructional at the end of the first week of the course:
At one point during today’s lesson, Amalia looked up at me and said: “You know,
you’re the only teacher in night school who makes us work.” She went on to
115 describe how they do next to nothing in the English class she takes before coming
to my geometry class. I asked her if it was good or bad that I’m making them
work. She indicated that she was happy – possibly felt that I respected her
intellectually – that I am expecting them to work. The other young women around
her seemed to agree. (Instructional Diary, November 20, 2003)
While I hesitate to state that I was the only teacher with high expectations, when I passed by other teachers’ classrooms I observed that their students were most often sitting at their desks apparently either listening to the teachers, quietly doing individual work, or with their heads resting on their desks. One of my colleagues who filmed my class on several occasions also worked in Guevara as an advisor to several of the regular day school teachers. He said that he struggled to get the Guevara teachers to set up regular routines and use student-centered activities like the ones he observed me using. Finally, it is also important to note, as Chapter 6 indicates, despite the tight security, inferior resources, and generally low academic expectations, the majority of the night school students conveyed, through their behavior, in their written work, and in interviews, that they were in the class to learn secondary mathematics.
3.5.7 The Night School Routine
The night school course ran for two hours a night from Monday to Thursday for nine weeks. During the first hour of class, we generally began with a 10-minute activity that I called the “opener.” This was followed by a related 40- to 50-minute standards- based reform activity. In the second hour, after a ten-minute break, we generally considered a second 40- to 50-minute reform mathematics activity. If there was time, we
116 ended the session with a short closing activity or summative discussion. The longer mathematics activities either came from the Interactive Mathematics Program (IMP)
(Fendel, Resek, Alper, & Fraser, 2000) or were CM activities I designed. The openers were shorter activities that built on and extended the IMP or CM activities that were the focus of each class. (I begin Chapter 4 with a sample activity and discourse used in an opening activity. An example of a closing activity is included in Appendix A.)
3.5.8 Gaining Access to the Guevara Night School Program
I gained access to the night school setting in a roundabout manner. I originally planned to teach geometry in the night school program at Park Vista, where I had previously taught. About three weeks before it was scheduled to begin, due to lack of funds, this night school program was canceled. I had to scramble to find another night course program for my study. A Park Vista administrator told me that the director of the
Guevara night school program was looking for a geometry teacher. When we met, I made it clear to the Guevara administrator that I wanted to teach the course, but also to research
CM. I explained that CM would make up some 20% of the curriculum. While he voiced skepticism about using innovative pedagogy with this population of students, he needed a teacher. He was willing to trust me, perhaps because as he said, he had heard “good things about my mathematics teaching” from an administrator at Park Vista as well as from a teacher-friend at Guevara. After securing the position, the night school administrator arranged a meeting with the Guevara principal, in part, so I could float my research agenda by her. When we met, I was a bit taken aback, because she welcomed
117 my overtly political research agenda with open arms. She told me something to the effect of, “we need to reach these kids – so, let me know how it goes.” Although I invited her and a second administrator to visit my classroom whenever they liked, I never saw her again.
3.6 Sources of Data & Data Collection Procedures
I collected nine distinct data sets for this study: (1) sixty-eight hours of videotaped lessons from the night course, (2) detailed transcriptions of spoken discourse from eight hours of these videotaped lessons, (3) participant observation fieldnotes, (4) an instructional diary, (5) formal beliefs-related documents, (6) my own critical mathematics curricular materials including the I&A unit as well as two published geometry texts used for comparison with the I&A unit, (7) twelve audio-recorded pre- and post-interviews with night school students, (8) samples of students’ written coursework, and (9) pre- and post-exams. I describe these data sources in detail in the sections that follow.
3.6.1 Transcriptions of Videotaped Instruction
I produced transcriptions from eight hours of videotaped night school lessons. In total, this amounted to more than 200 single-spaced pages of classroom discourse. These transcriptions allowed me to analyze micro-level patterns in discourse and discursive student participation in the critical and reform lessons I taught. Without these transcriptions, discourse analysis would have been difficult, if not impossible. I created these transcripts by watching classroom action on the video portion of the videotapes and listening to the audio portion of videotaped lessons. Following the recommendations of
118 Duranti (1997) and Gee (2005), I represented such things as overlapping speech, rise in pitch and volume, and other evidence of such things as enthusiasm or boredom. The following is an excerpt of a transcript that I produced:
Me (to Deno): alright / you wanna tell em / to get to work //
[do work //
Deno (to class): [hey // everybody / get in their se:at! // we gonna start work /
now // break is o::ver //
I laugh. I make eye contact with Jaime seated in front. He also laughs.
Deno (turns to class): break is // goddamn-it // over ///
Conversations quiet down, pencils begin to sharpen, and students head back to
seats.
The single slash mark (/) was a very short break in discourse while the double-slash marked a full stop (//) or approximately half-second to second long break. A triple slash mark (///) indicated a break of a second or more. Bold speech was louder and “!” indicated further emphasis in volume. Underlined speech indicated a raise in tonal pitch.
The “:” indicated the temporal extension of a vowel. When speech was overlapping I used the “[“ mark. It should also be noted that these transcriptions were detailed enough to get at non-discursive communication and visual data as eye contact, posture, and movement around the room. On average, one hour of transcribed lessons took approximately 22 hours to transcribe.
Perhaps because I was painstaking in making sure I accurately captured details, the production of transcripts was labor intensive. Therefore, I transcribed only eight of the sixty-eight hours of videotaped lessons. The primary purpose of this data set was to
119 allow me to use critical discourse analysis to address my second research question, namely: what is the nature of classroom discourse and student participation in a CM classroom and how does this evolve over time? I transcribed two hours of videotaped CM activities and six hours of reform activities. Two CM lessons were selected because they were representative of the seven critical activities in which my students were discursively engaged.
As I discuss further in Chapter 7, there were five critical lessons in which students were less engaged or, essentially, disengaged. Four of these “less engaging” CM activities occurred towards the beginning of the night course and point to the fact that relationships have to be built with and between students before CM is effective. I chose six hours reform activities that “surrounded” the CM activities; that is, they occurred on the same or next day. These reform lessons were typical, hence representative of the entire corpus of reform activities captured on videotape. The six reform activities allowed me to examine my typical pedagogy and typical student reactions. By choosing activities from Week 2 and Week 7, I was able to examine how student engagement with reform mathematics changed or failed to change over the timespan of the nine-week course.
3.6.2 Participant Observation Fieldnotes
As a participant-observer, I took fieldnotes of participant actions and occurrences in the night school setting (see Emerson, Fretz, & Shaw, 1995; Spradley, 1980). As discussed earlier, although I collected ethnographic-type fieldnotes over an extended period of time, my study was not an ethnography according to Spradley’s (1980) definition, because I started the study with specific (but flexible) questions about CM that
120 I wanted to answer. I stuck to my plan of making concrete observations of the night school setting, the Guevara school facility, the neighborhood surrounding Guevara, the staff who crossed paths with us, and my students. As the night course progressed, I began to make more structured observations directed at the following classroom issues: (1) the roles of my students in particular activities, (2) value statements made by me or my students about mathematics, (3) value statements about learning and my students as learners, (4) references to political issues, (5) references to the students’ home cultures,
(6) the nature of classroom activities, (7) in-class and out-of-class conversations with students, and (8) levels of student engagement.
During the nine weeks, I continuously asked students about their lives inside and outside of school. While many of these observations and conversations were captured on videotape, others were recorded as fieldnotes. While I started with a set of research questions, I attempted to put these on hold so I could be aware of emerging themes and ground my emerging theories about the reality of the night course. There was some degree of overlap between the fieldnotes and my curriculum diary because my objective observations and subjective reflections as a teacher were mutually informing. Separating them was somewhat artificial, as Heaton (2000) points out.
3.6.3 Instructional Diary
The idea of keeping a diary as a practitioner-researcher in mathematics education came from Lampert (2001). For additional methodological guidance, I turned other researchers concerned with methodology, such as McKernan (1991), Anderson, Herr, and
Nihlen (1994), and Hobson (2000). These scholars note several purposes for a teacher
121 researcher to keep a diary. McKernan, for example, notes that such diaries can be used objectively as a log of events or as a place for the researcher to take a reflective stance on curriculum design and instructional practice. My instructional diary served primarily as a place to record reflections about issues I faced in the design and implementation of CM activities and how I felt about my students’ response to CM. Following McKernan’s suggestions, I used them to: (1) make daily entries in the diary, (2) cross-reference diary entries to accounts presented in other venues, and (3) record both “facts” and interpretive accounts. I followed McKernan in describing both negative and positive features of being a teacher-designer.
While I optimistically intended to use the diary as a textual space to document solutions to the problems I encountered in the design and implementation of CM materials, it quickly became a place where I wrote about my evolving beliefs about secondary mathematics, critical education, and the characteristics of non-specialist mathematics courses in low-income urban schools. More specifically, I documented: (1) curricular and instructional decisions I made for planning purposes (i.e., plan better lessons), (2) objective structural constraints (i.e., structure of secondary mathematics, nature of urban schools), (3) personal subjective constraints I faced in having never taught CM previously, and (4) my beliefs about students, mathematics, learning, and school. In a nutshell, the diary provided ample evidence of the problems I encountered that forced a reconsideration of beliefs about school mathematics and re-conceiving mathematics instruction as an emancipatory tool.
3.6.4 Formal Beliefs-Related Documents
122 I included my draft literature review and final study proposal as a data set so I could explore how my views changed before initiating and after conducting the study.
This reveals that I took the risk of commencing the study before official acceptance of the proposal. I systematically examined these documents to compare my naive pre- instruction beliefs with my subsequent more informed post-instruction perspective.
3.6.5 Inequalities & Area Unit Text
This CM unit was the primary text used for the comparative textual analysis I present in Chapter 5. It was one of two CM texts that I designed. The Inequality and Area
(I&A) unit covered the mathematical topic of area and interwove such political considerations of economic justice as U.S. income distribution and Martin Luther King’s views on socialism. There were 27 titled activities in the I&A unit, 11 being more critical than mathematical in nature. The I&A unit required eighteen forty-five minute periods of instructional time (see the I&A unit in Appendix A).
3.6.6 The Bees and Rhoad Comparison Texts
I included two comparison texts in my curriculum analysis in order to examine how the incorporation of critical sociopolitical themes altered the dominant versions of mathematics curriculum. The first text was the first half of the “Do Bees Build It Best?” unit from the standards-based reform Interactive Mathematics Program (IMP) (Fendel,
Resek, Alper, & Fraser, 2000). The second text was Chapter 15 from the Geometry for
Enjoyment and Challenge text (Rhoad, Milauskas, & Whipple, 1991). For simplicity I refer to the first text as “Bees” and the second text as “Rhoad.”
123 I included the first 52 pages of the geometry unit “Do Bees Build It Best?” because they primarily dealt with area and not volume. My I&A text did not address volume. Three Bees pages that were of an organizational nature were eliminated, making a total of 49 analyzed pages. The following is a sample of IMP Bees text:
In this unit students work on this problem: Bees store their honey in honeycombs
that consist of cells they make out of wax. What is the best design for a
honeycomb? To analyze this problem, students begin by learning about area and
the Pythagorean theorem. Then, using the Pythagorean theorem and trigonometry,
students find a formula for the area of a regular polygon with fixed perimeter and
find that the larger the number of sides, the larger the area of the polygon. (p. xi)
The Bees unit was an optimal choice for my curricular analysis for several reasons: (1) IMP is a standards-based curricula, the constructivist type of curricula that is endorsed by the National Science Foundation and by prominent mathematics educators
(NCTM, 2000; Webb, 2003); (2) I had taught using the Bees unit text multiple times and had insights into its strengths and weaknesses; (3) IMP unit texts such as Bees were used as models for the I&A unit; and (4) both the Bees and I&A texts were equivalent texts in the sense that they covered the topic of area in problem-based or applied settings. Despite these similarities, the I&A and Bees unit texts had important philosophical, structural, and mathematical differences that made the comparative textual analysis worthwhile.
Most importantly, the I&A text had a critical-sociopolitical orientation that the Bees unit did not have.
Chapter 15 from Rhoad, Milauskas, & Whipple (1991), or “Rhoad Chapter” was the second comparison text. This text is typically used with “specialist” students being
124 prepared to have the option to further specialize in academic mathematics. I identified the
Rhoad text as a specialist text by asking personnel at three of the more elite suburban high schools in the Chicago area what geometry texts they use with their highest track or most advanced students. All three schools use the Rhoad text. In fact, two of the Rhoad” authors teach at New Trier High School which is located in the wealthiest school district in Illinois. This is the affluent district that Kozol (1991) made famous in his book, Savage
Inequalities. The third author teaches at the equally elite Illinois Mathematics and
Science Academy, which, coincidentally, was the third school that I called. Unlike I&A and Bees, the Rhoad text was both structurally and philosophically traditional. It was divided into eight chapter sections plus a review section that I did not include in my textual analysis.
Figure 3.2 shows structural and instructional variation in the three texts. Because it was the longest, Bees was at an advantage in my curriculum analysis. The I&A text, on the other hand, was at a disadvantage because did not include homework. This reflected the fact Guevara did not allow teachers to assign homework. Rhoad was organized around sections instead of sets of activities as I&A and Bees. The sections in Rhoad began with a pedagogical exposition where new, but relevant, principles (i.e., theorems, definitions) and sample problems were provided. Rhoad chapters ended with problem sets in which students were to solve mathematics problems similar to the sample problems. These textual differences were taken into account in my analyses.
125 Figure 3.2. Structural Characteristics of the Three Mathematics Text
3.6.7 Student Interviews
Two colleagues from Northwestern University and I conducted individual and focus group interviews before and after the night course. Students were not required to participate in the interview process. Those who did participate in the pre-interviews were selected by the night school administrator. Those who participated in the post-interview process were those present on the last day of class with the exception of three who opted out for different reasons (one was shy, one did not want to participate, one had to register so had a time conflict). I used this mixed approach of personally collecting and having colleagues collect both individual and group interviews, in part, because I realized the benefits and drawbacks of researcher and outsider interviews as well as individual and focus group approaches (Merriam, 1998). I conducted six of the interviews and my colleagues conducted three interviews each. I assumed that a benefit to having third-party interviewers might be that students would feel more comfortable being critical of my instruction with them. Probably because I had built trusting and comfortable relationships with my students, it turned out that the students I interviewed were as open and candid about the course as the students my colleagues interviewed. The primary purpose of the pre-interview was to understand my students’ past experiences in mathematics courses to compare with their descriptions of students’ perceptions of CM materials and instruction used in the night course. The student perspectives captured in interviews added depth and clarity to the discourse and curriculum analyses. Pre- and post- interview protocols are in
Appendix E.
126
3.6.8 Students’ Written Coursework
I provided the students in the night course with binders and told them to keep their course materials in these binders unless they had a better system– all chose to use my binders. I collected the binders at the end of each week to see how students were doing and communicate with them. I gave students the option of keeping their binders or giving them to me at the end of the course. Most gave me their binders. I photocopied the work in the binders of the ten students who chose to keep theirs. Three types of activities were included in the binders: (1) CM activities in which students typically wrote a one to three paragraph-long response to sociopolitical prompts, (2) reform activities that required mathematical explanations of one to three sentences, and (3) a dozen worksheets that allowed my students to rehearse the procedures and applications of concepts they were learning in reform and CM activities. Indeed, the instructional rationale behind including the critical and reform assignments was to provide my students the opportunity, as members of a learning community and as critically conscious members of U.S. society, to become more self-reflective learners. My inspection of these assignments helped me to understand my students as learners and better tailor my instruction. While there was considerable variation in the quality and quantity of students’ written work, many shared what I consider to be very personal information about their lives (e.g., gang membership, family/relationship problems, concerns about drug and alcohol usage). I believe that reading students’ work and responding to their critical and mathematical ideas helped me build stronger relationships with them than I would have had I not included these assignments. Finally, the written responses were valuable to me as a researcher; like
127 interviews, they provided insights about what my students thought about mathematics, school, and their lives outside of school.
3.6.9 Pre- and Post-Assessments
I administered a mathematic assessment at the beginning and end of the night course. These “exams” took about a half an hour to complete. The problems were created based on the skills and problem-solving components of tests that CPS used to assess geometry learning in 2000. Although this test is no longer being used, it was a relatively effective measure of what the students would be learning because, although the test has been discontinued, the required content standards for CPS Geometry had not changed at the time of my study. (Note that I devised a critical problem-solving component as a measure for the summer study that was not included in the night school study.) The assessments and assessment scores are in Appendix F.
3.6.10 Post-Activity Data Management
Following the advice of one of my dissertation chairs, I created an excel file that allowed me to inventory and keep track all of the data I collected in the night course. I created the file a week after the night course was over while the course and data collection were still fresh in my mind. Up to that point, I had electronic (digital) data that was stored on my computer, hard data (student work, printouts of fieldnotes), and a set of
CDs and Digital Hi8 tapes of the videotaped instruction. At the same time, I backed up the data and put it in forms that were easier for me to access. For example, I made a digitized copy of the 68-hours of videotaped lessons from the night course. I burnt these
128 onto compact discs. One hour of videotaped instruction fit on each disc. With help from an assistant, I transcribed the audiotaped recordings of the twelve student interviews from the night course. Because I had to prepare for the summer course, I did not have time immediately do a full analysis of the night course data. When I came back to the data six months later, however, I had gained a somewhat more distanced perspective. I saw more good in the instruction and whole class discussions than I had previously.
3.7 Data Analysis: Analytical Frames and Techniques
I conducted three primary analyses for my study, namely: (1) a self-analysis of my approach to the design of CM materials and my beliefs related to being the teacher- designer-researcher, (2) a comparative textual analysis of my I&A unit with the Bees and
Rhoads texts, and (3) an analysis of classroom discourse and participation from transcriptions of ten hours of videotaped lessons from the night course. In the next sections I provide detailed descriptions of the three primary analyses. I begin with the theoretical ideas that framed each analysis and then move to the particular techniques used to produce the results from my data analyses. I then summarize analyses of my fieldnotes and student work that I used for triangulation and thick description, although this is done in somewhat less detail.
3.8 Self-Analysis of My Critical Instructional Design Process and Related Beliefs as a Teacher-Designer
In this section I outline my approach to the analysis of my instructional diary and beliefs-related documents. I began my self-analysis by focusing on my curriculum diary,
129 however, once I realized I that it was important to describe the evolution of my beliefs about CM, I expanded this analysis to include my draft literature review and my subsequent literature review for the final proposal. For purposes of triangulation and adding depth, I also include the findings of my analyses of student interviews and written work samples.
3.8.1 Situating the Self-Analysis of Curriculum Design and Beliefs
My self-analysis resonates with current discourses about the importance of teacher knowledge of, and beliefs about, effective mathematics instruction and standards- based reform (Ball & Cohen, 1996; Lerman, 2002; Lloyd, 1999, 2002; Kloosterman,
2002; Sherin & van Es, 2003, 2005; Wilson & Cooney, 2002). As mentioned previously,
I follow the lead of several mathematics education scholars who have researched their own teaching (Ball, 1993; Chazan, 2000; Gutstein, 2003, 2005; Lampert, 2001;
Magidson, 2002; Romagnano, 1994; Schoenfeld, 1994). My self-analysis resonates with work by scholars who argue that effective mathematics instruction for African American and Latino/a students requires different knowledge, beliefs, and orientation to students than many white teachers have (Gutierrez, 2002; Gutstein, Lipman, Hernández, & de los
Reyes, 1997; Ladson-Billings, 1997, 1995; Martin, 2000; Moses & Cobb, 2001).
Zevenbergen (2003) shows the importance of teachers’ beliefs about working class students in shaping what teachers think is possible in their mathematics classrooms.
Sleeter (1997) similarly argues that the majority of the mathematics teachers she worked with in a professional development program failed to create a more culturally relevant
130 mathematics curriculum because these teachers failed to see mathematics as cultural (or socially constructed) and were therefore unable to humanize mathematics and make it relevant to students.
3.8.2 Iterative, Inductive Coding of My Curriculum Diary and Proposal Material
As stated earlier, I primarily used the following material or data sets in my self- analysis: (1) my instructional diary, (2) the first draft of my proposal literature review, and (3) an edited version of the literature review. For simplicity, I refer to these three documents collectively as “beliefs-related documents.” I included examinations of student interviews and collected student work to challenge or support my findings and interpretations as a method of triangulation for credibility purposes.
The analysis of my instructional design and implementation, and teacher beliefs
(reported in Chapter 4) began with an inductive and iterative examination of the instructional diary that I kept over the course of my study. While I read it over several times before teaching the summer course at Park Vista, I conducted a rigorous analysis of the diary as soon as I finished the summer school teaching. I began by first reading this approximately 200 single-spaced page document in its entirety. I did this to get a good sense of the whole document. I then went through the diary a second time and began coding each sentence with one or two short descriptors (e.g., real world context, tracking, design issue, belief about students). I often created a descriptor by quoting verbatim from the sentence under consideration. This open-coding process essentially produced a list of several hundred descriptors, which was obviously too much for a set of codes that might help me get a handle on what was in my diary. I therefore condensed this list by sorting
131 and collapsing these descriptors into twenty or so general categories (e.g., design, mathematics, student-action, student-statement).
I went through my curriculum diary a third time with this new, condensed, list of codes. I noted two things at this point: (1) inconsistencies within this set of codes or my application of them, and (2) that the codes failed to capture some themes or did not distinguish between issues. Indeed, I combined some of the first round of codes into a single category while eliminating others altogether because they overlapped. I also created new, more nuanced, categories to capture different meanings that the first set of codes failed include. I settled on the following general categories: (1) My Beliefs &
Identity, (2) Race/Class, (3) Discipline & Power, (4) Curricular Relevance, (5) Student
Stances (Resistance, Conformity, & Agency), (6) CM Design Issues, (7) CM
Implementation Issues, (8) Teacher Student Relations & Interactions, (9) Critical and
Constructivist Educational Theory, (10) Context (police presence at Guevara), (6) Power
(e.g., authority as a teacher, power of school and state standards, authority of such disciplines as mathematics), (11) Student Positioning (statements I made about students),
(12) Mathematics, and (13) Other. With this finalized topical categories, I was able to sort data in my beliefs documents into relatively exclusive and meaningful categories.
There were, of course, statements that might have been placed in more than one category.
The various iterative passes at coding gave me a deeper understanding of my curriculum diary than I had after only skimming through it. And, I had a useful set of categories for closed coding.
I recoded the entire diary once more with these finalized categories and began to produce a larger set of reflective memos (Emerson, Fretz, & Shaw, 1995). To be clear, I
132 had already produced several memos, however, I now was more purposeful about the activity. Specifically, I wrote one set of theoretical memos for each code. Some reflective memos came close to being ten pages long (e.g., beliefs/feelings, design) while others were shorter (e.g., mathematics). The closed-coding and memo writing process allowed me to clarify my subjective beliefs about critical mathematics, and non-specialist courses, and the evolution of these beliefs over time. The story about my beliefs and their evolution was as important to the study as the objective design-implementation-redesign process I set out to describe in the curriculum diary. I had not realized this fully before I completed the final, exhaustive coding process. Indeed, when I conferred with Professor
Sherin about my study’s progress, she suggested that I write up the analysis of my curriculum diary as a self-analysis of my beliefs as a teacher-designer rather than as a set of distanced and concrete observations about patterns in my approach to curriculum design.
I put this self-analysis on hold for seven or eight months, while I conducted my textual and discourse analyses. As depicted in Chapter 4, the textual analysis resulted in a second shift in my beliefs about CM (see Figure 4.1); specifically, at this final
(analytical) stage, I had come to question the potential of constructing a critical version of secondary mathematics because I worried that it would not be empowering in terms of their learning mathematics as principled cultural or educational capital (Bourdieu, 1977,
1984; Dowling, 1998). In particular, I worried that the utilitarianism (i.e., real world relevance) of reform mathematics and the political focus inherent in CM distracted student attention away from a disciplined and disciplinary study of secondary
133 mathematics. Together, my analyses of my curriculum diary and textual analysis indicate that the story of the evolution in my beliefs about CM was a central to my study.
After six months, when I returned to analyze my instructional diary, I was quite interested in the belief-related statements and how they might or might not have changed.
I used the closed coding described previously to separate belief-related statements into the following sub-categories: (1) utility and relevance of secondary mathematics, (2) critical education or theory, (3) students, (4) how empowerment could be integrated into non-specialist mathematics curriculum (critical-utilitarian, discipline-centered, student- centered, exam-centered), (5) urban schooling, and (6) the nature of non-specialist courses. At this time it became clear that my beliefs about CM had remained stable while
I taught the summer school course at Park Vista. This meant it was important to focus on the course at Guevara to document the evolution of my own beliefs.
The examination of beliefs led to a search for additional documentation beside that in my instructional diary. A comparison revealed that I had added more than a dozen sentence- to paragraph-long qualifications to the first draft of the proposal literature review after the night course was completed. These questions and qualifications pointed to two shifts in my (stated) beliefs, namely: (1) the utility of secondary mathematics and its potential to address critical sociopolitical concerns; and (2) the nature of an empowering secondary mathematics course for low-income urban students in remedial or non-specialist courses. While these shifts were also exhibited in the instructional diary, they were not as transparent as in the proposals’ literature reviews. After conferring with
Professor Sherin, I included these documents in my self-analysis to both support and challenge the evolution of beliefs that I constructed based on the instructional diary.
134
3.9 The Comparative Textual Analysis
My textual analysis, unlike my self-analysis, involved a deductive analysis of my
CM curriculum (see Appendices A & B). In particular, I used the theoretical “language of description” developed by Dowling (1998) to analyze the I&A, Bees, and Rhoad texts and contrast them with each other. Dowling’s sociological language of description offered the vocabulary and techniques to study the effects of incorporating critical themes into traditional and reform curricular materials. In addition to providing a framework for analyses, Dowling’s book influenced my thinking about the meaning of educational equity.
3.9.1 Dowling’s Sociological Language of Description
My aim in this section is to faithfully present the aspects of Dowling’s (1998) theoretical language of description that emerged as salient in my comparative textual analysis. Note that I do not present his entire theory here but only what is relevant to my work and findings.
There are three levels to Dowling’s analytical framework, namely, the structural, the textual, and the textual resource. The structural level essentially concerns human activity and its social organization. The textual level concerns representations of structure in texts. The textual resource level concerns the use of textual objects (e.g., graphs, alpha- numeric text) in texts. There is interaction between these levels; while mathematics textbooks comprise a part of the textual level, they are also reproduced by taking part in organizing activity at the structural level. Dowling describes the dialectical relationship
135 between the social activity at the structural level and representations of it at the textual level as follows:
In simple terms, I shall propose that the social comprises relations and practices
which are organized as empirically definable ‘activities’. School mathematics is
describable as an example of such an activity. An activity is a structure of
relations and practices which, essentially, regulates who can say/do what. It
constitutes positions which are always hierarchical (although not necessarily
simply hierarchical). The practices of the activity are distributed within this
hierarchy. Activities are produced by and reproduced in human subjects – who
move, routinely, between activities – and by texts. I shall try to signify the
dialectical nature of production/reproduction through the use of the expression,
(re)production. If texts are to (re)produce activities, then they must establish
textual positions (voices) with respect to each other and distribute textual
practices (messages) in relation to the structure of positions. Texts must constitute
positioning and distributing strategies. (p. 20)
As indicated above, Dowling splits the textual level, considering messages and voices on the one hand and textual resources (images, alpha-numeric texts) on the other.
3.9.2 Dowling’s Structural Level
The structural level, the first level of Dowling’s language of description, concerns human activity; that is, both social practices and the hierarchical positions occupied by people within social practices. While structural level activity can be represented in texts,
136 activity is not realized within mathematics texts and “practices and positions, which are features of the structural level are to be inferred from the [textual] analysis” (p. 142).
However, because mathematical activity is represented and structured in mathematics texts (e.g., via math tasks), this, in turn, (re)produces activity at the structural level, such as that taking place in mathematics classrooms.
Structural level practices regulate the hierarchical positions people occupy within human activity. Dowling names four educative positions that people can occupy within a social practice, namely: the subject, apprentice, dependent, and objectified positions. In the case of mathematics, the subject position is occupied by mathematicians, whereas students can occupy the other three positions. Concerning my analyses, classroom practices and texts either serve to “apprentice” mathematics students towards maximum subjectivity (i.e., specialist preparation in academic mathematics, preparation for math- related fields) or create “dependency” by positioning them in an inferior, objectified, or even object, status with respect to academic or disciplinary mathematics. In Chapters 4 and 5, I discuss subjectification of students as mathematical empowerment with respect to academic mathematics. Following Dowling, I claim that if mathematical principles are not made explicit and illustrated connectively (or conceptually) in pedagogical discourse, learners who may believe they are being apprenticed into academic mathematics (being prepared for college mathematics) are disempowered by diluted or distracting content.
Mathematics texts take part in creating subject positions at the structural level in mathematics classrooms by regulating “who can say what.” Of course, mathematics texts are only part of a larger educational system that regulates what students can say and do.
Mathematical texts do take part in regulating the content (what is said), its expression
137
(how it is said), and who can say it. As Figure 3.3 illustrates, activities at the structural
level, and those represented in mathematics texts, vary with respect to the academic
nature of mathematics content and expression. Dowling names four domains of practice
that describe the potential variation in mathematical activity and its (re)presentation in
texts, namely: the esoteric, expressive, descriptive, and public. Mathematical practices in
the esoteric domain are the strongest in academic content and expression, whereas those
in the public domain are the weakest.
content(signified) strong classification weak classification
Esoteric (High Discursive Descriptive
classification Saturation) (signifiers) strong expression Expressive Public classification weak
Figure 3.3 The Domains of School Mathematics Practice
138 The esoteric domain includes apprenticing learners into academic mathematics.
Both the content and its expression in esoteric domain activities are classified as having a strong disciplinary orientation, they are recognizable as (preparation for) disciplinary mathematics. In Dowling’s words, “the esoteric domain may be regarded as the regulating domain of an activity in relation to its practices” (p. 135, my emphasis).
Academic mathematics is the regulating domain for school mathematics.
The esoteric domain includes practices in which the mathematical apprentice
(e.g., high school algebra student) is taught how to solve equations such as 0.4x – 4.6 = -
2.0 (cited by Dowling, p. 133) and is shown its larger disciplinary purposes.
Mathematical activity in this domain is largely self-referential, references to the material and social world are downplayed.
Esoteric domain activity is also characterized by (high) discursive saturation.
Dowling explains, “[t]he crucial distinction between practices exhibiting high and low discursive saturation is the extent to which their regulating principles are realizable within discourse” (p. 138). Indeed, activities that exhibit low discursive saturation are those “that are commonly (although not necessarily) referred to as ‘manual’ [hands on] activities” (p. 139). Disciplinary mathematics is not a manual activity, instead it is regulated by the discourse of specialist practitioners. The principles (e.g., theorems, definitions, postulates, less formal conventions) that regulate disciplinary mathematics are realized in discourse that is transmitted to the apprentice in the apprenticeship.
At the secondary level, activities in texts used with elite students in college preparatory courses are in the esoteric domain. Such texts necessarily (re)produce discursively saturated esoteric domain activity because that is where, “the principles [and
139 discourses] which regulate the practices of the [mathematical] activity can obtain their full expression” (p. 134). In contrast, activities in texts used in dead-end courses for students who take basic or life-skills math fail to exhibit discursive saturation and generally fall into the expressive, descriptive, and public domains of practice.
Activity in the expressive domain of practice presents rigorous (academically- oriented) mathematical content but expresses this content in a language that mathematicians would not use, such as the terminology used in carpentry. As a result, such activities limit access to principled mathematical discourse. The “Rectangles Are
Boring” activity from IMP (Fendel, Resek, Alper, & Fraser, 2000) is representative:
Rancher Gonzales’s nephew Juan has appeared on the scene just as she has
decided to build a square corral. Juan thinks this is a totally boring idea. “Why
always rectangles?” he asks. “Why not be different? How about a triangle?
1. If Rancher Gonzales uses her 300 feet of fencing to build a corral in the shape
of an equilateral triangle, what will the area be?
2. How does that result compare to building a square corral with the 300 feet of
fencing? (Year 2 IMP Text, p. 242)
While clearly the mathematical content in this activity would be rigorous for many secondary students, its expression is in terms of a (contrived) real world problem. The real world concerns of fictional people motivate the need for the mathematics. In other words, it is not academic mathematics that motivates students but the “real world.” While the mathematics at hand might be engaging and meaningful, the real world expression limits access to disciplinary discourses, principles, and connections. In such activity, the relatively arbitrary disciplinary expressions and mathematical connections are essentially
140 left to students or deferred. While individual activities of this sort pose no problem to a subjectifying apprenticeship, long stretches of such activities deny access to disciplinary and college preparatory mathematics.
Structural level activities, and those represented in texts, can also fall into
Dowling’s descriptive and public domains of practice. These differ from the esoteric and expressive domains in that the content under consideration is comparatively less rigorous.
Activities in these domains are more clearly structured by everyday practices and are – like the expressive domain – discursively unsaturated. The descriptive and public domains of practice differ from each other primarily in the way the content under consideration is expressed. A textual activity in the descriptive domain uses an academically oriented language to express content that is weakly classified as academic mathematics. Dowling cites this example from the SMP 11-16 text (Book Y1, p. 32):
A café orders p white loaves and q brown loaves every day for r days. What does
each of the following expressions tell you?
(a) p + q (b) pr (c) qr (d) (p + q)r
(Dowling, 1998, p. 134)
The mathematical expressions in this example have a disciplinary aspect but the content under consideration is not rigorous or academic in orientation. In other words, the content in this activity is not structured by academic considerations and fails to apprentice the reader into academic mathematics. Instead of having a disciplinary purpose, this activity is rhetorical; its purpose seems oriented towards convincing the reader that mathematics is part of the fabric of their everyday worlds and that they need it in order to optimize their lives.
141 A textual activity in the public domain, on the other hand, presents weakly- or non-mathematical content using non-academic discourses. Dowling cites the example of a word problem that asks secondary students to compute a bill from a shopping excursion. While this activity is quantitative and computational, it does little to expose the student to the content and expressions necessary for competent academic mathematical performance. Computations in a shopping excursion bear almost no resemblance to the practices of academic mathematicians. As the Rancher Gonzales and café examples indicate, references to public domain objects such as corrals and loaves of bread in mathematics texts do not necessarily imply public domain activity. Real world objects and phenomena are often referenced in other domains of practice. It is how “real world” objects are used in such activities that matters – that is, if they are for the purposes of disciplinary preparation or not.
There is nothing entirely wrong with lessons that fall into the non-esoteric domains per se. Dowling clarifies that mathematical apprenticeship begins in the public domain, with what the apprentice understands, and works its way through the other domains in order to arrive at the esoteric domain. In terms of mathematical empowerment or subjectivity, the problem arises when the textual activity never fully arrives at the esoteric domain of practice.
3.9.3 Dowling’s Textual and Textual Resource Levels
The second level of Dowling’s framework is the textual level. According to
Dowling, this level concerns the way in which mathematical texts distribute messages and position voices. As Dowling states, there is “a weaving of textual strategies which
142 position voices and distribute messages” (p. 143). Textual voices and messages correspond with the social positions and practices at the structural level of activity. A text has identifiable authorial, reader and object voices that roughly correspond with the subject, apprentice, dependent, and object positions that exist at the structural level.
Mathematics texts distribute mathematical messages about the nature of mathematics to its reader voice at the textual level. When a text expands esoteric mathematical content using principled, saturated discourses that exhibit connective complexity, the reader’s voice is apprenticed. When the text provides limited content or unprincipled discursive access, however, it limits potential mathematical subjectivity and creates a mathematically dependent reader. Clearly, then, message distribution correlates with how readers are positioned by texts.
There is considerable variation in the discourses that texts use to accomplish the expansion of mathematical topics. Mathematics texts vary particularly with respect to how explicitly they state disciplinary principles and make disciplinary-oriented connections between, for example, theorems and sample problems. Dowling uses the term principling to describe textual discourse that makes disciplinary principles explicit and the term metonymic (or metonymy) to describe textual discourses that exhibit connective complexity. A text that goes beyond stating principles and procedures by illustrating and interconnecting them exhibits principled metonymy. Alternatively, a text that states a procedure such as the division of fractions simply as “turn upside down and multiply” without providing conceptual rationale would not.
In addition to differences in the nature of textual discourses, mathematics texts differ in their expansion of mathematical content. Dowling notes that mathematics texts
143 vary with respect to how intensively and extensively they develop particular mathematical topics (e.g., area, solving linear equations). Of course, mathematics texts might also distribute non-mathematical (e.g., historical, political) content as is clear in the
Bees and I&A units.
The textual resource level is the third level of Dowling’s framework. It concerns the signifying mode that a text uses to get its messages (e.g., mathematical, public domain) across. Dowling divides the resource level into three modes of signification: (1) the iconic mode (e.g., photographs, cartoon images); (2) the indexical mode (e.g., graphs, charts); and (3) the symbolic mode (i.e., “alpha-numeric” script). The “signifying mode describes a form of the relationship between expression and content that is implicated in sign production” (p. 151, author’s emphasis). Moreover, the signifying mode used is a good indicator of the type of mathematical activity being (re)produced by the text. One would expect discursively saturated esoteric domain activity, for example, to foreground the symbolic mode over the iconic mode and public domain activity to do the opposite. In terms of my analysis, I was primarily concerned with what the I&A text foregrounds and how it distributed esoteric domain messages to the I&A reader.
Finally, it should be noted that the three levels of Dowling’s framework (i.e., structural, textual, resource) interact as the previous paragraph indicates. Again, structural level activity is (re)produced by texts (the textual level) and vice versa. The resource level is likewise dialectically determined by but also shapes textual and structural level phenomena. Hence, findings at each level would be expected to support each other.
144 3.9.4 My Methodological Approach to the Comparative Textual Analysis
The first dimension to my comparative textual analysis concerned the
(re)production of the structural activity (i.e., esoteric, expressive, descriptive, public) in the three curricular texts. In particular, I used individual unit activities and chapter sections as my unit of analysis for this “domain message” analysis. I coded particular activities in the I&A and Bees units and chapter sections in Rhoad as (re)producing one of the four domains of practice by looking at both the content under consideration (i.e., whether disciplinary-focused) and its expression (whether disciplinary-focused).
The second dimension to my comparative textual analysis concerned the nature of the mathematical discourses distributed in the three texts. I essentially examined how the three texts used textual discourse to accomplish pedagogic goals. To do this I looked at the presence or absence of Dowling’s ideas of discursive saturation and principled discourse. First, recognizing that the three texts were roughly equivalent in terms of numbers of pages and instructional time required, I calculated the average number of phrases (mostly sentences) per page.
I then went through the texts and coded individual sentences as being either principled or not according to whether they stated mathematical definitions, theorems, and postulates. I coded individual phrases as “principled” when they explicitly developed or addressed mathematical theorems and the like and as “non-principled” when they did not. I also (double) coded within the second category of non-principled phrases as “task- oriented,” “context-oriented,” or both. As their names suggest, task-oriented phrases were concerned with setting up mathematical tasks and context-oriented phrases were concerned with setting up public domain contexts. These codes (or terms) were emergent
145 in that they were not part of Dowling’s language of description. While they triangulated with findings from the first dimension, they also allowed me to gauge the extent to which the curricular texts in this analysis exhibited the connective complexity (i.e., metonymy) discussed above. It can be noted that texts that were both non-principled and heavily task- and context-oriented failed to exhibit connective complexity precisely because the relationship between the tasks or contexts and the principles that regulate secondary mathematics was not stated clearly or fully, hence was difficult to illustrate and interconnect. Finally, I gauged connective complexity by observing whether or not the texts illustrated (e.g., by using examples) and drew connections between principles.
The third dimension to my textual analysis concerned the distribution of mathematical content in each of the texts. I used individual activities in I&A and Bees and chapter sections in Rhoad as units of analysis. I began by synthesizing two lists of
“content covered” that came from in the teachers’ guides that accompanied Bees and
Rhoad. I created this synthesized list by counting area-related topics that were the same or very similar (e.g., trapezoidal area formula) on the lists from each text. I also included area-related topics covered by one text but not the others (e.g., Pick’s Theorem in Bees).
There was a high degree of correlation between the topics covered by I&A and Bees.
This was probably because my knowledge of Bees influenced my design of I&A. All three texts focused on developing area-related content, so there was a relatively high degree of overlap in the content under consideration. After completing the synthesized list, I evaluated the extent to which the listed topics were covered both extensively and intensively in each text; that is, I counted the number of activities or chapter sections in which topics under consideration figured prominently.
146 Finally, for the purposes of triangulation and further description, I analyzed how the three texts used textual resources (e.g., graphs, charts, alpha-numeric text). This was an extension of the analysis of discursive saturation in each text. I made observations about the signifying modes (iconic, indexical, symbolic) each text used to get messages across. Dowling claims that the way texts use these communication modes is fairly easy to measure, and suggests using square centimeter grids to calculate how much a given text uses, or foregrounds, a particular signifying mode. Based on Dowling’s theory, it might be predicted that texts for privileged students use the indexical and symbolic modes differently than reform texts for “all” students. In order to get at textual differences, I measured the number of sentences per page (i.e., discursive saturation) and looked at which mode was featured in each activity/chapter in the three texts.
3.10 Analysis of Classroom Discourse and Participation
In this section I present the framework for the analysis of discourse and participation that produced the findings presented in Chapters 6 and 7. In Chapter 2, I defined discourse and discourse analysis, drawing from the work of well-known scholars.
In this chapter, I situate the importance of discourse and discourse analysis in the two disciplinary areas that informed my study, namely, mathematics education and critical education. I provide the rationale for my approach to discourse analysis and describe my approach to data collection, transcription, and analysis of the findings presented in
Chapter 6 and 7.
3.10.1 My Method for Discourse Transcription and Analysis
147 As noted previously, I had various assistants videotape almost all of the lessons I taught in the night school course. I wore a wireless microphone and set up two pzm plate microphones strategically on students’ desks in order to record their conversations as they worked in small groups. After the night course, I watched two hours of videotaped instruction from each week of the nine-week course. I was interested in how students participated in reform and critical activities over the timespan of the course and the relationship between how they participated in both types of lessons. After viewing all of the videotaped lessons, I decided to focus my discourse analysis on activities that took place in Weeks 2 and 7. I chose to analyze four hours of activity from these weeks because the first three hours of both weeks were reform and the fourth hour was critical, and also because they were far enough apart to allow me to examine changes in student response and participation.
My approach to Discourse analysis was both deductive (i.e., I used frameworks developed by others) and iterative (i.e., I developed my own frameworks as I examined the data). I first attempted to apply the ideas of Pruyn (1999), Mehan (1979), and Gee
(2005) to transcriptions of classroom discourse. A trial run at coding made it clear that an approach using a combination of ideas would lead to interesting results.
I conducted a second round of coding by systematically breaking up the transcribed videotaped discourse data from the eight lesson into topically related sets
(TRS) (Mehan, 1979). As the name suggests, a TRS coheres around a topic or theme.
When a topic under consideration changes, a new TRS begins. I next coded individual turns within TRS’s as Initiations, Responses, or Feedback (IRF) (Mehan, 1979, Pruyn,
1999). Breaking up the discourse into TRS’s and IRF’s allowed me to separate salient
148 participation structures in the classroom discourse. I was able to see, for example, who initiated new topics and when. I was able to code five general types of TRS: (1) about critical interpretations, (2) about mathematics, (3) about instructional tasks (e.g., “should we copy the figures into our notebooks?”), (4) about classroom management (e.g., “it’s time to get your binders”), (5) about Third Spaces (e.g., “students discussing a party or immigration during class or break). The “other” code was used if a TRS did not fit the five types, however, I rarely had to use it. This breakdown allowed me to examine, for example, the time spent on various classroom management or mathematical tasks.
I next used Pruyn’s (1999) framework for critical classroom discourse analysis to further code the eight lesson transcripts (see discussion of Pruyn’s ideas in Chapter 2). I coded my (teacher) utterances as related to student subjectification, objectification, or neither. Subjectification included my encouraging students to think for themselves about mathematics. Objectification was used when I shut down student ideas or indicated they were not capable of reasoning about a particular problem. I coded student utterances and communicative behavior (e.g., putting their heads down, initiating a topic) as exhibiting agency/engagement, resistance, conformity, or none of these. While Pruyn’s breakdown of critically-related themes was particularly useful in capturing ways students participated and how participation changed over time, it did not cover other important concepts. First, it allowed no way to distinguish between elaborate mathematical engagement and typical on-task type of engagement. Second, it did not capture the differing beliefs about the nature and practice of school mathematics that emerged in classroom discourse that explained students’ orientations to, and participation in, reform and critical activities.
Finally, it did not capture how students switched between different registers of speech
149 (e.g., vernacular, school-based) that I felt indicated ownership of ideas–or lack thereof-- as well as sociopolitical affiliation.
To address the first observation about engagement in mathematical activities, I
(re)coded students’ mathematical contributions as indicating “elaborate student engagement” when they measured more than two lines of text on the transcript page. This was particularly powerful in terms of measuring changes in student participation over the time in the night course. In addition, having read Gee (2005), I realized that his constructs of Discourse models and social languages would enable me to get at the second and third observations about beliefs and code-switching (see Gee’s ideas in Chapter 2). When students and I made our beliefs about school mathematics or sociopolitical matters explicit in classroom activities, I coded this as referencing Discourse models. When students switched out of using a school register of speech (e.g., it doesn’t work) and spoke using a vernacular social language (e.g., it don’t work) I coded the TRS or IRF accordingly.
I should point out that I wrote reflective memos about discourse and participation while coding the data. The memoing process allowed me to reflect on the salience and importance of findings and themes that arose around the nature of student participation in reform and critical activities.
Finally, I had to make decisions about how to present the discourse analysis findings. Because the combined length of my memos was more than thirty single-spaced pages, I decided to split my findings about reform and CM activity into two separate chapters. The question of what discourse data to select for presentation also surfaced. I chose to discuss excerpts that efficiently represented my general findings about discourse,
150 classroom power relations, equity, and the nature and quality of student participation and mathematical learning. I deliberately chose segments that did not necessarily illustrate best practice. In other words, my selections represent the themes of the study and certainly do not fall into the genre of celebration of effective teaching. I was pleased that
I had decided to do a self-study, because I could turn the critical eye on my own instruction and not on the flaws of others.
3.11 Analyses of Ethnographic Fieldnotes, Student Interviews, and Student Work
In addition, to the analysis of my instructional diary, curricular materials, and classroom activity transcripts described above, I did separate analyses (read, reread, coded the data) of ethnographic fieldnotes, student interviews, and student work collected during the night course. I call these “secondary analyses” because I did not write separate chapters based on the results, instead used these data sets to triangulate (i.e., challenge, support, enhance, validate) findings from the primary analyses, and to add descriptive power to Chapters 4-7. Fieldnote analyses centered around observation of contextual settings, including: Guevara and other CPS schools, surveillance and discipline, and the night school classrooms, staff, and students. My analyses were similar to the inductive approach to coding I did on my instructional diary (Emerson, Fretz, & Shaw, 1995).
In analyzing student interviews, I mainly focused on student statements about: themselves, society and justice, school mathematics, the night school context, and their experiences in my mathematics course. Specifically, coded statements were about students: (1) lives outside of school, (2) post-secondary ambitions, (3) sense of themselves as learners of mathematics, (4) ideas about the nature of school mathematics,
151 (5) evaluations of this and past mathematics courses, (6) political views, and (7) feelings about Guevara and the night school program. I created biographical sketches of students from their elaborations about their personal lives and their orientations to school and school mathematics (see Appendix D). Statements about the night course and previous mathematics courses were used to triangulate findings in Chapters 6 and 7, as were students’ written responses to CM activities.
In analyzing student work collected in binders and the pre- and post-tests, I focused on two things: (1) student responses to critical prompts and (2) student understanding of the core concepts I taught in the night course. These analyses helped me understand what students learned and how they felt about it (see Appendices D and F).
3.12 General Methodological Issues
In this final section, I briefly discuss general concerns about my study, in particular, and with qualitative design more generally.
3.12.1 Researcher Role and Positionality
In qualitative research, the researcher is the primary instrument (Emerson, Fretz,
& Shaw, 1995). On this topic, Eisner (1991) remarks:
The self is the instrument that engages the situation and makes sense of it. It is the
ability to see and interpret significant aspects. It is this characteristic that provides
unique, personal insight into the experience under study. (p. 33 in Eisner, cited in
Piantanida & Garman, 1999, p. 140)
152 My role in the study was more expansive than typical for qualitative work. I not only studied a classroom setting, but designed and implemented the reform and CM curriculum. In some ways these instructional roles competed with the observational roles, because as a teacher my instructional obligations came first. I got around this problem by using assistants to make video recordings of my instruction and the classroom dynamics throughout the course. These recordings allowed me to stand back and observe classroom proceedings after instruction was over. The transcripts of auditory discourse and visual activity provided the opportunity for thorough analysis of the findings. I also took brief notes immediately following class and used these to type up more extensive electronic fieldnotes after instruction each evening. The collected student work and post-course interviews with students gave me another window into the student perceptions of my instruction and CM more broadly.
3.12.2 Dealing with Practitioner-Researcher Bias
Of course, my own biases figure prominently in the research. In a school system where students are sorted along racial and social class lines (Lipman, 2004; Oakes, 2005), attempting to assume a politically neutral or unbiased research stance would be problematic. Given the history of race and social class relations in this country, some level of racism and class bias is inescapable. My own biases had to be accounted for in my research for it to be trustworthy. The positions of (White male) teacher and researcher are privileged societal positions based on unequal social class, race, and gender relations.
My approach was not to try to rid myself of my biases, but rather to become more fully
153 aware of my own position as a teacher, researcher, and White male. Many entries in my curriculum diary elicited problematic local issues and describe my attempts to work through them. I grappled with issues, including my social class privilege, institutional authority, pacifism ( and whether to challenge students who had chosen to serve in the military to reconsider), and the advice for college to give undocumented students.
3.12.3 Consent and Ethical Concerns
I had an undergraduate assistant videotape activity beginning on the first day of the course. I was assured by the Guevara principal that this was not a problem. All of the students at Guevara had signed video release forms, which is typical in CPS. These consent forms allowed Guevara teachers to videotape their classes for instructional purposes. I distributed the (IRB) consent forms for my study to students on the fourth day of class. About half of my students were over 18 years old and were able to sign for themselves as adults. Twenty-nine students, or their parents, signed these consent forms in the next week. Four students stopped showing up after the first week and one of these had signed a consent form as had two students were dropped from the course in the second week due to attendance problems. (The first week was not the focus of my study.)
In addition to allowing me to videotape for research purposes, consent forms allowed me to interview students and analyze their work. All of the students consented to being videotaped, a few did not consent to have me analyze their in-class work, so I eliminated their work samples. When students asked about the video camera, I responded that it was for research purposes, assuring them that they were not going to be on national television or the local news.
154 Finally, I was keenly aware of ethical issues regarding teaching and researching. I realized that my authoritative position as someone who assigns grades meant that some of my students may have felt compelled to agree to be a part of my study. In the first week of the course, I told my students that they could choose not to give consent to be included in my study, and could withdraw that consent at any time during the course. Some may argue that because these students, as low income urban students of color who have likely already received an inferior education (Kozol, 1991; Lipman, 2004), that trying new pedagogical methods with them is tantamount to testing experimental medications on relatively powerless, unwitting people. In response, as stated in my proposal, I promised to abandon my critical pedagogical aims if I observed that they were doing more harm than good. Indeed, there were several occasions that I chose not to do planned CM activities and instead used reform activities for precisely this reason (see Chapter 4). On the other hand, my rationale for choosing students who had experienced past failure in mathematics was that these students were likely to benefit from a critical curriculum that, for example, allowed them to use mathematics (or data analysis) to question why they, and students like them, receive an inferior education in inferior schools with inferior resources. Ethical decisions are inherently complex and decisions about doing the right thing are never easy.
3.12.4 Credibility Considerations Related to Self-Study
Because I was interested in my own teaching, and how students experienced it, I made no claims to being a disinterested or neutral observer. The self-study approach I adopted posed a unique set of credibility problems. For example, because I hoped that my
155
CM instruction would be effective/inspiring, why should my report of results be trusted?
Due to this and related concerns, I understood, among other things, that my methods for data collection and analysis needed to rigorous and transparent so they could be scrutinized by outside observers. For the sake of validity and credibility, I often dialogued with members of my committee, other educational scholars, fellow graduate students, former teacher colleagues, and my research assistants.
3.13 Conclusion
In this chapter I discussed my methodological approach to the study and my methods of data collection and analysis. I began by addressing the rationale behind my selection of a qualitative design and decision to do practitioner-research. I next discussed the setting where my study took place and the sources of data analyzed. I outlined the three major analyses, including: (1) my instructional diary, (2) my critical mathematics curricular materials, and (3) classroom discourse and student participation. These were supplemented by fieldnotes, student interviews, and student work samples.
156
CHAPTER FOUR: TEACHER REFLECTIONS ON THE DESIGN AND
IMPLEMENTATION OF CRITICAL MATHEMATICS
In this chapter I describe my instructional design approaches to critical mathematics (CM) and how my beliefs about CM evolved as I conducted my study. As detailed in Chapter 3, the process reported in this chapter was produced primarily by inductive and iterative analyses of the instructional diary I kept while designing and teaching CM. In addition, I synthesized analyses of my literature reviews, study proposal, and CM materials to supplement the diary analyses. While I had originally intended for the instructional diary to be a textual space to document observable approaches used to develop curriculum, I also, rather unexpectedly, used it as a textual space to record my beliefs about: (1) the utility of secondary mathematics, and (2) what empowerment (i.e., critical agency, critical consciousness, learning college preparatory mathematics as cultural capital) in secondary mathematics courses should look like for “non-specialist” students in remedial or lower-track classes. I was compelled to include these reflections precisely because my attempts to design and implement critical activities for students in such classes significantly altered my beliefs about CM, and these altered beliefs cycled back into the curriculum design process. In retrospect, I admit that I had not expected my beliefs to change as significantly as they did over the course of the study.
I need to make two considerations clear before proceeding. First, my goal is to describe my experiences and thinking as a mathematics teacher who taught from a critical perspective for the first time. I am not trying to prove anything about the general nature of CM. Second, I am largely concerned with curriculum development, therefore, “in-the-
157 moment” enactment of critical pedagogy is not a focus. I discuss my instruction in-depth in Chapters 6 and 7. While there is a cost in focusing on written rather than enacted materials, my assumption is that much can be learned about CM from a close inspection of the instructional design process and resulting materials.
4.1 Overview of Chapter: Model of Design Approach and Beliefs
Figure 4.1 is a model that I produced from an analysis of my diary and other written materials (i.e., literature review, study proposal, curricular materials) to describe the evolution of the design approach and beliefs about the utility of mathematics and the nature of student empowerment during the instructional and data collection phases of my study. These came before the analytical phase. Each cell in this model represents a period of time when my approach to curriculum design and my beliefs were relatively stable.
The vertical cell walls mark the approximate times my approaches and beliefs underwent substantial shifts as documented by analyses of the textual materials. Finally, my beliefs about CM also changed and this last shift is reported at the end of this chapter.
158
Figure 4.1. Evolution of My Design Approach and Beliefs
The chapter sections that follow correspond roughly with the nine cells in Figure
4.1. I first outline the two approaches that I took to develop my CM curriculum and the rationale behind my decision to shift from the first to the second design approach. After that, I elaborate on my beliefs about mathematical utility and student empowerment as I developed and implemented my curriculum. I end with the analytical component of this study (8/15/04 – 5/15/06) and clarify how I came to see students’ utilitarian and political empowerment after thoroughly analyzing them. I conclude with a discussion of the educational literature that informed my initial beliefs about the utility of secondary mathematics and the possibility of reconceiving it as a critical literacy.
4.2 Approaches to the Design of Critical Mathematics Materials
The following two sections cover my development of the CM curricular materials during my study’s planning and instructional phases. I took different design approaches before, during, and immediately following the night course.
4.2.1 Initial Design Approach: Critical Recontextualization of Geometry
I finished a draft literature review for my study two months before the night course began. This review documents my initial conceptual and theoretical assumptions about CM prior to teaching CM in the night course. Because it also captured my early beliefs, I expanded my analysis to include my instructional diary.
159 Having taught with a standards-based reform curriculum while a high school mathematics teacher, I began my literature review by arguing that standards-based reform instruction as outlined by the NCTM (1989, 1991, 2000) was superior to traditional instruction in terms of fostering both conceptual understanding and student engagement
(Schoenfeld, 1988, 2000; Skemp, 1978; Webb, 2003). At the same time, I was skeptical – as I still am – that standards-based reforms in mathematics would have a significant effect on mathematics education in places like low-income secondary schools in Chicago
(Anyon, 1997; Apple, 1992; Kozol, 1991; Lipman, 2004). I noted that such schools tended to have less qualified mathematics teachers (Lankford, Loeb, & Wyckoff, 2002;
Oakes, 2005). I emphasized that a primary reason (published) reform curricula and instruction might fail to take hold in urban schools was because they were not in touch with the lived experiences of urban students. Drawing from Gutierrez (2002) and
Gutstein (2003), I theorized that the sociopolitical relevance of CM would make it more empowering than reform mathematics for low-income students of color. While commenting on the power of the curriculum, I also noted that students who fit my students’ demographic profile were mostly disempowered by a school system that placed them in inferior, racially segregated schools and low-track (non-specialist) mathematics courses (Ladson-Billings, 1995; Lipman, 2004; Oakes, 2005; Tate, 1997a). I believed mathematics teachers needed to make political issues that affected urban youth an explicit part of the curriculum, and argued that current texts ignore the situations of urban students’ lives.
Although I had yet to actually develop my own critical curricular materials when I wrote the draft literature review, I was oriented towards CM to tackle the curriculum I
160 expected to develop. My plan was to design CM materials by substituting politicizing contexts for the real world problems that existed in current versions of reform mathematics. A passage from my first literature review illustrates how I proposed to radicalize reform curriculum:
Though they provide a foundation, it is clear that the NCTM principles and
standards are not a sufficient guide the organization of instruction in a [critical
mathematics] classroom. … I will [therefore] attempt to draw on work by the
aforementioned [critical] mathematics educators [i.e., Frankenstein, Gutstein,
Gutierrez, Skovsmose, Tate] as well as work done by the “Rethinking Schools”
teachers (e.g., Christensen, 2000), to create a set of guidelines that help guide the
curriculum and instruction component of my study. (Original literature review,
available upon request.)
My plan was to create a wealth of CM activities, ideally offering a “classroom” where critical themes permeated the reform mathematics curriculum. Using Gutstein’s (2005) terms, the ultimate goal was to “reconceive mathematics as a critical literacy.” From my initial perspective, then, if secondary mathematics were to be refashioned as a critical literacy, a few critical activities scattered throughout a mathematics course would not suffice. On a more pragmatic level, though, even from the beginning, I realized that I only would be able to replace a fraction of the dominant geometry curriculum I was required to teach with critical activities. Still, I felt that my study would be a step in the
“right” (critical) direction, even if it did not constitute a final step. My plan was to use the night course at Guevara to pilot an initial set of CM materials, which I would redesign and extend for official use in the summer program at Park Vista (see Appendices A & B).
161 I did carry through with the plan to revise the CM unit and collect data on instruction for summer school. I focus on the night course because it was here that my beliefs about CM underwent the most change.
Following Freire (1971), initially I argued that my curriculum and instruction would be developed to the fullest extent possible in response to, as well as through collaboration with, the night course students that I had yet to meet. As I wrote:
The students should contribute to the construction of the [critical] curricular
component of the course; thus it cannot be entirely pre-planned. Once the course
has begun, a serious attempt to dialogue with students about themes from their
lives will be undertaken [by me] in the hope of avoiding superficial student input.
I will then work to “problematize” aspects of these life situations so they can
analyze them mathematically. (Draft literature review, p. 42)
In sum, then, before teaching the Guevara night course, I presented idealized sketches of approaches to critical curricular design that I expected to later take up.
After completing my draft literature review, I worked intensively for two months to infuse critical elements into the geometry I was to teach in the night course. In my instructional diary, I documented my attempts to recontextualize mathematics problems in the standards-based reform Interactive Mathematics Program (IMP) (Fendel, Resek,
Alper, & Fraser, 2000). I had taught with these materials and knew them well, however, recontextualization was more difficult than I had anticipated:
I spent most of the morning and afternoon in my office looking at IMP to see
where I might alter the problem settings to make them justice-related. The
problem … was that both Shadows (similarity) and Bees (area, volume) [unit
162 texts] are set in physical/naturalistic settings that are not easily adaptable (by me)
to the sociopolitical. … In total, I spent a good four hours examining the problem
contexts of the IMP units and individual problems (different contexts). I did not
make much progress. Perhaps I lack imagination? After that, and relatedly, I went
online and searched for political themes or images that might help me imagine
alternative [problem] contexts for Shadows and Bees. In fact, I spent at least 4
hours online searching Google and Google Images … for usable data (i.e.,
representations). … [In the end,] I did not see any clear connections between [the
images and data] I was [downloading] and adapting to IMP. Some of the themes
or topics I considered dealt with: black farm ownership in the South, toxic waste
sites near African American communities, distribution of wealth in the U. S. and
the world, and segregation in Chicago. (Instructional diary, 9/26/03, p. 5)
I encountered problems mapping representations (e.g., graphs, tables, images, stories) of “real world” data (e.g., statistical reports, news-related themes) onto “real world” problem contexts in the IMP curriculum. When I stepped away from the IMP curriculum and considered how I would meet the required standards governing geometry topics, I faced even more problems. As the above excerpt indicates, during the initial design period, I found topics of critical interest and relevance to urban students of color, however, I generally was not able figure out how real world themes might be applied meaningfully to topics in secondary mathematics.
During this time period, I made several visits to four Chicago high schools. Two were college preparatory magnet schools that served predominantly middle class urban students and some “high achieving” lower SES students. I went to these schools as part of
163 work on a research project headed by one of my dissertation chairs. The third school was
Guevara and the fourth Park Vista–both low-income neighborhood high schools that housed the night school program and the summer school program, respectively. I had taught at Park Vista prior to attending graduate school. In terms of resources and treatment of students, the stark contrast between the schools for privileged students and those for non-advantaged students unnerved me (see Anyon, 1997; Kozol, 1991, 2005;
Lipman, 2004). Although, as a former CPS teacher, I already was aware of these differences, going back and forth between the four schools reminded me of the extent to which disparities and indignities were normalized in schools in low-income urban areas.
My reaction was to spend several weeks attempting to build CM activities around the systemic educational inequities I observed. My hope was that I could construct activities that were mathematically rich and would raise students’ critical consciousness about educational access and resource distribution in the U. S. school system. In this passage I touch on my thinking as I developed activities:
There is one thing that I am using as a quasi-principle for critical [mathematics]
curricular design, namely, the closer the data is to home the better. If it’s an issue
that affects CPS [Chicago Public School] students then that’s good. … I’ve
developed two activities (sets of questions really) that have to do with data that
I’ve gotten about CPS schools – “Drop Out Data” and “Race and Recess.” The
first has to do with drop out rates at Guevara. The other has to do with the
relationship between resources and race/class in CPS schools. (Instructional
Diary, 11/12/03, p. 9)
164
I expected the “Drop Out Data” and “Race and Recess” (see Appendix B) activities to foster meaningful whole-class discussions about the institutional and economic barriers faced by my (future) night school students and their peers; that is, I believed they were politically relevant. At this time, I became increasingly concerned because I felt that so far I had failed to produce sociopolitically contextualized activities that directly addressed the geometry topics I was required to teach. CM activities such as “Drop Out
Data” and “Race and Recess” were about analyzing data of the newspaper variety and did not “cover” such topics as the Pythagorean Theorem. While I believed them to be critically engaging, these activities did little to further the development of secondary mathematics knowledge, despite my attempts to reshape and improve them. Attempts to extend this data to traditional mathematics considerations seemed forced. Fortunately, data analysis was a required topic in the geometry curriculum according to the Chicago
Academic Standards (2003-2004).
165 Figure 4.2. Drop Out Data for High School Similar to Guevara
The chart shown in Figure 4.2 was the basis for my “Drop Out Rate” activity.
(The Drop Out Rate activity is available upon request.) It is an example of one type of representation that I found online about extant data from CPS schools. As a chart, it had a mathematical aspect and it also was promising in terms of promoting a critical conversation. Because it met the data analysis requirement for Chicago geometry, I felt justified in including the activity in the to-be-implemented curriculum but I was dissatisfied with it as geometry. I also realized that the algebraic topic of exponential growth (or decay in this case) was relevant to aspects of the drop out rate phenomena, however, exponential growth was not part of the required geometry curriculum.
Moreover, because CPS students in schools like Guevara had first hand understandings of the drop out rate phenomenon, using exponential equations to “mathematize” or “model” the situation seemed unnecessary and it may not have deepened mathematical understandings as some scholars assert (Lesh & Doerr, 2003). To be clear, the fact the drop out rate phenomenon is only approximately exponential might confuse budding student understandings of exponential functions.
Despite the fact that I had official cover for the inclusion of some off-geometry activities (i.e., 10% of the required geometry curriculum was data analysis), I still was uncomfortable with the critical data analysis activities I produced in this period. My dissatisfaction came through in an instructional diary entry from October 2, 2003: “I can’t just present students with data [or data analysis]… [t]here needs to be traditional [i.e., conventional] mathematics embedded in [these] problems” (p. 5). These issues and problems notwithstanding, I remained hopeful that I could begin to resolve the tension
166 between mathematical instructional goals and critical goals. I continued to search for a design method that would allow me to produce mathematically rich CM curriculum.
However, six weeks into the design process, such methods continued to elude me as this excerpt indicates:
On October 2nd I claimed that I was using [Gutstein’s] South Central [activity] as
a model for my [CM] problems [see Brantlinger, 2005]. This is not entirely the
case. Instead, I’m getting on the web, searching for a topic, and trying to find
data, charts, or graphs to develop a worksheet or semi-scripted activity of some
sort. … Once I find promising data it’s still difficult to mold into a critical
[mathematical] activity. … I’m not really sure what my image of a good critical
activity is but it’s there in some semi-conscious realm. And, [Gutstein’s critical
activity] “South Central” certainly informs this model as do his teaching for social
justice principles [Gutstein, 2003]. Of course, so does my past teaching
experience and knowledge of IMP. … When I reflect on [the CM] materials I’ve
produced so far I see things creeping up that are not reducible to design or social
justice principles. … [For example, w]hy are two- or three-page
activity/worksheets the end-result? Why do I tend to ask 8 to 12 questions? Why
not ask fewer more open-ended questions? … Indeed, what are my preconceived
notions about “remedial” students … that are coming into play? (Instructional
Diary, 11/2/03, p. 6)
This passage makes it clear that not only did attempting to design CM materials force me to reconsider my design approach, but also preconceived notions about empowering curriculum for students in non-specialist courses. It raised such questions as: How did my
167 beliefs about low-income urban students play out in the development of my CM? Was I reading my future students’ non-elite institutional and social class status in a biased way?
Was I projecting lesser mathematics goals onto through my CM curricular alterations? I return to these questions when I discuss student empowerment later in this chapter.
In total, I spent over 120 hours developing CM materials as I planned for the night course and developed seven teachable critical activities. Each CM activity was for one
45-minute period. This calculates, on average, to approximately fifteen hours to produce one period-long critical activity. While I had anticipated difficulties, the labor-intensive nature of designing CM materials was a great surprise to me. I had designed non-critical, relatively conventional, mathematics activities as a full-time teacher and had never spent that amount of planning time for individual lessons. I conjectured that only extraordinary secondary mathematics teachers might have the time and energy to develop critically responsive materials during the school year. I also knew such materials were not commercially available and there were few examples in the CM literature.
While I was a bit disheartened at the low-rate of my pre-course productivity, I continued to develop CM materials over the span of the night course. While teaching, I developed four more period-long CM activities. I also taught four periods worth of CM materials provided by Professor Eric Gutstein from the University of Illinois at Chicago.
Given that I had 72 hours of instructional time, CM activities made up 14 hours or approximately 20 percent of the night course curriculum and reform activities from IMP made up the bulk of the remaining 80 percent. While I had not yet reconceived secondary geometry as a critical literacy, I had a foundation on which to build.
168 4.2.2 Latter Design Approach: Interweaving Separate Critical and Mathematical
Activities
I designed my critical Inequalities and Area (I&A) Project about halfway into the
Guevara night course. The development of this three-period-long project marked a fairly substantial shift in my approach to CM design. Having now taught several CM activities,
I began to understand that critical and secondary mathematical goals, understandings, and discussions were generally, if not always, only tangentially related to each other – at least as I had designed and enacted them. Due to repeatedly experiencing the disjunction between critical and mathematical goals in my lessons, I began to understand that I did not have to fully integrate critical and mathematical tasks in CM activities as long as they cohered around an interrelated set of real world issues or problem contexts. Hence, when
I designed the I&A five weeks into the night course, I left its critical non-mathematical and mathematical non-critical activities, themes, and goals relatively uncoupled. The relatively separate critical and mathematical tasks of the I&A Project cohered around two related themes: (1) the critical theme of economic justice and (2) the mathematical theme of calculating an area-related measure of economic fairness. Exploring the first did not necessarily require or further an understanding of secondary mathematics. Calculating a measure was just that – it quantified injustice but did not necessarily further students’ understanding of social justice.
I must also admit that at the time I was not yet fully conscious of the disconnect between the “real” (e.g., critical content) and the mathematical. I still believed that critical and mathematical activities, while separate, cohered around the same theme of economic justice rather than around distinctive mathematical and critical themes. It was
169 not until the analysis phase of my study that I saw this distinction more clearly. Of course, the boundary between the mathematical and non-mathematical is of fundamental importance here, and I noted that I had begun to draw a tighter boundary around mathematics. I came to see (and worried) that the distinction I was drawing conformed to the relatively rigid boundaries set by university mathematicians, policymakers, and testing and textbook companies. Developing the more bounded definition of secondary mathematics was largely in response to a vocal subset of my Guevara night school students who advocated for me to teach the mathematics they would need in college. As I discuss in Chapter 7, they seemed to think CM was generally a waste of time and not legitimately included in a mathematics course.
Not attempting to fully synthesize critical and mathematical in my activities constituted an imperfect design solution that I resorted to in order to get CM to work for me. It did not resolve the “dialectical tension” between the critical and mathematical that
Gutstein (2005) and Gutierrez (2002) discuss. In the diary entries I wrote during this second design phase, I vacillated between blaming my own shortcomings, blaming secondary mathematics, and blaming CM as it had been theorized to account for my failure to resolve that tension. I was painfully aware that I was not measuring up to the successes reported for elementary-aged students in articles by Tate (1995) and Gutstein
(2003). I suspected that doing CM at the secondary level was more difficult than it might be at the elementary level.
To clarify how I chose to deal with – perhaps avoid – the dialectical tension between the critical and mathematical in this second design phase, my goal in developing the two critical activities that opened and closed the I&A Project was to deepen my
170 students’ historical, economic, and political understandings about the increasingly polarized and unjust distribution of income in the U. S and elsewhere in the world. My goal in designing the three mathematical activities was to have students use an area- related approach to calculate a statistical measure of economic fairness referred to as the
Gini coefficient in economics. While critical and mathematical activities often followed each other, I essentially left them uncoupled. Note also that the theme of economic justice via the Gini coefficient idea for my I&A Project did not emerge from Freirean analyses of students’ lived experiences, rather from a chapter written by the statisticians and critical scholars Williams and Joseph (1993). Based on my recent exposure to disparities between schools in wealthier and poorer CPS areas, I believed the topic of economic justice was relevant to my lower-income students even if it had not emerged from a deep
Freirean analysis of their actual life situations per se.
After designing and teaching with the I&A Project in the night course, I solidified my new design approach of interweaving uncoupled critical and mathematical activities.
Rather than attempting to pack critical and mathematical themes and goals together in one task or activity, I designed longer unit texts that interwove, rather than fully synthesizing, mathematical and critical tasks. Unlike period-long activities, and two-day projects (Appendix B), these longer unit texts were to be taught over a period of a dozen or more lessons (Appendix A). To summarize, my second, or new, design approach to
CM materials was distinguished by three features: (1) longer textual formats (i.e., projects and units instead of individual activities), (2) interweaving of relatively uncoupled mathematical and critical activities around related but distinct critical and mathematical themes, and (3) an emphasis on required data analysis and measurement
171 related topics (i.e., area, volume, proportional reasoning) over what might be categorized as less utilitarian mathematics (i.e., angle sums, Pythagorean Theorem). Note that this third point is discussed later in a description of my shift to a semi-utilitarian stance on secondary mathematics. Note also that I continued to design according to these features when developing materials for the summer courses.
In addition to transforming the I&A Unit, I also developed “School Problems” and “Volume of Pollution” units from shorter activities I created for the night course. The completed School Problems unit consisted of a collection of data analysis activities related to extant conditions in CPS schools and segregation in Chicago. It incorporated the Race and Recess and Drop Out Data activities. While School Problems allowed me cover the required topic of data analysis, it did not address secondary geometry as well as
I&A had covered area. I used the I&A and School Problems unit texts in the summer school program at Park Vista School, but the Volume of Pollution unit was not finished enough to implement. Again, I do not include summer school findings in this report mainly because my design approach and beliefs did not change significantly during the summer instructional period, but were mainly intensified and reinforced.
When I analyzed my CM curriculum after teaching both courses, my beliefs about the potential of developing mathematically empowering CM for the secondary level underwent a second major shift. Before addressing this final belief shift, I first discuss how my beliefs about the real world utility of secondary mathematics and mathematical empowerment changed as a result of designing, implementing, and
(re)designing my CM materials during the instructional phase of my study.
172 4.3 Beliefs about the Real World Utility of Secondary Mathematics
The real world utility and relevance of secondary mathematics were recurring, and related, themes in my instructional diary, draft literature review, and study proposal.
To be clear about terms, I use utility as a stronger version of relevance. Real world relevance requires mathematics to map onto aspects of the non-mathematical. Utility goes further in requiring mathematics to affect change in what are apparently (or believed to be) non-academic or non-mathematical settings or domains. I focus primarily on utility here because CM as I understood it required more than real world relevance, it was aimed at school mathematics “making a political difference” in school and non-school spheres.
Discussions of mathematical utility riddle my instructional diary. My conceptions of the non-academic use-value of secondary mathematics topics informed my approaches to curriculum design and vice versa. If topics in secondary mathematics were useful in people’s everyday lives outside of school, then it would seem that critical themes could be woven into the secondary mathematics curriculum with relative ease. On the other hand, if secondary mathematics topics were not particularly useful in non-academic
(sociopolitical) contexts, then it would be virtually impossible to directly tie geometry topics to critical real world themes; such attempts would come off as forced and artificial.
My beliefs about mathematical utility underwent a fundamental shift while conducting the study, in particular: I held utilitarian beliefs at the outset of the study, semi-utilitarian beliefs midway through my study after teaching CM in the night course, and became a skeptic of secondary mathematics utilitarianism after conducting my textual analysis. I describe my utilitarianism and semi-utilitarianism in the next two sections and my current skepticism about utility at the end of the chapter.
173
4.3.1 Initial Utilitarianism or Conceptions of Secondary Mathematics as a Tool Subject
As I began the study, I could appropriately be called a mathematical utilitarian.
This is revealed in my repeated discussions of secondary mathematics as a powerful political tool subject in the draft literature review I wrote before teaching the night course at Guevara. I repeated this mantra to such an extent that one of my dissertation chairs,
Professor Carol D. Lee, wrote “utilitarian” on the cover of my literature review after reading it. This excerpt illustrates my initial utilitarianism:
[T]eachers who teach mathematics for social justice … see mathematics as a tool
that students can learn to use to (a) analyze and reflect on their situation in the
world and (b) ultimately use to transform the world.
Thus, like several CM educators in the U.S. (Frankenstein, 1989, 1991, 1995; Gutierrez,
2002; Gutstein, 2003) I conceived of secondary mathematics as a social justice tool subject. The idea was (or is) to reconceive of school mathematics parallel to Freire’s
(1971) (re)conceptualization of reading and writing literacy as a sociopolitical tool for the oppressed to “read and write the world” (Gutstein, 2005). In terms of the discussion above on curriculum design, I believed that secondary mathematics was genuinely useful in non-academic spheres and concluded that critical geometry could be developed so that students could effectively read (understand) and write (change) the world.
Another example of my stated mathematical utilitarianism was my discussion of claims by Skovsmose’s (1994) and Niss (1983) that technical-mathematical processes are increasingly used to “format” society. I argued that CM would help my low-income students read the effects of math-based technology, noting:
174
Perhaps the key notion that Skovsmose develops … is the idea that …
mathematics is woven into spheres of life (e.g. technological, educational) …
Skovsmose cites Niss (1983) who claims that those who lack such a meta-level
understanding of how mathematics works become “victims of social processes in
which mathematics is a component” (Skovsmose, 1994, 57). (Draft Literature
Review, p. 22)
I argued that students could learn the conventional mathematics curriculum while also reflecting on the math-related processes (e.g., in the computer codes of financial and educational institutions, school drop out rates) that shaped their everyday lives. If school mathematics was increasingly woven into spheres of life, then weaving “life” into secondary mathematics was also possible. I suggested that students’ understandings of some spheres of real life and school mathematics would be mutually reinforcing.
While some might consider the social utilitarianism I was putting forth about school mathematics to be extreme and overly idealistic (e.g., Dowling, 1998; Ernest,
2001; Pimm, 1995), similar discourses about the (social) utility of school mathematics pervaded the views of prominent mathematics educators at the time of my study. Social utilitarianism was not particularly new or radical. For example, in the Standards 2000
Documents, the politically mainstream National Council of Teachers of Mathematics claims that mathematics education helps people become part of the “intelligent citizenship” that knows how to use their mathematical understandings developed in school to “[vote] knowledgeably” (NCTM, 2000, p. 4). I revisit the NCTM’s utilitarian rhetoric and its effects on my curriculum and instruction later in this chapter.
175 Relatedly, and in addition to Gutstein (2003) and Gutierrez (2002), my draft literature review drew on a number of scholars who called for mathematics curricula to be more relevant to the lived experiences of students of color (D’Ambrosio, 1997, 1985;
Moses & Cobb, 2001; Secada & Bermann, 1999; Tate, 1995). For these scholars, sociopolitically relevant curricula and utilitarianism are closely related. Gutierrez (2002) described the relationship between cultural relevance and CM this way, implying that school mathematics could be a powerful tool to “take up political issues in society”:
[C]ritical mathematics is mathematics that squarely acknowledges students are
members of a society rife with issues of power and domination. It takes students'
cultural identities and builds mathematics around them in such ways that doing
mathematics necessarily takes up social and political issues in society, especially
highlighting the perspectives of marginalized groups. (p. 161)
Hence, I believe my mathematical utilitarianism was in line with scholars who discuss politically and culturally relevant mathematics pedagogy. With these theories and justifications in mind, I proposed to design a geometry curriculum to empower students mathematically, politically, culturally (with the inclusion of culturally relevant themes), and in terms of dispositions (e.g., agency, engagement, interest). I believed the emphasis that my critical curriculum would place on sociopolitical utility would support, rather than distract from, academic mathematical preparation. I was convinced that, if done right, CM would help students see the relevance and power of the geometry required for graduation. Following Gutstein (2003) and Gutierrez (2002), I argued that, in perceiving mathematics as relevant and politically useful, students might be more motivated to learn academic mathematics.
176
4.3.2 My Shift to a Semi-Utilitarian Stance
During the six weeks of designing materials for the night course, I had given consideration to each geometry topic I was required to teach, including: similarity and proportional reasoning, congruence, properties of polygons, measurement topics such as area and volume, Pythagorean theorem, and right triangle trigonometry. I also was to integrate algebra when possible. The IMP curricular materials allowed me to cover many of these topics powerfully and in what appeared to be highly contextualized and generally complex set of real world problems. In addition to content standards, IMP also helped me address the “reasoning” and “process” standards put forth by the NCTM (2000). At the end of the design phase, I had separated the required secondary geometry curriculum into topics I categorized as utilitarian and those that were not. From my perspective, spatial measurement and similarity were far more useful in the material world, but not necessarily in the social world, than topics such as the Pythagorean Theorem and angle sums of polygons.
Teaching a class of students who tended not to embrace secondary mathematics reinforced my perception that a lot of what I had to teach was not seen by students as relevant to their lives. Hearing some students critique mathematics before I had been given a chance to “reach” them was also frustrating. In the first week, Leroy wrote:
One of the thoughts I think about when I’m in school while the teacher is teaching
is “man, when is this class over with,” “why is this class so boring,” and ”what
does this have to do with me or does for me in my near future?” I just think some
177 subjects the teacher teaches does not relate to us in our life. Not all, but some.
(Student Work, November 19, 2003)
Even though he did fairly well on the pre-assessments I gave my class, Leroy did not come willingly to secondary mathematics (Appendix F). As Leroy made clear, many students feel the required curriculum is not relevant to their lives. Apparently, the burden then falls on secondary mathematics teachers to relate the subject to students’ lives.
Public perceptions are that it is entirely possible to make mathematics relevant so that students will engage with it. They also believe this relevance comes at no disciplinary cost. Popular policy and educational discourses imply that knowledgeable mathematics teachers can show the relevance, importance, and utility of secondary mathematics. When students disengage, the argument goes, it is at least partly because texts and teachers fail to make it relevant (see Chazan, 1996). With these messages floating through my head, I worried what my struggles making mathematics relevant meant about me, a teacher with a Masters in Mathematics working on his doctorate in mathematics education. Written a week into teaching the night course, this diary entry captures this very point:
[T]here seems to be a consensus among some of the night school students that a
lot of what they are learning in this class and in math classes in general (a) is not
very interesting and (b) not particularly useful. … My problem as a teacher is that
I have the same issues with much of school mathematics. … I do think some of
the ideas we’ll cover should be taught as long as we’re going to require several
years of [secondary] math for every student. Area and proportions seem
particularly useful or relevant in people’s lives outside of school. But other topics
are more problematic, such as angle measures and the sum of the angles of a
178 triangle. How do I turn the properties of triangles into an opportunity to think
critically about the world? … Area, on the other hand, would seem to lend itself
more readily to a political contextualization. (Instructional Diary, 11/23/03, p. 20)
As this excerpt indicates, early (failed) attempts at CM design compelled me to draw distinctions between geometry topics that were or were not based on everyday utility.
Hence, I had already transitioned from my original utilitarianism to a semi-utilitarian stance on secondary mathematics. As might be expected, my semi-utilitarianism fed back into my development of newer CM materials; as noted above, I began to focus my design efforts only on geometry topics that seemed to have leverage in the material or social world (i.e., measurement topics, proportional reasoning).
The above passage makes it clear that I link utility with student interest: if students see mathematics as useful, they will be more interested in it. Pimm (1995) claims that many mathematics educators make this link reflexively and rather unreflectively. While acknowledging that many students see secondary mathematics as irrelevant (Boaler, 1997), I increasingly wondered whether placing a heavier stress on real world problem contexts was a simple, cost-free, solution to engagement and understanding for “all” students (see Lubienski, 2000; Wiliam, Bartholomew, & Reay,
2004). I wondered whether upping the realness or authenticity of problem contexts would translate directly into students’ increased engagement and improved understanding as some imply (e.g., Chazan, 1996; Edelson, 2001; Lesh & Doerr, 2003; Newman, Secada,
& Wehlage, 1995). In my instructional diary, I began to ponder such questions as: Is repackaging secondary mathematics something needed to get “all” students to engage with it? If repackaged, is secondary mathematics what is being engaged with? I began to
179 think that the curriculum could only be relevant and engaging for students learning college preparatory material. I also knew it was difficult to convince students to see long- term life benefits from taking mathematics, no matter how it was packaged. In the diary, as in the literature review, I discussed larger sociological and economic forces (e.g., tracking, family wealth, racial isolation, unclear or unattainable post-secondary plans, opportunity structures) that shape student interest in secondary mathematics. Perhaps because they are seen as beyond teachers’ control, these forces are rarely discussed in the mathematics education literature.
Another issue emerged once I began teaching that triggered me to confront my initial utilitarian beliefs about secondary mathematics. As I taught using the first set of four or five CM activities, I noticed that students focused on the critical real world theme
(correlation between students’ race and recess opportunity) without fully understanding the real world data in a mathematical sense. This happened in several CM activities as I show in Chapter 7. Perhaps the critical real world problem contexts were interesting enough that students chose to bypass the mathematics at hand. Conversations stemming from real world data essentially supplanted the math-related discussions I attempted to have (rather forcefully at times). Unfortunately, at the time, I was generally unaware of a substantial body of literature that indicates overdeveloped or overly “real” problem contexts can interfere with learning, or assessing knowledge of, school mathematics
(Boaler, 1993, 2003; Lerman & Zevenbergen, 2004; Maier, 1991; Taylor, 1989; William,
1990). Lerman and Zevenbergen build on work by Bernstein (2000) to suggest that realistic seeming mathematical contexts might pose particular difficulties for working class students who tend interpret them as real situations rather than as game-like
180 situations from which mathematics should be abstracted. While the argument may be a bit too reductionist in terms of social class, it resonates with my experiences teaching
CM.
Having taken several courses on the “design of learning environments” while in graduate school at Northwestern University, I was also painfully aware of the fact that my initial approach to critical curriculum design contrasted with the more theoretical and systematic approaches to curriculum design described in the educational literature (e.g.,
Edelson, 2001; Wiggins & McTighe, 1998). I reviewed a draft version of Edelson’s
(2001) “Learning for Use” curriculum design model during this initial design period. I hoped that Edelson’s framework would help me devise a more systematic approach to the design of CM materials. In my instructional diary I noted that this framework might be particularly helpful to me given the emphasis that it placed on the real world utility of school content such as mathematics. As Edelson (1999, draft) states, “the first step in designing activities that create a natural demand for content objectives is to identify tasks for which the knowledge is useful” (p. 14). After carefully considering the critical potential of each geometry topic, I concluded, perhaps mistakenly, that many secondary topics were simply not useful outside of the discipline of mathematics. I reasoned that non-utilitarian topics were included in the secondary curriculum for disciplinary rather than utilitarian purposes. I believed it was possible that they could not be refashioned into useful real world tools as Edelson’s framework suggested. In hindsight, I believe that such design models fail to adequately address the degree of relationship between disciplinary and utilitarian content. These models appear to be based on the assumption that many, if not all, required disciplinary topics transfer or relate to fields outside of the
181 disciplines in which they were developed. To be fair, Edelson is largely concerned with science education and it may be simpler to find natural contexts and motivations for science than in secondary mathematics.
Immediately after teaching the night school course, I remained a semi-utilitarian in that I continued to believe that the dialectical tension between the critical and mathematical could be partially, if not fully, resolved. The following part of a “note to students” prefaces the final version of I&A instantiates my semi-utilitarianism:
Many of you have probably been asking your mathematics teachers ‘when are we
ever going to use this stuff?’ for years now. The truth is that many of us math
teachers ask ourselves the same question. Unfortunately – or fortunately
depending on your perspective – very few people will use high school
mathematics on the job or in other real life situations outside of school. And, if
they do use mathematics on the job, they can almost always learn it there.
This unit is an attempt to rethink what mathematics we teach and how we
teach it. Rather than asking you to solve “bogus” word problems, you will be
asked to use mathematics as a tool to investigate our economic system.
Ultimately, we want to think about ways we can improve it to create a more just
world. As such, you will be expected to do things that go beyond what you
typically think of as mathematics. Namely you will be asked to write, talk, argue,
and, most importantly, think critically. And, again, you will learn mathematics
along the way. (Inequalities & Area, p. 2)
Interestingly, in this passage, my semi-utilitarianism is contradicted by other statements about the lack of utility of secondary mathematics. “Very few people will use high school
182 mathematics… outside of school” reveals an anti-utilitarian perspective. In the next paragraph, I took a utilitarian stance in asserting that in the I&A readers would “be asked to use mathematics as a tool to investigate our economic system,” thus reversing my position to state that secondary mathematics could, in fact, be useful in raisings critical consciousness about matters outside of school. The promise that one can “investigate” sociopolitical topics and still learn required secondary mathematics “along the way” is significant. In Chapter 5, I show that, in the case of my CM materials at least, this claim is not entirely accurate.
In addition to redesigning the CM instructional materials after finishing teaching the night course and before starting the summer courses, I wrote the second draft of my dissertation proposal, including an edited version of the draft literature review written several months before the night course. While the draft version reflected my initial utilitarian beliefs about CM, the edited version captured my new semi-utilitarian beliefs.
When, I compared these drafts for purposes of triangulation, I found I had added over a dozen new questions and qualifications, ranging in length from one sentence to one paragraph. These pointed to my concerns about mathematical utility and the nature of empowering mathematics curriculum. For example, I added a qualification (in italics) to the original discussion of Skovsmose (1994):
Perhaps the key notion that Skovsmose develops for TMSJ educators is the idea
that they should help students see how math is woven into spheres of life (e.g.
technological, educational) that affect them. Skovsmose cites Niss (1983) who
claims that those who lack such a meta-level understanding of how mathematics
works become “victims of social processes in which mathematics is a component”
183 (p. 57). What this means in practice is not particularly clear. (Second draft of
proposal, p. 21)
The fact that I now was questioning and qualifying assertions, indicated my growing skepticism about the utility of secondary mathematics. In contrast to the relatively ungrounded and idealistic utilitarian statements in my first review, after struggling with night course instruction, I began to ask what, if any, secondary mathematics could be found in social, and even technologically regulated, “spheres of life.”
Teaching with my redesigned materials in the summer school courses did little to challenge my newfound semi-utilitarianism. It was not until I analyzed my CM materials after teaching that I became further bothered about mathematical utilitarianism. I began to see a focus on utility in curriculum and instruction as generally distracting to the development of academic mathematics. It was not that I had decided that certain required topics (i.e., proportional reasoning, measurement) could not, at times, be useful real world tools. Rather, I was anxious that my focus on relevance obscured the within- discipline mathematical purposes of topics such as proportional reasoning. I worried that students who learned mathematics as a real world tool might not learn it as a “language of power” that provides access to further schooling and eventual middle class jobs. It was particularly troubling that only certain students – those at low-income schools – were focused on “everyday” purposes while others – students in high income schools – were clearly prepared to meet academic goals. Recognition of this bias made the issue ethically problematic. This triggered my thinking about how curriculum differentiation was tied into wider sorting problems posed by tracking and school segregation. I frequently returned to these in my instructional diary.
184
4.4 Beliefs about Critical and Mathematical Empowerment
In this section, I consider my beliefs about what empowerment means for students in non-specialist courses. While mostly combining and conflating them in my instructional diary and other documents, I began to take more seriously the idea that there were distinctive forms of empowerment, namely: critical empowerment (i.e., consciousness raising, agency) and disciplinary empowerment (i.e., academic mathematics apprenticeship, acquisition of educational capital). The fact that I linked discussions of empowerment to those of utility throughout these written artifacts reveals that I thought the possibility of secondary mathematics as an empowerment tool subject hinged on its political utility in non-academic settings. In the next sections, I detail beliefs about critical and mathematical empowerment for non-specialist students and the relationship between them.
4.4.1 Initial Additive Version of Critical-Mathematical Empowerment
In the draft literature review I wrote prior to teaching the night course, I asserted, with varying degrees of explicitness, that critical and mathematical empowerment could be combined together additively in curricular materials and that, during instruction, students would also see them as unified. I indicated that critical literacy, or the ability to read and write the world with mathematics, came at no real cost to disciplinary mathematics. I argued that mathematical analyses could deepen students’ critical consciousness, possibly leading them to political activism (with mathematics). At the
185 same time, I implied that critical consciousness-raising in secondary mathematics courses might help students acquire more positive and active stances towards the subject of mathematics. (In Chapter 7, I include evidence both that supports and undermines this belief.) In the following excerpt I argued that CM could set the best ideas (or problems) of reform mathematics in critical contexts without a mathematical cost:
A social justice curriculum that does not make a claim to be the best
[mathematics] curriculum for its students – especially oppressed students – would
be problematic in terms of equity. Therefore students need to become proficient in
the dominant mathematics that is required by such institutions as universities
while at the same time engaging in mathematics problems that heighten critical
awareness. (First Draft of Literature Review, p. 24)
It is clear that I was convinced that CM fit well within, and contributed to, the acquisition of dominant mathematics. I glossed over the idea that CM might be a distraction that ultimately would lock students out of post-secondary programs that valued, and tested for, a particular type of school mathematics. I asserted that well-designed CM problems could simultaneously heighten critical and disciplinary understandings.
It also is noteworthy that I followed Freire and Macedo (1987) in using the term
“oppressed” in the above excerpt. Positioning my lower-income students of color as uniformly “oppressed” was related to my sense that they are generally disempowered in schools and society, hence need a more empowering curriculum than they generally receive (Anyon, 1981; Gutstein, 2003; Ladson-Billings, 1995; Lipman, 2004; Moses &
Cobb, 2001; NCTM, 1970; Oakes, 1990, 2005; Secada & Berman, 1999). Positioning my future students in such a uniform manner makes some sense given that I had yet to meet
186 them when I wrote the draft. There is evidence of my worries about designing curriculum for hypothetical students in early entries in my instructional diary. On November 4, 2003,
I state, “[it’s difficult] to design [CM] materials for students I’ve yet to meet … I have to operate on assumptions I do not feel completely comfortable making.” Nevertheless, in lumping all low-income CPS students of color together as “oppressed,” I clearly had ignored the agency that many students in low-income schools exhibit (Martin, 2000). I de-emphasized the substantial within-group variation I would later encounter as I taught the night course, when I discovered the wide range of orientations to school mathematics and future schooling among the night school students.
In my defense, when I wrote my draft proposal, I knew that the night course was for non-specialist students who had previously failed geometry. I felt that most probably had negative experiences in previous mathematics courses and, therefore, held negative orientations towards the subject:
Based on my experience as a high school math teacher in Chicago, I believe that
lower-income students in low track courses who do not have the same
opportunities to go to college as middle and upper class students are more likely
to reject school mathematics (reform or traditional) as irrelevant, precisely
because they know they are less likely to use it in future academic settings. (Draft
Literature Review, p. 11)
Positioning my past and prospective students as having uniformly negative secondary mathematics backgrounds allowed me to argue that they would be more engaged and empowered by a politically relevant version of the secondary mathematics curriculum than they had been by the dominant version. Again, what I discovered was that many
187 though not all of my students, despite past failures, expected me to prepare them for college mathematics (see Chapters 6 & 7). They were all seniors and most had a clear sense that learning mathematics was important for academic reasons. They showed little concern about the curriculum’s real world relevance. Thus, it can be said, that students generally were more concerned with learning mathematics as cultural capital than as a politically relevant tool subject. Some might argue that if CM materials were well designed they could accomplish both.
4.4.2 Shift Away from the Position of Additive Critical Mathematical Empowerment
Mixed and unexpected experiences and results in designing and teaching from a critical perspective led me to question the assumption that critical empowerment came at no cost to mathematical empowerment. First, I observed that critical and mathematical learning were generally only tangentially related in the planned and enacted CM lessons.
Second, while there were some exceptions, I observed that enacted CM activities did not foster more student engagement than reform activities in any clear or uniform sense. I found, instead, that night school students who were active during reform IMP activities, tended to resist CM activities. A second group of students participated more actively and verbally engaged during critical activities (see Chapter 7). However, while some CM activities ignited student interest, this interest did not necessarily foster cognitive engagement with secondary mathematics.
Based on my actual night course teaching, I came to believe that mathematical learning and critical consciousness-raising were only tangentially related to each other.
188
Evidence of this belief change was that I mostly, and rather unconsciously, discussed mathematical and critical empowerment separately once the night course began:
I need to try to get my students … to do “reform” things. Becoming less passive
and more active is essential for math learning and for “reforms” to work. I [also]
need to try to make conversation and discussion more central to the classroom – a
conversation that isn’t directly focused on the teacher’s ideas or filtered through
the teacher. (Instructional Diary, 1/13/04, p. 61)
In this passage I did not imply that mathematical empowerment required critical empowerment to precede it. The passage shows my realization that I had to reposition the students more authoritatively in terms of exploring, developing, and discussing their own mathematical ideas in reform activities before I ventured fully into CM. This was somewhat ironic because I had initially theorized that CM might change students’ orientations to reform activities rather than the other way around. It was also ironic that my pedagogical attention to mathematical empowerment using the apolitical standards- based IMP curriculum undermined the opportunities I might have taken for critical empowerment of students. At the time I saw this as setting priorities and putting critical pedagogy on hold:
[M]y number one priority is getting the kids to learn most of the dominant
material they need to learn. This is what my students seem to expect – I mean I
get that impression. … My [critically-oriented] research is going to have to take a
back seat for the moment. (Instructional Diary, November 19, 2003)
189 Indeed, in several diary entries I note that implementing my predesigned CM curriculum was not all that mattered to me. Such entries indicate that I was beginning to see CM as a distraction or burden, but this was not how I wrote about it in my draft literature review.
Once the night course began, I noticed that the meaning of critical empowerment for a diverse class of non-specialist students was not straightforward. I had glossed over within-class diversity in my literature review, but was becoming aware that student diversity posed some problems to successfully implementing CM. I found, for example, that it was difficult to find critical themes that all of my students could rally around:
In the Freirean world, issues are supposed to emerge from the students …
[However,] the [night school] students are diverse, from diverse communities,
with diverse needs and a diverse set of things that “oppress” or affect them …
(Curriculum Diary, November, 19, 2003)
Despite the remedial or repeat class nature of the night course, many students had an immediate need for college preparatory mathematics while others did not. Some students, for example the several young men who had already enlisted in the military, were unlikely to take another mathematics course in the near future and made that clear to me.
Others shared their plans to attend universities or technical schools within the year. Those who wanted to be construction workers or beauticians tended to complain that school mathematics would be irrelevant to those careers. It became obvious that my original one-size-fits-all view of CM or mathematical empowerment was mistaken. Clearly, what might be of interest and empowering for one student might not be empowering for others.
The ethnic diversity and range of orientations toward school that I encountered apparently was different from classrooms described by Gutstein (2003, 2005) and Tate
190 (1995). In later chapters, I return to the important theme of multiple layers of student diversity in my urban classroom. In general, I appreciated this diversity, despite the problems it posed for the success of my utilitarian, CM agenda.
During the design/instructional period I came to see critical and mathematical empowerment as tangentially related rather than additive as I assumed previously. My newer CM materials essentially fostered critical empowerment alongside mathematical empowerment in an uncoupled fashion. Further evidence that actual teaching resulted in a reconceptualization of student empowerment is revealed in this passage:
I recently [taught] geometry for social justice to a group of about 30 high school
students. … I found that the social justice component of the course, while
interesting, seemed like an add-on to the rest of the mathematics activities. …
[Moreover], [t]he [critical] activities were not always well received, especially in
the beginning of the course when the students and I seemed to be negotiating
routines and types of activities that would be acceptable to both of us. That said,
there were several successes from my point of view. … I am still optimistic that
building a more coherent [critical mathematics] curriculum is possible and that it
can foster mathematical understanding, critical consciousness, a willingness to
use mathematics for social justice ends, and be engaging. I am presently in the
second iteration of designing a TMSJ curriculum for secondary geometry.
(Second Draft of Study Proposal, Spring, 2004, p. 5)
Despite belief shifts, I held out hope that I would be able to develop an empowering CM curriculum for non-specialist students. I still felt the problem was primarily in my design of CM materials and not in theories of critical mathematics. With
191 this orientation, I redesigned most CM materials I had prepared for the night course, particularly confronting the issues of tracking and segregation, and taught with them again in my summer courses. Although by now I recognized the mathematical cost, I reasoned that the costs might be worth it in terms of critical consciousness raising, engagement, and fostering whole class discussions. While I understood that CM activities distracted attention away from learning college preparatory mathematics, I still thought that improved CM materials would pay off in terms of “engaging the disengaged.”
4.5 Analysis Phase and Current Belief Status
In this section, I turn to my thoughts about the possibility of constructing an empowering, critical-utilitarian version of secondary mathematics. At the analysis stage, of course, I had evidence of the consequences of my instruction and the nature of students’ reactions to CM. I particularly reflected on the consequences of stressing real world relevance only in secondary mathematics texts and courses aimed at non-specialist students. I began to ask Gramsci’s (1971) essential “who benefits?” question regarding the Guevara and Park Vista settings. I considered the impact of messages sent about their lives on students already sorted into inferior schools and remedial mathematics courses.
At this time, I turned from self-blame, to confront the reality that if I, an experienced teacher with Masters degrees in mathematics and education and a proven track record of successful teaching, could not design and implement solid CM instruction, the problem may lie in the process (or the theory) and not in the person. At the same time, in doing a self-inventory, I became aware that I personally did not use any mathematics beyond arithmetic, measurement, and proportional reasoning in my own everyday life.
192 Nor did I use mathematics in political spheres of my life. I began to see that it might be irrational to believe students would need advanced mathematical knowledge in theirs.
4.5.1 Shift to a More Profound Skepticism of Mathematical Utility
An important event at some point after completing instruction for my study, I learned of Dowling’s (1998) work on the sociology of school mathematics. Dowling’s work had a powerful impact on my beliefs about the utility of secondary mathematics, the portrayal of utilitarian values in mathematics texts, and the meaning of empowerment for students in non-specialist courses. His ideas caused me to shift even further away from my former optimistic utilitarian and semi-utilitarian views about mathematics until I arrived at what can be called an anti-utilitarian perspective. I applied Dowling’s concepts to textual analysis of I&A (see Chapter 5). His sociological perspective gave me insight into my curriculum design struggles.
Reading and applying Dowling’s ideas altered my beliefs about the purpose, sociological function, and utility of secondary mathematics. Dowling claims that a central purpose of his book The Sociology of Mathematics Education was “to challenge the core myth that practices which are organized and developed within one activity can be effective, even transformative, within another” (p. 24). In making this claim, Dowling rejects the pervasive assumption among mathematics educators that school mathematics
(i.e., procedures, reasoning, problem solving) are effective in non-academic settings in a powerful, utilitarian manner. In fact, he labels the idea that mathematics can participate powerfully in everyday affairs “a myth.” Regarding school mathematics, Dowling states,
193 “utility, such as it is, is inevitably deferred and uncertain: a half-hearted promise” (p. 17).
Dowling was not alone in challenging mathematical utilitarianism or the use of school mathematics in non-academic contexts (Ernest, 2000; Lave, 1988; Pimm, 1995).
4.5.2 Viewing Critical and Mathematical Empowerment as Oppositional
My textual analysis featured in the next chatper shows that the critical and mathematical activities that made up the I&A unit text took place in different “domains of practice.” After reading Dowling and seeing the results of my textual analyses, the disjunction between critical and mathematical goals became crystal clear to me. I&A activities with a justice-related utilitarian focus did not further within-discipline mathematic connection-making, nor did mathematic connection-making tasks further sociopolitical understandings. Moreover, mathematical principles were not easily stated or coherently illustrated, while in the midst of politicized activity. When the I&A text had readers engage in mathematical connection-making (e.g., calculating of trapezoidal areas), critical real-world connection-making took a back seat. Discussions of justice- related concerns (e.g., income inequality, racism) in the midst of mathematical development (e.g., learning to measure surface area) simply did not make sense.
Not only was I increasingly questioning my initial utilitarianism, but I came to question the utilitarian discourses about school mathematics in popular, educational, and policy venues that had influenced my study. I wondered if the effects of such discourses on mathematics curricula for non-elite students were as subtractive as Dowling implies.
Specifically, utilitarian rhetoric seems to help justify inferior, non-disciplinary, utilitarian versions of secondary mathematics (e.g., basic math, vocational math, checkbook math)
194 for students in non-specialist courses. To be sure, my textual analysis made it painfully clear that, in targeting certain students for “mathematics for political empowerment,” I was helping to (re)produce the social order. I was relegating the low-income students of color in my non-specialist course to a mathematics curriculum that failed to deliver the fullest preparation possible for college mathematics. The crux of problem was that I&A and other CM materials made far more “real world” connections outside of mathematics than principled connections within it. Hence, students would have to pay considerable attention to realistic (non-mathematical) content if their critical consciousness about sociopolitical affairs was going to be raised. While my intention as a teacher-designer was to facilitate additive CM empowerment, I worried instead that I had produced a subtractive form of mathematics disempowerment.
In his own analyses of a British textbook series (i.e., SMP 11-16), Dowling
(1998) shows that working class British students were far more likely than their more elite peers to be targeted with a vocational form of utilitarianism that prevented their access to academic mathematical discourses and principles. Dowling’s conclusion is telling: “The Teacher’s Guides to the ‘low ability’ books make a great deal of the need for relevance. It is suggested that the teacher might make the work more directly relevant to the actual lives of the students” (p. 34).
Dowling shows that, in fact, real world contexts are emphasized to a much greater extent in these “low ability” books than in texts for more elite students. Furthermore, the real world contexts carry a clear social class message concerning the mathematics needed for efficient shopping and bricklaying. In fact, the implications for my own work with
CM was so disconcerting that I wrote “YIKES!!!” in the margin of Dowling’s book by
195 this quote. I had made a “great deal of the need for relevance” and utility for low-income students in lower track courses in my CM study. I assumed, among other things, that an increased dose of real world relevance would result in increased student motivation. I did not fully understand that I was failing to relay disciplinary messages, purposes, principles, and understandings to my students. As a naïve realist, I believed students would implicitly develop these understandings through engaging in highly contextualized or realistic reform and CM activities. I am not alone in making these assumptions.
Along these lines, I also included two comparison texts in my textual analysis.
One text, in particular, was a “specialist” text designed to be used with honors students.
Wealthy Chicago area schools used this text with their honors track courses. My analysis revealed that this text made few utilitarian connections to the real world, focusing almost exclusively on making principled, within-discipline mathematical connections. It sent the privileged reader a clear message about the academic purpose of secondary mathematics and students’ ability to participate in academic fields such as engineering and physics.
The specialist text provided access to educational capital by preparing its privileged readers for rigorous college courses. That said, I believe it would be difficult to teach with the specialist text in lower-track courses or highly segregated schools, in part, because lower track students in racially and economically isolated schools have been repeatedly told in a variety of ways (e.g., texts, tests, teachers, tracking) that they were not capable of the study of mathematics or related careers and should not consider them.
Dowling’s observations paired with my analyses, allowed me to see that textual or pedagogical attention to “real world” links outside of secondary mathematics came at the cost of making structural links within secondary mathematics. It is precisely these links
196 within secondary mathematics that need to be made to prepare students for more advanced study of mathematics (Dowling, 1998; Lakoff & Nunes, 2004; Pimm, 1995).
The specialist textbooks in both Dowling’s and my curriculum analysis made disciplinary links while limiting utilitarian links to the real world. These texts provided privileged specialist readers with principled mathematics as cultural capital. In contrast, the utilitarianism in the non-specialist texts, whether critical or vocational, interfered with the principled connection-making necessary for teaching mathematics as educational capital.
The assumption was that non-specialist students needed a utilitarian version of secondary mathematics for motivational purposes while specialist students did not. Another explanation is that specialist students see themselves represented in the specialist curriculum, although I question this assumption.
4.6 Discussion
This chapter is about how my own internal debates surfaced in my instructional diary as I engaged in the design, implementation, and redesign of CM materials. In fact, teaching low-track and remedial secondary courses can conjure a considerable number of questions and dilemmas for mathematics teachers (Chazan, 2000; Romagnano, 1994).
When I began writing up my findings, I was surprised to discover that debates about utility and what mathematics non-specialist should receive are historic, tracing back at least as far as the beginning of the twentieth century (NCTM, 1970). These debates have yet to be resolved in mathematics education. According to Dowling (1998) and Pimm
(1995), currently there is considerable rhetoric about mathematical utility on the part of mainstream mathematics educators (e.g., the NCTM) and policy makers. This rhetoric
197 dwells alongside a highly stratified, segregated and inequitable school system (Kozol,
1991, 2005; Lipman, 2004; Oakes, 2005). As Dowling indicates, utilitarian rhetoric helps to justify giving real world relevant versions of mathematics to the mass of secondary students in non-specialist courses.
In the discussion that follows, I present arguments against mathematical utility as a remedy for student engagement and empowerment in low track courses. These arguments reflect my current beliefs about the nature and sociological effects of school mathematics. After that, I briefly trace the history of utilitarian rhetoric and how this is used to justify targeting non-specialist students with real world relevant and academically disempowering versions of school mathematics. I end with a consideration of how this history continues to play out today, including a discussion of the utilitarianism in the scholarship of proponents of critical mathematics.
4.6.1 Academic Mathematics as a Non-Transformative, Self-Referential System
The inclusion of mathematics in the secondary curriculum is often justified on the grounds of being useful in the material and social world (Dowling, 1998; Ernest, 2000;
Kline, 1959; Paulos, 1988; Pimm, 1995). Note that it might also be justified in terms of its value as an academic discipline, cultural heritage, or means for personal psychological introspection (Pimm, 1995). A number of contemporary scholars reject utilitarian justifications for teaching school mathematics (Dowling, 1998; Ernest, 2000; Pimm,
1995). The well-known British philosopher of mathematics and mathematics education,
Ernest (2000), for example, puts it this way:
198 [T]he utility of academic and school mathematics in the modern world is greatly
overestimated, and the utilitarian argument provides a poor justification for the
universal teaching of the subject throughout the years of compulsory schooling.
Thus although it is widely assumed that academic mathematics drives the social
applications of mathematics in such areas as education, government, commerce
and industry, this is an inversion of history. (p. 2)
By inversion of history, Ernest means that academic mathematics is more of a by-product of technological innovation than an engine for it, as is often assumed, implied, or stated.
Indeed, anthropological and psychological studies that investigate the relationship between the mathematics learned in school and quantitative skills needed in everyday contexts (e.g., on the job, in the market) indicate this relationship is not straightforward
(Lave, 1988; Lave, Murtaugh, & de la Rocha, 1984; Nunes, Schliemann, & Carraher,
1993; Rogoff & Lave, 1984; Scribner, 1984). According to Smith (2001), in the 1920s
Thorndike and his colleagues conducted a study that set out to uncover how people used algebra in work settings. The study led Thorndike to conclude that algebra was used minimally and only in jobs of a small minority of people. Whether this finding was due to mathematical ignorance on the part of workers or the rigid and self-referential nature of academic mathematics is not clear. I suspect it was the latter.
More recent studies of everyday problem solving show that when people are faced with quantitative problems in the real world, they tend not to use the mathematical techniques and reasoning taught in school. This appears to happen for several reasons.
First, people develop more efficient and relevant ways to carry out sophisticated quantified reasoning to solve contextualized or situated real world dilemmas (Lave, 1988;
199
Lave, Murtaugh, & de la Rocha, 1984; Nunes, Schliemann, & Carraher, 1993; Rogoff &
Lave, 1984; Scribner, 1984). Second, everyday problems can rarely, if ever, be reduced to purely logico-mathematical problem statements. Everyday problems are much more complex and, in contrast to academic mathematics problems, involve making value-laden decisions. Simple transfer of school mathematics to real world contexts outside of school rarely, if ever, happens (Lave, 1988; Lave & Wenger, 1991). The problem, then, is not that people do not understand when to apply secondary mathematics as some imply (e.g.,
Paulos, 1988), but rather that applying it is not a particularly relevant or efficient way to proceed in non-academic realms.
Dowling (1998) claims that the disconnect between secondary mathematics and the everyday has its origins in the self-referential nature of mathematics as an academic discipline. Following Bernstein (1991), Dowling argues that there are relatively clear boundaries between the discourses and activities of academic disciplines and non- academic (everyday) discourses. He describes the discontinuities between mathematics and the (apparently) math-related practice of shopping, in this way:
[T]he principles which regulate mathematics and those which regulate shopping
constitute distinct systems. One may recruit elements of the other: a shopper may
use a memory of a multiplication table; a mathematics textbook may incorporate a
domestic setting. But precisely what is recruited is regulated by the recruiting
rather than the recruited practice. Mathematics is not about shopping because the
shopping settings which appear in mathematical texts are not motivated by
shopping practices. (p. 16)
200 As one progresses from elementary to secondary mathematics, the subject becomes less tied to the material world in which it was abstracted and increasingly self-referential. This explains, in part, why it is that secondary mathematics problems bear little, if any, resemblance to problems encountered in their mundane existence (see Love & Pimm,
1996). This also explains why it is difficult to make within-discipline mathematical connections while engaged in the politically-oriented types of tasks I was concerned with as a CM educator. While it certainly is political on many levels, mathematics is not fundamentally regulated by political activity of the everyday variety. I would suggest that mathematicians attempt to insulate themselves from politics and non-aesthetic values in their professional practice.
This is not to suggest that the non-mathematical is not important in learning mathematics. Effective mathematics learning certainly begins with what novices already know, including concrete experiences in the material world. Dowling makes it clear that the ideal (i.e., Vygotskian) apprenticeship begins with people’s everyday understandings and discourses and moves to the “esoteric” domain of mathematical abstractions and discourses. In the more advanced domains of school mathematics, the ideal subjectifying apprenticeship would increasingly take place in the esoteric realm of practice. This is where mathematics largely builds upon itself in a self-referential manner. While there is room for (generally and necessarily artificial) applications in a course such as calculus, the traces of the concrete or real are generally left behind. Lakoff and Nunes (2004) similarly suggest that mathematical learning begins by building metaphorically on an externally-oriented, concrete real world, yet the closer one gets to expert disciplinary performance, the more one builds metaphorically on structures within mathematics.
201 This suggests that the recontexualized real should be used in mathematics texts only to the extent that it furthers disciplinary learning (Dowling, 1998; Pimm, 1995).
Such is not the current case in mathematics texts designed for non-elite students. As noted earlier, non-specialist texts often attempt to motivate non-specialist readers with
(necessarily) artificial real world applications (Dowling, 1998; Pimm, 1995). Dowling’s analyses of British secondary texts show a correlation between a textual emphasis on accurate representations of the real and lack of access to academic discourses, principles, and practices. References to real world contexts in (non-specialist) mathematics texts often (or always) direct attention away from within-discipline connection-making. Love and Pimm (1996) call the (over)emphasis on the real as highly spurious and problematic:
The deictic relation of mathematics text questions the material world is a highly
problematic one. The questions in texts seem to be referential, yet they actually
call the world of which they seem to speak into being. Mathematics texts can be
viewed not as speaking of an external world but instead as offering substitute
experience that is largely self-contained. In consequence, a lot of this, “other-
directedness” in mathematical text is spurious. (Love & Pimm, 1996, p. 381)
From this perspective, therefore, it would seem that links between mathematics and the non-mathematical material world are created with far more ease and regularity in mathematics texts than in the world outside of texts.
At the same time, I realize the boundaries between mathematics and what is taken to be the non-mathematical can be drawn in different ways. This is one reason I have chosen to pair the term mathematics with qualifiers such as “school,” “secondary,” and
“academic.” Some may argue that in drawing restricted boundaries I am, in fact, taking
202 an elitist view of mathematics. Authors, such as Powell and Frankenstein (1997) might point to the fact that I simply exclude the quantitative reasoning and problem solving that people use in their everyday lives from my definition of secondary mathematics. They might point out that I am privileging a Eurocentric school curriculum over other mathematic versions and traditions that might be taught in schools. These arguments do not fall on deaf ears, however, I would counter that current educational policy binds teachers and their students to the conventional mathematics curriculum, a scholastic version that is fraught with elitist and Eurocentric features (D’Ambrosio, 1985). Teachers who stray far from the highly regulated mathematics curriculum run the risk of disempowering their students. Educational reformers tend to underemphasize the constraints the conventional curriculum places on teachers (Cohen, 1988; Lortie, 1975).
4.6.2 Rhetoric of Utility and Targeting Non-Elite Students with Utilitarian Mathematics
In what follows, I briefly outline the historical origins of the rhetoric of mathematical utility in the United States. I focus, in particular, on how for decades the assumed usefulness of school mathematics has justified teaching non-specialist students a particular non-academic version of school mathematics (e.g., general mathematics, business mathematics). Again, this is relevant to this chapter given that CM emphasizes the real in mathematics similar to vocationally oriented versions of school mathematics.
In the nineteenth century, it was widely held that learning mathematics instilled mental discipline (NCTM, 1970). Hence, it was argued that the inclusion of mathematics in the school curriculum was justified on the grounds that it improved students’ mental capacities and general ability to think rationally in academic and non-academic contexts
203 alike. At the turn of the twentieth century, the influential educational psychologist E. L.
Thorndike conducted a set of psychological experiments that mainly discredited the claim that learning mathematics or Latin instilled mental discipline (NCTM, 1970). According the NCTM (1970), after Thorndike, the idea that mathematics imparted mental-discipline,
“could no longer provide a basis for defending the inclusion of mathematics in the curriculum for purposes of utility and practicality” (p. 186).
Thorndike also worked to reconceive of school mathematics as a tool subject, along with scholars such as Franklin Bobbitt who were said to be in a “usage cult”
(NCTM, 1970). These scholars argued that mathematical topics with the greatest use- value in the everyday, economic, and industrial spheres of life should be taught in schools. They conducted a series of investigations developed to uncover the algebra that people used in everyday settings (NCTM, 1970; Smith, 1994; Thorndike, 1926). Their findings found less real world utility of mathematics (algebra) than they had assumed
(NCTM, 1970; Smith, 1994). In this same time period, there were also complaints by mathematicians and educational scholars that a curriculum constructed around real world utility would not be good preparation for academic mathematics or related fields (NCTM,
1970). Despite Thorndike’s findings and anti-utilitarian complaints from academia, calls for more relevant curricular materials for the general student continued.
In 1920, the progressive educator Kilpatrick headed the NEA’s Commission on the Reorganization of Secondary Education. The Kilpatrick Report, “hurled down the gauntlet of the needs of youth and the utility of mathematics as a curriculum determiners to replace mental discipline” (NCTM, 1970, p. 197). The fundamental question from the progressive standpoint became what mathematics, if any, to teach the non-specialist,
204 vocational students who were not-college-bound. Indeed, a range of vocational secondary mathematics courses emerged in the wake of the Kilpatrick report.
It is important to keep in mind that the question of what was useful was only considered to be relevant to the case of non-specialist students. Elite students continued to have access to the traditional college bound mathematics curriculum that supposedly pertained to their college or post college trajectories (NCTM, 1970). It should also be noted that in the first half of this century the survival of school/academic mathematics was under threat. Some scholars, politicians, and laymen openly questioned the value of including mathematics in the school curriculum, particularly for non-specialist students
(NCTM, 1970). It would seem, therefore, that the rhetoric of utility was necessary not only the expansion, but also the survival, of much of school mathematics. I believe this rhetoric continues today, in part, because it is a survival tactic for mathematicians and mathematics educators. We benefit from this rhetoric and its effects on policymakers and the general public. A mathematician recently told me that mathematicians regularly remind each other to emphasize the usefulness of mathematics to the general public.
4.6.3 The Current State of the Field and Utilitarianism
In the period before World War II, debates about the utility of mathematics and type of mathematics non-specialist students should receive often became rather heated.
For the most part, the historic debates discussed above have quieted since the Soviet launching of Sputnik. The rhetoric of utility won out. It is not surprising, then, that the popular perception held by many educators, policymakers and laymen today is that academic mathematics is an engine for national economic/technological progress
205 (Dowling, 1998; Ernest, 2000; NCTM, 1970; Pimm, 1995; Ravitch, 2005). At the same time, while some vocational courses have disappeared in name, non-specialist students still receive mathematics curricula purported to be useful in “the world of work.” The
NTCM continues to promote the idea that school mathematics is useful in people’s everyday lives. In the Standards 2000 documents, the NCTM maintains, “the need to understand and be able to use mathematics in everyday life and in the workplace has never been greater and will continue to increase” (NCTM, 2000, p. 4). Writers of the
NCTM Principles and Standards point to everyday examples where “quantitative sophistication” is used in, “making purchasing decisions, choosing insurance or health plans, and voting knowledgeably” (p. 4). This statement references the breadth and depth of mathematics knowledge needs:
Just as the level of mathematics needed for intelligent citizenship has increased
dramatically, so too has the level of mathematical thinking and problem solving
needed in the workplace, in professional areas ranging form health care to graphic
design. (NCTM, 2000, p. 5)
What is implied here is that mathematical problem solving in schools is the same as or analogous to problem solving in non-school contexts and that school based mathematical reasoning can help people optimize the mundane. The statement is a modern version of the century old theory that mathematics instills mental discipline. Yet, there is a dearth of empirical evidence to support claims that problem solving skills learned in school mathematics transfer to out of school settings. Existing evidence actually contradicts these claims (Lave, 1988; Lave, Murtaugh, & de la Rocha, 1984;
Nunes, Schliemann, & Carraher, 1993; Rogoff & Lave, 1984; Scribner, 1984).
206
The reform mathematics IMP curriculum that I used as my foundation curriculum for my dissertation study was utilitarian. In a “message to students” the IMP authors state:
Our goal is to give you the mathematics you need in order to succeed in this
changing world. We want to present mathematics to you in a manner that reflects
how mathematics is used and that reflects the different ways people work and
learn together. Through this perspective on mathematics, you will be prepared
both for continued study of mathematics in college and for the world of work. (p.
xviii)
Thus, the IMP curriculum aspires to create a dual mathematical apprenticeship, one with academic and the other with vocational goals in mind. Such a dual apprenticeship is untenable from Dowling’s (1998) disciplinary-based perspective. Real world training ultimately comes at the cost of disciplinary training. It is necessary to add the caveat that the use of real world contexts to illustrate mathematical applications is different from drawing on the concrete to facilitate learning. The NCTM Standards and curricula generally fail to distinguish between making external (real world, applied) connections and internal (disciplinary) connections. The NCTM (2000) position is that students can reconstruct (a lot of) disciplinary mathematics through engagement with a set of well- organized concrete experiences and real world problems. Little distinction is made between secondary and elementary mathematics in this regard. In fact, if concrete, student-centered experiences are organized and facilitated correctly, the reform teacher can supposedly recede into the background. As I argue in Chapter 2, and as noted by
207 Chazan and Ball (1995), the non-didactic role of the standards-based reform teacher is exaggerated by standards-based reformers. In a note to students, the authors of IMP state:
To accomplish your goals, you will have to be an active learner, because the book
does not teach directly. Your role as a mathematics student will be to experiment,
to investigate, to ask questions, to make and text conjecture, and to reflect, and
then to communicate your ideas and conclusions both orally and in writing. You
will do some of your work in collaboration with fellow students, just as users of
mathematics in the real world often work in teams. At other time, you will be
working on your own. We hope you will enjoy the challenge of this new way of
learning mathematics and will see mathematics in a new light. (IMP Teachers
Guide, p. xix)
Hence, the IMP authors maintain that the principles that regulate secondary mathematics do not have to be made explicit by the IMP text or teacher. Secondary mathematics is apparently “out there” in the material world awaiting student discovery. The teacher is notably, but not surprisingly, absent from the reform vision described above. Again,
Dowling and others (Pimm, 1995; Walkerdine, 1988) critique this position as untenable, at least as long as a curriculum and instruction can claim to maintain a disciplinary focus.
While student-centered exploration is certainly important for learning and understanding mathematics, a considerable amount of secondary mathematics that cannot be learned when students are engaged in student-centered problem-solving and mathematical modeling.
That reform curricula such as IMP construct a dual academic/vocational focus is not surprising given that it was designed with non-specialist students in mind (Wu, 1997).
208 (Wu claims that the IMP authors admitted this to him in private but may deny it in public.) Having taught with it, I do believe the IMP curriculum deals rather effectively with motivational problems created by tracking and the significant within group variation that generally exists in non-specialist courses. According to a variety of measures, IMP also does a better job educating the masses than more traditional curricula (Webb, 2003).
It is also important to note that some courses for privileged students adhere to reform methods. I know of at least one elite public high school that uses the IMP curriculum in conjunction with AP courses. Nevertheless, reform curricula appear to exchange disciplinary preparation for supposed real world preparation (Dowling, 1998; Pimm,
1995; Wu, 1997). Specialist texts do not make this trade-off as the “real” deemphasized and readers are not expected rediscover secondary mathematics in the absence of explicit pedagogical discourse (Dowling, 1998; Wu, 1997). Specialist students acquire mathematics as a set of exchange-values that can be used in institutional settings.
Differences that arise in social class-based curricular orientation are problematic because students in non-specialist courses do not receive the mathematics necessary for continuation in rigorous post-secondary mathematics courses, those courses which provide access to better jobs and further academic study (Moses & Cobb, 2001; Oakes,
1990, 2005; Pimm, 1995; Tate, 1997; Wu, 1999).
4.6.4 The Disciplinary Cost of Critical Mathematics
It is hard to imagine critical mathematics, which (re)conceives of school mathematics as a sociopolitical tool subject, could have emerged in the absence of utilitarian worldview and rhetoric that permeates discussions of mathematics education
209 such as found in the NCTM Principles and Standards (1989, 1991, 2000). I first read about critical mathematics in articles by Gutierrez (2002) and Gutstein (2003) that appeared in two top tier mathematics education journals. These scholars and others rely on utilitarian assumptions (Frankenstein, 1991, 1995; Powell & Frankenstein, 1997;
Skovsmose, 1994). Consider the following quotation in which Gutstein (2003) maps his version of Freire’s concept of “reading and writing the world” onto school mathematics:
Reading the world is akin to developing a sociopolitical consciousness of the
conditions and context of one’s life, but I am speaking here of doing this with the
specific use of mathematics. In my view, reading the world with mathematics
means to use mathematics to understand relations of power, resource inequities,
and disparate opportunities between different social groups and to understand
explicit discrimination based on race, class, gender, language, and other
differences. Further, it means to dissect and deconstruct media and other forms of
representation and to use mathematics to examine these various phenomena both
in one’s immediate life and in the broader social world and to identify
relationships and make connections between them. (p. 44-45)
Gutstein highlights the need to make externally oriented connections from mathematics to “phenomena in one’s immediate life” and/or “the broader social world.” School mathematics is assumed to be powerful in the sense that it can make real world connections and raise critical consciousness. Real world contexts are assumed to facilitate mathematical learning and, at the same time, mathematics is assumed to optimize political activity. Similarly, just as I conceived CM at the start of this study, CM proponents and other scholars (Lesh & Doerr, 2003; Swetz & Hartzler, 1991) often fail to
210 distinguish between internal disciplinary and external real world connection-making or fail to explain the relationship between the two. Current discussions of mathematical modeling (Lesh & Doerr, 2003) make external connection-making seem more or less equivalent to internal connection-making within mathematics. However, if Dowling and others are right (Ernest, 2000; Lakoff & Nunes, 2000; Pimm, 1995) disciplinary mathematics is essentially a socially constructed, semi-arbitrary, self-referential system.
Thus, mathematics is not “out there” in the material, sociopolitical world awaiting discovery. Real world explorations, problem solving, and modeling do not map onto disciplinary mathematics in any natural or straightforward manner. Interestingly, in discussing utilitarianism in mathematics education, Pimm (1995), singles out CM proponents as putting forth a relatively extreme version of mathematical utility.
Somewhat coincidently, perhaps, he discusses the same quote I first discussed (and bought into) by Skovsmose in my literature review, namely:
Skovsmose has gone so far as to talk of society being ‘formatted’ (as with a
computer disk by mathematics, and to conclude that the primary role of
mathematics education in a democracy should be to alert and educate pupils to its
effects. However, while I agree with Niss’s [1983] and Skovsmose’s [1994]
contentions that being in a position to examine the role that mathematics plays in
society is one important reason for teaching it, and while I support the
development of more critical attitudes as one goal (of any school subject), surely
the sole or even most important purpose of mathematics education for our society
cannot be to criticize it. My prime purpose is providing thoughtful access to the
211 ways of seeing which mathematics affords, while not being blind either to the
costs involved or to presumptions about it offering the right or best such ways.
Pimm claims to be sympathetic to the emancipatory goals of critical education – as I continue to be – but finds the utilitarianism inherent in CM problematic. While I come down hard on CM here, I argue that critical education should be a part of the secondary curriculum, however, secondary mathematics should not be reconceived as a critical literacy (see Chapter 8). I also believe that inclusion of occasional critical applications is different from recreating the curriculum as a critical literacy. I have no problem with the inclusion of some critical activities (e.g., Brantlinger, 2005) long as academic empowerment is kept in sight. I also take seriously Pimm’s warning that we not be
“blind” to the costs of developing critical attitudes in students in school mathematics.
4.7 Conclusion
This chapter can be seen as a documentation of my own internal debates about what constitutes the most empowering mathematics curriculum for my students of color and from low-income and working class communities. I began my study by believing that critical-utilitarian mathematics would be the most empowering curriculum. These beliefs were based on the utilitarian rhetoric that pervades the field of mathematics education. I came to believe that the emphasis my critical mathematics curriculum placed on real world relevance came at the price of educational or mathematical capital. My current position is that targeting students in non-specialist courses with utilitarian mathematics is problematic in terms of equity, in particular, because students in non-specialist courses tend to be from low-income African-Americans and Latino communities.
212 CHAPTER FIVE: TEXTUAL ANALYSIS OF A CRITICAL MATHEMATICS UNIT
This chapter includes the findings from a comparative textual (curriculum) analysis. I contrast my critical mathematical Inequalities & Area (I&A) text with two other geometry texts that are similar in length, required instructional time, and in area- related content. The goal of the textual analysis was to address the research question:
How does the incorporation of justice-related themes into the conventional secondary math curriculum transform this curriculum? More specifically, how does inclusion of CM activities influence students’ access to academic mathematics? I examined how the three texts distribute: (1) messages about the nature of mathematical practices and their relation to everyday practices, (2) mathematical discourses and mathematical principles in discourse, and (3) mathematical content. I found that, while my critical I&A text might have been politically empowering, the text’s attention to critical themes meant that it failed to provide full access to the discourses, content, and principles of disciplinary mathematics. While recognizing design problems in I&A that might have led to these findings, I point out more generalized trade-offs between the critical and mathematical in
CM at the secondary level.
5.1 The Three Curricular Texts
The primary focus of this comparative textual analysis was my I&A unit text. As a critical mathematics text, the I&A unit differed from the typical secondary geometry texts in tying together explicit sociopolitical and mathematical concerns. I describe the development of my CM curriculum, including I&A, in Chapter 4. I provide a complete copy of I&A in Appendix A. The two comparison texts in this analysis are: a standards-
213 based reform text and a traditional text. Because neither of the comparison texts were critical in nature, comparisons allowed me to examine how the infusion of critical sociopolitical themes into the I&A text affected more conventional versions of the secondary mathematics curriculum. The reform text used for comparison was “Do Bees
Build it Best?” (Bees) from the NSF supported Interactive Mathematics Program (IMP)
(Fendel, Resek, Alper, & Fraser, 2000). I include only the first half of Bees that focused on area-related topics in my textual analysis. I use the term Bees* to distinguish between the area-related text and the full Bees text. This is how IMP authors describe the first half of the Bees text:
In this unit students work on this problem: Bees store their honey in honeycombs
that consist of cells they make out of wax. What is the best design for a
honeycomb? To analyze this problem, students begin [in the first half of this unit]
by learning about area and the Pythagorean theorem. Then, using the Pythagorean
theorem and trigonometry, students find a formula for the area of a regular
polygon with fixed perimeter and find that the larger the number of sides, the
larger the area of the polygon. (p. xi, IMP, Author’s Emphasis.)
On the surface at least, the biggest difference between the Bees* and I&A texts was that the former used the honeycomb problem to motivate the development of area-related topics while the latter used economic justice to motivate the same, or similar, topics.
Finally, and importantly, the I&A and Bees* texts both were designed for use with non- specialist students (see Wu, 1997); that is, “medium to low achieving” mathematics students who have historically been tracked into lower-track mathematics courses
(NCTM, 1970; Oakes, 2005). I realize, of course, that IMP is also used with high
214 performing students – though I suspect rarely. I also realize that the IMP authors have never stated publicly what Wu (1997) claims they told him in private about having designed IMP for non-specialist students. With this caveat in mind, I refer to I&A and
Bees* collectively as the non-specialist texts.
The second comparison text that I include in my textual analysis is the chapter on area from the textbook Geometry for Enjoyment and Challenge by Rhoad, Milauskas, and
Whipple (1991). Unlike the I&A and Bees* texts, “Rhoad” is designed for use with privileged students being prepared to (have the option to) specialize (further) in academic mathematics. I identify Rhoad as a “specialist” text because three school personnel from suburban Chicago-area high schools with the highest achievement scores on standardized mathematics exams use this geometry text with their highest track students. As it turned out, two Rhoad authors taught at New Trier High School, located in the wealthiest school district in Illinois (see Kozol, 1991). The third author was on staff at the Illinois
Mathematics and Science Academy, a school that enrolls some of the highest achieving mathematics students in the state of Illinois.
In part because Rhoad was designed for specialist students, I held it up as an exemplar in my textual analysis. I realize, however, that this text would be seen as repressive by many mathematics educators, including myself, because it leaves no space for readers to informally discuss their mathematical ideas or “do mathematics” in a student-centered sense (see Stein, Grover, & Hennigsen, 1996). In identifying Rhoad as an exemplar, my intention was not to celebrate traditional mathematics, rather it was to discern potential flaws with Bees* and I&A’s assumptions about their readers.
Furthermore, while I critique the IMP Bees* text for not making mathematical principles
215 fully explicit, there is evidence that the reform IMP curriculum is in certain ways superior to traditional curricula for general, non-specialist students (Webb, 2003). Having taught with it, I believe IMP is superior to other non-specialist curricula in terms of general student motivation and learning. Furthermore, I would use IMP as my foundational curriculum if were I to return to the secondary classroom. I would be reluctant to use
Rhoad in a non-specialist course.
Figure 5.1 shows some of the structural and intended instructional variation in the three texts. Because it was the longest, Bees* was at an advantage in my textual analysis.
The critical I&A text, on the other hand, was disadvantaged because it did not include homework. This reflected the fact that I was not allowed to assign homework in the remedial night school program at Guevara. Finally, Rhoad was organized around eight sections instead of “sets of activities” as were I&A and Bees*. Each Rhoad section began with a pedagogical exposition of principles (i.e., theorems, definitions) and sample problems. It ended with collections of actual problems, similar to introductory sample problems, that Rhoad readers were expected to solve. I take these textual differences into account in my analyses.
Text Pages Instructional Periods Required Titled Activities
I&A 40 18 Periods (No Homework) 27 Activities
Bees 49 20 Periods (Homework) 35 Activities
Rhoad 42 13 – 15 Periods (Homework) 8 Chapter Sections
Figure 5.1. Structural Characteristics of the Three Texts
216 5.2 Distribution of Domain Messages
The three dimensions to my textual analysis were informed, first and foremost, by
Dowling’s (1998) “sociological language of description” for mathematics texts (see
Chapter 3). As Dowling (1998) notes, the content and expression of activities in mathematics texts vary in terms of their academic orientation. Dowling defines four domains of practice that capture much of this variation, namely, the esoteric, expressive, descriptive, and public. When mathematics (re)produce esoteric domain activity, both the content under consideration and its expression are easily recognizable as (preparation for) academic mathematics. In the expressive domain, the content but not its expression is recognizable as (preparation for) academic mathematics. In the descriptive domain, the expression but not the content is recognizable as (preparation for) academic mathematics.
And, finally, in the public domain, neither the content nor how it is expressed is within the boundaries of disciplinary mathematics. I used individual activities and chapter sections as units of analysis to examine the domain messages sent by the three texts. I coded particular activities or sections as (re)producing one of these four domains of practice by looking at the content under consideration (i.e., disciplinary-focused or not) and its expression (disciplinary-focused or not). This allowed me to examine how mathematical activity is (re)produced in texts and the messages each text sent students about the nature of the discipline and their potential as learners of mathematics.
5.2.1 Findings about Distribution of Domain Messages
As Figure 5.1 indicates, there was considerable variation in how the three texts
(re)produced mathematical activity. While the specialist text, Rhoad, exclusively
217 (re)produced esoteric domain activity, the reform Bees* generally (re)produced expressive domain activity, and the critical I&A essentially alternated between
(re)producing expressive and public domain activity. None of texts (re)produced descriptive domain activity.
I&A (27 Activities) Bees* (35 Activities) Rhoad (8 Sections)
Esoteric 7.4% 11.4% 100% Expressive 51.9% 77.1% 0% Public 40.7% 11.4% 0%
F igure 5.2. Distribution of Domain Message in the Three Texts
Rhoad was by far the most disciplinary-focused of the three texts. It always
(re)produced esoteric activity in which the mathematical content and its expression were within the bounds of preparation for disciplinary mathematics. Rhoad de-emphasized the public domain to the point of non-existence. For example, only 5 of the 167 problems in
Rhoad referenced public domain objects (e.g., a dog, a ski lift), and the references were brief and anecdotal. In the few cases when the “real” was referenced, it played absolutely no role in structuring the mathematical tasks at hand. Therefore, it would be difficult for specialist Rhoad readers to believe they were being prepared for something other than further disciplinary study of mathematics.
The non-specialist I&A and Bees* texts, on the other hand, (re)produced activities in which real world problem contexts figured prominently. To begin with, both
(re)produced a large number of expressive domain activities in which the mathematical content, but not its expression, was recognizable as (preparation for) academic
218
mathematics. Approximately three-quarters of Bees* and half of I&A fell within the expressive domain of activity. The “real world” played a significant role in structuring the tasks and in discussions of the problems being addressed.
Consider the “Tiling the Room” activity shown in Figure 5.3. It is an example expressive domain activity from I&A, though it is not dissimilar from expressive domain activities in Bees*. The first thing to note about Tiling the Room is that, while the mathematical task (i.e., find the best deal on floor tiles) is challenging to many students, it is not expressed the way similar mathematic problems are expressed in the esoteric domain; that is, as mathematics problems for disciplinary purposes. Tiling the Room illustrates how non-specialist texts use the public domain to motivate the need for, and expression of, real world topics in secondary mathematics. Mathematical objects (e.g., squares, polygons) are expressed as public domain objects (e.g., floor tiles, triangular paintings) and mathematical problems are expressed as everyday problems experienced by real characters. An important consequence for non-specialist readers is that mathematical ideas are left in terms of the real world problem context and not in terms of mathematical principles.
219
Figure 5.3. An I&A Expressive Domain Activity, Tiling the Room
To the extent the non-specialist texts have disciplinary goals, they are masked in expressive domain activities such as Tiling the Room. The assumption of Bees* and I&A appears to be that in order for non-specialist readers effectively engage in mathematics, secondary mathematics has to be dressed up as being about something other than disciplinary preparation. In fact, Bees* and I&A never suggest that their non-specialist readers might be learning mathematics for disciplinary purposes. Although subliminal
220 messages relate to the practical relevance of mathematics, Bee*’s authors are careful to avoid limiting their expectations for non-specialist students. In a “note to students” at the beginning of the second year text that includes Bees*, the IMP authors claim to provide the mathematics necessary for both the world of work and college.
In addition to expressive domain activities, the non-specialist texts (re)produce public domain activities in which the content and its expression are, at best, weakly associated with (preparation for) disciplinary mathematics. In public domain activity, the real world context and not secondary mathematics structures the tasks at hand. As Figure
5.2 indicates, the (re)production of individual activities in the public domain is much more pronounced in I&A than in Bees*, with 40.7% of its activities falling into this domain in contrast to 11.4% in Bees*.
It is important to note that the 11 critical activities in I&A all fall within the public domain. In fact, I&A text (re)produces two types of critical public domain activities, namely, those about: (1) sociopolitical analysis of quantitative data and (2) (non- quantitative) critical-consciousness raising. Figure 5.4 provides examples of the two types of critical activities found in I&A. The first, “Math, Equity, and Economics,” requires the I&A reader to conduct a political analysis of quantitative data. Math, Equity, and Economics was in a book published by Rethinking Schools (1994). I included it in my I&A text with some minor modifications. While it appears mathematical in the sense that it deals with quantitative data, the activity falls within the public domain. This is because, with the exception of the question, “how could we go about testing these hypotheses?,” nothing in this activity is recognizable as preparation for further disciplinary study of mathematics. Hence, contrary to what the title of this activity
221 suggests, Math, Equity, and Economics is structured according to principles that regulate the political rather than the mathematical.
Figure 5.4. Two Critical Public Domain Activities in the I&A Text
The “Civil Rights and Economic Equality” activity is the second non-quantitative type of public domain activity featured in I&A. It is clear that the goal of this activity is critical consciousness raising and not mathematical preparation. That said, however, my inclusion of this type of activity in I&A text is meant to send my students the message that secondary mathematics can participate powerfully in the struggle for economic and social justice. Indeed, as discussed further at the end of this chapter, consciousness raising apparently has to take place in the (non-mathematical) public domain. Moreover, as the Civil Rights and Economic Equality activity illustrates, both non-specialist texts use images of the real (i.e., photographs, drawings) to send the message that secondary mathematics can participate meaningfully in the public domain. The non-specialist texts use images (i.e., the iconic symbolic mode) to a greater extent than did Rhoad.
222
Figure 5.5. Domain Trajectories Featured in Three Texts
The way the non-specialist I&A and Bees* texts sequence activities also sends a domain-related message about the nature and purpose of secondary mathematics. Figure
5.5 illustrates the different “domain trajectories,” or perhaps intended apprenticeships, created in all three texts. Note that I&A oscillates back and forth between public and expressive domain activities. This oscillation, along with the inclusion of public domain images, serves to further link the critical and the mathematical. The interweave sends the message that even if the mathematics at hand is not immediately relevant to sociopolitical concerns, it is likely to become relevant by the end of the unit. Again, it reinforces the idea that non-specialist students should study secondary mathematics for political rather than disciplinary purposes. Bees*, on the other hand, creates a clear trajectory from the public to the expressive domain. While unlike Rhoad, Bees* never fully arrives at the esoteric domain, Bees* prioritizes the mathematical over the public domain in a way that
I&A fails to do.
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5.3 Distribution of Mathematical Discourse
Dowling (1998) notes the considerable discursive variation in mathematics texts.
First, they vary with respect to how much they use discourse to accomplish pedagogical goals. Dowling uses discursive saturation to capture the extent to which mathematics texts rely on alpha-numeric text. Texts, then, exhibit high or low discursive saturation.
Second, mathematics texts vary with respect to how explicitly they state disciplinary principles and make disciplinary-oriented connections between, for example, theorems and sample problems. Dowling uses the term principling to describe textual discourse that makes disciplinary principles explicit and the term metonymic to describe textual discourses that exhibit connective complexity. A text that goes beyond stating principles and procedures by illustrating and interconnecting them would exhibit principled metonymy. Alternatively, a text that states the division of fractions procedure as “turn upside down and multiply” without providing any rationale would not.
I used the individual phrase or sentence as the unit of analysis for studying the dimensions of discourse. I code individual phrases as “principled” when they explicitly develop or address mathematical principles and as “non-principled” when they do not. I also (double) coded within this second category; coding non-principled phrases as “task- oriented,” “context-oriented,” or both. As the names suggest, task-oriented phrases are concerned with setting up mathematical tasks, whereas context-oriented phrases set up public domain contexts. These codes (terms) were emergent in my study, which means they were not planned in advance and are not derived from Dowling’s language of description. They allow me to gauge the extent to which the curricular texts in this
224 analysis exhibit metonymy – an important construct in Dowling’s language of description. Specifically, texts that were both non-principled and heavily task- and context-oriented would fail to exhibit metonymy precisely because the relationship between the tasks or contexts at hand and the principles that regulate secondary mathematics are not stated clearly or fully.
5.3.1 Findings about the Distribution of Mathematical Discourses
As might be expected based on my findings about domain messages, discourse in the specialist Rhoad text was more saturated, more principled, and more metonymic than in the non-specialist texts. As Figure 5.6 illustrates, discourse saturated the specialist text and did not saturate the non-specialist texts. Rhoad uses approximately one paragraph of discourse (i.e., 4 to 6 phrases) more per page than the non-specialist texts. I&A Bees Rhoad Sentences per Page 5.9 7.9 11.8 Principled Discourse 4.0% 9.0% 36.0% Non-Principled Discourse (96%) (91%) (64%) - Context- & Task-Oriented 28.2% 30.0% 0.6% - Task-Oriented Only 43.3% 28.5% 53.9% - Context-Oriented Only 19.8% 28.7% 6.4% - Other (Organizational) 4.7% 3.8% 3.1%
Figure 5.6. Discourse Features of the Three Mathematics Texts
225 Discourse in Rhoad was principled in explicitly stating postulates, theorems, definitions, as well as in illustrated formal reasoning. As Figure 5.6 shows, phrases in
Rhoad are four times more likely to be principled than those in Bees* text and nine times more likely than those in I&A. In fact, Rhoad is deliberate about making disciplinary principles explicit at the beginning of each section. The preface for section 11.1, for example, states and illustrates four property of area postulates (e.g., the additive property of areas). In the following section, Rhoad states the first of twelve area-related theorems in the chapter, namely:
The area of a parallelogram is equal to the product of the base and the height.
A ‘ bh, where b is the length of the base and h is the height. (p. 516)
Rhoad also provides a formal, five-line proof of this theorem. This proof builds metonymically, in a step-by-step fashion, on previously stated definitions and postulates.
Indeed, this proof is representative of the way that Rhoad generally makes close metonymic connections between principles, formal reasoning, and exemplars.
Discourses in the non-specialist Bees* and I&A texts, on the other hand, are less dense, less principled, and less clearly metonymic. I&A and Bees* exhibit low discursive saturation, with an average of only 5.9 and 7.8 sentences per page respectively, whereas
Rhoad averages 11.8 sentences per page. Discourse in the non-specialist texts are less principled with four- and nine-percent of mathematics principle-referenced statements, respectively, in I&A and Bees* compared to 36 percent of phrases in Rhoad. Non- specialist texts limit discursive access to the formal reasoning and principles that undergird disciplinary mathematics. Discourses in I&A and Bees* also fail to make the close metonymic connections in discourse as Rhoad did. Both non-specialist texts have
226 sequences of interrelated tasks that stretch out over several activities or class periods.
Explicit discursive connections could have been made between the tasks or between tasks and mathematical principles, however, these connections were either left up to non- specialist readers or deferred. Again, as Figure 5.6 makes clear, the non-specialist texts are primarily concerned with setting up mathematical tasks and public domain contexts rather than making principled metonymic connections.
Figure 5.7. A Textual Learning Environment in the Non-Specialist Texts
Consider the “Nailing Down Area” activity shown in Figure 5.7. It is one of two area-related activities from Bees* that I include, albeit with slight modifications, in the
I&A unit. To begin with, Nailing Down Area exhibits low discursive saturation, using a minimal amount of alpha-numeric text to accomplish its pedagogic goals. The discourse in this activity is also mostly non-principled (the second sentence is an exception, given
227 that it introduces the square unit as a measure of area). Instead of relying on principled discourse, the activity foregrounds textual representations of geoboards (i.e., indexical symbolic mode) to create a textual “learning environment” where non-specialist readers explore the topic of area. As Figure 5.7 suggests, icons (e.g., photographs, cartoons) and mathematical indices (i.e., geoboards, polygons, graphs) are used to a much greater extent in the non-specialist texts than in the specialist text and, in some sense, these textual objects replace (principled) discourse.
The Nailing Down Area activity was context-dependent in a way that tasks in
Rhoad are not. First, in Nailing Down Area the mathematical ideas are to emerge through explorations with virtual or concrete geoboard-manipulatives. In this activity, as in Tiling the Room (Figure 5.3), non-specialist readers might get the (false) sense that the material, rather than the social, world regulates secondary mathematics. Second, the image of a hammer and nails links metaphorically to the name of the activity and to nail-like pegs on a geoboard. This image, along with the title of the activity, suggests that mathematics is an active – and possibly hazardous – endeavor that participates in the real world of work.
Like the hammer, mathematics is a tool that gets things done in the real world. Note that I include this image in my version of Nailing Down Area in I&A, yet it is not part of the
Bees* text, however, similar references to the world of work are included in the Bees* text.
While the discourses in all three texts are predominantly task-oriented, there is a difference in how they set up and use mathematical tasks. The Nailing Down Area activity illustrates that non-specialist texts ask readers to do such things as “explore,”
“create,” “construct,” “discover,” “generalize,” and “explain.” This discourse positions
228 non-specialist readers to take charge of their own learning and, therefore, is empowering.
However, given that this student-centered positioning is used to the virtual exclusion of principled discourse, it is simultaneously disempowering because it lacks direct connections to esoteric or disciplinary mathematics. In contrast, Rhoad readers are rarely asked to explore and explain mathematics in a student-centered fashion, so might be considered disempowered by the text in this regard. Instead, students are asked to “find” answers to tasks, essentially rehearsing the application of previously stated principles and procedures, in order to “master” secondary mathematics. Figure 5.6 supports this assertion by illustrating how few of the task-oriented phrases in Rhoad have a context- or non-disciplinary orientation. In sum, then, tasks in the non-specialist tasks are
(apparently) structured by the material world or by non-specialist readers, whereas tasks in the specialist text are clearly structured by a more expert other.
Finally, while in many ways they much alike, discourses in the non-specialist texts differed somewhat with respect to each other. As Figure 5.6 indicates, discourse in
Bees* is about twice as principled as in the critical I&A text. Most of the principled discourse in Bees* take place in two esoteric domain activities in which parallelograms and trapezoids are defined. As the findings about domain messages indicate, the fact that
I&A was less principled than Bees* is because it extends attention to critical activities in the public domain where mathematical principles are not coherent, in any disciplinary sense, unless somehow the sociopolitical phenomena map onto the disciplinary structures of mathematics. Relatedly, I&A also exhibits less metonymy, or connective complexity, than Bees*, again, because the critical themes it develops in the public domain are not clearly connected to the secondary mathematics it develops in the expressive domain.
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5.4 Distribution of Mathematical Content
The third dimension of my findings relates to distribution of mathematical content in the texts. Dowling notes that mathematics texts vary with respect to how intensively and extensively they develop mathematical topics. Of course, mathematics texts might also distribute non-mathematical (e.g., historical, political) content, as was the case with
I&A. Only the I&A includes such non-mathematical content and, indeed, only I&A has social and political consciousness-raising as a goal.
Once again, I used individual activities as the unit of analysis to measure distribution of content. Specifically, I began by synthesizing two lists of the content covered from the teacher guide that accompanied the Bees* and Rhoad texts. I created the synthesized list (the left-hand column of Figure 5.8) by counting area-related topics that were the same or very similar on the lists from each of the comparative texts (e.g., trapezoidal area formula). I also included area-related topics covered by one text but not the others (e.g., Pick’s Theorem in Bees*). There was a high degree of correlation between the topics covered in I&A and Bees*. This, of course, was because I modeled the I&A text, in part, on Bees*. Actually, there was a relatively high degree of overlap in the content under consideration in all three texts. After completing the synthesized list, I then examined the extent to which the listed topics are covered both extensively and intensively in each text; that is, I counted the activities or chapter sections in which the topic under consideration figured prominently.
5.4.1 Findings about the Distribution of Content
230
Figure 5.8 (next page) summarizes the results of my findings about how mathematical content is distributed in the three texts. A single star (*) means the content is covered in one activity or chapter section, two stars (**) means that the topic is emphasized in two or more activities or text sections. As Figure 5.8 makes clear, I&A distributes only about three-fifths of the area-related topics that the comparison texts distribute in similar amounts of textual space. Moreover, I&A does not reach many of the more rigorous mathematical subtopics (first column of Figure 5.8) that are developed in the other (non-critical) texts. Bees* treats circular area in another unit while Rhoad covers applications of trigonometry elsewhere. To be fair, I&A does not include homework activities which means it was at a disadvantage in this analysis. However, this fact alone does not entirely account for the significant difference between I&A and the other two texts. I&A fails to expand similar amounts of mathematical content precisely because it is busy – in 11 of 27 activities – expanding critical sociopolitical themes in the public domain. Put simply, critical activities in I&A are subtractive in terms of mathematical content.
As one might expect, area-related topics are covered in the non-specialist texts that appear to be too elementary for the Rhoad reader to consider. For example, the topic of computing area by counting square units (Figure 5.7) is not fully developed in Rhoad, whereas it figures prominently in several activities in the non-specialist texts. In fact,
Rhoad concisely tells its specialist readers that counting squares is “neither the easiest nor the best way to find the area of a region.”
231
Topic I&A Bees Rhoad
Measuring Area by Counting Units ** **
Alternative (Non-Square) Units * *
Real Area Measures (e.g., cm2, in2, ft2) ** *
Additive Property of Area ** ** **
Perimeter & Area ** *
Areas of Rectangles ** ** **
Areas of Triangles ** ** **
Areas of Parallelograms ** ** **
Areas of Trapezoids ** * **
Area of Regular n-gons ** **
Area of Semi- or Non-Regular Regions ** ** *
Areas of Kites **
Circular Area **
Pick's, Hero's, or Brahmagupta's Formulas * **
Ratios of Areas **
Trigonometric Applications & Area **
Special Triangles & Area * **
Non-Area Math Development * ** *
Computational Technology & Area **
Figure 5.8. Distribution of Mathematical Content by Text
232 5.5 Discussion
The results of the textual analysis presented in this chapter indicate that non- specialist texts stress real world relevance – critical, vocational, or otherwise – comes at the cost of developing principled discourses and, in the case of I&A, mathematical content. I illustrate how I&A’s critical goals (e.g., consciousness raising, cultural relevance) require extended textual (or discursive) attention in the public domain. This focus is subtractive mathematically-speaking, which is a concern given that I&A is designed for students historically underserved by school mathematics.
It must be stated up front that these particular findings about I&A do not necessarily point to a more general problem with CM. I am aware that tension between the critical and the mathematical that surface in I&A, might have been the result of my inappropriate or incomplete curriculum design. That said, while I cannot rule out inadequate curriculum design, the fact that I&A was subtractive mathematically speaking points to a general theoretical problem with CM. Evidence for this claim comes from three sources, namely, that: (1) the reform Bees* text fails to be principled in a manner similar to the critical I&A text, (2) other critical arithmetic texts (e.g., Frankenstein,
1989) also attend disproportionately to the public domain, and (3) theoretical, empirical, and philosophical scholarly work point to a disconnect between academic (mathematics) and everyday activity in utilitarian-focused instruction (Bernstein, 1990; Dowling, 1998;
Ernest, 2000; Lave, 1988; Love & Pimm, 1996; Nunes, Schliemann, & Carraher, 1993;
Pimm, 1995).
First, as my textual analysis suggests, tension between “real world” and mathematical orientation that surfaces in I&A also is evident in the reform Bees*, albeit
233 to a lesser degree. Both made a series of externally oriented connections from secondary mathematics to the real world, while they generally failed to make principled within- discipline connections. As Figure 5.5 indicates, CM turns to the public domain whenever its main concern is with raising readers’ sociopolitical consciousness.
Second, my informal analysis of Frankenstein’s (1989) critical arithmetic textbook indicates that its critical activities (i.e., expositions, problems) fall within
Dowling’s public domain. Indeed, many of her critical problems are similar to the I&A
Math, Equity, and Economics activity (see Figure 5.4). These problems are structured, first and foremost, by sociopolitical phenomena (themes, data) rather than arithmetic.
This assertion is true of major portions of Frankenstein’s text (see Lee, 1990 for a discussion of similar ideas).
Third, theoretical, empirical, and philosophical work point to a disconnect between mathematics as a discipline and the non-academic world. Bernstein (1990) argues that academic disciplines, such as mathematics, deal with specialized, and thus well bounded, practices and discourses. Academic discourses and practices are very different from everyday discourses and practices. Most people have no difficulty recognizing the boundaries between the two. With this in mind, Dowling (1998), a protégé of Basil Bernstein, argues that mathematics texts cannot fully develop public domain themes without sacrificing mathematical development in the esoteric domain of practice. Indeed, this displacement was what happened to the content of my critical I&A text.
Hence, while I&A, or the way I designed it, might magnify some of the problems inherent in critical (secondary) mathematics, these problems appear to reflect a more
234 general problem with CM, which is that its proponents (Frankenstein, 1991, 1995;
Gutierrez, 2002; Gutstein, 2003; Skovsmose, 1994; Tate, 1995) conceptualize (school) mathematics as a tool subject to use to help subordinated people fight for justice rather than as disciplinary preparation. Although I subscribed to this orientation to school mathematics at the beginning of my study, I now believe the use of secondary mathematics as a political tool is flawed. It promises potential political use-values rather than exchange-value (i.e., mathematics as a credentialing subject valued in educational settings). It is difficult to see how most secondary topics (i.e., Pythagorean Theorem, quadratic equation) might be powerfully applied in a political sense. As discussed in
Chapter 4, the problem with teaching secondary mathematics as a political tool rather than a set of disciplinary ideas is that it ultimately fails to provide students with principled secondary mathematics that serves as the cultural capital that opens doors to elite universities and desired careers.
While they may think my assertions are exaggerated, I hope that CM educators will acknowledge my concerns about emphasizing real world contexts in school mathematics. They have to a limited extent. For instance, Gutstein (2005) notes a similar tension between the critical and mathematical:
There is a dialectical relationship between developing mathematical power and
teaching students to use mathematics to study, and potentially change, structural
inequality. The two processes can facilitate each other, under certain conditions,
but there is a tension between them. To learn rich mathematics, students at some
point have to leave the situation in which the mathematics is embedded and focus
on the mathematical ideas themselves. … [W]hen learning mathematics for social
235 justice, context and content are important, as important as mathematics, and so
teachers need to explicitly reconcile the interrelationships between the different
curricular components.
Conversely, to prioritize the sociopolitical context, teachers need to draw
students into studying reality and at some point leave mathematics to the side,
even though the mathematics may be the entry point into the investigation and
may be integral to understanding the complexities. There are ways to make the
connections between mathematical and social justice more obvious, for example
by having students explain concretely how they used mathematics to understand
particular issues. … But it should be clear that when students discuss why
immigrants come to the United States leaving behind families and homes, analyze
the genesis of bad credit, or discuss neighborhood gentrification, at some point,
mathematics may drop out of the conversation. This is inevitable and not
necessarily a problem, but it is a tension teachers have to purposefully negotiate.
Despite students learning that mathematics can help them understand “what
surrounds” them, there are indeed significant ideas on which mathematics can
shed only so much light, even if, as I mention earlier, mathematics is important to
understanding political issues. (108-109)
Real world contexts as highlighted in CM respond to the political end of having students understand and fight structural inequality, especially if the problem contexts in which mathematics is set are authentically related to their concerns. As argued previously, I now see that assumptions about the relationship between school mathematics and the real world are not straightforward. Mathematics does not emerge from real world contexts
236 naturally as implied by Gutstein and many other mathematics educators (Fendel, Resek,
Alper, & Fraser, 2000; Lesh & Doerr, 2003; NCTM, 2000). While it has an empirical basis, mathematics is primarily a social construction that does not correspond closely to structures in the material world (Ernest, 1998; Hersh, 1997; Lakatos, 1976; Rotman,
1987, 2000). While mathematics can be mapped onto some mathematical situations, it is not clear that such mapping facilitates learning (Wiliam, Bartholomew, & Reay, 2004).
Some might argue that, like Dowling, I restrict the boundaries and purposes of secondary mathematics to too great an extent. They might note that the pursuit of careers in mathematics, physics, or engineering is not the sole goal of secondary mathematics instruction, in particular for working class youth (see Chazan, 1996). Although there do not need to be clear, indisputable boundaries around mathematics, nevertheless, it must be recognized that what counts as secondary mathematics is shaped by what powerful decision-makers (e.g., university mathematics departments, policymakers, textbook and testing companies) say is important.
I now believe that the relationship between mathematics and sociopolitical contexts is weaker than I previously thought, so I am fairly certain that secondary mathematics cannot be reconceived as a powerful tool to help students understand, and potentially change, structural inequality. Data analysis (of the newspaper variety) may be helpful in raising critical consciousness, but it should not be mistaken as secondary mathematics nor as particularly good preparation for post-secondary mathematics. Real world data rarely conform to the rules that regulate secondary mathematics. At the secondary level, critical data analysis would seem to fit better in a course on sociology than in a college preparatory mathematics course.
237
Finally, I add two notes of caution. First, the examination of written curriculum is different from examining how groups of students and teachers enact it. It is possible that an experienced or skilled critical pedagogue might teach I&A so as to avoid the problems
I found with it as written. It is possible that the mathematical and sociopolitical facilitate each other in practice in ways they do not in linear written texts (Gutstein, personal communication). I would argue, however, that written texts both constrain and embody what is pedagogically possible. If one cannot synthesize the dialectical tension between the critical and mathematical in texts, then it is hard to imagine how they can be synthesized in practice.
The second cautionary note is that the Rhoad text has many flaws. I find it repressive and would difficulty teaching with it in any context. I admit, however, that it has a clear disciplinary focus that the other texts lack. My personal belief is that students in specialist courses deal with repressive texts of this sort precisely because they know it is what universities value and stand to gain from submitting to this type of learning.
Students in lower track courses are less likely to submit to repressive traditional texts and instruction because they are less certain that they stand to gain from it. That said, I think that the reform IMP text would benefit from a more disciplinary-focused approach with preparation for college as the primary goal. One can present relatively open and conceptually rich mathematics problems in ways that students find engaging without heavy real world contextualization.
5.5.1 Responding to Ravitch and Clarifying my Views
238 In a June 2005 article in the Wall Street Journal, Diane Ravitch critiqued critical mathematics and ethnomathematics for “dumbing down” the mathematics curriculum.
Because I am critical of CM at the secondary level, my findings may be seen as supporting her critique. I want to dissociate myself with her position while clarifying my own. After blaming progressives for all that is wrong with mathematics education,
Ravitch concludes her opinion piece with the following statement:
It seems terribly old-fashioned to point out that the countries that regularly beat
our students in international tests of mathematics do not use the subject to steer
students into political action. They teach them instead that mathematics is a
universal language that is as relevant and meaningful in Tokyo as it is in Paris,
Nairobi and Chicago. The students who learn this universal language well will be
the builders and shapers of technology in the 21st century. The students in
American classes who fall prey to the political designs of their teachers and
professors will not.
This passage is representative of Ravitch’s broader critique of progressive and critical education in what it omits and oversimplifies. Contrary to Ravitch’s assertion that politically progressive educators are the real problem with mathematics education and hence the lack of economic competitiveness of subordinated students, or a threat to U. S. economic competitiveness, it is clear that currently most mathematics instruction adheres to the traditional model that she and other conservatives advocate (Stigler & Hiebert,
1999). Study after study shows that the “back to basics” approach she endorses, apparently on behalf of all mathematicians, is the primary culprit in dumbing down the
U.S. mathematics curriculum (Boaler, 1997; Schoenfeld, 1988; Stigler & Hiebert, 1999).
239 Ravitch implies that more politically mainstream or conservative educational scholars, unlike their progressive counterparts, simply want teachers to teach mathematics. In claiming that only left-wing educators have political designs, she invokes the ideology of non-ideology and implies that more conservative educators have no political agenda and adhere to no ideology (see E. Brantlinger, 1997). Ravitch fantasizes about a world that is calculable, rational, and devoid of (progressive) politics.
Her claim of political neutrality – even in mathematics instruction – is untenable. As I argued previously, traditional mathematics education, while not explicitly political, implicitly reinforces the status quo by, for instance, presenting mathematics as fact, and by being a critical filter that keeps a range people out of academia (Ladson-Billings,
1995; Schoenfeld, 2002; Sells, 1978).
Unlike Ravitch, CM educators (e.g., Gutierrez, 2002; Gutstein, 2003, 2005) acknowledge and respond to pressing problems in mathematics education and society.
Those of us who are interested in CM are concerned with social inequality and the role of mathematics education plays in maintaining it. We disdain the unequal social and school contexts in which students are asked to learn mathematics (Anyon, 1981, 1997; Kozol,
1991; Lipman, 2004). We are concerned that students fail to see the everyday and cultural relevance of what they are learning in mathematics courses. Ravitch ignores these issues in her critique of CM and, in so doing, implies that all students have realistic chances to use school mathematics as a steppingstone to “participate as builders and shapers of technology” (for differing views on this issue see Apple, 2000; Chazan, 1996; Popkewitz,
2004; Usiskin, 1995). Ravitch ignores the fact that, when compared with higher SES students, lower SES students have limited access to the resources (e.g., high quality
240 teachers, family wealth) that might allow them to become mathematicians, scientists, and engineers (Darling-Hammond, 2004; Kozol, 1991, 2005; Lankford, Loeb, & Wyckoff,
2002). Even if more students than currently were able to gain mastery over the “universal language” of mathematics, how many would realistically participate as builders and shapers of technology under the current socioeconomic order? Jobs in which people build and shape technology are limited and increasingly are outsourced to cheaper labor markets. Unlike some educators (e.g., Moses & Cobb, 2001; NCTM, 2000; Swetz &
Hartzler, 1991) I do not believe the current omnipresence of computational technology translates simplistically into the argument that people “need” more or better school mathematics to fully participate in politics and the workplace, or arm oneself against it
(Skovsmose, 1994). Rather than a mastery of secondary mathematics, under the current social system, students need credentials to have access to higher education. It is troublesome that Ravitch disregards these issues in her rush to blame progressives for the poor state of mathematics education.
5.6 Conclusion
In this chapter, I presented the results of my comparative textual analysis of three mathematics texts. I conducted this analysis as a means of understanding the effects of incorporating CM into the dominant secondary mathematics curriculum. I show that I&A devotes a considerable amount of textual space to the (re)production of public domain activity in order to accomplish its critical goals. Relatedly, I document that I&A trades principled, within-discipline discursive connection-making for external, real world connection making. In addition, I show that I&A develops a relatively limited amount of
241 area-related mathematics content compared to the other texts. I also provide evidence to support my claim that problems with I&A point to more general problems with CM. The focus on real world relevance means that I&A fails to (re)produce secondary mathematics as mathematical cultural capital valued by university mathematics departments and policymakers.
242 CHAPTER SIX: DISCOURSE AND PARTICIPATION IN REFORM ACTIVITIES
In Chapters 6 and 7, I present findings from the critical discourse analysis I conducted on transcriptions of eight hours of videotaped lessons from the Guevara night course. Systematic inspection of this data set allowed me to understand the nature of classroom discourse and student participation during standards-based reform activities.
Because these lessons came from early and late weeks in the night course, I was able to examine how students’ discursive participation changed and failed to change over the time span of the course.
In this chapter I discuss my implementation of, and my students’ participation in, the non-critical, reform activities that comprised some 80% of the night course. One of my goals as a reform teacher was to position my students authoritatively, or subjectively, with respect to school mathematics. The reform instruction analyzed in this chapter lays the groundwork for the analysis of my critical mathematics (CM) instruction in Chapter
7. Because CM as theorized by Gutstein (2003, 2005) and Gutierrez (2002) builds on standards-based reforms, it is important for me to provide a detailed analysis of what reform mathematics looks like in my classroom. As described in Chapter 2, these scholars maintain that authoritative positioning of students within mathematics is necessary, but not sufficient, for CM instruction to be successful.
In sections 2.2 – 2.5 of Chapter 2, I provide definitions of terms I use throughout
Chapters 6 and 7, including: hegemony, student subjectification, student objectification, critical agency, resistance, conformity, reform instruction, traditional instruction, social language, Discourse model, and Initiation-Response-Feedback (IRF) structure. It might be wise for readers to review these definitions before reading this chapter. The methods
243 of discourse transcription, in Section 3.6.1 of Chapter 3, might also be reviewed. I use
“authoritative positioning” rather than the equivalent term “subjective positioning” here because the former is used in the reform mathematics education literature. At several points in this chapter I compare reform activity from my classroom to the idealized vision of reform activity described by the National Council of Teachers of Mathematics
(NCTM) (1989, 1991, 2000). Its vision is:
Imagine a classroom, a school, or a school district where all students have access
to high-quality, engaging mathematics instruction. There are ambitious
expectations for all, with accommodation for those who need it. Knowledgeable
teachers have adequate resources to support their work and are continually
growing as professionals. The curriculum is mathematically rich, offering
students opportunities to learn important mathematical concepts and procedures
with understanding. Technology is an essential component of the environment.
Students confidently engage in complex mathematical tasks chosen carefully by
teachers. They draw on knowledge from a wide variety of mathematical topics,
sometimes approaching the same problem from different mathematical
perspectives or representing the mathematics in different ways until they find
methods that enable them to make progress. Teachers help students make, refine,
and explore conjectures on the basis of evidence and use a variety of reasoning
and proof techniques to confirm or disprove those conjectures. Students are
flexible and resourceful problem solvers. Alone or in groups and with access to
technology, they work productively and reflectively, with the skilled guidance of
their teachers. Orally and in writing, students communicate their ideas and results
244 effectively. They value mathematics and engage actively in learning it. (NCTM,
2000, p. 3)
NCTM evaluates this vision as “ambitious” but “essential.” Indeed, as I described in
Chapter 3, the night school setting was far from ideal and Guevara students did not come close to having the same access to a high quality education as students in more privileged settings. In this chapter, however, I document some progress as classroom practices moved toward the NCTM vision. Despite recognition of some success, this chapter is not about a celebration of my pedagogical accomplishments. Rather, I attempt to understand the implementation of reform activities by continuing to apply the critical stance adopted in Chapters 4 and 5 to the discourse analyses in this chapter and the next. As a critical scholar, I understand critique is a mechanism to advance scientific knowledge
(Habermas, 1984; Kuhn, 1962; Popper, 1968).
6.1 Overview of Four Findings Discussed in This Chapter
The discourse analysis presented here is both descriptive and comparative. To produce the findings in this chapter, I conduct a fine-grained analysis of reform activities.
I also compare students’ response to reform activity at the beginning of the course (week
2) with what occurred at the end (week 7) because I was interested in how students’ discursive participation changed over time. My analysis shows that the following four interrelated issues were salient in reform mathematics: (1) positioning students authoritatively with respect to mathematics required frequent overt pedagogical attention at the start of the course that was not needed as much by the end of the course; (2) students took on a range of complex stances (i.e., agentive, resistant, conformist)
245 throughout the course, but these shifted noticeably toward agentive engagement during reform activities; (3) students’ collective traditional Discourse model of what secondary mathematics should be surfaced time and again throughout the course, but with less frequency and intensity toward the end of the course; and (4) students and I constructed a problematic and persistent hierarchy of mathematical learners which was reinforced in reform activities, particularly at the outset of the course, but did diminish over time. I discuss these findings in this section.
First, my analyses show that, as a reform teacher, I had to work hard to position my students authoritatively with respect to mathematics at the beginning of the course.
While I continued to struggle with turning mathematics authority over to them throughout the course, I needed to do so less overtly and regularly by the end of the course. This progression was desired because my overall goal was to use reform activities to create discursive and mathematical spaces where students could develop and discuss their own mathematical thinking. I implemented student-centered classroom practices in a variety of ways:
1. I used the reform Interactive Mathematics Program (IMP) (Fendel, Resek,
Alper, & Fraser, 2000) as the foundational curriculum for the night course. It comprised about 80% of the curriculum with critical activities making up most of the remaining
20%. IMP curriculum usage allows students to regularly “do mathematics” as mathematicians might (Stein, Grover, & Henningsen, 1996). IMP stresses that students should reason for themselves about relatively novel mathematical tasks.
2. I generally responded indirectly, if at all, to students who asked me to tell them answers or precisely how to think. At the same time, I encouraged students to speak up
246 when they were confused and to see mistakes as a natural and valuable part of learning mathematics.
3. While didactic occasionally, I generally kept my authoritative teacher voice quiet in order to open a space for students to explain their mathematical thinking, show themselves as competent regarding school mathematics, and initiate, respond to, and evaluate their own and each other’s mathematical statements.
4. While the night school administration did not allow me to arrange the desks into a group format, I encouraged students to work collaboratively with their neighbors by explicitly stating, for example, “Ask Lupe,” or “I need you to help your classmates.”
Following these pedagogical guidelines was difficult for me in the first several weeks of the night course. As I illustrate, this was primarily because my students did not generally share the goal of transferring some mathematical authority from me to them. In interviews and casual conversations, students indicated that they were not accustomed to being asked to explain, explore, and reason for themselves in mathematics and did not necessarily embrace my requests to do so with open arms. On the second day of the course, when I outlined how I would assess student “collaboration” and
“communication,” many stared at each other in apparent disbelief, while others put their heads down on their desks. Obviously, instilling reform practices requires more than a pep talk (Cohen & Lotan, 1997). Positioning students authoritatively in mathematical activity required constant attention, assessment, and reinforcement.
In the reform lessons I analyzed, students exhibited a complex mix of stances from resistance (not immediately starting work), to conformity to traditional practices
(demanding mathematical answers from me), to engagement (calling me over to explain
247 their mathematical thinking). It is important to point out that most students engaged with the reform activities for considerable periods of time in every lesson. It was rare for a student to do absolutely nothing mathematical in an hour period. And, as I document, students engaged more readily and actively with reform mathematics in Week 7 than in
Week 2.
Third, and relatedly, students and I held competing Discourse models of school mathematics at the outset of the course. My Discourse model was generally in sync with the reform instruction described in the NCTM vision (2000). In contrast, most, if not all, students held a traditional Discourse model of school mathematics. In interviews conducted before the class began and in classroom conversations, to varying degrees, students stated or otherwise indicated the following beliefs about mathematics:
1. Students should not encounter an unfamiliar idea without first receiving preparation from mathematics teachers.
2. Mathematics teachers should immediately address mistakes, misconceptions, and uncertainty with direct answers or advice.
3. Students should be able to solve mathematics problems in a relatively short period of time.
4. Lectures, notetaking, individual deskwork, and frequent summative testing are the norm.
5. The teacher (and text) are authoritative, and mathematical knowledge flows unidirectionally from these sources.
6. The students’ job is to internalize mathematical methods and facts that they’ve received from the teacher and text.
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7. Students cannot learn much mathematics from (most) peers, therefore, they need not pay attention to other students’ mathematical ideas.
8. As a quantitative subject, mathematics involves minimal reading and writing.
9. Mathematical “ability” is innate, so school mathematics grades and test scores are indicators of innate intelligence. (Note that in pre-study interviews, several students rejected the idea that mathematical ability is innate, with a few arguing that mathematical ability and performance were based primarily on individual interest and effort. However, as Excerpt 6.1 shows, the correlation between in-class performance and innate ability was invoked by students in whole class activities, contradicting their claims in interviews.)
10. With the exception of pacing, curriculum and instruction do not vary across contexts; students in elite and less elite settings receive the same content which indicates that social class and race play no role in determining the nature of the secondary mathematics curriculum.
11. Schools are meritocracies; individual distinctions, rather than discrimination or social inequality, account for educational outcome differences.
With the exception of the last two points, I refer to this set of beliefs as the
“traditional Discourse model for school mathematics.” The last two points are part of a related Discourse model of society or social inequality that I discuss in depth in Chapter
7. The clash between my reform goals and my students’ collective Discourse model for school mathematics is clear. When students invoked the traditional Discourse model in whole class discussions, whether or not they realized it, they worked against my attempts to position them and their peers more authoritatively.
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Fourth, analyses of activity from Week 2 and Week 7 indicate that the students and I developed a problematic and persistent hierarchy of mathematical learners. This hierarchy was particularly salient at the beginning when verbally active students positioned themselves and their peers along a stratifying line of mathematical achievement and performance. There were, in fact, considerable differences in my students’ background preparation prior to entering my course; in the pre-tests five students scored in the 35%-50% range, five others had 15%-25% correct, while the remaining 14 answered less than half of the fifteen questions entirely correct (see
Appendix F). Several students did not take the pre-test, including two who were as strong as the higher scoring students. Excerpt 6.1 makes it clear that my students became aware of each other’s mathematical differences early on in the course. I must confess that I was generally unaware of this student differentiation phenomenon while teaching the night course. Only after I had time to view the videotaped lessons and reflect on the interactions did I see how powerful this differential validation of students was in undermining classroom equity.
In the three major sections that follow I first present two excerpts from Week 2 that illustrate the four findings discussed above. I provide quantitative summary data from my discourse analysis that indicates a shift in student participation toward the
NCTM’s reform vision. I then present two excerpts from Week 7 that further illustrate my assertions about change in our discursive practices.
6.2 Initial Student Reactions to Mathematical Subjectification
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In this section, I discuss two excerpts of discourse from reform activity in Week
2. Excerpts 6.1 and 6.2 came from the first hour of the first day of Week 2. In this first hour, I followed up on an activity begun in Week 1 in which students used protractors to find patterns in the angle sums of triangles and quadrilaterals. I began the lesson with the mathematical opening activity shown in Figure 6.1. The goal of this opener was to review angle sums in simple polygons and extend these ideas to pentagons, hexagons, and general n-sided polygons. After the opener, the lesson moved on to a reform activity called “Angular Sums” from the IMP curriculum that, as the title indicates, deals with the same ideas. Excerpt 6.1 is part of the transcript of the opening activity and Excerpt 6.2 includes discourse from the end of Angular Sums.
I selected Excerpts 6.1 and 6.2 from my discourse transcripts primarily because they highlight how the four findings previously identified emerged in Week 2 activity.
They show how: (1) I attempt to shift mathematical discourse and authority from myself onto students; (2) students simultaneously adapt to, resist, and challenge reform practices;
(3) students’ collective traditional Discourse model for school mathematics emerges to challenge reform practices; and (4) students with stronger mathematics backgrounds generally had more access to mathematical subjectivity so they and I (re)produced a hierarchy of mathematical learners.
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Figure 6.1. Isosceles Triangles Opening Activity
Figure 6.1 presents the first two of three problems included in the opener in the first hour of class in Week 2. Excerpt 6.1 is of students working on these problems.
Opening activities were part of a daily routine process implemented early in the course.
They had several pedagogical purposes: first, to have students work on mathematics as soon as the bell rang; second, to remind students of mathematical ideas we had considered previously that needed rehearsal to prepare for the current day’s activities; third, to extend the reform mathematics to more formal or traditional expressions of similar content; and fourth, as a forum for students to present mathematical ideas.
In Excerpt 6.1, two students, Diamond and Deno, came to the board to present their solutions to the problems shown in Figure 6.1. Of course, inviting students to the board is not subjectifying in and of itself, hence is only a surface feature of reform mathematics. While the opening task was more procedural than conceptual, from the students’ perspective, my instruction, including getting students to come to the board and explain their ideas, was a departure from the traditional instructional norm featured in urban schools that serve lower SES communities (Lipman, 2004). It should be noted that
252 in Week 1 and earlier on in this activity, in fact, I had described what it meant to explain mathematics at the board and in written responses. I revisited this idea a minute before this activity began when I told Diamond, “okay, yeah, if you could ex-plain it” emphasizing the word “explain.” Of course, at this point in the course, I was aware that most students did not understand that an explanation of their thinking was different from the “telling” of traditional mathematics teachers. I understood that developing my students’ understandings of explanation would take time.
6.2.1 From Beginning of First Hour of Week 2: Analysis of Excerpt 6.1
1 Diamond (mumbles at overhead): so I divided / (inaudible)
2 Deno gets up from desk and sits at desk in front of room to better see.
3 Me (to Diamond): okay // yeah / if you / could explain it
4 Me (to class): okay // so she’s // Diamond’s / going to explain the / second one here
5 /// I won’t do this / al/ways / on the opener / I won’t always / uh //
6 Diamond: alright // it’s one-eighty // subtract by thirty // is one/fifty // and then you
7 / divide that // by two / and then you get seventy-five //
8 Diamond abruptly leaves overhead and takes her seat in back of room.
9 Lupe: [that’s i:t? //
10 Deno: [okay so // so / that mean / that one-eighty go in the / one over there? //
11 Diamond: // no: /// cause / the whole thi:ng // equals to / one-eighty // got it? //
12 Deno (laughs out loud): hih / hehhehheh / hehheh //
13 Efrain (quietly to Deno): you / stupid //
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14 Lana: that’s my friend / right there // that’s my / be:st / friend //
15 Efrain looks at Lana, laughs, and says something inaudible.
16 Lana: wha::t?
17 Efrain: no:thing //
18 Lana: oh //
19 At same time, on the other side of the room, I am talking quietly with Lupe who
20 has pointed to her notebook.
21 Me: you doin’ okay / Lupe? //
22 Lupe: (inaudible)
23 Me (point to Lupe’s paper): I don’t wanna / sound like a / math nut // but //
24 Deno approaches the overhead to do the other problem.
25 Deno writes 35º for y on overhead instead of the anticipated 33º.
26 Me (to class): so // y is easy // right? // is everybody clear why / y / is thirty three
27 / here? // you okay with that // Lana? //
28 Lana (squinting): iyeah // I know tha:t // I’m tryin’ to / write / that down //
29 Me: but you can’t / see it? // you know you can / walk up //
30 Deno is writing at overhead
31 Diamond (to me): how’d he get / seventy? // I got / one-fourte:en //
32 Deno (at board): cause // oh / that’s thirty / three // da:mn //
33 Kampton (from back of room): stu:pid //
34 Shannon laughs and covers mouth with elbow
254 35 Me: yeah // it’s a / thirty-three // it’s hard to see / yeah //
36 Deno: I thought / thirty/five //
37 Princess: It doesn’t matter // [it still // gonna be (inaudible) //
38 Lana (to Deno): [boy / you better / hang in there //
KEY: (1) Bold – louder; (2) Underline – rise in pitch; (3) “:” extended vowel; (4)
“/” short pause; (5) “//” longer pause; (6) “[,“ “[[,“ “[[[” simultaneous speech
Excerpt 6.1. From Opening Activity at Beginning of Class on Monday of Week 2
Note, to begin with, the discussion that students and I co-produce is much looser and more open to student contributions (structurally speaking) than the rigid Initiation-
Response-Feedback (IRF or IRE) structures found in scripts of traditional classrooms cited in the literature (Cazden, 2001; Gutierrez, Rymes, & Larson, 1995; Mehan, 1979).
This already was typical of student presentations and class discussions in Week 2. After
Diamond begins and until lines 26 and 27, I step aside and keep out of the way of the discussion, both discursively and physically. I begin to position students authoritatively
(subjectify them) by opening the floor to them and allowing them to show themselves as competent with respect to school mathematics. Both Diamond and Deno exhibit a degree of mathematical engagement, what the criticalist Pruyn (1999) calls critical agency, in response to this authoritative positioning by taking considerable control of the mathematical activity at hand. Although the opening activity is not particularly conceptual, students begin to present their ideas and ask their questions relative to the task. By the second week, this pattern had already taken hold despite the fact that many appeared unfamiliar with taking charge of learning at the outset of the course.
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At the beginning of Excerpt 6.1, Diamond contributes positively to the flow of mathematical activity by volunteering to present her solution to the class. At the same time, she exhibits conformity to the traditional Discourse model of school mathematics when she writes down the procedural steps for her solution and reads them off without adequately explaining her reasoning (lines 6, 7, and 11). Diamond seems to have learned that the final solution to mathematics problems is more important than the solution process. If she had not internalized the traditional Discourse model, Diamond might have built on the work we had done the previous week to argue that the sum of angles in any triangle is 180º. Then she could have explained how her equation represented this idea. I doubt that Diamond was consciously working against my reform plans, instead she reverted back to fit the traditional mathematics model she had internalized. I would have to make my goals as a reform teacher more explicit and began to do so in Week 2 of the course.
Reform mathematics was unfamiliar terrain, as revealed by students’ participation early in the course and interview comments. Many, if not all, relied on a traditional model of school mathematics to make sense of the reform practices I implemented in the first weeks of the night course. It was apparently the only model most students held.
Diamond’s actions represented in the excerpt above indicate that students filter their understandings of “explain” through their traditional Discourse model; that is, they interpret explain as a command to regurgitate the teachers’ methods. In a reform IMP lesson from Week 2, another student, Princess asked me, “but listen, but what do they mean by explain?”
256 Deno, like Diamond, also exhibits mathematical agency in Excerpt 6.1 by volunteering to present his ideas about the second problem. Deno, however, is less certain about his answer than Diamond was of hers. Because I see mistakes as valuable, I deliberately encourage Deno to go to the overhead. I could have called on a more accomplished student in order to prevent Deno from making a public display of his mistaken ideas about the problem. Part of the problem is that Deno was absent the previous day and appears not to understand what most of his peers know. Deno arrives at an answer of 70º for z (Figure 6.1) by first misreading 33º as 35º and next assuming correctly that y has the same value as the first angle. He mistakenly assumes that z is equal to the sum of the other angles. On the short run, Deno needs direct assistance to understand the principles and complete the problem. Yet, he seems willing to work at it and get advice from classmates and me to get it right.
Diamond invokes another aspect of the traditional school mathematics Discourse model when, instead of asking Deno to explain his thinking, she defers to me, the math expert in the classroom, with “how’d he get / seventy?” (line 31). This discursive action objectifies Deno as not capable of explaining his own mathematical thinking. Diamond’s use of “he”(instead of “you”) excludes Deno from the conversation and perhaps leaves him feeling as if he were on display at the overhead. By deferring to me, Diamond calls on me to take up the authoritative teacher role that I am attempting to avoid. My relative silence is at odds with her expectations regarding the role of a mathematics teacher.
By not immediately responding to Diamond’s question, I give Deno a chance to explain. When he states, “cause – oh, that’s 33, damn!” (line 32), Deno admits his confusion. At this point in the lesson, Deno realizes that he has been exposed to the
257 evaluative gaze of teacher and classmates, and is visibly positioned to be judged as low ability by those who feel school mathematics performance equates with innate intelligence. Not surprisingly, given the competitive and failure-oriented nature of traditional urban schooling, a few students jump in to judge Deno. Kampton and Efrain call Deno stupid (lines 13 & 33). In do doing, they objectify him as they invoke the evaluative aspect of (traditional) school mathematics. In calling him stupid, Efrain and
Kampton remind Deno, and others watching, that a primary purpose of (traditional) school mathematics is to assess students’ mental capacity and plan subsequent interactions with them accordingly.
Whether or not they do so intentionally, Diamond, Efrain, and Kampton use this opening activity to construct a hierarchy of mathematical learners, placing Deno at the objectified (“stupid”) end and Diamond a mathematically-competent, subjectified position. When Diamond shoots her classmate’s questions down by asserting a mathematical fact (i.e., that the sum of the three angles of any triangle is 180º) (line 11), she also positions them at the objectified end as might be done by a traditional teacher.
Nevertheless, Deno’s laughter (line 12) shows some resistance, and perhaps agency, as he indicates that Diamond did not adequately respond to his question. To shore up her superior position, the more confident Diamond establishes herself as mathematically authoritative relative to classmates by asking me to confirm her own accuracy (line 31).
Kampton’s comments about Deno support hierarchy construction. However, as a cynical onlooker and commentator, Kampton attempts to locate himself outside the classroom intellectual hierarchy. Efrain’s objectifying comments to Deno are not loud enough to be heard by many. (At the time I was conducting this lesson, I did not hear either comment,
258 but definitely heard them in the playback of the videotaped lesson. I had placed a plate microphone near these students’ desks.) To make matters worse, when Deno returns to his seat from the overhead, Efrain quietly asks him, “you in a special ed class or somethin’?” to which Deno responds, “man, my math off today.”
Whether or not it is intentional, Diamond and Deno are building relationships with me and with each other as mathematics students. Diamond establishes herself as a student who can be relied upon for correct mathematical answers. Deno is brave enough to risk coming to the overhead even when not sure of the answer. However, when I directly intervene in Deno’s attempts to explain mathematical principles, I reinforce the traditional model of promptly supplying mathematical answers and explanations when students struggle rather than waiting for them to figure it out or seek classmates’ help
(Stein, Grover, & Henningsen, 1996; Stigler & Hiebert, 1999). I also interfere with
Deno’s presentation in a way I had not with Diamond. Hence, I send the class the message that some students can be trusted to present mathematics independently while others need assistance. My dilemma is whether to keep quiet and let Deno flounder further or to help him along and make his knowledge gaps less noticeable. I further reinforce Deno’s inferior position in this hierarchy when I refer to the problems he attempts as “easy” (line 26), even though he eventually fails to solve it. In my defense, the rise in pitch at the end of my statement “so, y is easy?” can be interpreted to mean something like, “is this question indeed easy for you all?” At this early point in the night course, I am not entirely familiar with students’ mathematics skills and knowledge.
My attempt to subjectify students poses a dilemma. I hoped to be a facilitator who encourages student-mathematicians to create and publicly explain their mathematical
259 ideas. Unfortunately, this pedagogy puts student competency on display, which conflicts with my alternative goal to develop a comfortable classroom milieu where students feel safe and protected. What emerges in the discourse transcripts is that positioning students authoritatively opens a space for peer comparison to reinforce learner hierarchy. These conflicting outcomes of positioning interferes with my potential to assume the supportive teacher role and help less efficient students learn mathematic principles and calculations, while being protected from humiliation. To the extent that I open up control of classroom discourse to students, I succeed in subjectifying them. Yet, it is clear my students need more practice with explanations and, perhaps, need me to model this for them. At this early point in teaching from a reform perspective, I have not had sufficient opportunity to model elaborated explanations or supportive outreach to all students that might have prevented the put-downs of Deno.
My attempt to value student contributions, whether wrong or right, conflict with students’ traditional Discourse model of school mathematics. Creating a discursive space for student ideas causes problems regarding Deno’s status as a learner in my class. Deno, like others, is unaccustomed to explaining his thinking in front of his classmates. It is definitely possible that Deno’s classmates could help him understand why his solution did not work, on the short run, his lack of understanding creates status problems for him that I hoped to prevent. Part of the problem undoubtedly stems from the fact that my students had not been socialized into classrooms where mistakes are valued and classmates are respected and encouraged regardless of their academic and social status.
Again, the emergence of a traditional Discourse model for school mathematics, exemplified by Diamond’s answer-oriented explanation and deference to me, indicate
260 this common socialization in the past. In interviews conducted with students, it became clear that most students had never been with a teacher who used reform curriculum to promote student thinking, explanation, and collaborative support.
While I did not want to reinforce the traditional model, I had trouble avoiding its gravitational pull. I discussed this issue regularly in my instructional diary. I too had been socialized to be the mathematics expert with authority to controls classroom discourse.
Given the pressures of traditional Discourse on the students and myself, it seems inevitable that, at least at first, I would tend to intrude in activities and explanations that I had designed to be student led (see lines 26, 27, 35). Romagnano (1994) experienced similar problems in his attempts to make practice in low-track classrooms more student- centered. Another reason I inconsistently allowed students to proceed independently was that I was nervous about teaching mathematics to students in the remedial course from new perspectives (i.e., reform for them, critical for me). I was preoccupied with making
CM and its reform grounding successful and my attention to my own actions seemed to suffer as a result. In addition, Week 1 had been stressful for me; the class was overcrowded and attendance had already become a problem for about ten of the students.
I believe that the questions I insert in the class discussion are an indication of my nervousness and uncertainty going into Week 2. Cazden (2001) identifies teacher statements such as “is everybody clear?” as “pseudo-questions.” They appear to call for a real response but, because few students interpret them that way, they do not act as real questions. Instead, such rhetorical questions may silence and objectify students. Cazden claims that pseudo-questions are part and parcel of traditional instruction. My discourse excerpt illustrates that no one responds to the pseudo-questions I ask (see lines 26 and
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27). Worse still, I use a pseudo-question to single out Lana as the student for whom
Deno’s problem might not be easy. I believe I did this because Lana had arrived late to class that day as she did frequently in the first few weeks. I believe I used this pseudo- question as a technique to make sure she was on task. When I viewed the videotaped lesson, it became clear that Lana was either not paying attention or could not see the board. It might be noted that “is everybody clear?” becomes a real question in classrooms where students trust their teacher. We had not built such trust at this early point in the night course.
6.2.2 Students’ Initial Traditional Model of School Mathematics: Analysis of Excerpt 6.2
1 A few minutes before break at the end of the first hour.
2 I walk up to Tara, who’s had a hand raised, in left front of classroom.
3 At same time Malik is waving hands vigorously in back right vying for my
4 attention.
5 Tara (whining): I sti::ll / need help / measuring / this thing! //
6 Me (to class): okay // so / one thing guys // is / there’s too many people / sayin’
7 I need help // and / and [you /
8 Deno (waving hands): [hey look! // Mister B!
262 9 Me: and you can / sit next to someone / who can help you //
10 Deno: Mi[[ster / B! //
11 Me: … [[like // like Le:ro:y //
12 Princess: but I / need the e:qua:tion! // I need the // I wanna kno:w / the answer
13 // I don’t / wanna be // stu:ck //
14 Deno: you / you / you: s’posed to be the te:acher / Mister B! //
Excerpt 6.2. Reform Activity at end of First Hour on Monday of Week 2
Excerpt 6.2 covers an activity that followed the opener featured in Excerpt 6.1, namely, “Angular Sums” from the IMP curriculum. I use Angular Sums to have students continue to explore the polygon sum patterns they began exploring on Day 4 of Week 1.
It requires students to develop a generalized rule that allows them to quickly calculate the sum of the angles of any polygon (e.g., S = 180n – 360 or S = 180[n – 2], where n is the number of sides of the polygon). Excerpt 6.2 points to two particular things about reform activities early in the night course: (1) the salience of the competition between the students’ and my Discourse models for school mathematics; and (2) how I subtly, but unintentionally, reinforce the mathematical hierarchy already under construction by naming certain students as helpers (i.e., Leroy) (line 11).
The competing Discourse models emerge rather forcefully in this excerpt. I state frustration with students’ over-reliance on me when I make the statement, “there’s too many people saying I need help when you can sit next to someone who can help you.”
While repeatedly telling the class my expectation that they work together and rely on neighbors when they have mathematical difficulties, many fail to see the instructional logic behind my advice because they perceive me as the only reliable source of help in
263 the classroom. In past courses, they have learned to assume that mathematical ideas flow from teachers and texts and not from peers. They convey that it is my job as a teacher to provide steady evaluative feedback about the correctness of their work as well as answers when students are stuck. Although students begin to understand aspects of my reform discourse model by the second week, apparently they still disagree with the logic behind it. Some vociferously resist reform practices and my subjective positioning of them.
Deno does not appear to be listening to my statements about my expectations when he somewhat comically exhibits conformity to traditional practices. He interrupts and frantically asks me to come over to evaluate work he did during the class period
(lines 8 & 10). Deno is not alone in his insecurity about new expectations. Others exhibit a similar reliance on the traditional Discourse model by strongly deferring to my teacher authority and reluctance to validate their own or other students’ mathematical subjectivity. Princess’ statements that she “needed the equation” and does not “wanna be stuck” (lines 12 & 13) clearly invoke aspects of the traditional Discourse model. Deno sums up the feeling of classmates when he states, “you s’posed to be the teacher, Mr. B”
(line 14.) Boaler (1997) and Schoenfeld (1988) discuss how students in traditional classrooms expect to be lead by cues in texts or from the teacher to answers. They do not expect to have to figure them out without direct teacher assistance.
Excerpt 6.2 also illustrates how I again unintentionally reinforce the mathematical hierarchy that was under construction in Excerpt 6.1 by singling out Leroy as a student who could provide mathematical help to his neighbors. I position Leroy as mathematically competent and a potential substitute for me and my mathematical expertise. Reflecting on this now, I realize that in naming Leroy, I was subjectifying him
264 and some other students as “helpers” while objectifying others as “in need of help.” I communicate that some students are capable of providing help (e.g., Diamond, Leroy) and others, who remain unnamed but “known,” should ask for help rather than figuring out the solutions themselves or with any other nearby student. Hence, I imply that it was fine for some students (e.g., Deno) to be “taught” by classmates, even though they have not been trained to “teach,” rather than by the teacher. This way of turning over authority to students appeared to be validating for students with stronger mathematics backgrounds, but it undermines the sense of competence and status of students with weaker backgrounds.
It is realistic to expect some degree of achievement diversity in mathematics classrooms. Mathematical hierarchies appear to be part and parcel of group-based education, even in reform mathematics classrooms. Of course, hierarchies also surface in traditional classrooms. While their effects can be exacerbated and also reduced (Cohen &
Lotan, 1997), it is difficult to displace these hierarchies even with the best intentions and best curricular materials. Despite student distinctions, teachers can model respectful and caring interaction with lower achieving students so they are not socially ostracized or rejected for seeming inadequacies in mathematics or other subjects. I should also point out that I do not advocate tracking or other homogeneous classroom arrangement. It does seem reasonable to acknowledge that student-centered pedagogy is subjectifying for some but perhaps not necessarily for all students in the reform classroom.
In sum, then, many students are willing and able to participate in reform mathematics activities. While, at the outset of the course, a few indicated that they would do the minimum required and these few generally resisted full participation, most night
265 school students wanted to do well in my course and most participated. However, it would take considerable effort on my part, and reflection on what was or was not subjectifying, to arrive at the NCTM’s reform vision. Years of traditional schooling made the shift to reform practices difficult and also raised equity concerns related to mathematical hierarchy. Three additional issues related to excerpts 6.1 and 6.2 that should be clarified before concluding this section: (1) some reform activities – even in the first two weeks of the course – went smoother than the one reported here; (2) interactions with the whole class tended to be more strained and activity in small groups was generally more positive;
(3) to some extent, I already had positioned my students authoritatively with respect to mathematics despite the resistance by some. Indeed, in the section that follows, I provide evidence that this was the case.
6.3 Evidence for a Valuable Shift in Students’ Discourse Models and Discursive
Participation in Reform Activities
Classroom activity shifted rather dramatically in a reform direction over the time span of the nine-week night course. Comparing coded lesson transcripts from Week 7 with Week 2, it is apparent that my students came to: (1) invoke the traditional Discourse model for school mathematics less often, (2) exhibit elaborated discursive engagement
(i.e., lengthy mathematical statements) with more frequency, (3) be more reliant on their peers for support, and (4) resist my reform instructional goals less frequently. I first discuss changes in students’ Discourse models (stated beliefs about school mathematics) and then address the shifts in my students’ discursive participation.
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The analytical unit of my discourse analysis is a topically related set (TRS)
(Cazden, 2001; Mehan, 1979). When topics in the videotaped class discussions changed or when I moved from one group to another, I coded this section of the transcript as a new TRS (see Chapter 3). The typical TRS was between 4 to 10 lines of discourse. A few were as short as three lines while others are close to 20 lines in length. For example, I broke Excerpt 6.1 into 6 TRS segments (i.e., lines 1 – 8; 9 – 12; 13 – 18; 19 – 23; 24 –
29; 30 – 38). Finally, there were over a hundred more TRS’s in Week 2 activities than in
Week 7 (i.e., 443 and 317, respectively). The difference in TRS’s per lesson is largely due to the fact that by Week 7 students relied more on their classmates and did not constantly call me over to help as they had in Week 2.
6.3.1 Changes in Students’ Discourse Models about School Mathematics Student Teacher Models Topically Invokes Invokes Arise in Related Sets Traditional Reform Conflict in (TRS) Model in Model in Same TRS TRS TRS Week 2 Hr1 174 25 8 6
Week 2 Hr 2 105 12 10 6
Week 2 Hr 3 164 28 17 15
Week 2 Totals 443 65 (14.7%) 35 (7.9%) 27 (6.1%)
Week 7 Hr 1 82 5 4 1
Week 7 Hr 2 98 2 7 2
Week 7 Hr 3 137 6 6 5
Week 7 Totals 317 13 (4.1%) 17 (5.4%) 8 (2.5%)
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Table 6.1. Counts of Discourse Models about School Mathematics
Table 6.1 exhibits the frequency with which my students’ traditional and my standards-based reform Discourse models emerged in the coded lesson transcripts from
Weeks 2 and 7. It also provides counts of the number of times these two models emerged simultaneously in the same TRS, indicating a direct conflict between the models. As the second column in Table 6.1 makes clear, my students were more than three times as likely to invoke the traditional Discourse model for school mathematics in Week 2 than in Week 7. In other words, later in the course, my students were much less likely to challenge me when I failed to respond directly to their mathematical questions or to complain when I asked them to explain their mathematical reasoning.
As the third column indicates, I made reference to my reform Discourse model less often in Week 7 than in Week 2. I no longer had to explain myself when I did not respond directly to students’ requests for immediate answers. By Week 7, students generally understood what I expected of them. It was also the case that our models were more likely to conflict more overtly with each other in Week 2 than in Week 7. An example of such a conflict is when, in Excerpt 6.2, Deno’s asserts that I was “supposed to be the teacher,” after I asked the class to rely more on one another.
Student work and post-interview data provide further evidence that the night school students had internalized, even accepted, my reform Discourse model by the end of the course. The self-reported transition of one student, Sonny, is particularly telling in this regard. At the outset of the night course, Sonny was one of the more vocal opponents to my reform instruction. Sonny regularly questioned the idea that he could learn from his
268 classmates and was generally reluctant to help other students. He often complained that I was not sufficiently didactic. At one point in Week 2 he cheered “yeah!” when I told my class I needed to explain more about an activity than I had initially. The following is what
Sonny wrote to me in response to an opener prompt regarding things “we” could improve in Week 3 of the course:
I think that when you put a problem on the board you should at least say the
answer to it because some of us can’t stop thinking whether our answer is the
right one or not. Also, you shouldn’t take off points just because we don’t want to
explain to somebody else how to do a problem. I don’t like to explain to others. I
don’t get paid for it! (Student Work, December 8, 2003)
Sonny’s comment that he doesn’t get paid to explain to others is consistent with of
Deno’s remark in Excerpt 6.2 that I was “s’posed to be the teacher.” Both wanted me to assume a more authoritative position in relation to students. What is important is that
Sonny’s views on this matter had changed considerably by the end of the night course. In a post-interview I conducted with Sonny at the end of the course, I asked, “if you were a math teacher how would you change the way math or geometry is taught?” Sonny responded:
You know it depends. Cause there are some teachers that like – they just go up to
the board, put up a problem, and be like, ‘okay’ you know, ‘if you got it, you got
it, if you didn’t, oh well.’ So, I guess I’d try to be more like you too. Sort of like
this class. (Post-Interview, January 27, 2003)
This shift in stated beliefs reflects Sonny’s in-class actions. Whereas he strenuously resisted helping classmates early in the course, after a few weeks, Sonny started to help
269 several students, sometimes when I requested it, but other times he did so voluntarily. He often called attention to himself when he was helping a classmate. He wanted me to give him an “A” for the course.
While Sonny was perhaps the clearest example of a student undergoing a shift in acceptance of my reform Discourse model, he was not alone as Table 6.1 shows. The fact that students brought up the traditional model with much less regularity in Week 7 was a good indication that students’ Discourse model was evolving to approximate the reform vision. Responses to interviews that two colleagues and I conducted on the last day of the night course further support my claim that the night school students generally came to understand my reform Discourse model and accept the rationale behind it. In the post- interviews, many claimed to appreciate learning mathematics as it was taught in the night course. Several students who had to take more mathematics courses asked me to continue teaching in the Guevara night school. However, the shift was not uniform across my students. In the post-interview, Osvaldo suggested that I could improve as a teacher by teaching like one of his day teachers who, apparently, first shows his students how to solve a particular type of problem, assigns a set of similar problems, and then allows them to socialize when they finish the assigned problems. Of course, rather than rejecting my reform instruction, he may simply have been asking me to give he and his peers more downtime.
6.3.2 Changes and Continuity in Students’ Discursive Participation
In addition to a shift in Discourse models, there was also a positive shift in how students participated in and discussed mathematics in reform activities. Coded transcripts
270 indicate that three changes were particularly salient in this regard, namely, at the end of the course students were: (1) more likely to respond with and to initiate elaborated and engaged mathematical responses to mathematical problems, (2) less resistant in reform activities, and (3) less reliant on me for mathematical assistance and evaluation during such activities.
Topically Moderate Elaborate Student Calls for Related Engagement Engagement Resistance in Teacher Help Sets (TRS) in TRS in TRS TRS in TRS Week 2 Hour 1 174 83 15 45 69
Week 2 Hour 2 105 45 1 21 27
Week 2 Hour 3 164 96 2 47 46
Week 2 Total 443 244 (50.6%) 18 (4.1%) 113 (25.5%) 142 (32.1%)
Week 7 Hour 1 82 42 13 15 24
Week 7 Hour 2 98 45 16 17 33
Week 7 Hour 3 137 73 11 18 19
Week 7 Total 317 160 (50.5%) 40 (12.6%) 50 (15.8%) 76 (17.2%)
Table 6.2. Student Engagement in Reform Activities from Weeks 2 and 7
The most positive indication of shift in a reform direction is that my students were three times as likely to contribute to classroom discourse with elaborate mathematical statements about their mathematical ideas in Week 7 than in Week 2. As discussed in
Chapter 3, in analyzing the transcripts of classroom discourse, I coded a student statement as elaborate when it deals with mathematics and it is over a line and a half in length. As Excerpts 6.3 and 6.4 from Week 7 illustrate, students’ ideas came to play a prominent role in shaping discussions and activities toward the end of the course.
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What is particularly powerful in this regard is that, by Week 7, students regularly demanded that I listen to their mathematical explanations. For example, at the end of the first hour that week, three students (Amalia, Sonia, and Lucee) asked me to come over to their desks so that they could explain their solutions to a problem on similarity. Shortly after I announce the bathroom break, Amalia called out, “Mr. B, will you come and hear my explanation now?” This was followed by similar requests from Amalia’s neighbors,
Sonia and Lucee. It took Lucee several minutes to explain her ideas to me. This was a stark contrast to student participation in reform activities in Week 2. When they called me over at that time, rather than explain their thinking to me, they generally requested direct evaluation or assistance.
Table 6.2 illustrates that student resistance decreased by the end of the course.
This meant that my students were not as slow to start in on activities, less likely to goof off when I was not nearby, and more likely to complete activities in a timely manner.
They were also less likely to arrive late to class and the scheduled twelve-minute break between the two hours of class do not drag on as happened in Week 2. While student resistance continued, it was no longer as big an influence on my instruction as it had been earlier in the course.
Table 6.2 also illustrates that my students became less reliant on me for direct assistance and evaluation. Students called me over (i.e., “Mister B”) approximately 47 times per lesson in Week 2 and only 25 times per lesson in Week 7. Over time, and with my consistence in encouraging peer discussion, my students generally grew accustomed to working collaboratively with classmates or with spending time trying to figure out
272 solutions on their own. Of course, when groups were collectively stuck they asked for my assistance.
Despite shifts in stated beliefs and participation, from my teacher perspective, many aspects of students’ initial orientations to school mathematics did not change substantially or sufficiently over the time span of the course; that is, the reform direction of students’ participation was not as far reaching as I hoped. Resistance, in particular, was always present and somewhat inhibited my attempts to reach the NCTM’s reform vision. While students wasted less time in Week 7, many continued to take longer than I thought they should to begin and complete activities. Some indicated that, based on their past experiences, hard work in mathematics or other school subjects, should be rewarded with breaks or down-time. Osvaldo reminded me on several occasions that other Guevara night school teachers allowed students to watch TV on Thursdays. He persisted in regularly telling me, “if we do this activity you should let us watch TV.”
6.4 Reform Discourse and Participation in Week 7
In this section, I focus on two excerpts of discourse and participation in a reform activity that I implemented near the end of the course (Day 1, Week 7). It details the change and continuity in students’ participation in reform mathematics that I discussed in the previous section. The analysis shows: (1) I continued to have to position students subjectively with respect to school mathematics in Week 7, albeit less frequently and overtly than in Week 2; (2) students’ reactions to this subjective positioning shifted (i.e., some took on increased student agency) or failed to shift (i.e., some continued to resist
273 reform instruction); (3) students’ traditional Discourse model for school mathematics continued to surface and shape reform activity, although with less force over time; and
(4) the hierarchy of learners remained however students did not reference it as overtly. As was the case in the lesson from Week 2, the one presented here was more typical than exemplary in illustrating common rituals rather than preferred behaviors.
Figure 6.2. IMP Activity on Enlarging Shapes from Week 7
Excerpts 6.3 and 6.4 come from the IMP activity “Draw the Same Shape” (Figure
6.2) which deals with the topic of similarity. In this activity students are asked to explore their intuitive understandings of similarity by drawing a scaled-up version of a “house-
274 like” pentagon. Discussing question #2 (“Draw the Same Shape”), the IMP Teacher’s
Manual states:
At present, you should leave the question unresolved. That is, tell students that,
for now, they might disagree about what “same shape” means, but that they will
be working toward the formal definition that mathematicians use. No absolute
conclusions need to come out of this discussion … (IMP Shadows Teacher’s
Guide, p. 27)
As an informal introduction, I am to avoid leading students to “absolute conclusions” or
“formal definitions” of similarity as a reform teacher. As Excerpt 6.3 indicates, I struggle with how much to tell students who attempt to recall the conventional meaning of doubling an angle. Doing so might trivialize the task. As the IMP authors suggest, this
“formal definition” will be realized later on in the curriculum.
In attempting to engage in this activity, most students try to double the sides of the pentagonal figure. However, they run into trouble because the doubled version of the house does not fit on an 8.5”x11” sheet of paper. This means they have to figure out a less obvious way to fit a larger, scaled, version of the house onto their papers. After an impromptu whole class discussion initiated by the class, some students decide to add the same length to every side (e.g., 6 centimeters) while others decide to multiply the sides of the house by 1.5 as a means to fit a larger and similar version of the pentagonal house in their notebooks. It is important that this mathematical conversation and the suggested solutions is spurred on by the students themselves. This is a clear indication that, as a whole, the night class accepted a more authoritative positioning by Week 7.
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In fact, Excerpts 6.3 and 6.4 both point to positive changes in terms of achieving the NCTM vision of school mathematics. Excerpt 6.3, however, takes place as students work in small group settings on the Renata’s House task. While it shows positive change, it is a rather negative example in the sense that it shows how the traditional Discourse model for school mathematics continues to surface in Week 7, although with less frequency and power than in Week 2. Excerpt 6.4 takes place at the beginning of the whole class discussion and indicates that, as a whole, the night school students take on increased mathematical engagement (agency) by Week 7.
6.4.1 Small Group Work in Week 7: Analysis of Excerpt 6.3
1 Robi: Mister B //
2 Me: yeah //
3 Robi: could you like add // could you like / mult / multiply the angles by 2?
4 Me: yeah // the angles // the angles won’t change? //
5 Robi: the angles will / change //
6 Me: they might change / right? //
7 Sonny (sitting sideways in seat to face Robi and me): no they don’t //
8 Sonny looks up at me and smiles or smirks.
9 Robi (quietly now): yeah they wi:ll //
10 Me: that’s / the question ///
11 Robi: so would I just / double that? //
12 Me: yeah / try doubling things // what would you do with the angles? / would you
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13 double the angles? //
14 Sonny: nah //
15 Robi: I / would double the angles //
16 Me: yeah //
17 (Here I am somewhat encouraging Robi’s thinking while not telling him that he
18 may be wrong.)
19 Robi: would you / multiply that by 2? //
20 Me: you could do that // and see what happens //
21 Sonny (looks at me): cause no matter what / the area still has to be // I mean / the
22 angles // all gotta measure up / to a certain thing // no? //
23 Me (with my hand on chin, look at Sonny): yeah-ah // (exaggerated sigh)
24 Sonny: I mean a triangle // it doesn’t matter how big //
25 (I verbalize my intentions as a way to distract Sonny from trying to tease the right
25 answer out of me—he was persistent despite my refusals to respond directly.)
26 Me: I mean / I’m being // I’m purposely not answering this question / right now //
27 Sonny: ha:h //
28 Shannon, Yalitza, Adelric, Efrain have turned around to look up at me after this last
29 exchange with Sonny.
30 Me: I’m purposely / not // telling you guys the answer / right now // I know that
31 makes //
32 Sonny laughs, looks down at his paper.
33 Me: I want / you guys / to try to figure that out //
277 34 Sonny smiles and looks away, mumbles something, and looks back up at me
35 sideways still smiling. Shannon looks back down at her work. Yalitza looks
36 towards camera at front of room.
37 Me: I could / just / I could just tell you what // I mean / I would say / I would say /
38 for right now Robi // if you can’t figure something else out / I would go with
39 Sonny’s idea / which is to keep the angles the same / but try to double the sides //
40 see if that works //
41 Sonny (smiling and looking up at me): and then / that means it’s the wrong answer
42 // because you want us to do / more work //
43 I laugh.
44 Robi (laughs): yeah //
45 Me (to Robi): alright well // I like your idea / of / doubling the sides // just try to
46 draw it / with the sides doubled / see what happens //
Excerpt 6.3. Small-Group Discourse in Reform Activity from Week 7
Excerpt 6.3 illustrates my indirect manner of responding to student questions during small group work. This is representative of how I attempt to move student activity along without simplifying the mathematical task. Specifically, I restate Robi’s questions to me about how to proceed as an idea to think about as he pursues the answers himself
(lines 12, 13, 20, 37 – 40, 45, 46). Similarly, to encourage his independent problem solving, I agree with Robi’s suggestion that he double the angles (line 12). I make my reform Discourse model explicit when I state, “I’m purposely not telling you all the answer right now” and “I want you guys to try to figure that out” (lines 26, 30, 31, and
33). By this time in the course, my students generally accept such indirect responses and
278 no longer call out for me to respond directly to them. Excerpt 6.3 represents the interactions I have with small group in Week 7 in the sense that students now accept such responses as well as my positioning them as authorities (e.g., do it for yourself). The interactions in Excerpt 6.3 are evidence for the shift towards the NCTM’s reform vision over the time span of the night course.
Ironically, however, throughout the course as I create space for student-centered activity, I inadvertently provide opportunity for students with stronger mathematics backgrounds, such as Sonny, to work against the subjectification of classmates, hence
(re)inforcing the hierarchy of mathematical learners. Recall how Diamond worked against the subjectification of Deno (Excerpt 6.1). In Excerpt 6.3, Sonny behaves similarly when he jumps in and evaluates Robi’s ideas as a traditional teacher would.
Sonny apparently takes over where I leave off by supplying an immediate answer – at least, from a traditional perspective of school mathematics. Sonny dismisses Robi’s ideas as incorrect while failing to provide the conceptual (or conventional) explanation that he has in mind, thus, objectifying Robi (lines 7 & 14). When Sonny does explain his thinking, his explanation is directed at me (as the ultimate authority and judge) instead of at Robi (lines 21 & 22). In so doing, Sonny interferes with my subjective positioning of
Robi as someone capable of figuring out his own mathematical answers. While Sonny directs “no” statements at Robi, he directs conceptual, or argumentative, statements at me, hence he differentiates himself from both of us. That is, Sonny (re)establishes his position as mathematically knowledgeable with respect to Robi but positions himself as less knowledgeable than me. This occurs in such activities despite my attempts to become
– and to be seen as – a facilitator of learning rather than a mathematical expert. While
279 Sonny may not feel he is working against Robi per se, he is challenging my pedagogical approach. As mentioned earlier, Sonny is generally a more helpful classmate (from a reform perspective) at this point in the course than he appears in this excerpt.
Despite Sonny’s interference, Robi does not allow Sonny to silence him or his ideas as he might have in Week 2. Robi counters Sonny’s assertion that the angles do not change by quietly stating, “yeah they will” (line 9.) As Excerpt 6.4 indicates, Robi has become a vocal contributor to whole class discussions by Week 7. In the activity that followed Excerpt 6.3, Robi figures out, perhaps with help from his neighbor Stephie, how to deal with polygonal angles in scaling up Renata’s house. Robi either is told by a classmate (likely) or figures out (unlikely) the conventional meaning of doubling an angle without relying on my assistance.
I coded Excerpt 6.3 as a place where reform and traditional Discourse models emerge as being in simultaneous conflict. By Week 7, such Discourse conflicts are less powerful than they were earlier in the course. Instead of arguing with me as he might have in Week 2, Sonny simply laughs when I state, “I’m purposely not answering this question” (lines 26). It is significant that the eight nearby students who overheard this exchange, watch Sonny and me in apparent awareness of the competition between my reform Discourse model and the traditional one that they entered the course with (lines 28
& 29). They witnessed past exchanges between traditional conformists, such as Sonny, and me, and pay attention to the disagreement in this activity in a way that indicated their interest.
Finally, Excerpt 6.3 points to a topic that emerges in my textual analysis (Chapter
5), namely the relationship between explicit pedagogic discourse, mathematical
280 conventions, and intuitive mathematical understanding. If the lengths of the sides of the pentagonal house are doubled (see Figure 6.2), the line segments that co-determine each of the five angles double as well. With this in mind, it is reasonable to suggest, as Robi does (lines 3, 11, and 15), that the angles themselves have doubled. According to mathematical convention, however, the angle measures do not change because the degree of turn between the sides that determine them does not change. As I reread the transcribed interactions for this discourse analysis, it became clear that Robi’s confusion lay in discourse rather than in a lack of intuitive experience or prior conceptions. Robi apparently has not internalized the conventional definition of angle that I expected he would have when we spent time using protractors to measure angles in Weeks 1 and 2 of the night course. By the end of this activity, Robi seems to have worked through these issues with help from his neighbor Stephie. In all such similar instances, I was unsure with how much to tell students who struggle to recall and understand the conventional meanings. If I had reminded Robi of the conventional definition of angle, I would have oversimplified the task as it was stated. On the other hand, by not stating this convention,
I risked disempowering Robi. I would argue that this struggle is not uncommon for reform mathematics at the secondary mathematics because it is a domain that is shaped profoundly by conventions, notations, and symbolic expressions (Dowling, 1998). (In terms of the importance of notational systems on mathematical thought see Rotman,
1993, 2000.) My larger point, one that I took up in Chapter 5, is that positioning students authoritatively with respect to mathematics is one form of empowerment. At the same time, making sure they understand conventional mathematics, which is perhaps more likely through didactic teaching, is equally important. This is a point about secondary
281 mathematics that is not often discussed by mathematics educators, and particularly, those who wish to translate a Piagetian framework for learning mathematics for the entire school mathematics curriculum (Dowling, 1998; Pimm, 1995; Walkerdine, 1988).
6.4.2 Whole Class Discussion in Week 7: Analysis of Excerpt 6.4
1 I move to the overhead as a result of several students pointing out that a doubled
2 version of Renata’s House (Figure 6.2) does not fit on notebook paper.
3 Me: that / that doesn’t fit? //
4 Robi: no it don’t // cause //
5 Me: okay // I think it does / I think it does / fit on the paper //
6 Robi: nope // do it //
7 Me: okay / let’s try to do it up here then / maybe / together then // what are we /
8 what are we / going to try to do? // what’s the thing? //
9 Amalia, Robi, Others: double it //
10 Me: double it / and are we going to double the angles too / or / are we going to keep
11 those / the same? //
12 Amalia: ke[ep them the same//
13 Sonny: [keep the[[m the same //
14 Malik: [[the same //
15 I draw the bottom edge of Renata’s House on the overhead screen.
16 Me: oh so / maybe you guys are right / maybe it doesn’t fit / does it? //
282
17 Osvaldo: fi:red! //
18 Me: ah / so / it doesn’t fit on there / well this //
19 Malik: you maybe wan-na / move / the centimeters / (inaudible) / for instance/ like
20 / the bottom / the base is twelve on there / (inaudible) / and keep adding six
21 centimeters //
22 Shannon (to Malik apparently): it don’t / fhi:t! //
23 Me: okay / so / you guys are right // it doesn’t fit // now that I see that // so / so
24 doubling it isn’t going to work // we’re going to have to / figure something else
25 out // And Malik’s / making a suggestion //
26 Robi: it’s a trick / question //
27 Me: and / what’s your suggestion Malik? //
28 Malik: it’s //
29 Me: so // you tell me what it should be //
30 Malik: the base / is like twelve / right? // so / you could add six to it //
31 Me: so / it should be eighteen? //
32 Malik (hard to hear): (divide) / the (inaudible) in the middle / six centimeters //
33 Me: ah / ha //
34 Malik: (inaudible)
35 Me: so / you’re adding six to every side? //
36 Malik: yeah //
37 Me: does that make sense to people / what he’s doing? //
38 Sonny (who apparently wasn’t listening): ah / why-who? //
283
39 Me: would that work? //
40 Sonny (quietly again): what-[[who? //
41 Malik (did not hear Sonny): [[take / half the base //
42 (Several students with quiet responses or suggestions.)
43 Robi: I got an idea //
44 Me (to Robi): you got an idea? //
45 Robi: yeah / if / you’re gonna add six to all the sides / you’re [gonna get //
46 Sonny: [you’re gonna get
47 eight-point-f(ive) //
Excerpt 6.4. From Impromptu Discussion of Reform Mathematics in Week 7
Excerpt 6.4 comes from videotaped activity that takes place about five minutes after my interaction with Robi and Sonny (Excerpt 6.3). It begins when Yasmin and
Amalia explain that a doubled version of the pentagonal house does not fit on an 8.5” x
11” sheet of paper. Doubling is not suggested by the IMP text and is apparently the default idea that many of the night school students collectively latched onto. While I realize the doubled version of the house does not fit on notebook paper, I initially pretend to disagree with my students. I do so to let them try their own ideas without running them by me first. At this point in the lesson, I continue to insist that it does fit on the notebook paper. My purpose is to provoke them to speak up and to challenge me and other authorities when they know are wrong. Indeed, Yasmin, Lupe, Amalia and Robi (lines 4,
6, 9) quickly and assertively respond that the doubled version of the pentagonal house does not fit on notebook paper.
284 The conversational nature of the segment of the whole class discussion should be noted (Excerpt 6.4). Again, despite being led by me, the discourse structure is loose and the students are able to initiate mathematical suggestions (lines 19 – 20) and respond to me and to each other’s ideas (line 22). Excerpt 6.4, in this sense, was representative of whole class discussions in Week 7. This passage also points out students’ comfort in challenging my mathematical authority by Week 7. For example, Robi tells me to “do it”
(line 6) in order to prove to myself and other doubters in the class that my thinking is mistaken. Robi and others were beginning to accept that doing the mathematics for oneself was the ultimate arbiter of correctness rather than the traditional teacher or text per se. In a traditional class, my statement might have shut down such student-centered activity (line 5). Of course, I still maintain a considerable amount of control in this whole class discussion. I do not recede to the background as I might have done if the shift toward student-centered classroom were more complete.
In terms of increased critical student engagement (Pruyn’s critical agency), this excerpt points to two important changes from Week 2 to Week 7. First, students who had not participated voluntarily in whole class discussions in the second week (e.g., Malik,
Yasmin, Amalia, Robi) are volunteering freely in Week 7. These students, who do not necessarily have the strongest mathematics backgrounds, are also initiating and shaping the whole class discussions and the activity in ways they had not in Week 2. My analyses show that Yasmin, Amalia, and Robi would not have been so quick to speak up in whole class discussions and challenge me in Week 2 as they do in Week 7. Second, the whole class discussion in Excerpt 6.4 is spontaneous (I did not plan it in advance) and largely student-driven. This is not to suggest that I do not play a role in making it happen, but
285 rather that it could not have transpired without significant engagement on the part of my students.
In the discussion and activity that follows that presented in Excerpt 6.4, I listen to several student suggestions on how to proceed with the Renata’s House activity. My students, including Robi, offer variations on Malik’s idea of adding a constant (e.g., 6 centimeters) to every side. After some prodding or funneling on my part, Yalitza suggests the method of multiplying the sides of Renata’s house by 1.5. In closing this whole class discussion, I tell the class they could choose either Malik’s method of adding six to all the sides or Yalitza’s method of multiplying by 1.5 to see what happens to the pentagonal house. Thus, I am able to incorporate student ideas into the activity and, in so doing, was clearly positioning the students’ subjectively. This is something I did with more regularity in Week 7 than in Week 2. My point here is that I need students to meet me halfway if I am to position them authoritatively. Their engagement in this and similar activities from Week 7 allows me to do so. In responding to their authoritative positioning by contributing discursively to a greater extent than they did in Week 2, we came closer to achieving the NCTM’s vision for school mathematics.
In Excerpt 6.4, and in Week 7 more generally, I no longer have to justify my reform instructional moves to students (e.g., not telling, validating student ideas as right or wrong) as I had in Week 2. Interestingly, however, Osvaldo invokes the traditional
Discourse model when he jokes that I should be “fired” for having making a mistake (in line 17). At this point in the course, students and I regularly joked about the clash in our expectations (i.e., Discourse models) for school mathematics. Recall that Deno and
286 Princess did not joke when they called my student-centered pedagogy into question in
Week 2 (Excerpt 6.2).
Finally, for purposes of the discussion in Chapter 7, it is essential to note that there are two social languages at play in Excerpt 6.4 (Gee, 2005). I said “doesn’t (fit)”
(lines 3, 16, 18, & 22) with respect to the angle, whereas two students said “don’t (fit)”
(lines 4 & 22). At the end of the section, I switch into Robi’s vernacular register, when
Robi says “I got an idea” (line 43) I repeat his usage of “got” (line 44) instead of correcting or challenging him with “you have an idea.” Gee (2005) points out that such use of social language signifies affiliation, membership, and belief. My students’ use of
“don’t” here might signal comfort in my classroom, ownership of the ideas, or perhaps that they consider me to be on their side. (It might also be seen as a challenge.) My use of vernacular social language might be seen as an attempt to value my students and where they come from, to signal my preference of my students over school the institution. I convey that membership in their social world, and that their intuitive mathematical ideas, are more important than making sure they speak the dominant form of English. I make this point now to draw attention to the language usage in Excerpt 6.4, but return to an extended discussion of the students’ use of a vernacular (working class) social language in Chapter 7.
6.5 Discussion
In this chapter I describe discourse and participation in the reform activities that comprised between 75% and 80% of the night course curriculum. In terms of reform
287 relationship to critical pedagogy, the most important result was that I positioned my students more authoritatively with respect to secondary mathematics, an outcome that I considered essential to the success of critical math. Most students had generally accepted this positioning by the end of the nine-week course. This meant that I had empowered them to challenge my status as the expert to whom they must turn, and had succeeded in getting them to draw on their own intellectual and problem-solving resources. While activity in the night course often fell short of the NCTM's idealized vision, the discourse and interview data presented in this chapter indicate that my students came to see and practice school mathematics in a markedly different way by the end of my course.
While this is a positive finding, my analysis of reform activities also raises several concerns. To begin with, transitioning students who have internalized the traditional
Discourse model of school mathematics to a reform Discourse model is not a simple, power-free endeavor. It seems clear that my institutional authority continued and perhaps was a primary reason that my students became more accepting of reform mathematics over time. While the post-interviews that were conducted after I had turned in student grades suggest that my students did see the logic behind reform practices, it was also possible that many students were simply complying with instructional strategies in order to pass the course or get a better grade. High achieving students such as Sonny and
Shannon, for instance, made it clear that they were helping their classmates in order to get an A in my course. Hence, it was not necessarily that their social consciousness had been raised or that they considered providing mathematical help the right thing to do.
Nevertheless, the nine weeks of conditioning in this social reciprocity ethics and in
288 learning they could be independent problem-solvers was likely to have some long term effect on many of them.
My analyses also point to an important equity related concern that was operating on the level of classroom discourse - one that I was not fully aware of before conducting the analysis of transcribed classroom interactions. In particular, I showed how students who have internalized a traditional model of school mathematics – and perhaps even those who hold other models – can use their more empowered positions in reform classrooms to sort themselves by mathematical background or ability. While mathematics may be considered to be a "depersonalized" discipline, it can become very "personalized" for students who hold traditional (and perhaps reform) models. Students are generally very aware of their own and others' test scores on national exams and their grades in particular classes. They internalize messages these assessments supposedly relay about their mathematical or mental ability (Wiliam, Bartholomew, & Reay, 2004). Student identities as students in the night course appeared to be primarily contingent on how quickly they could come to know and whether or not they were "helpers" versus those who were "helped." As a result, getting the students with weaker mathematical backgrounds to contribute in reform activities cut both ways. In the long run it provided more subjectivity to more students than a traditional model would. It also sent the message that mathematical mistakes are appropriate and valuable in learning mathematics and getting all students to contribute their mathematical ideas. At the same time it appeared to create a clearer hierarchy of learners than if I had positioned them as listeners and passive doers of independent seat work in the class.
289
It seems that the mathematical hierarchies that the night course students and I
(unconsciously) co-constructed is a reflection on traditions in school mathematics rather than reform practices per se. While, there currently is a considerable rhetoric about mathematics for all, mathematics continues to be used as a sorting device (Dowling,
1998; Martin, 2003; Oakes, 2005; Tate, 1997). And, because mathematics as linked to distribution of social goods more broadly, it is not surprising that inequality is reflected at the micro-level classroom activity.
Over time, my students referenced the hierarchy of learners less explicitly and less frequently. I never again heard them call each other stupid after the lesson I presented from Week 2. I found this to be true even when I carefully reviewed the classroom interaction from all angles in the videotaped lessons. Frankly, there was a good deal of student posturing in the first several weeks of the course. This died down once they became more comfortable with each other (many of the students were not from
Guevara), me, and my instructional style. Over time, I believe my non-evaluative and relaxed pedagogical style helped to create a safe place for students to reveal gaps in their mathematical knowledge to each other and me. This was particularly true in small group work. With one or two exceptions, the students with weaker mathematics backgrounds began to rely on each other for help with more regularity. The students with stronger mathematics backgrounds were also more willing to help their neighbors out when they struggled. In conclusion, in terms of student empowerment and social learning, this short foray into reform mathematics instruction appeared to have a positive impact.
290
CHAPTER SEVEN: DISCOURSE AND PARTICIPATION IN CRITICAL
MATHEMATICS ACTIVITIES
In this chapter, I report the results of my analysis of discourse and participation in the night course critical mathematical (CM) activities. Critical education requires that students be positioned subjectively or authoritatively with respect to disciplines such as mathematics. As a teacher, I used the standards-based reform mathematics activities described in Chapter 6 to accomplish this positioning. CM takes the subjective positioning of reform mathematics further by positioning students to use mathematics to think about sociopolitical phenomena such as economic inequality that affect their lives.
As might be expected, the way that the students and I enacted critical activities was very different from the way we enacted reform activities. This was due, in part, to the overt political nature of CM and the distinctive expectation that students reflect on sociopolitical matters, something they were unaccustomed to doing in school and had not been required to do in reform activities.
7.1 Issues at Play in Critical Mathematics Lessons
In order to orient the reader, I begin this chapter with an overview of six results that emerged from my discourse analysis of the transcripts of videotaped critical lessons and inductive analysis of my instructional diary. These CM results include the: (1) changed nature of student discursive participation, (2) unique characteristics of Discourse models that CM evoked, (3) prominence of vernacular social language, (4) tension between critical and mathematical goals, (5) unpredictability of student response to CM,
291 and (6) my own insecurities due to my inexperience with critical pedagogy and transgressive teaching (hooks, 1994). I summarize the themes before reporting results in order to provide readers with an orientation to interpreting the results I report in this chapter.
7.1.1 Student Participation in Critical Classroom Discourse
As might be expected, student participation in classroom discourse was different in critical activities than it was in reform mathematics activities. Two results were particularly salient: First, a somewhat different group of students were discursively engaged in critical activities than were engaged in reform activities; in particular, critical activities woke up several students who tended to resist reform activities. At the same time, many, but not all, of the higher achieving mathematics students exhibited resistance to critical activities. Second, in a subset of the enacted CM activities, my students’ discursive contributions were far more elaborate and energetic than they had been in most whole class discussions of reform activities.
7.1.2 Strength and Resilience of Hegemonic Discourse Model for Social Inequality
In Chapter 6, I demonstrated how my students’ Discourse models about school mathematics emerged in reform activities. CM challenged students’ beliefs about school mathematics in a similar manner. Due to their overt political nature, critical activities also elicited students’ Discourse models about socioeconomic inequalities. As critical scholars would predict (Gramsci, 1971), most, but perhaps not all, students held a hegemonic
Discourse model of social inequality. For instance, in writing assignments, some of the
292 night school students described their classmates (and sometimes themselves) as being in need of forceful discipline or described their classmates or classmates’ families as being responsible for the social inequalities they experienced. In other words, some students engaged in a blaming the victim narrative about people in their own social class (see
Ryan, 1971). While I know that the internalization of blame and sense of personal inferiority are ways that hegemony works (Gramsci, 1971), I was surprised, perhaps naïvely, at the strength and resilience of the hegemonic Discourse model(s) my night school students held. I also was alarmed at how potentially damaging this Discourse model could be to their self-perception and agency. In turn, evidence of these attitudes reinforced my belief about the need for a Freirean form of consciousness raising that is one goal of CM.
7.1.3 Prominence of Vernacular Social Language
My students used vernacular social languages (street language) (Gee, 2005) to a much greater extent in critical activities than they did in reform activities. Student use of vernacular language indicated that they perceived CM as being different, perhaps less formal, than school-as-usual or even standards-based reform mathematics. Their use of vernacular also signaled their ownership the sociopolitical ideas in critical discussions that they did not seem to have when engaged in reform discussions.
7.1.4 Tension between Critical and the Mathematical Instructional Goals
The dialectical tension, or conflict, between fostering critical discussions of sociopolitical matters and rich mathematical problem solving surfaced for me when I
293 began teaching with the critical activities I had designed and gathered for the night course. This tension was particularly salient in the first CM activity from Week 2, in which my students’ discussions of their experiences in Chicago schools interfered with my plans to focus on the mathematical interpretation of the featured statistical data. The sociopolitical context in the lesson presented a distraction to the mathematics content. As
I discuss later, rather than pointing to a theoretical problem with CM, this finding may have resulted from my inexperience teaching CM.
7.1.5 Unpredictability of Student Response to Critical Activities
Student participation in the critical component of the night school curriculum was less predictable than in the reform component. While seven CM activities ignited interest and critical agency, five actually dissipated participation. When critical activities did ignite critical agency, certain students clamored to be heard in ways they never did in discussions of reform activities. When lessons did not ignite their interest, I struggled in ways I had not since my first year of teaching, both in terms of feeling personally insecure as a teacher and in unsuccessful attempts to stimulate students’ involvement.
The unpredictability of CM lessons likely had to do with several things, including the fact that the explicit political focus of CM was unique for students and led to complex and conflicting expectations in the context of secondary mathematics. In addition, I was inexperienced in facilitating critical discussions. Nevertheless, student participation in critical activities did improve over the time span of the course as it had in the reform activities (Chapter 6). While my CM instruction may have improved, it can be partly
294 explained by the fact that students had built more trusting relationships with each other and me by the end of the course.
7.1.6 My Inexperience with Critical Pedagogy
Although I was an experienced secondary mathematics teacher, critical pedagogy was new terrain for me. My struggles in CM instruction were more intense than in teaching reform IMP lessons discussed in the previous chapter. I had taught IMP previously and could gauge in advance how students were likely react to and participate in particular activities. Such was not the case with CM. A theme that I repeated in my instructional diary was that I was more comfortable teaching with the reform IMP materials that comprised approximately 80% of the night course curriculum. I often worried before teaching a CM lesson that my students would reject the lesson as non- mathematical (see Skovsmose, 1994, for similar results). My inexperience with critical pedagogy is evident when videotapes of reform lessons are compared with videotapes of critical lessons.
To illustrate these six conclusions about my CM instruction, I present and discuss a total of seven excerpts from transcripts of two videotaped lessons. The first three are from a critical activity in Week 2 and the second four are from a critical activity in Week
7 of the nine-week course. Both lessons took place in the second hour of each day, after the break and first hour reform lessons. While these activities embody and illustrate the more general findings of my CM instruction, they do so in different ways.
295 7.2 Charting New Critical Territory in Week 2
In the second hour of class on Monday in Week 2, I presented the class with their second critical mathematics activity “Race and Recess.” I obtained the chart used in the activity (Figure 7.1) from an online news article in the Chicago Reporter by Pardo
(1999). (For the full Race and Recess activity see Appendix B.) I believed the real world theme of Race and Recess was relevant to my students’ lives as CPS students. I hoped it would foster a meaningful sociopolitical discussion as well as ignite interest in the mathematics component of the lesson.
Schools with Recess Schools without Recess
30-100
5-29.9
Percent White Students 0-4.9
0 25 50 75 100 125 150 175 200 225 250 275 300 325 Number of Schools
Figure 7.1. Chart in Race and Recess Handout
To review and contextualize, in terms of the curriculum analysis presented in
Chapter 5, Race and Recess could be categorized as a public domain activity (Dowling,
1998). It relied on data analysis from a news article (i.e., public domain) that largely failed to call for applications of secondary mathematics, at least as mathematics is currently shaped by requirements of university mathematicians and general education
296 policy. Nevertheless, the data analysis required in this activity could be considered part of the content that I was required to cover in geometry according to the local Chicago curriculum standards. The activity could not be judged to be unwarranted by the district.
The critical Race and Recess activity takes place in the second hour of the course, following the reform Angular Sums activity featured in Excerpts 6.1 and 6.2 (Chapter 6).
My plan for the Race and Recess activity is simple: I would first ask the students to explain what the chart means mathematically and then have them think about why the racial inequities presented in the chart exist. I hoped that the first task would engage them in mathematical reasoning and second would lead to an interesting sociopolitical discussion. However, the Race and Recess lesson does not transpire as smoothly as planned because several students are ready to immediately discuss the sociopolitical themes and many have difficulty understanding the chart. I did not anticipate these responses, so I spend much of the activity time trying to hold off students’ non- mathematical contributions in order to have them clarify the chart’s mathematical meaning.
While some students immediately jump in to talk politics, others puzzle over the meaning of the data in the chart. Students appear to have had little practice reading and interpreting the types of statistical charts commonly found in newspapers and magazines.
That means that they come to the activity lacking the mathematical knowledge I expect them to have. That my students struggle with the mathematical meaning of the chart, of course, points to the need for data analysis to be taught in schools (see Steen, 2004). My personal feeling is that, while data analysis fits in the elementary mathematics curriculum, at the secondary level, it fits more naturally in social sciences courses.
297 In retrospect, I think it is likely that I introduce this political lesson too early in the night course before building trusting relationships with my students. Were I to conduct my study again, I would wait a few weeks before teaching CM lessons of this sort. That said, however, I did not have such a luxury – the night course was only nine weeks long. Given that the focus of my study was on CM, it would have been inappropriate to push all the critical lessons I designed to the second half of the course.
Moreover, it is not clear how lack of trust might be at play in this lesson. Many, but certainly not all, students seem more than willing to contribute to the conversation.
7.3 The Race and Recess Activity in Week 2
The first three excerpts are from transcripts of the Race and Recess activity.
Taken together, to varying degrees, they illustrate the six themes that emerge in my discourse analysis as outlined above. Excerpt 7.1 illustrates how the critical and mathematical goals of the Race and Recess activity conflict with each other in ways that, due to my inexperience teaching CM, I did not anticipated. Excerpt 7.2 illustrates how my students switch to the use of a vernacular social language in critical activities, particularly when they draw on their lived experiences. It also shows that their Discourse models of social inequality are generally hegemonic. Excerpt 7.3 points again to my inexperience with critical pedagogy by exposing how I use the traditional teaching script to provide mathematical closure to Race and Recess. Although this may not be apparent to readers who are unfamiliar with the students, these excerpts indicate that students who are enthusiastic about Race and Recess are not the same ones who generally shine in reform activities.
298 7.3.1 Newfound Agency and Student Subjectification: Analyzing Excerpt 7.1
1 Me: but / these are elementary schools // is this // it’s still tricky / right? // are you
2 still?
3 Osvaldo: now percent / white / students
4 Me: percent white students / is w[here
5 Osvaldo: [it throws it / off
6 Me: it throws it off / basically // can anybody explain / that? // [[besides me?
7 Osvaldo (quietly): [[nah / man
8 Leroy pointing to the screen and explains something to Lucee behind him. Leroy
9 has not engaged discursively in most reform activities to date.
10 Sonny: it’s like // where there’s white people // there’s re[cess
11 Osvaldo: [I know / that’s what I’m
12 sayin’ // that’s what I’m thinking
13 Amalia start laughing, a few others quietly join in.
14 Shannon: it’s / always / the white / people
15 Osvaldo: here’s / what I’m thinking
16 Me (to Sonny): that’s part of it //
17 Osvaldo: the people that // the white people // the white people / are / real / real
18 // (slams arm to desk) // have better jobs and stuff // they / live [in a better
19 community //
20 Efrain (to me apparently): [just give us the
21 numbers //
299 22 Osvaldo: and [[so
23 Lana (referring to Osvaldo): [[oh my God // no / he didn’t just say that //
24 Osvaldo: they can afford to go to //
25 Me: but // this is all // this is all / Chicago Public Schools // so this is all / [in //
26 Chicago
27 Osvaldo: [oh // then
28 why is that percentage white in the school?
29 Shannon: [racist //
30 Me: because / some // okay so / I don’t / wanna explain all this / I want you guys to
31 figure some of this out yourselves //
32 Osvaldo (to me): look out / look out / look out
33 I was blocking Osvaldo’s vision of the overhead and he was asking me to move.
Excerpt 7.1. From Beginning of Race and Recess Activity
The above discourse takes place about five minutes into Race and Recess. Every student has a handout of the lesson. I intended for students to work in small groups to interpret the chart (Figure 7.1). About five minutes into the activity, so many students express their confusion that I change my initial instructional plans and lead an impromptu whole class discussion about the chart’s mathematical meaning.
It is important to point out that if this had been a reform activity, I would have allowed my students to struggle with it in small groups without intervening. I rarely interrupt what is designed for small group work in such a fashion in reform activity.
However, due to my inexperience and lack of comfort with critical pedagogy, I am anxious about the Race and Recess activity being successful and so diverge from my
300 usual pedagogical techniques aimed at subjectifying students. I do not want my students to associate too much confusion and frustration with CM in the early weeks of the course.
I want the lesson to be engaging so students will be receptive to my continuing to teach from a critical perspective. I also do not want students to give up in frustration as some appear to be about to do.
By changing my instructional plans and moving to a whole class discussion, I work against student subjectification. I twice verbalize my own authority (“can anybody explain that besides me,” “I don’t want to explain all of this”), signaling that my students need me to help them do what I present as an activity for them to do on their own. Were this to happen repeatedly, my students would learn that they could get me to intervene upon the first signs of confusion (see Stein, Grover, & Henningsen, 1996; Stigler &
Hiebert, 1999). At the same time, I do not immediately tell them what the chart means.
Instead, as Excerpt 7.1 illustrates, I subjectify students by asking them to “explain” and
“figure out” the chart for themselves. In addition, this whole class discussion is relatively open to student contributions and several students respond with critical agency by offering their own interpretations (e.g., lines 17-19). Sonny, who initially sits quietly at his desk looking around, probably to gauge his peers’ reaction, becomes involved when he offers a fairly accurate mathematical interpretation of the chart (line 10).
In terms of certain students being energized by CM, Osvaldo and Lana exhibit newfound critical agency when they help maintain the discussion with eager and relatively elaborate responses (lines 17-19), student-driven initiations (line 10), and evaluations (line 23). Unlike Sonny, these students had not yet participated in whole class discussions of reform mathematics at this early point in the course. In fact, several
301 students who tend to resist active participation in reform IMP activities, are highly engaged during Race and Recess. Kampton, Leroy, Lucee, and Eddie laugh, respond briefly, get out of their seats, and generally encourage their more vocal peers in ways they had not done in reform activities. The (taboo) sociopolitical issues that emerged in
Race and Recess appear to be relevant to their lived experiences. It may have been that personal relevance made the activity exciting, or they may have been eager because racism is rarely discussed in the official curriculum, particularly in mathematics.
Excerpt 7.1 highlights the difficulty I have synthesizing the critical and mathematical in CM activities such as Race and Recess. On the one hand, the real world theme of this lesson opens up a space for students’ lived experiences to count as relevant.
On the other hand, their discussions of their lived experiences relative to race, recess, and schools distract attention away from interpreting the mathematical message of the chart.
For example, Osvaldo draws directly from lived experiences and perceptions of racialized distinctions to interpret the chart, stating, “the white people have better jobs and stuff – they live in a better community and so they can afford to go to [schools with recess].” He later asks me if schools that were majority white actually exist in Chicago (lines 27 &
28). He suggests (to me) that the Race and Recess data is not consistent with his own observations of the highly segregated Chicago school system. As a first generation
Mexican-American, Osvaldo has probably seen few white students in schools he attends.
Guevara, for instance, is approximately 1% white. The fact that there are public schools in the city that are majority white contradicts Osvaldo’s, and others’, lived experience.
Conflicts between what they know and what the chart states may be another reason for students’ difficulty interpreting the mathematical message of the chart. It also can be
302 argued that the CM context is so relevant that Osvaldo (and others) have trouble recognizing, or caring about, the underlying mathematical task.
At this early point in the Race and Recess lesson, many students appear to be unclear about the mathematical meaning of the statistical chart (Figure 7.1). There are a few exceptions. Sonny offers a fairly accurate interpretation, when he states, “where there’s white people, there’s recess.” To be more mathematically accurate, however,
Sonny could have said, “where there’s white people, there tends to be recess.” I therefore reply, “that’s part of it” to Sonny. My hope was that he and others would elaborate on his interpretation. It might also be noted that Sonny’s statement does not come out of his lived experiences in any straightforward way. Although Sonny is not middle class, like the middle class students cited in Lerman and Zevenbergen (2004), it appears that Sonny can ignore the real world contextualization enough to get to the underlying mathematics in a way that Osvaldo cannot. It seems that Osvaldo and many peers need more scaffolding than Sonny, and certainly more scaffolding than I provide in the activity.
7.3.2 Lived Experience & Deno’s Hegemony: Analysis of Excerpt 7.2
1 Sonny: where / there’s white / people / there’s recess //
2 Me: is it telling you / where there’s black people / there’s recess // is that what
3 that’s telling you? //
4 Osvaldo: no //
5 Me: that the school is mostly black // is that what it’s telling you? //
6 Male: no //
303
7 Deno: hell [nah! //
8 Princess: [that [[the school is mostly black // so they don’t have recess! // black
9 kids can’t go outside! //
10 Male: [[I think it’s //
11 Amalia, Princess, Sonny and several other students laugh.
12 Deno: they cause / too much damn / trouble! // that’s [[why they can’t have
13 recess!
14 Me (meaning Latinos): [[it’s not just black kids at
15 these [schools //
16 Princess: [well / white people too // (meaning white youth cause trouble too)
17 Lana: we supposed to be playin’ in the (inaudible) – we all (inaudible) //
18 Kampton (to Princess): they [i.e., middle class white students apparently] be
19 blowin’ up shit and //
20 Princess: right! // they be experimenting with the //
21 Lucee laughs louder than the non-bolded speech presented here.
22 Me: is that what this chart is / telling you? //
23 Princess: it’s a ra[cial //
24 Osvaldo: [I / know! //
25 Princess: it’s a [[racist //
26 Osvaldo: [[it’s a / racial-as[[[s // chart [[[[man //
27 Kampton: [[[no offense / Mister B //
28 Princess: [[[[bar /graph! //
304 29 Kampton: it’s tru:e //
30 Osvaldo (laughs): it’s // it’s // you should just / take it off // before //
31 Princess (to someone in front): they’re saying (inaudible) //
32 Deno: Mister B! / Mister B! / [Mister B! // can I say (inaudible)? //
33 Princess: [white people / (inaudible) / [[goin’ outside
34 Me: [[okay // hold on one
35 sec //
36 Kampton (to me): can / we have a / discussion about this / Mister B?
37 Princess: … outside! /// and who are / non-white // [[[can’t go outside!
38 Me: [[[okay / let’s // okay
39 [[[[Princess / hold on one sec
40 Princess: [[[[the minorities! //
41 Lucee and Amalia laugh out loud. Several other students smiling and laughing.
Excerpt 7.2. About Ten Minutes into Race and Recess Activity
In contrast to excerpts from reform mathematics presented in Chapter 6, Excerpt
7.2 illustrates how students take over this critical discussion and are animated, amused, and somewhat indignant about the activity. They express critical agency in closing me out of the discussion, in condemning the chart as potentially racist (some appeared to be joking about this), and in interrupting each other to express their views. Despite my attempts to elicit mathematical explanations (lines 2, 3, & 22), the first half of discussion in Excerpt 7.2 is dominated by student contributions that are more clearly grounded in their lived experiences than in the mathematics at hand. Princess begins with a statement
“the school is mostly black, so they don’t have recess” (lines 8 & 9) that is consistent
305 with the mathematical message of the chart. However, she later joins Kampton to suggest that white suburban students, apparently, “be experimenting” and “be blowin’ up shit”
(lines 18 & 20). While I do not ask them for clarification, I remember noting at the time, that they were referring to the shootings at Columbine High School. The point seems to be that white students receive recess no matter how they behave, but the same is not true for black and Latino students. These students are aware of this stereotyping and injustice.
Both Princess and Kampton are African-American.
Students’ hegemonic Discourse model for social inequality emerges in the activity when Deno, also African American, maintains that black students “can’t have recess” because they “cause too much damn trouble” (lines 12,13). While it may seem that Deno is being facetious or ironic, his serious tone of voice and demeanor (e.g., his gaze was fixed on me) indicate that he appears to believe this or perhaps he thinks that this is what
I – the white teacher – wants to hear. He may be echoing what he has heard school officials state. Evidence that Deno is expressing his actual beliefs comes later in the discussion when he repeats and defends this hegemonic, and problematic, position in terms of his negative perceptions of the way some of his former African American classmates behave (see Fanon, 1967).
While pleased that Race and Recess sparks an animated discussion, pedagogically speaking, I am torn between the differing goals of fostering critical agency and fostering mathematical understanding. I want students to engage with the sociopolitical aspects of the activity, but want them first to understand the mathematics that the Race and Recess chart presents. In fishing for a mathematical explanation, I imply that students’ lived experiences are not “good enough” to count as mathematical interpretations (lines 2, 3, 5,
306 14, 15, 22). While I do not explicitly evaluate students’ attention to the sociopolitical aspects of the lesson as off-task, I imply that they are straying from the mathematical message of the chart. I also know that if I ask students to put discussions of their lived experiences on hold, I risk turning them off to this and future CM activities.
In an interesting display of student authority, my students turn my evaluation of them back onto the sociopolitical content of the Race and Recess activity by claiming it is too “racialized” even “racist” (lines 23 – 28). At this point, their relative control of the discussion, especially in comparison to their behaviors in the reform activities from Week
2, signals three things: (1) their use of a vernacular social language that is distinct from the institutional discourse of most teachers and schools indicates their personal comfort and ownership of the lesson; (2) that they have fun with the activity, calling it racist while laughing about it, indicates their comfort with themselves in this activity and me in my classroom, and (3) my relative lack of discursive presence means I am becoming more at ease when students do take charge.
In terms of students’ comfort level in this discussion, it is important to notice the way Princess switches codes between (what might be called) scholastic and vernacular social languages. Princess begins with a mathematical explanation of the chart using the social language that I, the white teacher, would use (lines 8, 9 & 16). She then switches to a vernacular language (line 20), one shared by many of her peers, in order to agree with
Kampton’s grounded statement about dangerous suburban white students (lines 18 & 19).
In other words, when Princess directs a mathematical interpretation at me she uses the scholastic social language and when she makes a sociopolitical comment mainly directed at her peers, she uses vernacular.
307 Deno, Kampton, and Osvaldo switch between vernacular and scholastic languages in a similar manner. Deno and Kampton clearly use “their” vernacular social language when they make their grounded statements in lines 7, 12, 18, and 19. However, when they address me in lines 30 and 36, they use a scholastic social language. In Excerpt 7.1, note that Osvaldo uses a scholastic social language to explain his thinking to me, however, in Excerpt 7.2 he says, “racial-ass” activity, apparently to affiliate with his classmates. Finally, Osvaldo switches back into the social language of school when he tells me, “you should just take it [i.e., the chart] off [the overhead].” I believe the discursively active students’ use of vernacular social language is further evidence of the fact that they take ownership of the Race and Recess discussion; that is, they display critical student agency. This code-switching also illustrates how students use discourse to affiliate with each other and distinguish themselves from me. I am, after all, the institutional authority despite my engaging them in social justice activity that might not be condoned by the institution where I teach (i.e., Guevara).
Finally, it is worth noting that the discussion becomes racialized, but not necessarily in the way that the students suggest. With the exception of three Latino/a students (i.e., Osvaldo, Sonny, and later Amalia), the discursive participants are predominantly African American (i.e., Kampton, Princess, Deno, Lana, and later
Shannon). It might be argued that race is an issue that my African American students feel more passionate about or more comfortable discussing. Kampton makes it clear that he would like our impromptu critical whole class discussion to continue (line 36) despite, or because of, the fact that the discussion is racialized and not mathematics as usual. The
Latino majority in the class do not necessarily resist the activity; a group of Latino
308 students located in left front of the class encourage their classmates by making it clear that they are following the discussion by making eye contact and laughing out loud (lines
11 & 21). Also, the African American and Latino students are dispersed throughout the room so that students of the same race are not all seated together. My students generally do not separate themselves along ethnic lines in their choices of where to sit and who to work with in small group activities. Nevertheless, there is some noticeable resistance to this activity on the part of some Latino students and one or two African American students. This resistance manifests itself, at the outset of the activity in particular, when students seem poised to give up trying to interpret the mathematical chart unless I intervene. (In Chapter 6, I noted that at this early point in the nine-week course, many students still requested more direct instruction than I generally provided.)
Similar to most CM activities, there is noticeable student resistance at various points in the Race and Recess activity. Just before the dialogue presented in the next excerpt, two students take jabs at the activity. Efrain asserts that the critical Race and
Recess activity is “not what we’re here for,” and Lucee adds, “it’s goofy.” These students are not convinced that CM is appropriate or worthwhile, although both appear to change their position on CM later in the course. Hearing Lucee’s evaluation of Race and Recess,
Amalia, one of the more active participants in CM activities, quickly jumps in to defend it, saying, “it’s not goofy, it’s just – I dunno – it’s confusing.” Amalia exhibits agency
(unsolicited by me) when she returns to the mathematical interpretation that is frustrating many students, and states, “when there’s less white kids there’s less [recess],” this begins the sequence in Excerpt 7.3.
309
7.3.3 Providing Closure Through a Traditional Teaching Script: Analysis of Excerpt 7.3
1 Me: so // Tania / Lucee // I wanna make sure you guys are / with me on this //
2 okay? /// so // this school right / here // are there more white kids / in these schools
3 // at the bottom // or more white [kids here //
4 Lucee: [Mister / B
5 Me: yeah //
6 Lucee (who just called this activity “goofy”): I wanna (talk) with you //
7 Me (to Lucee): huh? //
8 Lucee (quieter): I’ll tell you later //
9 Me (I point to chart and address class): um / right here // which of these / schools
10 has more / white kids //
11 Shannon: the [bottom one //
12 Osvaldo: [the bottom [[one //
13 Amalia & others: [[the bottom one //
14 Me: so / there’s less white kids here / and // t[[[hen //
15 Amalia: [[[less / recess //
16 Me: less recess // and then about // he:re / there’s almost no: / white kids // hardly
17 any //
18 Osvaldo: and // there’s al[[most no recess! //
19 Amalia (with Osvaldo): [[and there’s no recess! / right there //
20 Me: so // what percent of these kids / got // get // these schools // maybe about
21 Sonny: ten-percent //
310 22 Me: maybe ten percent // of these schools / get recess // okay / so now // Kampton //
23 wants to have a discussion / about this //
Excerpt 7.3. From Twenty-Five Minutes into Race and Recess Activity
Excerpt 7.3 takes place approximately twenty-five minutes after the Race and
Recess activity was introduced. We have reached an impasse of sorts: I continue to insist on a mathematical interpretation while many of the discursively active students are more involved in discussing their lived experiences relative to race, recess, and schools. Most are not purposefully trying to thwart my intentions, rather they appear not to fully understand them. Again, I do want students to discuss lived experiences relative to race and schooling, but only after they prove they understand the mathematical message of the chart. The fact that there is this degree of confusion points to my inexperience teaching
CM activities or, in Gutstein’s (2005) terms, managing the dialectical tension between the critical and the mathematical.
In hindsight, I might have prefaced this activity with having students use real world data to build a similar graph to that in Figure 7.1. Because the mathematics in the chart seems beyond many students’ zone of proximal development, or beyond the extent of their willingness to struggle, I face two equally poor spur-of-the-moment instructional choices: (1) continue to leave the (discursively active) students to their own devices, or
(2) objectify students by interpreting the chart for them. As Excerpt 7.3 makes clear, I choose the latter approach – again, after repeated attempts to avoid making this choice. In order to transition out of this whole class discussion, I rely on a traditional initiation- response-feedback (IRF) instructional pattern (Cazden, 2001), beginning with an initiation (line 14) that Amalia finishes with the brief, correct reply I look for (line 15). I
311 repeat with similar IRF cycles of this sort with Osvaldo and Amalia (lines 16 – 20) and
Sonny (lines 20 – 23). I rely on Sonny and Amalia to come up with a right response that I can validate. These students have relatively strong mathematical backgrounds and earlier show they understand the mathematical message of the chart.
According to Pruyn’s (1999) framework, I objectify students when I use this traditional IRF script; I ask leading questions and start and finish student responses. In such a script there is little space for students’ own ideas to count meaningfully. Given that I had positioned the students subjectively as “knowers” in most mathematics lessons, and earlier in this lesson, it is somewhat self-defeating for me (a progressive teacher) to suddenly switch gears and objectify students by placing them in the position to need my expertise. On the other hand, I rarely use this discursive technique in exploratory activities and only use it here, as a last resort, when I feel my mathematical goals for the lesson are in jeopardy. In hindsight, I believe it would have been better to stop the discussion (what might be seen as a “guessing what the teacher has in mind” game) and give my statistical interpretation without the pretense that ideas are coming from students. Or, I could have postponed interpretation until I had given my students the opportunity to develop more of the requisite skills to interpret the chart. (Skills I believed they had coming into the activity.) Were I to teach this activity again, I would ask students to construct similar charts for themselves prior to asking them to interpret the chart in Figure 7.1.
After Excerpt 7.3, I open the floor to students’ to discuss their sociopolitical ideas unhindered by my earlier insistence on a mathematical interpretation. This is the non- mathematical whole class discussion of Race and Recess that Kampton calls for (line 36,
312
Excerpt 7.2). However, the conversation that follows never turns into a full critique that would link inequities in recess to institutional racism or economic inequality as I rather naively hoped it would. On the contrary, Lana asks Deno to repeat what he said earlier and, again, he invokes a hegemonic Discourse model for social inequality when he responds, “they banned recess at our school cause they was cussing up the classroom.”
This sets off a brief flurry of evaluations by Shannon and Lana that challenge Deno for blaming African American students for their lack of recess. While the two take on counter-hegemonic stands against Deno’s hegemonic claims, Shannon and Lana fail to offer a deeper (elongated) counter-explanation that would explain the glaring inequities in who receives recess and extend these observations to broader conclusions about systematic racial bias.
Unfortunately, I did not push Lana and Shannon to further clarify their positions when they were on the desired counter-hegemonic track. My (progressive) goal to recede into the background in this conversation means I do not urge the female students to explain further when they touch on the “correct” counter-hegemonic perspective in the same way I had just done to provoke students to embellish on the correct mathematical interpretation. Thus, while I want to support the counter-hegemonic position, I decide to keep quiet and allow students’ voices to be heard without the scaffolding that I might have provided. While my silence appears to allow students the subjectifying space they needed to contribute to the whole class discussion, it is problematic from a critical perspective because I do not challenge the hegemonic beliefs that students obviously internalized.
313 Another problematic aspect of this later discussion is that several students who are not directly involved are no longer paying attention. My comments to Lucee and
Tania at in Excerpt 7.3 indicate that I believe their disengagement or resistance relates to their perception of CM. Lucee calls the Race and Recess activity “goofy.” Tania and another student put their heads on their desks. Despite the active engagement by many students, several stage performances that signal their disinterest in Race and Recess.
Their messages of resistance are effective because, shortly thereafter, I decide to end the whole class discussion and close by asking the class to write a response to the two questions: “what is the point this chart is trying to make?” and “does this mean the CPS system is racist?” My decision to shut down the whole class conversation clearly bothers the ten or so students who had been actively engaged throughout the activity as well as several onlookers who want the discussion to continue. When I state that we need to
“move on” because the discussion “broke down,” Amalia states, “No!” And, Deno, practically shouting, asks me, “What do you mean?!” Meanwhile Osvaldo assures me,
“It’s okay! Mr. B!” (It should be noted that these students might not want to write responses to my prompts and for that reason request that we continue with the conversation.)
7.3.4 Reflecting on the Race and Recess Activity
Critical activities such as Race and Recess raised a number of issues for me as an experienced mathematics teacher but an inexperienced critical pedagogue. The Race and
Recess discussion was more open ended and unwieldy than reform discussions of mathematics I had led. Race and Recess was among the critical activities that raised
314 pedagogical questions, such as: To what extent should I pursue students’ ideas when they do not engage with substantive mathematics? Should I use my authoritative position to side with Lana and Shannon and challenge Deno’s hegemonic assertions or should I allow the conversation to continue in no apparent hegemonic or counter-hegemonic direction? More generally, if students never challenge the hegemony, how can I help them reflect the hegemonic nature of their position without directly challenging them or giving didactic lectures? How do I intervene as a pedagogue and not objectify students by taking an authoritative stand? How do I react to students who (initially) reject or resist
CM because they believe it does not belong in secondary mathematics? Do I try to convince (or coerce) students who never engage with the critical part of CM that it is an appropriate part of secondary mathematics? Should I allow them an out of CM activity by setting aside separate work for them? These questions reflect my worries as I ventured into the new terrain of CM for the first time.
7.4 The Inequality and Area Project in Week 7
The Inequality and Area (I&A) Project came on the second hour of the first day of
Week 7 in the nine-week night course. By this time, I had taught several more CM activities after teaching Race and Recess (Appendix C includes an overview of course activities). (After teaching I&A in the night course, I redesigned it as the I&A Unit shown in Appendix A and analyzed in Chapter 5.) The four excerpts that I discuss here come from the introductory hour that is the most critical and least mathematical of the three-hour-long I&A Project. The other two hours are far more mathematical than critical in nature. See discussion Chapter 5 for an analysis of the different types of CM activities.
315
7.4.1 Overview of the First Day of the Inequality & Area Lesson
We begin the I&A Project after finishing the reform activity, “Draw the Same
Shape” discussed in Chapter 6. Having learned from the confusion in earlier CM activities such as Race and Recess, I am now careful to initially keep my critical goals separate from my mathematical goals so that they do not interfere with each other. Hence, as I describe below, the first hour of the I&A Project is essentially critical while the second and third hours are essentially mathematical (see Chapter 4 for details on the I&A
Project). While two distinct types of activities comprise the I&A Project, I thought that the critical and mathematical would build on each other in a dialectical fashion and that, by the end of the project, the tension between the critical and the mathematical would be somewhat resolved. At this point in my study, I still believe that the mathematics in I&A would heighten my students critical sociopolitical understandings, while critical activity would heighten students’ mathematical understandings.
1. When President Bush claims that we're spreading democracy and freedom to Iraq and throughout the world he's talking about spreading political democracy … Yet there is also economic democracy where citizens receive a fair share of the economic output they help produce … Do you think the U. S. economic system is fair to its citizens? Why or why not?
2. One of the key concerns of the Civil Rights Movement was worker’s rights …
Has the distribution of income in the U. S become fairer since the Civil Rights
316 Movement? What do you think? What would your family members who work say?
Figure 7.2. Abbreviated Versions of First Day Questions from I&A Project
The first hour of I&A proceeds as followed: I first ask students to write individual responses to the two prompts in Figure 7.2. After that, I break students into five
“quintiles” of income bracket groups: the first quintile represents the poorest 20% of U.S. wage earners in 2001, the second represents the second lowest 20% of wage earners, and so on to the wealthiest quintile. I next give each group the portion of fifty small candy bars that correspond to that quintile group’s yearly earnings. We end the lesson with a whole class discussion about whether or not this distribution is fair and I ask them to conjecture about why U. S. citizens put up with such inequality.
7.4.2 Overview of Inequalities & Area Lesson as Enacted
This Inequalities & Area Project lesson touches on five of the six CM results outlined at the beginning of this chapter. First, student participation continues to be markedly different in such critical activities than it is in reform activities; there is a shift in who participates and how students participate. Second, many students continue to invoke a hegemonic Discourse model for social inequality; for instance, several assert that wealthy people are richer because they work harder than poor people. In other words, despite having been exposed to several critical activities (see Appendix C), students still struggle to name structural conditions and to challenge hegemony. Third, as in Race and
Recess, many students use a vernacular social language, indicating a degree of ownership not exhibited in reform discussions. Fourth, student response to CM lessons still fluctuate in somewhat erratic ways; there is considerable student resistance at the outset of I&A
317 and, while the activity eventually ignites critical student agency, there is no a guarantee that it will ignite in the way it did here. Fifth, and finally, I no longer attempt to accomplish both critical and mathematical goals at the same time or in the same lesson.
My new, more experienced approach to CM includes designing activities so that critical goals do not interfere with mathematical goals.
The next four excerpts come from activity at different points in the I&A Project lesson. Excerpts 7.4 and 7.5 capture the unpredictability of I&A and critical activities more generally. It points to how considerable resistance and critical agency can coexist in two separate areas of the classroom. Excerpts 7.5 – 7.7 show how students use their vernacular social language to take ownership of critical activities once they engage in the lesson. It also illustrates how their (generally hegemonic) Discourse models for social inequality emerges quite powerfully. I point out the discursively active students who are not generally active in this same way in reform activities as well as their counterparts who tend to resist critical activities but generally engage in reform activities.
7.4.3 Resisting the Critical I&A Writing Prompt: Analysis of Excerpt 7.4
1 I’ve asked students to write individual responses to the two prompts in Figure 7.2
2 that dealt with U S economic inequality and justice. Jayla is sitting and staring
3 towards the front of room as others around her begin to write responses.
4 I walk over and sit down next to Jayla.
5 Me (to Jayla as I touch her paper): Is there something about this / you don’t
6 understand // like // what I’m asking you to do? //
318 7 Jayla (laughs): I just / forgot what I read //
8 Me (I smile at her and): okay // well // read it again //
9 Jayla yawns and looks at paper as I look towards the back corner of classroom
10 where Adelric, Malik, and Efrain are also looking towards the front of the room but
11 not writing or even talking to each other.
12 Me (I look and nod at Adelric in back row): you know what you’re doing? // you
13 guys okay? //
14 Adelric: my pencil’s broke //
15 Me: what? //
16 Adelric raises pencil.
17 Me: you can get one / off my desk // do you want me to / get you one? //
18 Adelric nods.
19 Most students in classroom hunched over I&A worksheet and writing.
20 Me (I go get a pencil off my desk.): Adelric // Adelric
21 I toss pencil back to Adelric as I head back in their general direction.
22 Efrain (looks up from paper): hey // can you say // can you [tell me // what this is
23 asking? //
24 Osvaldo (in front of room): [o:h / shit! //
25 Me: basically // there are / people who are janitors // there are people / who are
26 teachers // there are people who are / doctors // there are people who / work at
27 McDonalds // everybody gets paid something // do you think / what people get paid
28 / is fair? //
319 29 Efrain: yeah / cause they / went to school for it //
30 Me: okay // but / then what about // there are a lot of people / who didn’t go to
31 school // like jan[[itors //
32 Efrain: [[exactly! // (leans forward waves hands for emphasis) // just
33 because they’re janitors / they’d be getting paid / (inaudible) //
34 Me: okay // so then / write that // okay // it sounds like / you think it’s fair //
35 Kampton, meanwhile, has been patiently holding out his paper for me. Like several
36 students, and unlike others, he has had his head hunched over his paper and had
37 already begun writing.
38 Kampton (to me): Can you read it / and see if you understand it? //
39 I read it, nod, and smile.
40 Kampton wrote the following: “I would have to say yes & no. Because the U. S.
41 economic system sucks. I believe every one should be paid equally. Because
42 regardless of how much you make the Federal Government take out the 15%
43 amount of taxes so either way it goes you working for less.”
Excerpt 7.4. Initial Student Response to the Initial I&A Writing Prompt
Excerpt 7.4 illustrates the initial student resistance I face to the I&A writing prompts about economic fairness (Figure 7.2) and how I visit with small groups of students to get them to engage with it. Their resistance takes the form of simply not writing, claiming excuses (e.g., broken pencils), and asking for further clarification even though the writing prompts are fairly obvious. While the excerpt does not indicate how the delay tactics are overcome, by interacting with students I was able to get most to begin writing. Although some students were often slow to start in on reform assignments
320 as well, several who are generally quick to start in on reform activities, exhibit initial resistance in the I&A activity. For example, in the sequence that occurs shortly after the activity in Excerpt 7.4 is finished, Stephie and Robi, both of whom received A’s in the course, continue a conversation they started during break. As I move to their area of the room, Stephie asks me “So, you just want a like….” I respond to her by impatiently stating, “I want an opinion.” After observing my frustration with them, Robi and Stephie quickly begin working. In fact, Stephie writes fairly elaborated written responses to the two I&A prompts. Perhaps because she senses my displeasure with her, Stephie writes the following accommodating response to the first prompt:
I think that teachers are underpaid. Teachers help us with our education. Teachers
have to put up with a lot of things. People who get overpaid are basketball,
baseball, and football players. Lawyers are overpaid too! I think we should waste
less money on basketball, baseball, and football players and waste more money on
teachers. They can also fix up the schools and buy better books. Schools are more
important than sports players! (January 13, 2004)
In contrast to some of their classmates, several students who tend to resist reform activities launch into the critical I&A activity. Kampton, who did not finish the reform activity considered in the first hour and who tends to resist school mathematics (see
Appendix D), finishes writing a response to the first I&A prompt before a his neighbor
Jayla begins to write hers. Apparently he is more interested in the critical I&A Project than dominant versions of school mathematics. He also appears quite anxious to have me read his response (lines 35 – 43).
321 Students’ hegemonic Discourse models for socioeconomic inequality emerge once they begin to engage with the I&A activity. Efrain gets the ball rolling in his small group when he argues that some people get paid more because, “they went to school for
[higher paying careers].” While there is some truth to this claim, it is also hegemonic because, from the critical perspective, salaries for non-educated workers should be more equivalent to those of individuals with more education. It is also hegemonic in the sense that Efrain clearly buys into the meritocratic and social mobility argument of neo-liberals and neo-conservatives (e.g., schools are fair, the wage structure is fair) (see Apple, 2001).
7.4.4 Critical Ignition in a Small Group Discussion: Analysis of Excerpt 7.5
1 At about the same time the activity in Excerpt 7.4 transpires, Lupe, Amalia, Eddie,
2 and Juan begin discussing how my use of the word “citizen” in the first writing
3 prompt excluded immigrants from the discussion of economic fairness.
4 Lupe (to Eddie): we’re not taking your jobs // you guys / are / just / lazy // (slams
5 pencil to desk) // like / I said before!
6 Amalia: lazy //
7 Eddie smiles, looks at his desk, and shakes head no.
8 Lupe: if you // if you / really think about it / and // if you see // [Mexicans are very
9 hard workers / okay?
10 Eddie smiles, shakes head and looks down at desk. Says something inaudible.
11 Me (to class): [okay // how much
12 more time / do you guys need to write / write down answers to / one and two?
322 13 Efrain: [[four more / minutes //
14 Jayla: [[we’re having a / group discussion here //
15 I walk over near Eddie, Lupe, Juan, and Amalia.
16 Amalia (to me): we’re having a discussion / and you’re not included //
17 Lupe (to me): yeah we’re discussing I // this is important //
18 Me (smiling): okay
19 Lupe (points to Eddie): he brought up / a good thing // a good point //
20 Me (to Lupe): what’d / he say?
21 Lupe: he thinks / that // Mexicans // are taking up / all of the
22 Amalia: immigrants / pe:riod! //
23 Lupe (points to Amalia): yeah // (smiles at me) //
24 Me: yeah // that’s something we // I don’t think / we’re gonna be able / to get / to
25 quite with our mathematics // but we’re going to be able to compare / [Mexico to
26 Lupe: [and I was
27 saying that / that’s not true // it’s just that / U. S. citizens are just / la:zy //
28 Amalia, Lupe, Juan, and I laugh.
29 Eddie: you forgot / (inaudible) / the work we do //
30 Lupe (looks at Eddie smiling): which is nothing // you don’t / do / nothing //
31 More laughs.
32 Me (to Lupe as I walk towards another group): okay // so write that down! //
Excerpt 7.5. More Initial Student Response to Initial I&A Writing Prompt
Excerpt 7.5 overlaps in time with Excerpt 7.4. The critical engagement and agency that Lupe, Amalia, and Eddie exhibit might be contrasted with the resistance I
323 encounter with such students as Jayla, Adelric, and Efrain who are seated in the other side of the classroom. It points to the variability of impact that critical activities have in that these two groups of students have remarkably different initial reactions to the same activity. The conversation topic of immigration is sparked by Lupe’s observation that immigrants are not included in the I&A project questions and later in the quintile data.
This observation leads to a small group discussion about the work that immigrants do and the low pay they receive. According to Lupe, Eddie has claimed, “that Mexicans are taking up all of the jobs.” She responds in defense that “U. S. citizens are just lazy.” Note that Lupe and Juan (who does not say anything here) are immigrants from Mexico, while
Amalia’s mother is a Mexican immigrant. Eddie is Puerto-Rican American and does not appear to see himself or his relatives as immigrants. In another conversation, Eddie provokes his Mexican heritage groupmates when he brags that Puerto Ricans do not have to swim or take boats to mainland U. S. because they have U. S. citizenship.
My response in line 24, “that’s something I don’t think we’ll be able to get to quite with our mathematics,” shows how I feel constrained by the secondary mathematics curriculum I am required to teach in the course of critical activities. While I want the
I&A Project to be relevant to the lived experiences of all of my students, the economic data I collected for the class to work on in the next two days does not include data on immigrants (see Appendix A). While I could encourage students to discuss the work that immigrants do and the low pay that many earn to make the mathematical component of the I&A Project culturally relevant according to the race and ethnicity of students, in the midst of the lesson, it would not be easy to redesign the activity to include immigration on short notice. As discussed in Chapter 4, I found it generally difficult to find public
324 domain examples to include in my I&A Unit that were both mathematically appropriate and culturally relevant to all of my students. If this had been a course on sociology or history, I would not have felt as constrained as I did as a mathematics teacher.
Despite my apologies for the need to focus on secondary mathematics, Lupe and her peers seem unphased. Lupe cuts my mathematical apology off to make the point that,
“U. S. citizens are just lazy!” Lupe is clearly enthusiastic about the chance to discuss immigration and does not seem to care if the mathematics she was required to learn is relevant. Her enthusiasm indicates the degree of ownership over the discussion that many students have in CM activity. Although she said it in a joking and friendly way, Amalia’s says, “we’re having a discussion and you’re not included.” She directly asserts her ownership of the topic.
7.4.5 Whole Class Income Distribution Activity: Analysis of Excerpt 7.6
1 Eddie: nah // he’s fittin’ to // he’s going to give me a [piece of crumb! //
2 Kampton (to Eddie): [you broke / bastards!
3 Eddie: you’re going ta / give me / a piece of crumb!
4 Lupe (to Eddie): I / know / ri:ght? //
5 Osvaldo (to me): nah // they should get like //
6 Eddie: we’re going to get the / left-overs //
7 Lupe (quietly to Eddie): the fuckin’ wrappers // (laughs)
8 Amalia: how many / do we get //
9 Lupe (points to screen): one! // (laughs) //
325 10 Amalia: damn! //
11 … a minute later …
12 Me: what do you guys think // am I going to be able to / have a conversation
13 around this / or not? //
14 Osvaldo: ye[s! //
15 Jayla: [ye:[[s //
16 Boy: [[yes //
17 Me: okay // let’s have a / conversation about it //
18 Kampton (to students in back of room): shut / u:p! //
19 Osvaldo (towards Malik and Adelric): !Cállate la boca / cabrón! [Shut your mouth,
20 bastard] // hey / that ain’t fair / man // they [one group] got twenty-five! //
Excerpt 7.6. Second I&A Task: Modeling Distribution of Income
Excerpt 7.6 covers the quintiles activity that follows the writing prompt and small group discussions in Excerpt 7.4 and 7.5. At this point, I finish sorting students into five quintile groups. I begin distributing candy to these groups in a way that models the 2001
U.S. distribution of income by quintiles. Dialogue in this excerpt begins as I distribute fifty candy bars unevenly to the specific social-class related groups of students. Each candy bar represents two percent of the total national annual income and each of the five groups represent a quintile group of U.S. citizen-wage-earners.
Excerpt 7.6 points to three of the five findings that relate to CM, namely: (1) an increase in student agency and lesson ownership of this phase of the I&A activity, (2) the use of vernacular social language in whole class discussions of critical themes, and (3) the continuing impact of my inexperience and insecurity as a critical teacher.
326 As in the Race in Recess activity, several students (e.g., Kampton, Eddie,
Osvaldo) participate in the I&A Project in ways they generally do not participate in reform activities. Osvaldo and Jayla, two students who often resist the reform activities, indicate they are eager to discuss the unfairness of the U. S. distribution of income (lines
14 – 16). Recall that Jayla initially resists the writing component of I&A but does not hesitate to engage in the whole class discussion. She and others take considerable control over this critical discussion. Osvaldo and Kampton show their support for CM by quieting down their classmates to help me transition into leading the whole class in a discussion of economic inequality.
As in the Race and Recess activity from Week 2, students’ use of a vernacular language in the I&A project indicates ownership of lessons. Eddie begins by telling his groupmates, “he’s fittin to” and then, perhaps realizing I was listening, modifies his language, switching from the vernacular “fittin’ to” to the more scholastic, “going to.”
Osvaldo’s use of Spanish “cállate la boca” (i.e., shut up) is also an interesting example of use of language and code switching. These student behaviors, and student comments, signal that critical activities are in a quite different disciplinary register for them than they use in reform mathematics. My students clearly recognize critical activities as being something different than regular mathematics or school-as-usual. They act and speak differently during critical lessons than they do in reform lessons.
It was not simply the political orientation that draws the discursively active students and others into the I&A lesson. The candy distribution activity is also fun; many students enjoy it despite the fact that they do not receive equal amounts of candy. In fact, the unfair distribution of candy clearly provokes animated reactions from many in the
327 class. Eddie is even more animated about this part of the I&A Project than he was in the earlier discussion of immigration. Immediately preceding Excerpt 7.6, he makes several statements about bringing “communism” and “Fidel Castro” to the United States.
Kampton’s comment “you broke bastards!” indicates a similar level of enthusiasm and involvement on his part.
Despite student interest exhibited in the quintile (sub)activity, my experiences with critical mathematics taught me that student response could head in unpredictable directions and that the engagement they were exhibiting at this point in the lesson might diminish or vanish rather suddenly. Thus, when I ask, “are we going to be able to have a conversation around this?” (lines 12 & 13), I make evident my continuing insecurity with critical pedagogy and my need to have them drive the critical discussion in a way that makes it worthwhile. Fortunately, as the excerpt makes clear, many students continue to be interested in the theme of economic inequality and so a productive discussion ensues.
Their interest in the I&A project went deeper than being given candy.
7.4.6 Elaborate Critical Student Engagement: Analysis of Excerpt 7.7
1 Osvaldo (sits up in his chair, grins): I think / we all should / just get paid the same
2 thing //
3 Kampton: Yeah!
4 Jayla: but [then we (inaudible)
5 Osvaldo: [you know / why? // not /
6 Eddie says something about “communists.”
328
7 Osvaldo: not paid the same // but the income // [[you know?
8 Lucee: [[okay / Mister B // my turn // I’ll
9 go //
10 Me: okay //
11 Lucee (to me): are you / ready? //
12 Me: yeah //
13 Lucee: I think that / its not fair / because // that if we // they give us / like // okay //
14 cause some people // don’t have / higher educations than other people // because
15 most people / they simply / just don’t have / enough money // and they have / either
16 a large family / or a real small family / that they gotta help out // either you got
17 someone at home / that can’t work / or whatever // or there’s issues / (inaudible)
18 whatever // and you gotta work // so you [can’t go to school //
19 Me: [yeah //
20 Lucee: so / all that stuff // but if someone gives like / someone else / like the
21 opportunity // and they develop the skills // they can just be / as smart as anybody
22 else / who went to school // they just didn’t have all the time to do all that stuff that
23 they did //
24 Me: yeah //
25 Lucee (scrunches body and face up and looks at desk): yeah //
26 Me: so / if you have less money // then you gotta spend more time working // and
27 then you / can’t go to school and / stuff like that //
28 Lucee: yeah //
329
29 Stephie raises hand in back of room.
30 Me: okay // go ahead / Stephie //
31 Stephie: okay / I don’t think that // (points to Lucee) // like she was saying that // if
32 you don’t / go to school / you can’t work / because // like / I’m going to use myself
33 as an example // like / I work // and I go to school // and even though / I don’t have
34 my high school / diploma // I moved up / little by little // and I became manager //
35 so // it’s not that // you just gotta push yourself / forward // and / you’ll do it //
36 Me: I think / both you guys agree / it took that idea / that you / have to work harder
37 // maybe than pe[[ople //
38 Stephie: [[it’s not that / hard // I don’t think it’s that much harder //
39 Kampton: but // for a person / with strong / willpower // [that person…
40 Efrain: [I believe that /
41 everything you get // everything you want / you need to work hard for it //
42 Robi (has hand raised): yeah //
43 Kampton: or // kiss a lot of ass //
44 Efrain: yep //
45 Me: next // huh ah // I mean / I’ll be honest // well // I’ll talk / I’ll take that up /
46 another time // (to Efrain) // remember you said / that //
47 Lupe: what [[comes easy // goes easy //
48 Robi: [[Mister B // I think / in a way / it is / fair // that like / rich people get
49 more // because if you / went that / extra step // to be / [like a / higher education / or
50 something //
330
51 Princess: [but / some people get it /
52 from their parents //
53 Shannon: right! //
54 Robi: you deserve / more than // somebody who / didn’t / take / that extra step / or
55 use it //
56 Princess: okay / but // [[some people // inherit //
57 Shannon: [[if you’re born rich you
58 Me: yeah // did you take / I mean / so // if you’re born to be rich // did you [ take /
59 an extra step?
60 Amalia: [you are
61 / going to stay / rich //
62 Robi: but / if you’re already born to / then what’s the point // of working hard any
63 more? //
64 A number of students talk over each other at this point.
65 Lucee: if any // well / you know / what he’s talking about // if you took that
66 extra step // there’s people / who / wanna take that extra step //
67 Shannon: right //
68 Lucee: but they don’t really/ have the opportunity // they can’t // [it’s not really
69 their decision // [[they have to / go
70 Efrain: [too bad for
71 them // [[it’s really their / way of life //
72 Lucee (turns towards Efrain): you just // fucked / up // saying that //
331
73 Lucee then covers her mouth and smiles. Many students laugh.
74 Me: nah // I // I mean / I definitely / hear what you’re saying // I mean // besides the
75 um // f-word / there // the rest of / what you’re saying // but // you // you don’t /
76 agree with that // Malik? //
77 Malik (who had just raised his hand): no /// (shakes his head)
78 Me: that’s okay / you [don’t have to
79 Lucee: [cause // I could go // I would’ve / taken those steps / but
80 they’re // there’s // you know // there’s things / that are // holding me back /
right now //
Excerpt 7.7. Critical Whole Class Discussion of Economic Fairness
The dialogue in Excerpt 7.7 occurs during the follow up discussion to the quintiles distribution activity. It is a three-minute slice from a whole class discussion that lasts over ten minutes. In this discussion, students express opinions about economic inequality in light of the quintiles activity about the unequal distribution of resources they just experienced.
Excerpt 7.7 points to three of the six results about critical mathematics activities outlined at the beginning this chapter, namely: (1) several students who tend to resist reform activities wake up to contribute with critical agency to the whole class discussion of economic fairness; (2) students hold both hegemonic and counter-hegemonic
Discourse models of social inequality with the former dominating the latter; and (3) students, somewhat ironically, switch back to the use of a scholastic rather than a vernacular social language during this whole class discussion. I think they do so because
332 they were communicating through me at this point in the lesson rather than with each other as they had earlier.
Excerpt 7.7 illustrates the shift in terms of which students participate and how students participate in critical activities such as Race and Recess and I&A. Lucee, and others, who regularly resist participating fully in reform activities, exhibit considerable critical agency in this whole class discussion. Not only do they participate voluntarily, but their contributions are far more elaborate than in whole class discussions of reform activities. Moreover, this excerpt points to a shift in the orientation of some students towards critical mathematics. Recall Lucee called the critical Race and Recess lesson
“goofy” in Week 2, whereas now she is eagerly involved.
In terms of students’ Discourse models for social inequality, Lucee takes up the counter-hegemonic position when she argues that there are structural factors that limit the life chances of students from low SES backgrounds (lines 13 – 18 & 20 – 23). Stephie disagrees with Lucee when she invokes a more hegemonic model stating, “if you push yourself forward you can do it.” To be clear, Stephie’s argument essentially recreates the hegemonic, status quo argument “that anyone who works hard will make it.” Perhaps because Stephie’s message is more hopeful, many students seem to support it. In the discussion that ensues, a small number (e.g., Lucee, Shannon, Princess, and myself) argue the counter-hegemonic position, while Stephie and the majority of discussion participants argue from the hegemonic position. For them, any degree of mobility requires a positive attitude and a good work ethic because were born rich and therefore would not have things handed to them. Their contributions reflect that social reality rather
333 than a critical consciousness of the unfairness of their circumstances compared to their white and Asian, affluent suburban counterparts.
Unexpectedly, perhaps, the tone of the discussion became more serious and students switched back to the scholastic social language from the vernacular social language just before Excerpt 7.7 began. Again, I think it was because they were directing their comments through me at this point in the discussion. While many in the class continued to exhibit considerable interest in the discussion, there was a sense in which the I&A Project had become “school” again and signaled that they should switch registers. They wanted to be heard by me and their peers. The exception occurred when
Lucee (jokingly) lashed out at Efrain telling him he was “fucked up” in response to his hegemonic statement that “its just their way of life” that keeps poor people down.
Lucee’s use of a non-scholastic register in response to Efrain was likely more effective in putting Efrain in his place than anything she could have said in scholastic register.
7.5 Discussion
The two activities discussed in this chapter show both the promise of critical education and some problems with including critical themes in secondary mathematics curriculum. First, while imperfect, there were many positive outcomes to the two critical lessons discussed here. The introduction of politicized content into the night course curriculum altered patterns of classroom participation. Issues of economic fairness and racism in schools resonated with several previously disengaged students in ways that reform secondary mathematics rarely, if ever, had in the course. Elaborate student
334 contributions and the use of vernacular social language suggest that many students felt ownership over topics that were discussed in critical activities in contrast to those raised in reform activities.
The night course, and the critical component of the curriculum in particular, appeared to change students’ beliefs about school mathematics. In particular, student interviews and samples of written coursework indicate that the inclusion of critical mathematics activities changed – or at least reinforced – student opinions about the relevance and utility of school mathematics. Consider this post-interview conducted by my Northwestern colleague, Ken Rose, with two of the night school students:
Jaime: And what I like about Mr. B. is that he did not only use math, well, he did
use math in every subject.
Ana: He used, like, charts and stuff.
Jaime: He used charts, statistics, and all that – of how income works. And how it
is in the United States, and how it is in Mexico. He would compare stuff like that.
Ana: Yeah, the value of the money that you use.
… [a bit later in the interview] …
Ken: So what is math and where does it come from?
Jaime: Well, like I said, math is a tool. Basically it came back, I think it comes
from way back. I think it’s ancient. It’s, I think that it’s a way of how they
survived in the past. You know. Cause they used math in real survival because of
the fact that like how I told you before Mr. B. always used like, this income tax of
how to divide things in categories. Like low-income high-income. And I think
that it was their source of planning how, how much food you’re gonna get
335
individually. And I think that’s how they used the math, in the back, in ancient
times.
Hence, the inclusion of critical data analysis (e.g., charts, statistics) in activities such as the I&A Project allowed my students to see school mathematics as a “tool” subject that can be relevant to important sociopolitical issues. To be clear, while this was one of my
CM goals, I have come to question whether such utilitarianism is wholly beneficial (see
Chapters 4, 5, and 8).
Teaching secondary mathematics from a critical perspective raised several additional issues that I struggled with when teaching the night course and my concerns continue as I think about whether to recommend CM instruction. First, in the CM activities where mathematical and critical content were not presented separately (e.g.,
Race and Recess), a conflict arose between the critical goal of valuing students’ lived experiences and the mathematical goal of teaching students the required secondary geometry curriculum. Moreover, as discussed in Chapter 5, it is not clear whether critical and mathematical goals can be met simultaneously in critical mathematics activity. It seems that many, if not all, critical mathematics lessons can be parsed into relatively independent critical and mathematical components. This means that critical activities inevitably displace secondary mathematics in CM classrooms.
Second, while many students seemed to feel more positively about critical activities at the end of the course than they had at the beginning, a few students never seemed to approve of critical mathematics. Two high achieving students, Juan and
Sonny, stated such things as, “this is not mathematics” in about the same way in Week 7
336 as they and others had early on in the course. Juan wrote the following in response to the
I&A activity:
I don’t think you should teach these because we’re wasting time studying things
that it doesn’t belong in this class. Instead of doing these you should teach us
math equations that we have never studied. That’s my point of view. I hope you
don’t take offense and lower my grade because that would be unfair. I’m just
being honest. By the way, yes, I like the [reform] work that you give us but I
don’t like this [critical] kind of work. I told you before. (January 21, 2003)
While I am sympathetic to Juan’s perspective – actually tend to mostly agree with him after conducting this study – it also might be that critical activities posed a threat to the students who enjoyed comparatively high status on the mathematical hierarchy within the classroom. Regardless of the motives behind such criticism of CM content, relatively strong opposition of this kind from the academically strongest students made teaching
CM stressful and difficult to for me manage. When students resisted reform activities I could generally rely on my institutional authority to get them to work. Even aspects of reform mathematics instruction such as collaborative groupwork might be considered transgressive when institution insists that students be in their seats facing the teachers who lecture and test. So, if students or parents staged a major protest to my technique of interactive teamwork and insisting on elaborate explanation of principles, the Guevara administration would be unlikely to support any non-traditional form of teaching if push came to shove. Because the Guevara administration only gave wary consent to my critical instruction, I could not rely on institutional backup when students opposed and resisted it.
While students worried about the possible effect of resisting CM on their grades, students
337 such as Sonny and Juan seemed to understand that my actions might be suspect and my position vulnerable. They knew it was transgressive teaching that would be unlikely to be endorsed by school administration or perhaps even the community in the school district.
Third, I realize now that when I ask students from subordinated communities to reflect critically on a sociopolitical theme that the counter-hegemonic view may not be stated or even held by students. Or, at least, students may be afraid to endorse counter- hegemonic views in formal school activities. Teachers cannot rely on students – even those from subordinated communities – to express the counter-hegemonic position without first educating them to this point of view. Critical teachers have to be prepared to make the counter-hegemonic position known one way or another as it does not spontaneously arise in students. At the same time, there is a tension because critical educators and other progressives want students’ ideas and opinions to be a central part of the curriculum. Critical educators have to walk a fine line between valuing student contributions and wanting to alter their outlooks. Taking time to explicitly educate students to the counter-hegemonic position seems to run counter the pedagogical
(Piagetian) constructivist worldview that I held as a reform mathematics teacher. From this perspective, my job was to build on student ideas and facilitate learning rather than tell or be the “sage on the stage.”
In some sense, educating students to the counter-hegemonic perspective runs against the idea that critical education is more culturally relevant than the dominant curriculum in any natural or simplistic sense as I argued in my initial draft of my study proposal. Students who have internalized hegemonic worldviews do not immediately recognize the counter-hegemonic position as being more relevant to their lived
338 experiences than the dominant curriculum to which they have been exposed. They have spent years in schools being indoctrinated into the hegemonic position and this shapes what they see as normal and right. If they are not discussing counter-hegemonic ideas outside of school, in their homes and churches, then they have to be reeducated to see it as superior, more relevant, and more just. The hegemonic Discourse model for social inequality was sufficiently strong among my students that undoing it would have required more instructional attention than I was able to provide as a secondary mathematics teacher and perhaps more expertise than I had at the time.
7.6 Conclusion: Knowledge-Base Necessary for Critical Mathematics Pedagogy
As this chapter indicates, despite my years as a successful secondary mathematics teacher, I was not sufficiently prepared as a critical pedagogue. Teaching critical lessons required a different set understandings and instructional skills than I possessed.
Therefore, in concluding this chapter, I discuss what might comprise the knowledge base necessary to teach CM at the secondary level.
Building primarily on correspondence with one of my dissertation chairs,
Professor Carol D. Lee, I would argue that the knowledge base for teaching CM would minimally include the following: (1) a relatively deep conceptual understanding of secondary mathematics and its relation to post-secondary mathematics; (2) knowledge of how secondary topics are effectively taught and learned; (3) theoretical and practical/experiential knowledge of standards-based reform curriculum and instruction;
(4) an understanding of external and internal factors that shape mathematics instruction and educational opportunity (e.g., large-scale assessments, economic policy); (5) an
339 understanding of the philosophy and history of academic mathematics and its relation to everyday practices and discourses; (6) a reasonable and critical understanding of the social sciences (in particular history, political science, sociology and economics), with an emphasis on what these social sciences reveal about structures of inequality and strategic moves for empowerment; (7) a pedagogical understanding of adolescent development and how adolescents develop conceptual understanding of and dispositions toward knowledge of the political and economic organization of societies; (8) knowledge of the particular demands of living in poverty and being a member of groups who are stigmatized by virtue of race, ethnicity, gender, sexual orientation, class, or handicaps; and (9) experiential knowledge from a long term involvement in activists movements.
Frankly, I believe my understanding of the first four areas of knowledge were relatively strong and that I developed a stronger understanding of the fifth area over the course of this study. Among other things, I have a Masters in Mathematics and
Mathematics Education. I also learned to use a range of teaching methods in my teacher education program and trained to use a reform secondary curriculum. Admittedly, however, I have less expertise in the last four areas. In the interest of space I discuss these and what researchers interested in building on my research might consider.
In terms of the sixth type of knowledge, my background in sociology and critical theory was relatively limited when I taught the night course. Most of what I knew I had learned in a couple of graduate courses, on my own, or through discussions with my mother, whose research deals with social class (E. Brantlinger, 1993, 2004). As a result, my conception of social justice was built up primarily around faith in modern socialism.
Again, this was pointed out to me by Professor Lee. In one note to me, she wrote: “it
340 might be argued that instead of preparing students for a socialist revolution, oppressed people need a very realistic understanding of the nature of capitalism, of the demands of participating in world economies, and the ability to build institutions and infrastructures that can address their needs (rather than simply be aware that there is structural unfairness and hopefully appeal to those institutional sources of oppression to make things better).” I believe she is right and would admit that I often wondered whether effective pedagogy designed around preparing students to overthrow the ruling classes could be if a socialist or Marxist revolution never occurs. Given Professor Lee’s skepticism that modern socialism will be much fairer to the oppressed than capitalism, her conception of social justice was different than the one that informed my curriculum design. That said, I believe Freirean instruction, and my own CM instruction and materials (Appendices A &
B), work towards some of the goals Professor Lee outlined, including fostering a deeper understanding of the nature and effects of capitalism among students.
Outside of a few readings, my knowledge of the seventh point, about how adolescents develop political and economic dispositions and understandings, was limited to intuitive knowledge I developed as a result of being a secondary teacher. That is, I believe I have a reasonable practical understanding of adolescent development as a result of having had informal conversations with my students about politics and the economic constraints and possibilities in their lives. However, my understanding of empirical studies and theories about adolescent development in this area was problematically limited. Were I to teach from a critical perspective again I would delve more into this literature base before teaching. It might be argued that one of the reasons that my students
341 struggled in activities such as Race and Recess activity featured in this chapter was precisely because I lacked knowledge of their developmental stage.
In terms of the eighth type of knowledge, like many mathematics educators who teach in low-income schools, I have observed some of the effects of poverty but have a lot to learn about the structural barriers such students face. At the same time, I am also aware of the fact that I must also take into account the sources of agency, resiliency, identity, and social capital that such communities have as resources (Lee & Slaughter-
Defoe, 1995; Noguera, 1993). While I grew up white, middle-class, my upbringing was atypical in many regards. I grew up in a biracial family with two adopted biracial siblings. I believe I had to struggle (meaningfully) with racialized realities much earlier in my life than many white teachers. I also was a mediocre student who was tracked into lower track English courses with mostly working class peers. (I was not tracked into lower-track secondary mathematics courses.) This, in combination with my mother’s critical educational research on social class (Brantlinger, 1993, 2004), compelled me to confront my own social class privilege at an early age. In addition, I believe my experience in lower track courses as a student helps me relate to non-specialist students in lower-track mathematics courses. I guess that these experiences make me very different from secondary teachers whose families are white, suburban, and who only experienced academic success as students.
While I have an activist side, I do not have the knowledge gained from years of political activism. As Professor Eric Gutstein pointed out to me, my relative lack of experience as an activist also likely affected my CM pedagogy. The period of time when
I grew up was “post-revolutionary” (McLaren, 1998) and people were not in the streets as
342 they had been in the sixties and seventies. That said, I grew up with an activist mother
(Brantlinger, 1993, 2004), and have been involved in protests ranging from nuclear disarmament, to U S military involvement in Central America, to anti-Gulf war protests, to gay and lesbian rights, and immigrant rights. I also believe my choice to teach in low income urban schools was fundamentally activist. Because of my dual Masters in mathematics and mathematics education, I had job offers from numerous suburban school districts where I could have earned considerably more money.
In addition to this knowledge base, the CM teacher would have to have a particular set of dispositions, attitudes, and beliefs. As several scholars have noted
(Haberman, 1991; Ladson-Billings, 1994), successful teachers of the urban poor and students of color are committed to the communities in which they teach. They so not work in urban schools because they cannot find a better job or school. I would argue that this political commitment is just as important, and fairly well correlated with, the types of knowledge discussed above.
343 CHAPTER EIGHT: CONCLUSION
In this final chapter, I first review the central themes and issues addressed in
Chapters 4 through 7. I then discuss the nature and value of my research design. I go on to address the ways my findings connect to larger educational discourses and list related questions that might be considered for future research. While discussing the study’s constraints, I make several recommendations for practitioners who are interested in implementing critical pedagogy and researchers interested in studying critical mathematics education. Finally, I describe my current beliefs and how they emerged as a result of conducting this study.
8.1 Overview of Results
In Chapter 4, a description of how I developed the critical component of my mathematics curriculum is provided as well as the design issues that I encountered in the process. The chapter included a self-analysis of my beliefs as a teacher and my personal transition from being a fairly uncritical advocate of critical mathematics (CM) to a researcher and mathematics educator who had come to understand some of the dilemmas that teaching mathematics from a social justice perspective poses. This chapter also documented how my views on utility of secondary mathematics and student empowerment, central concerns of CM, shifted over the course of my study. I conclude the chapter by linking my own internal conversations about CM issues to larger historical and current discourses in mathematics education and policy.
In Chapter 5, I drew on Dowling’s (1998) sociological analytical framework to compare one of my CM unit texts with a similar unit in a standards-based reform text and
344 chapter in a traditional mathematics text. All three texts focused on area-related mathematical content. An important result was that the traditional text was clearly designed to provide privileged students with full access to (esoteric) academic mathematics while the CM and reform texts were slanted so as to prepare non-privileged students for real world uses of mathematics. This utilitarian focus in the latter two texts actually displaced esoteric mathematics content and provided less direct access to advanced mathematics. Hence, the three texts, which were aimed at distinctive student audiences, bore the imprint of social stratification. Through influencing different audiences distinctively, the texts likely contribute to the reproduction of the unequal social order and would appear to have a negative impact on non-advantaged students’ academic outcomes. By emphasizing more utilitarian goals and quantitative data analysis of the newspaper variety, the CM text went beyond the reform text in limiting adequate preparation for post-secondary mathematics (i.e., the exchange-value of school mathematics). Although I made these serious assertions about CM, I also noted that results were influenced by the fact that I had to create my own CM texts because none were commercially available and only limited ideas for secondary instruction were included in the CM literature.
Taken together, Chapters 4 & 5 suggest that, secondary mathematics teachers are relatively powerless to make the curriculum more politically and culturally empowering
(see Steen, 2004 for a similar argument). Secondary teachers who stray from the prerequisites for calculus put their students at risk of not being prepared for college mathematics. This is contrary to what CM advocates imply – or, at least, they give short shrift to the notion of disciplinary displacement. As Dowling (1998) suggests, in
345 contextualizing mathematics, or conceiving of it as a set of use-values, curriculum designers and instructors fail to recognize the largely context-free and non-utilitarian epistemology of academic mathematics. Academic mathematics simply was not created for utilitarian and political purposes (see D’ Ambrosio, 1985; Ernest, 2000; Steen, 2004).
In Chapter 6, I examined empowerment from a different perspective than did previous chapters; in particular, I investigated classroom discourse and student participation in reform activities (NCTM, 2000). While these activities were not critical, reform mathematics positions students subjectively with respect to mathematics (Stein,
Grover, & Henningsen, 1996). It is conjectured that this subjective positioning is necessary, but not sufficient, for critical mathematics to be effective (Gutierrez, 2002;
Gutstein, 2003, 2005). With this in mind, I attended to the discursive spaces that opened up for students’ mathematical ideas and how students took up, or failed to take up, these student-centered mathematical opportunities. I documented how, at the beginning of the course, my students resisted and challenged reform instruction, often calling for such customary aspects of traditional instruction as teacher lectures, notetaking, individual deskwork, and periodic tests. My analyses pointed to issues of power and equity in reform lessons that appeared to have been carry-overs from traditional mathematics instruction (e.g., achievement-oriented sorting among students, relying on teacher’s expert authority). One of the results of the discourse analysis was the documentation of a positive shift in student stances towards reform practices over the time span of the night course. This evidence indicated that my students generally became more receptive to reform instruction. In particular, they exhibited more discursive engagement in
346 mathematical discussions and were less resistant to open-ended collaborative problem- solving exercises.
In Chapter 7, I presented an analysis of classroom discourse and student participation in CM activities. I outlined how CM activities built on the subjectification of reform mathematics by positioning students to use school mathematics to think about the social, economic, and political forces that shape their lives. Results of my analyses indicated that students participated very differently in critical activities than they did in reform exercises. While some differences were intended and expected, others were unexpected. One rather surprising finding was that some of the more active participants in critical activities were those who tended to resist reform instruction. Indeed, prior to
CM lessons, they showed little interest in mathematics activities. At the same time, these students were generally the ones who were most attuned to recognizing societal discrimination and racism and who were most enthusiastic about discussing social justice ideas in my classroom. A second finding was how many students’ hegemonic acceptance of social inequality emerged quite forcefully during critical activities. For example, in an activity that dealt with economic inequality, many of my students asserted that rich people were wealthier mainly because they simply put forth more effort in scholastic and economic arenas than others. Thirdly, and again similar to my findings about CM and reform mathematics texts, the experientially realistic and political CM activities distracted student attention away from grappling with college-preparatory mathematics.
8.2 The Value of the Research Design
347 My research fits into a number of qualitative design genres. It is fundamentally a case study covering the tasks of designing, implementing, and evaluating the impact of
CM curriculum. Although CM was the central focus, my scholarly gaze ranged beyond a narrow focus on CM and included analyses of reform mathematics and, to a lesser extent and indirectly, traditional mathematics.
8.2.1 Importance of Insider Research
In addition to being a case study, my research can be called teacher (practitioner, insider) research or self study because as a teacher I studied my own instruction. This positionality and perspective allowed me to go beyond simply addressing curriculum and feature the often-ignored voices of urban mathematics teachers and the lower-income students of color who attend many urban schools. As an insider to the teaching profession, I conducted the study as I taught students enrolled in urban mathematics courses. This is the first study I know of in which an experienced urban secondary mathematics teacher explored the implementation of mathematics reform in an urban high school. Although reform mathematics educators speak as if they know what is in students’ best interest, in reality, few have had little prolonged contact with students of color from poor communities (Martin, 2003; Valero, 2004). Furthermore, they often have not have field-tested the pedagogy and curriculum they recommend.
University faculties, who are often outsiders to the urban school system, conduct much of the research on urban schools. Their outsider status usually means that they look in at someone else’s teaching. There are several drawbacks to this approach. First, no matter how carefully they select and train teachers as their key participants, they still may
348 face the hurdles of lack of teacher buy-in to their curricular or pedagogical approach.
While the outsider perspective can provide important results, such researchers cannot get inside the teacher’s head, so may gain little understanding of teachers’ perspectives as they engage in implementing the researchers’ ideas. Understanding teachers’ views of the introduced curriculum and pedagogy is important, especially if it is hoped that a new method will be implemented after a study is completed or if teachers elsewhere will find the report of findings compelling enough to use in their classrooms. Again, as carefully conducted practitioner-research my study responds to these concerns. It should be noted that I also found practitioner-research to be difficult precisely because I had to teach and research my teaching at the same time.
A growing number of educational scholars point to teacher-initiated research as a key to educational reform (Carr & Kemmis, 1986; McKernan, 1991; Stigler & Hiebert,
1999). This is because top-down approaches to educational research design and dissemination have not been entirely successful in translating into practice in schools
(Kline, 1973; Schoenfeld, 2006; Spillane, 2004; Stigler & Hiebert, 1999), particularly for schools that serve the poor (Anyon, 1997, 2005; Berliner, 2006). Scholars who have done site-based research often are disappointed that their ideas are not sustained in schools.
Stigler and Hiebert (1999) believe that reform efforts fail precisely because they do not adequately conceptualize teachers or do not sufficiently involve and inspire them while requesting that they try out a curriculum or pedagogy. In contrast, practitioner research initiates reform from bottom-up. (To be clear, the bottom-up approach does not preclude drawing on government- and university-based research.) Rather than follow a traditional
349 top-down research model, I framed my study from the perspective of an experienced teacher.
In addition to wanting to control the curriculum, there were additional, related reasons for doing a self-study. First, I wanted to experiment with the secondary mathematics curriculum as the study progressed and initial findings came to light.
Second, although I was comfortable with self-criticism, I did not want to be critical of teachers who generously offer their classrooms for research. I was dismayed that some observational studies end up being a forum for teacher bashing. Sleeter (1997), for example, criticizes the mathematics teachers she worked with for rejecting the idea that the mathematics curriculum can be made more culturally or politically relevant. This critique was unfair precisely because Sleeter did not provide concrete examples of multicultural mathematics, yet faulted teachers who did not use such curriculum. There are many studies of teaching in which criticism is aimed at the person of the teacher rather than at the teacher as representative of a wider class of teachers or as someone experimenting with innovative curriculum for the first time (see Cohen, 1990). I chose to conduct a self-study partly because I did not want my research to boil down to a display of another teacher’s inadequacies.
To be clear, unlike the mathematics teachers she cited, I agree with Sleeter that mathematics is not a politically or culturally neutral body of knowledge. Largely as a result of my study and the literature I reviewed to prepare for it, I am increasingly convinced that academic mathematics is a set of institutionalized, largely elitist, and mostly non-utilitarian practices that are linked to the culture of power in ways that lessen the school achievement and life chance outcomes of urban youth. And while educational
350 scholars point to the lack of relevance of mathematics curriculum to students’ lived experiences, cultural backgrounds, and socioeconomic realities, such advocates of culturally relevant and critical approaches rarely are explicit about how experiential and political content might be effectively infused in mathematics instruction. This is particularly true of the secondary mathematics curriculum. While I agree that secondary mathematics currently lacks cultural relevance, scholars offer few concrete examples that practitioners could use to remedy the situation. Moreover, in situating the blame at the school level, critical theorists not only fail to empower teachers, but also may alienate them from school improvement efforts.
8.2.2 The Benefits of Field-Based Research
Another important dimension of my study is that it is field-based, and takes place in the actual urban setting to which results are meant to apply. Educational innovations are frequently tested under ideal or non-relevant conditions (e.g., honors classrooms, low teacher to student ratios) (Cremin, 1964). However, failing to test reforms in situations similar to the sites where they are to be used means that the benefits that occur in inauthentic settings may not transfer to the reality of complex classrooms. Indeed, if not first field-tested at actual school sites, practices based on idealistic reforms might make matters worse for the intended students.
My results chapters reveal that my students often resisted and rejected methods that I had expected them to appreciate. My coverage documents that students are socialized in traditional mathematics methods and hierarchical teacher-student relations and these interfere with their engagement in critical and reform mathematics. Despite
351 good intentions and enthusiastic introductions, students may not readily change their attitudes or happily comply with instructional innovations. Had I not conducted my study in a non-selective urban classroom, important phenomena (e.g., interferences with my idealized curriculum, students response to political themes) may not have occurred.
8.2.3 Grounding the Analysis in Theory
The analyses I conducted in the study are theoretically driven and draw on analytical models explored in previous research. In Chapter 3, I articulate the rationale for analytic choices and clarify how I conducted the curriculum and discourse analyses.
The discourse analysis is based on methods and frameworks created by researchers as
Gee (2005) and Mehan (1979). These widely used approaches allowed me to get beyond surface level appearances of “empowering” instruction and dig deep into complex power relations in both reform and critical activities. Relatedly, Dowling’s (1998) theoretical framework for the analysis of school mathematics texts allowed me to get beyond surface level appearances of student empowerment via political real world contextualization in my CM texts. I note here that, because few American educators are familiar with
Dowling’s work, my report is a credible source of information about his important ideas and provides a sound model for using his analytical schemas.
The data used in my analyses and to support my study conclusions is thick
(Geertz, 1973). It includes hours of videotapes of lessons and student-to-student and student-to-teacher exchanges, displays of curricular materials, transcriptions of interviews with students conducted by myself and others, and realms of field notes as well as a lengthy researcher diary. My data also is accessible and transparent. Direct quotations,
352 summaries of findings, and transcript displays of classroom interaction are included in the text, tables, figures, and appendices. In addition, an extensive amount of data is archived in paper and computer files, so can be provided upon request. In terms of rigor, my analytical approaches to the analysis of the written and enacted curriculum were theoretically grounded. They were intensive in telescoping in on particular occurrences such as student-to-student exchanges, and extensive in covering a wide range of classroom phenomena from my own uneasiness in difficult situations to the self-selected seating arrangements of students. My study provides a comprehensive and rounded look into an urban mathematics classroom.
8.2.4 Acknowledging Teacher Wisdom
As a former urban mathematics teacher, one of my aims is to remedy the power imbalance that exists between scholars and teachers as well as between abstract theory and site-based practice. I dispute the perception that knowledge produced in schools is inferior to that produced at universities. Moreover, it is important for scholars to understand curriculum design from a teacher’s perspective, especially when an emphasis is placed on the teacher’s role as a curriculum designer (Wiggins & McTighe, 1998). I worked hard to link my situated findings back to the research literature and develop educational theory in areas of discourse analysis, curriculum development, curriculum analysis, as well as critical and reforms. As was my aim, my study strikes a valuable balance between considerations of theory and providing a reference point for practice. As self-study, my study moves knowledge production out of the academy and into the classroom. It gives voice to urban secondary mathematics teachers. It addresses questions
353 that apply to teacher practice, including questions concerning the nature of secondary mathematics, the two-tiered mathematics curriculum, and schooling in urban, high- poverty areas. These issues have been under-explored or unexamined in mainstream educational research (Anderson, Herr, & Nihlen, 1994; Lagemann, 2002; McKernan,
1991). In addition to deepening the understanding of relationships between theory and practice in secondary mathematics, my self-study may serve to move the reform movement away from simplistic notions to developing programs grounded on empirical findings and realistic conceptions of what improving urban education and the life chances of low SES students entails (see Anyon, 2005; Berliner, 2006). Although I am uneasy about critiquing CM, my findings open up an opportunity for a much-needed exchange about the benefits and constraints of current mathematics reform agendas.
8.3 Connections to Mathematics Education Discourses
My study deals with many central issues in mathematics education specifically, and in urban education generally. In this section, I want to highlight the ways that my research contributes to our understandings of the teaching of learning of mathematics in urban schools.
8.3.1 Attempts to Improve Mathematics Instruction for Urban Youth
My study is timely given the increasing interest in the education of poor students and students of color among mathematics educators (Boaler & Staples, in press;
Gutierrez, 2002; Gutstein, 2003, 2005; Kitchen et al., 2007; Lubienski, 2000; Martin,
2003; Moses & Cobb, 2001; NCTM, 2000). Although studies show a consistent pattern
354 of social class and race-related disparate outcomes in educational achievement and other school variables, they also flag huge disparities in resource distribution and school quality (Anyon, 1981; Gamoran & Berends, 1987; Kozol, 1991; Oakes & Rogers, 2006;
Orfield, 1992). I have long endorsed an equal distribution of school resources rather than relying on district-based taxes for school funding. Although subordinated students’ achievement improves when they attend high-quality schools, little is known about how low-income students and students of color might perform in neighborhood schools with the same caliber and kind of educational resources. Despite the considerable information about school achievement disparities, we have made little progress how poverty and race affect students’ learning and teachers’ instruction. In particular, we still face many challenges in understanding how educational reform might look from the perspective of students from low-income families or students of color (for exceptions, see Gutierrez,
2002; Gutstein, 2003; Martin, 2003). Research focusing on students’ interests, perceptions, identities, and experiences related to mathematics is needed (Boaler, 1997;
Martin, 2000; Nasir, 2002).
One of my goals in studying the night course at Guevara was to give voice to students who are rarely heard. Results in this area were rich. Clearly I could have devoted my entire study to student voice. (See Appendix D.) Among my more interesting findings is that students who fit a similar demographic profile (urban, low-income, low track, non- white), and who all had failed geometry previously, vary widely in their orientation to school mathematics. Those who plan to go to college after high school see (esoteric) geometry as good preparation for college while those less sure about college appeared not to see the relevance of it to their futures. This resonates with arguments put forth by
355
Labaree (1997) and Collins (1979) that modern schooling is more about credentialing than learning. While this argument makes sense to me, I am not sure if the mathematics educators and policymakers who have little contact with urban students appreciate it.
Policy documents and government reports reveal that educators and policymakers believe secondary mathematics and science have the potential to improve people’s lives as well as enhance the economy. I counter that if they truly wish to improve the situations of the poor, reform-oriented policies must draw on the voices, concerns, and desires of underprivileged students and their parents (Anyon, 1997; Martin, 2003 Noguera, 2003a).
Specifically, regarding school disparities and student achievement, we need to consider what incentive might be offered to students who do not feel the credential benefits of school mathematics apply to them. We also need to consider the possibility of eliminating social sorting in schools. Again, this is a difficult problem given that school mathematics currently functions to produce differences in social capital.
8.3.2 The Need for Mathematics Skills and the Impact of Specializing Instruction
In my study I explored the effectiveness a mathematics curriculum that was meant to be empowering and enriching for urban students. Other mathematics education scholars have made similar attempts (Boaler & Staples, in press; Gutierrez, 2000, 2002;
Gutstein, 2003, 2005; Kitchen, DePree, Celedon-Pattichis, & Brinkerhoff, 2007; Moses
& Cobb, 2001; Stein, Grover, & Henningsen, 1996; Schoenfeld, 2002; Tate, 1995,
1997a). Over the past decades, the mathematics education community has come to share concerns about equity and access for diverse and disadvantaged students.
356 I considered issues concerning what, and how much, school mathematics students in lower track mathematics courses should learn and to what end (Chazan, 1996, 2000;
Romagnano, 1994). I asked if the curriculum for non-advantaged students should be the same or markedly different to that found in settings with privileged students (Gutierrez,
2002; Gutstein, 2003, 2005). I struggled to define the “best” mathematics curriculum for urban low-track students in particular and U. S. students more generally. I asked whether the best mathematics curriculum for any student would have a vocational, critical utilitarian, or esoteric disciplinary orientation.
I considered whether curricular differentiation (e.g., recommending CM only for urban youth) has reproductive consequences similar to those already produced by the highly segregated/tracked school system. I dealt with ethical issues concerning having markedly different educational goals and practices for lower SES students. With this in mind, I wonder why higher SES students are excluded from critical consciousness raising agendas. It may be that higher SES students fail to work toward an equitable society as adults precisely because they internalize a hegemonic worldview that is reinforced, and rarely challenged, in privileged school settings.
While beyond the scope of my study, certain questions need systematic investigation and so could launch valuable research efforts: Besides producing uneven levels of social capital in various students, how is (esoteric) secondary mathematics useful? What is behind current rhetoric about “mathematics for all,” the mathematical utility, and international competitiveness in the education literature and policy documents? What mathematics skills and understandings do university faculties expect students to be proficient with and why?
357
8.3.3 The Efficacy of Traditional, Reform, and Critical Mathematics Texts
In gauging the effectiveness of mathematics curriculum and instruction, my findings link to efficacy studies. Mathematics educators (Chazan, 1996; Lesh & Doerr,
2003, NCTM, 2000) and curriculum developers (e.g., Fendel, Resek, Alper, & Fraser,
2000) imply, or argue directly, that an increased dose of real world relevance makes mathematics more meaningful and accessible to students, particularly those in low-track courses. Others contend that emphasizing everyday uses in secondary school mathematics may impede rather than enhance mathematics learning (Boaler, 1993;
Dowling, 1998; Ernest, 2000; Pimm, 1995). Lerman and Zevenbergen (2004) claim that the use of realistic seeming problem contexts exacerbates the already inequitable social class outcomes in learning school mathematics. They found that when posed with realistic mathematics problems, working class students have more trouble recognizing and solving underlying mathematical tasks than do their middle class counterparts. They claim that working class students take the real world contexts at face value and fail to recognize the mathematics in contextualized problems.
While these class-based distinctions might suggest that working class students need more practice with contextualized problems, I concur with Lerman and
Zevenbergen that this interpretation is mistaken. I worry that teachers who instruct non- privileged students might infuse the secondary curriculum with projects and tasks that have far more to do with everyday quantitative practices (e.g., shopping, investing in the stock market, sports, carpentry) than university-required mathematics. Dowling (1998) reports that the watering down of secondary mathematics curriculum for working class
358 students is standard practice in Britain. The analyses I cover in Chapter 5 indicate that a similar phenomenon exists in the United States. As noted earlier, Dowling found that mathematics texts designed for British lower-track students are so highly contextualized that they are not really structured by the discipline of mathematics. He claims such texts refer to a mythological world where mathematics helps people sort through supposedly real world problems.
Teaching working class students a highly contextualized version of secondary mathematics frequently is justified on the grounds that students in low-track courses lack motivation (Chazan, 1996, 2000) or have impoverished ways of reasoning about everyday phenomena (see Dowling, 1998). While it is true that many students criticize the secondary mathematics curriculum as uninspiring and largely irrelevant to their lives
(Boaler, 1997), an overemphasis on contextualization limits the textual access that working class youth have to principled secondary mathematics, hence contextualization comes at a disciplinary cost (Dowling, 1998). Related to the judgment of how much displacement of academic mathematics is acceptable is the issue of whether culturally and politically relevant themes are additive in furthering disciplinary connection making and enhancing student engagement. Regarding these arguments, teachers and future researchers might consider these questions: Does utilitarian, critical, and culturally relevant content enhance motivation and facilitate engagement in school mathematics?
Might the positive effect (motivation, engagement) of real world contextualization outweigh the negative effect of content displacement?
8.3.4 Issues of Empowerment, Status, and Agency
359 My instruction in the study is aptly called “transgressive teaching” because it confronts mainstream thinking about schooling and attempts to counter the impact of traditional pedagogy and mainstream ways of relating to students (hooks, 1994). My research adds to mathematics education discourses by examining how students respond to political issues integrated into a mathematics course (Gutstein, 2003, 2005; see also
Valero & Zevenbergen, 2004). An indication of CM’s potential to empower students is that, while involved in CM lessons, students who rarely had entered mathematics discussions were suddenly enthusiastic participants in conversations about racism and discrimination. Videotape transcripts reveal that students were animated and often used vernacular (street) language in political discussions, behaviors not exhibited during reform instruction. I interpret such actions to mean that students felt ownership over certain CM activities.
Several questions occurred to me before, during, and after my CM teaching, which identify issues and suggest research that other researchers might consider, namely:
What emotional and social tensions surround the power differential of a (white) person in authority asking students of color from low SES communities to consider race, racism, social class, and achievement disparities as an official part of the mathematics curriculum? How is staging formal discussions of politics different than having informal conversations with students before or after class? Should CM educators use their institutional authority to enforce student participation as they would if students refused to do the required mathematics?
8.4 Recommendations for Teaching and Future Research
360 In the previous section, I generated a number of questions worthy of consideration. In this section I discuss constraints I faced in doing the study as well as ideas about how research might done to extend knowledge about CM. Due to public school schedules and graduate school pressures, I was unable to study CM mathematics instruction for an entire school year. Although several beneficial changes took place in my study, given the very unique instructional approach I took, nine weeks (a condensed semester) was not enough time to adequately test the effectiveness of CM at the secondary level. Because relationship building in low track courses takes time and is not guaranteed, future research on CM programs for lower-track students should last for a year or more.
I recommend that my I&A unit be redesigned for further use by teachers and researchers interested in critical mathematics at the secondary level. However, because
I&A would not suffice for a year’s worth of instruction, the teacher would need to add more critical material pertinent to secondary mathematics. If the current lack of CM curricular resources still holds, teachers may have to develop much of it on their own, although I suggest caution in this pursuit because CM curriculum design is difficult and time-consuming. I do not think full-time teachers have the time to engage in CM curricular design. It might be possible to collaborate with others to share the burden of curriculum development as well as to support each other in grappling with the instructional dilemmas that arise.
With the exception of too little time and lack of peer collaboration, I believe my research methods were well founded and rigorous. I was able to have others to videotape my classroom instruction and student activity. As stated earlier, my methods of data
361 collection and analysis were intensive, extensive, and theoretically driven. I would not hesitate to recommend that other researchers use the same methods to explore innovative teaching or evaluate instructional programs. I would also recommend that others conduct of student interviews as I did. Perhaps because of my teaching style, however, students did not appear to be intimidated to honestly critique my instruction in interviews I conducted. (I conducted some interviews and had colleagues conduct others.) I developed additional questionnaires and interview protocols for the summer course at Park Vista.
These findings will be presented as part of continued research.
When I designed my study, I considered using a reform control class. Researchers who wish to test the effectiveness of CM might consider collecting pre- and post-study achievement, student self-efficacy, and course satisfaction data to match with a comparison class. I did not have the time or finances to include a control classroom.
Furthermore, my CM curriculum was not complete enough and my pedagogy not polished enough to warrant the use of a control class. Were I to have documented considerable success with my CM instruction in this study, the use of a control class would have been a reasonable next step. Instead I think it is important to further develop and refine CM materials. Pre- and post- scores are included in Appendix F. The class’s academic gains were substantial. Nevertheless, instead of focusing solely on academic outcomes, I spent my research time capturing and analyzing classroom phenomena related to secondary CM instruction. This approach is legitimate given that CM curriculum (my own and others’) developed for high school is at a rudimentary stage.
Given the time and logistic constraints for typical teachers, if CM is to have wider application, effective CM curriculum must be developed, field-tested, and disseminated.
362 With their conservative political leanings (Metcalf, 2002), it seems unlikely that major textbook companies would take on this marketing task. Hence, such products will have to be developed and distributed outside the corporate world as was done in the volume edited by Gutstein and Peterson (2005). At the same time, given recommendations to use local, organic (bottom-up) content in instruction, CM materials produced off-site might not be responsive to the particular lived experiences and interests of local students.
As noted earlier, currently distinctive textbooks, instructional materials, and teaching practices are designed for specific subsets of students according to their ability group, track, and geographically separated schooling arrangements (Anyon, 1981;
Dowling, 1998). It must be acknowledged that distinctive designs may be based more on stereotypes and faulty assumptions than real variations among the needs and learning styles of students (Oakes, 1985; Rist, 1970). Social and physical distance between decision-makers and poor students is a problem (Noguera, 2003a). The textual analysis I present in Chapter 5 reveals the wide and important distinctions in mathematics artifacts designed for different groups of students. Although I see benefits to CM, it is essential to avoid it becoming a dead end curriculum that provides students from subordinated communities with little of value in educational marketplaces. While curriculum differentiation is widespread, research has not demonstrated that it is valid. Because of the profound impact of quality distinctions in schooling (Anyon, 1981), certainly the benefits and fairness of “different things for different folks” must be validated through rigorous research.
I worry that problems are likely to result from using specialized approaches only for subordinated students rather than the full range of students. Hence I recommend that
363 studies of critical pedagogy be conducted in elite contexts with privileged students in addition to schools where students and their families have less power. It would be interesting to see, for example, how student (or parent) resistance to CM might be similar or dissimilar in suburban locales. While genuinely integrated schools are hard to find, it would be important to examine how social interactions occur with CM teaching in such an ideal setting. It is known that when students are taught together and have access to a quality education and the culture of power, achievement gaps narrow (Orfield, Eaton, &
Harvard Project, 1996).
As far as critical pedagogy is concerned, among the most exciting findings of my study is how the critical sociopolitical discussions altered the power structure and participation dynamics in my classroom. While many students were more engaged with reform than CM activities, the opposite was true of others. Improvements in agency and empowerment demonstrated by the level and quality of student engagement was a sound benefit. This finding does point to the fact that all students are required to take secondary mathematics (for their own supposed benefit), yet many would rather be studying something else. This phenomenon needs further study
In addition to considering the effects of CM, my study raises the question about whether to introduce critical themes and activities into the general school curriculum and, if the answer is affirmative, then where to include it must be determined. These are value- laden questions that cannot be answered by empirical studies alone. If, for example, we value social democratic goals, then CM pedagogy might be far more useful than a secondary course constructed around esoteric mathematics. If Americans continue to believe that economic competitiveness depends on high achievement, and if economic
364 and educational competitiveness is valued above all else, then there may be little space left for critical pedagogy in mathematics education or in educational subjects. Personally,
I would like to see a shift towards national and global democratic goals and away from our current preoccupation with individual student and nationalistic economic competitiveness. Although it has is opponents (Berliner & Biddle, 1996; NCTM, 2000), the rhetoric that links school mathematics to economic competitiveness is supported
(Ravitch, 2006), and most Americans tacitly support it. This mindset distracts attention away from pursuing more social democratic ends in education (Collins, 1979; Labaree,
1997). While I question whether secondary mathematics should be reconceived as a critical literacy, I am convinced that critical instruction should be part of schooling. If it could be done without risking one group’s social mobility, it might make sense to give secondary students some choice between taking esoteric higher-level mathematics courses and taking courses in critical sociology or history.
Relatedly, mathematics educators are right to assert that secondary mathematics education can lack meaning, in particular for students from subordinated communities
(Boaler, 1997; Gutierrez, 2002; Gutstein, 2003). Many of the Guevara students were quick to point out how irrelevant secondary mathematics was to their everyday lives in their responses to journal entries. However, the literature on multicultural mathematics implies that, in contrast to their less advantaged counterparts, middle class white students find school mathematics relevant to their lives because it matches white middle class culture. While middle class students’ perceptions of school mathematics were not the focus of my study, I am skeptical of such a claim. I doubt that middle class students find secondary and college mathematics particularly relevant to their everyday lives. Middle
365 class students take rigorous mathematics courses primarily because they expect them to pay off in their adult lives as exchange-value (Labaree, 1997). That is, rigorous secondary mathematics allows some students – largely middle class students – to distinguish themselves from the pack. If it is true that lower SES students are less interested in school mathematics, it is likely to be because they are less certain about attending college. They also have heard the ubiquitous message that they are not cut out for college via tracking, school segregation, lack of availability of college preparatory curricula, and test scores.
There is considerable potential for future research that investigates issues related to student orientations to school mathematics.
8.5 Rethinking Critical Mathematics: Where I Stand Now
In the introduction chapter, I pointed out that research in CM had yet to show that it is possible to design critical secondary mathematics activities that students find believable, engaging, and relevant to their lives yet also are mathematically sound enough to provide preparation for college mathematics. I argued that CM could only be equitable and empowering for students from subordinated communities if both criteria were satisfied. As a mathematics teacher, I felt it would not be enough for students to develop a critical consciousness and a sense of political agency if they missed out on learning the same advanced mathematics as their counterparts at advantaged schools. I began the study believing that advanced mathematics instruction could be modified to raise students’ consciousness about the school and societal conditions that had historically been limiting to them. I earnestly thought the two goals could be met, and intended to show this through studying my own teaching.
366 In designing my study, I attempted to reconceive secondary mathematics as a critical literacy or social justice tool to encourage the mathematics and personal empowerment of non-advantaged students. Following other CM educators, I defined empowerment primarily as social and political enlightenment. I downplayed the role that secondary mathematics plays as a gatekeeper course. Although the critical component of my curriculum was compelling in terms of student subjectification and engagement, the
“realness” of CM came at a disciplinary cost. If the analyses of my CM curriculum and instruction generalize to CM more broadly, then, presents a trade-off of mathematics empowerment (exchange value capital) for political empowerment. Testing educational theory as a practitioner-researcher in a realistic educational context enabled me to deepen my own understandings of CM mathematics and secondary education generally. It is not surprising that my beliefs shifted as I tested educational theory in the complex setting of an actual classroom. Other teacher-researchers who investigated reform ideas in situ went through similar personal and professional transformations (Chazan, 2000; Romagnano,
1994). Because theorized reforms are difficult to manage and unwieldy in less than ideal instructional conditions, unintended results may be the norm. Although my study reveals constructive outcomes of CM instruction, I no longer see it as a panacea for the mathematics education of inner-city youth. My appraisal of CM might be condemned as simplistic and cynical, yet, I believe I have a clearer conception of what equity in mathematics education might entail. I realize that my increasingly uneasiness about diverging from the typical reform mathematics curriculum was because this approach was recommended only for low-income youth of color – “other people’s children.”
Leonardo (2004) argues that suburban white students are most in need of multicultural
367 (critical) education because they are naive about the reasons for unequal power relations in society and misguided in their negative perceptions of poor people and people of color.
Indeed, I would not be opposed to CM for urban youth if it were also used for suburban adolescents.
Despite my current reservations about recommending CM for urban high school students, I argue that I my study should not be viewed as practitioner-research gone wrong. Critical education can be very right. I retain my critical perspective in insisting that schools must change to achieve the equitable society that America promises.
Although we differ in our assessment of the power of CM, I agree with Gutstein (2003,
2005) that people should move beyond thinking about equity within mathematics classrooms to thinking about societal equity. When discussing school reform, like Anyon
(2005) and Berliner (2006), I always add the caveat that it should be one facet of a broad societal movement for social justice. It also might be argued that critical goals are more worthwhile than mathematical goals. Gutstein’s (2003, 2005) critical projects seem considerably more interesting, thought provoking, and important than those featured in dominant mathematics texts. Not only is critical pedagogy needed to counter the hegemonic thinking of subordinated students, it is necessary for all Americans because hegemony does cultural and psychological damage to everyone as it undergirds inequality in capitalist society (Fanon, 1967; Freire, 1971; Gramsci, 1971).
Partly because of my observation of students’ interest in the critical content of
CM, I recommend that it be an integral part of K-12 schooling for all students, not just urban youth. After all, privileged students arguably are most in need of enlightenment about discrimination and injustice in American schools and society because they are
368 better positioned than their subordinated counterparts to change society. While I believe critical perspectives should be integrated into many courses, I have come to see secondary mathematics as one exception. As a remnant of the elitist, bookish, and anti- utilitarian education of the nineteenth century, school mathematics eluded modern school reform (Steen, 2004). According to Steen, it is the only classical subject taught today in much the same form as it was taught a century ago. That school mathematics has changed so little may be due to the rigidity and verticality of disciplinary mathematics as much as it is to the inflexibility of schools and teachers. Furthermore, secondary mathematics roughly conforms to the practices of academic mathematicians whose day-to-day work has little explicit concern with worldly matters and political affairs. Hence secondary mathematics cannot undergo radical transformation without a modification of academic mathematicians’ practices or a shift in the value that society places on esoteric mathematics. Relatedly, if university mathematics departments and testing companies loosen their grip on secondary curriculum, school personnel might teach in a way that is more meaningful and enlightening to students.
It seems that some secondary school subjects have more modern roots in terms of being more flexible and less vertical. Language education courses can include literature that represents all cultures and a variety of political perspectives. Students can write papers that focus on critical themes. Social studies programs would benefit from reflecting a social justice perspective. Zinn’s (1995) history of the U.S. is a model text for this purpose. Including critical themes in history and the social sciences would not diminish their exchange-value because critical perspectives are part of professional
369 discourses in these fields. Depending on the discipline, students should be challenged to think critically when it does not come at a disciplinary cost as it does in mathematics.
Of course, the question of which type of mathematics course – critical or esoteric
– is value-laden. Answering it depends on what society, by local communities, and individual students value. If social democratic ends were truly valued in U. S. society, then trading mathematics goals for critical ones could prove worthwhile. Given the important role that mathematics currently plays as a gatekeeper course, this choice cannot be left up to individual mathematics teachers. Educators and community members need to value critical content and also decide the appropriate disciplinary placement for it.
While the occasional inclusion of CM activities in mathematics courses seems reasonable
(Brantlinger, 2005), I do not support rethinking school mathematics as a critical literacy
(Gutstein, 2005).
Regarding educators’ proclamations about progressive curriculum, the historian
Lawrence Cremin (1964) asserts that a “protest is not a program” (p. 348), challenging progressive educators’ denunciation of mainstream education and their unquestioning advocacy for progressive educational reform that had not been proven to be effective.
Cremin claims that recommendations based in critique do not readily translate into methods that can be trusted to be effective in schools. In a similar vein, current critical reformers present powerful and valid critiques of U. S. schooling, however, these do not necessarily or readily translate into educational solutions as is sometimes implied or asserted. My study makes this clear. I responded to Cremin’s critique by studying my critical curricular dreams in a real educational setting. I carefully considered the theory- to-practice dilemma by capturing the complexities and contradictions associated with the
370 actual site-based implementation of the recommended secondary CM reform. I expect that the issues in my study and discuss to be of central concern to those who teach mathematics education in higher education and are similarly pertinent to practicing teachers whether or not they teach in urban public schools.
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390 Spillane, J. P. (2004). Standards deviation: How schools misunderstand educational policy. Cambridge: Harvard University Press.
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APPENDIX A: INEQUALITIES & AREA UNIT
Inequalities & Area
Student Guide
A Critical Mathematics Unit
by
Andrew Brantlinger
394
Note to Students
Many of you have probably been asking your mathematics teachers ‘when are we ever going to use this stuff?’ for years now. The truth is that many of us math teachers ask ourselves the same question. Unfortunately – or fortunately depending on your perspective – very few people will use high school mathematics on the job or in other real life situations outside of school. And, if they do use mathematics on the job they can almost always learn it there. This unit is an attempt to rethink what mathematics we teach and how we teach it. Rather than asking you to solve ‘bogus’ word problems, you will be asked to use mathematics as a tool to investigate our economic system. Ultimately, we want to think about ways we can improve it to create a more just world. As such, you will be expected to do things that go beyond what you typically think of as mathematics. Namely you will be asked to write, talk, argue, and, most importantly, think critically. And, again, you will learn mathematics along the way. Ideally this unit will make mathematics more meaningful and relevant. But, it’s not perfect. Your opinion matters as a math student matters and you will be asked to think about ways it can be improved along the way.
395
SEQUENCE OF ACTIVITIES
Your History with Area Quintiles & Income Distribution Representing Data in Tables Representing Data in Graphs Letting the Gini Out Democracy and Economic Inequality Nailing Down Area (IMP) Shape Shifting Ten Chairs of Inequality (Rethinking Schools) Areas of Triangular Regions Patterns, Tables, Graphs & Equations #1 A Measure of Inequality Federal Taxation: Regressive or Progressive? Trapezoidal Areas Math, Equity, and Economics (Rethinking Schools) Quadrilaterals, Parallel Sides, and Area Formulas In2 versus cm2 Approximating Area (IMP) Organic Goodies (Rethinking Schools) Patterns, Tables, Graphs & Equations #2 The Tiling the Room Project The Job Gap Tiles within Tiles Civil Rights and Economic Equality Learning Excel Regularity in Irregularity Putting the Gini in Excel Formatting Society Gini, Gini in Excel, Who’s the Fairest of them All? End of Unit…
396
Your History with Area
The primary mathematical goal of this unit is for you to learn more about the concept of area and how it can be applied. Of course, it helps your mathematics teachers teach better if they understand a bit about what you know about area already and what kinds of experiences you’ve had learning about area before. You should address three points in this “personal history”: • Write about your past experiences learning about area. (Try to be specific if possible. How did you learn about area? How did your teacher teach you?) • Write down everything you know about area – feel free to use drawings. • And what confuses you about the topic of area?
397
Quintiles & Income Distribution
For the rest of the unit we will be investigating the distribution of wealth and income in the U. S. This table is a representation of the distribution of income in the U. S. in 2001.
U S Income Distribution
Poorest 20% 3.7
2nd Quintile 9.1
QUINTILES. 3rd Quintile 15.2 A “quintile” is a fifth of a given population or a segment of society (wage earners). If 4th Quintile 23.4 there are 150 million wage earners then each quintile would have 30 million wage Richest 20% 48.6 earners.
What does this table mean?
What, if anything, surprised you about the “candy distribution” activity that your class took part in?
398
Representing Data in Tables We know from the candy distribution activity that the following is the distribution of income data for the United States in 2001. (Note: We are looking at income and not wealth here.)
Quintile Poorest 20% 2nd Quintile 3rd Quintile 4th Quintile Richest 20%
Income 3.7 9.1 15.2 23.4 48.6
(1) (a) Enter in a fairer distribution.
Quintile Poorest 20% 2nd Quintile 3rd Quintile 4th Quintile Richest 20%
Income ______
(b) Do you think it would be possible for the U. S. to have this distribution?
(2) (a) The data in the first table above can also be represented in the following “cumulative” way. (You will have to fill in the missing values.) What is the difference between this table and the above one? [Hint: 3.7 + 9.1 = 12.4.]
Poorest X% Poorest 20% Poorest 40% Poorest 60% Poorest 80% Everyone Income 3.7 12.4 28 ______Share
(b) What does this table mean? (For example, what does the 28 represent?)
(3) Reformat your invented (fairer) table from question #1b so it is “summative” like table 2(a)
Poorest X% Poorest 20% Poorest 40% Poorest 60% Poorest 80% Everyone Income ______Share
399 Representing Data in Graphs
100 In the previous activity – Representing Data in Tables – you 80 essentially reformatted the data so it could produce a graph like the one 60 to the right. Graph your “fairer” 40 distribution from table #3 in the Y% of Total Income previous activity in the blank graph 20 below. 0 0 20 40 60 80 100 Poorest X% of Society
100 100
80 80
60 60
40 40 Y% of Total Income Y% of Total Income 20 20
0 0 0 20 40 60 80 100 0 20 40 60 80 100 Poorest X% of Society Poorest X% of Society
QUESTIONS: (1) The third graph (the dotted line) is called the ideal line. Why do you think it is called the ideal line? What would the quintile table look like?
(2) The graph of your data (above) should be “closer” (visually) to the ideal line than the graph of the real U. S. income data from 2001. Why is this the case?
400
Letting the Gini Out
In the previous activity we saw that the closer a distribution curve is to the ideal line the fairer the distribution of income is. Another way of looking at this is to measure at the amount of space or area between the two curves.
100 The Gini coefficient is a measure that tells does just that; that is, it 80 tells us how far apart a distribution curve is from the ideal line by measuring (or calculating) the area 60 in the region created in between Ideal Line the ideal line and the distribution 40
Sums of Income curve. Curve When we look at these two 20 graphs we see that the “gap” is bigger in the top graph than the 0 bottom graph. The Gini coefficient 0 20 40 60 80 100 Quintiles of for the top distribution is correspondingly bigger than the 100 one on the bottom. (The Gini for the first country is about 0.42 and the Gini for the second country is Curve 80 about 0.26.)
The Gini coefficient is nice Ideal line 60 because it allows us to quickly look at a country’s distribution of 40 income at a given point in time. Sums of Income
20 The goal of this unit is to use area to develop a formula for the Gini 0 coefficient. Do you have any idea 0 20 40 60 80 100 Quintiles how to do that at this point in
time?
401
Democracy and Economic Inequality We’ve been looking at the distribution of income in the United States. This assignment asks you to write your opinions about this distribution or, more generally, our economic system.
When President George W. Bush claims that he’s spreading democracy and freedom throughout the world he seems to be talking about spreading political democracy - 1 person, 1 vote. He does not, however, discuss economic justice – where the citizens of a country (or the world) would have a fair(er) share in the economy.
(1) What is (or should be) the relationship between political democracy and economic justice?
(2) What does this cartoon mean to you? [Does it accurately represent the reality you know?]
402
Nailing Down Area (From IMP Curriculum)
Again, computing the Gini coefficient involves calculating area. Thus, we need to learn more about area.
4 In this activity, the unit of area will be the smallest square on the 3 geoboard, such as this one.
2 1. Construct each of the figures A
1 through M on your geoboard and find their areas. Record your
0 results.
0 1 2 3 4
4 4 4 A C E
3 3 3
2 2 2
F
1 1 1 B D
0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
4 4 4
4 4 4
3 3 3
3 3 3
2 2 2
2 2 2
1 G 1 1 I
1 1 H 1
0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
403
4 4 4
3 3 3
2 J 2 2 K L
1 1 1
0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 4 4 4 4 4 N 4 P 3 3 3
3 3 M 3
2 2 2
2 2 O 2
1 1 1
1 1 1
0 0 0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 0
0 1 2 3 4 0 1 2 3 4 0 0 1 2 3 4
2. Create triangles on the geoboard whose area is two units. Find as many different shaped triangles with this area as you can and record your results on the geoboard paper. (You might look back at what you just completed for examples.)
4 4 4
3 3 3
2 2 2
1 1 1
0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
4 4 4
3 3 3
2 2 2
1 1 1
0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
404 Shape Shifting
(1a) Draw as many rectangles of area 6 as you can in the area below. Remember the vertices of your figures need to lie on a peg.
......
......
......
......
(1b) Can you draw a square of area 6? Why or why not?
(2) Draw as many triangles of area 3 as you can in this region.
......
......
......
......
405
(3) Draw as many (non-rectangular) parallelograms of area 6 below.
......
......
......
......
(4) Now draw as many (non-parallelogram) trapezoids of area 8 below.
......
......
......
406
Areas of Triangular Regions
You have already looked at the area of triangular regions in two activities. In this activity we want to find, explain, and apply a formula that will give us the area of a triangle with a given base and height.
Using a geoboard or geoboard paper create triangles with horizontal bases and vertical heights. For each triangle you make record the length of the base and height and find the area. After you’ve created and “measured” several such triangles, investigate the relationship between the base, height, and area. Attempt to find a rule that describes this relationship.
407
Ten Chairs of Inequality (From Rethinking Schools)
PART ONE: Your class recently had a discussion on the distribution of wealth in this country. What, if anything, did you find surprising about that discussion?
PART TWO:
(1) Imagine that you are part of the super rich, the richest 1% of our country. Who are you and what is your life like? [Remember, the richest 1% of Americans owns 40% of the nation’s wealth. Do you actually know any super rich people?] (2) Imagine that you are a member of the super poor, the poorest 1% of our population. Who are you and what is your life like? Do you know anyone who is super poor? (3) Will these people ever meet? If so, when, where, why and what happens? And if not, why not?
408
Patterns, Tables, Graphs, & Equations #1.
1. Consider the pattern below. Use it to fill out the table where the x-column is the number of the element and the y-number is the area of the figure. x y
1
2
3 1 2 3 4 4
5
…
10 2. What is the equation that describes the relationship between the x- and the y-column? … 100
3. Graph the relationship (in #1 & 2) below.
13 4. What happens to the equation
12 and its graph if we add two
11 squares to the each element of the pattern? Try to write the 10 new equation and graph it to the 9 left (same chart). 8
7
6
5
4
3 2 3
2 1 1
0 -1 0 1 2 3 4 5 6 7 -1
5. Would it be possible to take away two squares instead of adding them to each element? If so, how would that affect the equation and graph? Would it be possible to take 3 or 4 squares off instead?
409
7. a) Consider the pattern below and use it to complete the table x y
1
2
3
4
5
…
10
… b) What is the equation that describes the relationship between the 100 x (the number of the element) and y (the area) variables? c) Try to draw the sixth element of this sequence (on a separate sheet of paper).
8. a) Graph y = 3x – 1 to the right.
9
8
7
6 b. How does the equation and graph for #8a relate to 5 the equation you found and graphed in #2? 4
3
2
1 0 -4 -3 -2 -1 0 1 2 3 4 c. Draw a sequence (patterns) that would fit the -1 equation y = 3x – 1=. -2 -3
-4
-5
410
A Measure of Inequality
Can we take what we know about area so far to figure out how to compute the Gini coefficient? That is, can we think of a way to compute the area between the ideal line and the graph of a country’s distribution?
100
80
60
40
Y% of Total Income 20
0
0 20 40 60 80 100 Poorest X% of Society
411
Federal Taxation: Regressive or Progressive? A progressive tax is one that takes a larger percentage from the income of high-income people than the income of low-income people. These are the current rates for 2003. A regressive tax (system) is on that does the opposite. (1) Take a look at the first chart. What do you think, can President Bush claim we have a fair (progressive) tax system?
PART 2: The second chart (below) shows the rates at which the following income levels were (or would have been) taxed in 1963, 1973, 1983, and 1993. (2) What has happened over time?
US Federal Marginal Income Tax Rates 1960 - 1997
Tax Rate on Interest Income by 1992 real income
Year GNPD 5000 10000 20000 40000 100000 200000 400000 1000000
1963 9.89 0 0 0 20 19 34 53 71
1973 22.14 0 0 0 16 28 44 58 69
1983 56.28 0 0 12 18 40 48 50 50
1993 105.02 0 0 14 14 31 44 39 39 (3) Is there a problem? And, if so, what can we do about it? (4) If you look at most of the income levels taxation has shrunk. Doesn’t everyone benefit here? (5) What does “by 1992 real income” mean?
412 Trapezoidal Areas b 2 Trapezoids are four sided figures. They look like the figure on the right. They have height h, a top and h bottom base, b1 and b2. Every combination of h, b1, b2 produces a new area, A. b1
1. Find b1, b2, h, and A of the following four trapezoids
b1 = b2 = h = b1 = b2 = h = b1 = b2 = h =
h = A = A = A =
b1 = b2 = h = b1 = b2 = h = b1 = b2 = h =
A = A = A =
413
b1 = 6 b2 = 5 b1 = 5 b2 = 3
h = 4 A = h = 3 A =
3. Organize your data into a table and see if you can find a relationship (or a formula) between b1, b2, h, and the area, A. If you want to you can make more trapezoids using geoboard paper.
b1 b2 h A 5 units 3 units 4 units 16 units2
414
Math, Equity and Economics (From Rethinking Schools)
Median Weekly Earnings of Full-Time Workers 1993
600
500
400
300
200
100
0 1 2 3 4 5 6 7 8 All men All women Black men White men Black women Hispanic men White women Hispanic women
Source: U. S. Department of Labor.
1. How do these statistics make you feel? 2. List hypotheses (possible explanations) for the dramatic inequality in earnings. Which of these seem most reasonable? How could we go about testing these hypotheses? What historical information would we want to know? 3. How might these economic realities contribute to hostility between different racial groups? Or women and men? 4. What more would you want to know?
415
In2 versus cm2 – Units for Area
On this side you’ll find the area of the triangle and trapezoid in square inches. (Draw square inch grids to justify (and check) your answers.)
4in
3in 3.4in 4in
This is a square inch or an in2.
416
Now use square centimeters to find the area of the (approximately) same (congruent) triangle and trapezoid from the previous page.
cm2
5cm
5√2 or about 7.1cm 5cm
10 cm
417
Quadrilaterals, Parallel Sides, & Areas
A quadrilateral (four sided polygon) can have two pairs of parallel sides, one pair of parallel sides, or no parallel sides at all as shown below
We can find the area of these figures by measuring (counting square units in) the interior space. In the case of the trapezoid we also have a formula that we can use if we know the measurements of the bases and the height.
The first question you want to investigate here is whether or not the trapezoid formula works to find the area of the parallelograms below.
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
0 0 0 1 2 3 4 5 6 7 8 9 1 0 0 1 2 3 4 5 6 7 8 9 1 0
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
0 0 0 1 2 3 4 5 6 7 8 9 1 0 0 1 2 3 4 5 6 7 8 9 1 0 There is a simpler formula that also works or parallelograms – do you know what it is? If not, can you figure it out?
418
Approximating Area (From IMP)
We typically measure area using square units. However, we could use other figures or units as well. In this activity you will use a parallelogram to measure the amount of 2-dimensional space in two “bigger” regions.
Find the area of the figures using the parallelogram
The following square has equal area to the parallelogram above. If we use the square to measure the area of the figure what would that area be?
419
2. Use the same parallelogram to find – or approximate – the area of this second irregular figure.
BONUS: How many parallelograms of cloth would it take to make up your shirt?
420
Patterns, Tables, Graphs, & Equations #2
x y 1. Consider the pattern. 11 1 10
2 9 The first “element” of the “sequence” is 3 8 a triangle. The second element is three 7 triangles. (The sequence is infinite – it 4 continues on and on without end.) 6 5 5
A. Use it to fill out the table to the 6 4 right. … 3
2 B. Find the equation that relates the x- 10 1 input & the y-output. … 0 99 -1 0 1 2 3 4 5 6 C. Graph this relationship (in the table) -1 to the right. (The ones that fit.)
2. The table to the right could also be x y 16
15 constructed given the same sequence. 1 3 14
2 7 13
12
3 11 11 A. How was this table constructed? 10 4 __ (Once you know you should fill in the 9 blanks.) 5 __ 8 7 6 __ 6 5
… … 4
3 10 __ 2 B. How does this change the related … … 1 0 graph and equation that describe this -1 0 1 2 3 4 5 99 __ -1 relationship? (Find them. .) -2
421
3. Complete the table, find the equation and graph the solutions that fit.
15 In the space provided, make up a x y 14 sequence (pattern) that fits this 13 1 relationship. 12 2 11
10 3
9 4 8
7 5
6 6 5
4 …
3 10 2
1 …
0 0 1 2 3 4 5 99
4. Use the sequence below to and use it to complete the table.
1 2 3 4
x 1 2 3 4 5 6 … 10 … 20
y 2 5 … …
5. Which of equation(s) below works for the data in the table above?
i. y = 3x - 1 ii. y = (x2+3x)/2
iii. y = x3 – 1 iv. y = 3x2 - 1
422
Organic Goodies [Write Up] (From Rethinking Schools)
• What did you personally do to try to stop your teacher’s efforts to divide people?
• How effective were you?
• Were there actions you considered, but didn’t take? Why not?
• If we were to do this simulation again, what different actions would you take?
423
Tiling the Room Project
PART 1: Your teachers – Mr. B, Mr. A, & Mrs. N – are going to use all the money they are earning this summer to put some new square tiles on the floor. Your goal is to help them find the best deal. There are three sizes of tiles that they can choose from at the hardware store: 1ft by 1ft, 6” x 6” and 4” x 4”. The prices per tile are listed below. The more tile he/we buy, the cheaper the deal.
4" x 4" 6" x 6" 12" X 12"
Less than 100 tiles $0.24 $0.50 $2.00
Between 100 and 999 $0.20 $0.44 $1.80
Between 1000 and 3999 $0.18 $0.40 $1.60
Between 4000 and 9999 $0.16 $0.35 $1.40
Over 10000 tiles $0.15 $0.33 $1.30
In your write up of the best solution explain what you and your groupmates did to find the best deal.
PART 2: Once you’ve figured out the best deal for your classroom. Figure out the best deal for the following situations (i) if the room were one-tenth the area, (ii) if the room were one half the present size, and (iii) if the room were double the size (in area).
424
The Job Gap
In the Organic Goodies activity we looked at a mini-economy where there were not enough jobs to go around. To what extent do you think this happens in our economic system?
Estimated Job Growth in Low- Low- Skill Occupations Low-Skill Job SkillJob Number of from 1998 to Gap (or Seekers per City Job Seekers 2003 Surplus) Job Boston 11,506 9,300 2,206 1.2 Chicago 97,874 20,920 76,957 4.7 Detroit 55,485 2,350 53,135 23.6 Milwaukee 11,419 3,740 7,649 3 New York City 217,535 33,870 183,665 6.4 San Francisco 8,186 33,590 -25,404 0.2 St. Louis 11,258 9,420 1,838 1.2 Washington-DC 9,465 4,280 5,185 2.2
This table shows data related to the Job Gap in 8 U. S. cities. What does it show? What does the last column mean and how was it computed? What do you think about the job gap? What is up with Detroit?
425
Tiles within Tiles
How many of the following fit in a square foot? • 1” x 1” • 2” x 2” • 3” x 3” • 4” x 4” • 6” x 6” • 12” x 12” • 1.5” x1.5” • 0.5” x 0.5”
Why weren’t you asked to find the number of 5” x 5” squares that fit? Can you find out the answer?
Complete the table using what you did above and find the relationship between the IN and OUT columns.
IN OUT
12 1
6 4
4 9
3 ___
2 ___
1.5 ___
1 ___
0.5 ___
Create a graph of this relationship in Excel or using a graphing calculator.
426
Civil Rights and Economic Equality
One of the key concerns of the Civil Rights Movement was workers’ rights – increased pay, unionized jobs, and dignity for all working class Americans. In the book Where Do We Go From Here: Chaos or Community? Dr. King called for a “guaranteed income” where all Americans would be employed and receive an income. (Of course, this hasn’t happened in the U. S. yet.)
(1) Do you think the distribution of income in the U.S. has become more or less fair since the Civil Rights Movement?
(2) What would we need to know to find out?
(3) One of the things the mass media and many textbook companies tend to forget is that Martin Luther King was a socialist – he wanted a much “fairer” distribution of income in addition and guaranteed jobs for all Americans. Why do you think history has begun to forget this – or not teach this – about MLK? What do you think about a guaranteed job and income for people living in the United States?
427
Learning Excel
In this exercise you need to make sure you can.
• Enter in a table of x- and y- values (about 10 values) and create a corresponding graph.
• Enter a formula such as (y = x2 + 1) in the x y y-column (or row) of your table and fill 7 50 down (to –7). Create a graph. (See right.) = B3-1 = (B4^2 +1) • The table you just created has an x- column that decreases from 7. How can you get the x-column to increase from – Fill down Fill down 7 upwards to 7 without typing in all of the values one by one?
• See if you can figure out how to create two graphs at once.
428
Regularity in Irregularity Find the areas of the “irregular” (or semi-regular) figures below.
3in
2in
3in
2in
1in
2” 2”
1”
5”
How are the units used in the first two figures different?
429
Find the area of the pentagon below. You are given that EC = 7.5 cm and that AB ll EC and that BC ll AE.
B A
5cm
9cm
C E
D
How could you approximate the area of the figure below? What units would you use and why?
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Putting the Gini in Excel In this activity we are going to try to program a spreadsheet, or a calculator to quickly calculate the Gini coefficient from a given distribution of income. That is, the goal of this assignment is for you to create a spreadsheet program that can compute the Gini coefficient for quintile distributions such as (3.7, 9.1, 15.2, 23.4, 48.6).
You might begin by seeing if you can create a graph like the one below, using the distribution above. Note that to create a two-dimensional graph you will x- and y-values.
If you can create something similar 100 to this graph you are halfway home.
80 From here, you should think about what numbers you would use to 60 calculate the areas (of the trapezoids) in your diagram. [Hint: 40 Instead of using one cell and creating Y% of Total Income 20 a big formula for areas use several cells to compute the areas of the 0 different regions in the graph.] 0 20 40 60 80 100 Poorest X% of Society
WRITE UP: (1) Briefly describe what you and your partner(s) did to create your spreadsheet program? (2) How would you describe the experience of programming in Excel overall? (3) What would you do to improve this assignment?
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Gini, Gini in Excel, Who’s the Fairest of them All?
Below is a list of the income distributions of several countries from around the world. In this activity we will use the Excel programs we recently created to calculate the Gini coefficients of each of these countries in order to see which country has the fairest distribution. But first, before we do that you should try to rank each country by “eyeballing” this data.
Country Low: 0-20% 20-40% 40-60% 60-80% HIGH: 80-100% India (2000) 8 14 16.5 22.5 39 Japan (1995) 10.6 14.2 17.5 22.5 40.2 Mexico (1995) 4.1 7.8 12.5 20.2 55.3 South Africa (1994) 2.9 5.5 9.2 17.7 64.8 Italy (1991) 7.6 12.9 17.3 23.2 38.9 Sweden (1992) 9.6 14.5 18.1 23.2 34.5 U. S. A. (2001) 3.5 8.7 14.6 23 50.2 Canada (1994) 7.5 12.9 17.2 23 39.3 Canada (1996) 4.6 10 16.3 24.7 44.5 [The data from Canada just shows how difficult it is to (a) measure something like the distribution of income and (b) get accurate/calibrated data off of the internet.]
We can use the computer program we developed to calculate the Gini coefficient for each country and will help us think about the situation. You simply have to go back into your program and plug in the new values into the quintile tables you created. Once you have computed the Gini coefficients for each country, rank the countries from fairest to least fair.
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APPENDIX B: RACE AND RECESS ACTIVITY
Race and Recess
Schools with Recess Schools without Recess
30-100
5-29.9
Percent White Students 0-4.9
0 25 50 75 100 125 150 175 200 225 250 275 300 325 Number of Schools
1. Explain the data shown in the figure above. What is going on?
2. How many schools were involved in this study?
3. Is there a connection between the two charts in the above figure? Explain.
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These graphs represent the percentage of CPS students (ages 6 – 11) who are overweight by population group – Black (in black), White (in pink), and Mexican (in blue).
Girls ages 6 to 11 Boys ages 6 to 11 16 20
18 14 16 12 14 10 12 8 10
8 6 6 Percent overweight 4 Percent overweight 4 2 2 0 0 1960 1965 1970 1975 1980 1985 1990 1995 1960 1965 1970 1975 1980 1985 1990 1995 Year Years
1. Interpret the following graphs. What do you notice? What is going on?
2. How do these graphs relate to the chart from part 1?
3. If you were going to send a letter to CPS, say Arne Duncan, what would you say to him? Is there more information that you’d like to have to make your argument?
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Recess Activity Continued …
Girls ages 6 - 11
30
25
20
15
10 Percent overweight
5
0 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 Year
Yesterday we looked at similar graphs from CPS students. This is a subset of the data. Today we’re going to look at the mathematics to see if it offers anything new.
1. What would happen if the lines continued?
2. Would they ever intersect? When or why not?
3. How obese will each population subgroup be in year 2002? Can you say for sure?
4. What does breaking a population into subgroups offer us? What is problematic about it?
435
APPENDIX C: COURSE OVERVIEW
DATES HOUR 1 HOUR 2 CORE TOPIC WEEK 1
Day 1 Opener: Pre-Test PATTERNS; (11/17/03) IMP Activity: Describing Patterns IMP Activity: In-Out Tables EQUATIONS
Opener: Sequences & Tables Supplement 1 Continued Day 2 Activity: Another In-Outer IMP Activity: Checkerboard PATTERNS; (11/18/03) Supplement: Linear Patterns 1 Squares EQUATIONS
Opener: Tables, Sequences IMP Activity: A Protracted PATTERNS; Day 3 Continued Engagement ANGLE (11/19/03) Critical Activity: School is Hell Supplement: Linear Patterns SUMS
Opener: Patterns, Tables IMP Activity: Find Angle Sum PATTERNS; Day 4 IMP Activity: Pattern Blocks for for Quadrilaterals ANGLE (11/20/03) Angles Supplement: Linear Patterns 2 SUMS WEEK 2
Opener: Working with the Triangle Sum "Rule" ANGLE Day 5 Activity: Angle Sums Table and SUMS; DATA (11/24/03) Equation Critical Activity: Race & Recess ANALYSIS
Opener: Regular Polygons IMP Activity: Diagonally PATTERNS; Day 6 IMP Activity: An Angular Speaking ANGLE (11/25/03) Summary Supplement: Linear Patterns 3 SUMS
Activity: More Equations Day 7 Opener: Decagons & In-Outs Writing Activity: About Night PATTERNS; (11/26/03) IMP Activity: Bags of Gold School. EQUATIONS
Note that IMP stands for Interactive Mathematics Program (Fendel, Resek, Alper, & Fraser, 2000). See Chapter 6 for examples.
Opener Activities were relatively short and took approximately 10 minutes per day. See Excerpt 6.1 in Chapter 6 for an example.
Critical Activities were those that I designed or adapted from Professor Eric Gutstein. See Appendices A & B and Chapters 5 & 7 for examples.
Supplemental Activities were designed to allow students to practice procedures (with meaning). See “Patterns, Tables, Graphs, & Equations” Activities in Appendix A.
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DATES HOUR 1 HOUR 2 CORE TOPIC WEEK 3
Opener: Triangle Sums & IMP Nailing Down AREA Cont... AREA; Day 8 Equations Supplement 4: More with PATTERNS; (12/01/03) IMP Activity: Nailing Down Area Equations EQUATIONS
Day 9 Opener: Finding Areas AREA; (12/02/03) IMP Activity: Approximating Area IMP Activity: An Area Shortcut PERIMETER
Opener: Area Day 10 IMP Activity: How Many Can You AREA; DATA (12/03/03) Find (Area) Critical Activity: SAT Data ANALYSIS
Opener: Where will you be in 5 & IMP Activity: Parallelogram & AREA; FOR Day 11 20 years? Trapezoids COMMUNI- (12/04/03) Activity: Ins and Outs of Area Activity: The Counterfeit $20 Bill CATION WEEK 4 Opener: Self- and Course Day 12 Evaluation IMP Activity: Having Your Way AREA; (12/08/03) Activity: Areas of Polygons Activity: Organizing Portfolios ASSESSMENT Opener: Using Triangle Formula Day 13 Activity: Finding a Trapezoid Area Critical Activity: Handgun AREA; DATA (12/09/03) Formula Violence ANALYSIS Opener: Apply Trapezoidal AREA; Day 14 Formula Tiling the Room Continued … PATTERNS; (12/10/03) Activity: Tiling the Room Skills Worksheet EQUATIONS
Day 15 IMP Assessment: How Big is It? Supplement 5: Area of Triangles, AREA; (12/11/03) IMP Activity: Forming Formulas Quadrilaterals, and Combinations ASSESSMENT WEEK 5 Opener: Find Areas of Polygons Day 16 IMP Activity: An Area IMP Activity: Flat Boxes (Nets for (12/15/03) Approximation? Cubes) AREA SURFACE Day 17 Opener: More nets IMP Activity: Another In-Outer AREA; (12/16/03) IMP Activity: Flat Boxes Activity: Area of Irregular Region VOLUME SURFACE Day 18 Opener: Surface Areas & Volumes Voluminous Task Continued AREA; (12/17/03) IMP Activity: A voluminous task Supplement 7 VOLUME
Day 19 Movie & Partial Attendance Movie & Partial Attendance (12/18/03) Before Winter Break Before Winter Break OTHER
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DATES HOUR 1 HOUR 2 CORE TOPIC WEEK 6
Day 20 Opener: Volumes & Surface Areas IMP Activity: More on Pythagorus PYTHAGO- (1/05/04) IMP Activity: Tri-Square Rugs Supplement 7 RUS; AREA
Opener: Review Day 21 IMP Activity: Any Two Sides Critical Activity: Gutstein's PYTHAGO- (1/06/04) Work (Pythagorus on Geoboards) Mercator Map RUS; AREA Opener: Pythagorean Theorem Problems Day 22 IMP Activity: Leslie's Fertile Critiical Activity: Gutstein's PYTHAGO- (1/07/04) Flowers (Pythagorean) Mercator Map RUS; AREA
Opener - Pythagorus, Area, Etc... PYTHAGO- Day 23 IMP Activity: Ins and Outs of IMP Activity: The Power of RUS; (1/08/04) Boxes Pythagorus 1 & 2 VOLUME WEEK 7
Opener: Mercator Map Debate PROPOR- Day 24 IMP Activity: Draw the Same IMP Activity: Statue of Liberty's TIONAL (1/12/04) Shape Nose REASONING
SIMILARITY; Day 25 Opener: Similarity Practice AREA; DATA (1/13/04) IMP Activity: How to Shrink It? Critical Activity: I&A Project ANALYSIS
Opener - Similar Houses SIMILARITY; Day 26 IMP Activity: Is there a AREA; DATA (1/14/04) Counterexample? Critical Activity: I&A Project ANALYSIS
Opener: Similar Houses 2 SIMILARITY; Day 27 IMP Activity: Triangular Critical Activity: I&A Project AREA; DATA (1/15/04) Counterexamples Activity: Supplement 9 ANALYSIS WEEK 8
IMP Activity: Make It Similar SIMILARITY; Day 28 IMP Activity: Why Are Triangles PROPOR- (1/19/04) Special? IMP Activity: From Top to Bottom TIONS
Opener: Review SIMILARITY; Day 29 IMP Activity: Are Angles IMP Activity: Inventing Rules PROPOR- (1/20/04) Enough? Supplement 10 TIONS
SIMILARITY; Day 30 Opener: Review PROPOR- (1/21/04) IMP Activity: What's Possible? IMP Activity: Inside Similarity TIONS
IMP Activity: Ins & Outs of SIMILARITY; Day 31 Proportions PROPOR- (1/22/04) Critical Activity: South Central Critical Activity: South Central TIONS
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DATES HOUR 1 HOUR 2 CORE TOPIC WEEK 9
Day 32 (1/26/04) Post-Test (Grades Due) Supplement: More on Graphing ASSESSMENT
Day 33 (1/27/04) Interviews, Games, & Pizza Interviews, Games, & Pizza ASSESSMENT
Day 34 Class Canceled by Night School Class Canceled by Night School (1/28/04) Administration Administration NO CLASS
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APPENDIX D: THIRTEEN STUDENT PROFILES
If I were asked to draw the “reform student” I would paint a being that looks like
an outer-space visitor, with a big head, probably a little heart and a tiny chunk of
body. That being would be mainly lone and mostly talk about mathematics
learning, and would see the world through this school mathematics experience.
(Valero, 2004, p. 40).
Carol D. Lee (personal communication), Martin (2003, 2000), Nasir (2002), and
Valero (2004) indicate that researchers in (mathematics) education often provide simple demographic statistics (e.g., “60% low income,” “urban school”) to describe school context and community. In so doing, they essentially ignore the identities of participants and historical context. In this appendix, I briefly describe the character of the night class as a whole and then move to individual profiles of approximately half of the students – that is, given that I have already contextualized the night school setting in Chapter 3.
These student profiles are meant to give the reader a sense of the individual identities of some of the night school students and some of the issues that they were dealing with in and outside of class. It should help the reader understand how I saw students.
These portrayals are based primarily on my analyses of students’ written work, participant observation, interviews, and informal conversations that took place before, during, and after instruction. (See Chapter 3.) While I attempt to let each student speak for his- or herself, these portrayals should be viewed as constructions over which I had primary control.
440 Guevara draws students from several rival gang territories and is well known throughout the city for its gang activity. Approximately ten of the students in my course claimed to be in gangs currently or to have been formerly. It was pretty clear, however, that the majority of the students in the night course were not in gangs, nor had they ever been. Seven of the students either openly discussed past or present gang involvement with me (or loud enough for me to hear) and three to five others signaled possible involvement in other ways (e.g., dating a gang member, fighting after class). Indeed, at least two students had chosen or been told to attend night school because of gang-related problems during day school at Guevara. It was not until the third week when I saw tattoos across one young woman’s hands and her neighbor openly joked about them, that I realized that many of the female students were either in gangs or had once been in gangs.
Three of these young women openly discussed past gang involvement with me as the course progressed.
Over time the students sorted themselves in clusters within the classroom. While I made two seating charts in the first two weeks, there were so many absences and tardies – which lead to seven students being dropped from the course – in the first few weeks that I had to move students around on a daily basis so the students with weaker mathematics backgrounds could be near someone with a stronger background in school mathematics.
By the third week of the course the night school students were working well enough together that I allowed most of them to sit where they wanted to. There were, however, approximately six students who consistently sat in the back corner of each side of the classroom who were generally slower to start on small group activities than their peers in the middle and front. Shor (1996) calls this spatial phenomenon “Siberia” and the
441 students who congregate there “Siberians.” The most resistant (from my perspective as a teacher) four male and two female students in my class might be described as Siberians.
These students sat in the back of the room – unless I made them move – and were often slow to start in on small group work. While these students seemed comfortable socializing with me, their Siberian orientation towards school mathematics did not seem to evolve much over the timespan of the course. (My discourse analyses support this claim.)
Individual Students
I now introduce thirteen of the night school students. Ten of these students appear in the excerpts of classroom discourse I provide in Chapters 6 and 7. I include three of the
“non-discursive” participants in order to highlight non-participation, which is a significant factor in teaching mathematics in remedial and low-track courses (Chazan,
2000; Romagnano, 1994). Non-participation in whole class discussions was not necessarily resistance although it had the same effect. There were also active (“A”) students who seemed shy and were generally hesitant to speak up in front of the whole class. I will begin with three of the students in the back of the room in Siberia and work my way up. Finally, as might be apparent, some students chose their pseudonyms for this study and I generally honored their choices.
Lucee. Puerto-Rican American. A Siberian. Lucee had been in a selective (“honors”) high school but switched schools when she decided to go to the less prestigious neighborhood high school with her friends. Like many of the students in the course,
442 Lucee had been expelled from day school and had to attend night school program in order to graduate. She had apparently been expelled for gang involvement and fighting. Lucee was unhappy about having to attend night school instead of the regular day school at
Guevara. From my perspective as a teacher, Lucee was trying to do the minimum amount of work in order to get course credit, though her discussions with me towards the end of the course about her low grade indicated that she did not see it that way. Lucee often relied on her neighbor, Princess, for mathematical answers. She frequently had non- mathematical conversations during break that spilled over into class-time. Although she could be friendly, Lucee had a tough life. One day she discussed getting jumped by another night school student with Princess and others around her:
Princess: So you beat her up?
Lucee: Yeah. But there was no blood on her face. I was steady looking at her like
– damn! – cause, she was on the floor and I don’t know how she got on the floor.
I guess she fell – I don’t know what happened. All I know is that she was on –
laying straight on her back with her legs straight …I was looking at her and I
wouldn’t be kicking her so, and I just went down on her and I was punching her.
When she was trying – and then when she was trying to get back up. [Her
boyfriend] had tackled me and I got away from her and I grabbed hold of her
again and I hit her … I don’t know what happened. That’s when she got my hair.
Though I labeled her a Siberian, Lucee became very involved in the last critical discussion which seemed to indicate that she, in contrast to some of the students with stronger mathematics backgrounds (e.g., Princess), thought critical topics were far more relevant and interesting than the (apolitical) reform geometry I was teaching.
443
Leroy. Half Puerto Rican, half White American. A tall and athletic looking Siberian with a shaved head. I also am tall and have a shaved head and he liked to point out his and my similar appearance. Like Lucee, Leroy’s mathematical performance was erratic. He attended a magnet school during the day and had a fairly strong background in mathematics. He scored higher than the majority of his peers on a pre-test I gave the class. Leroy was occasionally very helpful to those around him. At other times, however, he sat in the back and hid in the last row; that is, he engaged in Siberian behavior. I began to suspect that he was high on those occasions. In general, Leroy was bored with the reform geometry we were doing. In the first week of the course he wrote the following in response to a cartoon from the “School is Hell” cartoon book by Groening (1987).
One of the thoughts I think about when I’m in school while the teacher is teaching
is “man, when is this class over with,” “why is this class so boring,” and ”what
does this have to do with me or does for me in my near future?” I just think some
subjects the teacher teaches does not relate to us in our life. Not all, but some.
Indeed, when I asked students to write about their everyday lives in critical assignments,
Leroy generally wrote elaborated responses – responses that eclipsed his terse responses to many of the more mathematical (IMP) assignments that he was given. On one opener I asked students where they thought they’d be and what they’d be doing in five and twenty years. He wrote far more than his peer. His response to the first prompt was:
Wow! In five years, I would be chilling in my base, wherever the Marines is
going to send me. I would be coming back home on certain days to see my family
and friends. I would be cruising around the city of Chicago with my Cadillac
444 Escalade. This is my dream truck. One day I’ll be the happiest I ever been, only
because the man upstairs had blessed me with a special talent.
As this excerpt indicates, Leroy enlisted in the Marines shortly before the night school course began. He claimed to have done this, in part at least, to escape from his involvement in a prominent Chicago street gang. Leroy was candid about his gang involvement; in the second week of the course Leroy told me that he had been caught by the police seconds before he was about to shoot at rival gangbangers who had shot at his
“boys” several days earlier. He claims it was good luck to get caught by the police before he had become a murderer. I believe Leroy was relatively high up in the command chain, because other students in the school followed Leroy around during breaks and after class.
Martín. Puerto-Rican American. A Siberian. Attended night school full time and claimed to wake up every day around 2 PM to come to a vocational class (i.e., food preparation) that began before our geometry class. Martín also wrote the following “brief” description of teachers in response to a Groening (1987) cartoon from School is Hell:
Why are [teachers] so boring? Because all they talk about is thing[s] that we don’t
care about. If they would talk about something that we like we would not be so
bored.
Given that Martín claimed that he would attend college upon graduation I believe that hidden behind this statement is the fact that he found school mathematics to be painful.
He was labeled as “LD” (learning disabled) in the official attendance book I was given. I observed that he struggled with mathematical problems that others around him sailed through. I often observed him sheltering his answers from his peers and, at times, from
445 me. When he did speak up during class, he was generally made explicit what I read in his generally reticent mathematical stance. The following is an excerpt of an exchange between he and I caught on videotape, where he was encouraging me to talk more so he could “sleep:”
Me (at front of room to whole class): Alright. I’m going to talk at you all for five
minutes. You all ready for that?
Martín: You can talk longer if you want to. That’s alright. You can talk the whole
period. (11/25/03)
Martín did not necessarily see himself as a Siberian or even “resistant.” In a post interview conducted by a female colleague he suggested that he preferred night school to day school, stating, “you got two hours here instead of forty-five minutes; more time to do your work and finish it; more time to understand what you gotta do.” He also claimed to be impressed that I could build good relationships with he and his peers stating: “Not a lot of teachers could relate to their uh students or anything.”
Princess. African American. Princess had been a student in a selective program at a high school where I used to teach—not the same as Guevara. Several of her friends had had me as a teacher and I believe that the positive things she had heard from them was the primary reason that Princess opened up to me from the beginning of the night course.
Princess often came early to class to tell me about her life. As she explained, she had “ messed up,” “lost interest in school,” gotten pregnant, had an abortion, and moved to the
South for a year to get away from Chicago. In an interview she claimed her future career plan was now to:
446 I just wanna go to college. So I can be a lawyer. So I can make money – good
money. I don’t wanna live from paycheck to paycheck.
Princess had an Afro-centric orientation. She believed, like many people, that the U. S. government was intent on tearing the African American and Latino populations apart:
You don’t see what it’s doing to our community? If people are dying, don’t you?
They put that [crack] out there so we could die – for minorities to die, really. They
brought it – the drugs came from the government
She also had enjoyed going to school in the South because it was the first time she had been in an all Black school. She claimed that she might prefer to be a “motivational speaker” instead of a lawyer because of the “influence” she could have on students (of color) like her.
I could be a motivational speaker – talk to kids about teenage pregnancy, and
diseases, and things like that. What we go through.
Princess was also the spark that got many reform mathematical discussions and activities going. She would persist in working on the assignments I gave the class until she understood how to do everything. She helped several students around her, including
Lucee, when they struggled. While Princess was involved in the first few critical mathematics activities, she did not participate fully in the last two critical projects (one featured in Chapter 7) I gave the class. I interpreted Princess’ lesser participation in these compared to her involvement in mathematics activities as an indication that she did not see critical mathematics – ironically perhaps given her Afro-centric orientation – as relevant or appropriate.
447
Sonny. Sonny came to U. S. from Honduras when he was 8. At the time of the course,
Sonny was 19 and had recently married. He worked full time during the day at a large computer store during the day. He attended my course at night. He described his typical day as follows:
Go home, go to sleep, eat, wake up, go to sleep again, wake up again, go to work
again. It’s bad. At work, sometimes I get a chance to take a shower, get a change
of clothes.
Sonny expected to attend a technical/vocational college after finishing up night school where he would study radiology or, if that did not work out, auto repair. Sonny, like
Princess, had a stronger background in mathematics than many in the class and often helped to move along the whole class mathematical discussions. On the other hand,
Sonny often took what Pruyn (1999) might call a “conformist” stance, by attempting to get me to teach more directly than I did in reform activities. Sonny also questioned the idea that he could learn from his peers through the interactive group work and was reluctant to help other students around him, at least for the first several weeks. He wrote the following to me in Week 3 in response to an opener prompt regarding things “we” could improve in:
I think that when you put a problem on the board you should at least say the
answer to it because some of us can’t stop thinking whether our answer is the
right one or not. Also, you shouldn’t take off points just because we don’t want to
explain to somebody else how to do a problem. I don’t like to explain to others. I
don’t get paid for it!
448 That said, in a post-interview I conducted with Sonny and one of his peers at the end of the course, Sonny had the following response to the question: if you were a math teacher how would you change the way math or geometry is taught?
You know it depends. Cause there are some teachers that like – they just go up to
the board, put up a problem, and be like, ‘okay’ you know, ‘if you got it, you got
it, if you didn’t, oh well.’ So, I guess I’d try to be more like you too. Sort of like
this class.
Thus, there is evidence that Sonny’s views on the role of mathematics teachers and students had changed as a result of participation in my reform mathematics class.
Kampton. African-American. Kampton began the course at the back of the room in
Siberia. Kampton did the minimum amount he needed to do to pass the course and made it clear that he would be happy with a D – and he cheered when he found out he passed my course. When, in a post-interview, my colleague asked him to describe his typical school day, Kampton responded: “boring, tiresome, irritable.” He attended regular day school at a neighboring school. Kampton also claimed he was “nervous” about coming to night school because he was “black” and did not know people in the night school – which was majority Hispanic. I think he was nervous about coming to Guevara because he was also in an African American gang and had to cross into rival gang territory. Kampton had spent time in prison for gang related activities and even claimed to have witnessed his girlfriend shot and killed in a drive-by shooting. On the second day of the night course
Kampton intimidated an undergraduate assistant who was helping me by videotaping the night school course. I believe Kampton was trying to show his classmates and me how
449 tough and in control he was. This undergraduate was so bothered after this incident that he never returned to the school. Kampton was also one of three young men in “Siberia” who came to class drunk and high during the first several weeks of the course until I realized, in Week 3, that this is what these three were doing. In the third week of the course, I had to pull Kampton into the hallway after he called me “bald-headed asshole” and drunkenly threatened to fight me. Once we were alone, he backed down and told me
“don’t take it personally Mr. B, I’m drunk.” After that encounter, and threatened with expulsion, Kampton began coming to class fifteen to twenty minutes early. He began to open up to me about his life. Kampton and I developed a relationship that went beyond the relationships I have had with most students. In the post-interview my colleague conducted with him, Kampton claimed that my patience with and interest in he and his peers was important to this relationship:
Not to be too sensitive or nothin’, but it makes us feel connected to him. It makes
us feel that he’s one of us.
After this incident, I also moved Kampton to the front of the room – out of Siberia – to work with Osvaldo. Kampton improved as a mathematics student although not as much as I would have liked. When, in the post-interview, my colleague asked him, “what’s something that you’d rather just not do [in the course?]” Kampton laughed and responded, “the math, the math!”
Osvaldo. Mexican American. Like many students, he found ways to push at the limits of dress code and seemed to put a lot into his appearance. From time to time he had
“pointers” for me about my wardrobe. After high school, Osvaldo thought he might go to
450 a city college to study “computer graphing … cause I like messing around with computers.” Osvaldo was absent three of the first four days. Night school students were officially allowed 3 days but knew that they could miss 4 days before being expelled from the class. In the beginning of the second week, after missing 3 days in the first week, Osvaldo proceeded to do absolutely nothing mathematical (or school-related).
When I asked him how he expected to pass the course, he told me not to worry. He claimed that the first two weeks and the last week of night school were essentially
“vacation.” On the fourth day of each four-day week, he would demand that I put on the
TV like “all of the other night school teachers.” Osvaldo explained to me at the end of the course that I might want to teach more like a (traditional) teacher he had – with more incentives or extrinsic rewards:
Osvaldo: Here’s what he does. He goes up to the board, he tells you what page
number and everything and does a few examples. He tells us to pick an example
from the book – from the page that we’re going to do – and goes over it and like
keeps on, until we get it. It only takes one time for him to do it and then, unless
people don’t understand – and like the process …
Me: So, is that a good way to teach, like the way he teaches?
Osvaldo: Cause then it gives us motivation, you see? Cause, he tells us, if you
guys finish the work, you all can have the rest of the period – you know?
Me: Oh. So he gives you the rest of the period off.
Osvaldo: So it’s motivation. Just if you finish it.
Me: Are you allowed to work together?
Osvaldo: No.
451 Osvaldo began working harder as the course progressed. He completed all of the assignments after Week 2 – not always without trying to negotiate with me. After the second week, he often sat with Shannon, an A student who always finished her work. As mentioned above, I moved Kampton next to him so they could work together.
Stephie. Puerto Rican-American. At 19, Stephie lived on her own with her son and her older sister. Stephie expected to get an A in the geometry course and generally worked hard. Unlike Osvaldo (apparently) she seemed to favor the reform approach to geometry.
In Week 3, I asked her what I could do to improve the course and she responded,
“Nothing. I think this course has taught me a lot!” That said, it was more difficult to gauge her interest in the critical activities. At the outset of the Inequality & Area unit she and her partner, Robi, seemed disinterested. However, her contribution to the whole conversation was elaborate and instrumental as I show in Chapter 7. The following is one of two responses she wrote to prompts in the I&A Project:
The I&A in the U. S. hasn’t become fairer because a lot of people … work harder
and don’t get paid enough. My dad would say that he works his butt off but he
won’t complain about his pay. He knows that he will get a raise when his time
comes! But when he thinks of me and all the work I do & I don’t paid enough.
Stephie worked full-time as a manager in a movie theater in a wealthy suburb and had a two hour commute to night school. She often arrived to night school exhausted.
Lupe. The daughter of recently arrived immigrants from Mexico. Lupe had a stronger background in mathematics than most students in the night course. I often relied on her.
452
Lupe was one of the sparks that would get about 5 or 6 students nearby her started on activities; She would help several of the students around her when they were struggling, including Efrain (below). Lupe was also one of the minority of students who consistently stated counter-hegemonic positions and she often came early to class, apparently in order to discuss politics with me. Lupe understood the ways in which that the U. S. society was unfairly structured against (Mexican American) immigrants. She wrote the following in response to the I&A Project (Chapter 7).
Is the U. S. economic system fair to its citizens? To its citizens, maybe it is fair.
Some people (citizens) get paid good for doing almost nothing. On the other hand,
you have all of these other people working hard [undocumented workers from
Mexico] and actually making a difference in this world providing us with the type
of things we need in order to survive and regardless of their hard work they are
still getting paid minimum wage.
At the end of class one day, Lupe told me that she wanted to become a nurse to help her
“own people.” She laughed and said she would not want to work with white people
“cause most of them think they’re all that!”
Efrain. Puerto Rican. Efrain was tall, had a goatee, and looked like he was 23, although he was only 19. Efrain had dropped out of high school but returned a year later and claims that his disposition to school has changed because, in his words:
I need a high school diploma. I need an education. And I found [that] I wanna
keep on studying. I wanna go to college now.
453 He stayed after class a few times to talk with me and told me how good a writer he was.
He felt differently about mathematics; in an interview with a colleague of mine he stated,
“To me, math is just a waste of time.” That said, however, he seemed to like working with his peers, the longer class period, and my reform approach to mathematics:
Efrain: Me and math never got along. I used always get F’s in math classes.
Anneeth: You didn’t understand it, or?
Efrain: No. It’s like people could be telling me it, but it didn’t go into my ears.
It’s just – I’m just – I’d be more paying attention looking at a fly, than paying
attention to …
Anneeth: So now, what changed?
Efrain: Cause like in day school, we didn’t have enough time to teach what – And
like with Mr. B – Mr. B was cool, like, he explained it real good and he made like
all the students help you around. And you have two hours here, like almost two
hours here. It really got into details.
Efrain became a hard working student in my class and earned a B. After I moved him out of Siberia, Efrain latched on to Lupe (above) and his performance improved. Like most of his peers, Efrain had a lot to navigate when he was not in class. In the second week of the course, I went to the boys’ bathroom before class and heard some commotion in one of the stalls. I was not sure what was going on but Efrain emerged with two other young men. Several days later, Efrain stayed after class to tell me that he had been selling an
“eight-ball” of cocaine (approximately $120). He wanted me to understand that he was not a drug dealer but that, from time to time, he helped his uncles who were drug dealers.
In fact, Efrain claimed at the outset of the post-interview (cited above) that he wanted to
454 become a “cop.” He thought that it would be an exciting career where he could “make a difference.”
Robi. Mexican-American. Robi attended the regular day program and was making up credit in the night school program to graduate on time. Robi had a good sense of humor and seemed to always be in good spirits. His attitude helped to keep things light and moving along. This attitude was very welcomed from my perspective as a teacher. Like
Kampton and Leroy, Robi had enlisted in the military before the night school course began. Robi was also in the ROTC and occasionally came to class dressed in an olive- colored military uniform. Robi claimed that he had no “real life” role models but that Al
Pacino’s Scarface character might serve as one:
Scarface. Cause, I think he’s like a good role model, even though [in] the movie
they portray him [as] kind of like a bad person or being a drug dealer. But, the
way I see it, he got what he wanted, even though if it was a bad thing – drug
dealing – but, he wanted something [and] he pursued it until he had it.
Like this description of Scarface, Robi was driven, but not in any nefarious manner. He did all of the assigned work and generally tried to complete it quickly. That said, like
Sonny, Robi was generally reluctant to help those around him who struggled more than he did. Instead, he preferred to work with Stephie (above) and Diamond who were two of the more mathematically advanced students in the course.
Lana. African American. Lana, like many of the night school students, attended a regular day school elsewhere. I did not get to know Lana very well. She entered the course a
455 week late and then either came late or missed class altogether several times in the next two weeks. Lana was dropped from the course in the fourth week of class after she missed four days. While she was in the course, she was energetic, often contributing to whole class discussions. She reacted negatively when I indicated to her (in a small group setting) that she was too slow to start the assigned mathematics but she was friendly in non-mathematical conversations during breaks and outside of class. She had trouble working with her neighbors and seemed to have alienated several of the young women around her. By Week 3, they rolled their eyes when I asked them to work with her. I blame myself, in part, for not helping Lana and her neighbors adjust better to each other.
Deno. African American. Deno was in a regular day school at a predominantly African
American high school on the West Side of Chicago. Like several of the students, he had an hour-long commute to the night school. Deno was a big, easy-going young man who claimed to have been on his high school basketball team until he failed geometry. Like
Martín (above), Deno had “LD” marked next to his name in the attendance book I received. Unlike Martín, Deno interacted with me willingly and frequently. In fact, he demanded constant attention from me during the first two weeks. After some coaxing he began to work better with his neighbors. However, he missed two days in the third week and never seemed to catch up. When he missed his fourth class in Week 4, Deno was dropped from the course. He told me that his mother was going to “kill” him when he found out he was dropped. I believed him and felt bad for him.
456 Ana. Ana had moved to the United States from Mexico when she was eight years old and spoke English well but with a slight accent. She wrote interesting responses in English and Spanish to critical assignments I gave the class. One of the responses she wrote to the
School is Hell activity was:
Why is school run like a jail? Because we can’t do anything about it and because we
feel so guilty for being here.
The powerlessness and guilt that she claims to have felt seemed to sum up how many in the course felt for as night school students and for having to retake geometry. Ana blushed when I called attention to her during whole class discussions and would keep her whole class responses to a minimum. Ana worked diligently on small group work and she seemed to work comfortably with her neighbors. Ana often arrived early and would “chit- chat” with me and two or three neighbors. She never revealed very much about herself nor did she ever ask me about myself. Perhaps this was out of respect for teachers.
The Representative Nature of these Student Profiles
I tried to choose a range of students to present here. Their various past experiences with and orientations towards school mathematics was representative of those in the larger class. Some of these students were some of the loudest (Princess,
Sonny, Osvaldo) and most verbally active or quietest (Martín, Ana) students. It is also significant that the students in my night school classroom had failed geometry once before and many had a history of failing mathematics as the profiles below illustrate.
Indeed, many of the night school students described mathematics as irrelevant, “boring,”
457 and discouraging in interviews. Many, but not even the majority, were likely to be the students who in day classes were less engaged in activities and less likely to participate in whole class discussions.
458 APPENDIX E: PRE- AND POST-INTERVIEW PROTOCOLS
Pre-Interview Questions
Why are you enrolled in this course?
What do you want to get out of this night school course?
What do you want to get out of school (in general)?
What do you plan on doing after high school?
Do you think math will/can have anything to do with your future?
Do you think you can make a difference in the world? Explain.
Complete the following sentence: When I think about learning math I feel…
Why do we teach mathematics?
(Why do we teach geometry?)
Why do we go to school?
Is school “political”? If yes, why?
Is school fair? If yes, why?
What is math?
Where does math come from?
What kinds of problems can you (or other people) study with mathematics?
Do you use math in your life outside of school? When? Where? What kind of math?
Do you use geometry in your life outside of school?
459
Where do people in your family use mathematics?
Does anyone else you know use math in their everyday life? Where?
Who uses math outside of school?
What types of people are good at math? How do you explain this?
Describe the typical math class. (If you are in this class where do you picture yourself and what are you doing?)
Describe the typical math teacher.
Would you ever think about becoming a teacher? Why or why not?
If you were a math teacher how would you change the way math ( or geometry) is taught?
Who do you consider smart/intelligent? Why?
Do you have any role models? Who are they? Are they intelligent? Are they intellectuals?
Is being political a good or bad thing?
Do you consider yourself to be political?
Should teachers discuss politics in school?
Should teachers try to address racism in schools? Sexism? Homophobia? Social class bias?
Have you ever been taught something that you know was wrong? Explain.
460 True or False:
“I’ll never be able to learn math.”
“Smart people do math fast or in their heads.”
“I can learn from a student like me.”
“The teacher is the only one who can tell me the answers.”
“There’s only one right answer to every math problem.”
“There’s only one correct way of solving a math problem.”
“Education is important.”
“Math is important.”
“Math is boring.”
“Math is useful.”
“Math is everywhere.”
“Math has nothing to do with other subjects like history.”
“Math is political.” (Explain.)
“History is political.”
“Teachers like me.”
“Everybody has an equal chance in schools.”
‘Standardized tests are fair.”
“Standardized tests measure how smart I am.”
“School makes me smarter.”
461
“School teaches facts.”
“School teaches me the truth.”
Math teachers think I’m … (Why?)
School is …
Standardized tests are …
I need to pass this course to …
462 Post-Interview Questions
Describe the typical Monday, Tuesday, Wednesday or Thursday where you came to night school. (What were you doing before you arrived? Were you in day school or work before you came? Did you take another night school course or just Geometry?
How did you feel about coming to a 2 hour night school math class 3 or 4 times a week?
How is night school different from day school? (If at all.)
What do you plan on doing after high school?
Do you think you can make a difference in the world? (Be specific.)
Do you think math will/can have anything to do with your future?
Complete the following sentence: When I think about learning math I feel…
Why do we teach mathematics?
Why do we go to school?
Have any of your views of mathematics changed as a result of being in this course?
Do you use math in your life outside of school? When? Where? What kind of math?
Do you think you’ll use the math you learned in this course outside of school?
What is math and where does math come from?
Who uses math outside of school? Where and how?
Where do people in your family use mathematics?
Does anyone else you know use math in their everyday life? Where?
463
What types of people are good at math? How do you explain this?)
Would you ever think about becoming a teacher? Why or why not?
If you were a math teacher how would you change the way math (or geometry) is taught?
Describe the typical math class. (If you are in this class where do you picture yourself and what are you doing?)
Describe the typical math teacher. In what ways is Mr. B like this typical teacher? In what ways is he different?
Is school fair? If yes, why?
Is the American school system fair?
Is the world fair? Why or why not?
Is school “political”? If yes, why?
Should teachers be political in class?
Should teachers try to address racism in schools? Sexism? Homophobia? Class bias?
Have you ever been taught something that you know was wrong? Explain.
464 True or False:
“I’ll never be able to learn math.”
“Smart people do math fast or in their heads.”
“I can learn from a student like me.”
“The teacher is the only one who can tell me the answers.”
“There’s only one right answer to every math problem.”
“There’s only one correct way of solving a math problem.”
“Education is important.”
“Math is important.”
“Math is boring.”
“Math is useful.”
“Math is everywhere.”
“Math has nothing to do with other subjects like history.”
“Math is political.” (Explain.)
“History is political.”
“Teachers like me.”
“Everybody has an equal chance in schools.”
‘Standardized tests are fair.”
“Standardized tests measure how smart I am.”
“School makes me smarter.”
465
“School teaches facts.”
“School teaches me the truth.”
Math teachers think I’m … (Why?)
School is …
Standardized tests are …
I need to pass this course to …
Do you have any role models? Who are they? Are they intelligent?
Who do you consider smart/intelligent? Why?
466
APPENDIX F: RESULTS OF PRE- AND POST-ASSESSMENT
This appendix presents a summary of the class results from the pre- and post- assessment. There were fifteen questions. Summary of results of pre- and post-tests by each question are shown below. A copy of this three-page assessment is provided in subsequent pages.
Pre-Test Post-Test
1. Similarity (Use Similar Figures to Find 3 Missing Sides) 11% 93%
2. Similarity (How much bigger is the larger figure?) 11% 63%
3. Linear Equations (Graph y = 2x – 3) 18% 89%
4. Linear Equations (Find the Slope of y = 2x – 3) 16% 55%
5. Linear Equations (Find the y-intercept of y = 2x – 3) 11% 63%
6. Area (Use Gridded Map to Find the Area of Chicago) 18% 76%
7. Distance (Find Distance Between 2 Points on Grid) 21% 11%
8. Area (Find Area of Triangle) 11% 76%
9. Pythagorean Theorem (Find Missing Hypotenuse) 24% 79%
10. Volume (Definition of Volume) 18% 53%
11. Volume (Find Volume of “3-D” Object) 18% 76%
12. Surface Area (Definition) 3% 61%
13. Surface Area (Find Surface Area of “3-D” Object) 3% 37%
14. Solve Equations (ax + b = cx + d) 29% 55%
15. Solve Proportion (x/b = c/d) 47% 84%
Figure F.1. Class Results on Pre- and Post-Examination
467
PRE- AND POST-ASSESSMENT
1. Figure 2 is an enlarged version of Figure 1. Find the missing sides. 5
2
5 7.5
Figure 2
Figure 1
10
2. How many times larger is Figure 2 than Figure 1 featured above?
3. Graph y = 2x – 3 on the chart to the right. 7 6
5
4
4. What are the slope and y-intercept of the line you 3 graphed? 2
1
0 -2 -1 0 1 2 3 4 5 5. Draw a parallel line on the graph. What can you -1 say about this line and its equation? -2 -3
-4
-5
-6
-7
468
Use the map of Chicago to answer questions 6 and 7. The side of each square on the grid is 1 mile in length. So the grid is 26 miles by 16 miles
6. Approximate the following the area of Chicago (include units).
7. What is the distance from point A to point B?
Questions 8 – 10 concern the triangle on the left
8. What is the area? (Include
units)
x 6 ft
ø
8ft
9. What is x?
469
10. What is volume? (Not of the figure below, but your definition.)
11. If one block has a volume of 8 cm3 then the entire figure has a volume of what?
12. What is surface area (in general)? Your definition.
13. What is the surface area of the figure above? How did you figure this out?
Solve for x in problems 14 & 15
14. 3x + 5 = x – 7 15. - x = - 5 4 20