Mathematics and Teaching (Reflective Teaching and the Social Conditionis of Schooling)

MATHEMATICS AND TEACHING

REFLECTIVE TEACHING AND THE SOCIAL CONDITIONS OF SCHOOLING A Series for Prospective and Practicing Teachers Daniel P. Liston and Kenneth M. Zeichner, Series Editors

Zeichner/Liston Liston/Zeichner Maher/Ward Commins/Miramontes Meyer/Manning Anderson Crockett

• • • • • • •

Reflective Teaching: An Introduction Culture and Teaching Gender and Teaching Linguistic Diversity and Teaching Reading and Teaching Religion and Teaching Mathematics and Teaching

MATHEMATICS AND TEACHING Michele D. Crockett University of Illinois at Urbana-Champaign

First published 2008 by Routledge 270 Madison Ave, New York, NY 10016 Simultaneously published in the UK by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-Library, 2008. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”

© 2008 Taylor & Francis All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Crockett, Michele D. Mathematics and teaching/Michele D. Crockett. p. cm.—(Reflective teaching and the social conditions of schooling; 7) Includes bibliographical references and index. ISBN 978–0–8058–4419–1 (pbk. : alk. paper)—ISBN 978–0–203–93021–2 (e-book) 1. Mathematics—Study and teaching—Social aspects. I. Title. QA11.2.C76 2008 510.71—dc22 2007041683

ISBN 0-203-93021-5 Master e-book ISBN

ISBN10: 0–805–84419–8 (pbk) ISBN10: 0–203–93021–5 (ebk) ISBN13: 978–0–805–84419–1 (pbk) ISBN13: 978–0–203–93021–2 (ebk)

I dedicate this book to my beloved mother, Precious Jean Crockett, 1943–2005.

CONTENTS

Series Preface ix Preface xv Acknowledgments

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Introduction 1

I. CASES AND REACTIONS

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Introduction to Case 1 9 Case 1: “Race and Teacher Expectations” 10 Reader Reactions to “Race and Teacher Expectations” 13 Reactions to “Race and Teacher Expectations” 14 Reader Reactions 22 Summary and Additional Questions 23 Introduction to Case 2 24 Case 2: “Mathematics for All?” 24 Reader Reactions to “Mathematics for All?” 28 Reactions to “Mathematics for All?” 29 Reader Reactions 36 Summary and Additional Questions 37 vii

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CONTENTS

Introduction to Case 3 39 Case 3: “Culture and School Mathematics” 40 Reader Reactions to “Culture and School Mathematics” 43 Reactions to “Culture and School Mathematics” 44 Reader Reactions 50 Summary and Additional Questions 51 Introduction to Case 4 53 Case 4: “Politics and School Mathematics” 54 Reader Reactions to “Politics and School Mathematics” 58 Reactions to “Politics and School Mathematics” 59 Reader Reactions 68 Summary and Additional Questions 69 Reader Reactions to the Four Cases 70 Reactions to the Four Cases 71

II. PUBLIC ARGUMENTS

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A “Conservative View”: Mathematics for Global Economic Leadership 79 Comments and Questions 86 A “Liberal View”: Equality of Opportunity and School Mathematics in a Democracy 88 Comments and Questions 94 A “Radical Multiculturalist View”: Mathematics for Developing Critical Dispositions for Social Reconstruction 96 Comments and Questions 103

III. CONCLUDING REMARKS, SOME REFLECTIONS, AND RESOURCES FOR FURTHER REFLECTION 107 The Centrality of Culture: Understanding the Historical, Social, and Political Dimensions of School Mathematics 107 Resources 109 Exercises 111 Bibliography Index 121

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AN ESSENTIAL SERIES INTRODUCTION Whereas many readers rarely read introductory material, we hope you will continue. The success of this book depends, in large part, on how you use it. In what follows we outline some of our key assumptions and we suggest ways for approaching the material in each book of this series entitled, Reflective Teaching and the Social Conditions of Schooling. First, we identify some of our reasons for creating this series. We then relate a bit about our dissatisfaction with how teacher education is usually conducted and how it can be changed. Finally, we outline suggestions for ways to best utilize the material in this and subsequent texts. Some years ago we were asked to develop further the ideas outlined in our book Teacher Education and the Social Conditions of Schooling (Liston & Zeichner, 1991). It was suggested that we take our basic approach to teacher reflection and our ideas about teacher education curricula and put them into practice. The proposal was attractive and the subsequent endeavor proved to be very challenging. It never seems easy to translate educational “shoulds” and possibilities into schooling “cans” and realities. But we think (and we hope) we have made progress in that effort by designing a series of books intended to help prospective, beginning, and experienced teachers to reflect on their profession, their teaching, and their experiences. We are pleased and delighted to have the opportunity to share this work with you. We hope you will find these texts engaging and useful. ix

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We are two university teacher educators, both former elementary teachers, who have worked in inner-city, small town, and suburban elementary and middle schools. We are committed to public schools as democratic institutions, as places of learning in which people of all walks of life come to learn how to live together in a democratic society. Although we are personally committed to ways of working and living together that are much more collaborative than exist today we are educators first, realists second, and dreamers third. It is our firm belief that an education that engages prospective and practicing teachers’ heads and hearts, their beliefs and passions, needs to be fair and honest. We have neither written nor encouraged others to write these texts to convince you to see schools and society in a particular light, but rather to engage you in a consideration of crucial issues that all teachers need to address. Once engaged we hope that you will be better able to articulate your views, responses, and responsibilities to students and parents, and come to better understand aspects of your role as a teacher in a democratic society.

IMPACTS OF THE SOCIAL CONDITIONS OF SCHOOLING Prospective teachers need to be prepared for the problems and challenges of public schooling. Sometimes the focus in schools (departments and colleges) of education remains strictly on the processes that occur within the classroom and inside the school walls. At times, teacher education programs emphasize instructional methodology and the psychology of the learner in university course work and underscore survival strategies for student teaching. These are certainly important elements in any teacher preparation and ones that cannot be ignored. But classrooms and schools are not isolated environments. What goes on inside schools is greatly influenced by what occurs outside of schools. The students who attend and the teachers and administrators who work within those walls bring into the school building all sorts of cultural assumptions, social influences, and contextual dynamics. Unless some concerted attention is given to those assumptions, influences, and dynamics, to the reality of school life and to the social conditions of schooling, our future teachers will be ill prepared. Over the last 10 years, teacher educators have paid greater attention to the social conditions of schooling. But a consensus of opinion on this issue does not exist. The professional aspects of teacher education, including attention to the social conditions of schooling, have been criticized by scholars and politicians who believe that content knowledge alone is

SERIES PREFACE

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sufficient to teach. While we recognize the importance of teachers’ content knowledge, this view is inadequate. Students need teachers who have the professional preparation necessary to teach a diverse student population to achieve high academic standards. Content preparation alone does not suffice. We hope that the books in this series will contribute to a richer and more nuanced professional preparation. We are living in a time of remarkable change, a time of social and political transformation. In an era that is rife with social controversies and political difficulties, in which public schooling has increasingly come under attack, during which we are seeing marked changes in the cultural demographic make-up of America, in which there are great pressures to transform public schools into private, for-profit enterprises, in this era we must educate well our teaching workforce. Future teachers cannot, on their own, solve the many societal issues confronting schools, but they should certainly know what those issues are, have a sense of their own beliefs about those issues, and understand the many ways in which those issues will come alive within their school walls. Poverty and wealth, our culture of consumerism, what seems to be an increasing amount of violent behavior, and the work pressures of modern life affect the children who attend our public schools. Public attitudes about competition and excellence, race and ethnicity, gender roles and homosexuality, and the environment affect students inside and outside of schools. One can be certain that the issues that affect all of our lives outside of schools will certainly influence students inside their schools.

EXAMINING THE SOCIAL CONDITIONS OF SCHOOLING Probably the best way to begin to examine contextual issues such as these is to be attentive early on in one’s professional preparation, to experience features of the social conditions of schooling, and then to examine the experience and what we know about the social and cultural context of schooling. We encourage prospective and practicing teachers to do this. But teacher preparation programs often are not organized in a fashion that would encourage the discussion and examination of these sorts of shared experiences. What traditionally are called social foundations courses are typically not school-based, but set apart from some of the more realistic, practical, and engaged dilemmas of schooling. In schools of education we frequently teach what the sociology or philosophy of education has to say about schools but we tend to teach it as sociologists or philosophers, not as teachers struggling with crucial and highly controversial issues. Thus, in

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our own work with prospective and practicing teachers, we have developed ways to examine contextual issues of schooling and to enable ourselves and students to articulate our ideas, beliefs, theories, and feelings about those issues. The books in this series attempt to utilize some of these insights and to pass along to others the content and the processes we have found useful. When students and faculty engage in discussions of the social and political conditions of schooling and the effects of these conditions on students and schools, it is likely that the talk will be lively and controversies will emerge. In this arena there are no absolutely right or wrong answers. There are choices, frequently difficult ones, that require considerable discussion, deliberation, and justification. In order for these discussions to occur we need to create classroom settings that are conducive to conversations about difficult and controversial issues. The best format for such discussion is not the debate, the (in)formal argument, or dispassionate and aloof analysis. Instead the most conducive environment is a classroom designed to create dialogue and conversation among participants with differing points of view. There isn’t a recipe or formula that will ensure this type of environment but we think the following suggestions are worth considering. It is important for individuals using these texts to engage in discussions that are sensitive and respectful toward others, and at the same time challenge each other’s views. This is not an easy task. It requires each participant to come to the class sessions prepared, to listen attentively to other people’s views, and to address one another with a tone and attitude of respect. This means that when disagreements between individuals occur, and they inevitably will occur, each participant should find a way to express that disagreement without diminishing or attacking the other individual. Participants in these professional discussions need to be able to voice their views freely and to be sensitive toward others. Frequently, this is difficult to do. In discussions of controversial issues, ones that strike emotional chords, we are prone to argue in a way that belittles or disregards another person and their point of view. At times, we try to dismiss both the claim and the person. But if the discussions that these books help to initiate are carried on in that demeaning fashion, the potential power of the works will not be realized. A discussion of this paragraph should occur before discussing the substance raised by this particular text. It is our conviction that when a class keeps both substance and pedagogy in the forefront it has a way of engaging individuals in a much more positive manner. From our own past experiences, we have found that during the course of a semester it may be quite helpful to pause and focus on substan-

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tive and pedagogical issues in a conscious and forthright manner. Such time is generally well spent.

UNDERSTANDING AND EXAMINING PERSONAL BELIEFS ABOUT TEACHING AND SCHOOLING It is also our belief that many educational issues engage and affect our heads and our hearts. Teaching is work that entails both thinking and feeling; those who can reflectively think and feel will find their work more rewarding and their efforts more successful. Good teachers find ways to listen to and integrate their passions, beliefs, and judgments. And so we encourage not only the type of group deliberation just outlined but also an approach to reading that is attentive to an individual’s felt sense or what some might call gut level reactions. In the books in this series that contain case material and written reactions to that material, along with the public arguments that pertain to the issues raised, we believe it is essential that you, the reader, attend to your felt reactions, and attempt to sort out what those reactions tell you. At times it seems we can predict our reactions to the readings and discussions of this material while at other times it can invoke reactions and feelings that surprise us. Attending to those issues in a heartfelt manner, one that is honest and forthright, gives us a better sense of ourselves as teachers and our understandings of the world. Not only do students walk into schools with expectations and assumptions formed as a result of life experiences but so do their teachers. Practicing and prospective teachers can benefit from thinking about their expectations and assumptions. Hopefully, our work will facilitate this sort of reflection.

ABOUT THE BOOKS IN THIS SERIES The first work in this series, Reflective Teaching, introduces the notion of teacher reflection and develops it in relation to the social conditions of schooling. Building on this concept, the second work in the series, Culture and Teaching, encourages a reflection on and examination of diverse cultures and schooling. In Gender and Teaching, the third work in the series, Frinde Maher and Janie Ward examine the central role of gender in both teaching and schooling. In the fourth volume in this series, Linguistic Diversity and Teaching, Nancy Commins and Ofelia Miramontes focus on issues of linguistic diversity, instructional practices, and public schooling. In the fifth volume, Reading and Teaching, Rick Meyer and Maryann

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Manning offer a fruitful and illuminating introduction to beginning reading instruction. In Religion and Teaching, Ronald D. Anderson considers some of the complex questions that teachers face in their interactions with students, parents, administrators, and fellow teachers around the intersection of educational practices and religious beliefs. In this volume Michele Crockett explores the many and complex issues surrounding the teaching of math and culture. We are lucky to have Michele as our guide.

SERIES ACKNOWLEDGMENTS Two individuals have been essential to the conception and execution of this series. Kathleen Keller, our first editor at St. Martin’s Press (where the series originated), initially suggested that we further develop the ideas outlined in Teacher Education and the Social Conditions of Schooling (Liston & Zeichner, 1991). Kathleen was very helpful in the initial stages of this effort and we wish to thank her for that. Naomi Silverman, our current and beloved editor at Routledge, has patiently and skillfully prodded us along, attending to both the big picture and the small details. She has been remarkably supportive and capably informative. We are very thankful and indebted to Naomi. Daniel P. Liston Kenneth M. Zeichner

REFERENCE Teacher Education and the Social Conditions of Schooling, Daniel P. Liston & Kenneth M. Zeichner (New York: Routledge, 1991).

PREFACE

Like the other volumes in the Reflective Teaching and the Social Conditions of Schooling series, Mathematics and Teaching is intended for prospective and practicing teachers. It uses case studies to explore school mathematics and dimensions of culture. This book, like the others in the series, is organized into three parts. Part I includes four cases dealing with issues related to culture and teaching mathematics. These cases are not “authentic” in that they represent actual places and people. Rather, they are constructed from my experiences as a parent, public school teacher, teacher professional developer, graduate student, researcher, and university professor. The cases are representations of my lived experiences in all these roles and in a variety of circumstances that include interactions with colleagues at staff meetings, school improvement days, child study team meetings, parents in and out of school; conversations overheard in school corridors and faculty lounges; conversations with students in my undergraduate and graduate courses; and observations of print and television media, school board meetings, and sometimes participation in school board meetings. These cases are meant to be provocative, to strike a nerve, even discomfort, but mostly they are meant to promote discussion about important schooling issues. A range of prospective and practicing teachers, and other school stakeholders’ reactions to the cases accompanies each case. Part II elaborates the reactions expressed in Part I through the use of “public arguments.” The public arguments organize the claims and xv

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debates expressed through media and school boardrooms into three broad perspectives on issues raised in the cases. For this volume, conservative, liberal, and radical multiculturalist are the labels used to characterize the ways in which people think about the issues illustrated in the cases. Each will be elaborated later in the volume. Part III presents a concluding statement about some of the issues raised throughout this volume. Exercises for further reflection and a bibliography are provided so that readers may pursue further these issues.

THE CASE STUDIES The four case studies in Part I focus on issues related to the social, historical, and political dimensions of culture and teaching mathematics. While each case has a particular focus, each was written to capture the complexity of schooling phenomena. Therefore, various dimensions of culture are expressed in all the cases. Case 1, “Race and Teacher Expectations,” illustrates how a teacher’s biases influence the way she views an African American boy and how her biases may determine the quality of education he will receive. Case 2, “Mathematics For All?,” focuses on middle school teachers faced with a mandate to teach all their students algebra in the 8th grade. Who gets to learn algebra? This is the unstated question with which the teachers must grapple. Case 3, “Culture and School Mathematics,” challenges the reader to consider the relationship between mathematics and its cultural-historical development and how the mathematics taught in schools may disadvantage certain students. Finally, case 4, “Politics and School Mathematics,” emphasizes the impact of policies on mathematics curriculum and instruction. Each case study is followed by a set of reactions written by school stakeholders who were asked to read and respond to these cases. The reactions represent distinctively different ways in which people interpret the issues raised in the cases. It is anticipated that you will react to the reactions themselves. In this way, these varied and multiple interpretations serve to engage you—to help you articulate and clarify your own views about these issues. Between each case study and the reactions, and after the set of reactions for each case study, there is space in the text for you to write your own reactions or reflections. People approach this task differently. Some find it easier to write their reactions after reading the case study; others find it helpful to wait until they have read others’ reactions. You are encouraged to jot down your own reactions in both places. Your task is to examine

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your own beliefs, values, and knowledge about these issues. Such examination is a dynamic process. You will change your mind, see new perspectives, perhaps change your mind back again, or move in a different direction.

THE PUBLIC ARGUMENTS Whereas the case studies in Part I focus on the particularities and concrete renderings of problems and issues, Part II attempts to capture the broader and more general sense of the ways in which these cases can be interpreted. These public arguments are ways of tying together, in a somewhat coherent fashion, the distinct interpretations of the cases. Conservative, liberal, and radical multiculturalist are labels that serve as heuristics to help us think about the different orientations that one brings to the table when discussing culture and teaching mathematics. The word heuristic is important here, since these labels are only approximations of these views. Most likely, there will be disagreement as to what counts as conservative or liberal, and certainly, the meanings attached to the labels change from one cultural-historical moment to the next. Mary Frankenstein (1990), a critical ethnomathematician, fits squarely into the radical multiculturalist perspective as rendered in this volume. However, there may be considerable debate about which label to apply to James Banks (1993), a multiculturalist. Nevertheless, these three public arguments capture and express a synthesis of views that are regularly articulated in the public arena. “Education has become political. It shouldn’t be that way.” This is a common sentiment uttered in my undergraduate and graduate courses. This sentiment is understandable since politicians use educational issues to garner support for their candidacies. Whether we like it or not, education is inherently political; however, it is not solely a political enterprise. First and foremost, education is a moral enterprise, thus our values must inform educational decisions. As well, we must be cognizant of the larger contexts that shape what occurs in classrooms. The articulation of public arguments serves to highlight the distinct but general ways people come to view the issues associated with culture and teaching mathematics. The wiser we are about these issues and the beliefs and values comprising particular views, the more effective we can be in negotiating school contexts in ways that better serve all of our children.

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CONCLUDING REMARKS Part III offers some final thoughts about the issues associated with culture and teaching mathematics. As you will find, challenging taken-for-granted assumptions is of utmost importance in any effort to improve educational conditions. Included in this section are some of the many national organizations concerned with education and children’s issues. They offer opportunities to further elaborate your own developing perspectives about mathematics teaching and culture. Though labels such as conservative, liberal, and radical multiculturalist are used in this volume, these labels are not meant to convey ideological adherence to the issues represented in this book. Rather, they serve as intellectual tools with which to think about these issues. With this in mind, you are encouraged to take seriously the exercises, as they are ideas for teacher research and inquiry. This book does not provide an immediate or definitive resolution to the issues of culture and teaching mathematics. This text was created with the hope that it will inform and provide further direction to those prospective and practicing teachers who want to examine difficult and controversial issues. It is only the beginning of a discussion that should be a central aspect of professional practice. The cases represent difficult issues, but these are issues that competent teachers cannot ignore.

ACKNOWLEDGMENTS

While I was at the University of Southern California students in the elementary mathematics course that I taught during 2001–2002 were kind enough to share their reactions to the cases. Some of their responses and several graduate students’ responses are printed in this book. I would like to thank Etta Hollins for soliciting responses from her graduate students. Michael Eiland and Cheng-Yao Lin were most helpful in conducting library research needed for this book. Alishia Battle had the important role of ensuring that in-text citations were included in the bibliography. Sarah Lubienski graciously took the time to read and comment on an earlier draft of this manuscript even though she was in the process of relocating her family. Lecretia Buckley’s careful reading and attention to precision and consistency of language served me well. Raina Dyer-Barr, with great competence, prepared the index. Though their responses were not included in the book, many University of Illinois students and other school stakeholders were also gracious enough to take the time to respond to the cases. I would like to thank all of these people. For those who allowed me to include their responses and to acknowledge their contribution, I would like to thank them, too: Jasmine Ackerman, Cathy Andrew, Joy Braun, Laura Méndez Barletta, Carmen Cedano, Francine Crockett, Precious Jean Crockett, Cope Cumpston, Ruth Dewar, Kristen Droba, Amanda Eads, Susan Gregson, Mary Jo Heeren, Jay Hooper, Jenny Kim, Kate Meister, Deanna Long, Richard Long, Michael Orosco, Paula A. Schuler, Amanda Schwemin, Carla Sosanya-Tellez, Landon Stenger, Brooke Sternfield, Raol Taft, Rachel Teren, Barbara Trujillo, Lisa Verdick, Lisa Wachel, and Laura Wetzel. xix

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In this volume, school mathematics is the context for exploring the cultural dimensions of schooling. The exploration is a complex one. A work focused on “culture” and teaching mathematics is bound to be controversial as they present difficult issues to engage. For example, what is culture? What is mathematics? On the face of it, these questions seem easy to answer. But, the myriad ways in which these questions are answered give rise to controversy. Culture means different things to different people. For anthropologists it is a contested term and for laypersons common sense definitions abound. Discussions about culture are difficult since it is unlikely that anyone is talking about the same thing. For this volume’s purpose, however, culture refers to the ways in which individuals within communities negotiate their worlds in and through their daily routines, languages, and ideologies. How one thinks and how one goes about everyday activities are inextricably linked to the historically, socially, and politically embedded contexts in which we all live. This definition is meant to dispel some common misconceptions that arise in discussions about culture. Too often, culture and skin color, or, race/ethnicity are treated as synonymous terms. That is, many people believe skin color (or race) specifies one’s culture. While North Americans of African ancestry and South Americans of African ancestry may have the same skin color, they do not necessarily share the same cultures. However, here lies one aspect of the confusion. While skin color is not the same as culture, nevertheless, it is an aspect of culture because it carries 1

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with it a historical legacy that has social and political consequences for groups and individuals that delimit the ways in which they can negotiate their worlds. The black skin of people in the United States carries with it a historical legacy of slavery and White racial hatred encoded in laws, institutions, school curricula, and social discourse that continues to have an impact on the economic, political status and the psychosocial well-being of African Americans. African Americans continue to have fewer opportunities than other groups for high quality educational experiences, high paying jobs, and access to political systems that may operate to improve their social and economic conditions. These cultural circumstances constrain the ways in which African Americans envision themselves as participants in our democratic society. The confusion around culture does not stop here. Most of us had experiences in mathematics classrooms that convey mathematics as a static body of rules and procedures to be memorized. The mathematics, which is taught in schools, is presented as a value-free discipline that Burton (1994) views as “dehumanized, depersonalized and (of course) decontextualized” (p. 71). Our experiences with school mathematics have led us to believe that mathematics is culturally neutral and invented by White males, by which the entire discipline arose all at once many centuries ago (Volmink, 1994). But, what is mathematics? And, who invented it? For many mathematicians, mathematics originated in ancient Greece and they subscribe to a formalist view in which mathematics concerns itself with systematically defining structures such as propositions and ensuring that they are true by using logical arguments that include axioms, postulates, and formal proofs (Dossey, 1992; Zaslavsky, 1990). This “classic” Eurocentric view and the historical development of mathematics have been challenged. Joseph (1991, 2000) explains that this view places the Greeks at the center and the origins of mathematics, with its development interrupted by the “Dark Ages” of some one thousand years, only to be taken up by Europeans, who saw themselves as the rightful heirs of Greek intellectual heritage. This model is the consequence of European supremacist ideology and Europe’s domination in Asia and Africa and one that ignores and distorts the contributions of those outside of Europe. The European model of the historical development of mathematics is untenable, claims Joseph, as a considerable amount of evidence indicates that the development of mathematics was occurring in the non-European world prior to what the Eurocentric model leads us to believe. Joseph (1991, 2000) contests the historiographical bias of common renderings of the development of mathematics. Other scholars (e.g., Bishop, 1988; Burton, 1994) make apparent the cultural embeddedness of

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mathematics. Brazilian Professor Ubiratan D’Ambrosio popularized the expression ethnomathematics (email communication, August 22, 2007), a field that was largely concerned with the anthropological and sociological application of mathematics in the everyday lives of peoples without written expression. In other words, these scholars presume that mathematics is a human activity and study mathematics as manifested in cultural practices and the social functions that mathematics serve. For example, Zaslavsky (1990) has written about the mathematics of Africa, including its counting systems, currencies, record-keeping systems, and geometric forms in architecture. More recently, ethnomathematics has become concerned with broadening the view of mathematics and investigating its social and political implications. Knijnik defines an ethnomathematical approach this way: [This approach is] the investigation of the traditions, practices, and mathematical concepts of a subordinated social group and the pedagogical work which was developed in order for the group to be able to interpret and decode its knowledge; to acquire the knowledge produced by academic mathematicians; and to establish comparisons between its knowledge and academic knowledge, thus being able to analyze the power relations involved in the use of both of these kinds of knowledge. (Knijnik, 1997, p. 405)

For mathematics educators who advocate this approach, a goal of school mathematics reform is “reclaim[ing] the hidden and distorted histories of the contributions of all cultures to mathematics” (Frankenstein & Powell, 1997, p. 74). Another goal, as Knijnik (1997) expresses, is to interrogate the power relations inherent in academic knowledge and the knowledge of subordinated groups. Who gets to learn mathematics? How is mathematics used to perpetuate social inequities? Questions like these, claim ethnomathematicians concerned with power relations, should be addressed in mathematics classrooms. Given that conventional notions of mathematics are contested, it should be no surprise that the jury is still out on what counts as school mathematics, mathematics teaching, and mathematics learning. “Traditional mathematics” versus “reform mathematics” is a common way, albeit an oversimplification, to characterize the contentious debates in mathematics education. There continues to be much sparring between these two camps. Those in the traditional camp have referred to reformers as proponents of “fuzzy” mathematics. Those in the reform camp have referred to traditionalists as proponents for “drill-and-kill” mathematics. Nevertheless, the

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dominant approach to the teaching of mathematics in the U.S. is welldocumented (e.g., Stoldolsky, 1988; National Center for Education Statistics (NCES), 2003). In 2000, a commission headed by John Glenn was charged with investigating and reporting on the quality of mathematics and science teaching in the nation. The commission’s findings and recommendations are summarized in Before it’s Too Late: A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century (U.S. Department of Education, 2000). In this document, traditional practices are described as “numbingly predictable” (p. 20). That is, the dominant approach to teaching mathematics lessons usually proceeds in a fashion similar to the following: (1) a review of previous material and homework, (2) a problem illustration by the teacher, (3) drill on low-level procedures that imitate those demonstrated by the teacher, (4) supervised seat work by students, often in isolation, (5) checking of seatwork problems, and (6) assignment of homework. (U.S. Department of Education, 2000, p. 20)

This approach was evident in a recent videotape study of 8th-grade mathematics classrooms in the United States that was part of the Third International Mathematics and Science Study (TIMSS) (e.g., Stigler & Hiebert, 1999; National Center for Education Statistics (NCES), 2003). The problem with the traditional approach, according to reformers, is that it does not facilitate the development of mathematical reasoning and problem solving, nor does it address the conceptual underpinnings of the procedures we expect students to learn. An alternative view of mathematics teaching and learning is embodied in the policy documents produced by various organizations (e.g., U.S. Department of Education, 2000; National Research Council, 2001). Most attention has been given to the documents produced by the National Council of Teachers of Mathematics (NCTM), collectively known as the Standards (1989, 1991, 1995, 2000). A primary goal in reform mathematics is that students have conceptual understanding of mathematical ideas. For example, the pre-K-2 number and operations standard recommends that students “understand meanings of operations and how they are related to one another” (NCTM, 2000, p. 32). According to the Principles and Standards for School Mathematics (PSSM) (NCTM, 2000), this understanding comes through solving simple arithmetic problems with considerable attention to the strategies that young children employ when they solve these problems. Because they will generate a variety of

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strategies, these strategies should be shared and discussed. The PSSM offers this example: Bob got 2 cookies. Now he has 5 cookies. How many cookies did Bob have in the beginning? To solve this problem, young children might use addition and count on from 2, keeping track with their fingers, to get to 5. Or they might recognize this problem as a subtraction situation and use the fact that 5 – 2 = 3. Exploring thinking strategies like these or realizing that 7 + 8 is that same as 7 + 7 + 1 will help students see the meaning of the operations. Such explorations also help teachers learn what students are thinking. Multiplication and division can begin to have meaning for students in prekindergarten through grade 2 as they solve problems that arise in their environment, such as how to share a bag of raisins fairly among four people. (NCTM, 2000, p. 34)

The PSSM further explains that researchers and experienced teachers find that when children are encouraged to develop, record, explain, and critique one another’s strategies for solving computational problems, a number of important kinds of learning can occur [. . .]. The efficiency of various strategies can be discussed. So can their generalizability: Will this work for any numbers [sic] or only the two involved here? And experience suggests that in classes focused on the development and discussion of strategies, various “standard” algorithms either arise naturally or can be introduced by the teacher as appropriate. The point is that students must become fluent in arithmetic computation—they must have efficient and accurate methods that are supported by an understanding of numbers and operations. “Standard” algorithms for arithmetic computation are one means of achieving this [computational] fluency. (NCTM, 2000, p. 35)

These kinds of learning experiences, say math reformers, provide the foundation for understanding rational number concepts. If students understand operations on rational numbers, then students should have basic proficiency with the real number system that is encountered in high school mathematics. However, math traditionalists see these kinds of experiences quite differently. Mathematically Correct1 is a parents’ advocate group that

1 See www.MathematicallyCorrect.com.

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opposes the reforms embodied in the PSSM. On the homepage of their website, they dispute many of the reform practices explained above. They write: The advocates of the new, fuzzy math have practiced their rhetoric well. They speak of higher-order thinking, conceptual understanding and solving problems, but they neglect the systematic mastery of the fundamental building blocks necessary for success in any of these areas. Their focus is on things like calculators, blocks, guesswork, and group activities and they shun things like algorithms and repeated practice. The new programs are shy on fundamentals and they also lack the mathematical depth and rigor that promotes greater achievement.

The advocates support The Core Knowledge Sequence, which is a curriculum for preschool through grade 8. These curricula were developed by the Core Knowledge Foundation which was founded in 1986 by Professor E. D. Hirsch, Jr. Mathematically Correct advocates believe that “[e]very successful program for teaching math to young people follows these three cardinal rules for early mathematics education: 1) practice, 2) practice, and 3) practice.” In addition to general introductions to the topics found in each book there are specific introductions to the grade level materials as well as extremely detailed and example-filled descriptions as to what mathematics skills children should know at each grade. Specific notation is made of the fact that the material in these volumes is advanced relative to what many American schools require, but is aligned with what is taught in countries that are far more successful at teaching math than we are [italics in original text]. The material is clearly presented and contains a substantial number of examples that a child could reasonably be expected to do. Although the math section is extensive (e.g., 65 pages in the 3rd grade book), it is not a substitute for a complete curriculum.

The recent U.S. historical context for these alternative views and their subsequent controversies depicted in Mathematics and Teaching began with A Nation at Risk: The Imperative for Educational Reform (1983), a report produced by The National Commission on Excellence in Education. This report boldly proclaimed that “the educational foundations of our society are presently being eroded by a rising tide of mediocrity that threatens our very future as a Nation and a people” (p. 5). This influential report gave rise to two waves of reform. The first wave sought to reverse the “rising tide of mediocrity” with a focus on increased standards. It

INTRODUCTION

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brought recommendations that included core curricula, computer literacy, and lengthening the school day and academic year. The second wave of reform addressed a wide variety of issues including teacher salaries, leadership programs for school administrators, parental choice, school report cards, school accountability, and poor families. Two decades of school reform culminated in the first federal mandate to improve the education of all students in U.S. public schools with its focus on “standards-based learning” and “accountability” to eliminate the racial and socio-economic gap in academic achievement. This is a noble and democratic goal. It is consistent with the goals of reformers who have addressed inequitable schooling practices that have had negative consequences for women, the poor, and students of color. For these groups, these practices have limited opportunities to pursue advanced mathematics courses in high school and college, which, in turn, limit employment prospects. “Opportunity for all,” “opportunity to learn,” “mathematics for all,” and “equity and excellence” are catchphrases that pepper policy statements addressing academic disparities in school mathematics. Yet, despite the call to ameliorate inequitable practices, academic disparities between groups still persist in the U.S. Results from the National Assessment of Educational Progress (NAEP) (Lubienski & Crockett, 2007) indicate that while all racial-ethnic groups have experienced positive gains in mathematics proficiency, a disturbingly large gap remains between White students and Black or Latino students. Though democratic principles are compelling reasons for reenvisioning school mathematics, the reform movement has proceeded with little attention to the historical, social, and political contexts in which all issues of schooling reside. Since language or socio-economic differences alone cannot account for the persistent and chronic low achievement of certain groups of students, the lack of attention to cultural contexts may explain, in part, the disturbing achievement gap between White students and Black or Latino students. Many questions arise: “What mathematics should be taught in schools?” “Should all students take college preparatory mathematics?” “If not, then who should and why?” “What does culture have to do with teaching mathematics?” “Whose culture will be facilitated?” “Does it matter?” These questions point to multicultural education and social justice concerns and their implications for schooling in a democracy. Do we continue to teach a view of mathematics that perpetuates the myth that European males were the sole creators of mathematics? Or, do we teach a kind of mathematics that has been reconstituted with its cultural origins? If so, what do such curricula look like? If we limit access to advanced

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mathematics, what are the implications for democratic schooling practices and the culture at large? These questions are not easily answered and the answers to these questions are not neutral. These answers have cultural and historical origins that explain particular beliefs and values about the purposes and outcomes of schooling. Since stakeholders bring different sets of values to the decision-making process, decisions about school mathematics are moral, ethical, and political, fraught with contention and strife.

I CASES AND REACTIONS

INTRODUCTION TO CASE 1 In the United States, The Education for All Handicapped Children Act (Public Law 94-142) was passed in 1975, requiring states to provide “a free, appropriate public education for every child between the ages of 3 and 21 regardless of how, or how seriously, he may be handicapped.” In 1990 the Individuals with Disabilities Education Act (IDEA) was passed as an amendment to Public Law 94-142. The amendment extended services for students with disabilities. For example, it provided services for students with ADHD; and it replaced the word “handicapped” with “disabled.” These laws were intended to address the educational rights of children long neglected by our public school system. However, these federal guidelines have unintended consequences. Minority students are over-represented in special education programs throughout the United States. Misdiagnosis is a widespread practice (Harry & Anderson, 1994). Policymakers assume that “clinical” decisions that label children for special education services come about in some rational way. In fact, the decision-making processes are far less clinical than many of us care to acknowledge. Case 1 is about Benjamin, an African American boy in the 3rd grade. It illustrates how his teacher’s beliefs about race, class, and gender interact in ways that produce labels that may not operate in Benjamin’s best educational interests. 9

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CASE 1: “RACE AND TEACHER EXPECTATIONS” As soon as the dismissal bell rang, Benjamin Sykes, an energetic African American boy, darted for the door, backpack in tow, nearly taking out Maisha who sat in the desk near the exit. Mrs. Carlton sighed. She knew Benjamin’s action warranted a reprimand, but it was Friday. She was exhausted and ready to go home. From the moment he walked into her 3rd-grade classroom, Mrs. Carlton knew he was trouble. It seemed he couldn’t sit still for more than two minutes and she often placed him on “time out” for being out of his seat. It bothered Mrs. Carlton that his 2ndgrade teacher had not referred him for special education services. Now, she was stuck with another low achieving student. This was particularly annoying, since considerable district pressure was being placed on the teachers to raise test scores. Thank God for Anna, Mrs. Carlton muttered to herself as the last of the bunch noisily vacated the classroom. Anna seemed to have a calming effect on Benjamin and four other little Black boys in her classroom. Anna had been placed in Mrs. Carlton’s 3rd-grade class three weeks ago. It was the first trimester of her student teaching practicum. For two hours each morning, she observed the teacher’s instruction, management strategies, and assisted students with their work. With each passing week, her observations became more astute as her university professors posed questions about the experiences. Before leaving campus, she spent about thirty minutes writing down her thoughts and trying to address questions raised by her professors. The assignment she liked best was developing a case study focused on the math experiences of one student. The case study involved gathering information about the student’s math achievement in school, as well as the mathematics the student encounters in the lunchroom, on the playground, and at home. The professor suggested that the prospective teachers spend one or two evenings at the student’s home interviewing parents, siblings, and shadowing the student. Initially, Anna worried about getting permission to go into a child’s home. However, this proved easier than she thought possible. The child she decided to study was Benjamin. Benjamin’s mom was especially agreeable once she found out that Anna was the “new teacher” at school, the one who had been tutoring her son. Anna had become accustomed to the complaints Mrs. Carlton aired during the morning recess. At first, Mrs. Carlton’s complaints seemed reasonable. Anna could see that some of the children were often off-task, not following directions, and not completing assignments. Some were even rude, yelling out answers and talking out of turn. Yes, Anna agreed

CASE 1: “RACE AND TEACHER EXPECTATIONS”

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that this was inappropriate behavior. How could one teach under such circumstances? Little by little, however, Anna began to recognize that Mrs. Carlton’s management skills left something to be desired. She contributed to the bad behavior that left her so exasperated. Anna observed a math lesson on multi-digit addition with regrouping. As Mrs. Carlton explained step-by-step how to add the digits, few of the students seemed to pay attention. Two students were poking each other with pencils, another secretly, so she thought, played with her Barbie doll, while several others stared out the window as a custodial crew painted white lines on the playground’s black top. Mrs. Carlton continued as though unaware of the students’ lack of attentiveness. It wasn’t until she assigned them problems from the textbook that chaos ensued. Some students didn’t have their books and many complained that they didn’t understand the word problems. “Weren’t any of you listening?” she shouted. “This stuff is easy. Just go step-by-step starting in the ones place. You can do the word problems if you just read them!” Mrs. Carlton was more than agitated. Mrs. Carlton frequently complained about parents. “These parents aren’t like the ones we used to have. If they would discipline their children and see to it that their children did their homework every evening, my job would be a lot easier. These parents don’t care. I’m not trying to be insensitive, but if the parents are low quality then their children are low quality.” Anna guessed that the “low quality” parents were Black parents. It was likely that Mrs. Carlton and other teachers in the school weren’t used to having Black children in their classrooms. In recent years, professional and working-class parents fleeing the pressures of urban life moved to the fast-growing suburban community where housing was more affordable. For the first time, the community saw an increase in its Black residents. Nevertheless, Anna was growing weary of Mrs. Carlton’s constant indictments of the children and their parents. Many of them flew in the face of her classroom experiences and knowledge of the children’s backgrounds. Because of the case study she was developing about Benjamin, Anna wondered if Mrs. Carlton knew anything about the lives of the students she taught. For example, Mrs. Carlton insisted that Benjamin was a low achiever. She seemed to think he didn’t know his basic facts. When Anna examined his cumulative file, his test scores from 2nd grade indicated that he was at grade level. In fact, the grade equivalent score was 6.2 and 4.8 for mathematics basic skills and problem solving respectively. This information was consistent with her tutoring experiences with Benjamin. He resisted doing the daily math assignments, but with her urgings, he always successfully completed them. He knew his basic facts like the back of his hand.

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His resistance, Anna speculated, was because he found it boring to do 20 computation problems a day. On two occasions, Anna observed Benjamin on the playground. Basketball seemed to be his game. When Anna talked to him about it, she marveled at the numerical data he kept in his head. He knew the scores of each team for every day that week, how many points each team member made, how many fouls were committed and by whom. He talked about who was the best free throw shooter and who could make the best lay ups. Anna spent her first evening with Benjamin’s family several days ago. It was laundry night. Anna followed Benjamin and his mom into the basement of their apartment building. She watched Benjamin count out 12 quarters for three loads of laundry. As his mother loaded the clothes into the washers, Benjamin carefully poured liquid detergent into a measuring cup. Anna asked, “How much are you pouring in?” “One fourth cup,” he replied, putting the cup close to her face before pouring it over the clothes in the washer. Anna was fascinated by Benjamin’s family life. She was pleased when Mrs. Sykes agreed to a second visit. Mrs. Sykes was a single parent, but from what Anna could tell, Benjamin’s home life was reasonably stable. He had an after school routine that included homework and chores. Once a week he helped his mother do laundry. Her growing knowledge about Benjamin and his family caused her considerable concern. Benjamin demonstrated mathematical competence in tutoring sessions, on the playground and at home. Why was he not engaged in Mrs. Carlton’s math lessons? Why did Mrs. Carlton insist that he had limited capacity to learn? She threatened to refer him for special education services. Based on his school records and her observations, she could not see how Mrs. Carlton could justify such a referral. As far as Anna was concerned, what Mrs. Carlton needed was an attitude change. And, to put it mildly, her teaching was uninspired. She wondered if she should say something. Should she share her case study with Mrs. Carlton once it was done? Maybe she’d think better of the students after reading it, or, at least of Benjamin.

READER REACTIONS TO “RACE AND TEACHER EXPECTATIONS”

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REACTIONS TO “RACE AND TEACHER EXPECTATIONS” There were a wide variety of responses to this case. Many respondents grappled with Anna’s dilemma. How and to what degree should she serve as Benjamin’s advocate? For some, it was clear that Anna should share her findings with Mrs. Carlton. Some thought she should proceed cautiously, given her tenuous status as a classroom observer. Other respondents strongly identified with the events in the case as they paralleled their own student teaching experiences. Some respondents pointed to classroom management issues as the reasons for Mrs. Carlton’s difficulties with children like Benjamin. One respondent who identified Mrs. Carlton’s attitudes as racist said it was the result of “teacher burn-out.” Two veteran educators commented on the challenges that Benjamin’s situation presented to the educational system and what might be done.

Anna’s Dilemma Many respondents thought that Anna should share her case study with Mrs. Carlton, emphasizing an approach that is both professional and cautious. One respondent explains that sharing the case is the right thing to do if Anna has any integrity at all. I definitely think that Anna should share her findings with Mrs. Carlton. Who knows how many students there are like Benjamin who may simply require extra attention or a more intriguing way of learning in order to succeed in the classroom, but instead are being cast off to the side and given up on? It is certainly a teacher’s job to help her students understand what is going on and make use of constructive approaches to accomplishing a lesson. If a teacher does not care whether her students take anything from the lesson, why should the students bother trying to learn it? Furthermore, if a teacher gives up and reprimands for not understanding a problem the first time around, how is a student supposed to learn anything? These are questions that should be offered to Mrs. Carlton to address. —Undergraduate, Prospective Teacher

I think that Anna should most certainly show her observation results to Mrs. Carlton. Anna has come across a wonderful discovery that Benjamin is doing well with math at home and outside of school, and it is important that the teacher knows this. Anna should approach the teacher in a professional,

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non-accusatory manner in order to get this situation cleared up. It could very well be that Mrs. Carlton holds some prejudice against her Black students, or somehow stereotypes them into learning categories without examining the testing information. —Undergraduate, Prospective Teacher

Although it might be helpful for the student teacher to let the teacher read her paper, she needs to step lightly. If she were to show her the responses, it would contain a substantial amount of proof, as this teacher seems particularly stuck in her old ways. Additionally, it could be problematic during the rest of the semester if the master teacher felt threatened by Anna and her observations. Congratulations are in order for Anna for detecting such a small but important problem in this classroom. —College Senior, Student Teacher

Anna’s reaction to Benjamin Sykes is commendable. Her case study is extremely well documented to reveal a racial bias on his teacher’s side. The dilemma is should Anna make her master teacher aware of Ben’s ability outside of the classroom? I think if she really has integrity she should. But the way she presents the case study should be very low key. Perhaps she could ask for a meeting at the end of a “good” day and share her findings in a very positive manner. She would have to be careful not to make Mrs. Carlton defensive. —College Senior, Student Teacher

A Common Story For several respondents this case was similar to their own student teaching experiences. One student teacher acknowledges that race may have been a factor, recognizing that minority students are disproportionately represented in special education classes. Another considers the possibility that Mrs. Carlton needs to improve her teaching practices and brush up on classroom management skills. However, class is another social category that comes into play when teachers make decisions about their students. A retired family life educator tells about her student teaching experiences at an all White school, while another student teacher tells about her experience in a classroom with an African American teacher and an African American student she sought to help.

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The thought that there are teachers such as Mrs. Carlton, labeling children on the basis of ethnicity as being low achievers is frightening. I have witnessed this in the teacher’s lounge at my current placement for student teaching. While engaged in a discussion concerning how their students “just don’t get it” a few teachers have come close to making racist remarks. They stop short however, turn, look at me, and realize that yes; I have in fact been listening to their conversation. Teachers can use their authority to permanently scar children for the duration of their academic experience. —College Senior, Student Teacher

Unfortunately, I think that this happens with a significant number of minority students. Students who do not belong in special education are referred or placed there because the teacher does not understand why students are behaving or reacting the way they do. Teachers often make the mistake of labeling students as low achievers without really formally assessing their abilities. All Mrs. Carlton had to do was look in his file to see that he was not a low achiever. But she already had her mind made up the moment he walked in the class. She said that automatically she knew he was trouble. She is letting her own stereotypes cloud her view of this student. It seems like Mrs. Carlton had low expectations for the minority students in her class and she blamed their behaviors on everyone but herself. If she were to look at some of the principles of learning like motivation, feedback, and transfer, she would see why the students were not responding. Instead of just referring a “problem child” for special education, and looking for a way to get him out of class, she should have looked at how she could help this student. Also, she would greatly benefit from a refresher course on classroom management. Classroom management can sometimes be the most important variable in whether or not students learn. —College Senior, Student Teacher

These case studies remind me all too much of my student teaching days. My cooperating teacher seemed to believe her task was to PROVE to those awful students just how STUPID they really were! The coordinating professor seemed to agree with and reinforce her attitude. Here is a direct quote, “Just LOOK at all I have taught and THEY have resisted!” It seemed to me that a teacher’s job was to find the best way to reach the students; to start where they were, and reach out to them; but I was just a lowly student teacher. How was I to know anything? The issue was not race [in my student teaching experience] as there was de facto segregation; hence, an all White school. Mrs. Boyle really HATED having lower I.Q. students in her science classes, and tried to direct them into other things like home economics for the girls and industrial arts for the

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boys. More stereotyping!! Teachers of those classes were furious that their classes were used as a “dumping ground!” And all this BEFORE teachers were evaluated based on their students’ test scores!! The complaints about parents! I’ve heard them all! Again, an all White school, but the teachers could hardly blame and bad mouth the parents enough. Child rearing is the most difficult and complex job a person could ever have; yet there is a silly notion that everyone should know exactly how to do it instinctively! This may be the main reason why parenting education and support became my calling, if you will. Parenting education became my main specialization. Anna could allow Mrs. Carlton to read her case study, but if she were to suggest the problem COULD be Mrs. Carlton’s uninspired teaching, Anna could fail her internship and not ever get a good recommendation from her student teaching supervisor. —Retired Family Life Educator

I have to say that this case study fascinated me from the beginning because this could easily be one of my students in my old master teacher’s classroom. Benjamin could just as well be Roy, and Mrs. Carlton could be my master teacher. Benjamin sounds just like Roy, who knew all his math facts, but couldn’t sit still during class. My master teacher only used worksheets, and Roy would never complete them. My master teacher wanted to refer him to special education because of his lack of attention. I think she thought he had attention deficit disorder or something to that effect. I think he just did not like doing worksheets. I tried to be his advocate, much like Anna was to Benjamin. I tried to defend him when he did work with me, but in some ways it wasn’t really my place, and now that I am not even in that classroom anymore, I don’t know what good I did. The major difference between this case and my own, is that this case seems to be a racial issue, whereas in my case, both parties were African American. While my master teacher hinted that the parents were “low quality” it wasn’t because of race. It was simply because these parents didn’t place education as a top priority of their children. In this case, it does sound like it could be a race issue because she says, “these students aren’t like the ones we used to have” implying that the ones they used to have were White. —College Senior, Student Teacher

Discipline or Pedagogic Problem? The research literature is replete with studies documenting school failure between Anglo, African American, and Latino students. But as this case

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illustrates, mere statistics do not provide an understanding of how the failure is achieved. The mathematics lesson described in the case is an all too common occurrence in classrooms. In the first response that follows, the respondent believes that Mrs. Carlton facilitates failure in a variety of ways. She does not encourage students to be successful, blaming the students themselves for their academic failure. The second respondent is troubled by Mrs. Carlton’s special education referral as a first response to Benjamin’s problems. The respondents’ comments suggest that his problem is a pedagogic problem rather than a discipline problem. Teachers either give their students permission to fail or demand success. It seems the first problem to look at in this case is Mrs. Carlton giving her students permission to fail, especially Benjamin. Mrs. Carlton does not demand success from her students, meaning she does not encourage them to perform, as Anna does. When students get loud, confused, and are shouting out questions, Mrs. Carlton simply says, “Weren’t any of you listening? . . . This is easy stuff.” She puts the blame for lack of understanding on the children, and she never considers that she may be the one at fault. When Anna encourages Benjamin, he can complete his assignments, and he does so with ease. The case states, “He resisted doing the daily math assignment, but with her [Anna’s] urgings, he always successfully completed them.” Maybe these students just need some encouragement, like Benjamin, perhaps they also need to see that Mrs. Carlton cares. Anna obviously cares about Benjamin, and she wants to see him succeed, for this is why she urges him on. Mrs. Carlton needs to stop giving her students permission to fail by not reviewing and answering their questions, and she needs to start demanding success from her students as Anna did with Benjamin. —Graduate Student, Prospective Teacher

Why do teachers want to resolve their management problems by referring students to special education? I raise this question after reading this case study because I think that many management problems rely on the teacher not being able to implement management and discipline within the classroom. It seems to me that some teachers use special education referrals as their first resource to solving problems with one or two students because they do not want to deal with these students as in Mrs. Carlton’s case where she was upset that Benjamin Sykes’s 2nd-grade teacher did not refer him to special education services. —College Senior, Student Teacher

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One strategy for Mrs. Carlton is to learn more about her students by observing their lives outside of school. Although this may seem like “too much work,” it is important to keep in mind that it is dangerous not to do so. From one perspective, not seeing how kids think and act in non-school settings may account for many unnecessary referrals to special education programs, thus decreasing the likelihood of children succeeding in school. From another perspective, if the child is simply ignored in class and not referred to special education, it may dramatically increase the child’s chances of dropping out of school due to boredom and/or feeling out of place. A second strategy would be for Mrs. Carlton to change her teaching techniques to create a more collaborative classroom environment. Finally, Mrs. Carlton needs to learn to distinguish learning disabilities from cultural differences. This can be achieved by taking courses to learn about disabilities and by taking time to learn about students’ lives, family, community, and culture. —Doctoral Student Benjamin is a failure because his behavior does not fit into Mrs. Carlton’s socio-cultural paradigm. This is a fine example of how children from different socio-cultural backgrounds are looked upon in America’s school system, instead of the teacher trying to change. This is an ongoing dilemma in public education because many classroom teachers, who tend to be White, ignore their students’ socio-cultural backgrounds and only impose their Eurocentric views. —Doctoral Student

Teacher Burn Out One respondent attributed Mrs. Carlton’s racist attitudes to “burn out.” He offers several recommendations, including retirement. My reaction to the racist expectations of Mrs. Carlton is extreme pity. I feel sorry for the students she expects so little of and also I feel sorry for her. I can’t imagine what teaching would be like if you really hated it as much as Mrs. Carlton obviously does. She sounds like an older teacher who is burnt out and should retire. However, in her defense, maybe her classes have become “lower” and require the level of energy to keep their attention that she no longer possesses. Maybe she should have a permanent aide to help control her classroom management problems and allow her to interact on a more personal level with fewer students. —College Senior, Student Teacher

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Educational Challenges and Possibilities The situation in Mrs. Carlton’s mathematics classroom represents two major challenges to the educational system. First, there is the challenge of the urgency of providing a high quality education to Benjamin that allows him to be successful in both mathematical process and content. He is clearly a mathematical problem solver and has procedural skills, as well, as seen by his ability to count change, compute facts, and measure in everyday settings. Benjamin does not have time to lose in his trajectory toward being successful mathematically, because he is already receiving messages that he cannot be successful at school. The student teacher has data suggesting that he needs to be challenged to do higher level mathematical thinking, employing the skills he already has in place to solve more complex problems. I feel a sense of urgency for Benjamin, and children like him for whom formative assessments are not in place and whose ability is not recognized because the classroom environment and activities may be designed to overlook them. In this sense it is more than the responsibility of the teacher, Mrs. Carlton. It is the responsibility of the principal/district to bring to bear whatever resources it can to change the way this mathematics classroom functions by training teachers to deliver a standards-based curriculum in an environment conducive to critical thinking, problem solving discourse, and constructivist learning. The second challenge, then, is how to do that. Professional development takes time and money. Mrs. Carlton needs to know that there is another way and that the methods and approach she is using are not meeting the needs of all students. At the same time, she will need the tremendous support of a collaborative community in which she can come to understand her own mathematical understandings and beliefs. —Retired Teacher and Principal, Dual Language Elementary School

Any educator who holds on to deficit models does so to mask fear or to insulate themselves from knowledge they themselves hold that they are not reaching students as before. Even with this behavior, most teachers care about their students and wish to do well by them. Ever hopeful, I believe that we need to be “bridge builders”—bridging the gap between teacher and student realities. In my experience as a principal, I have been able to conduct a kind of “peer mediation” between agitated students and mystified teachers. This loving act seeks to help each see the other’s point of view and/or reality. Together with extended training opportunities, this kind of simple but embedded practice can begin to shift the culture of a school—or of school for our students of difference. These challenges have no quick fixes. To realize transformative learning in schools will take the will and belief to assume that all students reach their potential. It

REACTIONS TO “RACE AND TEACHER EXPECTATIONS”

will demand that we learn from and with our students, and so much more. Finally, all of us who love education must be learners at our own level of influence. If a cafeteria worker can learn some Spanish and try to reach out to a hungry child, then that act will have a ripple effect. The power of one caring act cannot be underestimated. Cumulatively, caring, loving acts consistently chosen are our best act in our quest to create a beautiful democratic education for all students. —Principal

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SUMMARY AND ADDITIONAL QUESTIONS

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SUMMARY AND ADDITIONAL QUESTIONS “Race and Teacher Expectations” raised a number of issues. Respondents grappled with Anna’s dilemma. Should she share with Mrs. Carlton her knowledge of Benjamin’s mathematical competency both in and out of school? The respondents related their own personal stories that acknowledged the all too common practice of referring children for special education services without ever considering teaching practices as part of the problem. How does race, class, and gender play a role in who gets referred for special education services? One particular concern cut across many of the responses: Was this a discipline problem or a pedagogic problem? Here are some other questions to consider. 1 What should Anna do? Why? 2 On what grounds did Mrs. Carlton base her decision to refer Benjamin for special education services? Were these reasonable grounds? Why or why not? 3 What might be the educational and socio-emotional consequences if Benjamin receives a special education label? 4 What does Mrs. Carlton need to know and do if she wants Benjamin and children like him to be successful in her classroom? 5 What is the administration’s role in helping Mrs. Carlton become an effective teacher for Benjamin? What is the administration’s role in ensuring that Benjamin is academically successful? 6 What might be the knowledge and experiences that pre-service teachers need to have if they endeavor to be successful with all children? What is the democratic rationale for this set of knowledge and experiences?

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INTRODUCTION TO CASE 2 While mathematics achievement is one of the nation’s priorities, improving the quality of mathematics teaching and learning has been an enduring effort. In recent years, this effort has also addressed the need to provide opportunities for all children to learn high quality mathematics. Many states in our nation have focused on algebra since it is the primary gatekeeper for college-preparatory mathematics and science. Many students of color lack adequate preparation for algebra. Many who enroll are unsuccessful. Those who never enroll and those who are unsuccessful are faced with limited educational opportunities. Despite recent efforts, the results from the National Assessment of Educational Progress indicate that the mathematics achievement gap between White and Black or Latino students remains disturbingly large (National Center for Education Statistics [NCES], 2001; Lubienski & Crockett, 2007). In this case, the teachers face a state mandate that requires them to teach algebra to all eighth graders. It is often assumed that saying it will be so, will make it so. What is often overlooked about policy making at the federal, state, or local level is that such decisions involve socio-cultural processes, which quickly bring to light stakeholders’ belief systems. These belief systems play a significant role in the forms that mandates take in practice. The teachers in this case make manifest this process as they face the implementation of the state mandate. The decision to teach algebra at the 8th grade raised a number of issues for the teachers. In “Mathematics for All,” equity and race-ethnicity are expressed as entangled concepts as the teachers grapple with their beliefs about their students’ ability and algebra learning.

CASE 2: “MATHEMATICS FOR ALL?” For years Orange Valley Middle School had been regarded as the city’s equivalent of a private elite school. It boasted of its high test scores, its rigorous curriculum, and the fact that it was a feeder school into the city’s prominent high school that served primarily the White children of doctors, lawyers, and professors of the local university. Orange Valley’s students had come from middle class homes, some of modest wealth. Until recently, the school had served a predominantly White student population. Even today, its teaching staff comprises mostly White teachers. In recent years, the community has undergone what old timers in the neighborhood and veteran teachers perceive as rapid demographic changes.

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In reality, the changes were more gradual. Elderly members began moving into retirement communities and many families began to “move up,” buying into more expensive neighborhoods. As these families moved out, working- and middle class African Americans and Latinos, and some White first-time buyers moved into the neighborhood. At the same time, the school district instituted boundary changes to accommodate enrollment increases in other parts of the city. The new boundary included Latinos from poorer parts of town. As a result, Orange Valley saw an increase in the number of students who are African American and Spanish-speaking immigrants. Orange Valley teachers expressed concern about these changes as they saw a drop in achievement scores. While the achievement scores were far from abysmal, the teachers viewed the decline as a sign of impending doom. Orange Valley’s stellar reputation began to wane. Even though White students were 60 percent of the school’s population, some community members and school professionals began calling Orange Valley the “minority” school. What seemed to complicate matters for teachers was the recent onslaught of legislative mandates. These mandates called for class-size reduction, new promotion standards and required mandatory testing and accountability systems for districts and teachers; and they called for an end to bilingual education. The mandates produced a flurry of district policies from which the teachers were reeling. On top of everything else the state insisted that all eighth graders take algebra I. Mr. Wilson was particularly agitated by all of this. He’d been teaching for thirty years and taught honor mathematics for eighth graders, an algebra I course. Since he was the chairperson of the mathematics department, he was responsible for developing a proposal addressing how the department was going to ensure that all 8th-grade students met the algebra requirement. He called a meeting and instructed his student teacher, Tara, to be there. As she was leaving for the day, Mr. Wilson said, “This meeting will give you a first hand look at how education is going to hell in a hen basket.” Tara arrived promptly at the meeting, soon joined by eight other 6th-, 7th-, and 8th-grade math teachers. Despite Mr. Wilson’s cryptic comment from the previous evening, she anticipated a discussion on how best to restructure the curriculum so Orange Valley students could be ready for algebra by 8th grade. Instead, she witnessed something altogether different. Mr. Wilson began speaking. “As you know, we will have to start teaching all eighth graders algebra I. I think this is a crock! How in the world can you teach all students algebra?” Other teachers chimed in. One of the 6th-grade teachers complained that many of her students didn’t know their basic math facts. Another concurred adding that she had

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students who couldn’t even speak English. Several teachers wanted to know that if students don’t know their basic facts and can’t speak English, then how could they be taught algebra? The discussion continued. Mr. Jones taught 7th-grade mathematics. Like most of the teachers in the room, Tara found him pleasant and genuinely concerned about his students’ needs. Usually, he wasn’t one to speak during meetings. I have trouble with what the state wants us to do. I think it will be very harmful. Not all kids are capable of learning algebra, just the lower level skills. It’s harmful to their self-esteem to force algebra on them. Take Rojelio, for example. He’s a kid in my class, very average. But his father is a migrant farmer. This kid isn’t going to college. He’s going to end up picking lettuce just like his father. It’s unfair to make him take algebra. He’s never going to use it.

All the teachers seemed to agree with Mr. Jones. They raised issues of self-esteem, increasing the dropout rate, and the drain on school resources as the negative outcomes to implementing the state’s new requirement. Without any discussion on a new curriculum proposal, Mr. Wilson concluded the meeting saying, “At the rate we’re going, the only thing we’ll be able to offer here is remedial math.” There was some mumbling from teachers, seemingly in agreement, as they made their way out the door. Tara listened to the entire discussion without comment. She was both taken aback and fascinated by what the teachers said. She tried to make sense of the teachers’ very negative reactions to planning an algebra program for all eighth graders. She realized that the teachers’ complaints centered on minority students. She wondered if the teachers had ever taken bilingual methodology courses. Although, if they had, she pondered, bilingual education had never been an issue at the school since its population had always been mostly White and English speaking. She wondered, too, had they learned ways of developing algebraic thinking in the early grades? On the other hand, if they hadn’t learned these things, then they had a good point. How do you teach algebra to children who don’t know their facts or speak English? Even though these issues had been addressed in her courses at the university, she realized that she had no practical experience dealing with them. After all, she had attended predominantly White schools, could not recall having a teacher of color, and she was doing her student teaching in Mr. Wilson’s honors 8th-grade class. The students were all White. Perhaps, she really didn’t understand the realities about which the teachers spoke so emotionally. Tara began the meeting thinking it was a good idea to teach algebra to

CASE 2: “MATHEMATICS FOR ALL?”

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8th-grade students. What was so special about algebra anyway? Surely, every educated person in the United States should be able to solve a basic algebra problem. Could Mr. Jones be right? Is it the case that some children are incapable of learning algebra? But, she also wondered if the teachers were concerned about teaching algebra to minority students or to all students? Not all White students took algebra. Only honors students did. But then again, most college bound students would take it in 9th grade. If we make all students take algebra will education be “going to hell in a hen basket?” Just what did that mean? “Wow,” thought Tara to herself as she gathered her things. “I have a whole lot of thinking to do.”

READER REACTIONS TO “MATHEMATICS FOR ALL?”

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REACTIONS TO “MATHEMATICS FOR ALL?” Many respondents found it reasonable to teach algebra at the 8th grade and raised many concerns about the teachers’ attitudes. They found the teachers’ stereotypes and assumptions to be racist and were concerned with how these assumptions influence the teachers’ decisions about their students’ future opportunities. Many were taken aback, even angered, by Mr. Jones’s belief that since Rojelio was most likely to become a lettuce picker like his father, making him take algebra would be unfair. One respondent questioned Mr. Wilson’s professionalism. She believed he set a poor example for Tara, the student teacher. Some respondents sympathized with the teachers, recommending professional development activities as a means to address the issues the teachers faced. Finally, one remained unconvinced that algebra is an essential part of the school curriculum, though she believed that the opportunity to learn algebra should be available for all students. Teachers’ Assumptions and Stereotypes As a student who began taking pre-algebra in 6th grade, I was surprised by this case. I did not realize that making eighth graders take algebra could be so controversial. However, I am certainly familiar with the attitudes of most of the math teachers at Orange Valley Middle School from personal experience. Growing up in a mostly White, middle class community and attending a school system with the same demographics, I have witnessed teachers’ negative perceptions towards their few minority students. Though there is no definitive proof, I believe that the teachers at the school would have a different opinion regarding the teaching of algebra had it been instituted when the school was still mostly White, middle class students. Mr. Jones’s remark about Rojelio supports this belief, as he comments that the student will end up picking lettuce like his father. Before making this statement, Mr. Jones indicated that Rojelio is an average student at what is still an above average school, yet he allows his racial and cultural perceptions of Latinos to affect the way he views his students. —College Senior, Student Teacher

It is completely unfair that Mr. Jones has predetermined what one of his students will be doing in the future. His idea that the student would not need algebra to become a lettuce picker is prejudiced. Students should not be discriminated against because of their background. His teacher should determine no one’s destiny. Mr. Jones’s attitude toward this student could

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have a major impact on his academic success. All students should be given equal opportunity to learn with no preconceived ideas of what they can and cannot do. There are plenty of opportunities for scholarships and financial aid for students like Rojelio to go to college. He should not be discounted just because he does not have money, nor should his education be limited. —College Senior, Student Teacher

The above responses represent the way in which a number of respondents reacted. These respondents focused on the teachers’ assumptions and the impact it could have on their students’ destinies. Some respondents addressed more directly the negative content of the teachers’ discussion. This is represented in the following response. Much of the conversation that took place in the meeting represented the teachers’ sentiments of what the students could not do. For example, “the students cannot speak English” and “the students don’t know their basic math facts.” Rather than considering how to meet the students’ needs and assist them in learning about math, encouraging them to think like mathematicians, and allowing them to understand the concept behind a math fact, the teachers were simply resigned to the impossibility of teaching all students algebra I. Specifically demonstrating the negative language that the teachers were using about the students’ abilities was Mr. Jones. Mr. Jones’s comments about all students not being capable to learn beyond lower level skills were of great concern. Furthermore, the racist comment that he made in terms of his student, Rojelio, not being able to go to college because his father was a migrant worker and he would also be a farmer like his father, clearly demonstrated the low expectations that the teacher had for the student. —Former Kindergarten Teacher, Doctoral Student

A Lack of Professionalism I question Mr. Wilson’s professionalism in making the snide remark to the student before the meeting, and then presenting the new information to the math staff in such a negative way. I’ve heard many teachers say “been there, done that” when new ideas are presented. Having taught about 35 years, I have been through many pendulum swings in a variety of venues. When basic sweeps are made (often due to politics or popular ideas on how things should be done), some will get enthusiastic, but unless the group buys into the ideas, it can be devastating. This one dropped as a bombshell, and the idea of ALL taking ANYTHING can be disputed, but when the state

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dictates policy changes, it’s best to see what must be done, and work with it. What an attitude Mr. Wilson had. And now all the teachers are fired up to be against it. There had to be some good reasons to try the new program. We know about computation versus algebra differences, but there had to be more to be presented than his shorthanded hell journey. I’m sure some of it came from resentment over the school losing its high ratings and gaining two minority groups that weren’t the “joys” to teach. It was always a joy to get a new Spanish-speaking student who didn’t know English, but boy could he/she do the math. It always amazed me how limited so many of the staff could be. —Retired, K-12 Bilingual Teacher

Understandable Concerns, Unacceptable Actions Some respondents sympathized with the teachers’ frustrations and concerns, while not necessarily agreeing with the teachers’ actions. They attributed the teachers’ frustrations to ignorance or fear of change, rather than racist attitudes. These respondents believed that a solution to the problem was to provide the teachers with professional development opportunities to learn about teaching algebra to students with backgrounds different from their own. After reflecting a while I saw that the frustration from the teachers was understandable, but their actions were unacceptable. I agree with the math teachers that certain students would not be ready to learn algebra 1 by the 8th grade. However, there are ways to ease the transition. The teachers failed to address the fact that they are going to have to prepare all of the students. They could have come together as a team and developed a plan on how to get students ready and expose them to pre-algebra in simple forms. There are ways to bring algebra concepts down to a concrete level which less advanced students need. Instead of just giving up on the students that they believe cannot learn algebra, the teachers should make it their mission to find ways of teaching to help the students succeed in math. It is not the teachers’ job to decide that they cannot do it. They should not decide their students’ fate. —College Senior, Student Teacher

Perhaps, the teachers should not be blamed. They might not have been given the opportunity to learn how to teach different students who come from different backgrounds and have different learning capability levels. These

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teachers need to learn to be less negative about the situation, be made aware that before they can lower the expectation of the other 40 percent of the school population, they need to begin to make changes within themselves. They need to take courses or workshops about how to teach algebra to lower grade levels, and learn how to teach a variety of different students. Teachers need to unify with other staff members and students to work towards a similar solution to the problems at hand. —College Senior, Student Teacher

The problem here is not necessarily prejudiced or racist issues against minorities. Rather, their opinions seem to stem from bitterness toward the political decisions made by their authorities, a fear of change, and the unknown. Change is often difficult, especially when it is against your will. District, state, and national standards are hard to follow to begin with, especially when they are changed drastically. The teachers also seem afraid of teaching kids that they are not accustomed to teaching. Like Tara, these adults are primarily used to teaching White, middle class children. They probably have not had much exposure to working with minorities, if any at all. Therefore, they feel very incompetent to teach kids from a different background, which they are expressing through anger and frustration. Because of this reality, the district or state should be in charge of providing the teachers with a workshop or some sort of guidance to help them cope with the changes. By receiving information as to how to teach English language learners and children from low socio-economic backgrounds, they will have more confidence teaching such students and will be able to have a more positive outlook on the transition. —College Senior, Student Teacher

Impact of Policies A few respondents commented on the negative impact of the numerous policies that teachers are responsible for implementing. Below, the respondents point out that policy implementation is a “two way street.” On the one hand, the teachers have to deal with students who are underperforming in mathematics. On the other hand, the teachers have been failed by policies that do not take into account that students may not be prepared to take algebra in the 8th grade. The policies do not make provisions for providing both teachers and students with support for curriculum changes.

REACTIONS TO “MATHEMATICS FOR ALL?”

The state mandates called for seem unrealistic unless they are backed with additional funding. Reduced class sizes are a luxury we can’t afford in [our district]. We are seeing the increased testing and “accountability” standards for teachers, although I’m doubtful of whether these are being developed in a way that actually increases professional growth and encourages more individualized teaching to different levels of students. —School Board Member

I think that this case raises several important issues facing middle and high school teachers, and I can identify with the student teacher’s conflict. My student teaching experience was in an inner city predominantly African American school with a very high drop out rate. Fifty percent of the students in the class I “inherited” had dropped out by the time I started my midyear assignment. My class was full of students who did not know their basics. Several could not read, and many had noticeable difficulties with reading. Only 5 of my 15 students understood how to create a bar graph. Yet I was expected to teach out of an extremely complex textbook and somehow compensate for their lack of basics—a tall order. This was the first time I began to understand how a student could graduate from high school without learning how to read. It is a two way street. These teachers, frustrated over and over again by the system, seem to have developed a negative attitude to policy in general. Instead of trying to work with the system, their automatic response was to say that it couldn’t be done. The teachers had valid concerns. I disagree with the point that not all students can learn algebra. However, I agree that if they have never had pre-algebra or courses that emphasize algebraic principles, it may be impossible. —Graduate Student, Student Teacher

An issue that struck me was the idea of making a general curriculum that forces every single student to take X class in Y year of school. Although the ideas of ability tracking are not to my liking, the idea of forcing a student into algebra who is still struggling with basic mathematical facts is even more preposterous IF the student does not receive supports. It seems as if the state has blatantly disregarded this issue. If the state is to make such a recommendation, then certain practices must be in effect. Tara brings up valid points such as the fact that the children may not have been brought up to think algebraically in elementary schools. When students are required to take classes such as algebra, these students need to have supports from an early age that will help them to benefit from this instruction. —Graduate Student, Prospective Special Education Teacher

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Socio-economic Status and White Children Socio-economic status or class is another notion often entangled with race-ethnicity, ability, and equity issues when curriculum decisions are made. The tendency is to view all African Americans or Latino students as poor and possessing limited capacity for learning certain kinds of school knowledge. Similarly, the tendency is to view White children as middle class, high achievers, and capable of learning the high status knowledge available in college preparatory courses. Though the respondent does not explicitly state it, he hints that class issues come into play when teachers talk about Rojelio. By confusing race-ethnicity with class and ability, we may further complicate matters by ignoring the needs of other groups of children. This respondent points out that the teachers’ focus on socioeconomic status ignores the struggles with mathematics that White children may have. The most obvious issue the case raises was the notion of teachers having cultural biases. It is wrong to assume that children are incapable of learning algebra based on the socio-economic status of their parents. Children must be given opportunities to succeed and by assuming that a student is just “average,” or “too poor to move up in the world,” teachers make an unfortunate error. It is interesting to see that all the teachers were only centering their concerns on the minority children. The possibility that White children could have an equally difficult time with algebra did not cross their minds once. —Undergraduate, Prospective Teacher

Unconvinced Finally, though unconvinced that algebra should be taught in the 8th grade, one respondent did express her certainty about making opportunities available to all students. After reading the case, many things raced through my mind. At first, I agreed that not everyone needs algebra to get through life. Actually, I wish I never had it as a high school student. But as I continued reading, I started to feel like it was not needed to teach junior high school students algebra. I was not sure why upper division math instruction (such as algebra) would need to be taught earlier than high school. Why would eighth graders need to learn algebra? As I continued reading I came upon Mr. Wilson. Again, at

REACTIONS TO “MATHEMATICS FOR ALL?”

first I agreed with his statement. “We will have to start teaching all eighth graders algebra. I think this is a crock!” Personally, I do not feel it is necessary to teach algebra at such a young age either. I think the idea pushes them past their developmental level. But, then I came to Mr. Jones’s reasoning for why he did not want to teach all eighth graders algebra. I was appalled at the statement. “He’s (Rojelio) going to end up picking lettuce just like his father . . . He’s never going to use it.” I was amazed that the teacher could place such a stereotype on a student. How does Mr. Jones know what Rojelio will end up doing? As I finished the case, I felt a lot like Tara, the student teacher. I had a lot to think about. I still do not know whether or not it would be a good idea to teach eighth graders algebra, but I do know that all students should have the opportunities in school that they deserve no matter what culture or ethnicity they come from. —College Senior, Student Teacher

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SUMMARY AND ADDITIONAL QUESTIONS

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SUMMARY AND ADDITIONAL QUESTIONS For the most part, respondents found understandable the concerns of the Orange Valley Middle School mathematics teachers. But, they found their assumptions and their lack of action unacceptable. One respondent remained unconvinced about the value of teaching algebra at the middle school. Reformers concerned with equity issues can be heartened by several studies reviewing large data sets that show all students benefit from taking algebra (Gamoran & Hannigan, 2000). A review of large data sets also indicates that “[e]arly access to algebra has a sustained positive effect on students, leading to more exposure to advanced mathematics curriculum and, in turn, higher mathematics performance by the end of high school” (Smith, 1996, p. 148). Certainly, research interventions like Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) (Silver & Stein, 1996) and grassroots movements like the Algebra Project (Moses et al., 1989; Moses & Cobb, 2001a, 2001b) attest to the value of teaching algebra in the middle grades in high poverty schools with large numbers of children of color. Despite the long effort to improve the quality of mathematics curriculum and teaching, and the recent attention to equity issues, the achievement gap has not narrowed between White and Black or Latino students. “Mathematics for All?” intends to capture the sensemaking of wellmeaning teachers when they are expected to implement well-meaning policies. While policy makers assume that curriculum implementation is a rational/technical process, the teachers’ meeting illustrates the inevitable socio-cultural processes at work in curriculum making. These processes bring to the surface a number of factors that constitute the ways that mathematics curriculum take shape in classrooms: teachers’ beliefs and assumptions about their students’ race-ethnicity and ability; the role of teachers’ beliefs in curriculum decisions about students and the impact of their decisions on students’ life chances; and the impact of multiple policies on teachers’ work lives. Some of the questions that arise from the case follow: 1 Should all eighth graders take algebra? Why or why not? What is the democratic rationale for your position? 2 What notions of “ability” manifest in the teachers’ discussions about the state’s mandate? How do their understandings of ability help or hinder their discussions? 3 What obstacles do the teachers need to overcome before they

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consider a plan for teaching algebra to all of their students in the 8th grade? 4 How might the teachers go about ensuring that their students can be prepared for algebra by the 8th grade? 5 What information and resources do the teachers need to implement such a plan? 6 What is the role of the state department of education and/or the district in the implementation of the mandate?

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INTRODUCTION TO CASE 3 In “Mathematics for All?” the state’s response to inequities of opportunities to learn challenging mathematics took the form of legislation mandating that all eighth graders take algebra. In the following case, curricular reform is the response to similar inequities in the district’s high school mathematics courses. Many children in the United States experience school mathematics as a culture-free body of knowledge. Rarely is the cultural-historical development of mathematics discussed. And, when it is, it gives credit to Anglo European males as the creators of mathematical knowledge. Many children leave school with the intractable belief that God handed to ancient Greek men on clay tablets all the mathematics there is in the world. Children have no sense that mathematics is a human and cultural production and that mathematics continues to be invented. The suggestion that mathematics is neither culture-free nor solely a European invention shakes us to our very core. It challenges deeply held assumptions about our view of mathematics. Scholars of many persuasions believe that mathematics is a cultural activity and product in the same way that language, religious beliefs, and food production techniques are cultural products. School mathematics exposes us to “universal truths” like “the circumference of a circle is 2 × π × r” and “2 + 2 = 4.” However, to say that such abstractions are culturefree is to confuse the “universal truth” of these mathematical ideas with the cultural basis from which the abstractions arise (Bishop, 1988). What is π and where does it come from? Why is 2 + 2 = 4? Why can’t it be equal to 5 or 6? And, who says so? Those of us who have spent time in classrooms with children know that these are the kinds of questions that children ask. These questions get at the cultural history of the mathematics that we expect students to learn. They are also the kinds of questions that many of us cannot answer, much to our students’ frustration. One aspect of the ethnomathematician’s work is to study the mathematics implicit in cultural activities. This work is not unlike the case study that Anna developed when her instructor asked her to investigate Benjamin’s mathematical competency in places other than school. Ethnomathematicians, as well as multiculturalists, and critical scholars offer ways of rethinking mathematics curriculum focused on restoring its cultural and historical components that give meaning to the mathematics that children learn. According to D’Ambrosio (2001), when educators do this, there is the possibility for reaffirmation and the restoration of children’s cultural dignity. These perspectives will be further elaborated in Part II of this volume.

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In the case that follows, Mr. Robinson, the central character, is annoyed when one of his colleagues uses the word “Eurocentric” to describe the school’s traditional mathematics curriculum and instruction. By Eurocentric his colleague means to convey that the conventional curriculum proceeds as though it is a European invention and is built upon the experiences of Anglo Europeans. Ethnomathematics, critical mathematics, and multicultural mathematics are terms that would also irritate Mr. Robinson. All three terms describe an orientation to the nature of mathematics that promotes pedagogies responsive to the dispositions, experiences, and knowledge that children bring to the classroom. While the aforementioned sentence would strike many as common sense, the problem, according to D’Ambrosio (2001), is getting teachers to understand the role of culture in their students’ learning. This is most certainly Mr. Robinson’s problem, and he is not alone. The case illustrates the difficulties that arise when Mr. Robinson’s taken-for-granted assumptions about mathematics bump up against an alternative view that includes restoring the historical and cultural aspects of mathematics.

CASE 3: “CULTURE AND SCHOOL MATHEMATICS” Mr. Robinson had been teaching high school geometry for three years. For the most part, he enjoyed his colleagues and his students. From Mr. Robinson’s perspective, his high school was a good place for a teacher to be. The school boasted high academic standards with over 40 percent of its seniors completing requirements for admissions into the state’s universities. Its sports and music programs were highly regarded by the students and the community. Last year, both the basketball and football teams ranked among the top ten schools in the state. As well, it was not uncommon for the marching band to compete in state and national competitions. Parents participated in fund raising activities and attended school events in large numbers. Mr. Robinson marveled at the school’s ethnic diversity. Forty percent Hispanic, 35 percent Anglo, 15 percent Asian, 8 percent African American, and 2 percent Pacific Islander/Native American represented the school’s ethnic profile. Mr. Robinson was proud of the fact that he had a decent command of the Spanish language. On the rare occasion that he had to communicate with Spanish-speaking parents, he did so with relative ease. It seemed to win him brownie points with these parents and their children. Mr. Robinson was comfortable with the school’s ethnic diversity.

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From the accounts he heard, the district and the school has been responsive to changing demographics. For example, several years before he was hired, professional Latino parents complained to the school board that its social studies curriculum lacked Latino contributions to the making of the United States. The school board responded swiftly, calling for the adoption of a K-12 social studies curriculum that included not only the contributions of Latino Americans, but other groups as well. As far as he could tell, the social studies department seemed to have embraced the district mandate. Certainly, there weren’t any more parent complaints. At the last staff meeting for the year, the principal announced that she would be meeting with the chairs of all departments throughout the summer. The principal’s call for the meetings was the result of the district’s efforts to be proactive about the community’s growing diversity. Most likely, the district’s self study motivated the superintendent to act. The report showed that Blacks and Latinos were underrepresented in high track mathematics and science courses in their high schools. The purpose of these meetings, the principal told the teachers, would be to determine how to embed multicultural issues throughout the high school’s academic programs. This seemed fine to Mr. Robinson, but for the life of him he couldn’t figure out what culture or race had to do with mathematics, or how it would motivate Black and Hispanic students to take advanced mathematics courses. What Mr. Robinson liked about teaching mathematics was that it was objective. Usually, there was either a right or wrong answer. As far as he was concerned, there was nothing subjective about it at all. Teaching mathematics was relatively straightforward. After going over homework, he began each lesson by focusing on a procedure or problem type, explaining to his students how to use the procedure or solve the problem. He provided some examples that he and the students would work together. Then he would assign a set of similar problems. If the students didn’t finish, they were expected to finish the rest at home. Instruction really was that straightforward. Of course, in geometry, students had to memorize properties of geometric figures, theorems, axioms, and trigonometric functions, although, many times, he provided them for use on examinations. How do you embed multicultural issues in mathematics? There is no such thing as “Black mathematics” or “Asian mathematics.” Mathematics is mathematics. Pure and simple. He spoke Spanish. Does this mean he is supposed to teach lessons in Spanish? Does it mean that he invite a Black mathematician to class on Career Day? Should he be teaching the Roman or Maya numeral systems even though neither has anything to do with the geometry his students should be learning? The Career Day thing might be

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nice for getting his students to take geometry more seriously, thought Mr. Robinson, but it certainly didn’t teach the students anything about mathematics. If he were supposed to do these kinds of things on a frequent basis, then the course would become watered down and not worth teaching since the students wouldn’t learn geometry. Heading back to his classroom, Mr. Robinson bumped into Mr. Lawson as he was leaving the building. Mr. Lawson taught algebra II/trigonometry and advanced placement calculus. “So what do you think of what’s happening with these summer meetings?” asked Mr. Robinson somewhat tentatively. “Oh you mean the multicultural thing? I think it’s pretty cool. It’ll be interesting to see what they come up with.” “Wait a minute. I understand how that all works in something like social studies. Mathematics is a different kind of subject matter. There’s nothing multicultural about mathematics. Mathematics is mathematics no matter where in the world you live.” “Yeah, it certainly seems like it. You feel that way because of how you were taught math. Actually, the mathematics we teach kids in this country takes a particularly Eurocentric view. Hey, spend a little time in the library this summer. It’s interesting reading. Gotta run.” Mr. Lawson sprinted down the corridor before Mr. Robinson could get him to elaborate. Eurocentric? What in the world did that mean? It sounded like one of those words that some of those political ideologues spouted, making people nervous about everything they say. Was political correctness now contaminating mathematics? There’s nothing cultural about mathematics, nor political, for that matter. It’s objective and politically neutral. Math is math. He was sure of it. He was so sure of it that he was going to spend his summer developing an argument for why a multicultural approach to mathematics was harmful. He couldn’t wait to present it to the math department in the fall.

READER REACTIONS TO “CULTURE AND SCHOOL MATHEMATICS”

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REACTIONS TO “CULTURE AND SCHOOL MATHEMATICS” The responses to the cases reflect the entrenchment of the ahistorical and culture-free mathematics experienced in U.S. schools and the confusion that arises when our taken-for-granted assumptions about mathematics and culture are challenged. Not surprisingly, many respondents agreed with Mr. Robinson. They could not see anything cultural about mathematics teaching and learning. Though many agreed with Mr. Robinson, the respondents’ positions were expressed in various ways. For example, some respondents dismissed outright the relevancy of culture in mathematics. Others, while agreeing with Mr. Robinson, acknowledged that it was important to pay attention to individual children’s learning styles, rather than culture. There were several who appreciated how mathematics could be taught within a cultural-historical context, but they did not articulate what that means with respect to curriculum or teaching practices. “Culture” was a confusing concept for many respondents. Even more confusing was the relationship between mathematics and culture. This is a testament to deeply ingrained beliefs about school mathematics traditions. As you read the responses, ask yourself, “What did ‘culture’ mean for this respondent?”

Math is Math In recent years, there have been well-meaning attempts to address directly the racial, ethnic, and linguistic diversity in our nation’s classrooms. Where previously only White children have been represented, textbook publishers now include pictures of African American, Asian, and Latino children. Names like Mary and John have been changed to Maria and Juan and basal readers include stories about children who live in the world differently than White middle class children. Many schools have a “Multicultural Day” where children dress in clothing and eat foods representative of their cultural heritage. More common at middle and high schools are “Career Days” which provide schools an opportunity to bring in professionals from diverse racial/ethnic backgrounds to talk to students about career opportunities. Certainly there is value to “Multicultural Day” and “Career Days.” But, if we aim to change curricula in ways that restore non-European contributions, then this approach is not sufficient. The following response represents precisely this common orientation to addressing culture and school mathematics.

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I just don’t see how culture can be tied into an everyday mathematics lesson, except maybe in the way it was explained in the case study where a teacher can bring in mathematicians or other people in some field of work where math plays a big part to be a guest speaker in the classroom. For example, scientists, astronauts, engineers, or architects all of which are some minority group whether it be Latino, African American, Asian, etc. to maybe encourage the students to take and excel in math classes so that someday they will have a job like the ones these professionals have. In addition, I feel that it would be a great supplement to math lessons to integrate history and social studies in order to enrich an activity and also break away from the daily routine. Math for me is already a difficult subject. Thus, I agree with Mr. Robinson when he says, “there is nothing multicultural about mathematics.” —Undergraduate, Student Teacher

Multiculturalism is Just Another Excuse The following response captures the confusion that arises about culture and its relation to mathematics. Culture is often confused with race/ ethnicity skin color, or in this case, language. Because of the confusion, the primary thrust of the respondent’s reactions seems to be that a Eurocentric curriculum is not problematic. Instead, multiculturalism is cast as an excuse for poor achievement. According to the respondent, individual students should take responsibility for their learning and stop blaming “outside factors.” After reading this case, the way I was feeling reminded me of my parents. They are always saying that nowadays, everyone has an excuse. If a child is not doing well in school, it is the teacher’s fault. If a child is a bully, his/her parents are not raising them well. Everyone has an excuse and they blame other people for what is not going right. This case does both of these things. I agree with Mr. Robinson in this case. I do not feel mathematics is a cultural subject. It says in the case that mathematics holds a Eurocentric view. Well, I feel all our subjects do. We work with Hispanic kids in bilingual programs to teach them English so they can do well in other subjects and succeed in America. They are taught history, science, even physical education with a Eurocentric view. Why should mathematics be any different? Math is a difficult subject for many people. I include myself in this. I find math to be very challenging and intimidating. I do not feel that if math was taught to me or is taught to me now with a Portuguese (because I am

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Portuguese) emphasis, that I would learn any more than what I have already learned. I feel people should take responsibility for themselves and stop making these excuses why they are not doing well in math and blaming outside factors. Everyone is capable of learning math regardless of how it’s presented. —Undergraduate, Student Teacher

Culturally Sensitive Teaching/Individual Learning Styles The next three respondents also agreed with Mr. Robinson. For them, mathematics is not a multicultural issue. However, the first respondent offers examples of how the teacher can be culturally sensitive while teaching the conventional curriculum. The second and third respondents point out that teachers need to address individual learning styles. I hate to say it, but I unfortunately agree with Mr. Robinson. Math is extremely objective and straightforward. I can understand how a language based subject such as reading, writing, and social studies can be multicultural. I can even see how I could make science multicultural, but math is math. Facts are facts, and there really is nothing multicultural about it. There are however ways to teach that can assess and react to students’ needs. As a teacher, I would obviously not use examples that were not part of my students’ culture. The same way I would not ask my first graders to subtract the number of candles on a menorah from the twelve days of Christmas. I would just have to use common sense. I agree that teachers have to be culturally sensitive, but there are some commonalities, such as fruit, or money that people living in any culture understand, and teachers just need to focus on those, and just be careful as to what they say and teach. Math is math to a certain extent. There is no way to make 2 + 2 = 4 into a multicultural issue. —Undergraduate, Student Teacher I agree with Mr. Robinson that math is math and I find it hard to see any multicultural connection to it. Therefore, I think that rather than focusing on cultural differences, teachers should look at the way students learn. If students are struggling, one of the problems could be that the student is simply not comfortable or benefiting from the teaching style. It might be more beneficial towards the student if the staff reevaluate their curriculum and manner through which they relay the information and adjust it to children’s individual learning styles. Maybe a teacher could offer more group

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activities or vary the way he lectures the class, incorporating more student involvement. —Undergraduate, Prospective Teacher

I sympathize with Mr. Robinson. I, too, believe that math is culturally neutral and that the motivation for taking math ought to be separated from the actual taking of math. (That would make me push for Career Days for motivating the taking of math, but not for inclusion of cultural factors into the teaching of math.) Where I sympathize with Mr. Lawson is that I don’t think that math is “learning-style” neutral, that students have different styles of learning math (visual, tactile, aural, etc.), and that it is there that math instruction (and Mr. Robinson) ought to pay attention. Addressing different math learning styles can help students with learning styles that may be influenced by culture. —Parent and Mathematics Ph.D.

Mathematics is Culturally Bound The next set of reactions indicates that the respondents believe that mathematics is culture-bound. One respondent’s comment suggests that Mr. Robinson will have a difficult time fulfilling his summer task. Another attributes Mr. Robinson’s attitude to the fear of change. Another respondent explains that he seeks to exploit the diversity of children’s thinking when they solve problems. Finally, the last respondent expresses her frustration for the lack of support when teachers are asked to change their practices. Mr. Robinson, a high school geometry teacher, resolutely sets off in search of data to support an argument that “math is math”—a neutral, acultural, apolitical discipline. As a middle and high school mathematics teacher, I am curious as to how Mr. Robinson will proceed. Where will he look for mathematics devoid of culture? Will the definition of mathematics on which he hopes to base his argument be limited to topics typically covered in K-12 mathematics instruction in the United States? What is typical and who decides? To whom will he look for data to support his argument? K-12 teachers? Mathematicians? Business leaders? Parents? Politicians? My experience tells me that Mr. Robinson is unlikely to discover consensus on the definition of mathematics, the mathematical topics to be taught, or the best approach to teaching them. If “math is math” why are there so many differing views as to what mathematics should be

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taught, how it should be taught, and to whom? Culture must certainly play a role. —High School Mathematics Teacher

Fear of Change Mr. Robinson’s attitude needs an adjustment. Although he advocates multiculturalism and loves the students in his school and the people in the community, he is not open to the idea of change in the classroom. Mr. Robinson should be open to the suggestions given at the meeting. If his students could possibly benefit from some changes in the curriculum, then it is worth the effort. He is afraid of what he does not know. Teachers in general should be open to change. The world around us is constantly changing. As society’s values and morals change, education in turn will change and work toward a better, more appropriate education for all. Education in American culture tends to rely more on books as the main source of learning. Other cultures may use different strategies for teaching. Some may use cooperative learning or take a research perspective on learning. These strategies must be taken into account when teaching children who come from different cultures. Most of the schools in America are Eurocentric in nature. The faculty consists mostly of White females and the administration consists mainly of White older males. African American, Hispanic American, Asian American, etc. students may find it hard to relate to someone who is not like them. It is impossible to have a teacher of every ethnicity in schools and someone will always feel left out and like they cannot relate to their teachers. This is something of which teachers should be aware. It can be very hard to relate to someone who comes from a different culture and holds different beliefs and values than them. It can also be hard if the school children are in taught in a style that is different from their culture. It then becomes important to bring multiculturalism into every subject area in schools so that no children will be left out. —Undergraduate, Prospective Special Education Teacher

Historical Development of Mathematics Culture diversity can be a difficult concept for some teachers to grasp because they might not be sure of how to integrate it within their current curriculum. I feel that any subject should have some kind of multicultural component to it. There is more to math than just numbers and facts.

REACTIONS TO “CULTURE AND SCHOOL MATHEMATICS”

Teachers could introduce mathematical inventions and creations by individuals from various ethnic backgrounds and discuss those in class. Even the ways mathematical problems are solved involve diversity. I feel that showing students exactly how to solve something deprives students of an authentic problem solving opportunity. By allowing students to brainstorm and figure out how different kinds of math problems are solved not only involves creativity, but also diversity in the way they are thinking. Teachers should take the time to appreciate and acknowledge their students’ achievements. —Graduate Student, Prospective Special Education Teacher

On a practical level, Mr. Robinson has it all wrong. Bringing in a Black mathematician to class on Career Day is a lame band-aid solution. He does not need to teach an entire unit on Roman numerals either. Instead, he could learn more about the historical nature of the geometry he is teaching and make that part of his class. Math was not created by individuals in isolation. What was the evolution of the ideas behind the geometry of Mr. Robinson’s math class? What cultures were involved? I don’t know the answers to those questions, but I’d be interested. —Middle School Science Teacher

Lack of Teacher Support I see this as a case study that points to hope for solving some of the problems pointed to in “Mathematics for All?” In this school, you have a diverse population and the teachers are comfortable with it. When asked to teach math and embed multicultural issues in the lessons a teacher has difficulty in understanding how to do it. The problem is not the teacher. He commits to researching the issue over the summer and plans on proving that it can’t be done. At least he will look up information and probably discover that you can teach math in different cultural contexts. The problem is a school that mandates a change with no support structure to help teachers incorporate that change. While the school administration is to be applauded for seeing a need for this they should have taken it one step further and provided the instruction to the teachers so that the teachers could feel good about being able to make the changes in their classrooms. This is a problem in the whole educational system today, mandated changes without any support structures to train the troops in the front lines to make the needed changes. The teachers are being left out of the equation. —Graduate Student, Prospective Special Education Teacher

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SUMMARY AND ADDITIONAL QUESTIONS

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SUMMARY AND ADDITIONAL QUESTIONS In “Culture and School Mathematics” the district’s goal to embed multiculturalism throughout the curriculum perplexes and frustrates Mr. Robinson, the geometry teacher. His reaction is not surprising since the district’s efforts challenge the prevailing view of mathematics. Most of us experience mathematics as a fixed body of knowledge that exists separate and apart from cultural and historical contexts. Just as Mrs. Carlton did in case 1, “Race and Teacher Expectations,” many teachers teach procedures for problems without any real world context nor historical appreciation for the mathematical ideas that underpin the procedures. Many reformers argue that the existing curriculum is so disconnected from children’s experiences that it undermines their ability to perform successfully in school contexts. For example, in “Race, Retrenchment and the Reform of School Mathematics,” Tate (1994) writes about this very point. He shares a story about a group of middle school teachers who noticed that a large number of African American students at a predominantly African American urban school responded incorrectly to an item on a district mathematics assessment. The test item was: It cost $1.50 each way to ride the bus between home and work. A weekly pass is $16. Which is the better deal, paying the daily fare or buying the weekly pass?

For the students, the weekly pass was the better deal. Tate explains that the students missed this seemingly easy problem because it assumes a 5day workweek. Furthermore, says Tate, the problem also assumes that ordinary people have only one job. The district test designers did not take into account the lives and experiences of the African American students attending the school. That is, they did not consider that many African American students had parents who worked 6 or 7 days a week and had more than one job to make ends meet. In “Culture and School Mathematics” the district sought to address these considerations and others through embedding “multiculturalism” throughout the curriculum. It represents another approach to improving teaching and learning, and increasing access to college preparatory mathematics, especially for underachieving populations. The situation presents a set of difficult tasks. Here are the most obvious ones. First, the mathematics teachers must grapple with “Eurocentric” curriculum. What does this mean? Second, what is culture? At the outset of this volume, I said

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that culture has to do with the ways that one negotiates one’s world in and through daily routines, languages, and ideologies, which are inextricably linked to the historically, socially, and politically embedded contexts in which we all live. I also emphasized the tendency to confuse race or skin color with culture. They are not the same things, though skin color carries with it a historical legacy that has social and political consequences for groups and individuals that delimit the ways in which they can negotiate their worlds and envision their participation in a democracy. In other words, a person’s cultural circumstances give meaning to skin color (or race). How very difficult it is to wrap our brains around the distinction between cultural circumstances and race. Furthermore, to appreciate the role of culture in the teaching and learning of mathematics, the teachers also must reflect on their beliefs about the nature of mathematics, its cultural-historical development, and the experiences that children bring to the classroom. In addition to foundational curriculum questions—What mathematical ideas ought we teach?—there are other questions that teachers must discuss if they are serious about educating the nation’s youth. 1 How reasonable was Mr. Robinson’s response to the district’s mandate? 2 How does (or does not) a Eurocentric view of mathematics disadvantage students, particularly non-European students? 3 How might doing ethnomathematical studies of the students (as Anna did with Benjamin) inform teachers’ understandings about teaching their particular students? 4 How might mathematical ideas from other cultures be used to enhance the curriculum? 5 How does (or does not) a mathematics curriculum, reconstituted with its historical and cultural origins, improve opportunities for all students to learn mathematics? 6 How does (or does not) a mathematics curriculum, reconstituted with its historical and cultural origins, improve opportunities for students to take college preparatory mathematics?

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INTRODUCTION TO CASE 4 For several decades, school mathematics has been the target of an avalanche of recommendations bent on improving teaching and learning in U.S. schools. For reformers, understanding the conceptual ideas of the mathematics we expect children to learn requires that children explore, conjecture, and reason logically about mathematical ideas embedded in problem solving contexts (NCTM, 1989, 1991, 2000). These recommendations are a response to traditional mathematics curriculum and instruction focused on basic arithmetic and memorizing procedures. Unfortunately, the debates that have emerged in policy-making decisions about mathematics curriculum pit reformers’ visions for school mathematics against the visions of what traditionalists think the curriculum ought to be. Aptly known as “The Math Wars,” improving school mathematics has been a contentious issue, to say the least, with various stakeholders using legislative or political means to have their views and values represented in state curriculum policies. In the following case, teachers are faced with teaching to content standards and making sense of “problem solving versus basic skills,” pitting problem solving against basic skills—the district’s unintended rendering of school mathematics. Further complicating matters is the state’s new accountability mechanism, a single standardized test. Not only do the teachers have to address content standards within a confused curriculum, they also have to ensure that their students perform well on the state test. The school principal has targeted place value understanding as an area for improvement. This is an important focus. The foundation for understanding place value lies in grouping activities. Through these activities children learn that numbers can be decomposed and represented in many ways, including representations of items as sets of objects, which is required for understanding the structure of our base-ten number system. Place value understanding is necessary for reasoning about rational numbers and the operations performed on them. The university professor she hires to assist her with improving the teaching of place value understanding observes how teachers make sense of teaching and learning in a highly charged political context. Her dilemma is to report findings that the teachers and the principal will find useful given the mixed messages that the teachers are receiving.

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CASE 4: “POLITICS AND SCHOOL MATHEMATICS” The elementary teachers of Eastside City Unified School District found themselves sitting at yet another district-wide in-service. This time, the focus was on aligning the district’s mathematics curriculum with topics present on the state’s new mandatory test. The facilitator began by identifying weaknesses across all grade levels as indicated by last year’s test scores. She passed out and talked about literature on teaching students test-taking strategies. The teachers were also given examples of items represented on the test. Next to each item appeared the content standard and the pages in the mathematics textbook that addressed the particular content standard. Many of the items marked as meeting the “problem solving” content standard confused the teachers since the problems looked like low-level exercises. After some discussion about curriculum alignment, one teacher asked, “What are we supposed to be teaching? Problem solving or skills? The reason why we can’t figure it out is because the politicians can’t figure it out. They can’t decide between using manipulatives or basic skills.” Other teachers mumbled similar sentiments. The teachers’ confusion over which mathematics to teach was understandable. Unwittingly, the district sent mixed messages in an attempt to comply with policies coming from the state department of education. The state was embroiled in a “Math War.” Political conservatives outnumbered liberals on the state’s curriculum framework committee. Battle lines were clearly drawn between political conservatives vying for a mathematics curriculum that reflected a back-to-basics approach and reformers who wanted a curriculum focused on problem solving. Conservatives believed that learning basic skills like addition, subtraction, multiplication, and division were absolute prerequisites for higher mathematical thinking. Reformers insisted that problems needed to be embedded in meaningful contexts and that problem solving facilitated the development of the conceptual underpinnings of the basic skills that political conservatives insisted upon. The bickering between curriculum framework committee members emerged as an either/or proposition—mathematics as basic skills or mathematics as problem solving. In the end, the conservatives on the framework committee won out. Their political maneuverings resulted in a back-to-basics framework, although they proclaimed that they had achieved a “balanced” approach to mathematics curriculum. The state’s political debates impacted the district and its teachers in unintended ways. The resulting policies attempted to seek a “balanced” approach. In doing so, these policies mirrored the dualistic rhetoric that pervaded the state’s debates about mathematics. For example, in response

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to the teacher’s concern about which mathematics to teach, the facilitator reminded her and the others that they needed to develop mathematical ideas by going from the concrete to the abstract. The facilitator’s recommendation is consistent with what is typically taught in elementary math methods courses. Children should first use concrete objects to create equivalent representations of numbers. These numbers can be represented pictorially (showing sets of three, five, ten, etc.). Pictorial representations can be represented symbolically using expanded notation, scientific notation, and the conventional form in which numbers are represented in the Hindu-Arabic numeration system. However, the facilitator did not provide specific examples of what she meant by “going from concrete to abstract.” Then she passed out a pie chart. It was titled Eastside City Unified School District Math Daily Time Management. The largest piece of the pie chart indicated that 30–45 minutes should be devoted to the district-adopted textbook. Five minutes should be devoted to basic skills review, which included mental math, skill counting, memorization of facts, and skills worksheets. Ten minutes should be devoted to problem solving including problems-of-the-day/week and test preparation for the newly mandated state test. The facilitator’s chart for daily time management presented basic skills and problem solving as separate aspects of daily instruction. It was no wonder that the teachers could not see them as interrelated components of mathematics. At the same time, the district implemented its own test focusing on basic skills. All fourth graders had to take and pass with 80 percent accuracy a test on addition, subtraction, multiplication, and division, as well as a test with addition and subtraction problems with regrouping. The teachers were as stressed about these sets of tests as they were about the standardized test. Concerned about the falling test scores, Mrs. Provines, the principal at John F. Kennedy Elementary School, enlisted the help of Professor Jordan, a mathematics educator from a local university. Mrs. Provines explained that she’d sent her primary teachers to a series of workshops on developing place value understanding. While she anticipated improved teaching, she wanted an expert opinion and someone who could offer assistance. Professor Jordan agreed to observe place value lessons in 2ndgrade classrooms. To some degree, all of the teachers used the districtadopted textbook. The professor took particular interest in Mr. Barker, since his principal considered him to be the best mathematics teacher. At first, it seemed like an innovative lesson. The students were arranged in groups. There was an assortment of manipulatives, counters, rubberbands, small plastic cups, and the like. He asked his students, “How many different ways can you make 23 using your counters.” The students set about

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problem solving. The professor noted the variety of ways in which the second graders represented 23. After about five minutes, Mr. Barker asked three students for their solutions. They drew their representations on the board. After congratulating the students and with no further explanations, Mr. Barker drew a place value chart on the board showing the ones, tens, and hundreds places. He put a 2 in the tens place and a 3 in the ones place. Pointing to the chart, he then asked, “How many are in the tens place?” The students, responding chorally, “Two!” “How many in the ones place?” “Three.” “And, what does this number mean? “Twenty-three!” “Good job! Twenty-three is two tens and three ones.” Mr. Barker passed out worksheets. He told the students to write their names on the worksheet and read the directions: “For each problem, write the value of the blue digit on the line.” The worksheet included an example problem that Mr. Barker went over with the students. After asking if anyone had any questions, he told them to begin working. The professor recorded a few more observations, but then spent the rest of the class period trying to formulate her thoughts. Mr. Barker was a successful teacher in many respects. It was easy to see that he liked his students and enjoyed teaching. The students seemed to be eager learners and were on task during the lesson. The worksheet assignment indicated that they worked well independently. Although he began the lesson by presenting a problem, he fell short of helping students get a handle on the place value meaning of the problem solving activity. He did not connect what were really grouping activities to place value. The worksheet required that the student simply memorize the places and the number of zeroes to add to the digits. It was clear to the professor that the teachers were trying to make sense of district policy. Mr. Barker did allocate a portion of his time to problem solving when asking his students to find multiple ways of representing 23. However, there was no opportunity for students to learn how the grouping activity was related to the worksheet he assigned after the problem solving task. Why is the one’s place the one’s place? The ten’s place? And so on. The same was true for the other teachers. The other teachers followed the textbook more closely. These teachers engaged the students in several grouping activities, instead of one. But, like Mr. Barker, they did not explicitly link the activities to the exercises that followed them. They spent virtually no time exploring the relationship between grouping objects and

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place value. Instead, they resorted to using a place value chart. As one teacher told the professor, “There are a few more lessons in the book devoted to place value. But, that takes too much time. We need to cover other stuff. They’ve got to memorize the chart and learn the trick so that they can do well on the state test.” Professor Jordan wondered about the contents of the report she would submit to the principal. How would she explain that the teachers, despite the time they’d already spent in workshops, still needed to improve their teaching of place value understanding?

READER REACTIONS TO “POLITICS AND SCHOOL MATHEMATICS”

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REACTIONS TO “POLITICS AND SCHOOL MATHEMATICS” “Standards” and “accountability” are seemingly innocuous words. It seems reasonable to have standards and it seems reasonable to hold persons and systems accountable for educating our nation’s youth. Mandatory testing has accompanied the standards-based zeitgeist. Most often, a single standardized test serves as a school system’s accountability mechanism. However, is it reasonable to assume that standards and mandatory testing improve the quality of teaching and learning in our nation’s schools? What are the effects of these policies? Who gains? Who loses? And, under what circumstances? By and large, the respondents viewed negatively the political context in which the teachers must do their work. They expressed a number of concerns regarding the effects of policy decisions on teachers and students. They commented that these policies yielded time constraints that impacted the way in which teachers covered the curriculum. Since covering the curriculum became the teachers’ main objective, this resulted in a superficial curriculum where students never acquired command of any mathematical topic. The respondents also raised issues about political rhetoric, competing goals, and how policies can result in de-professionalizing the practice of teaching.

Time Constraints and Curriculum Coverage The following respondents voice a similar complaint about the quality of teaching and learning mathematics. They comment that time constraints affect both teachers and students in negative ways. The time management plan described in the case forces teachers to “jump from one topic to another” in ways that compromise learning substantive mathematics and relating it to their children’s lives. The respondents’ responses suggest that the teachers would teach place value understanding more substantively if they were not pressured with time constraints and high stakes testing. One thing I don’t like about this case is the fact that there is a “mandated daily math time management pie chart” telling teachers what to do. A key problem in the way they have the layout of allotted times is that the teachers are unable to see them as “interrelated aspects of mathematics.” How can only five minutes be given to something so crucial as learning basic math skills? So much time is spent on covering the “district-adopted textbook.” I

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personally don’t like the district mandated mathematics program where I am now student teaching. After observing it for a while, I have come to the conclusion that it is so confusing at times because there is too much of a jump in topics and it seldom relates to the children’s lives. Some of the vocabulary used in the mathematics program is either too advanced or so unfamiliar to the children that they end up learning nothing. The mathematics curriculum I have observed fails to make a connection to children’s lives. There is no personal meaning of fun in mathematics for them anymore like I had when I was in elementary school, which I think is very sad. —College Senior, Student Teacher I think one interesting fact brought up in the case is the idea of effective time management in the classroom. This made me think of my 6th- and 8thgrade math teacher who would always complain about the limited amount of time he had with us each day (only 38 minutes he would constantly remind us), and he would always have to jump from topic to topic to fit everything into his very short class period. He felt constrained by this shortened class period, not to mention the amount of material he had to teach us before our state tests. It is hard for any teacher to follow a time management plan everyday, let alone teach everything they are supposed to before those ever-looming state tests approach. Time management plans are lovely things to make, and as a teacher you can always have good intentions to follow them, but 4 days out of 5 you are going to be behind and rushing to cover everything. I think this is why Mr. Barker and the other teachers breezed through their teaching of place values. As one teacher says in the case, “They’ve [the students] got to memorize the chart and learn the trick so that they can do well on the state test.” In this instance the teachers are relying less on solid, formal instruction and more on a memorization tactic. I believe the reasoning behind this is time: give the students a quick way to memorize, and we can move on to the next state test topic. I don’t think it should be like this. I think teachers should be able to take their time and teach their students effectively. —Graduate Student, Prospective Teacher

Curriculum Impact The following respondent directly addressed the mathematics curriculum that the teachers faced. According to this respondent, the district’s adoption of a textbook that represented both reformist and traditionalist views resulted in a curriculum “a mile wide and an inch deep,” a characterization of school mathematics in the U.S.

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The conservatives demand a more back-to-basics approach, while the more liberal members are more influenced by reformers calling for more problem solving and critical thinking skills. The district, in its attempt to reach a middle ground has selected a text that accommodates both view points, thus resulting in a curriculum with too broad a scope and does not allow for in depth study of either. Likewise the text presents basic skills and problem solving as separate and distinct concepts that are not integrated. Legislators tend to reach agreement on the use of assessment as a tool by which accountability can be measured, thus multiple assessments are mentioned here as being in place. The state here has required a “new mandatory test,” along with two additional district tests now requiring 80 percent accuracy to pass on differing areas within the curriculum. Teachers, as demonstrated by Mr. Barker, are concerned with the amount of material to be covered as well as teaching the “trick(s) to do well on the state test.” Once again one sees the concept of curriculum being a mile wide and an inch deep so that all stakeholders feel as though they “got their way.” Yet in fulfilling this need, students never learn any aspect of the curriculum well enough except that which is on the test. —College Senior, Student Teacher

De-Professionalizing Teachers Both respondents in the following comments indicate that policies may act to undermine the teachers’ expertise in making instructional decisions. For these respondents, accountability serves to deprofessionalize teachers by not allowing them to consider how mathematical ideas should be developed or make necessary modifications that facilitate students’ learning. The second respondent indicates that policy makers show a profound lack of understanding about what really goes on inside a classroom. I realize the difficulty of trying to cover all of the material that most districts require in a school year. However, by breezing through most major concepts how effective are teachers really being by just “skimming the surface” on some topics? Another thing to consider: Are students really learning when teachers just tell them to memorize a formula, or by just giving them an answer to save time? I know teachers need to be concerned with issues related to accountability, but students’ learning shouldn’t be compromised in order to save time and get through the entire curriculum. If a student is having problems understanding a certain concept or idea, then it might be a good idea to figure out a way to present the material differently or make

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modifications to it in order to make it easier to comprehend. Isn’t the whole purpose of teaching to teach students? —Graduate Student, Prospective Special Education Teacher

I thought the daily time management chart was humorous as it highlighted the lack of understanding of what really goes on in a classroom. What if the students need more than five minutes working on their basic skills? What if the entire day, week or month were devoted to problem solving skills—is that a bad thing? Every class has its own flow, and not every student will respond well within a time limit set externally. What if the textbook is horrible? What if the teacher knows a better way to teach place value than the textbook? Shouldn’t these things be left up to the teacher? After all, s/he knows the class better than anyone else. —Graduate Student, Prospective Teacher

De-Professionalizing Teachers or Poor Teacher Preparation? The following responses raise an interesting question. Do policies mandating testing act to de-professionalize teachers; or, do teachers struggle with teaching in the context of mandatory testing because of poor teacher preparation programs? In the first response, the respondent suggests that teachers who know how to teach well end up sacrificing good instructional practices because of the pressing political factors. The second respondent believes that the problem is not high stakes testing. Rather, the root of the problem is the generally poor preparation that teachers receive in their certification programs. As I read this case, I was immediately struck by the similarities of this case in terms of mathematics and the current state of literacy instruction. High stakes testing is now occurring, often in the name of teacher accountability. Teachers are expected to raise test scores and this often leads to students experiencing learning in the context of what is going to be on the test. While teachers in this particular case had once taught in the context of higher level thinking skills and problem solving, they are now being asked to focus more intently on skills. In terms of this case, it is interesting that when the professor is asked to come in and observe in order to provide some guidance and assistance, she observes who was hailed as the best mathematics teacher by the principal. Yet, upon observation, the professor notices that the teacher does not allow for a lot of problem solving. Instead, he allows for time allocation that he received during the district-wide

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in-service. The teacher became devoted to what would be covered on the test and planned instructional time based on what would be on the test as well as what the district asked in relation to the test. I think it is an interesting phenomenon that even when teachers are aware of and have practiced good teaching strategies, in any subject, they often negate what has been effective for them in order to answer to a mandate or teach to a test. I wonder what influences teachers in sacrificing good instructional strategies that support their students’ needs and their philosophy of learning and teaching in order to support the test. —Doctoral Student

I think that this case points out a very glaring fact. A lot of teachers out there are not teaching their subjects effectively. A lot of teachers are not even teaching in their content areas. A teacher who is not trained in these areas particularly affects math and science instruction. The professor saw the problem not in the state mandates but in the simple fact that the teachers (the best one in the school) need to improve their teaching skills. This points to what I consider one of the most important issues we must deal with in our educational system. The system is failing to fully train teachers to be the most effective teachers they can be. We are sending out individuals who are not prepared to deal with the environments that they are being thrown into. That is the fault of the system that prepares them. Too much theory and not enough real practical strategies and techniques are being taught in our programs. Teacher training is lacking at all levels and needs to be improved if we are to see a change in our schools. —Graduate Student, Prospective Special Education Teacher

Which Goals? Whose Goals? The problem, for the first of the following respondents, is that various stakeholders have different goals. Because the goals represent competing tensions, high quality math lessons are unlikely as different stakeholders aspire to different goals. Also concerned about goals, the second respondent cast the problem in a different light. For this respondent, long-term goals are sacrificed for short-term goals. When focused on short-term goals, that is, to raise test scores, the students lose out. The dilemma presented in this case is very damaging. The state, principal, and the teachers all have different goals. The state wants a back-to-basics approach. The principal wants to incorporate skills and problem solving but

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does not present that well. The teachers are motivated by the standardized tests. The different and unclear presentation of the different goals expected by the different parts of the school system presented in this case cheats the students from good math lessons. The students are being cheated of understanding concepts in order to generate good test scores. If these concepts are taught in order for the students to retain the concepts they will be able to quickly expand their basic knowledge into other areas and learn even more. Obviously, everyone in this case has different goals and the goal of having students learn may be lost in the politics going on in the background. I’d like to think that teaching does not include power struggles and only concentrates on the importance of teaching the students the content necessary. —College Senior, Student Teacher

We have settled for short-term goals rather than long-term benefits in our quest for higher test scores and better schools. Evidence of this is the multitude of teachers who instruct, not from curriculum guidelines, but from test booklets. The concrete pressure put upon teachers from their principal, school district, state, parent groups, community, to raise test scores is intense and it stands to reason if their performance as a teacher is associated with a numerical value, there will be more energy put into that which is a direct threat to their profession. Thus we have a situation in which the best intentions have gone awry. School children are given instruction relating to test problems in order to prepare them for three days in April, instead of teaching them critical skills that will provide a foundation from which to further their intellectual discovery and enhance their cognitive ability for the rest of their life. Test preparation is counterproductive in the long term. The skills necessary to perform adequately on many state run tests are in direct contrast to skills needed for problem solving assignments that require a considerable amount of creativity, practice, and review. Problem solving takes time and thought and dialogue; it’s a repetitive process of thinking that reflects how to arrive at a certain conclusion instead of narrowing the focus on memorization and fact gathering. We have reduced it to a race, this mania to have the best test scores, to be the best school in relation to the data; students ultimately pay the price. —Parent

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Both, Balanced, or Integrated? The following respondent expresses his dismay at what seems to be the ongoing tendency to polarize debates in mathematics education. According to the respondent, the answer is never one or the other. For the first and second respondent, the approach includes aspects of both arguments. Similarly, the last respondent addressed the policy language that produced seemingly irreconcilable notions of what counts as mathematics. Policy language that described mathematics as a balance between “basic skills” on the one hand and “problem solving” on the other perpetuates what often translates in classroom practice as two competing beliefs about what mathematics is. The respondent suggests that policy language that uses the word “integrated” rather than “balance” may facilitate a conception of mathematics that sees its various components as interrelated. Aligning the curriculum with the state test is a good idea. Deciding whether math should be taught as skills or as problem solving is a waste of time and an example of the dangerous tendency in education to see teaching choices as black/white, either/or. Clearly, children need to learn the basic skills of math. Clearly, children need to learn problem solving. How can we educators still be arguing these questions? Let’s encourage the facilitator, the teachers, the principal, and the math professor to all loudly yell, “Both!” and then proceed immediately to figuring out how to keep both in daily lessons. Speaking of “both,” the teacher who said “They’ve got to memorize the chart and learn the trick so that they can do well on the state test” has a valid point. Of course we want children to understand concepts. Still, students who are not gifted mathematically may do poorly on tests if their math instruction focuses too much on understanding and fails to give them the “tricks” that can help them remember how to solve a problem that they no longer are totally clear on. For example, knowing the formula for area of a triangle is worthwhile even for students who can’t explain or remember why that is true. We must, again, stop arguing which approach to teaching is better, agree that “Both!” is the answer, and spend our time and energy on figuring out how to do some of each day after day. —Special Education Resource Teacher

One of the lessons seen here is that having students memorize charts for the purpose of doing well on standardized tests neither supports educators’ efforts to improve their teaching methods nor helps students to obtain higher-level thinking skills. In the end, a balance must be struck between promoting high exam scores and creating good educational experiences that can potentially improve students’ professional opportunities. Teachers must

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make every effort in order that students know the meaning of a mathematical equation and know how to best approach and solve it. Having students memorize problems or charts (as adult students do in private test-prep courses for graduate school entrance exams) ultimately converts learning into a series of shortcuts and nifty tricks designed to beat a test. Even with high standardized test scores, students can become turned off to mathematics (or any other school subject) and any sort of real intellectual engagement. This may cause students to tune out or simply drop out of school. Teachers should employ a richer, more socially contextualized curriculum and instruction methodology. In the end, such a change might even improve students’ test scores as it raises graduation rates. —Doctoral Student Instead of thinking about all the different aspects of mathematics (basic skills, problem solving, new material from the textbook), and how all three of these can be taught in one short mathematics class, the teachers should be thinking about how they can integrate all the new knowledge with the students’ basic skills of math and with problem solving. Mr. Barker seemed to be doing this on the surface, but when analyzed, one can see that he was not integrating the problem solving to actually learning the material. I do like what the mandates say about having balanced mathematics education, but I would rather it be called “integrated.” By using the terminology of “balance,” it gives the idea that the structure of mathematics should have problem solving and basic mathematics as two aspects of instruction. However, using the terminology and concept of integration, both basic math skills and problem solving can be integrated to better deepen the acquisition of the basic math skills and help students improve on their ability to problem solve. —Graduate Student, Prospective Special Education Teacher

Professor Jordan’s Dilemma Virtually all respondents focused on the negative impact of high stakes testing on teachers and students. However, one respondent directly addressed Professor Jordan’s dilemma. Her recommendation suggests that improving outcomes on standardized tests and helping students to learn mathematics can both be accomplished. Professor Jordan should probably sit down with the principal as well as the teachers in order to discuss with them different possibilities and teaching strategies that would promote long-term understanding on the students’

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part. I think that if she is straightforward and clear in her explanations, then hopefully the teachers and principals will take information into consideration in order to improve the students’ results on the test while at the same time help the students to really “learn” from the lessons they are taught. —Undergraduate, Prospective Teacher

A Prospective Teacher’s Dilemma Finally, one respondent related to her own situation the questions raised in the case. The multiple issues raised in the case resulted in confusion about what she should really do if she were a teacher in this situation. The problem that occurs in this situation is that there is not much you can do. You can try, as an educator, to do your best in simply teaching what the state wants you to teach. All too often, however, you find that this is more difficult than they assumed. But because you are given so many boundaries and so many specific guidelines, you are pretty stuck. I personally do not know what I would do in such a situation. I would think that one option might be to present your issues and a solution to the school board. Another way would be to creatively come up with ways to incorporate all the important details into your lesson plans so that you are doing the most adequate job you can. This brings up a really deep issue. How do you work around legislation that has been proposed and enacted? How do you make sure you abide by all the rules that are given to you? It is definitely a very difficult problem to consider and face as a teacher. —College Senior, Student Teacher

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SUMMARY AND ADDITIONAL QUESTIONS The case was intended to illustrate how policy initiatives that mandate “accountability” and “standardized testing” impact the ways in which teachers think about and do mathematics teaching. Respondents to this case raised many of the issues documented in a study conducted by Linda McNeil (2000). In Contradictions of School Reform: Educational Costs of Standardized Testing, McNeil provides a vivid account of what happens to policy initiatives at the level of practice in real classrooms. She argues that accountability by a single standardized test only serves to “reduce the quality and quantity of what is taught and learned in schools” and “over the long term, standardization creates inequities, widening the gap between the quality of education for poor and minority youth and that of more privileged students” (p. 3). There is some empirical evidence that policies embodying some form of “accountability” can have a positive impact on academic outcomes (Barnes et al., 2000; Cohen & Hill, 2000). Nonetheless, her findings should alert us to the role that policy makers, teachers, and other school participants may unwittingly play in perpetuating educational inequities. Here are some questions to ponder: 1 How does the political context illustrated in the case impact how math lessons are delivered in classrooms? 2 How might policies, like those mandating high stakes testing, contribute to reform failure? 3 How might policies, like those mandating high stakes testing, promote reform success? 4 When mathematics education reform policies fail, who gains? Who loses? What are the implications for access to college preparatory mathematics, particularly for poor and minority students? 5 In a society that espouses democratic principles, what does “accountability” look like if we are interested in promoting students’ mathematical understanding in substantive ways? 6 How might teachers negotiate politically charged contexts in ways that preserve their professionalism and promote substantive learning about mathematics?

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REACTIONS TO THE FOUR CASES The following commentaries are about the cases as a whole. Respondents contributed remarks that are largely personal reflections on teaching mathematics from the perspectives of parent, principal, student, grandparent, teacher, and school board member. These were my thoughts as I read all the cases. First, the teachers did not know how to teach. That’s evident with most teachers in America. Second, the teachers have preconceived notions about minority students not being able to learn difficult curriculum of any type. Poor math teachers become defensive and blame the students and the parents. My grandson started Jefferson Prep, a 7th- and 8th-grade middle school, known as the district’s public/private school since one has to apply and be selected through a lottery process for admissions. Grades are important but the primary criterion is good behavior. The school only has a handful of Black students. On his first day of school, our first encounter with the principal and vice principal was very negative. They immediately saw my grandson as “one more trouble maker and underachiever” without even looking at his previous performance. They gave themselves away when the VP saw my grandson talking to his White friend who was a well-behaved student with high grades, and then asked ME how did my grandson know this student? I was furious! I guess I could go on for days about teachers, but one must remember that they too don’t get any help from the administration, just demands placed on them, while they are already struggling with their teaching skills. I agree with the teacher in “Culture and School Mathematics” who said, “Math is math.” As for the middle school teachers in “Race and Teacher Expectations,” non-English speaking students can learn math as long as it is taught correctly. A teacher’s attitude along with enthusiasm can go a long way. Students need to be challenged, made to be interested in a subject in a fun way. Why should it be so much easier for a teacher to teach White students than it was for him or her to teach Black students or non-English speaking students? It’s all about attitude and teaching skills! —Grandmother

These are interesting scenarios for a non-teacher. I often wonder what goes on in classrooms in our school district and wish there were easier ways to assess the quality of individual teachers without threatening them. It seems to me that our schools are badly in need of fresh air, of attitudes

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like the student teachers in these case studies, and that the older generation needs to realize that demographics and student needs have changed, and it is not effective to blame families and students for their failure to succeed at the standards being thrust at them by state boards of education. I’d like to see a wide ranging curriculum available for all teachers and staff to open the way for conversations about how school can be more interesting, challenging, and effective for all students. My own highachieving daughter is bored to death by most of her classes and finds high school an extremely negative experience. —School Board Member

Here is what I thought about as I read the cases. They pretty much made my blood run cold as those examples pointed very much to the areas of my failures as a teacher. My teachers in middle school and high school were all White, had perfect control of the classes without any apparent (or memorable) methods—the students just seemed to behave and do their work. The mathematics classes were always conducted the same way: go over last night’s homework, pass it in, have a lecture on the “next topic,” see examples on the board, do a few examples from the book and start on the assignment with the time remaining. It seemed to me like a good way to earn a living. I would get to stand in front of an audience and get paid for what I liked to do best—solve math problems. From about two months into my geometry class in my sophomore year of high school I knew what I was going to do until I retired. My experiences as a student in middle school and high school were limited as we were “tracked” and I seldom had any contact with students who weren’t “college bound.” I knew I was “going to college” from the time I was two years old (“You need to learn to tie your own shoe laces because who’s going to do it for you when you go to college?”, etc.). My student teaching consisted of five weeks in a predominately White school teaching chemistry classes that had been in session about six weeks before I got there. I gave my lectures and they did the work—a piece of cake. I learned nothing. My methods of teaching “worked” for me as long as I was given the algebra and the classes above algebra. When I “got stuck” with some middle school math classes which, to make things worse, had some non-English speaking Cuban exile students in them, my usual “methods” failed me and I was an embarrassment to myself, my students, the rest of the faculty, and the administration. I had nothing to fall back on to help me recover the year. June eventually arrived and everyone sighed with relief. As time passed and I had taught more classes of that nature, I developed some methods of diversifying my lessons and getting the students more

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involved in the classroom work. Madeline Hunter2 lectured at our district in about 1980 and I kept a 3 × 5 card with the basic steps of her method written on it in the top right hand drawer of my desk from then until I retired. Ideas like that helped me but I still had problems when my classes were large and I had students who didn’t walk into my classroom with an attitude of being prepared to sit quietly and soak up knowledge. (“I ain’t no schoolboy!”— the response from a Black student in one of my basic math classes when I urged him to pay attention—“Schoolboy” being a derogatory term used by his peers and the last thing he wanted to be known as. I had no answer to that.) About the only great experience I had with classes that were other than the “cream” was in Austin, Nevada, where my classes ranged anywhere from 3 to 16 students. I could individualize and work with each student who needed help every class period. But with 35 to 40 students in “those classes” I would sometimes be just hanging on for dear life. In California in the mid 90s I took classes in cultural knowledge and teaching techniques and even received a little certificate. My district got some kind of credit that I had done that but to actually use those methods in my classroom was pretty much beyond me as I was already maxed out on time and energy and simply did not have the energy or organizational skills to implement many of the ideas presented. —Retired, K-12 Mathematics and Science Teacher The case studies illustrate many relevant and realistic issues for the mathematics teacher today. Each case paints a different aspect of the issues of teaching mathematics in changing times of globalization. These matters are relevant not only to the American schools, but schools throughout our multicultural world. The case studies reminded me that as a teacher, I must be changing with the times, which comes in the form of changing students, changing reforms to teaching, and changing expectations of schools. My view of students must not be affected by the cynicism perpetuated by the profession and prior held prejudice. I was also reminded that mathematics knowledge is not limited to certain groups of people, but accessible for all people. Therefore, even if reforms are non-traditional and unfamiliar, it is worth providing quality mathematics instruction to all students. As the policy makers and administration try to address the needs of changing communities, I realize that the demands of a teacher given by higher authorities are heavier. However, I also think that collaboration and accountability among the education community must be implemented. If

2 Madeline Hunter is most known for her seven-step lesson design that included objectives, standards, anticipatory set, teaching (input, modeling, check for understanding), guided practice/ monitoring, closure, and independent practice.

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reform can become reality, reading and discussing the issues raised by these case studies are necessary. —Middle School Mathematics Teacher

In reading the cases, I am struck by the fact that teacher attitudes about race and culture have changed very little in the 22 years since I graduated from high school. When I was a student, it was assumed that Black students would not be placed in the higher math courses and only when a parent protested, was a Black student placed in an appropriate class. It saddens me to know that these attitudes persist. The cases brought memories of the hurt and confusion that I felt in some of my interactions with my teachers that I now recognize as racist behavior. A White teacher tried to transfer me out of his class to another that was predominately Black. It seemed that all of the Black students at that school were in that class which was taught by a Black teacher. The only reason I was not placed in that class was because the Black teacher refused to accept one more student in her already overcrowded classroom. She taught only remedial courses. I remember being good at math—and enjoying it—in grade school. When I reached 7th grade, that positive attitude turned into a math phobia. At that time, we were given standardized tests to determine at which class level we were to be placed: remedial, regular, accelerated or gifted. I did very well on the standardized tests, scoring high above average in Language, Social Studies, and the Sciences. I was placed in the classes for “gifted” children in all of those subjects; however, my math scores were well below average for my grade level. At some point, I had missed learning fractions and could not complete the test. Instead of investigating why my math scores were so low, the decision was made to put me in the accelerated math class—algebra I. I did poorly in that class. My teacher assumed that I was being lazy and inattentive and was very punitive in her dealings with me. She never contacted my parents to discuss my difficulties and gave me detentions when I could not complete my homework. Never did she ask me if I understood the subject matter. She never gave me the extra tutorials that she gave the other students. The White students were nurtured and supported while I was punished. I was the only Black student in the class. My self-esteem took a nosedive. As a result, my performance in my other classes declined. They kept me in the gifted classes because they needed at least one Black student to keep up appearances. They even took pictures. In my junior year in high school, the White students in my college prep classes were meeting with the school counselors and deciding to which universities they would apply. A school counselor never contacted me. When I went to the counseling office to pick up college applications, a counselor steered me toward the junior college applications. The White

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students were taken by bus to an SAT preparation class during our Civics course, while the one Latina student in the class and I were sent to the library for a “free hour.” I only learned where my fellow students were going when one of my friends showed me his SAT prep book. I had no idea that such a thing existed. I took the SAT without any preparation and without solid math knowledge. I scored in the top 5 of my senior class. A recruiter from Boston University came to our school to interview another student and also asked to see me. The school administration was flabbergasted. I finally received attention from the counselors and the assistant principal credited the school for my high score. I was paraded in front of the Superintendent of Schools. No one mentioned the fact that I scored relatively low on the math portion of the SAT—except the recruiter from Boston University. I know now that my 7th-grade algebra teacher was terrible at teaching math and could not answer the conceptual questions I had about algebra. I was labeled as an “underachiever,” a label that would follow me through high school. The math phobia that plagues me to this day feels almost like a disability. It very much influenced many of the decisions that I made regarding my undergraduate and graduate studies and even my career. —Social Worker Administrator

These cases resonate with my experiences as a White mathematics teacher in a variety of school settings where I have come to see culture as inextricably linked to teaching mathematics. In my experience, the comments and attitudes expressed by teachers in the vignettes are not atypical. Myths and stereotypes about students whose cultural backgrounds differ from the dominant White, middle class culture abound. If mathematics teaching is to improve for all students, teachers must realize that the attitudes and beliefs that they bring to the mathematics classroom are cultural and may affect their students in both positive and negative ways. Teachers need to spend time learning about the lives of their students, particularly if they are from different ethnic, race, or class backgrounds. Teachers need to work to develop classrooms where student differences can be seen as strengths, and a variety of approaches is seen as desirable. —High School Mathematics Teacher

What I have seen over my many years in schools is that all of us have comfort levels with mathematics and with risk taking. This is true for students, teachers, administrators, and parents. I have seen teachers who are risk takers but with weak conceptual mathematics backgrounds who have become remarkable mathematics teachers, inspiring students to become critical problem solvers. Often they say they learn by teaching. Sometimes

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these are the best teachers. Of course there are brilliant mathematicians. Unfortunately, this does not necessarily make them good math teachers since they cannot create the opportunity and environment for others to construct mathematical meaning. If students are given an opportunity to play with mathematics and discuss best practices for creating mathematics learning communities; and if administrators know how to support this; and if parents are honored in this mix; then there is hope for change. It is a big process. I have seen it work. It takes time and a dedicated leader who understands the urgency for children and a learning community that does not allow Mrs. Carlton to label Benjamin as a special education student without formative information, or Mr. Barker to never link grouping activities to the exercises that follow them, or never leave time for Mr. Lawson to really explain to Mr. Robinson exactly what he meant when he said, “Oh you mean the multicultural thing. I think it’s pretty cool.” —Retired Teacher and Principal, Dual Language Elementary School

II PUBLIC ARGUMENTS

In Part I, you read the responses to four cases that use school mathematics as a context for exploring the historical, social, and political dimensions of schooling. These cultural dimensions of schooling, particularly of teaching and learning, are absent from many teacher education programs. In addition to learning theories and designing lessons, prospective and practicing teachers ought to be engaged in on-going discussions about how cultural contexts influence the forms that mathematics curricula take and who gets to learn mathematics. Should all eighth graders take algebra? What is the role of culture in learning mathematics? How does a teacher’s race or class bias (unwittingly or not) play out in mathematics classrooms? How does high stakes testing affect the quality of educational experiences for teachers and students? Most importantly, what is the democratic justification for any educational decision? These questions are not trivial. Their answers have moral, ethical, political, and economic implications for our nation. The cases were intended to spark discussions that should remain the centerpiece of any teacher’s professional development and understanding about what it means to be a teacher in a democratic society. However, one must have a historical context for discussing these important questions. The history of schooling in the U.S. has largely consisted of the struggles for equality of educational opportunity. In 1868 the 14th amendment to the constitution gave full citizenship to former Black slaves. But it was not until 1954 in Brown v. Board of Education of Topeka that the U.S. Supreme Court ruled that separate but equal schools for 77

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Black and White children was unconstitutional. In 1974, U.S. Federal Judge W. Arthur Garrity ruled that the Boston School Committee was in violation of Brown v. Board of Education, resulting in violent desegregation wars. Hence, it should not be surprising that in the twenty-first century, students in poor urban schools endure decaying facilities, hyper-segregation, lack of monetary resources, and minimally qualified teachers. These conditions contribute to underperformance of students in these schools. These appalling conditions often have been characterized as an “urban” problem. These conditions and the poor achievement results they produce, however, are a symptom of a larger problem that extends to all schools, including those in suburban and rural communities. Notwithstanding the inequities of the larger culture and underperformance in other subject areas, the focus of this volume is the teaching and learning of school mathematics. The cases in this volume illustrate the cultural embeddedness of the issues involved in making decisions about mathematics curricula, teaching, and learning. Part II offers public arguments that comprise the claims and debates we encounter in print and television media, state legislatures, boards of education, school board meetings, and school hallways and corridors. Part II offers three different lenses in which to interpret the responses you encountered in Part I. Three labels serve as public arguments—“conservative,” “liberal,” and “radical multiculturalist.” They have been crafted so as to represent a broad spectrum of views, both across and within each orientation. In the conservative orientation mathematics for maintaining global economic prominence is emphasized. Equity and access to challenging mathematics is highlighted in the liberal perspective. The radical multicultural perspective emphasizes school mathematics as a way to develop critical dispositions for solving social problems. These views not only represent public arguments regarding culture, schooling, and teaching mathematics, they also capture features of how we view our social realities and the implications for our relationships with others in the world. It is hoped that an examination and discussion of these public arguments will enable you to make further sense of the claims heard and read daily, and that you articulate and understand better your own views. Your individual views will not fit easily into any one of these “slots.” We are complicated beings. It is more likely that you will find that you share ideas and sentiments with two and probably all three of the public arguments. To that end, you are encouraged to “enter” into each point of view understanding it on its own terms and to also look at each one with some distance and skepticism. Additional general and specific questions and issues appear after each

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public argument. Public arguments are linked to the case studies in Part I with the hopes of generating more discussion of the particular incidents. You are encouraged to explore further these positions. It is hoped that these linkages will enable further discussion of the particular incidents in Part I and the general claims made in Part II.

A “CONSERVATIVE VIEW”: MATHEMATICS FOR GLOBAL ECONOMIC LEADERSHIP Introduction In an increasingly technological age, it is crucial that our children can perform basic computation skills with mastery. In addition to basic computation skills, the nation’s children should be able to solve problems involving percents, ratios, and finding the areas of polygons. Unfortunately, the National Assessment of Educational Progress (NAEP) consistently documents that our students cannot perform these basic skills. The following problem was an item on the 1996 NAEP test. It was administered to twelfth-graders. The population of the United States is approximately 250 million, and the national debt is approximately 4 trillion dollars. If this debt were divided equally among the population, what would be the debt, in dollars, per person?

Only 39 percent of twelfth-graders, from a sample that included students from public and private high schools, were able to correctly answer this question. If 12th-grade students cannot correctly solve a long division problem, then our students are not being well prepared to pursue advanced math and science courses. If they don’t have the basic skills necessary for these courses, then there will be no students for the math and science professions that are an essential aspect of our thriving economy. Such a situation threatens our nation’s economic survival and global prominence. Progressive educators such as liberals, radicals, and multiculturalists worsen matters when they criticize us for wanting a mathematics curriculum that focuses on basic skills knowledge. They accuse us for not attending to children’s cultures and emotional needs. This is simply not true. We believe that in the United States every child, no matter race, creed, or color, has the opportunity to succeed. Though we believe that

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certain groups have unfairly suffered in this country, we also believe that with individual initiative and drive, any member of a group can rise above his or her conditions. Social progress is dependent upon the hard work of individuals in our society. We must remember that this great nation was built upon the hard work of many individuals escaping persecution or economic hardship in their native countries. It is true that we do not believe that infusing multiculturalism into mathematics improves student learning. Instead, we believe that common cultural values serve as the glue that holds together our society as one undivided nation. We believe in schooling that provides the most talented and hardworking individuals with the tools needed to become economically productive members of the society. In addition, schools should socialize children into adult roles to uphold the laws of the land and schools should transmit our nation’s history, traditions, and values that make this land great. Below, we elaborate our position about the role of mathematics in our schools.

Differences, Harms, and Wrongs If we want to maintain our leadership in the global economy, mathematics must be a core aspect of the curriculum. Progressive attempts to reform the nation’s mathematics curriculum threaten our economic leadership. Progressive educators criticize us for wanting a mathematics curriculum that focuses on basic skills knowledge. Progressive attempts to reform the curriculum, as exemplified in the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000), have resulted in “fuzzy” or “feel good” mathematics where students do not learn the basic facts and computational skills. In fact, the curriculum is bereft of any real mathematics knowledge. This is harmful to our youth and our nation. We think Dr. William G. Quirk, a mathematician, said it best: If our kids never learn the importance of remembered knowledge, and if they are programmed to think that memorization and practice are not necessary, then what happens if they somehow reach medical school and need to quickly memorize thousands of facts from Gray’s Anatomy? It’s difficult enough even with the traditional preparation of the mind. This is not just about kids who go to medical school. The current reigning educational philosophy is dangerous for all our children. If they are to be successful in life, they must effectively use the amazing knowledge-storing power of their

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brains. Are we really going to continue to let today’s educationists program our kids to believe that remembering specific knowledge is a bad idea, and that computer “tools” and “look-up skills” are the key to success in business, professional life, and personal knowledge-based interests? Who will build bridges in the 21st century? If current trends continue, the answer will be Asians and Europeans. They still believe in knowledge transmission and the critical importance of specific, remembered knowledge. They still stand in awe of the amazing knowledge-storing power of the human brain, and they’re leaving us in the academic dust, even though they typically educate 40+ students in their classrooms.3

Multiculturalists believe that schools should pay attention to the cultural differences of every possible group. With this attention, the curriculum becomes an endless list of harms and wrongs inflicted onto every group. This balkanization leads to social fragmentation; it divides us rather than unites us. Multiculturalists also believe that adding “culture” to the mathematics curriculum will improve achievement for students who have historically under-performed. The problem with this claim is that mathematics is an objective discipline comprising universal principles and facts that always hold, no matter a child’s culture. One’s national origin or race-ethnicity has nothing to do with a student’s performance. Instead, it is the responsibility of the family to instill the values necessary to succeed in school and it is the student’s responsibility to work hard. Schools are meritocracies that reward deserving students for their academic efforts.

The Acclaimed Antidotes: Self-Esteem and the Multicultural Mathematics Curriculum Many liberals believe that schools should be child-centered. That is, they believe that the school should develop a child’s self-esteem, promote a culturally responsive curriculum, and understand the forces that work against children. This represents a “child-as-victim” mentality. That is, the child does not share the blame for failure. We believe this focus is an incorrect view of the purposes of schooling and the processes of learning. Schools do not exist as institutionalized therapy sessions and learning should not focus on the learner’s self-concept or self-image. Claims that

3 See http://wgquirk.com/#who.

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minority members need to understand their cultural roots through learning more about their culture’s contributions to our subject matter and knowledge base are misguided. And if one persistently views and treats minority children as victims of a larger system of oppression these students will never become empowered to act in constructive ways. Before we elaborate our preferred educational program, we explore more fully the three themes of self-esteem, a “multicultural” mathematics curriculum, and children as victims. Self-Esteem. There have been many attempts to improve mathematics achievement by enhancing children’s self-esteem. The rationale for these approaches goes something like this: Minority students do not do well in school because “societal messages”—advertisements, media, school curricula—tell them that they are inferior to White middle class students. These students, so the rationale goes, cannot learn if their self-esteem is damaged. The solution has been to help students feel better about themselves. If they feel better about themselves, then they will learn. One approach has been to make all curricula accessible to all students, since tracking, it is believed by progressives, is harmful to low achieving students’ self-esteem. In the last fifteen years, many states have mandated that all students take algebra before graduating from high school. This involves a large degree of de-tracking whereby students get placed in courses with others representing a wide range of ability. Putting students into challenging mathematics courses, who do not have the ability, preparation, or inclination to be successful, merely exacerbates the problem. It is also harmful to high achieving students. Teachers have to slow down the pace of learning, which prevents high achieving students from learning at their potential. Dropping out or failing courses does not enhance selfesteem. The key to improved minority achievement is not to put low achieving students in rigorous mathematics courses to enhance their selfesteem. Rather, the key is a curriculum that challenges students to use their minds, disciplines their thinking, and rewards them for the work they have accomplished and the standards they have achieved. We cannot expect students to exceed beyond their potential. What we can do is provide all students with the necessary skills and the opportunities to practice those skills. This is done through high quality instruction. Teachers who set high expectations and who persistently push and prod students to reach their potential are needed. Because schooling is a merit-based system, those who work hard will be rewarded for their hard work. Therefore, low achieving students will be able to move into courses with high achieving students, if they work hard.

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“Multicultural” Mathematics Curriculum. Another aspect of building self-esteem, are attempts to make the mathematics curriculum “multicultural.” A common theme is that in a diverse society the curriculum needs to “honor” that diversity. Some of the more extreme adherents of the multicultural orientation maintain that our existing curricular content is essentially “Eurocentric.” Supposedly, such a curriculum enhances White people’s self-esteem while harming all others. It enables the White majority to do well in school but creates obstacles for children of color. According to this critique, the mathematics curriculum as currently constituted represents a widespread Eurocentric bias in the cultural and historical development of mathematics. This bias gives credit to White European males as the progenitors of mathematical knowledge. In doing so, it ignores the contributions of people of color. Multicultural proponents also ignore the necessity for learning basic facts. Instead, they prefer a curriculum based in “real world problems” where children “construct” their own mathematical knowledge. The problem with “constructing” knowledge is that students may construct incorrect understandings. Without direct instruction, they may never learn to do mathematical procedures correctly. We call this “fuzzy” mathematics, including constructivist teaching techniques. As far as we are concerned, not much of anything gets taught or learned. Teachers ought to stick with proven techniques for mastery learning: memorization and repetition. John Saxon (1982, 1984) is correct when he says that mathematics textbooks lack the opportunities for students to practice math skills. Teachers teach the skills. The students are tested for mastery. Then the teacher moves on to the next topic. Because students are not given an opportunity to practice and master skills, these skills have to be retaught every year. Constructivist teaching approaches and mathematics curricula focused on conceptual understanding only contribute to the sorry state of affairs regarding our students’ mathematics achievement. According to Saxon (1984), “[t]hese books [work] so hard to instill understanding that they [neglect] the teaching of skills whose mastery [require] considerable practice” (p. 12). Thus, student understanding is incomplete if they are unable to get the correct answer. Furthermore, no matter who influenced the development of mathematics, one must learn the basic math facts. The sum of 2 + 2 does not change with cultural contexts. One can know that the Egyptians and the Mayas had developed sophisticated mathematical understandings but still do quite poorly in math. Simply adding the contributions of Arabs, Africans, and women to the mathematics curriculum so as to have minority representation in the curriculum bespeaks the worst type of tokenism. The

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mathematics curriculum represents the basic skills and procedures that all students should learn to enable their participation in maintaining a unified culture and a strong economy. But all too often multicultural sloganeers are enforcing a quota system; they are not enhancing our cultural understanding. As a result, a multicultural mathematics curriculum is watered down, offering little substance and yielding little education progress. Child as Victim. Minority student as victim is the negative result of this unwarranted focus on self-esteem and the push for a multicultural curriculum. Multicultural literature provides a distorted view in which the world is populated by two types of people: the oppressors and the oppressed—the dominant class and the victims. The problem with this view is the victims are not active agents and therefore play no role in improving their conditions. We have already acknowledged that many minority groups have experienced hardships that others have not endured. And certainly even today some minority students face hardships that others do not have to confront. But when students begin to see themselves as victims they blame their entire situation on others—they claim that they are the victims of societal injustices. In doing so they shift the onus of responsibility onto others: They shoulder no blame and believe themselves to be innocent. With such a view of things, it is difficult to see how these students will better learn mathematics, take advanced mathematics courses, and gain admissions into colleges and universities. When students see themselves as victims, they will not achieve in school, let alone mathematics. It seems to us that much of the current focus on culture in schools serves little educational purpose. In fact it appears that many of the multicultural mathematics programs and proposals detract from rather than enhance students’ educational progress. Next, we outline briefly what is required of all schools and of all students (regardless of color) so that everyone has a fair chance of academic success.

Our Plan: Academics and Character for All We believe that academics, especially subjects like mathematics, are essential components of the core curriculum. We have argued that a focus on culture and self-esteem undermines the school’s mission, the child’s sense of worth, and ultimately the country’s economic future. In these increasingly changing times, much is at stake. National comparisons such as NAEP and international comparisons like the TIMSS indicate that our students perform poorly when compared to students from other countries.

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If we plan to remain a global power, this cannot continue. We propose a mathematics curriculum where every child acquires basic skills knowledge. We are not alone in our beliefs. Mathematically Correct, a parent advocacy group concerned with improving mathematics education, has outlined a curriculum that we find appropriate. For kindergarten through 6th grade, they recommend mastery in the areas of number sense, number facts, comparison and fractions, measurement, graphing, time and money, patterns and geometry. They also outline mastery standards for prealgebra, algebra I, and geometry, subjects typically taught at middle school and high school. We share with you a portion of the curriculum outline for kindergarten and 6th grade on number sense. Number Sense Kindergarten The student will . . . • K-1 count in various ways including counting objects up to 12, counting by ones up to thirty-one and backwards from ten, skip counting by fives and tens to 50 and by twos up to ten (2 to 10 and 1 to 9) • K-2 identify written numbers from 0 to 31 • K-3 select the correct numeral to indicate a quantity from 0 to 9, trace over the numeral, and write the numeral • K-4 select a reasonable order of magnitude from three given quantities—a one-digit number, a two-digit number, and a three-digit number (e.g., 5, 50, and 500)—for a familiar situation • K-5 identify ordinal positions from first to fifth using concrete objects Sixth grade The student will . . . • 6-1 read, write, and order positive and negative decimals to the nearest hundred-thousandth • 6-2 write decimals in expanded form and write numbers in expanded form with scientific notation • 6-3 round whole numbers to the nearest ten through million • 6-4 round decimals (and decimal quotients) to the nearest whole number, tenth, hundredth, and thousandth • 6-5 read and evaluate numerical expressions with exponents • 6-6 identify powers of 10 to 106 • 6-7 compare positive and negative decimals, mixed numbers, whole numbers and fractions with like and unlike denominators, using the signs , and =, including scientific notation4 4 The full text of the curriculum outlines can be retrieved from http://mathematicallycorrect.com/ kprea.htm.

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We believe, as do concerned parents, that every child should be able to add, subtract, multiply, and divide. College preparatory mathematics must be made available for the students who are the most capable and the most hard-working. Our economy is dependent on the intellectual talent of the students who have the capacity to pursue math and science professions. However, there is another essential component to our plan. While cognitive growth is important, so is character development. We would do better to focus on character development than focus on a multicultural mathematics for a child’s self-esteem. As we have argued before, a focus on multicultural mathematics curriculum in the interest of promoting a child’s self-esteem merely victimizes the child. We do believe that the school should keep a watchful eye on the child’s emotional well-being, but we think this is better accomplished through character education. Students must learn what it means to pull oneself up by the bootstraps if they are to achieve the American dream. They must be able to persevere in the face of adversity, delay the desire for immediate gratification, and be honest and respectful of authority. These character aspects are necessary for success in the workforce. The school’s focus on academic knowledge and character education will provide skilled workers enabling our nation to maintain global economic prominence.

COMMENTS AND QUESTIONS “The Conservative View and You” General Questions 1 What aspects of this public argument seemed to capture your understanding of the culture in mathematics debate? 2 With what aspects of the debate did you disagree? Why?

Specific Questions 1 What comprises the mathematics curriculum that conservatives propose? 2 Would a recognition of and value for cultural differences necessarily diminish mathematics curriculum? If so, how? 3 Does the route to self-esteem offered by conservatives entail a denigration of some students’ cultures? How so or why not?

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In the next few paragraphs, the case studies are briefly summarized and analyzed according to the conservative view just outlined. Do you think the analysis fits with a conservative view of each case? Why or why not? After considering each analysis, do you still feel the same way about the conservative view?

“Race and Teacher Expectations” Conservatives might say that Mrs. Carlton’s concern with Benjamin’s behavior and poor academic performance is legitimate. A child’s bad behavior is harmful to the learning environment. When the teacher has to address bad behavior, it requires a good deal of instructional time. This situation negatively impacts the learning of children who are well-behaved and want to learn. Therefore, it is important to remove behavior problems from the classroom as soon as possible. As for Anna’s dilemma, she should not share her case study with Mrs. Carlton. Benjamin’s mathematical competency outside of school is interesting. However, his performance inside the classroom is what counts.

“Mathematics for All?” Conservatives might say that it is unreasonable to expect all children to learn algebra. Since children have different strengths and abilities, the role of the school is to ensure that the best and the brightest students receive the necessary education to ensure the nation’s economic productivity. Some might support a stratified curriculum in algebra that accommodates students’ varying levels of achievement. With this scheme, all students would learn some algebra, while the best and brightest would learn college preparatory algebra. Many conservatives would say that it is most important for the school’s growing Latino population to learn to speak English and to learn about our common American culture.

“Culture and School Mathematics” Conservatives might say that Mr. Lawson is correct to be dismayed by the district’s proposal to embed multiculturalism in a discipline like mathematics. Conservatives believe that simply adding the contributions of non-European groups results in a watered-down curriculum, bespeaks

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the worst type of tokenism, and undermines the foundations of a unified culture and economy. All children need to learn basic math skills. Introducing “culture” in the mathematics curriculum will not achieve this goal.

“Politics and School Mathematics” Conservatives might support a “balanced approach” to mathematics teaching. They view standardized tests as useful tools to hold schools accountable for students’ learning. Therefore, it is a good idea to share test-taking strategies with teachers, the Daily Time Management chart, and content standards aligned with the district’s textbooks so that district scores improve. Most likely, they would not be supportive of Professor Jordan’s problem solving approach to mathematics teaching. The teachers should be most concerned about making sure that the students memorize the values of the places in our numeration system, since this level of understanding is regarded as basic skills.

A “LIBERAL VIEW”: EQUALITY OF OPPORTUNITY AND SCHOOL MATHEMATICS IN A DEMOCRACY Introduction Like the conservatives, we liberals, too, believe in a market capitalist economy. However, a market economy is prone to cycles of recession. We believe that if the free market is left unregulated, it will disadvantage the most vulnerable groups of our society. Therefore, it is necessary for our government to intervene. This minimizes the harmful consequences of economic downturns. Rather than focus on educating our youth to maintain global economic leadership and ensuring a “common” culture, we believe that schools should focus on providing equality of opportunity and honoring the diverse cultures of the children who attend our schools. Schools should socialize children in ways that promote a healthy and thriving multicultural and just world. We are most concerned with educating a citizenry for full participation into a democratic society. Unlike conservatives with their focus on individual initiative, we believe that individual effort alone is not always sufficient to overcome adverse life circumstances. If we are concerned about fair treatment for all citizens and economic stability, then the government is responsible for economic and

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social interventions. We liberals, unlike conservatives, seek a balance between the needs of the individual and the needs of the society.

Equality of Opportunity in the United States As liberals, we acknowledge that our democracy and capitalist economic systems are imperfect. Despite the flaws, we believe that our system has provided economic and social mobility far better than any other system in the world. Nonetheless, we must continue to improve. Our nation has taken great strides to do just that. The University of Iowa, in 1856, was the first to admit women. Shortly thereafter, Vassar, Wellesley, and Smith Colleges were founded. Historically Black Colleges and Universities such as Howard University in Washington, D.C. and Hampton Institute in Virginia were established with the help of the Freedman’s Bureau in 1868. At the same time, laws operated to limit civil rights. Plessy v. Ferguson, an 1896 Supreme Court decision, upheld a Louisiana law that segregated railway passengers by race. Known as the “separate but equal” doctrine, it wasn’t overturned until 1954 in the landmark Brown v. Board of Education when the Supreme Court ruled that legalized racial segregation of schools was unconstitutional. Unfortunately, the “separate but equal” doctrine contributed to producing a set of inequitable conditions that we continue to address to this day.

Equity and School Mathematics School mathematics is just one example of how far we still need to go to improve inequitable educational conditions. Tate (1997) examined National Assessment of Educational Progress (NAEP) trends assessments for 1973, 1978, 1982, 1986, 1990, and 1992. He reported good news and bad news. The good news was that all three racial-ethnic groups showed growth in mathematics proficiency. The bad news was that the data showed large differences between the achievement of White students and that of African American or Latino students at all age levels. Tate reported that on average, by age 17, no racial-ethnic group was performing at the highest performance level. This finding is especially disturbing since these trends assessments are largely basic skills examinations. The NAEP 1999 trends assessment reports on what seems to be a chronic achievement gap. In 1999, 17-year-old African American students performed significantly better than 17-year-old African American students in 1982. Though not

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significant there was a 6-point drop in scores between 1992 and 1999. However, the gap between the performance of White and Black 17-yearolds had widened. In 1990, the difference in scores between these populations was 21 points. In 1999, the difference in scores was 31 points. The gap has been reported as not statistically significant. The most recent long-term trends assessment (U.S. Department of Education, 2005) reports that while 9-year-old and 13-year-old African American and Latino students have made statistically significant gains in closing the racial achievement gap, 17-year-old African American and Latino students have not. Compared to the 31-point difference in 1999, there remains a 28-point difference between White and Black 17-year-old students. Compared to a 22-point difference in 1999, there remains a 24-point difference between White and Latino students. These gap differences are not statistically significant. The trends data do show that the gap has narrowed significantly since 1973. Nevertheless, the racial achievement gap continues to be a persistent problem. Opportunity to Learn Mathematics. To what do we attribute these differences in achievement scores? There are a number of things, but it should be no surprise that a student’s exposure to advanced mathematics is related to his or her performance on standardized tests such as NAEP. NAEP 1999 trends assessment documented that 13-year-olds taking algebra scored higher, on average, than students taking pre-algebra. Those taking pre-algebra scored higher than those taking regular mathematics. A similar relationship was documented for 17-year-olds. NAEP reported that since 1978 all groups are taking more advanced mathematics courses. There remains, however, some discrepancy between groups. NAEP 2004 trends assessment reported that 71 percent of White students had taken algebra II or calculus, while 68 percent of Black students and 63 percent of Latino students had taken the same courses. Opportunities to learn challenging mathematics are impeded by a common school practice called tracking. Implicit in tracking practices is the notion that making curriculum accommodations based on students’ differences in ability is a fair way to do schooling in a democracy. After all, we expect students to have different abilities and talents. To many Americans and certainly to conservatives, it seems to be a good idea. High school graduates eligible for admissions into four-year colleges and universities are beneficiaries of tracking practices, since they take college preparatory and advanced placement courses. On the other hand, tracking practices place a good number of students, mostly poor and minority, in non-college preparatory courses. The students, differentiated by race and

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class, are enrolled in programs and courses that at best prepare students for low-level occupations. Unfortunately, tracking differentiates curriculum, producing a variety of forms of school mathematics that preclude, for many, any real opportunities to learn high quality mathematics. Cogan and his colleagues (2001) used TIMSS data to examine U.S. eighth graders’ opportunities to learn mathematics. They concluded that the rationale for tracking–individual student differences—was not an adequate explanation for the curriculum differentiation that they observed in 8th-grade classrooms. The bottom line is that too many children do not have opportunities to learn challenging mathematics.

Our Plan: Affirming the Child, Diversity, and Democracy The Child and the Curriculum. Conservatives advocate a “social Darwinism” where groups and individuals succeed or fail according to their own intelligence and efforts. They also place little value in the child’s socio-emotional well-being and the experiences that children bring to the classroom. A liberal perspective places the child at the center of his or her educational experiences, a perspective that has its origins in the philosophy of John Dewey. In The Child and the Curriculum (1956), first published in 1902, Dewey wrote about the tendency for educative processes to pit “the child” against “the curriculum,” an idea that informs our approach to mathematics education. Dewey argued that the fixed and ready-made curriculum imposed on children stood in sharp contrast to how the child experienced his or her everyday life. The child lives in a world that he or she experiences through social experiences. It is a world of personal and social interests without the hard divisions of subject matter disciplines, which fractionate the world into bodies of knowledge, comprising abstractions and sets of facts. These facts, said Dewey, exist without reference to the original experience that gave rise to such facts. They are decontextualized and without any meaning in the child’s own experience. According to Dewey, “[t]he child is the starting point, the center, and the end. His development, his growth, is the ideal. It alone furnishes the standard [for curriculum decisions]” (p. 9). Dewey co-wrote with Georgia Alexander (Alexander & Dewey, 1921) an elementary arithmetic textbook. It lists a number of suggestions for teachers that are consistent with his ideas. Here are several of them:

• Introduce new ideas inductively and through situations that are vitally interesting to the child. This can be best be done by a

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generous use of dramatization. In dramatization throw the burden of suggestion upon the pupils. The ingenuity fostered when children transform schoolroom furniture or construct, at home, playthings for school use, is of inestimable value to them. Moreover, the lesson thought out by twenty-five little minds is far richer in suggestive material than one which only the teacher plans. Insist on play for the sake of number—not for mere amusement. Require the utmost courtesy at all times.

• Follow concrete and objective presentation with objective illustration. In beginning classes use sticks, seeds, shoe pegs, etc. to make all classes familiar with representation through group counters [. . .].

• Teach the child to estimate answers before beginning to figure, and to use his common sense on all occasions. What was the difficulty in the case of the pupil who gave 238 years as the answer to the following: “My uncle is 7 years older than my aunt, who is 34 years old. How old is my uncle?” Give frequent exercises in which the pupils do nothing but estimate answers in round numbers, or in which they merely indicate the process by which the answer is to be obtained.

• Put a premium on the original solution of a problem. Commend highly the child who has ingenuity enough to see two or more correct methods. (Alexander & Dewey, 1921, pp. ix–x)

With Dewey’s ideas in mind, we believe that it is important to teach mathematics that builds on the child’s prior knowledge and experiences. For example, young children learn to count before entering kindergarten. They know nursery rhymes and counting songs. Even when a young child can count to 100 with ease, the child’s understanding of number relationships still needs to be developed. When using concrete objects does the child understand that each spoken number corresponds to one object in the set? Does the child understand that 5 is one more than 4 and one less than 6? Can the child provide equivalent representations of a number? Building on what children already know about numbers can develop these ideas. Children can write their own counting song about the number 6, for example. They can do a math walk around their neighborhood to find objects that depict mathematical ideas. We claim that there are learning activities built upon children’s experiences that promote meaningful mathematical learning. We do not eschew worksheets or practice, as

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conservatives claim. Even Dewey’s “progressive” philosophy, often referred to pejoratively, calls for practice. “Follow the concrete introduction with objective illustration. Use large perception cards for this purpose” (Alexander & Dewey, 1921, p. ix). We merely believe that there are meaningful ways in which children can learn and practice mathematics. Diversity. We believe in a multicultural curriculum that respects differences, diversity, and honors the child. The child as the starting point necessitates that we attend to the child’s culture, be it class, racial origins, national heritage, or otherwise. How can we educate children without accounting for who they are? Conservatives reduce multicultural mathematics curriculum to the addition of mere cultural curiosities of the mathematics of ancient peoples. They claim that such an approach waters down the curriculum since basic skills do not get taught. Conservatives misunderstand our goals for mathematics education and schooling in a democracy. Like the conservatives, we believe that the curriculum is an important aspect of schooling. We don’t believe, however, in the inherited curriculum that conservatives advocate since it emphasizes only Western intellectual traditions. The curriculum should balance Western intellectual traditions with the intellectual traditions of other groups. In an increasingly multicultural society, such balance is important if we want our democracy to survive. Democracy. Like Dewey, we believe that both the curriculum and the child’s needs and experiences are equally important. Too often the child’s needs are left out of the equation. We believe that the teacher’s central task is to meet the needs of all students with a mathematics curriculum that honors the contributions of diverse cultures to the field. This can be done without sacrificing basic skills knowledge. In fact, we believe that providing a historical and cultural context for the mathematical rules, procedures, and algorithms that we expect children to know enriches the quality of their experiences in classrooms. In our plan, all children would have equality of opportunity to learn challenging mathematics. This may require compensatory programs such as summer enrichment camps for disadvantaged students; nevertheless, the goal must be to provide for all the opportunity to learn challenging mathematics. Meeting children’s needs entails democratic practice. For Dewey (1916/1944) democracy was more than how one characterizes a governance structure. Democracy is also shared and ongoing communication, a social pact among numerous individuals of diverse shared interests who act only after considering the implications of the ideas of others, actions that operate in others’ best

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interests. According to Dewey, this kind of democratic practice “is equivalent to the breaking down of those barriers of class, race, and national territory which [have] kept men [and women] from perceiving the full import of their activity” (p. 87).

COMMENTS AND QUESTIONS “The Liberal View and You” General Questions 1 What aspects of the liberal view seemed to promote your understanding of the culture in mathematics debate? 2 What aspects of the public argument did you find disagreeable? Why?

Specific Questions 1 What comprises the mathematics curriculum that liberals propose? 2 Would a recognition of and value for cultural differences necessarily enhance the mathematics curriculum? If so, how? 3 Will de-tracking promote opportunities to learn challenging mathematics? Why or why not? 4 How might tracking be used to provide opportunities for all students to learn challenging mathematics? In the next few paragraphs, the case studies are briefly summarized and analyzed according to the liberal view just outlined. Do you think the analysis fits with a liberal view of each case? Why or why not? After considering each analysis, do you still feel the same way about the liberal view?

“Race and Teacher Expectations” A liberal view recognizes that race is undeniably related to educational outcomes. In a society that continues to be highly stratified by race, Mrs.

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Carlton ought to consider the possibility that it may be her beliefs about children of color that play a role in her ineffectiveness with Benjamin. Anna should share her case study with Mrs. Carlton. This is not unreasonable since it is likely that Mrs. Carlton would be interested in what the university requires of its pre-service teachers. While Mrs. Carlton may not recognize her own bias as she considers the case study, it may offer a way for her to see Benjamin in a different light.

“Mathematics for All?” A liberal might say that all children can learn the algebra typically taught in high school regardless of race, gender, linguistic, or socio-economic background. It may be the case that some children, because of circumstances beyond their control, may require compensatory services to help them learn algebra. If children are learning English as a second language, the school should provide services in ways that do not undermine the substance of academic courses. All children must have an equal opportunity to succeed in school. A solid mathematics background is necessary for informed participation in a democracy.

“Culture and School Mathematics” Liberals might embrace a multicultural approach to mathematics curriculum and instruction. They believe that offering both a cultural and historical context for the basic skills that we expect children to know will enrich their mathematical experiences. Such an approach to mathematics may improve achievement of children of color, affirm the cultures of those typically not depicted in the curriculum, and make mathematics more meaningful. Multicultural goals are consistent with democratic goals.

“Politics and School Mathematics” Liberals, like conservatives, want to hold schools accountable for children’s learning. However, liberals might say that it should not be at the expense of high quality teaching and high quality learning. The impact of policy in this case is not acceptable since it reduces mathematics to basic skills. This amounts to an impoverished curriculum that inadequately serves democratic goals and equity issues in schooling. Liberals would be

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sympathetic to Professor Jordan’s plight. They would advocate a “balanced approach” to teaching mathematics, but not at the expense of problem solving.

A “RADICAL MULTICULTURALIST VIEW”: MATHEMATICS FOR DEVELOPING CRITICAL DISPOSITIONS FOR SOCIAL RECONSTRUCTION Introduction For years conservatives and liberals have been “reforming” schools. These reform efforts have had minimal, if any, impact on curriculum and instruction. Nor have they improved the quality of learning experiences and opportunity for the poor, minorities, and women. Our education system continues to fail these groups. It perpetuates inequality of both opportunity and achievement results. Conservatives and liberals place significant responsibility on the school to improve these conditions. We, radical multiculturalists, do not believe that the school bears sole responsibility for these failures. We attribute these inequities to larger structural issues. The urban school crisis, in particular, “is in large part a result of economic and political devastation in cities” (Anyon, 1997, p. 13). Schools will improve when our culture undergoes fundamental social change. That is, we need to create an economic system and a governance structure that eliminates the gross inequities that plague our society. In this regard, we call ourselves “radicals.” Like liberals, and even conservatives, we believe in a democratic government, though our version of government looks most like a social democratic government. We take a different view of the role of schooling. Conservatives believe that the role of schooling is to produce skilled workers for global economic prominence. Liberals believe that the school should educate citizens for full participation in a democratic society. Unlike conservatives and liberals, we believe that schooling, as it is currently practiced, reproduces the existing social order, a political and economic structure that serves the interests of those with the most wealth and power. Bennett (2001) elaborates four broad principles for multicultural education that we find useful in explaining our radical multiculturalist perspective: (a) the theory of cultural pluralism; (b) ideals of social justice and the end of racism, sexism, and other forms of prejudice and discrimination;

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(c) affirmations of culture in the teaching and learning process; and (d) visions of educational equity and excellence leading to high levels of academic learning for all children and youth. (Bennett, 2001, p. 173)

Like many liberals and multiculturalists, we believe in cultural pluralism, social justice and the end of all forms of prejudice and discrimination. We believe that children’s culture should be affirmed in the teaching and learning process. We believe that educational equity should lead to academic excellence for all children. However, we take things a step further than many liberals and multiculturalists. Since we cannot get to the heart of the matter, that is, change the economic system that creates our society’s abundant inequities, we do support efforts to empower teachers and parents, pedagogies that address diverse populations, and curriculum that sensitizes students to racism, sexism, classism, and the voices of the oppressed. Schooling’s primary function ought to be about educating young people into critical dispositions so that they can participate in the construction of a society that is just for all of us. At the school level, we believe, this is the democratic endeavor.

Our Plan: Social Justice and Mathematics Curriculum and Teaching Culture and School Mathematics Curriculum “Ethnomathematics” makes the most explicit connection between mathematics and culture. Ubiratan D’Ambrosio, a Brazilian mathematician, first used ethnomathematics in the late 1960s to describe the mathematical practices of identifiable cultural groups. Ethnomathematics, then, is often defined as the study of mathematics and the cultural practices in which it arises. As an anthropological endeavor, some ethnomathematicians like Claudia Zaslavsky (1990) concern themselves with descriptions of numeration systems, patterns and forms in art and architecture, and mathematics in games, as well as the mysticism associated with the mathematics that existed on the African continent. Such scholarly studies of culture and mathematics make obvious that mathematics is a cultural production. The notion that mathematics is not culture-free is a very difficult pill for many to swallow, despite the work of ethnomathematicians like Zaslavsky and other researchers who document the role of culture in mathematical thinking (e.g., Brown et al., 1989; Carraher et al., 1985; Lave, 1988; Nasir, 2000; Saxe, 1991). Culture matters. When conservatives say that 2 + 2 = 4,

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no matter what the cultural context, they misunderstand a great many things—the nature of knowledge, the historical development of mathematics, the role that culture plays in cognition, to name a few. We quote generously from Ascher and Ascher (1997), because the anecdote they present is a powerful illustration of how culture is the process of sharing symbolic systems among group members and how this cultural process matters as we interpret our daily realities. The anecdote tells of an exchange between a native African Demara sheepherder and someone else variously described as an explorer, trader, scientist, anthropometrist, or ethnologist. It is intended to show that the herder cannot comprehend the simple arithmetic fact that 2 + 2 (or 2 × 2) = 4. It describes how the herder agrees to accept two sticks of tobacco for one sheep but becomes confused and upset when given four sticks of tobacco after a second sheep is selected. Of course, the problem is not that the shepherd doesn’t understand arithmetic, it is rather that the scientist/trader doesn’t understand sheep. Sheep are not standardized units. Since the Demara herder finally agreed to the trade, his confusion could be attributed to the trader’s willingness to pay an equal amount for the second, different animal. Other than demonstrating two people talking past each other, the anecdote raises the issue of the difference between a mathematical concept and its application. When inferring mathematical ideas from concrete usage, one is always dealing with applications. It is recognized, for example, that 2 + 2 = 4 does not apply when discussing combining volumes of gases or combining elements of overlapping sets. Even more important, as was recognized as early as 1912 by Wertheimer, the applicability of even the most [sic] simplest of mathematical models becomes a question of cultural categorization. We say, for example, that 2 apples + 2 pears = 4 fruit, but 2 pants + 2 jackets = 2 suits. A while ago we might have said that 2 men + 2 women = 2 couples, but now we are careful to say that 2 men + 2 women = 4 people. Further, we believe there is meaning in 2 in. × 2 in. = 4 sq. in. but not in 2 apples × 2 apples = 4 sq. apples. A question that has been posed by Western scholars when trying to determine the universality of mathematics, is whether or not 2 + 2 always equals 4. Once 4 is recognized as the name for 2 + 2, the question becomes, as it is here, whether or not the model applies. That question can have different answers in different cultures and even different answers within the same culture. (Ascher & Ascher, 1997, pp. 29–30)

We believe that understanding the role of culture in mathematics is important. We also believe that the distorted history of the development of mathematics perpetuates the hegemony of white racial superiority.

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“Mathematics is perceived as an exclusive product of White men and European civilizations” (Joseph, 1987, p. 16). Joseph tells us that the common historical view is that pre-Greek mathematics lacked the intellectual characteristics of what gets called mathematics in the present day. Pre-Greek civilizations are viewed as “non-literate” peoples incapable of abstract thought. Specifically, there is no evidence that those who practiced pre-Greek mathematics had a well-defined concept of “proof.” According to this view, mathematics served mere utilitarian purposes “such as the construction of calendars, parceling out land, administration of harvests, organization of public works (e.g., irrigation or flood control) or collection of taxes” (Joseph, 1987, p. 23). These rules and procedures were carried out without “proof ” or even the recognition that “proof ” was needed. This view trivializes the mathematical contributions of non-White civilizations and represents an intractable historiographical bias that has led to the widespread myth that scientific progress can be solely attributed to White European males. Hence, some ethnomathematicians and radical educators take a critical view of the curriculum that children experience in schools. We believe that the traditional curriculum is racist, sexist, and classist. It ignores the culture and history of the oppressed. It operates to homogenize groups into a “common” culture and does not invite critical inquiry of the social problems that threaten our democracy. Ethnomathematics, then, is used as a vehicle to correct what is considered to be a distorted cultural view of mathematics and to uncover its hidden history. Mathematics, as is the case with any kind of knowledge, does not exist separate and apart from its historical and political contexts. Mathematics is used to stratify individuals by race and class as evidenced in most high schools across our country. Mathematics is also used to deceive the ordinary citizen. Even in traditional school settings, curricula focused on making students consumer-savvy have used statistics to teach students a form of quantitative literacy to show how the ordinary consumer can be misled by numerical data. We must always ask how mathematics is being used, for what purposes, and for whose interests. Frankenstein (1990) has attempted to address race, class, and gender in curriculum that mathematizes social problems. She calls this a critical mathematical literacy curriculum. Critical mathematical literacy involves the ability to ask basic statistical questions in order to deepen one’s appreciation of particular issues. It also involves the ability to present data to change people’s perceptions of those issues. This critical understanding of numerical data thus prompts

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individuals to question taken-for-granted assumptions about how a society is structured and enables them to act from a more informed position on societal structures and processes (in Frankenstein & Powell, 1990, p. 336)

Frankenstein has used real-life data with largely working-class urban adults. They were non-traditional students in that they were over thirty years of age, held full-time jobs, and most had families. As ways of learning the “basic facts” knowledge that conservatives claim is missing from mathematics curriculum, these students examined a variety of numerical data including statistics on the distribution of wealth, income by educational attainment, welfare for the rich and the poor. More importantly, these students acquired high-level reasoning skills that allowed them to see how social and political conditions deny members of certain groups access to worthwhile educational opportunities and careers. According to Frankenstein and Powell (1994), this critical awareness builds self-confidence and can lead to the social activism necessary to improve current conditions.

Social Justice Pedagogy As with Frankenstein’s critical ethnomathematical approach, teaching mathematics for social justice endeavors to imbue students with critical dispositions toward the social conditions in which they live. Gutstein (2006) theorizes that there are three mathematics pedagogical goals. They are “(1) reading the mathematical word, (2) succeeding academically in the traditional sense, and (3) changing one’s orientation to mathematics” (p. 24). However, these goals are not sufficient. Teaching mathematics for social justice entails three additional goals: “(1) reading the world with mathematics, (2) writing the world with mathematics, and (3) developing positive cultural and social identities” (p. 24). For Gutstein, reading the world with mathematics means to use mathematics to understand relations of power, inequities in economic and educational resources, the ability to deconstruct racialized myths and other forms of representation and to examine such conditions in light of one’s own circumstances. “Writing the world with mathematics means using mathematics to change the world” (p. 27). This is much like Frankenstein’s (1990) approach. She helps adult students use mathematics to become informed about their positions in the larger social context, which comprises a vast array of inequities.

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Culturally Responsive Pedagogy Ethnomathematical and social justices approaches are similar to culturally responsive pedagogies in that these pedagogies consider the role of culture in teaching and promoting critical reflection. Culturally responsive pedagogies are known by several names: culturally relevant, centered, compatible, sensitive, contextualized, reflective, congruent, synchronized, etc. There is no one “right” way or a set of prescriptions for teaching in ways that recognize and accommodate the experiences that children bring to the classroom. Gay (2000) defines culturally responsive teaching “as using the cultural knowledge, prior experiences, frames of reference, and performance styles of ethnically diverse students to make learning encounters more relevant to and effective for them. It teaches to and through the strengths of these students” (p. 29). Using Gay’s rubric, culturally responsive teaching is validating and affirming, comprehensive, multidimensional, empowering, transformative, and emancipatory. Gay borrows from Diamond and Moore (1995 as cited in Gay, 2000) as she discusses the implications for the teacher’s role and responsibility for structuring learning environments for underperforming students. Teachers must be cultural organizers. That is, they must attend to and understand how culture manifests in the quotidian realities of classroom life and create learning environments that promote high-quality learning for all students. Teachers must be cultural mediators. They must encourage dialogue about cultural conflict in ways that clarify and honor ethnic identity and deconstruct racist myths and stereotypes. Teachers must be orchestrators of social contexts for learning. This means teachers must acknowledge that children’s cultural heritages and experiences influence learning and seek pedagogical processes that build upon the sociocultural knowledge and frames of references that children bring to the classroom. Gay (2000), Hollins (1996), and others offer conceptual frameworks in which to think about what it means to teach in culturally relevant ways. Ladson-Billings (1994) makes concrete the dispositions and practices of successful teachers of African American students in The Dreamkeepers. Irvine (2002) presents a collection of scholars’ work that provides theoretical and practice-based explications regarding the influence of race and culture on African American teachers. Tharp and Gallimore (1991) provide a detailed theoretical and empirical account of teaching as assisted performance whereby the teacher’s approach to language development and literacy builds upon the experiences, backgrounds, and “talk-story,” the

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conversation style of native Hawaiian children. This conceptual and empirical work about productive pedagogies for students of color is but a small part of the socio-cultural work concerned with culture and school learning. Also, it is a subset of the research concerned with culture and cognition. For example, Lave (1988) challenges traditional psychological views of cognition, offering a socially-situated account as the analytic frame for making sense of adults’ practice of everyday mathematics. Brown, Collins, and Duguid (1989) make the case that the problem with school learning is that it is divorced from activity and cultural contexts. The result is that it produces knowledge that is not useful, nor particularly robust. Attending to cognition, cultural contexts, and activity, Geoffrey Saxe (1991) documented the socio-cultural processes entailed in the mathematical thinking that emerged in Brazilian children selling candy on the streets for economic survival. The work of these scholars contributes to our understanding of the central role that culture plays in school learning.

African-Centered Pedagogy We highlight Peter Murrell’s (2002) work because he advances a theory that attempts to unify scholars’ ideas in the previous paragraphs and similarly-minded others about educational practice in a diverse society. Murrell theorizes about a coherent and unified system of instructional practices that integrate the cultural dimensions of the African American experience in ways that maximize the academic and personal development of African American children. Murrell’s theory, African-centered pedagogy, serves as a tool for understanding and accommodating the actual lives of African American children, so that teachers can develop “accomplished practice” (p. x) with African American children. According to Murrell, accomplished teachers critically examine the larger socio-political context of schooling. Teachers must ask, “How does this practice or policy perpetuate the underachievement of African American learners?” (p. xvii). This kind of interrogation is a systemic approach to addressing the poor academic performance of African American students, since it requires that teachers must think beyond the child’s individual abilities or family background. It is also important to note that Murrell’s theory is a deliberate critique of multicultural education of the kind promoted by liberals. Murrell says that the purpose of multicultural literature is to help children appreciate differences and to embrace diversity. Such literature is designed for White children to help them expand their awareness of other ethnic and cultural groups. This approach does

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not meet the needs of many African American children. Multicultural education gets confused with “pedagogy for Black children.” When Black children have to learn about diverse cultures, it is usually done at the expense of learning about their own culture and the identity work that is critical to Black children’s academic and personal development. As radical multiculturalists, we agree. It is imperative that schools provide the space for Black children to recover their lost heritages and affirm their identities in ways that facilitate their personal and academic growth. Only then, can African American children re-imagine themselves as participants in a just democracy. As you can see there is nothing “feel good” or “fuzzy,” as the conservatives claim, about a radical multicultural approach to school mathematics. We do not reduce mathematics to mere cultural curiosities, nor do we believe that a multicultural curriculum can remedy social inequities, as liberals tend to think. Only restructuring the economic and governance systems can do that. We do, however, promote curriculum focused on correcting the distorted historical and cultural views of mathematics, that builds on children’s cultural backgrounds, as Tharp and Gallimore have done, and mathematizing social problems, as Frankenstein does, and that employ pedagogies that sensitize all students to racism, sexism, classism, and affirming their cultural identities. Such an approach educates students into a critical disposition. Such a disposition enables informed participation in activities aimed at reconstructing an economic and governance system that justly serves its citizens. As stated at the start, Bennett’s (2001) principles concerning cultural pluralism, social justice, culture in teaching and learning, and academic excellence for young people inform our radical multiculturalist stance. This is not a “feel good” approach to school mathematics. This is a rigorous approach, requiring considerable intellectual work and moral courage. There is much work to do. We must get on with it. COMMENTS AND QUESTIONS “The Radical Multicultural View and You” General Questions 1 With what claims made in this public argument did you disagree? Why? 2 What claims did you find compelling? Why?

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Specific Questions 1 Do you think it is unreasonable to think that schools and their teachers cannot change larger cultural and social structures? Why or why not? 2 What might be the impediments to using curricula and pedagogical practices that encourage critical dispositions? 3 What might be the benefits to using curricula and pedagogical practices that encourage critical dispositions? 4 Do you agree that a liberal notion of multicultural curriculum is harmful to African American students? Why or why not? 5 Is African-centered pedagogy exclusionary? That is, does it leave out non-African American students in its efforts to diminish schooling inequities for African American students? Why or why not? In the next few paragraphs, the case studies are briefly summarized and analyzed according to the radical multiculturalist view just outlined. Do you think the analysis fits with how a radical multiculturalist would view each case? Why or why not? After considering each case, do you still feel the same way about the radical multicultural view?

“Race and Teacher Expectations” It is Anna’s responsibility to share her case study with Mrs. Carlton. Teachers like Mrs. Carlton must face their assumptions about students, particularly students of color, their family backgrounds and their ability to learn. At least Anna is on the right track in recognizing that learning about her students’ backgrounds is an important aspect of educating them.

“Mathematics for All?” The good news is that Tara is showing a disposition toward critical mindedness in questioning both the pros and cons of the mandate to teach algebra to all eighth graders. The bad news is that she lacks the knowledge to deal appropriately with the dilemmas she faced in the teachers’ meeting. Was Tara aware of the racist beliefs that underlie many of the teachers’ concerns? Is she aware that school mathematics differentiates

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students, usually by race and class? If she leaves unquestioned a Eurocentric view of mathematics and the role it plays in school, it is likely that she will concede that the teachers are correct—a position that only reproduces and perpetuates social inequities.

“Culture and School Mathematics” Mr. Robinson’s anger typifies a common response to situations that shake a person’s foundation to its core. However, why is it that the responses are often angry and belligerent? Why is it that people like Mr. Robinson label honest efforts at inclusion or correcting the historical record as “political correctness”? Just because Mr. Robinson speaks Spanish, doesn’t mean he’s capable of meeting the needs of students, even Spanish-speaking students. Unless he sheds his Eurocentric view of mathematics, he will likely continue to disenfranchise many students.

“Politics and School Mathematics” It is already the case that traditional school mathematics distorts the cultural and historical development of the discipline. The current policy environment embodies a limited view of “standards” and “accountability” that places teachers in the impossible situation of meeting the social and educational needs of diverse students. In fact, it further distorts the mathematics that children learn in school. Professor Jordan ought to tell the principal how both the traditional approach and the current policy environment works against teachers and students. Teachers will not improve their practices. Students will not learn substantive mathematics in ways that honor who they are.

III CONCLUDING REMARKS, SOME REFLECTIONS, AND RESOURCES FOR FURTHER REFLECTION THE CENTRALITY OF CULTURE: UNDERSTANDING THE HISTORICAL, SOCIAL, AND POLITICAL DIMENSIONS OF SCHOOL MATHEMATICS If mathematics educators take seriously the goal of equity, they must question not just the common view of school mathematics but also their own taken-for-granted assumptions about its nature and worth. (George M. A. Stanic, 1989, p. 58)

This final section begins appropriately with George Stanic’s words since they capture a central problem to overcome if we are serious about equity and school mathematics. We hold many assumptions about mathematics and mathematics as it is practiced in schools. As discussed in this book, many of us assume that mathematics is a given, a static body of knowledge invented by White men many years ago. We assume no relationship between mathematics and culture, including the political ways in which it is used. We believe that only some students are capable of learning mathematics. And, furthermore, we assume that mathematics is a necessary part of the school curriculum since we believe that our increasingly technological society will require a mathematically literate citizenry. These assumptions arise from our cultural circumstances, that is to say, the historical, social, and political contexts in which we live our lives. These taken-for-granted assumptions inform the ways in which we think 107

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about mathematics, its nature, and worth. Case 3, “Culture and School Mathematics” elicits questions such as, What is mathematics? It is just one of many questions with which to grapple regarding the nature of mathematics. Its worth encompasses important questions ranging from its utilitarian to its moral purposes. When we grapple with these issues, it becomes clear that our cultural assumptions are difficult to overcome. For example, because we regard mathematics as high status knowledge, we think that only certain individuals are entitled to the mathematics offered in college preparatory, Advanced Placement, or gifted classes. The mere suggestion that this assumption is fallacious, inequitable, or immoral leaves us reeling since it is difficult to imagine an alternative way of thinking about who gets to learn mathematics. Equity, according to the Principles and Standards for School Mathematics (PSSM) (NCTM, 2000), is one of the six principles that constitute the features of a high quality mathematics curriculum. Touted as a core aspect of mathematics reform, the PSSM states that [a]ll students, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to study—and support to learn—mathematics. Equity does not mean that every student should receive identical instruction; instead it demands that reasonable and appropriate accommodations be made as needed to promote access and attainment for all students. (NCTM, 2000, p. 12)

According to PSSM we can accomplish equity when [t]eachers communicate expectations in their interactions with students during classroom instruction, through their comments on students’ papers, when assigning students to instructional groups, through the pressure or absence of consistent support for students who are striving for high levels of attainment, and in their contacts with significant adults in a student’s life. These actions, along with decisions and actions taken outside the classroom to assign students to different classes or curricula, also determine students’ opportunities to learn and influence students’ beliefs about their own abilities to succeed in mathematics. (NCTM, 2000, p. 13)

While an equity principle seems to be a good start, an implicit claim made in this book is that we can’t begin to address equity issues until we first acknowledge and understand the ways in which the inequities are produced in schools. The cases serve to problematize our assumptions

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about culture and school mathematics. “Mathematics for all” is a phrase that has peppered many reform documents in recent years. Unfortunately, phrases like these become mere slogans. Serious issues about equity are rarely addressed, and even more rare are substantive actions that improve learning opportunities for a great number of students. The case, “Mathematics for All,” illustrates this point. The teachers’ taken-for-granted assumptions regarding their students, the nature of mathematics and its worth remained unchallenged. If their assumptions remain unchallenged, then there is no problem. If there is no problem, then the teachers will proceed with business as usual. And this is precisely what happens. Across all the cases, the teachers and other school stakeholders must acknowledge and come to understand several things: the role of culture in school learning and how school mathematics is a translation (rather than a transformation) of the larger society’s structural features that stratify groups by race and class. School mathematics distorts, even hides the cultural and historical development of mathematics. School mathematics conveys a static notion of the discipline, comprising mostly rules and algorithms to be memorized. School mathematics is objective and neutral, existing separate and apart from the cultural contexts in which it is practiced. School mathematics existing in myriad forms stratifies students by race and class. These are just some of the pervasive, seemingly intractable cultural beliefs and practices that must be challenged before we can make the vision of equity in mathematics a reality. In the years since Brown v. Topeka Board of Education, not much has changed in providing equality of educational opportunity for children in the U.S. Will federal mandates exacerbate or improve the problem? What is certain is challenging our own assumptions is always a tremendous task. We will know that we have partial success when we can think differently and productively about the challenges we face in teaching mathematics to diverse populations in a democratic society. The process of facing the challenges will be like taking a long trip whose destination is unclear. But, such a trip is worth taking.

RESOURCES National Council of Teachers of Mathematics www.nctm.org The Math Forum www.mathforum.org

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Enlaces en español para la matematica www.matheformum.org/teachers/spanish/ Mathematically Correct www.mathematicallycorrect.com/ Mathematically Sane www.MathematicallySane.com Children’s Defense Fund 25 E Street N.W. Washington, D.C. 20001 Phone: 202-628-8787 E-mail: cdfi[email protected] National Center for Children in Poverty 215 W. 125th Street, 3rd Floor New York, NY 10027 Phone: 646-284-9600 Fax: 646-284-9623 E-mail: [email protected] www.nccp.org Center on Urban Poverty and Social Change Mandel School of Applied Social Sciences Case Western Reserve University 10900 Euclid Ave. Cleveland, OH 44106-7164 www.povertycenter.cwru.edu/ The Civil Rights Project 8370 Math Sciences Los Angeles, CA 90034 E-mail: [email protected] www.civilrightsproject.ucla.edu/ Rethinking Schools 1001 E. Keefe Avenue Milwaukee, WI 53212 Phone: 414-964-9646 www.rethinkingschools.org

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EXERCISES

Teaching Tolerance c/o The Southern Poverty Law Center 400 Washington Ave. Montgomery, AL 36104 Phone: 334-956-8200 www.tolerance.org

EXERCISES The following exercises are but a small number of activities to further your thinking about culture and school mathematics. Most of the activities require your powers of observation and interpretation. At first glance or even with long association, classrooms seem to be very familiar places. We tend to think that we know what’s happening in classrooms, since virtually all of us have spent a large part of our childhoods in classrooms. Because of its familiarity it is difficult to attend to how issues of race, class, gender, or politics play out right before our eyes. Record your initial impressions. Most likely they will express your biases. Then, record what you see as objectively as possible. After spending time in classrooms, interpret your observations in light of your initial impressions. Note other questions that arise and see what other practitioners and classroom researchers say about your observations.

Mathematics Inside and Outside School Do a case study or a mathematical profile much like the one that Anna in “Race and Teacher Expectations” did on Benjamin. Observe a linguisticminority child or a child identified (formally or informally) as underachieving. To the degree possible collect as much information about that child as you can—standardized tests scores, particularly in mathematics, and observe him or her in mathematical activities in class, on the playground, and at home. Analyze the data you’ve collected. Were there any surprises? Is there a discrepancy between the child’s mathematical performance in school and out of school? When and where does the child seem most mathematically competent? Is the teacher’s view of the child’s mathematical competence similar to the one you’ve compiled? How might your case study or mathematical profile serve as a tool for improving the quality of the child’s mathematical experiences both in and out of school?

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Mathematics and Diverse Learners Visit a mathematics classroom with a preponderance of Anglo students, one with mostly African American students, and one with students whose primary language is not English. In each classroom, observe the students, the mode of instruction, and the nature of interaction between teacher and students. Record your observations. Here are some questions to consider: What is the goal of the mathematics lesson? How would you characterize the lesson—skills-based, conceptually based, problem solving based, etc.? How would you characterize the nature of interaction between teacher and students—teacher-directed, teacher-facilitated, student-directed, etc.? Did the students seem to understand the material? How do you know? Compare and contrast your observations. What other questions arise from this experience?

Children of Color and their Mathematical Identities Choose two or three underperforming children of color and/or linguisticminority. Choose two or three high achieving students of the same age. Do an interview study regarding their experiences in math classrooms and their attitudes about mathematics. Compare and contrast their responses. Some possible questions: Are you good in math? How do you know? Do you think everyone can be good at mathematics? Why or why not? How would you define mathematics? Is mathematics mostly memorization? Is there more than one way to solve a math problem? Do you plan on taking more math when you get to high school/community college/university? Do you use it in everyday life? How? Do you understand your math lessons? What do you do when you don’t understand? What do you plan to do when you grow up? Why have you chosen this profession? Does the job require lots of mathematics?

Ethnomathematics and School Mathematics The cases presented in this book give you a sense of the contentious nature in decision-making processes involving school curriculum. Mathematics, as you have explored, is no exception. Do some library research to investigate the many sides of the debate around the issues that emerge when promoting ethnomathematics as part of the mathematics curriculum. How would you characterize the issues that emerge in this debate? On what

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grounds do those who oppose ethnomathematics make their case? On what grounds do those who promote ethnomathematics make their case? Are there intermediate stances on the issues? After researching the issues, conduct a debate with some of your classmates. Take opposing sides of any issue associated with ethnomathematics.

Policy and Mathematics Teaching Interview a teacher from an elementary school, a math teacher from a middle school and a math teacher from a high school. Talk to them about how mandatory policies influence what they teach and how they teach. Some questions to ask: What mandatory policies (formal and informal) most impact what you do in your math class? Are these federal, state, district or school level policies? Give me several examples of how it impacts what you do? What benefits does the policy afford? Are there disadvantages? What are they?

Children, Poverty, and Mathematics Learning The impact of poverty on the lives of children cannot be overstated. Many teachers, in their classrooms, deal daily with effects of poverty on children. In addition to social service agencies, there are a number of organizations concerned with children’s issues. Identify these organizations, using them to investigate the issues related to poverty. Discuss your findings with your classmates, paying particular attention to how poverty impacts children’s health, socio-emotional well-being, and school learning.

BIBLIOGRAPHY

The following list includes both works referred to in this volume and additional articles and books that might be beneficial for further inquiry. The list is intentionally short so as to keep it manageable. Please note that it is only a small part of a vast literature on the issues explored in this book. The items are arranged by category. Those marked with an * are cited in this volume. Culturally Relevant Pedagogy, Teaching and Teachers *Alexander, G. & Dewey, J. (1921). The Alexander–Dewey arithmetic elementary book. New York: Longmans, Green and Company. *Dewey, J. (1956). The child and the curriculum and The school and society. Chicago: The University of Chicago Press. *Dossey, J. A. (1992). The nature of mathematics: Its role and its influences. In D. A. Grouws (ed.), Handbook of research in mathematics teaching and learning (pp. 39– 48). New York: Macmillan. Foster, M. (1997). Black teachers on teaching. New York: New Press. *Gay, G. (2000). Culturally responsive teaching: Theory, research, and practice. New York: Teachers College Press. Gerdes, P. (1998). On culture and mathematics teacher education. Journal of Mathematics Teacher Education, 1(1), 33–53. *Gutstein, E. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York: Routledge. Gutstein, E., Lipman, P., Hernandez, P., & de los Reyes, R. (1997). Culturally relevant mathematics in a Mexican American context. Journal for Research in Mathematics Education, 28(6), 709–737. *Hollins, E. (1996). Culture in school learning: Revealing the deep meaning. Mahwah: Lawrence Erlbaum Associates. *Irvine, J. J. (ed.) (2002). In search of wholeness: African American teachers and their culturally specific classroom practices. New York: Palgrave.

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*Ladson-Billings, G. (1994). The dreamkeepers: Successful teachers of African American children. San Francisco: Jossey-Bass. Liston, D. & Zeichner, K. (1996). Culture and teaching. Mahwah: Lawrence Erlbaum Associates. McAllister, G. and Irvine, J. J. (2000). Cross cultural competency and multicultural teacher education. Review of Educational Research, 70(1), 3–24. Maher, F. A. & Ward, J. V. (2002). Gender and teaching. Mahwah: Lawrence Erlbaum Associates. *Murrell, P. C. (2002). African-centered pedagogy: Developing schools of achievement for African American children. Albany: SUNY Press. *Saxon, J. (1982). Incremental development: A breakthrough in mathematics. Phi Delta Kappan, 63(7), 482–484. *Saxon, J. (1984). The way we teach our children math is a disgrace. American Education, 10–13. *Stigler, J. W. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: The Free Press. *Stodolsky, S. S. (1988). The subject matters: Classroom activity in math and social studies. Chicago: The University of Chicago Press. Tate, W. F. (1995). Returning to the root: A culturally relevant approach to mathematics pedagogy. Theory into Practice, 34(3), 166–173. *Tharp, R. & Gallimore, R. (1991). Rousing minds to life: Teaching, learning, and schooling in social context. New York: Cambridge University Press. Zeichner, K. M. & Liston, D. P. (1996). Reflective teaching: An introduction. Mahwah: Lawrence Erlbaum Associates. Culture and Cognition *Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32–42. *Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3, 21–29. *Lave, J. (1988). Cognition in practice: Mind, mathematics, and culture in everyday life. Cambridge, NY: Cambridge University Press. *Nasir, N. (2000). “Points ain’t everything”: Emergent goals and average and percent understanding in the play of basketball among African American students. Anthropology and Education Quarterly, 31(3), 283–305. Rogoff, B. (1995). Observing sociocultural activity on three planes: Participatory appropriation, guided participation, and apprenticeship. In J. V. Wertsch, P. del Rio, & A. Alvarez (eds), Sociocultural studies of mind (pp. 139–164). Cambridge: Cambridge University Press. *Saxe, G. B. (1991). Culture and cognitive development. Hillsdale: Lawrence Erlbaum Associates. Multicultural, Multilingual, and Bilingual Contexts Adler, J. (1997). A participatory-inquiry approach and the mediation of mathematical knowledge in a multilingual classroom. Educational Studies in Mathematics, 33(3), 235–258. Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht, The Netherlands: Kluwer. *Banks, J. A. (1993). The canon debate, knowledge construction, and multicultural education. Educational Researcher, 22(5), 4–14.

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Banks, J. A. & Banks, C. A. (eds) (2004). Handbook of research on multicultural education (2nd edn). San Francisco: Jossey-Bass. *Bennett, C. (2001). Genres of research in multicultural education. Review of Educational Research, 71(2), 171–217. Gorgio, N. & Planas, N. (2001). Teaching mathematics in multilingual classrooms. Educational Studies in Mathematics, 47(1), 7–33. Licón Khisty, L. (1997). Making mathematics accessible to Latino students: Rethinking instructional practice. In J. Trentacosta (ed.), Multicultural and gender equity in the mathematics classroom: The gift of diversity—1997 Yearbook (pp. 192–201). Reston: NCTM. Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual mathematics learners. Mathematical Thinking and Learning, 4(2–3), 189–212. Political and Social Issues and Contexts (Race, Ethnicity, Class, Gender, Equity) *Anyon, J. (1997). A political economy of urban educational reform. New York: Teachers College Press. *Barnes, M., Clarke, D., & Stephens, M. (2000). Assessment: The engine of systemic curricular reform? Journal of Curriculum Studies, 32(5), 623–650. *Cogan, L. S., Schmidt, W. H., & Wiley, D. E. (2001). Who takes what math and in which track?: Using TIMSS to characterize U.S. students’ eighth-grade mathematics learning opportunities. Educational, Evaluation and Policy Analysis, 23(4), 323–341. *Cohen, D. K. & Hill, H. (2000). Instructional policy and classroom performance: The mathematics reform in California. Teachers College Record, 102(2), 294–343. Cooper, B. & Dunne, M. (2000). Assessing children’s mathematical knowledge: Social class, sex and problem solving. Philadelphia: Open University Press. *Dewey, J. (1916/1944). Democracy and education: An introduction to the philosophy of education. New York: Free Press. *Frankenstein, M. (1990). Incorporating race, gender, and class issues into a critical mathematical literacy curriculum. Journal of Negro Education, 59(3), 336–347. Frankenstein, M. (1997). In addition to the mathematics: Including equity issues in the curriculum. In J. Trentacosta (ed.), Multicultural and gender equity in the mathematics classroom: The gift of diversity—1997 Yearbook (pp. 10–22). Reston: NCTM. *Frankenstein, M. & Powell, A. (1997). Ethnomathematics: Challenging Eurocentrism in mathematics education. Albany: State University of New York Press. *Gamoran, A. & Hannigan, E. C. (2000). Algebra for everyone? Benefits of collegepreparatory mathematics for students with diverse abilities in early secondary school. Educational Evaluation and Policy Analysis, 22(3), 241–254. Gorgorió, N., Planas, N., & Vilella, X. (2002). Immigrant children learning mathematics in mainstream schools. In G. De Abreu, A. Bishop, & N. Presmeg (eds), Transitions between contexts of mathematical practices (pp. 23–52). Boston: Kluwer. Gutstein, E. (2003). Teaching and learning mathematics for social justice in an urban, Latino school. Journal for Research in Mathematics Education, 34(1), 37–73. *Gutstein, E. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York: Routledge. *Harry, B. & Anderson, M. G. (1994). The disproportionate placement of African American males in special education programs: A critique of the process. Journal of Negro Education, 63(4), 602–619. *Knijnik, G. (1997). An ethnomathematical approach in mathematical education: A matter of political power. In A. B. Powell & M. Frankenstein (eds), Ethnomathematics: Challenging Eurocentrism in mathematics education (pp. 403–410). Albany: SUNY Press.

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INDEX

accountability 7, 25, 33, 59, 61, 69, 105 achievement gap 7, 24, 37, 89–90 African-centered pedagogy 102, 104 Algebra Project 37 Banks, J. xvii basic math facts 25, 83 bilingual education 25–6 Bishop, A. 2, 39 Brown v. Board of Education of Topeka 77–8, 89, 109 Burton, L. 2 “Child as Victim” 81, 84 Conservative view 79, 86–7 see also public arguments content knowledge x, xi content standards 53, 88 contextual issues of schooling xii Core Knowledge Foundation 6 Core Knowledge Sequence 6 cultural dimensions of schooling 1, 77 cultural pluralism 96–7, 103 culturally responsive pedagogies 101 culturally sensitive teaching 46 culture definitions of 1, 51–2

D’Ambrosio, U. 3, 39–40, 97 de-tracking 82, 94 Dewey, J. 91–4 dimensions of culture xv, xvi Dossey, J. 2 Education for All Handicapped Children Act 9 educational system 14, 20 challenges to 20 equality of educational opportunity 77, 109 ethnomathematics 3, 97, 99 definition of 3, 97 work of ethnomathematicians 3, 39, 97, 99 Eurocentric curriculum 45, 51 Frankenstein, M. xvii, 3, 99–100, 103 critical mathematical literacy curriculum 99 Gay, G. 101 Glenn, J. 4 Gutstein, E. 100 heuristics xvii

121

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Individuals with Disabilities Act (IDEA) 9

National Commission on Excellence in Education 6 National Council of Teachers of Mathematics (NCTM) 4–5, 53, 80, 99, 108–9

Joseph, G. 2, 99

opportunity to learn 7, 29, 90, 93

Ladson-Billings, G. 101 Liberal view 88, 94 see also public arguments

Plessy v. Ferguson 89 policy initiatives impact of 69 political conditions of schooling xii Principles and Standards for School Mathematics (PSSM) 4–6, 80, 108 professional development 29, 31, 77 professionalism 29–30, 69 public arguments xiii, xv, xvii, 77–9 Conservative 79–88 Liberal 88–96 Radical multiculturalists 96–105

high stakes testing 59, 62, 66, 69, 77 high status knowledge 34, 108 Hirsch, E.D. 6

McNeil, L. 69 mandates 24–5, 109 mandatory testing 25, 59, 62 “Math War” 53–4 Mathematically Correct 5–6, 85, 110 mathematics as a cultural activity 39 assumptions about 107–8 constructivist teaching of 83 culture-free 39, 44, 97 cultural historical development of xvi, 39, 52, 83, 105, 109 curriculum 37, 39–40, 60 Eurocentric view of 2 formalist view of 2 “fuzzy” 3, 6, 8 historical development of 2, 48, 98 multicultural approach to 42, 95, 103 multicultural issues in 41, 49 origins of 2 pedagogical goals 100 reform 3, 4 self-esteem and 81–4, 86 traditional 3, 4, 40, 53 multicultural curriculum 84, 93, 103–4 multicultural education 7, 102–3 multiculturalists 39, 79, 81 Murrell, P. 102 Nation at Risk: The Imperative for Educational Reform 6 National Assessment of Educational Progress (NAEP) 7, 24, 79, 84, 89–90 National Center for Education Statistics (NCES) 4, 24, 90

Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) 37 Quirk, W. 80 Radical multiculturalist view 96, 104 see also public arguments Saxon, J. 83 school mathematics xv, 1–3, 7–8, 77 characterization of 60 culture and xvi, 39–40, 44, 109 curriculum 97 dimensions of 107 equity and 89, 107 ethnomathematics and 112–13 politics and xvi, 53 reform of 3 teaching and learning of 78 “Separate but equal” 77, 89 social conditions of schooling x, xi, xiii “Social Darwinism” 91 social justice 7, 96–7, 100–1, 103 stakeholders xv, xvi, xix, 8, 10, 24, 53, 61, 63

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INDEX

standards-based learning 7 Stanic, G. 107 stratified curriculum 87

Third International Mathematics and Science Study (TIMSS) 4, 84, 91 Tracking 82, 90–1

Tate, W. 51, 89 teacher biases xvi, 34 teacher burn out 14, 19

Volmink, J. 2 Zaslavsky, C. 2–3, 97