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"description_t":"For many, mathematics and social justice are perceived as incompatible. Several mathematics education researchers have noted resistance to social justice among mathematics teachers. However, mathematics education has a consistently negative impact on the education of students of color. This study seeks to better understand the nature of this resistance by studying how preservice secondary mathematics teachers grapple with understanding social justice mathematics education. For this study I draw on discursive understandings of the operation of power and Whiteness Theory in order to understand the ways in which the discourses of mathematics serve to exclude the discourses of social justice. The participants in this study were seven preservice secondary mathematics teachers in a master's degree program in mathematics with teaching certification. Class discussions were recorded and transcribed then analyzed using Critical Discourse Analysis and a Whiteness Theory lens to interpret the analysis. The findings are organized around three main themes. These themes include discourses of the abstract nature of school mathematics, teacher and student subject positions, and our struggle to engage with the concepts of social justice mathematics. At times we disrupted these discourses through playfulness, repositioning students, and embracing the struggle of incorporating social justice into mathematics. There are important implications for mathematics education, mathematics teacher education, and teacher education generally.",
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"ocr_t":"SOLVING FOR IRRATIONAL ZEROS: WHITENESS IN MATHEMATICS TEACHER EDUCATION by Trevor Thayne Warburton A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Education, Culture and Society The University of Utah December 2015 Copyright © Trevor Thayne Warburton 2015 All Rights Reserved The Univers i ty of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Trevor Thayne Warburton has been approved by the following supervisory committee members: Audrey Thompson , Chair 6/12/2015 Date Approved Rochelle Gutiérrez , Member 6/8/2015 Date Approved Verónica E. Valdez , Member 6/12/2015 Date Approved Leticia Alvarez Gutiérrez , Member 6/12/2015 Date Approved Frank Margonis , Member 6/12/2015 Date Approved and by Edward Buendía , Chair/Dean of the Department of Education, Culture and Society and by David B. Kieda, Dean of The Graduate School. ABSTRACT For many, mathematics and social justice are perceived as incompatible. Several mathematics education researchers have noted resistance to social justice among mathematics teachers. However, mathematics education has a consistently negative impact on the education of students of color. This study seeks to better understand the nature of this resistance by studying how preservice secondary mathematics teachers grapple with understanding social justice mathematics education. For this study I draw on discursive understandings of the operation of power and Whiteness Theory in order to understand the ways in which the discourses of mathematics serve to exclude the discourses of social justice. The participants in this study were seven preservice secondary mathematics teachers in a master's degree program in mathematics with teaching certification. Class discussions were recorded and transcribed then analyzed using Critical Discourse Analysis and a Whiteness Theory lens to interpret the analysis. The findings are organized around three main themes. These themes include discourses of the abstract nature of school mathematics, teacher and student subject positions, and our struggle to engage with the concepts of social justice mathematics. At times we disrupted these discourses through playfulness, repositioning students, and embracing the struggle of incorporating social justice into mathematics. There are important implications for mathematics education, mathematics teacher education, and teacher education generally. This dissertation is dedicated to my students at Battle Mountain High School who taught me about being a teacher, and to the seven mathematics teachers who participated in this study. CONTENTS ABSTRACT ....................................................................................................................... iii ACKNOWLEDGEMENTS .............................................................................................. vii Chapters 1 CHALLENGES IN TEACHING MATHEMATICS FOR SOCIAL JUSTICE.............. 1 Problem Statement .......................................................................................................... 2 Research Questions ......................................................................................................... 9 2 LITERATURE REVIEW .............................................................................................. 11 Teaching Mathematics for Social Justice ..................................................................... 11 Understanding Dominant Mathematics ........................................................................ 19 An Historical Perspective of Mathematics as Political and Practical ........................... 27 Conclusion .................................................................................................................... 31 3 DISCOURSES OF WHITENESS IN MATHEMATICS TEACHER PREPARATION ............................................................................................................... 33 Discourses ..................................................................................................................... 33 Social Justice Mathematics ........................................................................................... 56 Conclusion .................................................................................................................... 61 4 METHODS: CRITICAL POSTSTRUCTURAL DISCOURSE ANALYSIS ............... 63 Participants .................................................................................................................... 64 Data Collection ............................................................................................................. 74 Data Analysis ................................................................................................................ 84 5 ABSTRACT MATHEMATICS AND THE \"GOOD\" MATHEMATICS TEACHER 90 Playfully Disrupting Abstract Discourses ..................................................................... 95 vi Dominance of Abstract Mathematical Discourses ...................................................... 135 Conclusion .................................................................................................................. 168 6 SUBJECT POSITIONS: TEACHER CANDIDATES AS POWERFUL AND HELPLESS ..................................................................................................................... 170 Discourses of Responsibility and Authority ............................................................... 172 Dominant Discourses of Authority and Responsibility .............................................. 176 Teacher Authority and Responsibility to Students ..................................................... 203 Conclusion .................................................................................................................. 233 7 LIVING IN THE STRUGGLE .................................................................................... 235 Radical Research in Mathematics Education .............................................................. 240 Teachers, Students, and Authority .............................................................................. 244 Balancing Achievement and Identity .......................................................................... 257 Normative Goodness and Fitting In ............................................................................ 271 Conclusion .................................................................................................................. 277 8 CONCLUSION ............................................................................................................ 279 The Discourses We Use .............................................................................................. 280 Excluding Mathematics for Social Justice .................................................................. 283 Struggling in the Differences of Discourses ............................................................... 287 Implications ................................................................................................................. 290 APPENDIX ..................................................................................................................... 300 REFERENCES ............................................................................................................... 312 ACKNOWLEDGEMENTS I'd like to first thank Joanna and our children who have supported me during this long process. I also wish to thank Audrey Thompson for continually pushing me to dig deeper and my committee for their insight, guidance and encouragement; Rochelle Gutiérrez, Veronica Valdez, Leticia Alvarez, and Frank Margonis. Thank you all. CHAPTER 1 CHALLENGES IN TEACHING MATHEMATICS FOR SOCIAL JUSTICE Recent research suggests that secondary mathematics teachers have difficulty teaching mathematics for social justice and that when the attempt is made teachers tend to focus either on mathematics or social justice. Mathematics teachers, in particular, have difficulty integrating mathematics and social justice to achieve both mathematical and social justice goals simultaneously (Bartell, 2013; Brantlinger, 2013). Social justice and mathematics, for these teachers, seem to be like oil and water. Even when teachers bring the two together they simply will not stay mixed in the moment of teaching. This kind of divided thinking has led some teachers to give up on or delay efforts to teach mathematics for social justice (Brantlinger, 2013). As a mathematics teacher I have faced some of these same difficulties and felt that the mathematics would not allow me to teach in ways that would meet the needs of my students. I watched my students struggle with the mathematics, especially those learning English, and how disconnected the mathematics was from the things they really cared about. Despite my ability to understand the mathematics well, to teach clearly, and to make connections with my students, the mathematics, at times, seemed to get in the way. However, there are alternatives to an oil and water perspective on the teaching of mathematics for social justice. A poststructuralist discursive perspective may open up 2 greater possibilities to understand the difficulties involved in teaching mathematics for social justice and aid teachers in learning to teach mathematics for social justice. In particular an understanding of the ways in which the discourses of mathematics and the discourses of Whiteness intertwine may create new possibilities for mathematics teachers to more effectively teach mathematics for social justice. Problem Statement Difficulties in Teaching Mathematics for Social Justice While teaching for social justice is not an easy task in any field, it may be particularly difficult in mathematics, since the perception of mathematics as abstract, apolitical, and acultural may appear to form an insurmountable divide from the explicitly political and contextualized perception of social justice. One source of this divide between teaching for social justice and mathematics may lie within the way modern mathematics is conceptualized and understood. Modern mathematics is dominated by abstract, decontextualized problems written with formal, symbolic language (Walkerdine, 1988). Modern mathematics is further assumed to transcend the concerns and problems of the world to focus on those things that are universal (de Freitas, 2013). This way of thinking about mathematics has created a way of talking about and enacting mathematics that is focused on the static existence (in this fictional world) of mathematical objects (de Freitas, 2013), rather than on solving meaningful, contextualized problems or on the inclusion of alternative ways of thinking (except for professional mathematicians; Gutiérrez, 2012a; 2012b). One relevant effect of conceptualizing mathematics in these ways is to disconnect 3 mathematics inquiry from the inequities both students and teachers face in and out of school. With these discourses of mathematics as abstract and universal influencing teachers, it is perhaps not surprising that most mathematics education in U.S. schools is characterized by teachers teaching and students following rote procedures that lead to a single and previously known (to the teacher) answer (Gutiérrez, 2012a). In these traditionally taught classes there is often little room for creative uses of mathematics or for connections between subject areas (Boaler & Greeno, 2000). Skovsmose and Valero (2001) describe mathematics teachers in traditional mathematics classes as autocratic and students as passive. The lack of creativity and connection can be alienating to any student who does not identify as \"mathematical.\" Mathematics education has been specifically linked to the technological advancement of nations, and has remained so for over 50 years (Skovsmose & Valero, 2001). All student experiences, especially experiences that might be seen as culturally specific, are largely irrelevant to the class since the mathematics taught lacks contextual connection to student lives. In these classes students learn to see mathematics as unitary and certain, with minimal social dimension and minimal need to collaborate. Learning takes place on a very shallow level (i.e., only learning rote procedures and formulas) and students view themselves as passive learners. Some students go so far as to view thinking as only minimally necessary in mathematics, other than selecting the correct formula (Boaler & Greeno, 2000). In response to the disconnect experienced by many students in traditional mathematics classes, there have been significant reform attempts in an effort to incorporate meaningful problems and contexts and to develop multiple understandings of mathematics (Ellis & Berry III, 2005; McClintock, O'Brien, & Jiang, 2005). Other 4 reform efforts have tried to increase the cultural sensitivity of teachers and increase their willingness to use classroom practices that are equitable for female, racial minority, and/or language minority students (Ellis & Berry, 2005). Despite their significant contribution to an understanding of mathematics education, these efforts have met with limited success and significant resistance among mathematics educators (de Freitas, 2008; Gutstein, 2006). De Freitas (2008) has suggested that mathematics teachers may resist these reforms out of an interest to preserve the kind of mathematics that they themselves were comfortable with. In other words, they work to preserve the discourses of mathematics where knowledge is certain (Ernest, 1991) and static (de Freitas, 2013), in large part because they were taught in these ways and are comfortable and secure using these same discourses. Further, as these discourses also preserve and maintain White privilege there is inherent self-interest for the majority of mathematics teachers. However, Bartell (2013) notes that even mathematics teachers who are committed to teaching for social justice struggle to do so, because they end up focusing more on the mathematics despite their plans and intentions to do otherwise or, at best, they divide mathematics and social justice into separate parts of a lesson. Teachers, who want to teach in socially just ways, will be the focus of this study. It is possible that for these teachers the discourses of Whiteness and mathematics appear to be working against their desires to teach for social justice. Further these teachers may act in these ways without recognizing the contradictions of their behavior and desires. Yoon (2012) explains that Whiteness and the need to maintain White privilege can lead to behavior (in teachers) that is paradoxical and contradictory. In the case of mathematics teachers who plan to teach for social justice, but in the moment 5 are unable to, this may be the case. Whiteness, Mathematics, and Discourses Whiteness Theory operates on the assumption that the lives of all people in the US in particular (but elsewhere as well) are racially structured, including the lives of White1 people (Frankenberg, 1993; Frye, 1992). Since mathematical achievement plays out on clearly racial lines in U.S. K-12 schools (Stinson, 2004) I operate from the assumption that race (as a social construction) is a significant factor in determining mathematical success as traditionally measured within mathematics (e.g., correctness, following procedures, grades, standardized test scores). Whiteness is a system of privileges, power, and authority that primarily benefits White people. The ways in which these privileges are built into the institutions and thinking of society has created a power structure that favors and is maintained through discourses. Discourses are these ways of thinking and being in society (Gee, 2005; 2012). The discourses of Whiteness are the \"common sense,\" unquestioned (by most Whites, at least) beliefs and values about the way the world does and should work (Yoon, 2012). These discourses of Whiteness currently serve to deny the existence and relevance of racism and, as a consequence, preserve racist structures in the United States, as well as within schools and mathematics education. Just as discourses maintain Whiteness, there are also discourses of mathematics and mathematics education that describe and exemplify the ways in which mathematics 1 Here, and elsewhere, in this document I will use White (capital W) to draw attention to White as a racial category. One of the ways that Whiteness operates is by directing attention away from Whites by suggesting that only people of color have something called \"race.\" 6 should be taught and the ways in which mathematics should be learned. These discourses shape teacher and student understandings of what mathematics is. For example, if you are asked to think about mathematics you probably think of a school classroom, textbook, or equations on a paper or on a board. Each of these contexts (classrooms, textbooks, and equations) is dominated by common understandings of mathematics as abstract, apolitical, neutral, and acultural (Ernest, 1991; Walkerdine, 1988). In this context doing mathematics means manipulating symbols to arrive at a correct answer that has little to no meaning (Brown, 2001; Gutiérrez, 2012a). Even so-called \"application\" problems in school mathematics typically have little connection to contexts that are meaningful to students or teachers. Because of the disconnect between mathematics and their experiences students are often unable to make meaningful connections between the mathematics they are learning in school and their struggles and interests outside of school. Even students who \"correctly\" answer these questions often do so by ignoring real-world considerations that make their mathematically correct answer irrelevant outside of math class (Mukhopadhyay & Greer, 2001). More importantly for students of color these understandings of mathematics are based on the White, male, middle-class perspectives (Ellis & Berry III, 2005: Walkerdine, 1988; 1990), which work to deny the relevance of perspectives of communities of color (Thompson, 1998). Thus beyond irrelevance for students of color mathematics is also potentially alienating. These discourses of mathematics as abstract, apolitical, neutral, and acultural interconnect and overlap (these connections will be explored in more detail later) with discourses of Whiteness that deny the existence of racism and view the world through a White lens. The ways in which these discourses 7 work together exclude both the possibility and the necessity of teaching mathematics for social justice. Poststructural Perspectives on the Operation of Power From a poststructural perspective power is multidimensional and productive. It resists simple definition as essentially two-dimensional (the power of the elites to control vs. the power of the people to resist). In this perspective one power structure is merely replaced by another, which then becomes dominant (Wang, 2011). From a poststructural perspective power is diffuse and exists throughout society in interactions between people. Thus, the ultimate goal in teaching mathematics for social justice is not to simply replace dominant perspectives on race or on mathematics, but instead to create a new system that works in fundamentally different ways towards \"a positive relationship between mathematics, people, and equity throughout areas of the globe\" (Gutiérrez, 2002a, p. 148). This goal requires a different understanding of the ways in which power operates. Power as productive means that it produces, but rather than producing things, power produces events and structures relationships between people. As teachers learn to think critically about their own discourses and exercise of power they may use their power in different, more socially just ways. Teachers and students in a mathematics classroom together produce a particular event of mathematics education. This event (a moment of mathematics education) does not exist outside of the moment and efforts of these people; it is not an object. However, it also does not exist free of the influence of the context in which it occurs, which includes prior events of mathematics education (as engaged through discourses), understandings of what it means to be a teacher or a student, and understandings of race 8 and what it means to be White, Black, or Latina/o, etc., and understandings of what it means to be male or female. In this sense then each moment in a mathematics class is a recreating (as opposed to creating) from prior, historical moments in mathematics classes, but never the same as prior or future events and never completely formed (Wang, 2011). The discourses that teachers and students engage in this process influence the thoughts and actions of teachers and students in part through the beliefs and values embedded within them. Problems arise because when a teacher is familiar and comfortable with these discourses the underlying beliefs and values are mostly invisible (Fairclough, 2001). In the case of the mathematics teacher trying to teach for social justice the common-sense discourses of what it means to teach mathematics may appear to conflict with what it means to teach for social justice. Why Focus on Mathematics? There are multiple reasons to study the teaching of mathematics for social justice. Mathematics is a primary means of maintaining White privilege in schools through its gatekeeping role in determining who has access to high status fields (science, medicine, engineering, mathematics, physics, etc.) and higher education in general (Stinson, 2004). The maintenance of these privileges reinforces existing power structures with Whites in positions of power and with increased opportunities for wealth accumulation for White people. Further, mathematics may prove to be a powerful tool in the promotion of social justice (Gutstein 2006) by critiquing current inequities and exploring other ways of understanding and viewing the world. Mathematics as a subject area has been particularly resistant to efforts to promote social justice (de Freitas, 2008; Gutstein, 2006). While some subject areas have appeared to be more open to teaching for social justice, 9 mathematics, especially secondary mathematics, has not. The reasons for this resistance are only partly understood. A better understanding of the ways in which discourses of mathematics and discourses of Whiteness promote resistance to teaching for social justice is the primary aim of this study. Research Questions Traditional school mathematics is a problematic subject, particularly for historically marginalized students. School mathematics has been used to sort students into higher and lower tracks and to bar entry into higher education (Stinson, 2004). This sorting mechanism is accomplished through the discursive link between mathematics ability and intelligence. Through this connection mathematics achievement is used to create an intelligence hierarchy among students. Further, even those students who have been successful in school mathematics may have had to change meaningful aspects of their identities (Boaler & Greeno, 2000; Gutiérrez, 2012b) in order to advance. While social justice approaches to mathematics education have seen some success they remain only a small part of U.S. secondary mathematics education (Gutstein, 2006). The small impact of social justice mathematics on mathematics education is in part because of the consistent resistance to social justice education among mathematics teachers. But this resistance is not well understood (de Freitas, 2008) and may be due, in part, to the influence of the discourses of mathematics and Whiteness. As a consequence in this study I seek to understand how teachers' desires for equitable classroom practices are alternately facilitated and impeded by how they position themselves to teach mathematics for social justice and how they are influenced by the discourses of school mathematics in ways that work against their ideals of equitable practice. To this end this study seeks to 10 answer the following questions: 1) What discourses do secondary mathematics teacher candidates invoke when discussing social justice in their own teaching practice? 2) How do the discourses secondary mathematics teacher candidates use around school mathematics in the United States interact with the discourses they use around social justice mathematics? 3) How do secondary mathematics teacher candidates merge/manage and challenge the disparate discourses of mathematics and social justice during student teaching in a program that emphasizes preparation for teaching in culturally and linguistically diverse contexts? CHAPTER 2 LITERATURE REVIEW In this chapter I review some of the relevant research on social justice mathematics. I first explain the principal aims of social justice mathematics and the challenges of teaching mathematics for social justice. I then explain how the discourses of dominant mathematics contribute to the challenges of teaching mathematics for social justice. I demonstrate how the dominant discourses of the nature of mathematics are inconsistent with an historical understanding of the development of mathematics. This historical piece further demonstrates the potential political and practical nature of mathematics, and thus that mathematics does not have to be incompatible with social justice education. Teaching Mathematics for Social Justice Gutiérrez (2002a) uses the term \"dominant mathematics\" to describe the mathematics that is traditionally taught in schools, that serves dominant interests in maintaining the status quo of society and schooling, takes an uncritical approach to the structure of society and schooling, and fails to recognize the contributions and potential contributions of marginalized peoples. Drawing from her definition I will use two terms throughout this study. First, I use \"school mathematics\" to refer specifically to the mathematics that is taught in schools. School mathematics is part of, but not all of, 12 dominant mathematics. Second, I will use \"dominant mathematics\" to refer not only to the mathematics that is taught in schools, but also to mathematics outside of schools, as it is used by professional mathematicians as well as other professionals whose work depends on mathematics (engineers, quantitative scientists, actuaries, etc.). The need to improve the mathematics educational outcomes of historically marginalized groups of students is largely unquestioned, with various mainstream organizations calling for improvement in mathematics education for decades (Gutiérrez, 2002b; Secada, 1989). However, social justice mathematics advocates argue that social justice education must go beyond merely improved educational outcomes (Ebby, Lim, Reinke, Remillard, Magee, Hoe, & Cyrus, 2011; Frankenstein, 1990; Gutiérrez, 2002a; 2009; 2012a; 2013; Gutstein, 2003; 2006; 2007; 2009; Moses & Cobb, 2001; Secada, 1989; among others). Gutstein (2006) explains that even when students of color and other marginalized groups have been successful in mathematics classes, they have not learned to use mathematics to challenge the inequalities that directly affect them. The result is that the status quo of inequality is maintained for the majority of marginalized students. From this perspective a social justice approach to teaching mathematics is a necessary part of achieving more equitable outcomes in mathematics education and, more importantly, for the creation of a more just society. The work of Frankenstein (1990) was ground breaking in linking critical pedagogy and ethnomathematics to U.S. mathematics education, through what she calls critical mathematical literacy. Frankenstein's work became the foundation for what I refer to here and throughout the text as social justice mathematics. The examples of social justice mathematics as lessons that engage in social (both local and global) critique as shown in Gutstein's work (2006; 2013) and 13 Frankenstein's (1990) have become the most well-known approach to social justice mathematics. However, there are others who promote a broader view of social justice mathematics. These include Garii and Appova (2013) who distinguish between teaching in a socially just manner (equitable access through pedagogy), teaching about social justice (social critique), and teaching for social justice (making connections to students lives). Skovsmose and Valero (2001) use the term \"democratic\" mathematics education to discuss similar ideas to those of social justice mathematics. They define democracy in terms of collectivity (collective action), transformation (collective work for change), deliberation (as dynamic dialogue), and cofleciton (collective reflection). Skovsmose and Valero link these ideas to mathematics education by suggesting that we rethink what mathematics are needed for democratic citizenship, teacher-students relationship in mathematics classes, what mathematics education means in a school, the national role of mathematics education, and the global role of mathematics education. I draw primarily from the work of Gutiérrez (2002a; 2002b; 2009; 2012a; 2012b; 2012c; 2013; 2015) and Gutstein (2003; 2006; 2007; 2008; 2009; 2012). Gutstein's work is important, in part, because of the concrete examples of teaching mathematics for social justice that he gives and because it is perhaps the best known form of social justice mathematics in U.S. mathematics education. The work of Gutiérrez encompasses a variety of approaches that are relevant to the U.S. context and focuses on the school-teacher- student level that is most applicable to this study. A social justice approach to mathematics education has demonstrated the potential to addresses some of the current issues in mathematics education (Gutstein, 2006). Social justice approaches to 14 mathematics have been developed to address societal inequities and to increase access to advanced mathematics for women, students of color, and poor students. Gutiérrez (2002a) defines social justice mathematics as working towards a goal of: Coordinat[ing] (a) efforts to get marginalized students to master dominant mathematics with (b) efforts to develop a critical perspective among all students about knowledge and society in ways that ultimately address (c) a positive relationship between mathematics, people, and equity throughout areas of the globe. (p. 148) The first portion of this goal is in reference to the wide disparities in achievement in mathematics classes that have disproportionately affected students of color and students living in poverty. This part of her goal then is to achieve equitable outcomes (as evidenced by traditional measures of achievement) in mathematics courses. Many mainstream organizations such as the National Council for Teachers of Mathematics (NCTM) also push this goal (Secada, 1989). However, the second portion of this goal goes beyond outcomes to include teaching students how to use mathematics as a tool for social critique. Historically mathematics has been used to justify inequalities. Here Gutiérrez argues that students should be taught to use mathematics to argue against inequalities and critique the mathematics used in arguments that maintain or exacerbate current inequalities. The third portion is a more distant goal and, by her own admission, is a kind of mathematical utopia in which mathematics has been used to achieve full equality across the globe. Gutiérrez (2002a) notes that Eric Gutstein has developed a curriculum that perhaps best achieves the first two goals. Gutstein (2006) explains that his work in Chicago with low SES students of color has focused on working towards a more equitable society, achieved through greater academic success and access to advanced 15 mathematics and higher education for students of color. However, his goal goes beyond success and access to teach students to use mathematics to argue for a more just society as well as critique arguments for maintaining the status quo. To accomplish these goals he has developed curricular materials that make mathematics relevant to issues that the students face and connect them to more global issues. These local issues provide the rich, meaningful details that are lost in abstract school mathematics. He then works with students to use mathematics to understand and argue against the inequities that they face, such as poverty, racial profiling, and gentrification of their neighborhoods. In his own work Gutstein (2006) draws on the Freirian tradition of education for liberation and defines social justice mathematics as working towards both mathematical and social justice goals. The social justice goals can be summarized as saying that students should be able to use mathematics to better understand their world (especially inequities), construct mathematical arguments against those inequities, and counteract the effects of deficit racial perspectives in society. The mathematical goals can be summarized as saying that students should develop positive attitudes about mathematics and understand college-preparatory mathematics to achieve traditional academic success. Here Gutiérrez (2002a; 2012b) might suggest an additional goal of \"writing the mathematical word\" to acknowledge that students of color can also shape the field of mathematics and make contributions to mathematics. Otherwise their goals overlap in many aspects. Difficulties of Teaching Mathematics for Social Justice While Gutstein (2003; 2006; 2007; 2012) in particular has demonstrated impressive successes through social justice mathematics curricula, his work, as well as 16 the work of other social justice mathematics educators (Ebby et al, 2011; Gutiérrez, 2002a; Moses & Cobb, 2001), has received relatively little attention from the mainstream segment of mathematics education research (Gutiérrez, 2002a). Mathematics teaching as a whole has not seen widespread acceptance of mainstream reforms (McClintock, O'Brien, & Jiang, 2005) much less of social justice education (Gutstein, 2006). Both Gutstein (2006) and Gutiérrez (2009) explore the difficulties of teaching for social justice within a mathematics context. However, they do so in different ways. Gutstein (2006) lays out what he views as the characteristics and knowledge that a mathematics teacher should exhibit in order to effectively teach for social justice. He argues that mathematics teachers will need to go \"beyond the mathematics\" in order to build understanding of social and political forces that structure a seemingly straightforward or neutral situation. For example, in one project his students investigated the potential effect on home prices of a proposed gentrification project in their neighborhood. To go beyond the mathematics in this situation requires a knowledge of some of the local area history, the way home prices and mortgages work, and the history of racially segregated neighborhoods in the US. Both teacher and students then need political knowledge, historical knowledge, and economic knowledge in addition to the mathematics that are necessary to understand just the numbers of gentrification. Not only is this knowledge not part of a traditional teacher preparation program, but because it is perceived as nonmathematical knowledge mathematics teachers may see it as irrelevant to their teaching, or even inappropriate. Gutstein (2006; 2008) further advocates that teachers develop \"political relationships\" with their students. This includes taking a political stance in support of 17 students, but also requires making that political stance known. This may be difficult for teachers who view mathematics as inherently apolitical and particularly so if they value their relationship with mathematics over their relationship with students (Gutiérrez, 2009). But even when teachers have the necessary knowledge and work to develop political relationships with students they may have students who resist their efforts, because students also may have learned to view mathematics as apolitical and decontextualized, through their previous experiences with school mathematics. A final difficulty that both Gutiérrez and Gutstein (2006) explain is that, in the current education system, teachers must still find ways to prepare students to take and pass high stakes tests as well as prepare them for success in future mathematics classes. Without this preparation students' educational opportunities will grow more limited. This requires some kind of negotiation of the discourses of school mathematics and teaching for social justice. These negotiations may be facilitated as teachers begin to challenge the ways in which mathematics may push them away from teaching for social justice. Despite the scope of his work mathematics teachers tend to associate the work of Gutstein (2006) only with the social critique lessons that he presents. The work of Gutiérrez (2009) is useful in creating a more complete understanding of what it means to teach for social justice. As she explains, teachers must recognize the importance of knowing their students well, including their culture, history, and personal background, in order to incorporate and validate those experiences in the classroom. Simultaneously, teachers must also recognize that they can never truly know their students because their students are continually changing, not static individuals or essentialized representatives of a culture. This stance complicates the discourse of teacher as knower. Second, teachers 18 must take charge of the curriculum in the classroom, how it is presented to the students, and use all of their skills to make the mathematics engaging and inviting to the students, but they must recognize that it is ultimately the students' decision to participate or not. This tension recognizes that both students and the teacher have power in the classroom. Too often a textbook, department culture, school policies, district policies, or state policies dictate what a curriculum will be and how it should be taught, resulting in a disengaged teacher and/or disengaged students. Finally, Gutiérrez notes, as does Gutstein (2006), that there is a tension between teaching students (i.e., meeting their needs and interests in addition to seeking social justice) and teaching the necessary mathematics to meet standards or to prepare students for the next level of mathematics and college. This last point is problematic because not preparing students may limit their education, but the standards do not recognize the value of teaching for social justice. These tensions are not meant to be resolved; instead Gutiérrez (2009) advocates that teachers hold on to both sides of each in order to teach for social justice. These difficulties highlight the possibility that there is something about mathematics that influences teachers in ways that make teaching mathematics for social justice difficult. Various authors who write about the nature and philosophy of mathematics and mathematics education provide perspectives on what it is about mathematics that make teaching for social justice particularly difficult in this subject. Rousseau and Tate (2003) make one of the more direct links noting that the philosophy and foundation of mathematics discourage reflection on issues of social justice because they normalize those same issues. For example, by presenting mathematics as neutral (not biased) the consistent achievement gap between student groups appears to be natural. 19 However, this does little to explain what it is about mathematics that makes the normalization of social injustice possible (after all this happens in other subjects as well). Brown (2001) explains that mathematicians, like other groups, form a kind of community with mathematics as their language. Since the language of mathematics is created and re-created by mathematicians, the beliefs and values of those mathematicians are part of the discourses of dominant mathematics. In this case then dominant mathematics was created with the beliefs and values of and in the image of a select group of wealthy, White, males and is reinforced through abstraction, decontextualization, and the use of a formal, symbolic language (Walkerdine, 1990). Historically these wealthy, White, and (often) state-sponsored mathematicians have policed what was recognized as mathematics and viewed other approaches to mathematics as a threat to the bounded and absolute mathematics that they had created (de Freitas, 2013). This defensive behavior protected their own positions by excluding other possibilities and in the process they shaped mathematics as the absolute and apparently apolitical mathematics taught today, by closing off other possibilities. It is from this absolute nature of dominant mathematics that potential conflicts with teaching for social justice may arise. Understanding Dominant Mathematics Ernest (1991) claims that mathematics is constructed as the most certain of human knowledge. The dominant view of mathematics for millennia has been an absolutist view. Ernest defines this absolutist view as maintaining that mathematics is made up of certain knowledge and absolute truth. Further because these truths can be established without reference to empirical evidence they are the most certain of any knowledge. The absolutist view of mathematics has dominated mathematics education and is present in 20 most textbooks and classrooms in the United States. Brown (2001) specifies that in this absolutist world mathematical terms do not refer to anything tangible. For example, the symbol \"5\" represents a numerical value. While people may use the symbol to refer to the quantity five of something, the symbol on its own does not refer to anything concrete. This lack of referent is part of the certainty and abstractness created in modern mathematics. Further Brown (2001) points out that mathematical objects are imaginary. This is perhaps most clear in Euclidean geometry, which is literally an imagined world in which points have no dimensions and lines have no thickness. Everything is an idealized form that cannot exist in a tangible form. It is possible that the decontextualization and abstractness used to create these idealized forms make it difficult for teachers, students, and others to connect mathematics to the highly contextualized reality of social inequality that is required by social justice education. Dominant Mathematics Complicates Social Justice Efforts Mathematics education may be particularly resistant to social justice approaches, because mathematics teachers are dependent on the discourses of abstract certainty that arise from a discourse of mastery and certainty that is only possible within abstract mathematics (de Freitas, 2008). The work of de Freitas (2008) suggests that, in the case of mathematics teachers, resistance to social justice approaches to education may stem, in part, from the teachers' connections to and investments in mathematical discourses. In the case of mathematics this resistance may be facilitated by the dominant discourses around school mathematics that can cause conflict between a desire to teach for social justice and what they feel falls within the discourses of dominant mathematics. This could push mathematics teachers to resist explicitly political ideas, such as social justice, because 21 they pose a perceived threat to their mathiness-the mathematical part of their identity. Mathematics teachers may feel a particular need to defend their mathiness because they are juxtaposing a high-status field (mathematics) with a relatively low-status field (teaching; R. Gutiérrez, personal communication, May 13, 2013). A Perception of Mathematics as Apolitical The potential conflict between school mathematics and social justice may lie in the perceived apolitical nature of school mathematics. This apolitical characteristic of mathematics depends on a view of mathematics as neutral, abstract, certain, and absolute. Walkerdine (1990) refers to this understanding of mathematics as a kind of axiomatic, rule-bound world in which mathematical objects and forms have consistent definitions which can be depended on and proved. These conditions are necessary for the certainty with which mathematics operates. Despite the real effects for many students, this mathematical world is a fictional world, because of its lack of connection to anything tangible. To enter into this fictional world a problem needs to be stripped of the context in which it arose including the needs and perceptions of the people for whom the problem is meaningful (Walkerdine, 1988; 1998). In this way, a mathematical problem that addresses the lack of resources common in many urban schools is unlikely to be recognized as valid within the discourses of dominant mathematics because the numerical comparisons between schools are not mathematically interesting; instead they are politically meaningful. However, problems that are often challenging for students such as the now legendary \"If a train leaves point A at a certain speed and another train leaves point B at the same time at a certain speed when will they meet?\" is likely to continue as a regular part of the mathematics 22 curriculum because of its perceived mathematical value and despite its lack of relevance or meaning in the lives of students. When contextualized and meaningful problems are allowed in they are always on the periphery. It is only after meaningful problems have been generalized and abstracted that they become mathematically valuable (Walkerdine, 1998). However, this generalization and abstraction reinforce the view of mathematics as universal and apolitical, thereby hiding the way the dominance of these discourses excludes social justice. Problems Arising From an Apolitical View of Mathematics The dominant view of mathematics as absolute and certain grows out of an ancient Greek mathematical system that valued formal, abstract mathematics over practical, everyday mathematics (Cooke, 2013; Seife, 2000). The result today is that practical or applied mathematics is often not recognized as fully in the mathematical world despite the complexity of practical mathematics. Practical mathematics are influenced by so many variables that situations can never be completely predictable or certain. They are in this way a challenge to the certainty of absolute mathematics. Often those who enjoy school mathematics, enjoy this certainty (Boaler & Greeno, 2000). However, a common complaint about school mathematics is the refrain, \"when will I ever use this?\" The problem is that applying school mathematics to real contextualized problems is complex and does not usually look like the mathematics that is found in textbooks and taught in schools. As a consequence contexts that may heavily involve mathematics are not recognized as mathematical. Rose (2012), for example, describes the complex mathematics found in a vocational welding program, The central precepts of welding are travel-the speed of your movement of the 23 instrument-the distance of the instrument from the metal, the angle of it, and how hot you've got it. . . . Travel, angle, and all that are further complicated in some processes by the fact that the electrode conducting the current is being used up as you weld, so you've got to continually adjust your travel speed and angle and distance to keep things constant. (p. 9) Speed and distance are both directly related to algebra, while angle is a key concept in both geometry and trigonometry. Burn rates and temperature also involve complex mathematics. Yet what these welders do and what they are taught is not recognized as mathematics, sometimes even by those who do it. Rose describes the instructor of this welding course as saying that he doesn't know mathematics very well. The ideal, he believes, would be to have a mathematics teacher demonstrating the division of decimal fractions and calculation of volume, and explaining the why of what the class is doing, the mathematical principles involved. (p. 12) In a related way, Frankenstein and Powell (2002) explain that the problem of knitting the crook of a sock without the material bunching up is mathematically similar to the problem of creating a curved pipe without the metal folding in on itself. While the problem of a curved pipe associated with \"man's work\" is engineering, the knitting, seen as \"women's work,\" is not even mathematical. So why is it that the instructor does not consider what he does to be mathematics? Why is it that knitting a sock is not considered mathematical? It is likely that neither the welders nor the knitters are making mathematical calculations in their heads, nor are they solving equations; without these formal markers of abstract mathematics most of us are unable to recognize knitting or welding as mathematics. However, good knitters and welders take all of these factors into account and make adjustments as they go. Because both the mathematics of welding and of knitting socks are not recognized as mathematics, welders and knitters may be left with the idea that they are not mathematical and that they are not as intelligent (because 24 we equate mathematical ability with intelligence) as mathematicians. Clearly these decisions of what counts and what does not count as mathematical are political and they have political effects. As in the previous example from Rose (2012), mathematics teachers may be unaware of how an absolutist view of mathematics obscures their ability to recognize mathematics without its formal markers. In fact these teachers may also be caught up by discourses of abstract mathematics just as students are (Walkerdine 1998; Walshaw, 2013). Further, mathematics teachers may feel a need to preserve these discourses. Teachers of mathematics may enjoy the distinction that mathematics gives them over other teachers and over their students. They can feel a sense of control and mastery from their ability to solve mathematical problems and manipulate mathematical symbols (de Freitas, 2008; Walkerdine, 1998). Changing that (already political) world by introducing politics explicitly, to take a social justice approach, or students' experiences and perspectives, to take a culturally relevant approach, can threaten the control that teachers gain from formal mathematics and the system of White privilege that formal mathematics helps maintain. Teachers and Whiteness also benefit from the appearance of neutrality inherent in absolutist mathematics that allows them to hold themselves above the political opinions of others and other subjects. Making mathematics explicitly political can threaten this neutrality and mathematics would lose the illusion of colorblindness. By using mathematics to maintain differences between themselves and their students and between themselves and other teachers, mathematics teachers create a kind of authority, both literal and moral. The literal authority accompanies the traditional positioning of teachers. The moral authority derives from the perceived neutrality of 25 abstract mathematics. This neutrality (lack of bias) allows mathematics teachers to perceive and present themselves as fair judges in the classroom. This role of unbiased judge is one of the privileges that accompany and maintain Whiteness (Frye, 1992), because it allows White people to judge wrong and right, including what acceptable responses are in a mathematics class. The authority to judge right and wrong closely overlaps with discourses of dominant mathematics that portray mathematics as absolute and certain. The mathematics teacher, as representative of this certain world, is required by the discourses to judge students as right or wrong mathematically. This privilege of judgment then determines which students advance academically and shapes how students perceive themselves in relation to mathematics. However, this authority is created by stripping away the human contexts in which mathematics operates and from which it arises (Cooke, 2013; Walkerdine, 1988; 1998) and it is only within a fictional world that mathematical authority can exist; as such it is always precarious and must be reasserted. When this authority is threatened, those who have mastered the abstract discourse of school mathematics can retreat into the formal, symbolic language to reassert their supposed superiority. In the mathematics classroom a teacher can, without realizing it, use this power to keep students in their place and maintain their own authority. Teachers can use this power to justify and normalize the failure of students-especially minority and low-income students (Martin, 2007; Rousseau & Tate, 2003)-and in the process maintain a sense of fairness and justice based on the neutrality of mathematics. In this manner, teachers may use the authority of school mathematics to justify and rationalize the injustices perpetuated by inequitable practices in schools. Mathematics aids in this normalization process so that mathematics teachers may participate in failing students 26 along racial lines, while maintaining a self-perception as just and fair, because their judgment rests on the neutrality of mathematical discourse. Recognizing the role of mathematics and the role of teachers of mathematics in creating these injustices threatens the comfort of moral authority that teachers may gain from the fictional world of school mathematics. For these reasons, among others, the formal, symbolic language of mathematics continues to be privileged and the White, middle-class, male discourses of mathematics continue (Walkerdine, 1988; 1998). While skilled use of mathematical discourses grants access to power and authority within the real world, that access serves to divide students as mathematical or not (Skovsmose & Valero, 2001). These divisions create an over-representation of students of color and women as \"not\" mathematical. It is exactly the abstract within mathematics that is pointed to as the pinnacle of thinking that those perceived as not mathematical (whether White students or students of color) are unable to obtain (Walkerdine, 1990). Within this world alternative ways of thinking and solving problems are not viewed as mathematical, are looked down on, and ultimately discouraged (Gutiérrez, 2012b). Gutiérrez notes that often if a student in a middle- or high-school geometry course suggests that not all lines are straight (a perfectly logical conclusion from the real-world) that understanding is often quickly corrected to fit the constraints of classical, Euclidean geometry. However, mathematicians have used the simple understanding that not all lines are straight to explore other geometries (hyperbolic, spherical, taxi-cab, etc.), the existence of which is unlikely to be mentioned in secondary geometry courses. The result too often can be that students may stop thinking of alternatives, focus on the information given, and come to believe that their understandings and experiences 27 are not valid within the mathematical world (Boaler & Greeno, 2000). Students thus must \"become someone else\" (Gutiérrez, 2012b, p. 30) to succeed. Students are actively discouraged from using the kind of thinking that views knowledge as multiple and making connections in the ways that mathematicians do (Boaler & Greeno, 2000; Gutiérrez, 2012a; 2012b). These students may feel that to succeed in mathematics they must think and reason in ways that are more like an elite White male view. Just as mathematics, historically, was held back by an unwillingness to accept the concept of zero (Seife, 2000), Gutiérrez (2002a; 2012b) has argued that modern mathematics is held back by an unwillingness to recognize the potential contributions of women and students of color. While abstract mathematics need not be the only kind of mathematics, it continues to function as if it were. As a consequence, school mathematics has remained relatively unchanged despite decades of reform efforts (Gill & Boote, 2012). Both teachers' and students' thinking is constrained by the maintenance of the discourses of absolutist mathematics. However, this view of mathematics is inconsistent with an historical understanding of the development of mathematics. An Historical Perspective of Mathematics as Political and Practical A review of some of the history and development of mathematics helps to dispel some of the myths about mathematics that are perpetuated by the dominant discourses. In general, these discourses portray mathematics as apolitical, abstract, certain, and neutral. It is the most decontextualized of the sciences and gains its prestige from this abstraction and decontextualization (Gutiérrez, 2012b; Walkerdine, 1988). However, mathematics has not always been neutral, apolitical, and abstract. Cooke (2013) explains that 28 mathematics is created anytime people think about their world. In particular he credits accounting, surveying, astronomy, and kingdoms (including laws and theology) with creating the roots of mathematics. Joseph also links the development of Indian mathematics (1997) and Mayan mathematics (2008) to religion and astronomy. He further connects the development of aspects of Chinese mathematics to the surveying of land (1997). All of these practices are political, contextual, or practical if not all three. An extreme example may illustrate the political nature of mathematics: Hippasus of Metapontum stood on the deck, preparing to die. Around him stood the members of a cult, a secret brotherhood that he had betrayed. Hippasus had revealed a secret that was deadly to the Greek way of thinking, a secret that threatened to undermine the entire philosophy that the brotherhood had struggled to build. For revealing that secret, the great Pythagoras himself sentenced Hippasus2 to death by drowning. To protect their number-philosophy, the cult would kill. (Seife, 2000, p. 26; see also Joseph, 1997) That secret was the existence of irrational numbers3 and their existence went against Greek understandings of how the world worked. For the ancient Greeks number and shape were directly linked and formed a kind of religion. Those who followed it gained power and prestige, which was resented by others, eventually resulting in Pythagoras' death (Seife, 2000). Dominant mathematics, especially as represented in schools, has its roots primarily in this Greek system. This Greek influence is seen most clearly in the lasting impact of Euclid's Elements, which lays out his approach to the study of geometry (Cooke, 2013). The Elements is, in a literal sense, a fictional world in which a point has no dimension and lines continue on straight, and infinite. The world described in the 2 Whether he was actually killed is disputed; he may have been only banished from the society (Veljan, 2000). 3 Irrational numbers are decimal numbers that do not repeat and do not end. Some of the more commonly known irrational numbers are π, e, and 2. It was 2 that was particularly troubling to Greek geometers. 29 Elements bears only a passing resemblance to the world we live in, because that was not its purpose. According to Cooke (2013) what set Greek mathematics apart was its formalization, which is only possible through abstraction and decontextualization. The formal development of mathematics (as opposed to a practical or applied or musical, etc. development) has a continuing influence on the development of mathematics and has set the standard (in the Western world) of how mathematical knowledge is to be judged. Greek mathematics was clearly political and probably religious. The Greek system is probably best recognized currently in the form of the Pythagoreans, who viewed shapes as having characteristics of good and evil (Cooke, 2013). This understanding of mathematics is the foundation of modern school mathematics. In this system the abstract was venerated, while the practical was denigrated. It was this mathematical belief system specifically that pushed Western mathematical philosophy to reject zero, resulting in centuries of set-backs in mathematics, science, and economics. While zero was used to advance Indian and Mesoamerican mathematics, mathematics in Europe was hampered by the lack of a place value system that zero makes possible, among other mathematical advantages of using zero (Joseph, 2008; Seife, 2000). Political Efforts to Shape the Development of Mathematics An unwillingness to accept zero led to experiences that were somewhat comical. Personal and societal beliefs about zero at times resulted in ridiculous behavior, such as Greek mathematicians who translated problems involving fractions into the Babylonian systems in order to surreptitiously use zero, but then translated problems back to the Greek system to hide their use of zero (Seife, 2000). But there have also been more 30 serious consequences both for society and individuals. Political differences over the use of zero have arisen periodically in the history of mathematics (Joseph, 2008). Mathematical history is filled with power struggles such as that between Newton and Leibniz over the invention of calculus. Since the English chose to follow Newton, with his more awkward notation, they quickly fell behind the rest of Europe in mathematical prowess. The problem was caused in part because Newton held on to his work for years after its development. He feared the political-religious ramifications of work that depended so much on zeros and infinities. After his work the church of the day complained about this use of zeros and denounced it as heretical. Newton (like other mathematicians) was unsure about the zeros in his work and he tried to hide them away in his calculations, resulting in the conflict with Leibniz (Seife, 2000), which may have contributed to the awkwardness of his notation. The conflict over the origin of calculus and who to follow were not isolated incidents. Power struggles were not uncommon and resulted in one mathematician being committed to a mental institution in one case while both church and state tried to ban the use of zero in another (Seife, 2000). In many ways political, religious, and personal values shaped the development of dominant mathematics. Mathematics Arises From Practical Situations While dominant mathematics, with its Greek roots, idealizes the abstract, it is often through practical problems, rich in contextual detail, that mathematical innovation has developed. For example, Italian workmen used pumps to raise water out of canals, but despite their efforts they could not get the water over 33 feet. Pascal was later able to determine that this had to do with the pressure of the atmosphere pushing down on the 31 water; the pumps were not able to overcome this pressure. Johannes Kepler developed methods for finding the volume of three-dimensional objects in order to more accurately calculate the volume of wine barrels. Egyptian mathematicians developed ways to calculate areas of triangles and rectangles in order to place property boundary markers after previous ones had been washed away by the flooding Nile River (Seife, 2000). While Seife recognizes Egyptian contributions to mathematics (especially as the forerunner of Greek mathematics), he also denigrates it, because it was practical and did not embrace abstraction, as the Greeks would. These same Egyptians were likely among the first (and certainly before Pythagoras) to use what is now known as the Pythagorean Theorem, but because their use was practical and Pythagoras abstracted and generalized it, he is the one who gets the credit (Veljan, 2000). The idea of an average arose from the need to divide the losses of merchant ships equally among the investors (Rubenstein & Schwartz, 2000). The innovation of the vanishing point in art, is essentially a zero, and allows the representation of three-dimensional space on a two-dimensional canvas (Seife, 2000). The vanishing point is just one of many uses of mathematics in art. However, because we do not see any formal calculation in the process of creating art it is not recognized as mathematical. Conclusion Social justice approaches to mathematics education have potential to improve the educational and life opportunities for students who are perceived as nonmathematical. This is particularly true for students of color and women. However, the dominant perceptions of mathematics exclude the possibility of mathematics that addresses social justice, even though these perceptions are inconsistent with the practical and political 32 development of mathematics. These perceptions lead to mathematics teachers who are resistant to mathematics education that is clearly political and contextualized. Social justice mathematics is potentially threatening to mathematics teachers' position in society, to the structures that maintain White privilege, and to dominant perceptions of mathematics. Thus mathematics teachers who want to teach for social justice face numerous obstacles in understanding what it means to teach for social justice. For teachers who have been schooled in and grown up with a static, absolute view of mathematics, as is standard in school mathematics, this shift in perspective will be difficult and will involve unlearning some of what they already know. I struggled (and still struggle) in my own teaching to work with my mainly Latino immigrant students as they tried to balance learning what they needed to advance academically with making mathematics relevant and critical. CHAPTER 3 DISCOURSES OF WHITENESS IN MATHEMATICS TEACHER PREPARATION Discourses The concept of discourse is central to the questions that I am asking in this study. In focusing on discourse I draw first from the work of Gee (2005; 2012) and Fairclough (2001). Gee distinguishes between discourse (with a lowercase ‘d,' to refer to individual speech acts)4 and Discourse. Discourse (capital D) includes particular ways of speaking (such as, but not only, discipline specific speech), but also all the things that accompany speech that help us make meaning of the spoken word and of the speaker. These things include dress and appearance, the objects and tools that someone uses, the location or context of the interaction, and the ways in which people interact. Together these aspects of discourses allow someone to take on a particular role. In the case of a mathematics teacher then we recognize someone as a mathematics teacher not just through what she/he says but also by where the teacher is (in a classroom, in a school), the way the teacher is dressed (probably not formal, and not too casual, but somewhere in between), the objects the teacher has and uses (chalk, markers, red pen, worksheets, computer, etc.), 4 I do not maintain this distinction preferring to use the uncapitalized term since each time I use the term I use it in the capital \"D\" sense. 34 the way the teacher interacts with other people in the school (his/her own students, other students, administrators, etc.), and the ways others interact with the mathematics teacher. All of these aspects of discourse hold for teachers generally, at least in U.S. public schools. Mathematics teachers, however, have additional discourses to take up to get recognized as mathematics teachers and so to distinguish themselves from other kinds of teachers. They will be expected to use the symbols and language of mathematics. They will be expected to teach in specific ways, often ways that are not conducive to teaching for social justice. They probably will not be expected to discuss politics or bring political issues into the classroom (whereas a social studies teacher is, and other teachers might). Explicit politics are likely excluded by the discourses of mathematics education, in order to maintain a perception as apolitical. If this teacher brings social justice issues into the classroom she/he runs the risk of no longer being recognized as a mathematics teacher, potentially resulting in dismissal from the profession. It is important to note that most of what makes up discourses, like the discourses of high school mathematics teachers, is not explicitly taught. Instead someone who wishes to become a mathematics teacher must continually pick up these things through life experiences and observation, because the discourses are constantly shifting. This necessary work to be recognized as a mathematics teacher creates pressure and constraints on what a mathematics teacher can do. As Gee (2005) explains, \"The key to Discourses is ‘recognition' . . . . Whatever you have done must be similar enough to other performances to be recognizable\" (p. 27). As you interact with others and receive (implicit) feedback that they either recognize your discourse or not; you must continually make adjustments in order to maintain the successful use of 35 that discourse. Each moment of discourse use is a recreation influenced by, but never exactly the same as past moments. The need for similarity in successive reproductions of discourse use means that it is difficult to make significant changes in a discourse without your use of that discourse becoming unrecognizable (Fairclough, 2001). As a mathematics teacher if I deviate too radically from the discourses of mathematics teaching I risk being perceived as using other discourses altogether. However, if the changes are gradual enough (and include a large enough group), while engaging enough of the standard discourse then that discourse can begin to shift and change. As a consequence our ability to freely disrupt or change any discourse is constrained by what is recognizable to others-our shared understanding, which is influenced by how that discourse is historically understood. However, Gee (2005; 2012) does little to explicitly address the constraining power of discourses. Instead I turn to Fairclough (2001) to explain how power operates in and through discourse. While Fairclough does not use quite the same terms as Gee (2005; 2012) I will continue to use discourse as defined by Gee. As Fairclough (2001), drawing on Foucault, explains, the need to use discourses in a recognizable way forms a set of constraints that both restrict what is possible, but also enables action and interaction. Discourses Enable and Constrain The discourses of mathematics education both lay out the conditions of what it means to be a mathematics teacher and limit what a mathematics teacher can do and say and still be recognized as a mathematics teacher. In these ways discourses both enable and constrain (Fairclough, 2001). Discourses enable in the sense that taking up the appropriate discourses allows me to be seen and understood as a mathematics teacher. 36 However, they simultaneously constrain in that they set limits as to what speech, behavior, dress, actions, etc. are acceptable in being a mathematics teacher. This simultaneous enablement and constraint illustrate the operation of power through discourses. As we invoke various discourses we engage in a continual negotiation of power relations (Baxter, 2002). In the mathematics classroom this negotiation occurs most directly between the teacher and students, but they also include other members of the school community and society wide perspectives. These negotiations are implicit and dependent on a shared idea between a teacher and students of what \"teacher\" and \"student\" means. This shared understanding is necessary for a discourse to enable interactions. However, if a teacher tries to politicize mathematics or teach in a way that validates various, nondominant perspectives, methods, and ways of thinking he/she runs the risk of not being recognized as a mathematics teacher by the students, by their parents, by other mathematics teachers, or by administration. These other participants (parents, other mathematics teachers, and administrators, etc.) are also part of this negotiation of power even when they are not physically present. Consider this description of how discourses operate and how we use them. We are immersed in a sea of discourses through our continuous interactions with people, media, and institutions (see Figure 1). Some of them are dominant (capital \"D\") others are not dominant (lowercase d). Many of these dominant discourses are discourses of Whiteness since they maintain White privilege. The dominant discourses are connected via large block arrows (imagine varying sizes of arrows connecting discourses to the person) to show how through their ubiquity and regular, consistent repetition we 37 Figure 1 Discoures in Interaction 38 have much greater access to them. These discourses carry power with them that includes a sense of \"goodness\" (e.g., what it means to be a \"good\" mathematics teacher). The connections between us and the nondominant discourses are thinner and more tenuous. When we interact with someone we each draw on the discourses that we have access to and put them together to re-create in that moment our shared understanding of our topic of discussion (e.g., what it means to be a \"good\" mathematics teacher). To be able to discuss it meaningfully we need to have some shared discourses in order to understand one another (enabled), but this also means that we are limited (constrained) in our ability to imagine our topic of discussion in a way that differs from what is provided to us in the discourses that we know and have been exposed to. Importantly through discourses we re-create significance, activities, identities, relationships, politics, connections, and sign systems and knowledge. Gee (2005) calls these building tasks. Further when we talk and interact with others we are never engaging only one of the building tasks. Thus, as in the example in the figure above, if you and I are discussing what it means to be a \"good\" mathematics teacher, beyond the meaning that we give to this topic we communicate a particular identity as a mathematics teacher in our interactions with particular people and discourses. This identity is communicated through the discourses we draw on in that particular interaction. We also favor different sign systems or kinds of knowledge, we make connections between ideas, and connect with various political beliefs in the discourses that we use. Since our recreations are not exact reproductions of past interactions, and as we interact with enough other people and institutions, the discourses shift and change. Moments of playfulness and vigilance, especially when others will play with us, can help us see the cracks in the dominant 39 discourses and work through these cracks. Even though teachers are traditionally understood as the more powerful party in the classroom, the constraining power of discourses means that they do not operate freely. The norms of what it means to be a mathematics teacher are discursively and collectively enforced by students, other mathematics teachers, school administrators, teacher education programs, and society at large. In this way there is a constant normative pressure on mathematics teachers to take up the dominant discourses of mathematics education. In Fairclough's (2001) explanation there are at least three contexts that create this pressure. There is the immediate context (what it means to be a mathematics teacher or student in a particular classroom), an institutional context (what it means to be a mathematics teacher or student at a particular school), and a societal context (what it means to be a mathematics teacher or student in general). A mathematics teacher then needs to draw on discourses from all of these contexts in order to successfully re-create a recognizable although momentary mathematics teacher subject position. However, if there is a change in philosophy in any of these contexts the pressure can be greatly reduced or increased. For example, if a school adopts an explicitly social justice approach to education then a teacher would be supported in (or pressured to) politicize the teaching of mathematics. Dominant Discourses Dominance in discourses is one way in which power operates, accumulates and flows through discourses and becomes a means of limiting what is possible without the use of force. Dominant discourses are those discourses which exclude other ways of thinking and speaking by establishing an apparently static, universal truth. This casts 40 other discourses in the position of either not true, only partially true, or simply unimaginable. A discourse becomes dominant when it is widely accepted as common sense, and as a result, unquestionable. They can then be used to control, to some extent, the allowable content or topic of discussion; it can limit the possible ways of interacting, and it limits which roles are permissible in a given context (Fairclough, 2001). There are multiple discourses around mathematics that enable and constrain how teachers think about and practice mathematics education (Brown & McNamara, 2011). It is through these discourses that policy makers and others attempt to influence mathematics education. As an example of some of the many discourses that mathematics teacher candidates may engage with consider the following: the official, regulatory discourse of the Common Core State Standards which delineate what should be taught in various classes, the discourses in the curriculum materials based on those standards which specify what should be taught and sometimes how it should be taught, the discourses of the mentor teacher about how and what mathematics should be taught, the discourses of various mathematics teachers who have taught these teacher candidates through their K- 12 education and college, the discourses of university methodology courses, the discourses of educational foundations courses about what education means, discourses of what it means to be a teacher and a student, and discourses of equity and social justice. Each of these discourses is multiple and contradictory and accesses varying levels of power depending on the specific context. Even without direct supervision these discourses shape what happens in classrooms. Gutiérrez (2009) explains that a mathematics education research focus (which is also shaped by discourses) on the \"achievement gap\" in mathematics between 41 White students and students of color has changed the way teachers and schools understand what it means to teach mathematics and which students are most deserving of the teacher's time and attention. While there is little obvious evidence of policy changes to direct teacher's attention, teacher behavior is influenced by these discourses as they become dominant. When this research suggests that students scoring in the low-middle of an achievement test have the most potential to increase their scores, teachers face pressure to devote more time and energy to these students, potentially giving less attention to a significant portion of their class. This redirection of teacher time and energy occurs through self-regulation of behavior based out of a desire to support the school or department, out of a fear of getting caught doing or saying something outside what is perceived as part of mathematics education (dominant discourses), or even out of a belief that their efforts are what is best for the students. As the dominant discourses of mathematics education shifts so does the behavior of teachers (Gutiérrez, 2009). Thus even when no one is watching, the rules of these discourses are still followed. Discourses of Whiteness Thus far, I have only hinted at the role of discourses in gaining and maintaining systemic power. However, the pressure felt to conform to these discourses as well as the ways that the discourses place limits on what is possible to think and speak (and still be recognized as a mathematics teacher) maintain power structures. In the United States (and elsewhere) the dominant discourses (those which are represented as normal, common sense and unquestionable) are discourses of Whiteness. Whiteness Theory helps to uncover the ways in which discourses are used to maintain and promote White power and privilege. Whiteness Theory operates on the assumption that the lives of all people in 42 the U.S. in particular (but elsewhere as well) are racially structured, including the lives of White people (Frankenberg, 1993; Frye, 1992). Since historical mathematical achievement plays out on clearly racial lines in U.S. K-12 schools (Stinson, 2004), I operate from the assumption that race is a significant factor in defining mathematical success as traditionally measured within school mathematics. Racial and gendered patterns are evident in almost any measure of mathematical achievement including the numbers of PhDs in mathematics granted to women or people of color (Martin, 2009), people of color and women who complete STEM careers (Metcalf, 2010), test scores on a variety of standardized tests and college entrance exams (Lubienski, 2002), and placement into the various K-12 mathematical tracks (Stinson, 2004). Often overlooked in these narrow measures of achievement are other racial and gender impacts of school mathematics. These include the literal erasure of the contributions of mathematicians of color to the historical development of mathematics (Almeida & Joseph, 2009; Joseph, 1994) and the ways in which students generally and women and students of color particularly are pushed to conform to the dominant discourses of mathematics education (Boaler & Greeno, 2000; Gutiérrez, 2012b;). It is important to note that, while White students generally are privileged in mathematics courses, there are many White students who are also excluded by the discourses of school mathematics. Further students are not the only ones caught up in these inequities. Walshaw (2013) demonstrates how an analysis of power through discourses shifts the focus from the \"failings\" of the individual teacher to an understanding of how systemic discourses shape teacher actions, often in ways that deflect attention away from social inequities. Both teachers and students are caught up by the dominant discourses of 43 mathematics education. Despite these clear racial and gender patterns in mathematics education, most research on mathematics education has problematically only addressed race in mathematics by looking at students who are not White, usually in a way that frames students of color as inferior (Gutiérrez, 2002a; Martin, 2003; 2013); one problematic assumption in this research is that race is not a relevant factor in the education of White students. In this way the operation of the discourses of Whiteness is often invisible in mathematics education research. However, these dominant discourses exclude race talk from the discourses of mathematics education. I also suggest that the discourses of mathematics education overlap with the discourses of Whiteness in key ways which create a kind of doubling effect constraining the efforts of mathematics educators to teach for social justice. In general, Whiteness is a power that is often invisible to White people (although it may be highly visible to people of color) and it is through this power that Whites maintain privileges and rights not granted to people of color (Frankenberg, 1993). While it is often easiest to think of Whiteness (and race in general) based on genetic indications (skin color, etc.) this can also be misleading. Whiteness is a set of beliefs, cultural values, and privileges that are maintained in part through the normative discourses of society. These discourses are used to construct White ideals and perspectives as universal norm and anything else as deviant. These dominant discourses of Whiteness then become the norm by which cultures and people are measured (Fairclough, 2001: Frankenberg, 1993), while simultaneously delegitimizing or hiding alternative discourses and perspectives. These discourses allow Whites to benefit from a racist society while maintaining an 44 appearance as nonracist, fair, rational, unbiased judge, and, bound by rule and principle, as above reproach (Berlak & Moyenda, 2001; Frye, 1992; Pratt, 1984). Key to the success of these discourses in maintaining White privilege is the lack of modifiers that mark the discourses as White (as opposed to various linguistic means of marking discourses as belonging to or representing a particular group, such as Black, women, or immigrants, for example). This lack of markers presents the discourses as universal, rather than White. It is this particular claim to universality that maps onto key characteristics of school mathematics to the exclusion of social justice education thereby helping to maintain White educational privileges. To understand how this exclusion works I will use Whiteness Theory to analyze the ways in which the dominant discourses of mathematics education influences mathematics teacher candidates, particularly among mathematics teacher candidates who are otherwise committed to teaching for social justice. In light of the inequities within mathematics education Perreault's (1994) \"blind embrace\" of Whiteness is instructive. Perreault uses the term \"blind embrace\" to describe the ways in which discourses of Whiteness create a worldview that normalizes inequalities by allowing Whites to ignore history and dismiss current inequality: Much is lost in this blind embrace: both the facts of history, and the inequities of the present are erased-but the most glaring flaw is that the white speaker is once more claiming the right to define the parameters of ‘humanity,' or ‘universality.' (p. 235) But this \"blind embrace\" requires that Whites actively not see, thus rather than blindness it is a discursively enabled shutting of the eyes. Not seeing history or inequity then allows Whites to determine what is fair and claim to be unbiased. Similarly mathematics education maintains an achievement gap perspective that holds White students as the 45 norm by which all others are judged (Gutiérrez, 2009). With the achievement gap, mathematics education does not see its role in the creation of the gap. Further, dominant mathematics ignores both the historical contributions of people of color to mathematics and the potential contributions of students of color to mathematics now, contributions that are made less possible by the inequities of school mathematics. This eyes shut view, which does not see either history or inequity, is strikingly similar to Walkerdine's (1990) description of mathematics where such contextual factors (including history and inequity) are stripped from a problem in order to give it mathematical significance. As a result mathematicians are left to define the extent of their mathematical universe without regard for political concerns. As noted previously, an absolutist view of mathematics (Ernest, 1991) is currently dominant. This view of mathematics is characterized by a valorization of neutrality, abstraction, and formal symbolic language (Walkerdine, 1990). These characteristics coincide with the principal characteristics of Whiteness. Similarly Martin (2013) argues that mathematics education research is dominated by Whiteness in that it represents both White interests and White perspectives in defining what the problems and solutions worth exploring are within mathematics education. He also notes with some dismay that there have been minimal attempts to link mathematics and Whiteness in mathematics education research. Whiteness Theory provides a useful lens in this study because it provides tools and perspectives that can make visible the ways in which White privilege is maintained by the discourses of mathematics and the racialized effects of mathematics teaching and learning. These influences may have a norming effect on mathematics teachers, which serve to uphold the ways in which the education system privileges particular students. 46 Further, Whiteness Theory is essential to understand the push away from social justice and the pull towards neutrality and apparent apoliticality that seem to be valued so highly in both mathematics and Whiteness. Whiteness Theory has potential to explain at least some of the reasons that mathematics teachers are particularly resistant to social justice efforts in the classroom. The discourses of Whiteness maintain White privilege with the façade that White privilege does not exist (Yoon, 2012). In a traditional mathematics classroom one of the ways in which teacher power is asserted is by evaluating the correctness of student responses. Thus when school mathematics is constructed to emphasize unambiguously \"correct\" answers (as opposed to open questions where \"correctness\" is relative) teachers are set up to evaluate these answers in a manner which increases their classroom power. While teacher discourses are generally used to reinforce the power of the teacher, for the mathematics teacher this effect is doubled since the discourses of mathematics education reinforce this power, in addition to the discourses of Whiteness that position Whites as judges. Thus the mathematics teacher is positioned as the ultimate judge. Since mathematics is decontextualized and abstract there appears to be little room for a biased judgment. There is a ready-made position for the teacher to judge students as either correct or incorrect and in a greater sense as mathematically capable or not. Similarly Kidder (1997), working with expatriate Whites in India, found that many fall into these ready-made positions of privilege and power. Many of these Whites came from a middle-class background in the United States and Europe, but when relocating to India, found the expected role for Whites includes the explicit exercise of power over their Indian hired servants. 47 The privileges these expatriate Whites enjoyed were both embraced and denied. Kidder (1997) found that, as these White women met, their discussions often revolved around the precariousness of their privileges and comparing their (relative) lack of privilege by pointing to other more privileged Whites. The privilege of these White families in India suggests racism (and colonialism and classism); thus in order to maintain an image as a \"good\" (i.e., not racist) White they had to find ways to justify and/or deny their privilege. This more current example is similar to the historical example presented by Anderson (1994) of the necessity of justifying slavery in light of U.S. ideals of equality. Justification occurs in these cases by positioning African Americans as inherently inferior, in the historic case, and by positioning Whites as not racist, for the expatriates. Both parts are key aspects of the creation and maintenance of racism. This particular kind of White goodness, with its accompanying justifications or denials of privilege, are particularly important to progressive Whites. The specifics of this kind of White goodness will change and shift both over time and according to the specific context as (White) social values change. However, in general, goodness has consisted of some combination of \"earned\" wealth (see Lipsitz, 2006) and moral authority/superiority. This White moral authority is created through the portrayal of Whites as unbiased and not racist; this position is especially attractive for progressive Whites. In order to maintain this portrayal Whites must continually prove their \"goodness\" even if only to themselves. As Whites one way we can prove our \"goodness\" is by creating a hierarchy of racism or bias. As long as we can point to someone else as racist we can say that we are not racist like \"that person.\" The effect is that we no longer feel the need to listen to those voices that call into question our 48 nonracism, even when that voice is our own. Through this arrogant maintenance of our goodness we reinscribe Whiteness. To be clear I am not suggesting that Whites should not try to be good. Rather that there is a particular way of being good that is more about proving our goodness than it is about making a real difference in terms of racism and inequity. Further this kind of goodness extends to antiracist academics who judge the attempts of younger/less-experienced scholars as naïve, which is a more sophisticated (implicit) way of saying \"I am not racist like that person.\" (Thompson, 2003). Again this is not to suggest that we should not be critical of misguided antiracism, but that we not use criticality as a means to perform our own goodness. I call this kind of goodness normative goodness, because of the ways that it pushes people to adopt dominant norms, sometimes at the expense of other values. While I am drawing the term from Whiteness theorists I want to separate normative goodness from a White racial identity, which could imply that people of color do not also face normative pressures. Instead I argue that both White teacher candidates and teacher candidates of color are pushed by normative goodness, especially as they come in contact with the dominant discourses of teacher preparation that present an idealized \"good\" mathematics teacher. This then becomes the standard by which teacher candidates (and their professors, supervisors, and mentor teachers) may measure their efforts to become teachers. This normative goodness helps to maintain the authority of White people, especially insofar as we are able to portray ourselves as fair and unbiased. Teaching for social justice is at odds with this role of authority; it introduces contexts which the teacher may know less about than the students and which require both teacher and student 49 to make evaluations of various situations. This is a potential threat to the authority of the teacher since the teacher is required to recognize and validate the knowledge of the students. Because the role of teacher, especially that of mathematics teacher, is already molded with a discourse of authority (Gutiérrez, 2009), progressive teachers may find themselves taking on that authoritative role at times against their better judgment and desires. The teacher may use that authority in ways that promote White privilege and power by reenacting dominant discourses in the classroom. However, the teacher may use that authority instead to work alongside students to disrupt dominant discourses and for the promotion of socially just practices. Alignment of Whiteness and School Mathematics The discourses of Whiteness and school mathematics align in certain key areas. Currently the maintenance of White privilege requires the denial of the existence of racism and racial inequality in order to uphold the idea of normative goodness. This is achieved in part through the perpetuation of the following myths: meritocracy, neutrality, and colorblindness. There are, of course, many other ways in which White privilege is maintained. However, I highlight these three because of their important role and, especially for their potential for alignment with the discourses of school mathematics. While I will discuss each myth separately they are intricately interrelated and it is difficult to talk about one without the others. For example, meritocracy has to assume a system of judgment (to determine who has merit and who does not) in which color is not a factor since that would open up the potential for bias and destroy neutrality. The myth of meritocracy goes something like this: anyone regardless of race, class, gender, etc. through hard work and good choices can achieve success at X (school, 50 work, mathematics, etc.). This myth perpetuates the normalization of race, class, and gender inequality by blaming the victim for their lack of success and exonerating the mostly White men who have benefitted from the (supposed) meritocracy. In a mathematics class this comes into play because of the decontextualization that is common in school mathematics. Since mathematics problems are abstract and stripped of contexts mathematics teachers may assume they give accurate measures (neutral judgments) of students' ability without regard for race, class, or gender. Rousseau and Tate (2003) found that mathematics teachers used mathematics to rationalize and explain away the racial patterns of failure in their mathematics classes, using discourses of hard work (or lack of it) to explain students' failure. In other words meritocracy suggests that these students failed because they lacked the necessary merits and not because of other potential factors such as poor teaching, a disengaging curriculum, or systemic racism in schools, among other factors. This reasoning, built on the perceived neutrality of school mathematics, normalizes the failure of these students by blaming them and deflects critique away from the teacher, school system, or society. The myth (or idealization) of neutrality maintains that neutrality is both possible and desirable. Everyone, but especially authority figures, must maintain a neutral stance. If the authority figure's bias is revealed then he/she is labeled (racist, sexist, etc.) and removed from her/his position in order to maintain the appearance of neutrality, or rules are created to ensure that such bias does not affect decisions. If the status quo of a situation is challenged then the neutrality of the authority can be displayed to maintain the appearance of fairness that serves White privilege. While other fields have gatekeeping mechanisms, the perceived neutrality of school mathematics makes it an 51 ideal candidate as a gatekeeper for a number of high status disciplines as well as college entrance generally. Mathematics' position as gatekeeper places mathematics teachers in this position of authority to deny or admit students. To maintain the White ideal of neutrality then mathematics teachers must appear neutral in their decisions. The common view of school mathematics as neutral enhances this ability, while teaching mathematics for social justice is a potential threat to this appearance of neutrality. Thus some of the status of mathematics teachers depends on perpetuating the idea of mathematics as neutral. The neutrality of mathematics teachers gives credence to the tests, grades, and other markers of success that determine student advancement in mathematics classes or in college. The myth of colorblindness is the idea that it is both possible and desirable to not see the race of another person (Thompson, 1998). By actively not noticing color and then reporting to others how they or someone else didn't notice color Whites portray themselves as neutral, unbiased, and nonracist (i.e., \"good\"). This colorblind narrative then works to deny the experiences of people of color by denying the distinctness of their experience. From this perspective meaningful conversations about race are impossible to have since they would require noticing and talking about color. When race comes up in conversation it is sidelined or silenced and so race, and the inequalities connected to race, cannot be studied without violating the principle of colorblindness (Thompson, 1998). The perception of mathematics as universal and acultural makes it an ideal context for a colorblind perspective to thrive. Unlike other subject areas there is not recognizable Black mathematics, or Chicano mathematics, etc. in the same way that there is Black literature, Chicano history, or Indigenous art. However, the perception that mathematics 52 is acultural discourages teaching for social justice since to do so would violate colorblindness in general and the perception of mathematics as universal. Within the education system mathematics teachers enjoy a certain status connected to mathematics. Generally mathematics teachers have an easier time finding teaching positions, have more choice in which teaching position they accept, and are more likely to have received funding for their degrees or other forms of financial incentives in becoming teachers. These privileges are directly tied to the discourses of mathematics and the perception of greater intelligence that accompanies mathematics. These privileges in addition to the myths explained above provide incentive to mathematics teachers to maintain the discourses of school mathematics, even if it comes at the expense of socially just teaching. However, this also requires progressive mathematics teachers to justify their failure to teach for social justice. While there are certainly many discourses that justify not teaching for social justice the following are those that I have found myself using or have heard other teachers use. In a mathematics context these could play out as teachers deny their own privilege, possibly pointing to the relatively low status of teachers. They may express an inability to act on social justice desires out of fear of offending students, parents, or others. Teachers may point to student misbehavior or ways in which students are responsible for their own failures or struggles. Finally, as noted by de Freitas (2008), mathematics teachers may disengage by suggesting that mathematics is just mathematics, without leaving room for human context and injustice. These match what Perreault (1994) found as common responses to a challenge of White privilege including denial of privilege, paralysis of fear or guilt, anger that is misdirected at the victim of privilege, and disengagement from discussion of 53 privilege. Kidder (1997) found similar denials of privilege among White expatriates in India. Disrupting/Playing With Dominant Discourses Part of learning to teach mathematics for social justice is learning to disrupt the dominant discourses, including normative goodness. Applebaum (2010) suggests that one way to work against this kind of normative goodness is what she terms vigilance. For Applebaum vigilance includes uncertainty, humility, and Foucauldian critique. Uncertainty includes a willingness to question our own certainty and what we think we know, especially about racism and the experiences of the other. Implicit in this is a requirement to not make final judgments (which suggests certainty). Humility means that we are \"open to examining how our progressiveness might be oppressive in ways that we are not aware of\" (p. 186). For our work in teaching mathematics for social justice this means that we critically examine the work that we do and see how even our promotion of social justice may reinscribe dominant discourses. Key to this kind of critique is to question our own frameworks of knowing, especially those things we think we know for certain; to ask ourselves what our assumptions make impossible for us to know or question. This cannot be done individually as it requires listening carefully to those who question what we think we know and who ask questions we would never think to ask. Secondly, I borrow from Lugones (1987) the concept of playfulness as a potential way to disrupt the dominant discourses of mathematics education and Whiteness. While Lugones is specifically not writing from a poststructural perspective, I adapt this concept to fit within a poststructural frame. Lugones describes a kind of playfulness that creates opportunities for us to reform or re-create our worlds as well as to reform and re-create 54 our multiple selves within those worlds. While some worlds inhibit our playfulness, those where we can exist playfully allow us to create opportunities to change. The playfulness that Lugones describes can occur in any activity if we bring a playful attitude into the situation. Before defining playfulness I need to first describe how Lugones (1987) understands \"worlds\" and \"world-traveling.\" She draws these concepts from the experiences of people of color. Lugones explains that each of us inhabit multiple different worlds. For those who are powerful, navigating these worlds is relatively simple. For those who are not powerful, though, navigating these worlds can mean adopting/inhabiting different personas, some of which are restrictive or damaging to a sense of self. Lugones noted that in some of these worlds she could be playful and in others she could not. For her playfulness is uninhibited, unconcerned about rules, not taking yourself too seriously, a willingness to be a fool (or just to be wrong), and a willingness to be surprised. There are worlds/situations where we find it easier to be playful and others where we become rigid or defensive (not playful). For those of us in power we have to learn to be playful when traveling to other worlds, rather than imposing ourselves, which is manifest in many ways including the defensiveness that can arise in cross-race relations. While Lugones' (1987) work implies some kind of preexistent core (even if multiple) self, which is inconsistent with the discursive, poststructural approach I am taking here, the concept of playfulness may still be useful. However, if we think of the \"worlds\" Lugones describes as various discourses, some of which promote playfulness while others do not, then playfulness can become a means of disrupting the norms of that 55 discourse. It is in playful moments that we can ask questions about what we think we know that might otherwise be absurd or even unthinkable. Playfulness can reveal the fractures and fissures of dominant discourses. Additionally, Lugones (1987) describes playfulness as a primarily individual trait (even though the example she gives involves another playful person). However, I find it useful to think of play as a social activity. While it may be necessary for an individual to take the first playful step, others help to set the conditions that make that possible. Further, in order to create a disruptive moment others have to respond with playfulness, otherwise the dominant discourses are simply reasserted. We cannot play alone, and disrupt a discourse. Playfulness as described above is in many ways antiWhiteness, especially in moments where Whiteness could be challenged, and especially for progressive Whites. It is helpful in this description to think of the discourses of Whiteness as creating a set of rules or norms of what it means to be a \"good,\" progressive White. Imagine, for a moment, a progressive White man in a classroom with other Whites and students of color. A question is brought up that suggests the possibility of racism (even if race is never explicitly mentioned), on his part. How does he respond? Can he possibly be playful in this moment? The chances are slim. For most White men in this situation who they think they are, as \"a good White,\" has been challenged. He will likely now take himself very seriously as well as display increased concern about the (unwritten, unspoken) rules of cross-racial interaction. He is unwilling to be made a fool in terms of race relations (although he may be perfectly capable of this on other terms) and as a result unable to embrace the uncertainty, humility, and self-criticality necessary to listen to the other. 56 This is not to suggest that he should not take the implication of racism seriously, but instead that he should not take \"himself\" and his investments in being a \"good White\" too seriously. The example that Lugones (1987) uses to illustrate her conception of playfulness suggests a kind of joyful play. However, I suggest that there are other types of play that maintain the core ideas of her conception of playfulness. The key aspects of playfulness can also be present in a kind of serious play that may be more appropriate for the progressive White man in my scenario above. This playfulness can allow us to shift around, change and challenge the discourses of Whiteness and of school mathematics, even if only temporarily. For this to have lasting impact this play must be collective; we must both make and accept invitations to play. Social Justice Mathematics As a researcher and educator one of my goals in disrupting these dominant discourses is to create discursive space for imagining and implementing more socially just mathematics teaching practices with teacher candidates. To work against, play with, and decenter Whiteness in mathematics education requires a more robust and nuanced understanding of what it means to teach for social justice than the typical access (and sometimes achievement) focus advocated by the National Council of Teachers of Mathematics (NCTM) and other mainstream perspectives. To define what I mean by social justice mathematics I turn first to conceptions of social justice in education broadly and then more specifically within mathematics education. Multiple authors note the lack of clear, agreed upon definitions of social justice in education, even though the term is now widely used (Grant & Agosto, 2008; Hackman, 2005; Hytten & Bettez, 2011; North, 2008). In an attempt to deepen the theoretical 57 foundation of social justice in education these same authors have reviewed the existing uses of social justice in education. Cochran-Smith, Shakman, Jong, Terrell, Barnatt, and McQuillan (2009) in making their case for the inclusion of social justice in teacher preparation state, \"the bottom line of teaching is enhancing students' learning and their life chances by challenging the inequities of school and society\" (p. 350). This definition, is useful because of its simplicity and the potential for bridging various perspectives on education. However, its necessary breadth limits its practical applicability. I turn to North's (2008) review which lays out three broad tensions to grapple with in understanding social justice in education. These tensions are between redistribution and recognition, macro- and microlevel issues, and knowledge and action. In addressing these tensions North (2008) suggests that social justice must balance the potentially competing priorities of the redistribution of material resources and the need to recognize and embrace differences between groups. This tension should also recognize the fluid nature of group membership. Social justice will also need to address both macrolevel and microlevel (as well as in-between levels) of social injustice. Macro-level issues include broad national and global inequities and the policies that promote these inequities, while microlevel include the day to day happenings of a particular classroom and the needs of the students and teacher (both in and out of school) who come together in that class. The final tension suggests a need to balance learning about inequity (knowledge) and taking action. This tension recognizes the potential for damage if action is taken without adequate understanding of a situation as well as the fruitlessness of endless theorizing. To address social justice requires that we (and our students) engage in both learning and acting. In closing her article North (2008) suggests the need to continue 58 questioning and evaluating what social justice in education means, in order to avoid replacing one form of dominance with another. In this study, focused on the work of one small class of teacher candidates, our work tended to emphasize recognition, microlevel issues, and knowledge. This focus was in part due to the context of our work together as well as my perception of what was needed and possible for this group of teacher candidates. However, our discussions also addressed the redistribution of resources (including quality teaching), macrolevel issues (including discourses and standardized testing), and action (including what the teacher candidates could do in the process of becoming teachers) Within mathematics education the theorizing of social justice has drawn heavily from Freirian critical pedagogy (Gustein, 2006). From this perspective social justice mathematics is understood as having mainly to do with curricular changes that involve the use of mathematics for social critique (Gutiérrez, 2015). This view is perhaps the most well developed within mathematics education and has led to increasing interest in social justice among mathematics educators. However, this view, which tends to dominate teachers' understandings of what social justice mathematics is, has, perhaps, contributed to the limited adoption of social justice mathematical practices among teachers. Additionally, it appears to give priority to certain aspects of (rather than maintaining balance between) the various tensions of social justice in education (North, 2008). In order to expand this understanding of social justice mathematics I turn to two concepts from Gutiérrez. These include Gutiérrez's (2012c) four dimensions of equity and Gutiérrez's (2009) description of an equity stance. Together these concepts encourage a broad and multiple understanding of social justice mathematics and are more 59 in-line with the tensions suggested by North (2008) First, Gutiérrez (2012c) describes four dimensions of equity. These are access, achievement, identity, and power. She divides these into a dominant axis (access/achievement) and a critical axis (identity/power). Access deals with the resources that students have available to them, including technology and high-quality instruction. Achievement is measured in the traditional sense of grades and test scores. Both access and achievement generally leave the mathematics content relatively untouched. School mathematics is still the focus of instruction. However, this axis is necessary in order to provide students with the material resources and social capital to advance in the school system and to have the potential to impact the field of mathematics. Attending to identity means providing opportunities for students to draw on their own linguistic and cultural resources, becoming better by their own standards, and coming to understand themselves and their world in relation to mathematics. Addressing power includes addressing whose voice matters in the mathematics classroom (authority is part of this), using mathematics for social critique, questioning the nature of mathematics and mathematical ways of knowing, and making mathematics more humanistic (Gutiérrez, 2012c). For a mathematical analogy this might be thought of as a coordinate plane5 with each axis forming one of the axes of the coordinate plane. To deepen this analogy a complex coordinate plane6 may be more useful with the dominant 5 This idea was first proposed when I gave a presentation to some colleagues. One of the mathematicians in the group asked whether the axes could be thought of as a coordinate plane. My initial response was that they could not, thinking that presenting the axes in this way implied a trade-off, for example, between access and achievement, but after additional thought, and if not carried too far, I believe it is a useful analogy. 6 For those unfamiliar with the complex coordinate plane it functions similarly to the 60 representing the real axis and the critical representing the imaginary7 axis. Just as imaginary numbers are invisible when operating within the realm of real numbers, the critical axis is hidden from a mainstream perspective of mathematics education. Each instance of teaching can be thought of as a set of coordinates attending more or less to each of the four dimensions of equity, but over some lengthier period of time (a unit, a month, a semester) the teaching could attend to all four dimensions of equity. This framework is useful, in part, because it creates a wider view of what it means to teach mathematics for social justice. The image of the teacher regularly engaging students in the in-depth social critique lessons similar to the work of Gutstein (2006; 2009; 2012) is excellent, but it has come to be understood as the one and only way to teach social justice mathematics. This then becomes simultaneously too simple and too difficult. It is too simple in the sense that it leaves out other dimensions of equity as Gutiérrez's framework (2012c) makes plain and can risk perpetuating the idea of the (White) teacher as \"lone hero\" (Thompson, 2003). It is too difficult in the sense that teachers, especially young teachers, can feel inadequate if their ideas or lessons do not measure up to the examples given by Gutstein (2006). Gutstein does take up many of these issues. However, it seems that the best known aspects of his work are the excellent social critique lessons that he creates and teaches. Gutiérrez's conception of equity is further enhanced by her suggestion that teachers develop an \"equity stance\" (2009). An equity stance is an off-balance stance that standard coordinate plane with the difference that one of the axes represents imaginary numbers instead of real numbers. 7 The terms \"real\" and \"imaginary\" are mathematical terms that do not correspond to the ideas of existence, concreteness, or fantasy as they do outside of mathematics. 61 requires teachers to try to hold onto both sides of several binaries simultaneously. Maintaining this equity stance can help teachers avoid falling into the continuum of normative goodness. These binaries (explained more fully in Chapter 2) include knowing/not knowing your students, being in charge/not being in charge, and teaching mathematics/not teaching mathematics. The first part of each of these binaries (knowing your students, being in charge, and teaching mathematics) is a standard, common-sense concept of mathematics teacher preparation. The standard implication is that either you know your students or you do not, you are in charge or you are not, and that your responsibility is only to teach mathematics. However, as Gutiérrez (2009) pairs each one with its opposite the goal is to do each simultaneously. In stating and presenting these binaries Gutiérrez (2009) makes the limits of standard mathematics teacher preparation discourse more clear. Embracing this equity stance requires that mathematics teachers accept the uncertainty, humility and critique of vigilance (Applebaum, 2010) Conclusion Poststructural research has not commonly been applied to mathematics education research (Stinson & Bullock, 2012). However, poststructural understandings of the circulation of power through discourse can inform our understanding of the ways in which the discourses of mathematics education work to exclude social justice from mathematics education. Further, Whiteness Theory opens the potential to understand the racial dimensions of these discourses and the ways in which mathematics education serves to maintain White privilege in part through the progressive White desire for \"goodness\" that can exclude or minimize social justice. One potential means of working against this goodness is the idea of \"playfulness\" which may disrupt some of the rules of 62 White goodness. Finally, a more in depth and robust understanding of social justice that encourages vigilance in critiquing our own practice is necessary to trouble the dominant discourses of mathematics education. CHAPTER 4 METHODS: CRITICAL POSTSTRUCTURAL DISCOURSE ANALYSIS The site for the current study is an action research course I just finished teaching to a group of seven preservice mathematics teachers. These seven teachers took my course concurrently with their second semester of student-teaching, which provided the sites for their action research projects. The course met once weekly in the evenings. The first few course meetings focused on developing a shared understanding of what it means to be a mathematics teacher, the role of discourse in shaping our thoughts and actions, and on the role of social justice in mathematics classes. These classes were meant to challenge traditional thinking about teaching and to generate reflective thinking about the role of a teacher and the role of mathematics in our education system. These classes also prepared a foundation of social justice from which my students could select and investigate topics that address social justice in some way. Following these beginning classes (4 classes total), our focus shifted to explore what action research is, how it differs from other research traditions, and to the development of the preservice teachers' own action research projects. The selection of topics and development of action plans was a collaborative process through which we all offered feedback on research topics and methods (3 classes total). The remainder of the semester was spent with the students reporting on their progress on and adjustments to their projects and revisiting 64 mathematics teaching and social justice. Participants My interest in educational research grows out of my own experiences as a White high school mathematics teacher. I grew up immersed in and accepting of the dominant discourses that are all around us, especially discourses of Whiteness. I did not consider myself privileged and believed many of the myths of Whiteness, including meritocracy. I thought of racism as a problem of the past and had minimal interactions with people of color. Learning to speak Spanish and living for 2 years in southern Chile began to open my thinking to different perspectives on the world, racism, and discrimination. When I returned to the United States and reenrolled in college it was now with a goal of becoming a teacher. On completing a bachelor's degree in mathematics education and Spanish teaching as well as a master's degree in teaching English as a second language, I took my first teaching job at a public rural high school in Colorado. The high school had a student population of about 750 with 50% Latino students, mostly from Mexico and Central America. The school had a strong ESL program and was one of the few rural schools I found with a functioning sheltered mathematics program. As a 1st-year teacher I taught all of the sheltered mathematics classes and continued teaching every sheltered mathematics class that was offered during my 4 years. Overall my teaching experience was very positive and the relationships that I developed with my students continue to evolve. However, I was also aware that my teaching was not what I wanted it to be. I identified then (and now) first as a teacher and secondly as a 65 mathematics8 guy. What I found frustrating was the difficulty in teaching in a way that was so different from how I had been taught and in which I had little outside support. In addition, I felt that I could not see beyond the abstract, dominant mathematics in order to understand how to make the connections to students' lives that I felt were necessary. In this high school I witnessed first-hand the roles of race and class in the lives of students in our education system. There was a superficial harmony at the school between the wealthy White students and the working-class Latino students. However, there were clear divisions on race and class lines that determined which entrance to the school students used, what classes they took, which sports they participated in and supported, what cars they drove (or didn't drive), even where they parked their cars,9 and where and if they went to college. Mathematics classes were one of the key ways to maintain these divisions. Once these divisions were made (mostly in middle school) they were set. A student who began high school taking Algebra 1 would not make it to AP Calculus as a senior. As a new teacher I felt the pressure of maintaining my position as a teacher, and so supporting the school policies; I also wanted to better serve my students who were not being served by those same school policies. I wanted to teach mathematics in innovative and at times critical ways, but I felt the need to conform to traditional views of teaching mathematics. 8 I hesitate to use the term \"mathematician\" as I don't see myself that way. I have always been very good at school mathematics, but in my mind that is very different from being a mathematician. 9 During my final year of teaching the school implemented a policy, over my weak protest, in which all parking stalls in the student lot were assigned to students, but to get a spot students had to present a driver's license, proof of insurance, and pay a fee. Since many of my students either did not have a license or could not afford the fee they had to park in the mud parking lot of the park across the street. School administrators did not see the policy as discriminatory. 66 Perhaps more importantly I saw that the school was not meeting the needs of my students in a number of ways. There was a stark contrast between the privileged educational experiences that I had and those of my students. The combination of these factors led me to pursue a PhD in education, in order to better understand the education of Latino students in the United States and to improve the teaching of mathematics. Mathematics Teacher Candidates The participants in this study were all students enrolled at our university in a program that will lead to a Master's degree in mathematics with teaching certification. These students are funded by Mathematics for America, which seeks to improve the mathematics education of U.S. secondary schools. As part of this process Mathematics for America recruits students with Bachelor's degrees in mathematics fields, funds their further education, and pairs them with accomplished mathematics teachers for their student teaching (Mathematics for America, 2013). This is a nontraditional program that leads to teacher certification by following a condensed version of the standard education curriculum that most teacher candidates experience. During the Fall 2013 academic semester I supervised these teacher candidates during their early months of student teaching. I also attended monthly MfA meetings and an MfA re"}]
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