Problem Solving for Today's Social Context Holly K. Brewster Submitted in Partial Fulfillment O

The Teacher as Mathematician: Problem Solving for Today’s Social Context

Holly K. Brewster

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY 2014

ABSTRACT

The Teacher as Mathematician: Problem Solving for Today’s Social Context

Holly K. Brewster

A current trend in social justice oriented education research is the promotion of certain intellectual virtues that support epistemic responsibility, or differently put, the dispositions necessary to be a good knower. On the surface, the proposition of epistemically responsible teaching, or teaching students to be responsible knowers is innocuous, even banal. In the mathematics classroom, however, it is patently at odds with current practice and with the stated goals of mathematics education.

This dissertation begins by detailing the extant paradigm in mathematics education, which characterizes mathematics as a body of skills to be mastered, and which rewards ways of thinking that are highly procedural and mechanistic. It then argues, relying on a wide range of educational thinkers including John Dewey, Maxine Greene, Miranda Fricker, and a collection of scholars of white privilege, that an important element in social justice education is the eradication of such process-oriented thinking, and the promotion of such intellectual virtues as courage and humility. Because the dominant paradigm is supported by an ideology and mythology of mathematics, however, changing that paradigm necessitates engaging with the underlying conceptions of mathematics that support it. The dissertation turns to naturalist philosophers of education make clear that the nature of mathematics practice and the growth of mathematical knowledge are not characterized by mechanistic and procedural thinking at all. In these accounts, we can see that good

mathematical thinking relies on many of the same habits and dispositions that the social justice educators recommend.

In articulating an isomorphism between good mathematical thinking and socially responsive thinking, the dissertation aims to offer a framework for thinking about mathematics education in and for a democratic society. It aims to cast the goals of mathematically rigorous education and socially responsible teaching not only as not in conflict, but also overlapping in meaningful ways.

Table of Contents

List of Figures ...... v

Acknowledgments ...... vi

Chapter 1: Introduction ...... 1

1.1 The question ...... 1

1.2 Objectives ...... 6

1.3 Existing Lines of Inquiry: Mathematics ...... 7

1.4 Existing lines of inquiry: Education ...... 9

1.4.1 Pedagogy of the Oppressed ...... 12

1.4.2 Critical Mathematics Education ...... 14

1.4.3 Critical mathematics education on the ground ...... 16

1.5 Methodology ...... 19

1.6 Theoretical Framework ...... 19

1.6.1 Focus on the Privileged ...... 19

1.6.2 Deweyan Democratic Education ...... 24

1.7 Overview & Chapter Summaries ...... 26

1.8 On Radicality, Positionality, and the Role of the Philosopher ...... 27

Chapter 2: Problem Solving Skills Paradigm ...... 30

2.1 Distinction among mathematicians ...... 31

2.2 Problem Solving in Mathematics Education: Historical Context ...... 33

2.3 Problem Solving Skills Paradigm ...... 37

2.3.1 Problem solving skills in action: Khan Academy ...... 39

2.3.2 Further discussion ...... 44

2.4 Animating Ideology ...... 45

2.4.0 There are no myths in mathematics ...... 46

2.4.1 Mathematics is universal, unified, and transcendental ...... 46

2.4.2 Mathematics is objective and impersonal ...... 51

2.4.3 Mathematics is provable and certain ...... 54

2.4.4 Mathematics is a solitary achievement ...... 55

2.5 Counter Currents ...... 56

2.5 Generalizing Mathematical Thinking ...... 58

Chapter 3: Epistemic Aspects of Social Justice ...... 63

3.1 Introduction ...... 63

3.2 Thinking for Freedom ...... 65

3.3 Focus on the powerful ...... 71

3.3.0 A Note on Terminology ...... 72

3.3.1 Unconscious prejudice: Institutional Knowledge ...... 72

3.3.2 Unconscious Prejudice: Individual Level (Credibility) ...... 76

3.3.3 Objectification and Direction of Fit ...... 78

3.3.4 A Second Approach: Ways of being and phenomenology of privilege...... 81

3.4 Conclusion ...... 89

Chapter 4: Intellectual Courage and Humility ...... 93

4.1 Introduction ...... 93

4.2 Uncertainty, Open-mindedness, and Humility ...... 96

4.2.1 Unertainty and Open-mindedness ...... 96

4.2.2 Humility ...... 97

4.3 Courage and Surprise ...... 99

4.4 Virtues vs. Skills ...... 104

4.4.1 Judgment and the Will ...... 104

4.4.2 Learning virtues ...... 107

Chapter 5. Recasting Mathematics ...... 110

5.1 Fermat’s Last Theorem ...... 113

5.2 Standard Explanation ...... 114

5.2.1 Concerns about the Standard Explanation ...... 115

5.2.2 Philosophical Context ...... 115

5.2.3 Godel’s Theorem ...... 117

5.3 Newer philosophies of mathematics ...... 119

5.3.1 Open Questions and Mathematics Problems ...... 120

5.3.2 Discourse ...... 126

5.3.3 The Mathematical Community ...... 132

5.3.4 Mathematical Objectivity and Certainty ...... 135

5.4 Complicating the Ideology of Mathematics ...... 140

Chapter 6: Paradigm and Pedagogy ...... 143

6.1 Radicality: Reprise ...... 146

6.2 Four Classrooms ...... 146

6.2.1 Lockhart ...... 147

5.2.2 Lampert ...... 151

5.2.3 Parrish ...... 153

6.2.4 Brewster ...... 155

6.3 Criteria ...... 156

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6.3.1 The end in sight ...... 156

6.3.2 Growth is not additive ...... 157

6.3.2 Wrong answers and courage ...... 159

6.3.3 Authority and Humility ...... 160

6.4 Teacher beliefs ...... 161

6.5 Changing the paradigm: Teacher as Mathematician ...... 164

6.5.1 Math Circles ...... 164

6.5.2 The Teacher as Mathematician ...... 165

References ...... 169

List of Figures

Figure 1: Standards for Mathematical Practice (National Governors Association Center for

Best Practices & The Council of Chief State School Officers, 2010) ...... 36

Figure 2: Commitments of the Problem Solving Skills Paradigm ...... 38

Figure 3: Unit Cost ...... 47

Figure 4: Sum of odds ...... 49

Figure 5: Volume of a Cylinder ...... 49

Figure 6: Volume of a Cylinder Solution ...... 50

Figure 7: Purity (Munroe, 2008) ...... 52

Figure 8: 15 Cupcakes ...... 57

Figure 9: Intersecting Parabolae ...... 58

Figure 10: Proof of Base Angles Theorem ...... 120

Figure 11: Three classic constructions ...... 124

Figure 12: Natural numbers as groups ...... 148

Figure 13: Combining evens and odds ...... 148

Figure 14: Sequence of odds ...... 150

Figure 15: Making Groups (Lampert, 2003, p. 143) ...... 151

Figure 16: Richard's Response (Lampert, 2003, p. 147) ...... 152

Figure 17: Parrish 2010, p 238 ...... 153

Figure 18: 12 x 3 ...... 154

Figure 19: 8 x 25 ...... 154

Acknowledgments

Officially, this dissertation is the culmination of six years spent with the Philosophy and Education program at Teachers College, Columbia University, during which time I have had the good fortune to learn from more wonderful people than I could have imagined, and for whose influence I am deeply grateful. Practically, however, it brings together elements from the past eleven years of my life, and has been an opportunity to thoughtfully work through questions that have been with me since my first day as a mathematics teacher. The students, colleagues, and teachers who during this time have allowed me the space to grow and experiment as a scholar and teacher are too numerous to mention, but I am mindful that without that support, this project would not have been possible.

I thank the members of my dissertation committee, beginning with Megan Laverty, whose support, detailed attention and close reading of my work at every stage was crucial in pulling this dissertation together. Her willingness to wade through endless messy and ill-formed drafts was vital to the evolution of the argument, and the elevation of so many discrete parts into a cohesive whole. David Hansen’s role in the initial shaping of the project was invaluable. Though superficially I landed quite far from where I began, his guidance in the proposal stage served me well through all of the incarnations. I hope he can hear in my words the depth of influence he has had on my thinking about scholarship, philosophy, and schooling. Kurt Stemhagen has been exceptionally helpful in the final stages of this project. I am grateful for his thoughtful responses to my work, both in formal and informal conversation. I look forward to building our relationship as I move into the next stage of my career.

An earlier and abbreviated version of chapter 2 was presented at the Philosophy of

Education Society’s 2014 Annual Conference, under the title “Problem Solving as

Theorizing: A New Model for School Mathematics.” The feedback I received on the piece throughout the process, from the anonymous reviews, the response paper delivered by

Kurt Stemhagen, and the thoughtful questions raised by those who attended the session were helpful in bringing focus to the chapter, and I am thankful to the Society for the invitation to present, and to the commenters for their challenges and advice. I hope that your concerns have been addressed in full in the final document.

I very much appreciate the time taken by friends and family to proofread and edit sections of the dissertation; Tim Ignaffo, Marie Tracy, and Bethany Record all contributed to finalizing the project at a time when I could barely stand to look at it any more. Thank you for your support. I would also like to recognize my colleagues in the Philosophy of

Education program—Brian Veprek, Tim Ignaffo, Michael Schapira, Cara Furman, and Matt

Hayden, without whose company and conversation I most certainly would have lost hope, direction, or both. I am especially indebted to Matt, for his timely responses to my unending questions about the minutia of navigating the dissertation process. I am fortunate to such good and thoughtful people as my friends.

Finally, I am glad to have the chance to thank my girlfriend, Veronica Golden. She has supported me, cooked for me, coached me, forgiven me, and endured countless ups and downs, while navigating a graduate degree of her own. Thank you, Veronica; for everything.

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Chapter 1: Introduction

1.1 The question

Mathematics on its surface makes no reference to human or earthly concerns, which gives the impression that it is neutral in matters of social and cultural importance. It seems to even have an admirable multicultural quality about it that makes it accessible to everyone, because it does not favor any culture over another. Physicist Richard Feynman writes that mathematics and the hard sciences are the most egalitarian fields in academia because they are not imbued with the “intangibles” that permeate the humanities. These fields are more purely meritocratic, he writes, because students from disadvantaged backgrounds, like himself, can be admitted and supported based purely on achievement.

(Feynman & Leighton, 1997). Academic and professional mathematics culture reveals a persistent lack of diversity, however, which is at odds with its purported universality. This disconnect led researchers throughout the 20th century to address a woman question in mathematics.

The first iteration of the study of women in mathematics came in the early 1970’s, when, as Fennema (2000) explains, a rigid understanding of “sex differences” implied that disparity in educational outcomes is biologically determined, and thus not changeable. In the later 1970’s and 80’s, the differentiation between gender and sex in social and educational literature attention was turned to possible social and environmental causes for women’s struggles in mathematics and related fields. Initially, this meant that attention was turned to remediation, offering extra help and support to help girls overcome their deficiencies and underdeveloped skills (Boaler, 2002). Structurally, this approach places

blame for the gender gap in mathematics performance on individual women, framing the disparity in outcomes as a lack of ability on the part of female students (Willis, 1996).

One response, to that first wave of research was the proposal that boys are often given an advantage in mathematics classes, because the subject is taught in ways that encourage their natural thinking patterns (e.g. Buerk, 1982, 1985). In contrast, girls’ typical thinking patterns were shown to be at odds with the values of traditional mathematics education.

The Fennema-Sherman studies investigated affective factors contributing to success in mathematics and indicate that the disparity between male and female outcomes is more attributable to motivation and attitude than to innate capabilities (Fennema & Sherman,

1977, 1978; Sherman & Fennema, 1977). Both of these lines of inquiry have led researchers to propose alternative methods and pedagogies to make mathematics content equally accessible and meaningful to all students.

A further development in the consideration of women and mathematics is the possibility that the traditionally masculine and exclusive nature of mathematics, both in k-

12 and professional contexts, is not simply an unfortunate and contingent hurdle to be overcome. Instead, the discipline itself may play a role in maintaining broader gender inequality in society (e.g. Mendick, 2006; Willis, 1995). This is to say that there is a possibility that how we collectively understand mathematical knowledge and practice leads us to act in ways that perpetuate injustice along gender lines. This final possibility is the focus of the present dissertation.

Mathematics as a discipline can be difficult to circumscribe. In its present form, it is practiced by academics in a multitude of sub- and sub-sub-fields, like so many academic pursuits, maintained by university life, conference presentations, and journal publications.

A wider mathematical community is made up of scientists, computer programmers, engineers, analysts, and more, who are engaged in work that relies on mathematical concepts and computation but is not directed at mathematical ideas directly. Finally, the largest realm of mathematics practice and knowledge is the educational community. The only exposure many of us have to pure mathematics is in our k-12 classrooms, with perhaps an obligatory course at the undergraduate level. The interaction that most

Americans have with mathematics is mediated by the school environment and curriculum.

This is to say that for the most part, the mathematics encountered by the public is the mathematics taught in schools. Thus to understand the function of mathematics in the social order, it is necessary to take up the habits and practices that structure mathematics education.1

Researchers have sought to understand the social significance of mathematics education in a number of ways, in particular its potential contribution to persistent gender inequality. As a result, there is a steady stream of reform efforts designed to augment and change the way mathematics is taught so that it can be more equitable to all students and serve as a site of positive social change instead. In the day to day of teaching, though, a schism can develop between this general social concern and the more well defined and immediate drive to transmit content to all students. Too often the result is directly conflicting interests. The time pressed teacher, subject to the assessment demands of the school and district, might sincerely wish to teach in ways that empower her female students and let them connect with the material, but at the end of the day she will never be

1 Habits, practices, and the social order will all be articulated more clearly later in the dissertation. 2 Chapter 2 will articulate in detail what is meant by “school math” and the public understanding of mathematics. Chapter 5 will contrast this public conception with the 3

able to do it all, and she will have done pretty well to have simply taught the students some techniques.

Traditionally, teachers have filled a number of roles in American society, including helping newcomers assimilate into the mainstream, contributing to an informed public, helping prepare children to take up vocations and contribute to the economy, and helping children develop the discipline and values necessary to participate in democratic society.

Increasingly, however, the public understanding of the teacher’s duty is constrained to transmitting knowledge and subject matter to students (Biesta, 2006). The work of teaching is thus guided by the mandate to maximize content transmission, with the intent to raise student achievement. Concerns of social justice and of democratic education are supplanted by the need to convey information to students as efficiently as possible. The contrast between the two is especially stark in mathematics, because the subject matter is perceived as being characteristically extra-human and non-moral. Whereas in the humanities, addressing issues of gender might augment and enrich understanding of the content, mathematics is seen as wholly divorced from social concerns.

Even for a teacher that cares about and takes seriously the problem of gender and mathematics, concerns for social justice and democratic education remain separate from the end in sight, which is teaching students content. As long as those two projects remain parallel, the latter will overshadow the former. What is needed to truly enrich mathematics education may not be the addition of a social dimension to the intellectual project of doing mathematics but to reconsider what it is to do mathematics in the first place. It might be the case that reimagining what it is to do mathematics yields an understanding of

mathematics pedagogy that, rather than being divorced from social and democratic concerns, is integral to them.

Using gender as a lens through which to understand or critique mathematics can be precarious, because the question of women in mathematics presents itself at each juncture as already having been decided. In its most superficial iteration, it is either no longer relevant, with data no longer showing disparities in performance, or relevant only to a very small number of people, as a gender gap does still persists in the number of PhD’s and professorships awarded in mathematics. In a meta-analysis aggregating 242 articles and covering 441 samples of students, Lindberg, Hyde, Peterson, and Linn (2010) find that there is currently no significant performance disparity between the genders in United

States k-12 mathematics. In general, the authors report that the gender difference found in individual studies is small, and depending on the demographic and the outcome measure of the study (e.g. computation vs. complex problems, numerals vs. word problems), sometimes girls are favored and sometimes boys are favored. Melhuish and colleagues’

(2008) study corroborates this finding, by comparing nine different predictors of mathematics performance among 10 year old students. They find that of the following factors: birth weight, gender, socioeconomic status, mother’s education, father’s education, family income, quality of home learning environment, preschool effectiveness, and elementary school effectiveness, gender has the smallest effect size. That is, gender is a weaker predictor of performance than any of the other eight factors.

Despite equality in performance at the elementary and secondary levels, as a female mathematician moves higher up the ladder of institutional mathematics, she still finds herself increasingly surrounded only by men. The National Science Foundation (2009)

reports that in the United States, while women earn 57% of bachelor’s degrees and 51% of doctorates overall, they account for only 43% and 31% (respectively) of bachelors and doctorates in mathematics and statistics. Math-related fields fare no better, with women earning 31% of doctorates in physical sciences, 22% of those in engineering, and 22% of those in computer science. Women’s representation on university faculties is even lower; in 2006 only 8.6% of professors in these fields (including full, assistant, and associate professors, and instructors employed 35 hours per week in 2- and 4-year colleges and universities) were women.

The world of academic and professional mathematicians is rather small and an already accomplished and privileged group, and seems a marginal concern compared to the struggle for liberation and basic needs still experienced by women around the world.

Furthermore, the question of a feminist mathematics can seem to be an oxymoron from the start. It seems an impulse to politicize, or to link values and an ideological commitment to what should be the paragon of value free inquiry. It might be the case, however, that gender imbalances in mathematics are not themselves the problem, but rather a symptom of something deeper.

1.2 Objectives

This dissertation will bring conceptual resources that have been developed to address gender inequity to bear on mathematics education, and argue that feminist social epistemology can offer a different way of conceptualizing the relation between mathematics education and social justice. I aim to defamiliarize present practices in mathematics education and make them seem less self-evident, so as to open space for new forms of experience. In the process, a language will be offered that will help make a

conversation possible that departs from the narrow aims articulated by the dominant social and political voices. By calling the dominant paradigm in mathematics education into question, by articulating a relationship between knowledge practices and social oppression, and by characterizing the work of mathematicians as fundamentally different from that of mathematics students, I will make the case that math class in its current form tends to promote ways of thinking and knowing that perpetuate enduring social inequality in the United States. Further, I will propose that given a change in certain fundamental understandings, it has the potential to be a space that is not only socially neutral, but can help students engage in the kinds of thinking that is necessary for deep social change.

1.3 Existing Lines of Inquiry: Mathematics

There is a long history of researchers who recommend making school math more like professional mathematics, for one reason or another. Most notably was the rise and fall of the New Math movement in the 1960’s in the United States and some Western

European countries that advocated teaching arithmetic in the early grades through set theory and logical definition rather than through direct experience (Kline, 1974). The reasoning was that traditional mathematics curriculum was comprised of antiquated material (i.e. concepts developed before 1700), which therefore lacked credibility. Cutting- edge fields including set theory, abstract algebra, symbolic logic, and topology were adapted into curricula for use in elementary schools. The intention was to teach students to think abstractly from the start, rather than introducing them to concrete experiences of numbers in the world, which will only be supplanted by abstraction later. For a myriad of reasons, the movement was a very public failure, giving rise to a “back to basics” backlash that in many ways is still with us. The most lasting effect of the New Math movement is the

idea that elementary and school mathematics are different from higher mathematics and that professional and research mathematicians cannot make valuable contributions to thinking about lower level mathematics education (Kline, 1974).

Despite the failures of New Math, calls to make elementary mathematics more like professional mathematics do still exist, particularly recommendations to bring proof and proving to all levels of mathematics (e.g. Hanna & de Villiers, 2012; Hanna & Jahnke, 1997;

Hersh & John-Steiner, 2011; Hersh, 1998; Lockhart, 2009, 2012). Many of these voices originate within the professional mathematics community, and as such speak eloquently about the discipline and the recommendations they make are thoughtful and promising, but generally speaking, fail to find traction in education circles. Lockhart (2009) is an interesting case. Paul Lockhart is a research mathematician turned high school teacher, and he wrote what initially was his own pedagogic creed. The document was photocopied and passed around, gaining wide circulation among mathematics teachers, and was later revised and published as a monograph. By drawing comparisons to art and music, the author makes a vivid contrast between the mathematics that is compulsory in schools, and what he and others see as beautiful and compelling in the discipline.

Presumably, one reason that the scholarly community has not addressed this work is its unconventional style and non-adherence to disciplinary norms; that is, it is not written so as to engage with the educational research community. In addition, it is possible that it has not found an audience because it takes up the disciplinary underpinnings of the entire field and outlines a critique that amounts to an indictment of the totality of mathematics education. Lockhart and others seek to argue that “we’re getting it all wrong.”

The critique is directed at the education system, including teachers, policymakers, and

curriculum creators, but the ideology of mathematics that is at issue is upheld by the public and mainstream understanding of the discipline (Appelbaum, 1995). Because school math is most people’s only encounter with mathematics, and because the process-oriented, mechanistic characterization has persisted for generations, the public understanding and school math perpetuate each other.2 A critique this comprehensive is especially tenuous when it comes from an outsider, without any attendant concrete solutions.

One of the objectives of this dissertation is to bring the spirit of these critiques into the academic discourse, and to offer a specific recommendation for making change. I will identify as a critical point the understanding and formulation of the problem in school mathematics and argue that working to reinterpret the mathematics problem is a specific means to bring about a number of changes in mathematics education.

1.4 Existing lines of inquiry: Education

There are a number of ways in which scholars have sought to understand the link between mathematics education and democracy at large, the most pervasive concern being the way in which math class relates to widespread inequality on the basis of race, gender, and socioeconomic status. Livingston captures this view as follows:

As educational standards in mathematics become the rubric upon which the success or failure of teachers and schools are measured, it is important to consider whether the curriculum standards contain the seeds of social justice or hegemony. If mathematical standards convey an unconscious privilege to one group at the expense of another, then equity is at issue (2009, pp. 423–424). Equity in education is crucial if democracy is understood to run deeper than simple governmental systems. The concern for equity in math education is a response to the

2 Chapter 2 will articulate in detail what is meant by “school math” and the public understanding of mathematics. Chapter 5 will contrast this public conception with the work done by mathematicians in the field.

possibility that the institution might be perpetuating such barriers, leading to an increasingly stratified and antidemocratic society.

Sue Willis (1996) gives a loose taxonomy differentiating four perspectives that can be seen within this vein of work, categorizing them as remedial, non-discriminatory, inclusive, and critical or radical. I will use her taxonomy here to give a broad characterization of the existing lines of inquiry that take up gender inequity in the context of mathematics education. With a remedial perspective (e.g. Boaler 1997, Boaler 1998,

Moses and Cobb, 2001), the curriculum is taken to be given and objective. Disadvantage, and thus inequity, lies with the learner, who, often by virtue of race, gender, class, or socioeconomic or language background, lacks the ability to succeed in math class.

Solutions in this mode focus on remediation of particular students, offering them extra academic help and support for their underdeveloped skills. With a non-discriminatory perspective, researchers understand the problem to lie with pedagogy and assessment practices that privilege certain groups over others. A good example here is work on gendered learning styles, and the ways in which math pedagogy can be more accessible to either girls or boys, thus inordinately disadvantaging the other half of students. The solution is to improve implementation so that students of all backgrounds can access math content and have equal opportunity to succeed.

Willis calls the third perspective “inclusive”, wherein the mathematics curriculum itself is called into question, and the attempt is made to formulate a curriculum that is equally meaningful to students with different backgrounds (e.g. Boaler 2011, Villalobos

2009). The sequencing and the form of mathematics is called into question, and the typical learner that a curriculum serves is considered in formulating concepts and materials that

will be meaningful to different learners. It is acknowledged from this perspective that mathematics is a large and varied field, and that educators can choose what to teach and when based on social concerns. An important aspect of this work is an understanding of math education as a discourse in the Foucauldian tradition. This means that the language and communication surrounding math systematically forms it (Foucault 1982, p 79), positioning participants within networks of power and constraining options, disciplining them into certain ways of being. An inclusive impulse identifies features of this discourse that systematically disempower certain people, cornering them into less desirable positions, and seeks to make more options available.

Heather Mendick’s (2006) work on masculinity and mathematics, for example, examines the ways in which math discourse and what it means to be a mathematician can be starkly masculine. Mendick argues that this masculinization of the discipline has the effect of encouraging men to enter mathematics and its related fields and reap the benefits and repelling women. As she points out, the result of focusing on the discipline, as she does, is a “figure-ground switch, whereby the problem of gender and maths is not located within individual girls who make the ‘wrong’ choices or who understand their performance

‘wrongly’, but in the discursive context of gender, schooling and maths” (2006 p 18). This is to say that because math is publicly construed as the pinnacle of objectivity and rationality, and because those are characteristics classically identified with maleness, the field is more hospitable to individual men than it is to women. Following Mendick’s reasoning, once this is understood, efforts can be made to make the field more accepting and welcoming for women and to work toward a construction of femininity and of gender that does not exclude these highly valued characteristics.

1.4.1 Pedagogy of the Oppressed

A final category of research addressing gender and mathematics can be called

“critical” or “radical,” in which mathematics practices are seen as actively reproducing social inequality. Work in critical mathematics education has developed immensely since

Willis first laid out this taxonomy, and it now comprises the majority of work in the area.

For context, I will offer a short account of one of the seminal texts of critical mathematics education: Paulo Freire’s (2000) Pedagogy of the Oppressed.

Freire offers a compelling framework for understanding the relation between education and social justice. Writing in Brazil in the late 1960’s, Freire calls the wealthy, land owning, politically powerful class the oppressors, and the physically laboring, politically powerless, and resource-poor class the oppressed. While the most glaring contrast between oppressors and oppressed is the disparity, Freire’s primary concern is knowledge practices and the ability of people to be knowledge creators. He argues that the key to liberating oppressed people is not simply redistribution of resources, rather, the nature of their dehumanization is that they are unable to be knowledge creators and name the world according to their own experiences and values. What the oppressed need is to be empowered to become fully human, that is, complete epistemic agents capable of shaping their own understanding of the world.

Fundamentally, this means that the oppressed must liberate themselves. Any political or social action take on behalf of the oppressed, despite good intentions, threatens to reinforce their status as agents without efficacy thereby adding to, rather than ameliorating, their oppression. Freire writes: “Attempting to liberate the oppressed without their reflective participation in the act of liberation is to treat them as objects

which must be saved from a burning building; it is to lead them into the populist pitfall and transform them into masses which can be manipulated” (Freire, 2000, p. 65). But the oppressed are thus left in something of a conundrum: “If the implementation of a liberating education requires political power, and the oppressed have none,” Freire asks,

“how then is it possible to carry out the pedagogy of the oppressed prior to the revolution?”

(2000, p. 54). He explains that “certain members of the oppressor class join the oppressed in their struggle for liberation, thus moving from one pole of contradiction to the other.

Theirs is a fundamental role, and has been so throughout the history of this struggle” (2000, p. 60). Freire describes this movement, of people from the oppressor class engaging in the struggles of the oppressed as “entering into communion with” (e.g. 2000, p. 61), in contrast to working on behalf of or struggling for.

Freire’s work has been hugely generative for educators in the United States and elsewhere. The social structure in the United States is more complex than that in Freire’s context, with more than one axis of oppression. In Freire’s context, there are two easily identifiable social classes, a structure which he seeks to disrupt. In the United States and other modern capitalist democracies, power and resources are distributed more evenly.

Regardless, the oppressor-oppressed dynamic and the characterization of dehumanization have resonated with scholars of education, and are routinely applied to our more complex context.

Freire argues that just as the oppressed are dehumanized by educational practices that fashion them into passive receptacles of information, the possibility for liberation lies in radical educational arrangements. He recommends “problem-posing” pedagogy, wherein students are engaged in discourse that calls into question the oppressive

structures that affect them and are enable to critique them. This discourse is both a means to political and material liberation, as it enables collective social action, and it is liberatory practice in itself. In the author’s own words: “to surmount the situation of oppression, people must critically recognize its causes, so that through transforming action they can create a new situation, one which makes possible the pursuit of a fuller humanity. But the struggle to be more fully human has already begun in the authentic struggle to transform the situation” (2000, p. 47)

Because the condition of oppression is characterized by the inability to shape the world according to one’s experience, reclaiming that ability is in itself humanizing practice.

Accordingly, in the Freirean tradition of critical mathematics education seed to shape the mathematics classrooms such that they can be spaces for humanization and growth, rather than dehumanization. Figuring mathematics as one of many critical literacies (Giroux,

1988) that enable students to interpret the world, with particular attention to illuminating institutions and practices that contribute to the students’ own marginalization and oppression.

1.4.2 Critical Mathematics Education

Mathematics for social justice, alternately called critical mathematics, describes work that seeks to use the classroom as a site to engage students in thinking critically about oppressive and unequal social dynamics. In the Freirean tradition, and as a part of the larger fields of critical literacy and critical pedagogy, these educators advocate bringing social and political material into the mathematics classroom for the sake of effecting more meaningful education and social change. Working from the position that we presently inhabit an unjust and oppressive situation, and that liberation depends on consciousness of

one’s own oppression, critical mathematics educators seek to make the classroom a site for consciousness-raising among under-privileged communities. The mathematics classroom is thus envisaged as a site for developing critical consciousness, and mathematics itself as a means to consciousness raising about social injustice. Internationally, the field has been shaped by the early works of Scandinavian authors Stieg Mellin-Olsen (1977, 1987) and Ole

Skovsmose (1980, 1981), who laid groundwork for thinking about mathematics as a human endeavor, and mathematics education as inherently political (Skovsmose & Greer, 2012).

In the United States, Marilyn Frankenstein (1990) contributed a textbook for use with remedial college students that aims to put Freirean principles into action inside the mathematics classroom (Skovsmose & Nielsen, 1997).

Eric Gutstein, one of the most prolific scholars of critical mathematics education and one of the authors of the Rethinking Mathematics series, articulates the following objectives for critical mathematics education. Students should develop: (1) Sociopolitical consciousness, (2) A sense of agency, (3) Positive social and cultural identities, (4) More positive attitudes toward mathematics, and (5) Mathematical power (Gutstein, 2003). The first three of these objectives derive from Freire (2000), and the last two from the NCTM

(2000).

1.4.3 Ethnomathematics

A closely related line of inquiry was established by Ubiritan D’Ambrosio in 1984.

Ethnomathematics (D’Ambrosio, 1984, 1985; also see Giroux, 1997; Powell & Frankenstein,

1997) is a field which brings together (or perhaps straddles the boundary of) cultural anthropology and the history of mathematics. Taking math knowledge to be inherently culturally entrenched, ethnomathematics is concerned with the idea that traditionally we

valorize and canonize the cultural knowledge of a small privileged class. These scholars argue that by passing down a body of math knowledge that is associated with and characteristic of Western European privileged White culture contributes to the reification of the dominance of this group of people worldwide. In response, ethnomathematicians seek to both bring to light the diverse history of our shared math traditions (e.g. that the origins of symbolic algebra are in the Arab Middle East, not in Europe) and to question the practices that have worked to develop the mathematics of one people while stifling development elsewhere. Ethnomathematics has a gender component, as oftentimes the activities deemed mathematical in nature are traditionally women’s work. Giroux (1997) for example proposes a study of basket weaving, which opens the mathematical questions of pattern, geometry, and production, as well as social questions about divisions of labor and the perceived value of such work. A project like this doesn’t fit within the established disciplinary boundaries of mathematics, and accordingly, ethnomathematicians often work to legitimize ‘other’ mathematics so that they will be taken seriously by the established field.

1.4.4 Critical mathematics education on the ground

Brantlinger (2013) documents his attempt to enact a critical mathematics curriculum within a traditional classroom setting. Brantlinger is an experienced mathematics teacher who had done graduate study with Eric Gutstein, and was looking to put the principles outlined by Gutstein (2003) and Gutiérrez (2002) into action in a high school level geometry course. He describes his thinking as he entered the course:

I thought that critical and mathematical activities would be mutually informing, and that their synthesis would manifest in materials and instruction….In this initial stage, I understood that any such synthesis would hinge on the general political utility or social applicability of secondary mathematics and of high school geometry in

particular; it was only possible if aspects of Euclidean geometry could help students become aware of the biased structural phenomena that affect them and help them address social problems in and out of school. (Brantlinger, 2013, p. 1061)

Brantlinger depicts the difficulties he faced in trying to implement a critical mathematics curriculum. His first struggle was to find data appropriate to his mission: raw data from the world that both provided information about social injustice, and which provided a ground for exploring the mathematical terrain dictated by his course assignment.

Logistically, the absence of developed curriculum materials and detailed guidance left

Brantlinger to develop materials and lessons from scratch. In preparation for the semester, he spends over 120 hours researching content to bring into the classroom. While this might be expected with the development of any new pedagogy or curriculum, Brantlinger concludes that there is a real “lack of fit between the critical and the mathematical” (2013, p. 1075). This is to say, developing a course on the principles of critical mathematics is difficult nor simply because it is new territory, but because the content of high school mathematics is not amenable to integration with social concerns.

Once the course was underway, Brantlinger still struggled to enact the principles of critical mathematics. He describes, for example, a Race and Recess activity, wherein students examine statistical data that correlates recess opportunities with racial composition in Chicago Public Schools. Recounting the class discussion, the author shows that the social context—the racial composition of local schools—proved to be a distraction from the mathematical activity—interpreting bar graphs. Nearing the end of the class period, Brantlinger decides to offer his own “official” mathematical interpretation of the chart, pointing out the connections he had wanted the students to make. As he explains, this teacher-telling is counter to the intentions of the activity, as it disempowers students.

While his intention throughout the course was to integrate the mathematical and critical content, such that the “mathematical analysis of socially relevant data would flow seamlessly into critical social enlightenment and empowerment” (2013, p. 1069) the experience throughout however, revealed that the students and instructor both perceived an implicit difference between the social and the mathematical. The social content was an enticement to participate for some students, and a deterrent for others.

Brantlinger concludes his study skeptical that the content of high school geometry can be meaningfully integrated with social and political concerns related to oppression and injustice. It is questionable whether the attempt to integrate critical mathematics in this way is ultimately beneficial for the students. Because economic competitiveness is valued above all of the other ends of schooling, because access to vocational and educational opportunities is dependent on narrowly conceived high academic achievement, and because the sort of critical social content seems to come at the expense of mathematical content, it appears that integrating critical mathematics content may not be only be ineffective as a site of social justice, it may on the whole be harmful to the very students it is designed to empower.

I propose seeking an alternate way forward. In the coming chapters, I will make the case that liberatory mathematics education can be developed out of western mathematics itself. This is to say, I intend to show that bringing social concerns and data directly into the mathematics classroom is not the only way to conceptualize a mathematics pedagogy that contributes to lessening persistent inequality and oppression.

1.5 Methodology

The aim of this philosophical investigation is to become aware of something we had not noticed before, and which might explain or dissolve our puzzlement when articulated.

The portrayal of open questions as investigations warrants a different treatment than everyday mathematics problems, empirical, or scientific investigations. As I will explore, one way to understand mathematics problems is as invitations to clarify our thinking.

Problems in mathematics are valuable when, in the course of working them out, we come to a fuller understanding of the terrain we inhabit. The problem itself is prized as an indicator of where things might be amiss beneath the surface, where a concept may have been extended beyond its usefulness, or where a lack of clarity or cohesion is causing turbulence on the surface. The work of solving the problem develops better conceptual resources for moving forward. This dissertation is constructed in the same spirit. By rendering the current paradigm in mathematics education problematic, I will make room for new possibilities in how we think about school math. Conceptual resources external to education, including feminist social epistemology, virtue epistemology, and naturalist philosophies of mathematics will help construct an alternate understanding of mathematics, and rethink what the work of mathematics education should be.

1.6 Theoretical Framework

1.6.1 Focus on the privileged

To accomplish this goal, I will focus primarily on the less-studied pole of Freire’s oppressor and oppressed dyad. The majority of critical mathematics work aims to empower and work with the least well off in our society. I aim to argue that there is another important aspect of promoting and creating more equitable conditions, which is

that persons in positions of social and political power must cease acting in ways that maintain their status. Put differently, democratic conditions require that all members of society have equitable opportunity to participate in defining and naming the world and to be recognized as credible knowers. Given our particular legacy of denying people the opportunity to be knowledge creators, based on their race, ethnicity, socioeconomic status, and gender identity, working toward more democratic conditions is a two-fold endeavor: people historically denied the ability to interpret the world must be enabled to do so, and people maintaining the monopoly on knowledge creation must cede a significant amount of their power and learn to participate as equals rather than as oppressors.

I recognize that there are many students who are in direct and immediate need of better schooling. Many of our students are in underfunded schools, in violent or neglectful environments, and are not having their needs met in any number of ways. These students will benefit from school reform and innovative curriculum that can help them navigate the educational system and succeed. Critical interventions into these environments can help students empower themselves to challenge the institutions that marginalize and disenfranchise them. On a structural level, these same students need for the systems that maintain their marginalization to be deconstructed. That deconstruction requires that people in socially powerful positions work to interrupt their habits that reproduce marginalization.

It is important to problematize the binary of oppressor and oppressed, or privileged and marginalized. Many facets of identity and social positioning contribute to marginalization, and as individuals we inhabit a plurality of these characteristics at once. I, as a queer-identified, white, cis-gendered female, who is economically and educationally

privileged, find myself on different sides of the privileged-marginalized binary depending on the context and the people around me. It can be tempting to see these identities as immutable characteristics, and give in to the impulse to tally them up to assign a status as oppressor or oppressed. It is more helpful, however, to stay mindful that these identities are relational and context dependent. Drawing on different aspects of identity can help us understand how systems and institutions position us in relation to one another, marginalizing some for the benefit of others.

The intention to address social concerns in mathematics education is a frightful prospect to some. In response to Stemhagen (2006), which is written in a spirit similar to the present dissertation, one mathematician writes:

if there's one thing I'm sure of, it's that mathematics has nothing to do whatsoever with justice, or for that matter, with any aspect of the physical world. Sure— physicists and engineers have used mathematical tools with great success to build scientific theories and neat gadgets. But math lives in its own separate Platonic world and we mere mortals can only hope for an occasional peek inside. (Kontorovich, 2007)

He follows up later with: “to see [social justice] used in the context of mathematical education sends shivers down my spine” (Kontorovich, 2007). These responses are not unique, and they do not come only from people already skeptical of social justice purposes.

Many educators deeply committed to social justice and to democratic education are hesitant to import social concerns into the realm of mathematics, understanding the latter as a purely intellectual endeavor. This hesitation is part of the math wars that have persisted in the United States through the 1990’s and 2000’s.

Accepting that mathematics is a purely intellectual realm, however, education is always social and moral in character3. I am reminded of the distinction Dewey makes in

Moral Principles in Education. “Moral ideas,” Dewey writes, “are ideas of any sort whatsoever which take effect in conduct and improve it, make it better than it otherwise would be” (2009). These moral ideas are contrasted with “’ideas about morality’ [which] may be morally indifferent or immoral or moral” (Dewey, 2009). The distinction is important because we typically understand moral education to be teaching that engages directly with moral or ethical principles. Dewey argues, however, that any ideas that have the moving force to impact action and make it better are indeed moral ideas. “What the normal child continuously needs” he writes later “is not so much isolated moral lessons upon the importance of truthfulness and honesty… as the formation of habits of social imagination and conception” (Dewey, 2009).

At this point I depart from Dewey, as he introduces mathematics as a prime example of non-moral ideas, which have no moving power and no effect on social action. In chapter

3, I will argue that ideas about mathematics do impact social action, and should thus be understood to be part of the constellation of social and moral ideas. The mathematician may not care that ideas about mathematics impact social action, and he may prefer to remain focused on concepts internal to the discipline. Education, however, always has a moral character.

Mathematics education is thus located at an intersection or overlap of the intellectual and the social. Rather than prioritizing one of these aspects, I have found it

3 I am purposely eliding the terms social and moral through this section, using them interchangeably to refer to the world of human affairs, in contrast to the purely intellectual realm or the realm of ideas, which ostensibly does not bear on the human world.

more generative to turn to bodies of research that have taken up this nexus. Naturalist philosophies of mathematics, feminist social epistemology, and virtue epistemology all make important contributions to the conversation, and in the following chapters I will use frameworks drawn from each to consider the relation between knowing well and doing good, and what role the classroom might play.

Within the social and moral realm, my particular interest is in lessening persistent oppression. To flesh out and investigate the role that knowledge has in perpetuating oppression, I will turn to feminist social epistemology, and to a lesser extent, white privilege studies. Scholars in these fields have turned their attention to the interplay between social structures and knowledge systems. There is a particular focus on how power relations manifest epistemically, but avoiding reducing knowledge to power politics.

Feminist social epistemologists seek to understand how we can improve our knowledge practices, with a focus on developing normative epistemological accounts given the fact of systemic relations of power. Crucially, they understand improvement to have a moral component—indicating movement toward anti-oppressive practices—and an epistemic component—indicating movement toward increased reliability of truth claims. Though there is variation in methods and approaches, a common theme is the expectation that these two aspects of improvement are intertwined.

I am confident that parallel analyses can and should be undertaken which take as their beginnings racial and cultural disparities as well as the rigidity of unequal social classes, among others. I do not intend to claim that gender oriented analysis is more important than these other concerns. I intend to show that when we do seek to set our educational house in order and make it hospitable for people of all genders, we will find ourselves with

a pedagogy and curriculum that is more human, vibrant, and deeply mathematical than what we have now.

1.6.2 Deweyan democratic education

The overriding framework for the paper is Deweyan in nature (e.g. Dewey, 1909, 1916,

1938), meaning that it understands schools to be not so much places for accruing skills to be used at some later date as sites that manifest and perpetuate deep democracy.

Furthermore, it prioritizes curriculum because in the classroom, the content is always the end-in-sight. This is to say that curriculum is conceived not as being important for its instrumental quality of being useful later in life, but insofar as it is what educators focus on in the classroom, how they understand it will shape what they do. In turn, how educators choose, present, and engage with content shapes the experiences that their students have, and consequently forms the habits and ways of knowing that they develop. While Dewey did not provide us with many resources for managing social inequality, it is clear that he recognized its importance and the role of educational institutions in making it possible. In

The Need for a Philosophy of Education, (1934) Dewey writes that “the aim of education is development of individuals to the utmost of their potentialities…. An environment in which some are limited will always in reaction create conditions that prevent the full development even of those who fancy they enjoy complete freedom for unhindered growth…. [A] philosophy of education must make the social aim of education the central article in its creed” (p. 244). Bringing contemporary theorists into the conversation who have carefully considered the nature and functioning of social environments that limit some for the sake of others will help develop this sentiment fully for our present situation.

In the Deweyan tradition, “a democracy is more than a form of government; it is primarily a mode of associated living, of conjoint communicated experience” (1916, p. 87).

It follows that to educate for democracy is to make possible these conditions of communication and associated living. Thus while, as he writes, “the devotion of education to democracy is a familiar fact” (1916, p. 87), when democracy is understood as a set of social conditions rather than a governmental arrangement, democratic education takes on a more comprehensive and nuanced meaning. In the United States, one of the most persistent obstacles to the democracy that Dewey promotes has been entrenched inequality arranged along lines of gender, race, financial resources, and a number of other immutable characteristics. Gender inequality, racial inequality, and the like enforce barriers to what Dewey calls “free intercourse and communication of experience” (1916, p.

99). It follows that one task of the democratic educational institution is developing in students dispositions oriented toward mitigating social inequality and unjust political arrangements.

Much of the scholarship that has been informed and inspired by Dewey’s work has been concerned with the classroom arrangements and pedagogical methods that are used to deliver curriculum to students. Less attention is given to the knowledge that is delivered and created in those educational spaces. Especially in mathematics education, where the content at hand is understood to be asocial and apolitical, the Deweyan tradition is largely uncritical of what is taught, taking issue instead with the manner in which it is delivered.

There are two lines of inquiry from which I hope this project will remain distinct.

The first is the neo-liberal commitment to equitable access, and the assertion that we can move beyond past wrongs by establishing equal opportunity to obtain material goods. The

second is the idea that anti-patriarchal or anti-racist mathematics should be based in a politics of difference (and thus seeking to develop a ‘feminine mathematics,’ for example).

1.7 Overview & Chapter Summaries

This dissertation will be presented in three movements. The first offers a characterization of the dominant paradigm in mathematics education, with particular attention to the vision of mathematics that animates the paradigm. I will argue that a very specific understanding of solving mathematics problems serves to limit what can be meaningfully said about good mathematics education. The framing of mathematics as a set of skills to be learned pre-determines what happens in the mathematics classroom, precluding authentic mathematical practice. The intention here is to articulate and clarify the assumed ends of mathematics education, or, what we take ourselves to be doing when we teach and talk about teaching mathematics.

The focus will shift in chapters 3 and 4 to social justice, to argue that the pedagogy and practices supported by the current paradigm in mathematics education promotes ways of thinking and being that reproduce patterns of social oppression and injustice. For the mathematics classroom, this would seem to establish a conflict, between the social good and the task of teaching, or between the school’s role as a democratic institution and its duty to instruct. I will summarize literature from feminist social epistemology and white privilege studies that identifies an epistemic element to social justice, articulating the ways in which individual knowledge practices can either contribute to or work against persistent social oppression. Drawing from virtue epistemology, I will suggest that what is called for are the intellectual virtues of courage and humility. This is to say that disrupting unjust

social arrangements requires that individuals are both courageous and humble in coming to know the world.

In chapter 5, the dissertation will turn to mathematics itself, and argue that the dominant paradigm of mathematics education is supported by a flawed representation of mathematics. Specifically, it is supported by an antiquated and unexamined understanding of mathematics that has effectively been jettisoned by those in the field. A more contemporary vision of mathematics reveals that the intellectual humility and courage called for by social theorists is characteristic of mathematical practice.

To conclude, I will offer a short series of illustrations of mathematics teaching that I believe exemplify the sort of pedagogy promoted by the foregoing analysis. In offering brief narratives of teaching, I intend to substantiate the claim that such pedagogy is possible within the elementary or secondary mathematics curriculum.

1.8 On Radicality, Positionality, and the Role of the Philosopher

At the heart of this dissertation is a challenge to the problem solving skills paradigm in mathematics education, and a proposal to fundamentally change how we understand the project of mathematics education. As is any challenge to an extant paradigm, this is a radical proposition. This dissertation is not undertaken in the spirit of blowing up mathematics education, or of disavowing so much work that has been done. On the contrary. I see conversations in mathematics education, which, on account of theoretical disputes, are unable to move forward. To do mathematics education in a way that is socially responsive and contributes to a larger project of democratic education seems to require prioritizing either the social or the intellectual, and the two camps that result from making that decision will not be able to reconcile, because their priorities are different. As

a philosopher of education I believe that revisiting the premises of the conversation can help reveal a way out of that stalemate, by recasting the terms of the discussion.

I arrive at this dissertation having taught mathematics in several different contexts, including public and independent high schools, a remedial community college program, and American Waldorf schools. These teaching experiences, paired with my undergraduate degree in pure mathematics from a liberal arts institution, have shaped my engagement with mathematics and they have given me a sense of what is possible within the k-12 classroom. A bachelors degree in mathematics does not make me a professional mathematician, or someone who can claim to speak for the field. It has given me enough of an insider’s perspective, though, to know intuitively that the content and lessons that I have been asked to teach from my first year following a mainstream curriculum are fundamentally different from what mathematicians do. In teachers’ lounges and department meetings, this difference is no secret, and is regularly lamented by teachers with mathematics backgrounds. School math just is not real mathematics.

Taking a philosophical approach to school mathematics can provide frameworks for understanding why that is; for comparing school math to higher mathematics and identifying what is characteristically different. The objective of this theoretical inquiry is to defamiliarize present practices and to render them problematic by bringing them into conversation with external discourses. Because present practices appear so self-evident, there is rarely space to offer alternatives, or even to offer critique. By bringing theory to bear on mathematics education, I hope to offer a language and a framework for thinking otherwise, and an opportunity to consider possibilities other than those given by the most

dominant voices. In so doing, space can be opened for new modes of practice and new forms of experience.

Chapter 2: Problem Solving Skills Paradigm

“Solving problems is a practical skill, like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing. Trying to solve problems, you have to observe and imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.” George Polya (1957, pp. 4–5)

This chapter begins with a characterization of the current paradigm in mathematics education as dominated by a narrow conception of problem solving. If I can show that the dominant mathematics pedagogy is governed by a specific conception of mathematics, we may be able to achieve a critical distance from it. Once that distance is established, there will be space to evaluate the paradigm on its merits and introduce alternative conceptions.

In the coming chapters I will show how the current paradigm serves to support a lingering oppressive patriarchy. I will then claim that this shallow mode of thinking is not only bad for women, it is an inappropriate model of mathematics. If we reconceptualize what it means to do mathematics, following recent trends in the philosophy of mathematics, we can move into a new paradigm in mathematics education that has a positive social impact as well as a more holistic and authentic engagement with mathematics.

The chapter offers a short historical account of the proliferation of the current paradigm in mathematics education. Charting the historical emergence of problem solving skills in mathematics education helps to lay bare the contingency of the present paradigm.

That is to say, by showing that the situation in which we presently find ourselves was not always the case, I can show that it does not have to be the case. I will then articulate what I understand to be the core commitments of the present paradigm, and offer curricular examples to support my assertions. After making the case that the paradigm in which we find ourselves is driven by a particular understanding of problem solving, I will argue that

the paradigm is upheld by an ideology of mathematics itself. We do mathematics education in this way right now because this is how we understand mathematics to be. By identifying pervasive myths about the nature of mathematical knowledge, I will argue that a mythology of mathematics enforces, maintains, and is integral to the mathematics education paradigm.

I would like to be clear at this point that my intention in identifying the upcoming myths is not to reject them outright, or prove them to be false. Presently, my intention is to identify them and show how they are integral to maintaining the mathematics education paradigm. In chapter 5, I will complicate these myths, and offer a richer portrayal of mathematics than the following ideology can support. One aspect of these myths is that they come to us in forms such that embracing their opposites appears absurd. So for example, taking on the idea of certainty in mathematics, it would seem that to challenge mathematical certainty would be to embrace skepticism or agnosticism. It seems absurd to claim that we should never feel certain that 8 + 8 = 16, or that the Pythagorean Theorem is true. What is needed, and what will be offered in chapter 5, is a complication of these myths. Philosophers of mathematics will help show that mathematical certainty does exist, but that it is of a different nature than everyday certainty.

2.1 Distinction among mathematicians

A distinction has been made within pure mathematics that separates problem solvers from theorists. Gian-Carlo Rota, the Italian philosopher and mathematician, writes that:

to the problem solver, the supreme achievement in mathematics is the solution to a problem that had been given up as hopeless. It matters little that the solution may be clumsy; all that counts is that it should be the first and the proof be correct. Once the problem solver finds the solution, he will permanently lose interest in it, and will

listen to new and simplified proofs with an air of condescension suffused with boredom. The problem solver is a conservative at heart. For him mathematics consists of a sequence of challenges to be met, an obstacle course of problems. The mathematical concepts required to state mathematical problems are tacitly assumed to be eternal and immutable. (Rota, 1997)

This is contrasted with the theorizer, to whom

The supreme achievement of mathematics is a theory that sheds sudden light on some incomprehensible phenomenon. Success in mathematics does not lie in solving problems but in their trivialization. The moment of glory comes with the discovery of a new theory that does not solve any of the old problems but renders them irrelevant. The theorizer is a revolutionary at heart. Mathematical concepts received from the past are regarded as imperfect instances of more general ones yet to be discovered... (Rota, 1997)

British mathematician Tim Gowers notes the same distinction, and writes more succinctly that for theorizers, “the point of solving problems is to understand mathematics better”, and for problem solvers, “the point of understanding mathematics is to become better at solving problems” (2000, p. 65). He writes that the distinction is widely acknowledged among mathematicians, but also that it is something of an oversimplification, as mathematicians recognize the distinction and identify with both impulses.

We can describe mathematicians as problem solvers, or as theorizers, which gives us a sense of their intentions and priorities, just as we could describe them as being more intuitive or more rigorous in style. The categories are not fixed, however, and doing mathematics requires both in tandem. As mathematicians solve existing problems, they create fodder for new ones, and as they articulate new theories, they put together resources for solving existing problems.

2.2 Problem Solving in Mathematics Education: Historical Context

In mathematical communities, concerns of the theorizing sort are more highly regarded than solving persistent problems. As Hersh and John-Steiner (2011) point out, mathematicians are remembered for working out powerful new theories which reframe and generate new knowledge. Newton, Euler, Riemann, Gauss; all of the names we remember are those who have articulated new theories. In mathematics education, however, problem solving has come to be the central activity of mathematics, to the exclusion of theorizing. The process of solving problems has come to be understood as a generalizable skill. This development has been heavily influenced by George Polya’s work in generalizing methods of mathematical thinking and what he calls a science of problem solving.

Polya was a Hungarian-American mathematician working in the mid-20th century. He had a long and successful career as a professor of mathematics, first in Hungary and then in the United States, where he turned his attention to pedagogy. While he only formally taught at the university level, he argued that his methods were appropriate generally, and gave workshops and lectures for teachers of all levels (Appelbaum, 1995). In his widely used work How to Solve It (1957), he imagines himself in the place of the student and offers a heuristic and opportunities for independent problem solving. The method is made up of four basic phases, each of which is subdivided into a number of possible courses of action.

He aims to offer a short set of steps that are basic enough to assimilated as a mental habit, and are general enough to not be rigid or mechanical. “First, we have to understand the problem.” He writes, “we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain

the idea of a solution to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it” (Polya, 1957, pp. 5–6). Polya frames the classroom and the practice of mathematics as an activity of problem solving; it is his end as well as his means. Polya’s four step heuristic is still quoted widely by mathematics texts, as well as by teacher education texts (e.g. S. Beckmann, 2012, p. 43; Long, DeTemple, &

Millman, 2008, pp. 9–20; Walle, Karp, & Bay-Williams, 2012, pp. 33–35).4

Polya’s heuristic and the idea that mathematics can be conceptualized as a more general mental activity of solving problems resonates with the dominant political tides around the turn of the 21st century. The 1983 A Nation at Risk report had set in motion a series of policy decisions and movements toward national standardization of policy and curriculum, and it had been influential in framing public education as a primarily economic endeavor

(Mehta, 2013). The report linked the success of schools, particularly in mathematics and science, to the nation’s position as a global superpower. It named a crisis of education, warning that if the schools continued to be ineffective, the workforce would suffer, rendering the United States unable to compete with emerging world powers. The report was most influential not in direct policy recommendations but in framing the educational policy discourse around a singular problem. Whereas traditionally, schools had a number of functions (including integrating recent immigrants, providing for social mobility, creating an informed and democratic populace), after the report, public discourse about

4 It is important to point out that Polya would likely not have endorsed the overly rigid and systematic way in which his heuristic is currently promoted and employed. This chapter’s epigraph notwithstanding, Polya writes of mathematical thinking as characterized by flexibility and the antithesis of what is promoted by the traditional “skill and drill” pedagogy. His texts, however, have been interpreted as endorsing a skills-oriented pedagogy. I take issue with these contemporary interpretations and uses of Polya more so than the author’s original intent.

schools was largely limited to their economic functions and their ability to produce a skilled workforce.

Within mathematics education, there had long been tension between traditionalists and constructivists (Appelbaum, 1995; Dewey, 1902). The former largely embrace a Platonic realist view of mathematics, and advocate a return to basics and the need for students to learn an established canon of information. Reformers (the latter), for the most part embrace constructivist versions of mathematics, and emphasize student centered learning and growth over content knowledge (Ernest, 1991). Polya’s problem solving approach allowed educators to circumvent this conflict and avoid joining one side or the other. He gave the option of a value-free version of mathematics—one dedicated not to knowledge of material but to processing power. This value free approach fit well with the wider social calls for focus exclusively on producing a technically literate workforce for an increasingly technical economy.

As a result, Polya’s methods were widely embraced by the mathematics education community in the 1980’s, one author stating that “for mathematics education and for the world of problem solving, it marked a line of demarcation between two eras, problem solving before and after Polya” (Schoenfeld, 1987). In 1980, the National Council of

Teachers of Mathematics (NCTM) declared that problem solving should be the basic skill of mathematics, on which school mathematics should be focused (1980). Curriculum specialists took up this directive, publishing a host of studies and resources for teaching problem solving skills (Appelbaum, 1995). In the 1990’s, the NCTM went even further, placing all mathematics education in the context of problem solving. The most recent iteration comes with the 2010 publishing of the Common Core State Standards(National

Governors Association Center for Best Practices & The Council of Chief State School Officers,

2010), which lay out eight “process strands”, or norms of mathematical thinking (shown below). The first is, explicitly, problem solving, and the remaining seven implicitly figure problem solving as the primary activity in question.

Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problem situations…. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. 5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. 6. Attend to precision Mathematically proficient students try to communicate precisely to others…. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. 7. Look for and make use of structure. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Figure 1: Standards for Mathematical Practice (National Governors Association Center for Best Practices & The Council of Chief State School Officers, 2010)

There is much to appreciate in the CCSS, and these recommendations are an important aspect of the rubric that has been set up. I mention them here to help illustrate the emergence of the problem solving skills paradigm, and to show that the identification of mathematics with the act of solving problems has become so entrenched that it no longer needs mentioning. Currently in mathematics education, there is always already the assumption that attention should be turned to solving problems, instead of any other activity. The contemporary paradigm in mathematics education is one of problem solving skills.

2.3 Problem Solving Skills Paradigm

To understand how a narrow interpretation of problem solving can serve to limit our understanding of mathematics, it is helpful to delve into the nature of paradigms in general.

The notion of a paradigm has been extended to many different contexts since Thomas

Kuhn’s 1962 work The Structure of Scientific Revolutions. In the book, Kuhn tries to refute the common characterization of science as a constant progression towards ever better representations of the world, and focus on the role that overarching frameworks have on the work that scientists do. This framework he calls a paradigm. He does not define the term explicitly, however, and he uses the term in no fewer than 21 different senses

(Masterman, 1970). Rather than being an oversight, the avoidance of a strict definition should indicate the many manifestations and functions of a paradigm. Masterman characterizes the descriptions into three groups: metaphysical paradigms (paradigm as set of beliefs, new way of seeing, or an organizing principle governing perception itself; this is the aspect or type of paradigms most often referenced), sociological paradigms (paradigm as concrete scientific achievement or set of political institutions); and finally artifact or construct paradigms (paradigm as textbook or classic work, instrumentation, or grammar)

(1970, p. 65). Masterman’s characterization is helpful because she points to the multiple and interconnected aspects of a scientific paradigm. In doing so, she elucidates Kuhn’s original concept and reveals the deep and structural interplay between the objects we use, the experiments and achievements we hold up as exemplary, the grammar we use, and the way our beliefs are organized. In mathematics education, this construct of a paradigm can refer to the structure and grammar of curriculum materials and textbooks, the stories of mathematics achievement that get told and retold, and the way we understand success and

achievement. The totality of these factors prefigures the questions we ask and the way we interpret the results that we see.

Paradigms govern and regulate the generation and interpretation of questions and propositions. The problem solving paradigm in mathematics education establishes a bounded space within which only some statements can be meaningfully expressed. The problem solving skills paradigm relates all mathematics learning to the capability solve problems, so that the only learning that makes sense to talk about is mastery of specific types of math problems. The multifaceted characterization of a paradigm means that it is often difficult to reveal the paradigm we are currently in. I will rely on textbooks and online curricula as artifacts and elements of the current paradigm, on the specimens that we hold up as exemplars of mathematical knowledge, and the stories we tell about them.

The core commitments of the problem solving skills5 paradigm are listed below.

1. A focus on knowledge how over knowledge that 2. Themes and processes are broken down into measurable and articulable sub-processes 3. Unchanging nature of conceptual units—an indication of a good tool is that it does not need maintenance or upgrading 4. Focus on justifying and articulating steps to show mastery and understanding of the sub-processes 5. Learning happens by observing and modeling, with a focus on practice 6. Additive model of acquisition, where the learner’s task is to accumulate increasingly more 7. Universal assessability. There are transparent and universally accepted norms for executing procedures. 8. Transferability/generalizability. Figure 2: Commitments of the Problem Solving Skills Paradigm

5 This list draws from Steven Johnson’s (2010) definition of “skills” in learning, as distinct from content and from growth.

In the coming chapters, I will argue that these are not characteristics of mathematics itself, and that they are misplaced in mathematics pedagogy. First, it is necessary to substantiate the claim that they are guiding commitments of the dominant paradigm in mathematics education.

These elements are evident in the grammar and structure of textbooks. One such example is Tobey, Slater, Blair, and Crawford’s widely used series of texts for use in college courses by Pearson. Each book is organized with what they call building block organization, and is the familiar format. The content of each course is divided into chapters and subdivided into self-contained topics, estimated to be suited to a single class session. Each topic is further divided into objectives, each represented by a type of problem to be solved.

The text teaches by example, first showing an algorithm, then allowing space for the student to replicate it. Following the introduction of the new process will be a set of problems just like the examples, on which the student should practice executing the new process. The student will be asked to solve slightly different versions of the example problem, as well as to explain his steps to show that he understands the ideas. The following day in class, a new process will be introduced, likely a further development of today’s lesson, and the student will follow the same process, each day adding a little bit to his understanding.

2.3.1 Problem solving skills in action: Khan Academy

As a second example, consider the online open educational resources produced by

Khan Academy. Founded in 2006, Khan Academy produces free instructional videos on a number of subjects for students and teachers in grades k-12 (“Khan Academy,” 2014).

Increasingly, it is entering higher education as well, focusing in particular on community

and other two year colleges (Khan, 2010). Of over 5,500 available videos, approximately

3,000 of them are about mathematics, making mathematics a primary focus for the organization (Murphy, Gallagher, Krumm, Mislevy, & Hafter, 2014). Originally, the content was slated for use as supplementation to classroom work and formal institutions, but Khan

Academy is increasingly working closely with schools and teachers to integrate content into educational sites for use as extra practice, remedial intervention, enrichment, as well as assessment. As one of the most popular websites in the world, with approximately 10 million unique users per month (as of February, 2014), an estimated 12% of k-12 students in the United States access the site each month (Murphy et al., 2014; The Center for

Education Reform, 2014). This reach, coupled with Khan Academy’s rapid growth and almost complete coverage of k-12 mathematics curriculum as articulated by the CCSS, makes the site a tangible example of the curricular issues that I wish to problematize. I draw attention to the Khan Academy’s resources because they are artifacts of the problem solving paradigm as it stands.

Content is divided into standardized grade level areas, and subdivided into modules, as a traditional textbook series might be divided into chapters within books. Each module is further divided into a series of skills and definitions that comprise the topic. Under

Algebra I (2014), for example, we find modules titled Linear Equations, Linear Inequalities, and Quadratic Equations, among others. Linear Equations includes the topics “solving for a variable,” “converting repeating decimals to fractions,” and “age word problems.” Finally, each topic is comprised if short sub-topic videos and question banks. The content is very much akin to a traditional textbook, translated into interactive online environment. The student working through the material begins by watching a “how-to” video that shows an

explanation of a method for solving a given problem, and is asked to show mastery of the method by then repeating the method. Progress is demonstrated by effectively moving through the topics and modules, accruing the skills given in each sub-topic.

By returning to the core commitments enumerated above, I will show that the Khan

Academy mathematics program exemplifies the problem solving skills paradigm.6

1. A focus on knowledge how over knowledge that

Each subtopic identifies a skill to be learned. Methods are demonstrated and applied to slightly different versions of the same problem. Even in cases where conceptual understanding seems to be the end in sight, the videos still focus on solving the same sorts of problems, with the addition of extra diagrams or explanation. “Why we do the same thing to both sides: Two-step equations,” for example, which could be a gateway for understanding one of the fundamental concepts of algebra, is a set of steps for solving the equation 3� + 2 = 14, accompanied by an illustration of a balance (“Khan Academy,” 2014).

The focus is clearly still on a method for solving the equation, not understanding the relation in question.

2. Themes and processes are broken down into measurable and articulable sub- processes.

Khan Academy materials illustrate this commitment perhaps better than any written resource could. Progress is tracked in the system by a system of points, badges, and

6The distinction between mathematics problems and exercises (e.g. Schoenfeld, 1988) is illustrative here. A mathematics exercise is a task that requires practicing skills or concepts that one has learned. A mathematics problem entails an unfamiliar challenge. When presented with a mathematical problem, the student should not know a method for solving. This distinction will be drawn out in more detail in chapter 5 with the help of Ludwig Wittgenstain’s work.

achievement, awarding a relative point value to each sub-topic. Mastery of the totality of sub-topics is mastery of the overall content.

3. Unchanging nature of conceptual units

As with any prescripted and produced material, concepts and definitions are introduced by the videos, and are to be learned by the student. Definitions and subject matter are atemporal and absolute.

4. Focus on justifying and articulating steps to show mastery and understanding of the sub-processes

Khan Academy’s medium is likely the reason that it does not seem to highlight the justification and articulation of steps for executing each sub-topic skill. By necessity, it appears, the system is set up to accept and evaluate only final solutions to problems, and cannot evaluate a set of steps as a live teacher could.

5. Learning happens by observing and modeling, with a focus on practice

For each sub-topic, the student watches a video and repeats the process on her own. The video models the process by which she should use, and by which she can expect to derive correct answers to the given problems.

6. Additive model of acquisition, where the learner’s task is to accumulate increasingly more

Each sub-topic builds incrementally on the last. For each sub-topic mastered, the student is awarded points and badges, and makes progress toward the goal: total completion of the modules. Working through the modules, the student never is given or asked to consider a richer or more complete conceptual understanding of the topics, it is simply an accrual of increasingly more skills. A colleague, whose son uses the Khan Academy resources as a

home-school curriculum, described the experience as “a treadmill of tiny problems,” with no conceptual development.

7. Universal assessability.

The automated nature of the Khan Academy’s format renders the assessment of a student’s answers invisible. For each problem solved, the answer must be instantly marked right or wrong by the browser, so the only questions the system can pose are those with universally accepted norms for executing procedures, and those for which assessment can be automated.

8. Transferability/generalizability.

The idea that student will be able to transfer the skills they accrue in the Khan Academy system to other kinds of tasks, including other academic work and real world or vocational pursuits, is intrinsic to the structure of the program. There is no indication that moving through the exercises is an intrinsically valuable practice, as the integration of various reward modalities makes clear. The only value of moving through the system and accruing the skills is the idea that they will be useful for some other, outside endeavor.

Research has long showed that student s have trouble transferring their knowledge of school math to their regular lives, even to closely related pursuits like the hard sciences

(e.g. Gill, 1999a, 1999b; Jacman, Goldfinch, & Searl, 2001). The response to this difficulty has largely been to seek to understand transference better so as to develop better instructional practices (Mayer & Wittrock, 1996). In coming chapters, I will argue that better methods are not all that is needed. I will argue that many important and vibrant aspects of mathematics practice are not the sort of activities that we should expect transference from.

2.3.2 Further discussion

As a contrast, we might compare this model of mathematics pedagogy to art education.

Generally speaking, art education is conceived primarily as a space for expression and developing students’ own perception and engagement with the world. Technical skills are taught in service of this more general end. The core commitments of a meaning or expression oriented pedagogy might be:

• Exposure to a wide variety of modes of expression

• Understanding and engaging with social, cultural, and historical development of

context and techniques.

• Development of a critical aesthetic judgment.

“But!” We might respond “mathematics isn’t like art. If we are to teach mathematics, we ought to be faithful to the discipline, authentically representing and engaging with it.”

In forthcoming chapters I will take on this idea and argue that mathematics is in fact not well represented by this problem solving skills paradigm, and that the current representation of mathematics in educational institutions is not faithful to what mathematics is and historically has been.

While problem solving has proliferated as the primary mode of doing mathematics, it has also become invisible as it is always already assumed that mathematics is problem solving. Course content, which has been relatively stable over the past half century, is seen not as an end in itself to be learned, but as a means to the more general goal of developing problem solving skills. Appelbaum (1995) points out that images of mathematics education in popular culture over the same time periods have picked up problem solving as

the ultimate goal of mathematics education, mirroring and (he argues) perpetuating the evolution at the policy level.

Reductive and mechanistic problem solving is not the only type of mathematics education around; across the country there are reform efforts, innovative programs, and inspired teachers who are delivering rich and holistic curricula to many different kinds of students everyday. The majority of students in the United States are in classrooms and systems where a shallow problem solving skills approach to mathematics is all they see

(Stigler & Hiebert, 2009).

2.4 Animating Ideology7

The problem solving skills paradigm is animated by a pervasive and deeply rooted understanding of mathematics that gets reinforced each time we present mathematics to a new generation. As mathematics is taught and retold, students learn not just the content of the curriculum, but also they develop an understanding of the discipline. They are taught mathematics knowledge, as well as what sort of a thing it is and how it functions in the world in large part by the myths that are told about the practice. Myths of mathematics8 exist as stories that are told about mathematical history, as the characters that animate the tradition, and as unexamined characterizations that are affirmed generation after generation by the public and schoolteachers alike. Simply being myths does not necessarily

7 I use the term “ideology” here to describe the conscious and unconscious concepts that frame our understanding of mathematics. While I contest the accuracy of a number of the ideas that constitute the shared ideology of mathematics, I do not intend it as a pejorative term. I use it to refer to the shared understanding of the nature and content of the discipline, which has a normative component.

8 The notions of myth and mythology have a rich tradition in philosophy, and I am not referring to any specific thinker’s understanding or claims. I instead use the two terms in a layperson’s sense, to refer to legend, folk tale, or misconception.

imply truth or falsity of the characterizations, and my intention here is not simply to refute them. Because of the way the myths are constructed, and their maintenance of a total paradigm, they make outright refutation quite difficult and counterproductive. Instead, I hope to show that when they remain unexamined, these myths and the pedagogy they drive can have negative social effects. What we should want to do is not shoot down the myths but complicate them.

2.4.0 There are no myths in mathematics

The first myth of mathematics is simply that no such thing exists—that mathematics defines itself essentially by its lack of any such mythology. But this is simply a characteristic of any paradigm. The nature of paradigms is that they structure discourse in such a way that they render alternatives unintelligible, thus establishing themselves as simply the way the world is (Kuhn, 1996). In the sections that follow, I will articulate four such myths that animate mathematics education and show how they serve to impose limitations on human capacity.

2.4.1 Mathematics is universal, unified, and transcendental

The first myth of mathematics is that it is a cohesive whole, without internal disagreement or change over time. There is one body of knowledge properly called mathematics. It consists of propositions that are compatible with one another, whose truth-value is attributable to the Way the World Is, and is thus equally accessible to people in any culture or time. Accordingly, mathematical claims would be true whether or not anyone wrote them down or discovered them, and the details of their discovery are purely incidental. Furthermore, the truths of mathematics do not change. Seven and three has

always been ten. Because of mathematics’ status as objectively and unflinchingly true, the historical context in which it was discovered and developed doesn’t matter.

The universality of mathematics is, of course, a very old idea. At present, we see this myth manifested by two aspects of textbooks and curriculum: definitions and questions.

2.4.1.1 Definitions. The problem solving skills paradigm as presently constituted relies on a static world with stable categories and objects. These fixed bodies are presupposed by conventional mathematics pedagogy, which is evident in the grammar of textbooks and in the direct instruction of mathematical concepts. Textbooks start off new sections and teachers begin new lectures and lessons by introducing a new term with a formal definition, and proceeding to tackle problems involving the term and its concept. For a rather simple example, Aufmann and Lockwood’s (2010) text for basic college mathematics begins a section on consumer math with the following statement:

Frequently, stores promote items for purchase by advertising, say, 2 Red Baron Bake to Rise Pizzas for $10.50 or 5 cans of StarKist tuna for $4.25. The unit cost is the cost of one can of StarKist tuna. To find the unit cost, divide the total cost by the number of units. Figure 3: Unit Cost The new term (unit cost) is introduced before students have a reason to want such a term. It is just the next in a list of concepts to learn so the student can do the work and move on. Furthermore, it presents the concept as existing objectively, independently, and transcendentally. It just is. Immediately thereafter, attention is directed toward doing something with the concept or definition as stated. As in the example above, there is no space for a justification for the definition, an indication of a different concept that was used previously but has now fallen out of fashion, or an indication of why it might be a helpful concept to have. It just is and the student’s vocation is to operationalize it.

2.4.1.2 Authorship and framing of questions. School mathematics puts the student always in a posture of responding to questions posed, and accepting concepts as they are presented. Traditionally, direct instruction is the norm, wherein the student is shown a technique and expected to reproduce it. Typically, textbooks are set up so that a body of material is broken down into digestible themes, and then subdivided so it can be covered in a typical school year or semester. Each section of a textbook has a problem set at the back, which presents a series of problems designed to require the algorithms from the forgoing section. The section itself will have general conceptual information (formal definitions, theorems, derivations, etc.), as well as examples of problems that rely on those concepts for their solutions. Students reproduce varieties of those problems in the problems sets.

Polya’s problems, which he uses both as a curriculum and as a model, are no less pre- determined. A great number of the problems are essentially “guess what I’m thinking” questions, where the author offers carefully chosen bits of information and the student is to deduce the missing bits. For example, Problem # 4: “To number the pages of a bulky volume, the printer used 2989 digits. How many pages has the volume?” A few of these questions deliberately obfuscate, for example the first problem “A bear, starting at point P, walked one mile due south. Then he changed direction and walked one mile due east.

Then he turned to the left and walked one mile due north, and arrived exactly at the point P he started from. What was the color of the bear?.”9 Others of the problems are stated more

9 The polar bear problem is commonly used in geometry courses, in the context of spherical geometry. While on the surface, it appears to have nothing to do with mathematics, notice that the bear makes only two 90° turns, and returns at his starting place. On a planar surface, he’d have to make three 90° turns to end up where he started. The only way the conditions of the problem are possible are if the bear is travelling on the surface of a sphere, and given the directional details given, the bear must begin at the North Pole. Thus the bear is most likely a polar bear and should be white.

discretely, but are of the same form. Problem 18 states “consider the table shown below.

Guess the general law suggested by these examples, express it in a suitable mathematical notation, and prove it.”

1 = 1 3 + 5 = 8 7 + 9 + 11 = 27 13 + 15 + 17 + 19 = 64

21 + 23 + 25 + 27 + 29 = 125

Figure 4: Sum of odds “Perfectly stated” and “reasonable” problems, as Polya describes them, must have all of the necessary data, with nothing superfluous, and must have a sufficient, neither contradictory not redundant, condition (1957, p. 97). Textbook and exam questions are, for the most part, perfectly stated; the student is given the exact right information to solve the problem in the manner the text sees fit. For example, opening Hornsby et al to a problem set at random, we find (my annotations below):

# 93. Volume of a cylinder. The volume of a right cylinder [1] is jointly proportional [2] to the square of the radius of its circular base and to the height [3]. If the volume is 300 cubic centimeters when the height is 10.62 cm and the radius is 3 cm [4], approximate the volume of a cylinder with radius 4 cm and height 15.92 cm [5]. Figure 5: Volume of a Cylinder A student, reading the problem sentence by sentence, is likely to interpret the problem in the following steps:

[1] look up formula for volume of a cylinder: v = π⋅r2 ⋅ h [2] use z = kxnym instead [3] radius = x, n = 2, y = h. plug into formula: v = k⋅r2⋅h

[4] plug in numbers: 300= k ⋅ 32⋅10.62 300 ÷ (32⋅10.62) = k = 3.139 [5] plug in new numbers: v = 3.139⋅42⋅15.92 v = 799.566 Figure 6: Volume of a Cylinder Solution This problem has the particular feature that it begins from a concept that the student has some familiarity with—the volume of a right cylinder (students will at least know that a formula can be looked up). The problem then goes on to direct the student to do something else; something more complicated for no seeming benefit. The task of the student is to march through and interpret each piece of information as a directive to perform some action. When she gets stuck, she can simply flip back a few pages to where there is a fully solved problem of the same sort, and follow the same template. No information missing, and no information extra. As I will argue in chapter 5, perfectly stated-ness is the most salient difference between problems in school math and problems in higher mathematics.

My aim in highlighting a number of typical mathematics problems is not to provide a caricature of curriculum, or to erect a straw man to then pull down. Rather, it is to direct attention to the very mundane assumptions that are routinely built into school mathematics. One such assumption is that problems are perfectly stated. If I can assume that the problem before me is perfectly stated, I can infer certain things from its phrasing.

Answering the question is a matter of combining all of the numbers given according to certain keywords that I can decode.10 Problems in the world, however, do not come to us in

10 Early in my teaching, I worked with a group of students who had been taught by a previous teacher that the best way to work out word problems in mathematics is to read the problem backwards. By starting at the end of the problem and reading the words in

this perfectly stated manner, which Polya recognizes. He briefly remarks that “practical problems however are usually far from being perfectly stated, and require a thorough reconsideration of the questions discussed in the present article” (1957, p. 98). The difference is not mentioned again.

Students learn that doing mathematics well is a matter of following specific instructions, and that success is getting the same answer as what is printed in the back of the book or in the teacher’s manual. Schoenfeld (1985, 1989) reports that high school students expect mathematics teachers, more than teachers of other subjects, to “show the exact ways to answer the questions”, and that the way to succeed in mathematics class is to do exactly what the teacher says. When students are given something different, or asked to take on a task for which they have not been shown a method in advance, they often rebel, or report that the teacher is not teaching.

2.4.2 Mathematics is objective and impersonal

The second myth of mathematics, closely related to the first, is that mathematics is objective. This is to say that there is remarkable consensus over what is correct, and the truth of a proposition does not depend on the position from which one approaches it. If many of us begin from the same question, we are likely to end at the same solution, which endows it with an air of objectivity, or the idea that the solution we are all likely to find is somehow located outside of ourselves or beyond human subjectivity. Insofar as mathematical claims are outside or independent of us as subjects, mathematical propositions and mathematics knowledge in general is thought of as impersonal and objective. Objectivism has been contested for some time in other areas of knowledge reverse order, they could glean the numbers and keywords from the problem without being distracted by the sense of the question.

(there are compelling arguments for and against the existence of objective ethical knowledge, for example), but no such debate has taken hold over mathematical claims in the educational realm. While the notion of relativity in moral claims might seem false, inappropriate, or even dangerous to many who argue against it, the idea does merit a response. Relative truth for mathematical propositions, in contrast, is taken as absurd, and hence unworthy of response or debate.

2.4.2.1 Purity and exclusivity.

Figure 7: Purity (Munroe, 2008) A particularly pervasive image is of the purity of mathematics as a type of knowledge.

Insofar as mathematics is not about human affairs, it appears to us as free of the complications the fickle human world is saddled with. Historically, because it is not about what we see and experience directly in the world, mathematics often gets framed by negation: mathematics is not of the body, it is not of contingent affairs, it concerns that which is not changeable and contingent. As humans who are unavoidably embodied, changeable, and messy, we are excluded by this body of knowledge, and the tradition that claims to be what we are not.

The Kingdom of Number is all boundaries Which may be beautiful and must be true; To ask if it is big or small proclaims one

The sort of lover who should stick to faces.

Numbers and Faces—W.H. Auden (Auden, 1976) Auden’s piece illustrates this idea beautifully—if I have a question about the kingdom of number, I am revealed as inadequate. If I am unable to already grasp the obvious necessity of what the kingdom of number offers, I reveal myself as worthy only of the lesser, embodied types of knowledge. Perhaps I should stick to something simpler.

Bertrand Russell expresses a similar sentiment in the following two passages:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." (1918, p. 47)

The world of mathematics … is really a beautiful world; it has nothing to do with life and death and human sordidness, but is eternal, cold and passionless. To me, pure mathematics is one of the highest forms of art; it has a sublimity quite special to itself, and an immense dignity derived from the fact that its world is exempt from change and time … The only difficulty is that none but mathematicians can enter this enchanted region.(Russell & Griffin, 1992)

The mathematician is the only one allowed in to Russell’s enchanted world—he is the only one worthy of it.

Russell’s statements above exemplify the modern ideal of mathematics—he seeks to buttress the validity of mathematics as the most secure kind of truth by calling it pure, thus distancing it from knowledge derived from experience, and changing and volatile human affairs. Mathematics is defined by this very distance, and the rejection of the contingency and change of human life. The image of mathematics’ exclusivity has been documented in a number of studies (e.g. Buerk, 1982; Ernest, 1996; Maxwell, 1989; Mendick, Moreau, &

Hollinworth, 2008; Picker & Berry, 2000). Mendick, Moreau, and Hollinworth (2008) find that mathematical knowledge is often described in mainstream culture and by mathematics

students in terms of its negations. It is characterized by what it is not—it is unlike literature and language, it is not creative, emotional or imaginative—and most of the things it is not are associated with the feminine.

2.4.3 Mathematics is provable and certain

The first two myths are supported by the idea that mathematics is an elaborate network of propositions, all of which have been unequivocally proven. While this notion primarily concerns higher-level mathematics (as creating proofs is a significant part of what they do), it trickles into k-12 mathematics, as well. At times, proofs are offered as support for a given procedure—the derivation of the Quadratic Formula, for example, often plays this role. The student is presented with a heretofore unsolvable problem (an unfactorable quadratic equation), is then shown a series of algebraic manipulations which culminates in a general formula that can be used to solve the problem. The student has no reason to pay attention to the derivation, and instead scans to the formula, and begins plugging in numbers. The message is clear—the formula is right, it is supported by a chain of reasoning that is too complex for you to bother with, so just trust us. More common are notations in algebra and geometry texts that “the proof relies on calculus techniques” (and is thus beyond the student’s comprehension), or, worse, that the proof is simply trivial (it rarely seems so to the student).

As Kitcher (1977) points out, every beginning mathematics student learns that in mathematics, results are proven. The proof is how results are formalized and published in journals and how they are turned in as homework assignments. In a general sense, it is how a proposition can be established with absolute certainty, independently of its author and applications. The idea that logical proof is the means to mathematical certainty dates

back to Euclid’s time. In approximately 300 BC, Euclid gathered the mathematical ideas that were available to him and organized them in increasing complexity, in a work eventually published as Elements. In it, he begins from five basic postulates, and combines them, adding definitions along the way, so as to gradually prove by logic theorems about plane geometry, natural and rational numbers, and spatial geometry. Both the theorems he aggregates and his axiomatic and constructive method have been hugely influential, with the text itself being widely used as the primary mathematics textbook into the early 20th century (Boyer & Merzbach, 1968). While the textbook itself is generally not used in the classroom anymore, its material and methods still serve as the standard by which logical geometry is taught in Western curricula (Kline, 1980). The source of mathematics’ privileged epistemic position is the understanding that all of its statements and claims can be absolutely proven. This idea is actively taught through the curriculum, as students learn to show and explain their work and give reasons for its correctness. This ideal becomes more explicit when students begin studying formal geometry, learning techniques for proving basic geometric statements. Beginning with the concept of proof and the steps for constructing basic proofs from accepted premises, students begin with Euclid’s widely accepted premises and explore logical proofs for increasingly complex geometric assertions

2.4.4 Mathematics is a solitary achievement

The fourth myth underlying the problem solving skills paradigm is the idea that mathematical thinking is a solitary achievement. This is to say that mathematical thinking is a solitary act, contained by the mind of the individual. While we often see concerns for conversation and group work in mathematics pedagogy and curriculum, that discussion is

always in service of developing the skills of the individuals involved. Legends of mathematics exemplify the ideal of solitude: Gauss as a young boy, Archimedes in his bathtub, Descartes in his bedroom, Galileo with his telescope. The figure is always a man, always alone, and always in some sense inconsequential to the work. He is merely a vessel, or the one who figured it out and wrote it down.

A corollary to the myth of individual achievement is the assumption that those who succeed in mathematics have a natural or innate ability that predisposes them to mathematical success. In the everyday of teaching and learning mathematics, it appears that either one can or cannot do mathematics, and that this is a rigid characteristic of the individual.

These four myths together make mathematics attractive as an impartial indicator of achievement in schools. Because it bears no relation to one culture or another, and it does not entail subjective judgment, it can be universally assessed according to standardized criteria. Each of the myths is closely related to the others, and together they exert their power by remaining largely invisible and seemingly above question. The universal, transcendental, objective, provable, and solitary characterizations of mathematics reduce it to a discipline amenable to skills learning. Thus the impulse of the problem solving skills paradigm. Mathematics is given as an impartial means of processing information about the world, or as a set of tools that can be appropriated by the student. The problem solving skills paradigm derives from these characterizations of mathematics itself.

2.5 Countercurrents

Throughout the emergence of the problem solving skills paradigm, there have been calls to avoid the sort of superficial, rules oriented thinking that is at issue here. A number

of proposals have been advanced for doing problem solving differently. Notably, Schroeder and Lester (1989) differentiate between teaching for problem solving, teaching about problem solving, and teaching through problem solving. Their focus is on how the problem is used in curriculum, and has been influential in grounding new methods in teaching. The traditional model of mathematics teaching embodies teaching for problem solving. Content is introduced by the teacher in abstract terms, students practice techniques, and then are asked to apply them to solve story problems that require the new skills. This mode of learning conditions students to rigid thinking, and positions them as passive learners, dependent on authority for conceptual resources, which is detrimental both mathematically and socially. The authors recommend that teachers instead take time to teach about problem solving, following Polya’s (1957) framework, teaching strategies including drawing diagrams, making charts, writing equations, and guessing and checking.

Teachers are encouraged to avoid “proceduralizing” problem solving, instead encouraging students to solve it any way that makes sense to them.

Teaching about problem solving, Schroeder and Lester write, can support teaching through problem solving, an idea that has permeated curriculum in the past decades. In essence, the students develop mathematical knowledge out of solving problems, rather than the other way around. Abstracted concepts are refined from their constructive work, rather than being introduced in the abstract and applied to tasks. Van de Walle, Karp, and

Bay-Williams (2012, pp. 81–82) offer examples of such problems for the early grades:

Mary counted 15 cupcakes left from the whole batch that her mother made for the picnic. ‘We’ve already eaten two-fifths,’ she noted. How many cupcakes did her mother bake? Figure 8: 15 Cupcakes and for middle school:

Does the graph of � = �! ever intersect the graph of � = �! + 2? Would your conjecture hold true for other equations in the form of � = �! + �? Figure 9: Intersecting Parabolae

The intention is that students, in answering the questions, will come to mathematically significant conclusions. In most cases, however, teachers are unable to coordinate the discussions that make the links between the student activities and the abstract concepts, and revert to a traditional model, telling the students what they were supposed to discover from the activity or problem (Stein, Grover, & Henningsen, 1996; Stigler & Hiebert, 2009).

2.6 Generalizing Mathematical Thinking

At the same time as problem solving has come to dominate mathematics education, mathematical thinking is promoted as a structure for more general rational thinking, and the mathematical method is used as a template for any problem that can be reasoned out.

Polya (1957) for example, gives instructions for stripping away irrelevancies and “going straight to the heart of the problem”, arguing that mathematical thinking is generalizable and helpful for solving all sorts of problems. He notes in the preface to the first printing that “although the current book pays special attention to the requirements of students and teachers of mathematics, it should interest anybody concerned with the ways and means of invention and discovery (Polya, 1957, p. vi). Mathematics educators often argue that students should learn to “think mathematically” in all areas of life (Fleener, Carter, &

Reeder, 2004; Rhee, 2007; Skovsmose, 1994). In 1985, Schoenfeld found that students had absorbed this message, describing mathematics class as a discipline in which they learn to think well. Even so, the concept of mathematical thinking is at best loosely defined. The

NCTM (2000) defines the term mathematical thinking as including “making conjectures and developing sound deductive arguments” (p 15). While the Common Core State

Standards (2010) offer more criteria of mathematical thinking, it still seems to offer a list of characteristics that are ambiguous, possibly arbitrary, and more appropriately viewed as general characteristics of good thinking. Argyle’s (2012) survey of educational research literature reveals that there is neither common definition nor sustained attempts to establish one. The only commonality Argyle finds in the definition of mathematical thinking is that it is a kind of “sense-making.” In education research as well as in the standards teachers are asked to follow, mathematical thinking is blurred with simply good thinking.

This generalization of mathematical thinking is concurrent with an increase in mathematization of how we think about and operate within educational institutions. On one level, mathematics is and has long been used as a general measure of academic aptitude and a gatekeeper to college entrance and a wide variety of majors (Moses & Cobb,

2002). Since the early 1980’s there has been exponential growth in the collection, analysis, and deployment of data on education. At the same time as children are taught explicitly that mathematical thinking is the best and most effective thinking, this lesson is reinforced by their disciplinization into a system where mathematics (via statistics) is used to assess and understand their actions (Simons & Masschelein, 2008). Explicitly and implicitly, mathematics is the language and measure of objectivity in educational institutions.

The problem solving skills paradigm is appealing today for the same reasons it emerged in the early 1980’s. It provides a framework for an apolitical, value free curriculum compatible with an increasingly computer centered economy. Mathematical skills themselves are understood to be apolitical. Once acquired, they can be put to any number of uses, but a simple skill—the ability to process information in a certain way—

itself has no political implications (Davis, 1989; Gutstein, 2006). The paradigm itself presents mathematical knowledge as a commodity to be dispersed, for the good of both the student and the society. In so doing, the paradigm constrains questions that can be asked about justice, limiting us to notions of distribution.

A recent trend in mathematics education research is to conceptualize mathematics as a technology of thought. The idea is that while not a physical object, mathematical concepts are much like tools in the sense that they can be applied to the world in ways that help us get along and solve problems. This technological view tends to cast the tools themselves as apolitical—they can be put to certain uses, which might have social and political implications, but the decision rests with the user.

Kurt Stemhagen and Brian Warnick (2007) argue that mathematics is rightly understood as a technology, but that technologies are never neutral. Every tool is optimized for a certain purpose, they point out, and a technology’s design implicitly lends itself to certain forms of behavior. A hammer and nail, for instance, may not have an opinion about how I use them, but there are uses I can put them to that will be tremendously more successful than others. In this way, the hammer and nail themselves guide my actions. This is not to imply any bias on the part of the hammer maker, but to point out that technologies influence our deliberations and how we approach problems and tasks. Tools guide our actions in the ways that they can be applied to the world. In the same way, technologies of thinking encourage certain process and information gathering and discourage others.

We ask students to use these conceptual technologies to construct models of the world that only account for quantitative and measurable characteristics, and that yield

singular and unambiguous solutions to already existing problems. The authors remind us that mathematical tools are useful and productive because they distill information and allow us to focus only on certain data, but that thus distillation comes at a cost. Insofar as math class privileges one mode of decision-making (computational), and one set of criteria

(quantitative) over all others, math education advances particular habits of deliberation.

The context in which mathematics education is conducted (its prominence in the curriculum, and the social value attributed to success in the discipline) means that mathematical thinking is not just one technology among many, it is the exemplar. These habits, however, can serve to oversimplify complex moral situations and render us insensitive to qualitative information and interpersonal dynamics. Stemhagen and Warnick make an important contribution, and I think that they do not go far enough. Mathematics education conditions us to reason and think in certain ways that are reductive and insufficent for moral thinking and reasoning about social matters, but that within the problem solving skills paradigm, we can never escape that reductivism.

Because the end of education is the acquisition of certain skills, the only questions that make sense to ask are how to distribute shared resources in a just manner. The overwhelming majority of reform efforts in mathematics education can be characterized as prioritizing or seeking distributive justice: given the impact of a panoply of social conditions on educational achievement, just distribution is complex at best. As Robbie

McClintock (e.g. 2004, 2012) has argued at length, though, in education there are more complex issues of justice at stake than simple distribution of goods. In the coming chapters,

I will argue that the problem solving skills paradigm serves to further entrench damaging

social hierarchies, and suggest that a return to the forgotten half of the mathematicians’ dichotomy is a possible remedy.

Chapter 3: Epistemic Aspects of Social Justice

3.1 Introduction

Chapter 2 set out to show that the current dominant paradigm in mathematics education frames mathematics as a set of problem solving skills. We saw that the paradigm is enforced and manifested by the artifacts of mathematics education, and the exemplars held up as models of good practice and knowing. We saw that one consequence of the dominant paradigm is that it constrains dialogue by limiting the questions that can be asked about teaching practice and about social justice. Chapter 3 sets out to show how this paradigm and engendered pedagogy and policy serve antidemocratic and oppressive social interactions and arrangements. It will show that the problem solving skills paradigm inculcates poor epistemic habits for individuals and it entrenches a model of subjectivity and knowledge that is harmful.

The chapter will consider the relationship between knowledge and social oppression from a number of perspectives, with the overall goals of appreciating the role of the intellect in social justice, and for showing a common theme among the perspectives. A conclusion drawn from the bodies of literature presented here is that social justice requires people that can reconsider value judgments, rather than just working within the frameworks they inherit. As we will see, empowerment of oppressed people and ameliorating harmful practices of oppressors requires changing the terms on which conversations are taking place. The chapter begins by taking up the Enlightenment era promotion of individuals thinking for themselves. Though this tradition is rich and extensive, Immanuel Kant’s short essay “What is Enlightenment” will stand for the bulk of it.

The idea that liberation and complete citizenship, and ultimately the functioning of the

state, requires individuals who are autonomous thinkers has its roots in in the work of

Kant and his contemporaries.

The initial connection between thinking for oneself and social oppression will be addressed through the work of Maxine Greene. Notice that there will be echoes of the

Freirean framework outlined in chapter 1, as Freire and Greene share a concern for the role of the intellect and the creation of knowledge in overturning the oppression of people.

Greene offers finer detail and outlines more specifically the capacities she sees as central to overcoming oppression. In particular, she will identify the capacity and willingness to change the terms on which debates are happening in order to make room for new possibilities. Greene will show us that the sort of thinking that makes social change possible is altogether different from what is cultivated in the current climate in mathematics education. The chapter will then consider the roles and responsibilities of socially powerful or privileged people in mitigating oppression. A number of approaches will be considered, whoch all share the following theme: individuals who find themselves in positions of social power need intellectual virtues, not just cognitive skill, to mitigate their perpetuation of oppression.

Some of the comments from the introduction bear repeating at this point. One objective of this dissertation as a whole is to make a contribution to socially oriented mathematics education by reconciling the perceived conflict between social goods and intellectual goods. As I stated earlier, this conflict limits our thinking about mathematics education as it inherently demands that we choose one priority over the other. But again, mathematics education is unavoidably both social and intellectual in character. Feminist social epistemology and the other traditions explored in this chapter thus have something

to offer mathematics education. The work cited here can help us explore the interplay between the social and the intellectual while maintaining a commitment to both.

3.2 Thinking for Freedom

There is an epistemic aspect of social justice. On an individual level, this means that being a good thinker is integral to being a good citizen. I take “good citizenship” to mean contributing to and promoting the conjoint living of people in a society. Two aspects are important to articulate: the ability to participate fully, and restraint from impeding others’ participation. I take it as given that promoting inequality or the marginalization of some members of society, or actively working to disenfranchise others is bad citizenship. It is not necessary to make the stronger claim that there is an imperative to meet some minimum standard of proactivity, just that acting in ways that disenfranchise others is inherently antidemocratic. Many of us assume that we are acting neutrally in cases where,

I will argue, we are not. In fact, the assumption that we can speak from a neutral position is frequently the very problem at issue.

The link between independent thinking, individual freedom and flourishing, and a healthy public is not new. In 1784 Immanuel Kant called upon his fellow citizens to “Sapere aude! Have courage to make use of your own understanding!” (1999, para. 8:35). Without the ability and courage to act on one’s own understanding, one remains dependent on the direction of others—essentially a minor. When the public is made up only of minors, it can remain yoked by guardians who wish to remain in power, thus subject to tyranny and oppression. When individuals are called to think freely, however, the mentality of the public can change, forcing the government to accommodate. The idea that a free society relies on independently thinking citizens, and that a primary role of schooling is to

inculcate independent and autonomous thought has persisted through the centuries and continues to inform conceptions of education. A surfeit of books over the latter half of the

20th century took up this theme, warning of a public incapable of sufficiently independent thought (e.g. Bauerlein, 2008; Frank, 2004; Gore, 2007; R. Hofstadter, 1966; Jacoby, 2008;

Shenkman, 2008).

Maxine Greene complicates the enlightenment era ideal by offering a more nuanced interpretation of the kinds of thinking that contribute to freedom and democracy.11 In particular, she argues that self-determination, autonomy, and liberation from restrictions inevitably favor those already commanding social and monetary power. As a result, cultivating a free and democratic public requires more than encouragement to follow

Kant’s dictum. Greene tells a long history of the first part of this imperative and the role institutionalized education has had in promoting it. Herein, she recalls a tradition of advocating for freedom in thinking about philosophy and education that is distinct from the quest for autonomy, self-determination, and independence with which it is often conflated in public discourse. In The Dialectic of Freedom (1988) she cites a wide range of thinkers— from John Dewey to Hannah Arendt, from Michel Foucault to Jean-Paul Sartre to Karl

Marx—and finds a shared concern for the intellectual in the concern for human freedom.

She writes that “there is general agreement that the search for some kind of critical understanding is an important concomitant of the search for freedom” (1988, p. 4). Greene defines this critical understanding as requiring the imagination to think beyond the present situation toward a different future. This imagination, in turn, requires a consciousness of

11 Greene’s work is firmly rooted in Dewey’s. While the ways in which Dewey speaks to entrenched inequality can be difficult to reconcile with the register of today’s conversations, Greene develops a number of his frameworks to speak to the possibilities for social justice and parses details of specific historical movements toward equity.

the obstructions in one’s path—denaturalizing the current conditions to open spaces for new arrangements and ways of being. One role of the school is to provoke individuals and empower them to become mindful in this way (p 12), and education has the potential to help people think better about their lives.

Greene continues, examining the stories of those who, in her words, “could never take freedom for granted in this country” (1988, p. 55). Greene demonstrates that individuals' ability to problematize their situations is the prerequisite for social changes that have brought freedom to women, to racial and ethnic minorities, and to recent immigrants. Nineteenth century middle class women, for instance, who were confined to a domestic sphere to compulsory heterosexuality and motherhood could not abandon their social responsibilities and identities to claim independence in the public sphere. Because of the concrete particularities of their lives, they could not follow the examples set by men who had won and established freedoms earlier. They needed to see alternative possibilities for their own lives; viable ways forward toward different situations. For women who had been discouraged from using their minds in this way, such imagination was revolutionary. Before they could engage in radical thinking, however, women had to experience salient obstacles. Rather than perceiving the extant order as natural or ordained, they had to understand it as contingent and potentially transformable.

Likewise, the successes of the Civil Rights era are typically seen as legal triumphs, though they were not exclusively so. The legislative and judicial rulings that secured equal access to public accommodations like schooling, transit, housing and social spaces provided important material benefits. They accompanied a change in the public imagination of the possibilities for engagement and social action. Greene writes that Dr. Martin Luther King

was able to inspire so many so powerfully by evoking an image of a different way forward

(1988, pp. 100–103). Rather than appealing to abstract reason or to law, King inspired and led by appealing to the imagination and the possibility of a different future.

In the tradition of humanist education Greene turns to the school as the locus of possibility for nurturing the sort of thinking that can bring about freedom. Insofar as she has identified a critical awareness that is a prerequisite for enacting the freedoms possible in a democracy, she calls for “a concern for the critical and the imaginative, for the opening of new ways of ‘looking at things’ [which is] wholly at odds with the technicist and behaviorist emphases we still find in American schools” (Greene, 1988, p. 126).

It is not feasible to cultivate imagination and dialogue by conducting a unidirectional delivery of information and skills from teacher to student. Making room for perspectives and vantage points of students and other outsiders, and recognizing the possibility of as yet unknown ways of understanding an experience requires accepting that that no method or accounting can be complete (p 128). Keeping a classroom open to the possibility of new critique and inquiry, and inviting even this engagement, is not a matter of finding a method for settling or solving any issue that arises. Rather, it requires keeping questions open, inviting new response. Greene writes:

In the classroom opened to possibility and at once concerned with inquiry, critiques must be developed that uncover what masquerade as neutral frameworks…. Teachers, like their students, have to learn to love the questions, as they come to realize that there can be no final agreements or answers, no final commensurability (1988, p 134).

The values Greene outlines here are patently at odds with the mathematics paradigm articulated in chapter 2.

Greene’s concern is for the existence of a public realm within which freedom can be imagined and enacted. She turns to important achievements of freedom by oppressed groups to show that the precursor to enacting freedom occurs on the imaginative and epistemic level. If we are to maintain a democratic public, all people must be able to do this imaginative and participatory work. When we are unable to conceptualize oppression as contingent, we are unable to imagine a different way forward or a way out (Greene, 1988, p.

2).

What stands in the way of experiencing our oppression as such is, in many cases, social power. Relations of unequal social standing can skew the available conceptual resources with which we are to make sense of the world, meaning that those in power have the ability to define social situations and problems according to their own perspectives and experiences. This can leave the powerless with concepts and meanings that are ill- equipped for making their experiences meaningful. Feminists in particular have paid special attention to this dynamic, pointing out the ways in which women’s (in)ability to define their own experiences has figured in their subjugation and marginalization. The idea of marital rape, for example, was not recognized as a possibility in the United States before the mid-1970’s. It was understood that consent for sexual activity was given with the marital contract, and thus could not be revoked. Most rape statutes in the United States precluded the prosecution of spouses on these grounds, with variation in their ruling on separation and divorce. The spouses had agreed to be sexually available to one another at all times, it was argued, so a rape within the bounds of a marriage was a contradiction in terms.

When there exists a lacuna in the concepts and vocabulary effective for articulating and making sense of experiences such that one group is rendered unable to make their own lives intelligible to themselves, that group is at a disadvantage that strikes them specifically in their capacity to know. We exist in societies where knowledge production and proliferation has been monopolized by a small tradition of people in power, and only a small number of people have participated in the institutions and practices by which our social meanings are defined. The experiences of those on the outside are left either insufficiently theorized or cognizable only from the perspective of those at the center. In a case of spousal abuse, if the victim does not have the conceptual resources available to name the act as a crime or transgression, she (or he) has no recourse to protest. Lacking the concept to make meaning of the experience keeps the victim from taking steps to stop the abuse.

The Enlightenment era ideal of individual autonomous thought and the possibility for schools to produce such thinkers lingers in the public imagination and in education discourse. Greene shows us that good thinking—rich, imaginative, critical thinking—is necessary not just for throwing off the bonds of tyrannical rulers, but also for making social change toward more equitable and free conditions. For oppressed people to fully participate in public life, they must have access to epistemic resources, and they must be able to do the cognitive work of challenging and rethinking present arrangements.

The distribution of epistemic resources is not simple accident or misfortune. The nonexistence of a concept of marital rape derived from women’s nonparticipation in the formation of the pertinent concepts. Men, who, since rape laws were codified, had written the statues, were served by this conceptual gap. The inability to name and combat abuses,

including marital rape, allowed men to maintain their dominance in homes where it occurred. This also contributed more generally to relations of dominance in society at large. From the perspective of the victimized in this situation, the ability to create new concepts, naming an injustice rather than accepting given categories and arrangements, has been vital to bringing about social change. It is also important, however, to determine whose interests are served by maintaining the status quo. The following section will document research trends that seek to account for the ways in which socially powerful groups maintain and reproduce oppressive arrangements.

3.3 Focus on the Powerful

A concern for the well-being of the marginalized was the initial impulse driving scholarship on entrenched social justice in education. Much work was (and still is) done with the goal of improving institutions and practices for teaching our society’s least well off with the intention of providing social mobility and equality of opportunity. In recent years, the scope of such work has widened to include a focus on understanding and challenging the systems and norms that maintain these social inequities and contribute to the persistent marginalization of certain groups of people. Particular attention has been paid to the habits and practices that allow privileged people to maintain their social status, thus keeping them at the top of the social hierarchy. The recognition that the under-privileging of one group is maintained by an over-privileging of another reveals methods by which those in privileged groups maintain their advantage. This focus on privilege implicates the advantaged in an unjust system, threatening to name their status as unearned. As a result, it can be difficult to broach. There are two different approaches that scholars take, both of which are pertinent to this discussion. One approach focuses on unconscious prejudice,

and the other on ways of being in the world. Both approaches recommend similar remedies.

3.3.0 A note on terminology

Typically, scholars understand privilege to mean the garnering of unearned advantages due to (often socially constructed) characteristics including race, socioeconomic status, gender, and physical ability, among others (McIntosh, 1997). These unintentional benefits accrue regardless of an individual's ability to apprehend that benefits are being received, and cannot be simply denied or rescinded. As these identities are multiple, the majority of us find ourselves able to claim membership in certain dominant groups while being excluded from others. Rather than identifying entirely as marginalized or privileged, complex and intersecting identities abound. As Swalwell

(2013) points out, however, there are people who, “in general, garner unearned privilege in most situations and who more often than not can claim association with elite groups….

[They are] net beneficiaries of privilege” (2013, p. 7). In contrast, she writes, “there are people who dominant groups consistently marginalize and whose actions are severely constrained regardless of context—net malefactories” (2013, p. 7). Going forward, I will use this binary terminology of privileged and marginalized, oppressor and oppressed, remaining mindful of the complex and shifting nature of the framework, and the complicated positions each of us occupies in relation to the binaries.

3.3.1 Unconscious prejudice: Institutional knowledge

One of the most consistent themes in the feminist tradition is that knowledge purporting to be objective and value-free can be shaped by unexamined prejudice. When scientific knowledge is founded on social norms or other implicit values, it can have the

effect of adding legitimacy to already uneven social conditions. Individual prejudices shape everyday interactions and thus play a role in reifying social hierarchies. In some cases, prejudgments are amplified by the role of the knower. Scientists and other knowledge creators, vulnerable to the same prejudices, have the potential to magnify their effects. The model of good thinking as process serves to absolve the researcher of responsibility for such prejudgments with the presented chain of reasoning seeming to legitimate the claim that “any reasonable person would have arrived at the same truths”

3.3.1.1 Carol Gilligan. One such example is Carol Gilligan’s (1982) In a Different

Voice which takes on Lawrence Kohlberg’s (e.g. 1958, 1981) research on the moral development of children. In his widely influential work, Kohlberg established six distinct stages in the moral reasoning of children, ranging from an initial stage of punishment avoidance and obedience to a final stage (which many never reach) in which decisions are made according to universal ethical principles. Kohlberg found that women routinely reach a lower level of development than men do, adding scientific legitimacy to the idea that women are less capable of moral reasoning than men are. In a Different Voice challenged the orthodoxy of Kohlberg’s moral development theory by arguing that the author had inappropriately assumed that characteristically male subjectivity is the pinnacle of moral development. Gilligan presented evidence that females tend to develop morally according to a different trajectory, and tend to exhibit moral reasoning that, while different from typically men’s reasoning, can be described as highly systematic and refined.

Kohlberg’s linear scale placed women at lower levels of development, and consequently they appeared delinquent in comparison to men. The scale was then used in a triumph of circular reasoning as evidence that women are inherently less morally developed or able

than men. Gilligan proposes that the discrepancy signals deficiency of Kohlberg’s model, not of women. She points out that the participants in the initial study were largely male, and gives evidence of gendered tendencies in moral reasoning, contradicting Kohlberg’s initial assumption of a universal staged progression. Gilligan shows how basing a rubric only on male development, and using it as an objective standard by which to measure everybody will result in prioritizing the values and development of only that one group and marking the others as deficient.

Gilligan argues that the flaws in Kohlberg’s work are located in the basic assumptions and the interpretations on which it rests. Beginning with a sample of primarily men was reasonable in a context where the autonomous and independent man was the paradigmatic ethical agent. That women systematically scored lower according to the analysis made sense in a generation when women were generally assumed to be less reasonable. The mathematization of Kohlberg’s work adds to its purported objectivity and legitimacy as a measuring device, which in turn adds fuel to the rubrics’ low estimation of women’s capacities for reasoning.

Gilligan’s critique of Kohlberg is not unique in the field of science studies. There have been many critiques of similar structure that show how problematic initial assumptions (about racial hierarchies, about gender roles, or about innate qualities or supremacy of different groups of people) are given legitimacy by a misappropriation by the scientific process. 12 The critiques, like Gilligan’s, largely are not of a scientific nature, in that they fall outside the purview of the scientific method.

12 For example Elisabeth Lloyd (1993)has shown that when evolutionary biology was dominated by men, it was assumed that female sexuality is always and only related to reproduction. This resulted in inadequate and inaccurate understandings of women’s

Gilligan’s detractors have been many (e.g. Sommers, 2001; Weisstein, 1993), with the majority being complaints that the work is not statistically sound. Gilligan’s response now, as then, is that what the book indicates is something else—something not necessarily quantifiable. If you ask questions of how much, how many (the mathematical questions), you can do a statistical study. But those were not Gilligan’s questions (Graham, 2012). As she writes retrospectively in 2011, the book calls for a change in the terms of the conversation, and a challenge to the idea that the apex of moral maturity is detachment

(Gilligan, 2011). The questions Gilligan raises are ones which mathematics will never be able to answer. Instead, her critique is directed to the epistemic responsibility of the researcher, challenging the legitimacy of his ideals and criteria for moral development.

It is widely acknowledged by social psychologists that group stereotypes influence perception, interpretations, and behaviors toward members of that group, and that these responses operate beneath the level of consciousness (Devine, 1989; Eberhardt, Goff,

Purdie, & Davies, 2004; Greenwald, McGhee, & Schwartz, 1998) It is worth articulating, however, the link between privately held judgments and social responsibility and moral culpability, as the realm of thought is typically considered to be exempt from judgment as it does not directly cause any harm. I offer two accounts on which privately held prejudice inevitably shapes action in the public sphere, and is thus socially relevant. Differently put, social injustice is maintained in part by knowledge practices of those in power. What

anatomy, sexuality, and reproductive health. Despite evidence to the contrary, she argues, the inability to differentiate between reproductive function and sexuality resulted in empirically inaccurate data. She presents a number of studies on the female orgasm that come to empirically disprovable results because of their underlying assumptions. Had these researchers asked women about their sexual activity and experiences, their data would not have been flawed in this way.

follows are two accounts of mechanisms by which unjust social status is reproduced by individually held, often unconscious prejudice.

3.3.2 Unconscious prejudice: Individual level (Credibility)

It is not only people who are involved in producing public knowledge that must be responsible with their prejudices; our individual interactions can be influenced in ways that perpetuate injustice, as well. Miranda Fricker offers a framework within which to understand how individually held and often unconscious prejudice contributes to widespread social inequality. She approaches the matter as an epistemologist, and articulates ways in which social identity and complex power relations impact and are impacted by the epistemic landscape.

In daily life we are accustomed to receiving testimony from others and relying on it to function. My girlfriend tells me she has a dentist appointment, the bus driver tells me what the fare is, or the weather person tells me to expect rain; in each case I decide whether or not to believe the speaker and react accordingly.13 Fricker argues that

13 Epistemologists interested in testimony typically locate themselves in relation to inferential accounts and non-inferential accounts. Inferentialists argue that in order to believe P from being told P, the listener creates an argument for justifying P on her own. The testimony itself does not cause the hearer to believe P, it is merely an impetus to investigate and decide for herself. Non-inferentialist accounts figure that under normal conditions, the testimony itself causes belief, meaning that we operate with a default acceptance of what we are told. Fricker points out that these inferential theories tend to require too much from hearers, as we must acquire independent justifications for the truth of testimony before it is accepted as reliable. Non-inferential accounts tend to be too lenient, taking testimony as its own justification, and seem to bestow on the hearer no culpability for what is believed. Fricker takes a phenomenological approach to outline something of a hybrid approach to testimony. She writes that we generally communicate in ways that feel immediate and direct, but switch to more critical modes when a reason for doubt is presented. This means that the former mode, while it seems effortless and direct, is actually imbued with low-level critical perception, which can be made conscious and operationalized should a situation warrant it. In most cases, perception of the speaker himself is the grounds for confidence in or skepticism of what is said.

prejudgments of character, which are rooted in and shaped by public stereotypes, enter into the calculation of whether or not to trust a speaker’s testimony, in ways that threaten both the knowledge of the hearer and the well-being of the speaker.

Testimonial injustice occurs when credibility is inappropriately withheld due to prereflective prejudice or reasons having to do with identity. “Social imaginative ideas of

‘Negro’ or ‘woman’ distort the hearer’s credibility judgment, and this operation of identity power14 controls who can convey knowledge to whom and, by the same token, who can gain knowledge from whom” (Fricker, 2007, p 90). The primary wrong here occurs at the moment the credibility deficit is operationalized. When a hearer fails to attribute credibility to a speaker, the speaker becomes someone about whom things can be known, rather than a knower in herself. “The moment of testimonial injustice wrongfully denies someone their capacity as an informant,” Fricker writes, “testimonial injustice demotes the speaker from informant to source of information, from subject to object. This reveals the intrinsic harm of testimonial injustice as an epistemic objectification: when a hearer undermines a speaker in her capacity as a giver of knowledge, the speaker is epistemically objectified” (Fricker, 2007, p. 132-3). Fricker points out that the action of inappropriately reducing the credibility of a speaker is itself an injustice.

There are cases where speakers garner unequal levels of credibility, where no injustice occurs. In a conversation between a student and a teacher, for example, or a doctor and a patient, the earned authority and credentials of one participant legitimize a higher level of credibility. Fricker is careful to point out that what she is interested in are

14 Fricker defines identity power as “the strongly intuitive idea [of a] capacity we have as social agents to influence how things go in the social world” (p 19). She points out that identity power can operate actively or passively, and intentionally or not.

instances where credibility is inappropriately withheld, or withheld on account of illegitimate justification. Prejudgments made on account of gender, race, and other immutable characteristics are not a reasonable basis for dismissing a person’s testimony.

Insofar as the capacity to be a knower is an essential human quality, incursions on that capability, Fricker argues, should be considered primary injustices. The notion of epistemic objectification goes beyond the immediate interaction and affects a person’s constitution as a knower. While an isolated instance of testimonial injustice may not do harm to the wronged party, consistent and systematic objectification is damaging to the self-understanding and the capacity to speak authoritatively on one’s own experience.

3.3.3 Objectification and direction of fit

The epistemological idea of “direction of fit” is crucial to understanding how this is so.

The normal direction of fit, in epistemology, has come to indicate that the belief conforms to the world—that the subject believes P because P is the case. Most of our mundane beliefs about the world have the normal direction of fit: I believe that I have two siblings because evidence shows this to be the case; were the facts to change, my belief would change. Instances where P is the case because the subject believes it—where the world conforms to the belief—exhibit an abnormal direction of fit (Langton, 2000). As an example,

Lorraine Code suggests a situation where “If I choose to believe that you like me and behave toward you in the assumption that you do, my acting in this way may quite possibly be instrumental in bringing about precisely the kind of liking I believe exists” (1987, p. 80).

The norm of Assumed Objectivity (Haslanger, 1993) governs much of our reasoning about the world and how to navigate within it. It is our primary default setting, and functions properly when conditions are normal. Langton(2000) points out, though, that in

cases of social relations, the assumption that conditions are normal when in fact they are not helps turn prejudice into objectification, reifying and entrenching relations of power.

Langton, asserts that certain beliefs about other people are not normal because they do not exhibit the normal direction of fit. While the physical world is staunchly resistant to my thought about it, humans are not. People respond to one another and the expectations for their behavior. This phenomenon becomes a mechanism of oppression when the social world can come to be shaped by perception. For example, some men's beliefs about women have the potential to be self-fulfilling. Because some men see woman as subordinate, they can (intentionally or not) manipulate the environment, perception, and judgment of women so that women’s available options are diminished. Women are treated (by both men and women, both intentionally and unintentionally) according to how they are perceived. This treatment includes how they are talked to and about, and how their language and actions are interpreted. Marilyn Frye writes that “how one sees another and how one expects the other to behave are in tight interdependence, and how one expects the other to behave is a large factor in determining how the other does behave: (1983, p. 67).

This is to say that socially powerful people seeing women as subordinate and operating under the assumption of objectivity makes women subordinate.

While testimonial injustice is a wrong done by one person to another, objectification is a consequence of a plurality of interactions. The individual interaction is problematic insofar as it is complicit in a larger pattern. The assumption of objectivity, in the context of unequal social positions and in conjunction with prejudice has the unintended consequence of objectifying women. In acting on the assumption that we are objective, we play our parts in making the world conform to our expectations. Langton writes that

“being objective helps to make [us] successful objectifier[s]” (2000, p. 141). These authors do not take issue with the idea of objectivity or assert that it is inherently detrimental or androcentric. Instead, the point is that assumed objectivity, in the context of existing social hierarchies and prejudice, has negative consequences for women and others about whom prejudices and stereotypes abound.

Fricker’s remedy for testimonial injustice is a heightened reflectivity toward one’s own prejudices. She recommends that we, as hearers, accept that we may be subject to unintentional prejudice, and that we consciously correct for them. Insofar as testimonial injustice is enacted in our pre-reflective assessments of one another as credible witnesses before making a judgment on the content of testimony, it is to these assessments that we have a responsibility to attend. Because our perceptual habits are acquired and shaped by public stereotypes that figure some types of people as unreliable witnesses, and because people in positions of social power in any given interaction have the capacity to silence the less powerful on account of being unreliable, it is incumbent on the hearer to correct for the effects of prejudice on judgment. This action is distinctly reflexive, requiring the hearer to adjust her own judgments to achieve an appropriate level of credibility.

Fricker’s account is important because she shows that what is needed to overcome this particular brand of injustice is not just cognitive skill, it is intellectual virtue.

Individuals in positions of social power, who have the potential to do harm through their knowledge practices, can work against their inclinations by developing habits of reflectivity.

Thinking skills may aid in this endeavor, but as Fricker makes clear, what is needed is not only the ability to identify prejudice, it is the inclination to do so.

The stance of the mathematical knower is the paragon of the non-situated distanced standpoint. This is to say, mathematical practice is conventionally thought to be ahistorical ad uninfluenced by the identify of the mathematician. To do mathematics, it is understood, is to shed the contingencies of one’s own individuality, and take up a non-situated standpoint. This position is integral to doing mathematics and is one source of mathematics’ power as a conceptual scheme. As a social position, however, the assumption of objectivity is harmful to people in entrenched positions of inequality. 15 When shallow mathematical thinking is extended as an epistemic exemplar, students are being taught to operate under an assumption of objectivity.

3.3.4 A second approach: Ways of being and phenomenology of privilege

A concern has been raised of Fricker’s account over the extent to which we are capable of becoming conscious of and controlling or excising our prejudices (Alcoff, 2010).

Fricker’s remedy depends on our ability and willingness to accept that we harbor prejudices. For the most part, however, westerners see themselves as not being affected by group stereotypes, and thus not needing to do the work of compensating for them. Even if we are able, through critical reflection, to rid ourselves of the prejudices and stereotypes that negatively influence our interactions and, if we are able to hear one another as credible witnesses, a final group of scholars argues that it may not be enough. Turning now to racial domination, scholars concerned with white privilege argue that even when white people

15 Catherine MacKinnon (1987) argues that objectivity is the stance of the male knower, and that consequently, and that bodies become feminized—made female—by becoming the object of the masculine gaze. To be pithy, objectivity objectifies. That is to say, MacKinnon argues that the way in which knowledge is traditionally pursued and understood actively harms women. Sally Haslanger and Rae Langton are unwilling to commit to such a strong assertion, but agree that there is an impulse worth salvaging. Both offer detailed interpretations of MacKinnon that figure the assumption of objectivity as the detriment.

work to disrupt structural privilege, it can have the effect of reifying it. These scholars argue that a focus on prejudice is not enough, even if we try to manage unconscious prejudice.

A second approach to unconscious prejudgment at the individual level focuses less on the individual and highlights broad social patterns that organize people hierarchically.

This approach is particularly well developed in the study of racial oppression as it manifests in white supremacy, privilege, and complicity, and has come about in response to the realization that despite living in societies deeply divided along racial lines, the great majority of westerners consider themselves to not be racist and to not harbor racial prejudices (Bonilla-Silva, 2003).

Peggy McIntosh’s seminal 1988 article “Unpacking the Invisible Knapsack” describes white privilege as an invisible knapsack of unearned resources that white people carry with them. Entries like “I can criticize our government and talk about how much I fear its policies without being seen as a cultural outsider” and “I am never asked to speak for all of the members of my social group” are still salient for making visible the sorts of benefits that accrue to white people on account of the color of their skin (McIntosh, 1997). The simple fact of making whiteness a tractable characteristic is noteworthy, as the default position is that whiteness, like maleness, is the default position for humanity: it is the lack of a distinct social position.16 Frye writes that “it is important for a member of a dominant

16 Beverly Daniel Tatum (2003) notes that when asked to describe themselves in a few words, white people rarely use race as a descriptor, while people of color almost always do. Tatum argues that we choose those descriptors that differentiate us from the norm, so for white people, whose race is constantly reflected back at them, race does not feel like a salient aspect of identity.

group to come to know s/he is a member of a group, to know that s/he is only a part of humanity” (Frye, 1992, p. 117).

McIntosh’s knapsack metaphor is limited, however, because it implies that race privilege can be renounced. More recent scholarship is in general agreement that this is not the case—that insofar as we are always already raced and gendered, we cannot abdicate these positions. What is necessary is not striving for objectivity or a certain neutral subjecthood, but to acknowledge and work against the ways in which our being in the world reanimates a system of oppression. While this idea is clear in McIntosh’s original piece—“my culture gives me little fear about ignoring the perspectives and powers of people of other races”, “if I declare there is a racial issue at hand, or there isn’t a racial issue at hand, my race will lend me more credibility for either position than a person of color will have”—it is often ignored. The more concrete items on the list are far easier to grasp, and thus become the focal point of discussion (Applebaum, 2010).

Frye (1992) describes the totality of these habits as “whiteliness,” akin to masculinity.

The term ‘whitely’ to describe ways of being has become standard, to mean habitual ways of being that assume the universality of a white perspective on the world. Whitely ways of being are critiqued on the basis of what Applebaum calls “the white complicity claim.” This widely held claim, she writes “maintains that white people, through the practices of whiteness and by benefitting from white privilege, contribute to the maintenance of systemic racial injustice” (2010, p. 3). Further, traditional conceptions of subjectivity and epistemic ideals fail to expose white complicity and contribute to the normalization of denials of complicity that protect systematic racism from being challenged (2010, p. 4).

3.3.5.1 Certainty. One of the privileges granted by whiteness is the license to decide which speech acts should be granted an audience, and which should be taken seriously.

George Yancy (2008) describes delivering a lecture on the “interstitial ‘race’ dynamics” manifested in a specific personal experience, and being met with a call of “bullshit!” from a white audience member (p 227). The response, as Yancy points out, functioned to conceal his perspective and his argument, excluding it from the discourse. Testimony that challenges white supremacy is met not with argument or rebuttal, but with disqualification, which allows the white listener to not engage and to not risk feeling responsible for being complicit in racist social practices.

The assumption that a white person can achieve a neutral, raceless subjectivity from which to judge and be judged, is not only disingenuous, it is damaging. Alan David

Friedman points out that harm looks different when it is described from the victim’s perspective than it does from the perpetrator’s perspective. From the victim’s perspective, there is a need to describe the conditions of social existence, and to see the harms of an unjust system. From the perpetrator’s perspective, the focus is on individual actions and intents, which means that systematic, unintentional, and relational harms are unarticulable.

(Freeman, 1996, p. 29). If the assumption is that the perspective of the perpetrator is the only authoritative position, or that it can provide a universal descriptor, these systematic effects are rendered invisible. If one’s epistemology assumes that knowers are interchangeable, these differences will escape consideration on account of their epistemic irrelevance.

Looking around in the world, I can make any number of observations and judgments that do not seem to be raced or gendered, or in any other way impacted by my identity.

Observing patterns in the clouds, the feel of speed on a bicycle, or the sound of rain on a rooftop do not seem to be observations that I make specifically as white person. Likewise, discerning the tensile strengths of different cables or the seaworthiness of a boat are judgments that I should be making in my capacity as a knower, and not on account of my gender. One approach is to argue that all judgments and knowledge, even these just described, are socially situated. Standpoint Theory, an influential trend in feminist science studies promoted by Sandra Harding among others, is guided by this idea. For the sake of this project, it is a perilous path to follow as it quickly leads to claims that we should want to pursue, for example, a feminine mathematics or a subaltern mathematics, which is opposed to the established canon. While interesting work has been done in this area (e.g.

Frankenstein, 1997), it is counter to the intentions of this dissertation.

Attending to the role identity plays in observation and judgment does not require accepting that all decisions hinge on positionality or on our individual characteristics. The mistake that we too often make is to assume that these non-identity-related judgments should be the model for thinking about social and human matters. Returning to terminology introduced earlier, judgments become problematic when we do not recognize that they do not adhere to the normal direction of fit. It is not necessary to claim that all beliefs are abnormal in this way, and have the power to shape the world in their image, only to recognize that some are, and to develop the kind of nuanced judgment to discern those that do from those that do not.

A number of scholars (e.g. Ahmed, 2007; Applebaum, 2010; Blum, 2008; Hytten &

Warren, 2003; Mayo, 2004; Yancy, 2008) have reflected on the difficulty of engaging privileged (mostly white) students on issues of privilege and complicity by taking up ideas

of positionality, ignorance, and responsibility. A common theme is that oftentimes, white students refuse to engage with such material altogether. Hytten and Warren (2003) document this effect, arguing that privileged students have at their disposal a number of deflection strategies that enable them to direct conversation away from their own complicity and responsibility. Practices like focusing on their own connections, experiences, or feelings; and appealing to or advocating progress and action over understanding enable students to subtly change the focus of racial issues so that their own culpability is not at stake. Crucially, the authors argue that these are not simple, conscious attempts by students to shut down inquiry, rather they are strategies made readily available by public discourse. There are readily available frameworks for distancing ourselves from having to consider our own implication in an unfair system, which allows us to maintain our own moral purity and isolation.

But these distancing strategies aren’t just a refusal to consider their own complicity.

Long before a student steps into a college course on race and power, or on male privilege, she has been brought up in a system that teaches her that good knowing is appositional, and that responsible knowing derives from stripping away the trappings of particularity.

Distanced objectivity is not just a strategy for deflecting responsibility for complicity, it is the standard that has been held up for the entirety of her educational career. That the student has been taught, explicitly and implicitly, that social position is epistemically irrelevant, and that the most reliable knowledge is attained by reasoning through her own experience.

3.3.5.2 Uncertainty. A common concern in critical whiteness studies is that one of the tenets of white privilege is the ability to define what can and should be known in

and about any given situation—the license to determine the terms on which discourse will take place. Mayo writes that:

“Whites can make assumptions about their welcome, their dominant knowledge, the acceptability of their practices, and rarely need to worry about being challenged because they define the norm. Privilege, in other words, gives whites a way to not know that does not even fully recognize the extent to which they do not know that race matters or that their agency is closely connected with their status.” (Mayo, 2004, p. 309)

Theorizing whiteness and working to critique it as a mechanism of oppression can have the unintended effect of redeploying its privilege by fixing whiteness as the symbol around which attention is focused, excluding other voices and perspectives. In particular, she argues, the tendency in the study of whiteness is ultimately to recenter white agency, by making it the object of study. The critical focus on whiteness, or the attempt to repair whiteness or remake it, further establishes whiteness as the central symbol around which discourse is organized. Shannon Sullivan (2006) introduces a helpful concept: “ontological expansiveness.” By describing white people as ontologically expansive, she writes that they have the tendency to “act and think as if all spaces—whether geographical, psychical, linguistic, economic, spiritual, bodily, or otherwise—are or should be available for them to move in and out of as they wish…. The self assumes that it should have total mastery over its environment” (2006, p. 10).

Rather than further critique of whiteness, or a newer, better model that white people can strive to embody, what is needed is for the dominant group to turn away from the goal of certainty. Certainty and knowing are the typical goals in education, and most certainly in mathematics education. Mayo argues here, though, that socially, uncertainty is a more beneficial habit to cultivate. Mayo (2004) recommends centering a practice of uncertainty. She writes that “suspicion, uneasiness, and careful interpretive strategies can

be useful to white people attempting to work against racism” (2004, p. 308). In Mayo’s argument the motivation is personal and related to self knowledge—for a white researcher, reaffirming the white subject in the abstract is personally reaffirming, while to dissolve it can be existentially dangerous. What is needed is intellectual courage, which provides motivation to decenter the subject, face that personal risk, for the sake of better racial knowledge and accounting for more diverse subjectivities.

Yancy (2008) develops this idea further, arguing that white people should not only be open to uncertainty, but that the moments that call us to account can be most valuable for continuing to grapple with undoing whiteness. Yancy argues that, contrary to the implication of the knapsack metaphor, practices of whiteness are always with white people.

Whiteliness ambushes well-meaning white people when they let their guards down, and it flows through them. Even those who dedicate themselves to understanding and undoing white privilege and complicity unintentionally reassert it, unable to be aware of all of the ways in which whiteness asserts itself. Rather than resigning oneself to the seeming futility of undoing privilege, Yancy argues that we should be grateful for the moments that reveal our complicity, for they are what make it possible to maintain vigilance, and understand better.

Too often, people in positions of unearned power, upon receiving evidence that it is unearned, respond with denial. By denying involvement in an unjust system, individuals can escape the feeling of responsibility (Yancy, 2008). This avoidance takes many forms, but often relies on the notion of “the good white person.” The well meaning white person, upon learning that she is complicit in a dynamic of inequality, often seeks to separate herself from the real villains by calling them out as racist. She seeks separation from “those”

whites for the sake of repudiating them, and achieves self-absolution by bringing a different, more racist type of white person into focus. The public immolation of celebrities or politicians, for example, who are revealed to use racially offensive language behind closed doors (e.g. Paula Deen in 2013 and Donald Sterling in 2014) allows the majority of white people to feel morally innocent in comparison. The focus on “bad whites” reframes racism as a matter of individual prejudice and ill-will, once again rendering invisible the ways in which ordinary, well-meaning white folks are complicit in systems that maintain racial inequality.

What is needed is to forge new relations between groups and reorganize power imbalances. To do that, white people have to give up some of the certainty they have come to expect, opening a space for people of color to define problems from their own experience, and to look for and take seriously the instances in which they are ambushed by their own whiteness, as these are the experiences with the potential to transform.

3.4 Conclusion

This chapter has endeavored to show that there is an epistemic dimension of social justice, and that knowledge practices play an important role in sustaining oppression and in dismantling it. In contrast to Kant’s call to attend most importantly to one’s own understanding, I have made the case that we must also attend to our responsibility to others and to seek to know responsibly. I have shown that a well-functioning public requires citizens who not only think independently, but who think responsibly. By extension, schooling that takes social justice and democratic education seriously has an interest in promoting epistemic responsibility and socially responsive knowledge practices.

Insofar as one commitment of schooling is the creation of a democratic public, and insofar as participation in a democratic public requires access to conceptual frameworks with which to make sense of one’s experience and being received as a credible witness, schools and educators have a mandate to intervene to counteract epistemic justice where possible. Further, being that schools are by nature the location where we teach children how to be knowers, schools are the natural place for this intervention to occur.

On the surface, the proposition of epistemically responsible teaching, or teaching students to be responsible knowers is innocuous, even banal. In the mathematics classroom, however, it is patently at odds with current practice and with the stated goals of mathematics education. Crafting a socially responsive mathematics curriculum seems at the outset to be a contradiction. Solutions that have been advanced in other disciplines— incorporating multiple viewpoints and striving for inclusivity, or decentering the dominant narratives, for example—cannot get much traction in a discipline where the ideal is the rejection of subjectivity and positionality.

The ways of being advanced in this chapter as remedies for entrenched social injustice—adjustment for prejudice, accepting uncertainty, curtailing or reining in ontological expansiveness—are not compatible with the problem solving skills paradigm, which promotes skills and processing as paragons of good thinking. Skills oriented thinking posits a socially innocent thinker, framing appositional thinking as the apex of good thought. White privilege is maintained, in part, by this expectation that the best thinking ignores positionality. Problem solving thinking renders systemic oppression invisible, which is troubling because the way it survives is by staying concealed from those who perpetually enforce it. Problem solving thinking fails to expose institutional and

structural power imbalances and normalizes the denials of complicity that insulate injustice from critique. The assumption that socially innocent thinking is desirable or even achievable sets the well-meaning socially powerful person to the task of disavowing their social position to try to achieve it.

If we conduct our schooling such that students are not taught to be epistemically creative, so that they can adapt and expand existing concepts to fit the world they inhabit, and to forge new ones where gaps exist, we will raise generations ill equipped to make social change. Furthermore, the lack will disproportionally affect those already marginalized, simply because the wealth of concepts we inherit has been skewed to exclude them.

It follows from the arguments presented here that an antipatriarchal or an antiracist pedagogy is not limited to avoiding racist and antifeminist materials and methods. Because schooling has a deep impact on the knowledge habits and values we develop, which in turn have a tangible impact on inequality and oppression, the knowledge practices cultivated in classrooms are integral to a socially just and democratic education. Specifically what is needed to disrupt persistent oppression is not mechanistic and skills based thinking. From each of the perspectives outlined here, we have seen calls for changes in disposition and habit. This change in focus brings us away from skills type learning and recommends that we turn attention to the development of intellectual virtues. Of particular interest will be intellectual humility and courage.

Though I have been unable to find any meaningful engagement between research in mathematics education and in intellectual virtue, I intend to show that mathematics has the potential to be an ideal context for developing this virtues and habits of mind. Having

argued in chapter 2 that the current paradigm in mathematics education breeds mechanistic and procedural thinking, I will show in chapter 4 that professional mathematics is not characterized by skills thinking, and that in the work of higher mathematics, we can see evidence of intellectual courage and humility. First, however, I will offer a short orientation with respect to intellectual virtues in education, and attempts to promote them.

Chapter 4: Intellectual Courage and Humility

4.1 Introduction

In order to show that intellectual virtue can be a helpful framework for thinking about mathematics education, I will offer a brief overview of the literature as it relates to the current conversation. Recall that the focus of chapter 2 was on detailing the extant paradigm in mathematics education, which characterizes mathematics as a body of skills to be mastered. This paradigm, I argued, is supported by a superficial conception of mathematics. Chapter 3 argued that an important element in social justice education is the eradication of such process-oriented thinking. We should want people (including ourselves) to develop a habit or reconsidering value judgments and recasting the terms of conversations to open space for new possibilities. Doing so, however, can be difficult, because it oftentimes requires individuals in positions of social power to reconsider their own status and certainty for the sake of an other. This short chapter aims to elucidate contemporary understandings of intellectual virtues, humility and courage in particular, and to place them in respect to educational theory. The chapter lays the groundwork for arguing for the importance of these virtues in mathematical practice in chapter 5.

Intellectual virtues have received attention in philosophical circles recently as many epistemologists have become interested less in the verifiability of knowledge and more in characteristics of good knowers. Put differently, some attention has been turned away from the task of identifying which beliefs can or should be considered knowledge, which is much of the focus of traditional epistemology, and the traits and dispositions of knowers have ben brought to the center. The language of virtue is a good fit for this task, as it puts the focus on improving practice and acknowledges the interplay between the intellect, the

will, and the emotions. Furthermore, it is staunchly resistant to codification and systematization.

The initial appearance of the idea of epistemic virtue in education literature came as an effect of the promotion of epistemic virtue and responsibility in general. If epistemic responsibility is important, and if school is the place where we learn our habits of knowing, then it follows that in school we should turn attention to the development of ethics of knowing and belief. Code (1987) and Degenhardt (1998) are excellent examples of this reasoning. Ultimately both are arguing for the importance of epistemic virtue in moral life

(Degenhardt calling for an ethics of belief and Code calling for epistemic responsibility), which leads them to the role of the school in sustaining epistemic ideals. The majority of the work advocates teaching students to be epistemically virtuous, presenting virtue epistemology as an alternative to critical thinking. Two main concerns are raised: first, promoting critical thinking as such is simply not effective, and second, that it has unintended and undesirable social outcomes (e.g. Alston, 2001; Hyslop-Margison, 2003).17

The framework of intellectual virtue does constitute an intentional return to an

Aristotelian framework, which has a number of consequences. In particular, if courage is taken to be a virtue, it is assumed that: it is a mean between two vices, it is related to and made possible by other virtues, and that it is a disposition, developed not just by abstract learning, but by practice and emulation. Following MacIntyre (2007) there has been an understanding that our conception of and relationship to virtue and virtues has changed

17 This characterization does not go unchallenged, however, Bailin (2003) argues that that the dispositional component that Hyslop-Margison finds in virtue epistemology is in fact present in philosophical theories of critical thinking. She also questions whether a program of virtue epistemology is rigorous enough if it focuses solely on qualities of character. “Disposition by itself does not guarantee successful performance” she writes (Bailin 2003, 328).

since Aristotle’s time, on account of changes in basic subjectivity, the good, and human flourishing. Contemporary proponents of intellectual virtue consider Aristotle (1998) to be the progenitor of their work, but in different ways. Some consider the intellectual virtues to be equal counterparts to the moral virtues that Aristotle addresses at length, and some consider the intellectual virtues to be in service of the moral virtues. In both cases, attention is turned to the capacities, habits, and dispositions that contribute to intellectual flourishing (Baehr, 2004).

The Aristotelian frame is helpful because it recommends elucidating virtues by examining their attendant vices. In the realm of character virtues, for example, wittiness is defined as the mean between boorishness (an absence of wit) and buffoonery (an excess of wit) (Aristotle, 1998). We come to understand the nature of the virtue by pointing out and examining its related vices. In the contemporary consideration of intellectual virtue, the precision of Aristotle’s mean theory has been relaxed, but we still seek to define virtues in part with respect to their related vices. Following up on the discussion from chapter 3, there are two closely related virtues that I would like to outline here, followed by a short discussion of judgment in the intellectual realm.

It is worth mentioning that philosophical work on intellectual virtue can be roughly divided into two categories. The original work focuses on reliability of knowledge, and is thus concerned with effective knowledge gathering capacities—eyesight, intuition, memory, etc. (e.g. Greco, 1993; Sosa, 1993, 1999). A second trend focuses on the responsibilities of the knower, moral and otherwise (e.g. Code, 1987; Montmarquet, 1987; Roberts & Wood,

2007; Zagzebski, 1996), and takes up traits like generosity and humility. This chapter is consistent with the latter group, often called a responsibilist strand of virtue epistemology.

4.2 Uncertainty, Open-mindedness, and Humility

4.2.1 Unertainty and open-mindedness

Recall that one of the prominent themes of chapter 3 was the call to open- mindedness, and to remain open to uncertainty. From each perspective presented, the complacency of certainty allows us to remain in the situations in which we find ourselves.

Greene and Mayo in particular pointed out that displacing the expectation of certainty opens the opportunity for seeing new social possibilities.

As an intellectual virtue, the maintenance of uncertainty that Greene and Mayo recommend is most often called open-mindedness, and concerns the willingness to revise our previously held knowledge in light of new information (Sockett, 2012). William Hare

(e.g. 1979, 1985, 2009) has done extensive work promoting open-mindedness in education.

He describes an open-minded person as one who is “disposed to revise or reject the position he holds if sound objections are brought against it, or, in the situation in which the person presently has no opinion on some issue, he is disposed to make up his mind in light of available evidence and argument as objectively and as impartially as possible” (1979, pp.

8–9). While in common parlance we might consider ourselves open-minded about certain things (“I don’t normally like the symphony, but I’ll keep an open mind”), and not about others (“he is not going to reconsider, his mind is closed on this”), a general tendency toward open-mindedness or uncertainty requires a certain attitude toward oneself as a knower (Sockett, 2012). Acknowledging the possibility that I may be wrong about any number of assumed certainties requires an acceptance, an embrace, even, of my fallibility as a knower. Psychologists remind us, in addition, that as humans we tend to have a

“confirmation bias,” which is to say that we are predisposed to look more favorably upon

information that appears to confirm what we already know. Genuine open-mindedness, as a result must take a somewhat active role in correcting for such prejudgment. Recall

Fricker’s recommendation of a similar approach—taking preemptive corrective measures to mitigate unseen prejudice.

4.2.2 Humility

Hare points out that a call for such a quality might appear trite, but that for the teacher, holding uncertainty implies something of a contradiction. “In order to have something to teach, the teacher must be something of an authority on the subject or topic in question” (1992, p. 2). Relative to a different community, the teacher may be something of a novice, but with respect to the present students, it is the teacher’s certainty that enables her to take the role of teacher. At the same time, however, the zealot, the most certain among us, is widely acknowledged to be ill-suited for teaching, precisely because of this certainty. The result is that teachers come down on one side or the other, and either capitulate altogether to the knowledge of the students (rarely if ever pointing out faulty work or reasoning, and refraining from offering a statement of right), or expecting to never be surprised, and to understand teaching as the transmission of already settled information

(Floden & Clark, 1988).

The negotiation therein has been framed as the pursuit of humility in teaching. As a virtue of character, Hansen (2001) explores the humility of the teacher, illustrating how holding “tenacious humility” as an ideal can help navigate the complex role of the teacher.

Hansen accounts on the one hand for the expertise and intellectual authority of the teacher, and on the other, the acknowledgment that the teacher should be at once a learner. This negotiation is central to his later book, in which places intellectual humility at the heart of a

cosmopolitan disposition. Hansen’s (2011) treatment of humility is of particular interest here, because he addresses the virtue specifically as a necessity for cross-cultural communication. Humility is necessary for the balance of “reflective openness to the new and reflective loyalty to the known” that enables a person to learn from and be changed by an other without becoming subsumed (Hansen, 2011, p. 1).

As an intellectual virtue, humility is rightly understood in relation to its vicious counterparts, of which there are many. Humility is opposed to arrogance, egotism, pretentiousness, presumption, and the ontological expansiveness that was identified earlier. Roberts and Wood describe intellectual humility as “a disposition not to make illegitimate entitlement claims on the basis of one’s superiority, as a relative weakness of desire to be the author of other people’s minds, [and] as a disposition not to ascribe oneself a greater moral excellence than one possesses” (2007, p. 255). Recall that one of Fricker’s concerns was the way in which prejudice drives us to inappropriately devalue some people’s testimony based on their outward characteristics or social location. Intellectual humility is, in part, the tendency to refrain from overvaluing our own perspectives on account of a sense of superiority. Likewise, recall Sullivan’s identification of ontological expansiveness as the tendency for white people to assume that all spaces are available for their habitation and colonization (Sullivan, 2006). Roberts and Wood’s “weakness of desire to be the author of other people’s minds” is, in my reading the absence of this expansiveness. In Hansen’s (2011) terms, humility entails pushing back against a sort or tourist-like orientation, where one expects access to all parts of a different culture and everything is attainable for a price. Again, I read Hansen’s humility here to be a resistance

to Sullivan’s ontological expansiveness, and thus integral to a socially responsive consciousness.

In more poetic terms, Freire describes intellectual humility as “the quality of recognizing—without any kind of suffering—our limits of knowledge concerning what we can and cannot do through education. Humility accepts the need we have to learn and relearn again and again” (Freire, 1985). In the moral sphere, humility is the right relation to appearances and self worth; it is the ability to judge oneself fairly, instead of overly vainly or with self-effacement. In the intellectual sphere, it is subservience to facts and reason. In the words of philosopher and novelist Iris Murdoch, humility is “a selfless respect for reality” (2001, p. 93)

Insofar as this subservience to reason entails willingness to accept one’s own fallibility, and insofar as respect for reality must my selfless, humility is closely related to open-mindedness, and many scholars treat them as the same virtue (e.g. Hare, 1992). I think it is worth differentiating the two, especially as it will be useful for thinking about mathematics education. In acknowledgement of the convention to understand them as a singular virtue, however, I have classified them as two aspects of the same virtue, or as constituting a twofold virtue.

4.3 Courage and Surprise

Bound up closely with calls for humility and open-mindedness is the necessity of a certain type of courage in intellectual matters, including an orientation toward surprise.

The admittance that one is or could be wrong, which is central to the previous virtue, can be a fearful prospect. Being wrong has potential consequences for the self as a whole.

Perhaps accepting that I was wrong about the bus schedule, and revising my thinking to fit

the data is not likely to shake my world view or sense of self. Entering into genuine dialogue with another person, truly open to the possibility that a prejudice will be brought into the open, or that a core belief will be challenged, can be a risky endeavor that has the potential to catalyze a radical altering of my consciousness (Mulryan, 2009). Not every encounter reveals core beliefs to be faulty, or renders the world intelligible, but to engage in intellectual practice with genuine humility and open-mindedness necessitates being open to the possibility. Negotiating that prospect requires intellectual courage.

Insofar as it is the drive to face something that is fearful, courage entails a motivation to not face a given intellectual challenge. The courageous person, however, is motivated more strongly by the love of knowledge to face the challenge. Intellectually courageous actions then have a characteristic motivational pattern: the courageous person is good at acting with aplomb in the interest of significant propositional knowledge, acquaintance, and understanding, for himself and others, in face of perceived hazards of life.

These hazards include, but are not limited to: self knowledge (gaining or losing), knowledge of criticisms of others, acts which are painful to us, looking bad in front of others, believing falsehoods, misunderstanding, misleading others, damaging cognitive powers (Roberts & Wood, 2007, p. 219). The intellectually courageous person will have a tendency to seek true knowledge in spite of perceived risks.

One of the challenges of being intellectually courageous, and of opening up our foundational beliefs to criticism in general, is that it is never entirely clear where they need to be examined. None of us consciously knows our beliefs to be false or otherwise faulty,

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otherwise we would not continue to believe them18. So even with a healthy does of humility and a sense of our own fallibilism, it always comes as something of a surprise when we turn out to have been wrong. Schultz writes that “if fallibility is built into our very name and nature, it is in much the same way the puppet is built into the jack-in-the-box: in theory wholly predictable, in practice always a jarring surprise” (2010, p. 6). What is needed, then, an aspect of intellectual courage, is openness to surprises of this sort—those that reveal us to have been wrong, or our understandings to have been inadequate.

In this vein, Kerdeman (2003) considers the notion of being ‘pulled up short’.

Drawing the term from Gadamer (1989) to describe the experience of disappointed expectations, she finds it to be an integral aspect of learning and understanding in the world. Specifically, when our assumptions or expectations fail to materialize, when what happens is contrary to what we expect, we become conscious of the distance between ourselves and the world. This experience of alienation due to thwarted plans is what

Kerdeman refers to as being pulled up short. Gadamer introduces the term to describe interpretation of text, but Kerdeman is using it more broadly here to refer to interpretation of and absorption in the world at large.

She points out that calls for reflectivity in learning are commonplace, often suggesting that teachers enable students to examine and identify errors in their own thinking, as well as subject it to critique, and proposes that in addition to this conscious

18 James makes this point nicely when he writes of Pascal’s Wager. Pascal famously argued that absent a rational proof, it is rational to believe in God because of the consequences alone. If the reason Pascal is believing in God is on the grounds of a cost-benefit analysis (no loss if it turns out to be false, eternal salvation if it turns out to be true), it engenders a feeling of bad faith, as this is not what we mean when we say we believe in something. We believe things that we genuinely thing to be true or right, that is, we believe things because they are so.

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activity, there is a phenomenon whereby we are caught unawares, in which our thinking may be challenged by something wholly unexpected. She does not propose that this is a more accurate description of the way that thinking is routinely challenged, but rather that it is an experience that is central to the task of learning which is not captured by calls for conscious reflectivity. “When we are pulled up short, she explains, “events we neither want nor foresee and to which we may believe we are immune interrupt our lives and challenge our self-understanding in ways that are painful but transforming” (2003, p. 294). She draws from Gadamer’s characterization of the experience in Truth and Method to describe the way in which certain unexpected events can call into question underlying assumptions and challenge us to see the world and ourselves in new ways. Central to this experience, she argues, is the element of surprise, the precisely the fact that it is not possible to bring it about by practice or premeditation. Instead, what is necessary is a cultivated disposition that is open to it.

It is important to note that Kerdeman’s description of being pulled up short is not identical to more common calls for surprise. Adler (2008) and Garrison (2009) both address the pedagogical possibilities of surprise, both pointing out that the teacher can orchestrate surprising experiences to direct attention to areas where students’ expectations are at odds with the truth, or to arouse interest and create openings in seemingly closed structures. In both of these accounts, however, surprise is posited as a means to learning abut some thing in the world, whereas the experience Kerdeman describes is educative in itself. It is not the case the being pulled up short is useful for directing attention toward flawed assumptions, but rather that the experience of having expectations disappointed, alienation from the world in this way, is an experience of our

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own limitations. It is something of a gain in that it can bring clarity, honesty, or depth to self-understanding, but not in the way of being an accrual, or even of making us less susceptible to having expectations disappointed in the future. Ultimately what is gained is not knowledge but understanding of the limits of planning and expectation.

The courageous response to being pulled up short is to acknowledge and internalize the limitations that are revealed. This means accepting that, as Kerdeman writes, “despite our planning, life-events might unfold in ways we do not foresee or want. While this experience is painful, living through it can awaken us to choices we could otherwise not imagine” (298). Compare Kerdeman’s call to open oneself to the possibility of being called up short to Yancy’s recommendation that white people appreciate instances where they are ambushed by their privilege. In both cases, the situation is such that an individual is caught unaware, and her life is interrupted as a result of being shows that things are not how they seemed. These instances, which we cannot foresee, have the potential to be transformative by making possible deeper and fuller understandings of ourselves. For both of these authors, what is needed is not to avoid the unpleasantness of the alienation caused by being pulled up short or being ambushed. The recommendations are not to avoid such happenings, to wall ourselves off from them, or to attempt to predict or control them.

Rather, we are asked to cultivate dispositions that invite such experiences in, recognizing that it can be humbling and disorienting, these are intrinsically valuable experiences.

Intellectual courage entails cultivating this disposition with respect to surprise. By definition, we cannot predict when it will happen that we are pulled up short or ambushed by our own privilege, but we can develop a willingness to take on challenging situations, letting them call us to account.

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4.4 Virtues vs. Skills

4.4.1 Judgment and the will

When we depart the realm of skills learning and enter a context of cultivating virtues, the role of the will comes into view. This is one of the most important contrasts between the two, in fact, as in skills learning, there is no room for the individual’s judgment.

The idea that the judgment and the will do not have a role in mathematics contributes to the conception, described earlier, that it is asocial and unrelated to moral or human concerns. In a virtues context, judgment is central, and as we will see in chapter 5, judgment is crucial to the actual work of mathematicians.

As with the moral virtues, intellectual virtues require a foundation in finely tuned judgment to operate properly. The cultivation of judgment is a common theme in education literature, with authors such as, Halliday and Hager (2002), Hansen (2002), and

Wilson (2003) arguing for judgment as the ultimate end of education; that at which all schooling should aim. Hansen, for example, reports that the theme of judgment is long known, present even in the 16th century writings of Michel de Montaigne. In his essay “On

Educating Children”, Montaigne describes the teaching of a child, an education that is directed toward cultivating judgment rather than gaining knowledge. Just as bees pilfer pollen from all different flowers, only to use it to create something entirely their own, “the child will ‘transform’ all that he engages, from books to objects in nature” (Hansen, 2002, p.

237)(Hansen 237). Where the bee aims to make honey, the student aims to form his own judgment. He seeks to bring up a child who is able, rather than erudite (Montaigne 168).

While this education takes place through and by traditional means and curricular pieces,

Montaigne makes clear that the “everyday business of what today we call curriculum and

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instruction—for example, reading, writing, thinking, speaking, and interacting with others—boils down to an education in judgment” (Hansen, 2002, p. 242). The classics are read not for their intrinsic value or cultural worth, but for their capacity to raise questions and stimulate judgment. Like the majority of those after him, the judgment that Montaigne is concerned with developing is moral judgment—the capacity to make decisions and opinions about this action or that, or to choose a course of action in a given situation. He sees this moral judgment as a means to quelling much of the social and religious conflict that he saw around him.

A less prominent strand takes up the idea of what might be called epistemic judgment—that is, judgment about what to think or believe and how to makes sense of competing information. While moral judgment is the ability to make a decision or form an opinion regarding responsible and appropriate action, epistemic judgment is the capacity to make a decision or form an opinion regarding the appropriate treatment of knowledge.

Deciding whether a given input qualifies as valid evidence, for example, is an instance of epistemic judgment. This epistemic judgment is trained in much the same way as moral judgment is trained; it is honed by using it and trying it out.

Luntley (2004) argues that cultivating intellectual judgment is the ultimate educational task. He argues that learning concepts is a process of making sense of the world, or of ‘getting it right’. This is in contrast to the cognitivist model, which would say that learning concepts is a process of articulating or explaining the world. Luntley writes that we learn concepts not for either of these reason external to us, but simply for the internal experience of seeing the world correctly just as it is. “What the infant acquires with the concept of the doll”, for example, “is not primarily a cognitive device for explaining

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experience, nor a comforter that affirms its solidarity with others; what it acquires is a capacity to see things just as they are. It is the capacity to judge ‘That’s a doll’” (Luntley

2004). That judgment is what enables the child to generalize and consider other dolls.

This resulting judgment is entirely the child’s own, created by him out of a series of experiences.

More specifically, for history education, Levisohn (2010) writes that the fundamental activity in learning history is the negotiation of historical narratives. Like

Luntley, he is concerned with the notion of getting things right, and learning to see the world as it is. The case of historical narratives is not quite as simple as the definitions of midsized objects. Negotiating different narratives responsibly and well is historically virtuous, that is, the ability to understand and engage with partial, subjective, and often conflicting stories is itself an intellectual virtue, and engaging with this activity helps to develop the capacity and disposition for doing so. History education, then, ought to be aimed at improving students’ historical interpretations and at fostering those qualities that make them good interpreters.

These two pieces are pertinent here because they define a concept of specifically epistemic judgment—the ability to think well about the world and make judgments. Not for the sake of acting well, however, this intellectual judgment is taken to be important in its own right, as an element of virtuous functioning in the intellectual realm. Like moral judgment, it can determine the appropriate manifestation of individual virtues like courage.

It is what will determine what is worthy of being feared, to what extent, and in what way.

This epistemic judgment serves as the regulating body of intellectual courage. While the current paradigm in mathematics education leaves no space for judgment, as it collapses

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mathematical practice into the proper application of processes to data, in the coming chapter I will show that intellectual judgment is central to mathematics practice.

4.4.2 Learning virtues

A final consideration in taking up a virtues framework for education instead of a skills framework is the very different ways in which we develop intellectual virtues. Recall the skills model of mathematics, as represented by the Khan Academy materials. Learning for skill entailed breaking down complex ideas into sufficiently many sub-components each of which can be explained in abstract terms, and mastery of the totality of parts comprises understanding of the whole. Crucially, understanding is indicated by the ability to articulate and apply a general principle to particular situations. Virtues, to be slightly reductive, are learned in the opposite direction. As Hansen (2001) writes of tenacious humility, we do not need a fixed image of the virtue in advance of acting, nor do we need a plan of action for exemplifying it. Understanding of the virtue develops out of everyday practice and engagement with it. We come to know and inhabit the virtue by attending to the particulars of tenaciously humble practice on the ground. What is required for proper response is not meta-cognitive skill and abstract reasoning, but perception of the concrete particulars of a situation.

4.5 Conclusion

The intention of this chapter has been to offer a brief overview of an intellectual virtue framework, with particular attention to humility and courage. To this point in the dissertation, I have led the reader on what likely seems a wandering path, but having covered that ground, we are now ready to return to the world of mathematics. The dissertation began with a portrayal of the dominant paradigm in mathematics education. It

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was argued that mathematics in the United States is characterizable as skills-learning, which promotes rigid and mechanistic thinking, even as it tries to promote the generalizable skill of problem solving. Further, I argued that one reason the problem solving skills paradigm is so entrenched is that it emerges from widely accepted, yet unexamined, objectivist ideas about mathematics. While there are perennial efforts to dislodge and complicate the current methods, a widely held conviction that school math is an authentic representation of mathematics keeps it firmly in place.

Following the characterization of the current paradigm in mathematics education, I took up social justice concerns to show that the ways of thinking inculcated by the problem solving skills paradigm contribute to persistent social inequality. Specifically, I argued that when people occupying positions of social power are unable to challenge frameworks through which they understand the world, despite good intentions, they end up perpetuating marginalization of historically oppressed people and groups. The current chapter introduced a language of intellectual virtues as an alternative to a skills framework, through which to understand the recommendations of the social theorists. I suggested that two intellectual virtues that are crucial to lessening oppression (or, perhaps more modestly, doing less harm as a powerfully located individual) are intellectual courage and humility, and, by extension, intellectual judgment. These virtues were shown to be in direct contrast to the certainty and trust in inherited methods that are rewarded in the typical mathematics classroom.

The task of the next chapter is to look to mathematics to see if these dispositions find a home there. Because the current paradigm is supported by a mythology internal to mathematics, challenging that paradigm entails challenging the mythology and perhaps

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suggesting a new one. Fortunately, much work has been done in the philosophy of mathematics that considers just this issue.

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Chapter 5. Recasting Mathematics

What is a math problem? To a mathematician, a problem is a probe— a test of mathematical reality to see how it behaves. It is our way of “poking it with a stick” and seeing what happens. We have a piece of mathematical reality, which may be a configuration of shapes, a number pattern, or what have you, and we want to understand what makes it tick: What does it do and why does it do it? So we poke it— only not with our hands and not with a stick. We have to poke it with our minds. Paul Lockhart (2012, p. 5)

The previous chapters have sought to show that the present paradigm in mathematics education has negative social effects. The knowledge practices promoted by the problem solving skills paradigm value processing acumen over responsibility. In spite of persistent calls for reform, however, and objections to the sort of process-oriented and formulaic thinking promoted in the current climate, the problem solving skills paradigm persists. I argued in chapter 2 that the reason the paradigm is so resilient is that it is based in an ideology of mathematics itself. Calls from social justice perspectives to diversity or otherwise alter the curriculum are received as being in opposition to the goal of doing rigorous mathematics. Recall Brantlinger’s (2013) struggle to integrate social content into his mathematics classroom and execute a critical mathematics pedagogy. Challenging the paradigm in mathematics education requires telling a different story about mathematics.

Fortunately, a wealth of resources exist for doing so. Philosophers of mathematics have, for the past three decades, been rethinking the nature of mathematics, and have given a number of arguments calling into question the objectivist conception that presently animates mathematics education. But criticisms of objectivism and alternative philosophies of mathematics have been a part of the conversation within mathematics for a generation have nor found much traction in educational circles, because the objectivist framework is so firmly entrenched.

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This chapter will work to complete an isomorphism between the socially responsive thinking outlined in chapter 3 and with contemporary understandings of mathematical practice. I argued earlier that socially responsive thinking, particularly for people in positions of social power and privilege, requires intellectual courage and humility. While these dispositions are in opposition to what is rewarded in the problem solving skills context, they are both important aspects of mathematical thinking at higher levels.

Contemporary thinking about mathematics and its nature parallels that in feminist social epistemology, and I intend to show that, in putting them in conversation with one another, they can be mutually reinforcing.

We often do not feel an organic connection between socially responsible thinking and mathematical thinking, and so efforts to bring them closer together can feel forced and like a distraction from one or the other priority. The framework that has been built up here thus far, however, can serve to bring these two seemingly disparate areas into harmony, by directing our attention to the effects that our ways of knowing have on the social world.

Of particular interest throughout the chapter will be the role that the problem plays in mathematical practice. Recall from chapter 2, that a defining characteristic of a good problem in school math is that it is “perfectly stated,” with no information missing, and none extra. In higher mathematics, interpreting the question and filling in the requisite structure is a majority of the work that needs to be done. The nature of the mathematics problem is thus a critical point for this dissertation. This chapter will argue that rethinking the function of the problem in mathematics practice can make visible a number of elements of mathematical practice that are obscured by the problem solving skills paradigm. In the

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final chapter, I will introduce examples of teachers who use problems in this more authentic manner, opening possibilities for rich and holistic work in the classroom.

In the present chapter, I will propose an alternative model, drawing on more recent developments in the philosophy of mathematics. In short, I propose that mathematics is best conceived as a discipline of transformation rather than of increasing certainty, which proceeds by discursive and collaborative action rather than individual achievement. I will first legitimate the proposal by showing that the practice of mathematics is characterized by constant transmutation of ideas, rather than addition. I will then point out specific consequences that follow from characterizing mathematics as such, and finally, I will show how a paradigm of dialogue and transformation in mathematics and mathematics education helps resolve the epistemic issues raised in chapter 3.

This chapter looks to three philosophers of mathematics to show that the concerns raised in the previous chapter are resonant with work being done within contemporary mathematics. Ultimately we will see that while the discourses are rarely put in conversation with one another, the concerns raised by the feminist theorists in chapter 3 can find their answers in naturalist philosophies of mathematics. While they start from different beginnings, the concerns of the feminist mathematician are coterminous with those of certain philosophers of mathematics. Specifically, this chapter will show that the concerns raised in chapter 2 have been considered in a disciplined way within the fields of mathematics and philosophy of mathematics, and furthermore, that the conceptual resources exist to meet the challenge set out.

I will proceed by first recounting the most widely publicized mathematical result of the

20th century, and giving the standard interpretation, within the Problem Solving Paradigm.

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I will then turn to philosophy of mathematics to problematize the standard interpretation, and offer an alternative account of mathematics.

5.1 Fermat’s Last Theorem

Fermat’s Last Theorem appears as one of many marginal notations in Pierre Fermat’s translated copy of Diophantus’ Arithmetica. Following Fermat’s death, his notes in this book were incorporated into a new translation, many of which were unproven theorems and conjectures, and most of which have been shown to be correct. Next to the

Pythagorean Theorem, a note appeared that (as expressed in modern notation) there are no such integers x, y, z, n such that xn + yn = zn, when n > 2. Fermat famously wrote that he had “assuredly found a proof for this, but the margin is too narrow to contain it”(Singh,

1996). The conjecture is an analog to the Pythagorean Theorem, which has been proven in dozens of different ways and accepted since some of the earliest recorded mathematics.

But while many of the most accomplished mathematicians from the centuries since this comment was published have tried to fashion a proof for it, and tens of thousands of dollars offered in prize money for finding a proof, it remained unproven until 1995.

Mathematicians commonly assume that Fermat believed he had a proof when he made the note, but that he was mistaken, and that a proof would not have been possible with the concepts available to the author (Boyer & Merzbach, 1968). Alternately, he may have proven a weaker version of the statement, which would have met his standards for rigor, but not more modern ones. Eves notes that Fermat’s Last Theorem has the dubious distinction of being “the mathematical problem for which the greatest number of incorrect proofs have been published” (1964, p. 293)

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Andrew Wiles became famous for finding a proof for Fermat’s Last Theorem in 1995.

He was singularly focused on solving the problem at hand, having become fascinated with it at age 10. The proof brought together “virtually all of the breakthroughs in 20th century number theory and incorporated them in one mighty proof” (Hersh & John-Steiner, 2011).

Wiles created new techniques, and combined the existing ones in innovative ways. His work opened up possibilities for approaching a wealth of other problems.

Fermat’s last Theorem captures the imagination because it seems so easy to understand—on its surface, it appears that it can be either proven or disproved with basic mathematics and cleverness. And indeed, Andrew Wiles had become interested in the problem at age 10 for this very reason—it caught his attention for its seeming simplicity and he did not let it go until he had worked out a proof years later (Hersh & John-Steiner,

2011)

5.2 Standard Explanation

The standard account of problem solving is that there exist open questions in mathematics, and mathematicians work steadily on answering them. Once solutions are found, the questions are no longer live, and the answers provide ground for the solutions to further problems. Problem solving in school math follows this standard account—as students learn increasingly complex algorithms, they are given more difficult problems on which to practice, with no intention of or reason to return to earlier problems.

The standard account of problem solving is the framework that guides the problem solving skills paradigm. This account, however, renders invisible a number of essential aspects of mathematics. Notably, the standard account misses the fallibility of proofs, the

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role of exploration and so-called “false steps”, the role of invention, the possibility and necessity of changing the question, and the role of the mathematical community.

5.2.1 Concerns about the standard explanation

Skeptics of the power of proof brought up criticisms of the idea of a cohesive system of proofs. One concern, for example, is the necessary length of a proof of any but the most basic theorems if it has to start from axioms. The sheer length and complexity of such a proof is beyond the capability of a human to grasp all at once or in one pass. There is thus a possibility of gaps in the reasoning or internal contradiction going unnoticed. One of Wiles’ errors was caught in 1994, sending him back to revise, but there is no guarantee that there are no other inconsistencies lying in wait, either in Wiles’ own work or in the many theorems he invoked in crafting his proof. This complaint is often brushed off as a technicality, but the recorded history of mathematics has plenty of errors that have been overlooked for just this reason. A number of articles in the past several years have been dedicated to exposing errors in mathematics (e.g. P. Beckmann, 1976; Grunbaum, 2009;

Hersh & John-Steiner, 2011). In the most extreme cases, errors have been identified in work that had gone unquestioned for centuries and even millennia.

5.2.2 Philosophical context

Philosophy of mathematics as a discipline is conventionally considered to have been established by Gottlob Frege at the conclusion of the 19th century. Until this time it was widely accepted that mathematics was about the world in the same way that the natural sciences are. This Platonic view holds that there are mathematical entities that exist independent of human actions and thoughts, and mathematics is the body of knowledge that concerns these entities. Just as statements about cells and chemical reactions are true

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or false according to the properties of the objects they describe, statements about figures and sets are true or false in virtue of the objective properties of the objects (Linnebo, 2011).

Philosophers have rejected the Platonist framework for a number of reasons,19 leaving open the question: if mathematics isn’t true in virtue of the forms it describes, what makes it true? Trained as a mathematician, Frege worked to establish an epistemological foundation for abstract mathematics, once Platonism had been abandoned. He sought to show that there is a justification underlying the statements and objects of mathematics, ultimately arguing that math is reducible to logic. In the process of doing so, in his 1884 text (the most comprehensive of his statements), he establishes that the central task of a philosophy of mathematics is the foundationalist project. That is to say, the primary task is to articulate and display proper justification for the mathematical statements at issue

(Kitcher, 1988). The majority of the work in the field since Frege’s initial efforts has concerned itself with this foundationalist project, and aligns itself with one or more of the methodologies Frege outlined. Frege’s work is important not just for its endorsement of logicism, but because he defined the terms of the conversation about the nature of mathematics. Because of Frege’s initial volumes, the mainstream of philosophy of

19 Paul Benacerraf (1965), for example, explains two fatal contradictions within a Platonist conception of arithmetic. First, under a platonic understanding, mathematical statements refer to abstract objects; there is a form of “2” to which the numeral refers. But these abstract objects exist as mind-independent forms which do not interact with other objects, in particular, as they are described, they do not interact with human subjects. They thus cannot have causal relations with human subjects, and their existence cannot cause our belief in them. Secondly, the very definition of the abstract objects is problematic. The most common definition of the natural numbers, for instance, is done with reference to sets, where the number 1 is understood to be the set of all things that can be put into one-to-one correspondence with a single empty set. The number two can be defined in two different ways, though, with neither being preferable. This leaves the Platonist with a methodological directive to define the numbers, but with no means to do so.

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mathematics has accepted a number of premises, such as the idea that mathematics is objective, unified, and transcendental, and that it is not affected by subjectivity or individuals. Frege further entrenched the standard version of mathematics as the body of provable statements that can be rigorously proven from given sets of axioms.

5.2.3 Godel’s Theorem

While there had been disagreement over the nature of mathematics previously, until the early 20th century it was generally accepted that mathematics was a means of establishing absolute certainty. Individual theorems from the Elements have been scrutinized for correctness since the text was assembled. For the most part, this has meant clarifying definitions and supplementing Euclid’s initial work. The fifth of his initial postulates, however, turned out to be especially problematic. The “parallel postulate” is significantly more complex than the preceding four, and is not referred to in the text until relatively late (proposition 29), which conventionally has been interpreted as showing that

Euclid himself was uneasy about its status as a self-evident axiom (Boyer & Merzbach,

1968; Eves, 1964). The postulate is an important piece of plane geometry because it can be used to establish theorems about parallel lines such as even the most basic, “parallel lines can be extended indefinitely without intersecting”. In response to its perceived dubiousness, mathematicians beginning in the early 19th century tried to recreate Euclid’s structure without using that fifth postulate, and have found that drastically different geometries can result (Kline, 1980). These geometries are generally referred to as Non-

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Euclidean or neutral geometries, and they give very different and no less truthful understandings of space than we get with Euclidean geometry.20

The development of new geometries, among other influences, called into question the legitimacy of the Euclidean axiomatic system as a whole. After all, if simply changing the initial premises can engender an entirely new understanding of space, how are we to negotiate between the understandings? In what sense can a proposition be said to be true if in a different axiomatic system, it is false? Beginning at the end of the 19th century, logicians and mathematicians became quite interested in trying to answer these questions and articulate just what exactly proofs are and can do (D. R. Hofstadter, 1999). In 1931, however, logician Kurt Gödel permanently undermined the idea of a perfect axiomatic system by writing true but unprovable statements. He turned mathematical reasoning back on itself, in an attempt to make mathematics introspective, as part of a long tradition of mathematicians and philosophers trying to nail down for themselves what proofs really are. He ended up writing a mathematical analog of a paradoxical statement; in essence he wrote a statement that said “this statement has no proof.” He thus showed that there will always be statements in number theory that its methods are too weak to establish, and that provability and truth are not identical (with provability being a weaker notion).

Godel’s Theorem and the development of new geometries in the 19th century seemed to pose a bigger epistemic problem. For if mathematics is the paragon of perfect knowledge,

20 Projective geometry, for instance, pioneered by Gerard Desargues and taken up by Blaise Pascal in the late 17th century, accepts the first four postulates and rejects the fifth. Consequently it is assumed that parallel lines do intersect, at a point that must be infinitely far away. The notion of a point and line “at infinity” are incorporated, making possible a geometry that centers around relations of projections, in the same way that Euclidean Geometry centers around relations of translation and rotation (Boyer & Merzbach, 1968; Eves, 1964).

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and if mathematical truth turns out to be relative after all, we should be worried about the status and legitimacy of the rest of our knowledge. There has been consternation among philosophers about the meaning of this loss of certainty (Hersh, 1998; Kline, 1980), but this is an epistemological concern that derives from misunderstanding the nature of mathematical certainty. Among mathematicians, mathematical concepts are always up for revision given new information. This does not mean that they embrace uncertainty or skepticism over seeming facts like the sum of the angles in a triangle. It does mean that mathematicians do not balk at new concepts that curtail and change the old ones. More on this idea shortly.

5.3 Newer philosophies of mathematics

In contrast to the mainstream of philosophy of mathematics, a tradition also exists of naturalist philosophers of mathematics. Thinking about mathematics from a naturalist perspective means to observe and take seriously the habits and interactions of mathematicians as they impact the development of mathematical concepts. This is not a traditional methodology—typically, the social and historical conditions in which concepts are developed are taken to be immaterial. This is one of the aspects that makes mathematics unique—its seeming radical departure from its conditions of creation/discovery and one of the characteristics that imparts its purported superiority over other more contingent types of knowledge.

I will now draw from this naturalist tradition in the philosophy of mathematics to articulate a vision of mathematics that stands in stark contrast to the characterizations listed in chapter 2. I will first offer comments from Ludwig Wittgenstein who is not commonly associated with the naturalist tradition, but whose comments on mathematics

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embody the spirit and intention of the movement. I draw upon Wittgenstein as a place of entry, and to open up questions about mathematical work that will bring an alternate conception into view. Wittgenstein’s initial insights will be developed with the help of Imre

Lakatos, who is commonly understood as the progenitor of the naturalist strand in philosophy of mathematics (Kitcher & Aspray, 1988). Lakatos’ treatment of dialogue and discourse in the practice of mathematics will begin to answer the questions that

Wittgenstein opens up, characterizing mathematics as a practice rather than an abstractable body of knowledge. Finally, we will turn to Philip Kitcher, perhaps the most influential figure in mathematical naturalism to offer a detailed account of the growth of mathematical knowledge in light of the concerns that have been raised.

5.3.1 Open questions and mathematics problems

Consider a fairly straightforward theorem and its proof.

Theorem: The base angles of an isosceles triangle are congruent.

Proof: B Let ΔABC be isosceles with vertex B. Bisect angle B , and draw bisector BD AB is congruent to BC Definition of Isoceles BD is congruent to BD Identity

∠ABD is congruent to Definition of bisect ∠CBD ΔABD is congruent to Side-Angle-Side ΔCBD congruence property of triangles ∠A is congruent to ∠C Corresponding parts of A C congruent triangles are h D congruent Figure 10: Proof of Base Angles Theorem

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The proof is conventionally thought to show the logical necessity of the conclusion—show that the two properties (equal sides and equal base angles) are internally related.

Wittgenstein asks, however, whether we are really shown anything new in the proof, other than that the angles simply are congruent. That is to say, the proof doesn’t reveal anything deeper or more fundamental than we had before. What it adds is connection to an existing network of other assertions we know and accept about parts of triangles. It shows us that no matter what isosceles triangle we draw in the future, the parts will lie in the same relation, and the base angles will be congruent. Rather than uncovering some logical structure, it sets the parts in relation to one another in a way that wasn’t obvious before. As a result, the proof doesn’t have the capacity to convince.

Wittgenstein traces the confusion to a misuse of the word proof—using it in its casual sense in a technical setting. By exposing the misconception, he shows that we are mistaken to think that mathematics carries with it an absolute and objective logical necessity. This is important because it makes clear the difference between the way the word is used in everyday conversation and the way it is used in mathematics. Conventionally, to prove is to convince, or to demonstrate the truth of a proposition (Hersh, 1993). But this proof does not play that role—to accept the proof (and certainly to write the proof) one has to accept it as true already. This idea is familiar to mathematics teachers—in the classroom, the proof itself is ineffective as an explanation of an idea, it resonates only when it is presented after students have accepted the idea at hand. What does the convincing is the preliminary imaginative and exploratory work. These informal explorations can serve as a framework for writing or understanding a formalized proof, but they are two distinct phases.

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Traditionally, the imaginative and preliminary work that goes into convincing oneself of a result and of crafting a proof are thought to be inconsequential to the final product. The

‘messing about’ as Wittgenstein calls it, is contingent, and the Real Mathematics is the final result. While rarely explicitly stated, this traditional approach is evident through the history of mathematics. Published papers have none of the imaginative work, they tell no story of discovery of an idea or the author’s preference for one route over another.

Histories of mathematics report only the successful results, oftentimes divorced from any details about their authors, leaving out the popular ideas that ultimately fell out of favor, the resistance to incorporating unfamiliar concepts, and the changes in perspective that came from an expansion of meaning (e.g. Boyer & Merzbach, 1968; Eves, 1964). What is reported is only the continuous adding on of new concepts, expressed in modern notation.

Wittgenstein does not argue that the process of proof-building is unimportant, but rather than it gives a different kind of information than we tend to expect from it. In the usual sense of the word, the imaginative or preliminary work proved the propositions—it is what convinced the mathematician of the truth of the proposition. The formal proof shows that the proposition makes sense with the set of existing propositions and definitions. He states: “the proof places this decision in a system of decisions” (1983, p.

162).

In orienting the proposition with respect to the existing framework of accepted propositions and concepts, we are enabled to se it in a new light, and have the chance to see things which might otherwise be obscured or go unnoticed. Wiles’ proof of Fermat, for example, rids us of the notion that it is a simple property of elementary mathematics.

Säätelä (2011) points out that even if we do not understand Wiles’ proof, we have been

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shown that the problem is not what it seemed to be.21 Accepting the proof requires shifting

the way we see it, or a shift in our understanding of it. This is the crux of a

transformational theory of mathematics—that mathematical change is characterized by

transformation of existing concepts rather than simple addition.

The problem solving skills paradigm is resilient, however, and accepting that while

crafting proofs is a practice of meaning making rather than of establishing certainty we

should still be interested in this ability to solve problems skillfully, which can be taught by

practicing with mathematics problems. But solving mathematical problems is not

appropriately characterized as a skill. To substantiate this claim, it is helpful to divide

mathematical problems into two groups: those which we know how to solve and those for

which we first need to search for, discover, or invent a method.

Consider two problems:

A. Calculate 55

B. Construct a pentagon with a compass and straightedge

On first glace, these problems might be equally likely to appear in any high school or

middle school class or textbook. As with the examples shown earlier, the student would

have been shown other examples like these, and her task would be to reproduce the

process; to do the same. For problem A, she would have seen examples like:

25 = 2⋅ 2⋅ 2⋅ 2⋅ 2 = 32

35 = 3⋅ 3⋅ 3⋅ 3⋅ 3 = 243

€ 5 4 = 4⋅ 4⋅ 4⋅ 4⋅ 4 =1,024 €

€ 21 Reports on the progress toward proving Goldbach indicate that it, too, will be shown to be much more complex and rich than it first appears (Castelvecchi, 2012)

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65 = 6⋅ 6⋅ 6⋅ 6⋅ 6 = 7,776

She could follow the same pattern, substituting a 5, and write:

€ 5 5 = 5⋅ 5⋅ 5⋅ 5⋅ 5 = 3,125

When she begins, she doesn’t know what number she’ll find, but she has an algorithm for

€ calculating. For problem B, the student’s task is different. She could be shown the

following constructions:

Construction of

an equilateral Construction of a Construction of a triangle (3-gon) square (4-gon) regular hexagon (6-gon)

Figure 11: Three classic constructions

But “doing the same” is rather meaningless. Finding a construction of a pentagon does not

entail applying the same set of steps as the previous constructions, and even being facile

with the 3-, 4-, and 6-gons is no help in constructing a 5-gon. Furthermore, the search for a

solution is different, in that the solution to the problem is the process. She’ll try to find a

method for constructing the pentagon by drawing different shapes, by applying the method

she knows in different orders (e.g. bisecting angles, marking off arcs along the circle),

essentially messing about until she gets an idea about how she can proceed. She is not

looking for a solution that is out there somewhere, she is trying to invent one.

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This distinction that Wittgenstein points out will be helpful for thinking about school math. He argues that when we use the word “problem” to describe both activities— applying an already accepted method and searching for or creating a method—we get misled, because these are fundamentally different activities. He distinguishes mathematical problems—questions for which there is a known method of solution—from open questions—problems that lack a method and a solution (Säätelä, 2011). As with questions A and B above, the former is more akin to mechanical process, and the latter is not unlike a philosophical problem.

For the most part, the problems mathematicians work on are akin to problem B, not A.

They come in the form of propositions that have no established proof, like Fermat’s Last

Theorem. The theorem presents itself as a definitive statement, which has a truth-value that we cannot yet discern. It seems obvious to say that it either is or is not the case that there exist some real numbers such that x n + y n = zn . Some of the most compelling problems, like Fermat, capture the imagination because they appear so simple; giving the impression that they can be proven with ninth grade mathematics and a hearty dose of € pluck.

Without a truth-value, and without a rational connection to the rest of the conventionally accepted propositions, the proposition is without a sense. It appears to assert something, but it does not. The work of the proof is to set the statement in relation to the existing network of accepted propositions, thereby giving it a sense. The process of proving a proposition is a process of invention. Part of the wonder of Wiles’ proof of

Fermat is that such a simple sounding proposition would bring together such widely ranging and complex branches of mathematics. In this manner, the problem and the

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solution are connected—it is the solution that gives the problem its sense and meaning.

Until the proof is formulated, the formula is merely a structure without a sense (Säätelä,

2011).

Wittgenstein is helpful because he makes clear a linguistic sloppiness that leads to a sloppy understanding of the nature of mathematical activity. He gives a snapshot of the different roles questions play for mathematicians and challenges us to think beyond the superficial problem solving paradigm. It’s possible that we are left with the impression that simply differentiating between mathematical activities will set us right: perhaps adhering to Säätelä’s characterization of open and everyday problems, acknowledging the structural differences between them.

5.3.2 Discourse

If we turn to the history of mathematics, we can develop this idea further. By tracing the historical evolution of mathematical ideas we can see that this discursive process of articulating and clarifying statements, and of setting constructs in proper relation to each other is not simply incidental or a preliminary stage of what mathematicians do. The standard account of mathematical progress figures discussion as a means to clarity and better truth. That is, mathematics is essentially an individual pursuit, but given that we are fallible, we sometimes need to subject our reasoning to the scrutiny of others to help identify and eliminate errors. Furthermore, the competition that is fostered by individuals striving for the same prize functions as a motivator and drives progress. The means that discussion is one means among many to the objective, provable structures of mathematical truth.

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The standard interpretation of Wiles’ proof of Fermat fits this narrative nicely: since he was young, Wiles thought inwardly about the theorem. As a professional, he withdrew from the professional community, all but isolating himself for seven years, working on the proof independently. He first submitted the proof for publication in 1993, and was alerted to an error in his reasoning. He returned to his isolation, fixed the error, and published a corrected version a year later that was met with widespread acceptance

(Singh, 1996). Lakatos claims, however, that we should interpret this story differently.

His most influential work is an article in which he takes up the nature of informal mathematics and the history of a single conjecture. He had hoped to expand upon some of the underdeveloped ideas in the piece and to write a more comprehensive philosophy of mathematics, but was unable to do so in his lifetime (Larvor, 1998). Two of Lakatos’ former students sought to complete Lakatos’ project and assembled a small book titled

Proofs and Refutations, which brings together this initial essay, selected pieces from

Lakatos’ PhD thesis, and commentary of their own.

Lakatos’ primary concern in the lectures is the inadequacy of formalism as a philosophy of mathematics. His complaint is that it simply leaves too much out. Under a formalist conception, the only knowledge that counts as mathematics is presented in propositional form, properly backed up by formal proof, in Euclid’s style. The process of development of ideas is left out, as are creativity and changes in thinking, and Lakatos argues, so too is

“most of what has commonly been understood as mathematics” (Lakatos, 1976). In purging mathematics of the “impurities of earthly uncertainty, none of the creative or critical periods are admitted into formalist heaven” (1976, p. 2). Under the conventional formalist definition, for example, Newton’s work in calculus, wouldn’t merit being called

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mathematics until four centuries after he lived, when it was eventually translated into formal language.

The problem with the formalist conception is that, in leaving so much out, it purges the field of meaning. “Under the present dominance of formalism,” Lakatos writes, “one is tempted to paraphrase Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty” (1976, p. 2).

In abstracting so totally from human experience and values, mathematics becomes just that: divorced from meaning and value. In response, he turns to the methodology of mathematics, to show that in its practice, it is much more than the formalist conception claims. Specifically, he aims to show: “that informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number of indubitable established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations” (1976, p. 5).

The most substantial of the four essays in Proofs and Refutations is written as a dialogue taking place in a classroom. The class and teacher are working through Euler’s polyhedron formula, which relates the number of edges, vertices, and faces of polyhedra.22 The speakers in the dialogue advance positions that had been presented historically as the

22 The Euler characteristic was originally defined as x = V − E + F , where V= number of vertices, E= number of edges, and F= number of faces. Euler’s polyhedron formula states that a convex polyhedron has x = 2. For example a cube has eight vertices, 12 edges, and six faces, so x = 8 – 12 + 6 = 2 (Lakatos, 1976). €

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conjecture was discussed among the mathematics community.23 He calls the form a

“rationally reconstructed or distilled history” and notes the parallels to actual history in the footnotes. He doesn’t elaborate much on this form, but it implicitly highlights the interpersonal and dialogic aspects of mathematics.

He begins with an initial conjecture, which is open to refutation, after the refutation, the conjecture is reformulated and adapted, completing the cycle. In the course of students offering and responding to arguments using the concept ‘polyhedra,’ the concept itself is developed and refined.24 Lakatos is interested in the means by which this discussion improves mathematical concepts. In the first appendix, he articulates the pattern as follows: (1) primitive conjecture, (2) proof (a rough thought experiment or argument,

23 Characters in the dialogue represent epistemological positions that have been advocated by various mathematicians and philosophers of mathematics. The character Gamma takes a Wittgensteinian approach, arguing that: “I think that if we want to learn about anything really deep, we have to study it not in its ‘normal’, regular, usual form, but in its critical state, in fever, in passion. If you want to know the normal healthy body, study it when it is abnormal, when it is ill. If you want to know functions, study their singularities. If you want to know ordinary polyhedra, study their lunatic fringe. This is how one can carry mathematical analysis into the very heart of the subject” (Lakatos, 1976).

24 This sequence mirrors Popper’s (e.g. 1959) portrayal of the growth of scientific knowledge. The other major influence here is Hegel, whose fallibilism and dialectical opposition Lakatos adapted to mathematical thinking. In an earlier piece, he writes that “the proof procedure seems to me to be a remarkable example of the dialectic triad of thesis, antithesis, and synthesis.” (Lakatos, 1961), with the primitive conjecture, the refutation, and the final theorem corresponding to the parts of the dialectic, respectively. The other element that derives from Hegel is the inevitable historicity of thinking—that mathematical thought is embedded in a development of a human history. Larvor (1998) reports that Lakatos’ intention was to outline a complete dialectic theory of mathematics. Though these elements remain central to Lakatos’ thinking, in his later works he moves away from any reference to Hegel explicitly. The shift has been attributed largely to the political conditions in Europe at the time and the common association of Hegel with Communism, rather than philosophical reasons having to do with either of their theoretical work (Ernest, 1990; Larvor, 1998)

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decomposing the primitive conjecture into subconjectures or lemmas) (3) ‘Global’ counterexamples (counterexamples to the primitive conjecture) emerge (4) proof re- examined: the ‘guilty lemma’ to which the global counterexample is a ‘local’ counterexample is identified and is either eliminated or accounted for in the original conditions of the conjecture. At times, three other stages occur: (5) proofs of other theorems come into question or suggest themselves, (6) further consequences of the original conjecture are checked, and (7) counterexamples turn out to serve as new examples in their own right and new fields of inquiry open up (Lakatos, 1976, pp. 127–128).

His point is a descriptive one rather than a normative one, and he states that this method of proofs and refutations has been the unacknowledged standard (though not the only) mode of mathematical discovery since the 1840’s.

Lakatos shows that the discussion, the mistakes and the exploration are not just incidental to the discovery of mathematical concepts—they are essential to the nature of mathematics itself. To do mathematics is to engage in this dialogic interaction. There are thus two different ways in which to interpret Wiles’ proof, in light of Lakatos’ framework.

We might see Wiles as an outlier: that he did the majority of his work in isolation is not typical of mathematicians’ practice, as his contemporaries point out. His peers had never seen a period of isolated work like Wiles’ (Singh, 1996). The other interpretation is that even though Wiles was limiting communication about his work, he was still in an ongoing dialogue. One of the notable features of Singh’s documentary about the proof is that Wiles highlights the contributions of other people coming at just the right time to help him develop his proof.

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Philosophers of mathematics traditionally consider only what is said to constitute mathematical knowledge. Their treatises take up mathematical documents and arguments apart from their authors, audiences, and intentions. Lakatos convinces us, though, that every mathematical statement is made by a speaker to an intended audience—for every proposition spoken (or written), there is a speaker and a spoken to, and mathematics does not exist outside of this discursive context. Each proof and paper—the most formalized version of mathematical thinking—is written with an audience in mind that will interpret and understand the progression of thought. This reader is expected to interpret the argument through a framework of background knowledge, and only then does it mean anything. Katz reports that reading Wiles’ proof in penultimate form required frequent elaboration on the claims he was making. Katz, a colleague well known to Wiles, in a similar field, is one of the few people best suited to interpret the argument without additional explanation, and still he reports having to solicit clarification or other explanation at every page of the paper (Singh, 1996). The idea that a mathematical proof stands alone as a complete thought does not hold up.

Lakatos’ account is unsatisfactory, however, because it cannot account for the certainty and objectivity that we perceive in mathematical knowledge. That is, the aesthetic and discursive account compiled by Wittgenstein and Lakatos cannot account for the enduring nature of mathematical knowledge, for the unique resistance it has to positionality or point of view of the observer. I will now turn to Philip Kitcher to fill in the remaining gaps in the account. Kitcher’s work is essential in reconsidering the nature of objectivity in mathematics, in light of Lakatos’ work on its discursive nature.

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At this point we might be inclined to think that while the discourse that Lakatos outlines is an inescapable aspect of mathematical practice, it is rightly understood as a means to discovery of objective mathematical concepts.25 Or alternatively, it might appear that mathematics is simply a social construct, created by the whims and particularities of mathematicians.26 Kitcher responds to both ends of this polarity, first by addressing the role of the educational community in mathematics, then by accounting for change in mathematical knowledge.

5.3.3 The mathematical community

In education discourses, it is hard to imagine discounting the value of the community of knowers by which we come to learn, as the standard image is of a teacher conveying knowledge or subject matter to a student. But mathematics, like the natural sciences, has the peculiar feature of being externally verifiable and independently provable, (which is part of the reason why it carries the cultural weight that it does), which has led philosophers of mathematics to conclude that the source of mathematics knowledge is the math itself, not the human interaction. It is also the reason that mathematics curriculum is seldom the target of social critique. Educators and mathematicians alike take mathematics to be objectively and externally verifiable, and true by logical necessity.

25 The echoes of Popper’s language throughout support this interpretation of mathematical progress as an analog of a type of scientific progression.

26 An extreme of this social construction view came to light when, in 1897, the Indiana State Legislature ruled that, for ease of calculating, pi would henceforth be equal to 3 (P. Beckmann, 1976). This had the benefit of making calculations easier, and the detriment of also making them wrong. No amount of consensus, even if it had been more widely or more enthusiastically taken up, can change the ratio of a circle’s circumference to its diameter.

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Kitcher’s argument is that the initial educational interaction has a primary epistemic significance, turning our attention to the communities within which we practice and pass on mathematics. He points out that the way we learn mathematical concepts is in fact not by observing them in the world, or by deducing them from principle, but rather “in almost all cases, there will be a straightforward answer to the question of how the person learned the axioms. They were displayed on a blackboard or discovered in a book, endorsed by the appropriate authorities, and committed to the learner’s memory” (Kitcher & Aspray, 1988).

The educational encounter is often discounted by saying that while the student may learn a concept from the teacher in the first place, that original source is (or should be) soon replaced by the student herself when she realizes the truth of the statements. So while a student may believe her teacher when he says that that all equilateral triangles are similar (the teacher thus being the source of the knowledge), in interacting with this idea, putting it to use and testing it out, she will soon recognize the logical necessity of it. The warrant for her belief thus becomes that logical necessity, not the authoritative assertion.

That mathematical ideas are reinforced by the world around us, makes it seem like this mathematician has a way of independently, individually, knowing the truth she espouses

(Kitcher, 1988, p. 93).

Kitcher points out, however, that our mathematical beliefs at every level are causally overdetermined. He begins from the definition of knowledge as warranted true belief. He delineates between the cause of a piece of knowledge, or the process which produced it, and the reason it is warranted. Beliefs are reinforced by recollection of being taught, by a person or a book, perceptual recognition that a maxim holds, and cognitive recognition that a maxim is logically necessary. In the simplest cases (e.g. all triangles have three sides),

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there is an abundance of justifications. Apriorists, Kitcher argues, dismiss the wrong things for being irrelevant in bringing knowledge about, placing too much value on what are ultimately merely local justifications.

There are additional ‘local justifications’, like the meeting of certain expectations, a formula’s “working” when we use it, etc, that serve to overdetermine this knowledge.

Among the factors overdetermining is an axiom’s fitting into the formal proof structure, ie being provable from existing accepted precepts. “…the community supplements primary source (authorities) with local justifications, providing the student with ways of looking at mathematical principles which make them seem obvious. So it comes to appear that the mathematician, seated in his study, has an independent, individual means of knowing the basic truths he accepts” (1983, p 93). The implication is that while we may learn initially from teachers, ultimately each of us has logical truth and necessity as the warrant for our mathematical beliefs.

Kitcher claims that previous philosophers of mathematics have examined the determining factors and claimed the wrong ones to be irrelevant. That is, they take the authoritative assertion to be extraneous, nothing more than an impetus to find ‘proper’ warrant for the belief. While knowledge can be warranted by testimony alone, local justifications are a psychological process that is a dispensable (albeit useful) aid to knowledge (p 94). This is important because local justifications require background beliefs.

Thought experiments (long held as a valuable tool in mathematics for exploring new hypotheses and ideas) can serve pedagogical goals, and can fix belief more firmly than testimony can, because they have the power to aid memory and integrate beliefs and experiences. Thought experiments, however, cannot engender a priori knowledge.

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All this means that a majority of people can attribute all of their math knowledge to authoritative assertion, not to logical deduction. Kitcher effectively places the mathematical community and the math classroom at the center of the growth of mathematical knowledge

(rather than the mathematician toiling away in solitude) and connects the pedagogical question of the ways by which the individual comes to accept and have a warrant for the statements she accepts to the principles that govern the development of mathematical knowledge at large. Kitcher’s question is how the principles that govern the development of the field serve to enable the individual to have a warrant for the statements she accepts.

Despite the image and ideal of the autonomous mathematical thinker, the fact is that we are inducted into a mathematical community before we have the tools to decide to join. We do not freely choose mathematical communities on their merits, we get disciplined into them at a young age.

5.3.4 Mathematical objectivity and certainty

It’s difficult to grapple with a transformational conception of mathematics because everything around us, certainly all of school math presents it as an objective monolith. The impression of objectivity derives from the awareness that mathematical concepts and structures are independent of us as individuals. As Hersh writes, the mathematician recognizes that “the roots of a polynomial are what they are, regardless of what he thinks or knows” (1998, p. 42). Unreflectively, it is easy to assume that this objectivity is located outside all of human consciousness and experience, and it is tempting to assume that questioning the objectivity of mathematics entails embracing the opposite. But a subjective mathematics simply does not fit with what we experience of mathematical knowledge, and appears as patently false or disingenuous from the outset. Kitcher’s

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account of mathematical knowledge illustrates what objectivity and certainty mean in mathematics. He shows that while it is right to think that mathematics is objective and certain, these terms have more nuanced meanings within mathematics than they do in their everyday uses.

Our predecessors generally agreed that there is no number that, when multiplied by itself, equals -1. That is to say, in modern notation, that √-1 does not exist or that there is no x such that x2 = -1. Mathematicians today would disagree, showing that the unit i is such a number. It has been defined so that i2= -1. We do not tend to say, though, that the ones who came before us were wrong, like we might say that they were wrong about the sun going around the Earth. We do not say that they held false beliefs, because given the concepts available at the time (before the development of the complex numbers), there was no such number. With the creation of i, the referential of the word number has been expanded, and so we now wish to say that there is no real number x such that x2 = -1.

Even mathematical simples like numbers can evolve in ways that we might not expect.

Beckmann (1976) tells the story of the number π, and shows that that while the number π has been in use since before recorded mathematics began, it has had different meanings and different values for people in different times. One basic comparison is that schoolchildren today learn that it is a never ending decimal, which would have had no meaning to people working with roman numerals or other number systems. Most recently, the use of computers to calculate billions of digits of pi has been a part of the emerging role of computers in pure mathematics. Despite being a constant, π is still a changing idea.

Hersh (1997) tells a similar story of the evolution of the number two. He puts two in the contexts of expanding number systems, and shows that as number systems get more

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complex, the definitions of individual numbers gain more and more facets. Twoness is a different idea now than it was.27 With each new discovery about the number system and the forms and relationships of real numbers, the concepts of 7, 3, and 10 expand their reference potential, and the equation is placed in a denser and changed network of connections. It thus means something different when we assert its truth.

To be certain of a proposition in mathematics is thus not to preclude openness to alteration of the proposition in the future, given new concepts or information. In fact given the history of mathematics, one would be wise to expect that change. Certainty within mathematics is thus a more nuanced notion than it is in its everyday sense. Mathematical certainty is provisional, not absolute. Recall the discussion of intellectual humility and courage from the previous chapter. In the same way that for social thinking, the stance advocated was not one of skepticism or agnosticism, but a disposition “to revise or reject the position [one] holds if sound objections are brought against it” (Hare, 1979, p. 8), mathematical thinking requires maintaining a provisional working certainty that allows for the likelihood of future revision.

Just as mathematical concepts change and evolve, so to do mathematical questions, and they change in different ways. Some established questions are simply answered— mathematicians craft an appropriately thorough and targeted response that fills the gap highlighted by the question. Questions can also cease to be meaningful. In cases where presuppositions of a question are shown to be false, or are incompatible with newer classifications, mathematicians simply leave questions behind. Most interestingly, as

27 Lakatos’ character Delta exemplifies this aspect of mathematical change; he constantly proposes new limitations on the definitions of the terms in question. For each problem or contradiction that arises, Delta meets it with a proposed exclusion that could disqualify it.

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language evolves, new questions emerge. Commonly, as the reference potential of terms expand, new categories of questions are presented which are analogous to the old questions. With the development of complex numbers, for example, a whole host of questions presents itself—in what ways do complex numbers behave like real numbers and how are they different? Can complex numbers have logarithms? Can they be factored?

In Wiles’ description of his progress toward proving Fermat, we can see this sort of mathematical change. Wiles explains that he was initially inspired to take up Fermat by a number of developments by his contemporaries that put a solution within reach. The first was the Taniyama-Shimura conjecture, which states that every elliptic curve is modular.

This conjecture established a method of transformation between the two types of curves, enabling mathematicians to translate between the two structures. The second development was Gerhard Frey’s contrapositive, which states that if there exists a solution to xn + yn = zn, when n > 2, it would create an elliptic curve that is not modular, thereby disproving Taniyama-Shimura. Frey’s proposal (for which he initially gave a plausibility argument, but was later proven) meant that if Taniyama-Shimura was correct, then so too was Fermat. The question of proving Fermat’s Last Theorem was thus changed: the problem was no longer to prove the nonexistence of a solution to the equation, but to perfect the Taniyama-Shimura transformation method.

Another more subtle mode of mathematical change figures prominently. Wiles describes one of his most important breakthroughs as essentially changing the question at hand. He was able to convert modular forms into a different representation (Galois representations) thereby making them easier to catalog and count. Redefining the question in this way ultimately allowed him to answer it.

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Kitcher (1984) points out that the impression that mathematics is simply cumulative and does not meaningfully evolve likely stems from the historiography of mathematics.

With a few recent exceptions, histories of mathematics (e.g. Boyer & Merzbach, 1968; Eves,

1964) tend to be triumphalist, telling the story of a neatly unfolding body of knowledge.

They rarely go beyond chronicling names, dates, and modern formulations of important theorems: they tell the mathematical equivalent of a kings and princes story of history.

Rejected theories and inter-theoretical struggles are not recorded, giving the impression that mathematics is a body of a priori knowledge that proceeds by simple accumulation.

Even the premodern mathematicians are interpreted through an expectation that they exemplify the modern ideal. The stories told come to reinforce the model of the great solitary thinker, shaping the expectations we have for our own work.

Kitcher’s account of mathematical change rests on his likening mathematics to natural science, showing that mathematics is subject to the same sort of internal theoretical conflicts as science is.28 Stresses caused by conflicting theories and explanations are what provide the impetus to modify beliefs and structures. In comparison to physical sciences, mathematical theories enjoy an exceptionally high rate of survival. Euclid’s geometry, for example, is still taken to be a true representation of physical space even though entirely new geometries have resulted from the rejection of the 5th postulate. While in geology, for instance, competition between two theories will eventually result in the triumph of one and the death of the other, in mathematics it is understood that the two geometries describe

28 Note that one of Wittgenstein’s basic concerns is that we are misled by equating mathematics with natural sciences, which would appear to put Wittgenstein and Kitcher at odds. Given that they are referencing different aspects of natural science, however— Wittgenstein is concerned with data and evidence collection, and Kitcher with theoretical frameworks—I do not believe the two meaningfully contradict one another on this point.

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different things. The theories can coexist. The original theory must be reinterpreted, however, in light of the new one, which happens primarily by suggesting stipulations be placed the old theory. When the stipulations are shown to be useful and productive, they become generally accepted. Mathematics tends to be transformational, not additive. It grows by continually reinterpreting existing constructs in light of new relationships and ideas.

5.4 Complicating the Ideology of Mathematics

In light of this developing picture of mathematics as discursive and transformational rather than inert and additive, it becomes clear that the problem solving skills paradigm gets mathematics wrong. A few points are worth highlighting. When we are limited to a problem solving skills paradigm, we miss:

• The place of judgment in mathematical thinking. The ability to judge appropriate

criteria is context dependent and relies on subject and problem specific details

• Change and adaptation. Skills do not require adaptation or change. They get added

onto to respond to new situations, but in skill-learning we are not asked to

reconsider what we have already mastered.

• Mistakes and exploration. The only thing valued by skills learning is mastery.

Mistakes are values only insofar as they can reveal wrong thinking that can then be

corrected.

The problem solving skills paradigm derives from the general ideals of humanist education.

The intention is to conduct schooling in ways that support students’ quality thinking. We want them to be able to leverage centuries of human thought and achievement to make sense of the world, for their own fulfillment and well being as well as for their vocation and

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the betterment of society. This impulse goes wrong when we reduce good thinking to a set of processes or mindless routines that can be universally applied to the world. The remedy is to refrain from making this conceptual leap from skilled thinking to the possibility of isolatable training on the sub-components of more complex processes (Bailin & Siegel,

2003).

In order to do that, a different paradigm is needed; one that provides exemplars of skilled thinking, that establishes a terrain on which meaningful evaluations can be made, questions can be asked and interpretations of data can be made. Mathematics itself can provide this paradigm, provided we interpret it differently. I propose that essential to an improved paradigm for mathematics education is a more nuanced and accurate understanding of the nature of mathematical problems and what it means to solve them.

For all of the philosophers mentioned here, the mathematics problem has played a central role, and indeed, working problems is central to the practice of mathematics. Returning to the distinction made by Rota referenced in Chapter 2, however, there are different ways to understand the role of the problem—either as supporting the development of theory, or being supported by it. While the standard interpretation and the problem solving skills paradigm assume the latter—that mathematical knowledge is developed for the sake of solving ever more complex problems—these theorists demonstrate that the former is more often the case. Mathematics problems function primarily as impetus to challenge what we know and to expose limitations and contradictions in what we take for granted.

The mathematician is not in the business of constant forward motion, gradually adding on to his established knowledge, amassing an ever more powerful problem solving arsenal. He is rather in an iterative process of seeking out quandaries or contradictions,

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and adjusting his prior understandings to alleviate them. The mathematician’s object in seeking out problems is less to master or conquer than it is to be changed by them.

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Chapter 6: Paradigm and Pedagogy

Abandon the notion of subject matter as something fixed and ready-made in itself, outside the child’s experience; cease thinking of the child’s experience as also something hard and fast; see it as something fluid, embryonic, vital; and we realize that the child and the curriculum are simply two limits which define a single process. John Dewey (1902, p. 11)

The dissertation thus far has endeavored to present possibilities for a socially just mathematics education. The first task was to characterize the rigid and often superficial nature of the current paradigm in mathematics education, pointing out features that typically go unnoticed. I argued that while the intention behind the promotion of problem solving in school math is promising the way in which it is currently enforced is overly reductive, and turns mathematics into a set of skills to be mastered.29 As in other disciplines, when the content is reduced to an abstract set of skills to be learned and applied there is no space for the reflective thinking that can challenge or reframe concepts that are given. This problem solving skills paradigm, as manifested by the artifacts and mythology of mathematics education, disciplines students into shallow and reactionary habits of thinking, where the most prominent lesson they learn is pattern recognition and response to particular stimuli. Insofar as such thinking is conditioned to accept handed down meanings and categories and assumptions, to reproduce diligently what has been modeled, and is used as an exemplar for good thinking in general, it promotes and supports a limiting, patriarchal social arrangement. The superficial and mechanistic mode of teaching and learning mathematics has been challenged from a number of angles, the most prominent critiques in the pat decasde taking issue with the ways in which it contributes to

29 As was mentioned in chapter 2, while the current paradigm rests on Polya’s (1957) text, as it is currently enacted, is quite at odds with his intent.

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social injustice and inequality. What Lerman (2000) calls the “social turn in mathematics education research,” and what Gutierrez (2010) names the “sociopolitical turn in mathematics education” has turned attention to the ways in which institutions and systems of mathematics education serve to disempower various historically marginalized groups, and thus reify persistent social inequality. Within this body of work and the circles that seek to enact the principles being developed, there has been much criticism of the assumed certainty and lack of reflectivity that I have taken issue with here. My analysis has been oriented toward students in positions of social power or from communities of privilege, because it is at present, an area that is undertheorized. Though my approach is different, this project is complementary to the work being done by thinkers and educators whose primary interest is in empowering marginalized students. But while challenges to mathematical certainty, assumed objectivity, and a skills model of mathematics are widespread within the critical mathematics community they have not found much traction in the mainstream. Recall the discussion of the Khan Academy curriculum, which so perfectly exemplifies the problem solving skills paradigm. In the mainstream conversation, the objectivism is so well entrenched that social and political concerns appear as something other than mathematics. The conflict that results has cpntributed to the math wars for the past 20 years.

To circumvent this stalemate, I have taken a different approach. Rather than thinking of the mathematics classroom as a site for developing critical literacy, or for directly engaging with structural social inequality (as Brantlinger did, following Gutstein), I have illustrated an isomorphism between socially responsible thinking and genuine mathematical thinking. This entailed showing that the problem solving paradigm is

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inadequate even as a representation of mathematics as a discipline. I gave an alternate characterization of mathematics as unavoidably transformational, rather than additive; as discursive, rather than individual. Highlighting intellectual courage and humility in particular, I have endeavored to show that the dispositions necessary for working against one’s tendency to perpetuate oppression are central to the work of mathematics. The mathematics classroom, therefore, has the potential to be a space in which students develop these habits of mind.

The extensive work of establishing a causal link between a recommended pedagogy or method and some observable good is beyond the scope of this dissertation, and is a direction for future research. I will, however, offer a selection of portraits of teaching that embody a commitment to developing intellectual virtue rather than exclusively skills. As has been a theme throughout the dissertation, the conceptualization of the mathematics problem in the classroom will be central in this discussion of teaching practices. I will conclude by offering remarks on the possibility of changing the paradigm in mathematics education, and how we might begin.

It is not the intention that these lessons described here be copied directly or, or to offer concrete solutions that will lead to higher achievement in contemporary classrooms.

Instead, these portraits of classrooms are offered as suggestions of how mathematics class might be differently imagined. The foregoing theory offers a mode of differentiating between the classrooms, and criteria according to which we might choose new modes of teaching and learning over the ubiquitous accepted models. Further, the intention is to de- familiarise those ubiquitous practices, and to challenge their status as self-evident and necessary. Pure math tends to be quite resistant to quantification and measurement (for

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reasons that will be explored), while applied math is amenable to it, so in this sense the project will be working “upstream” against the current of assessment and quantification. It will not present nor make claims to effective methods for teaching math, nor will it present quantifiable evidence that the understanding it promotes will be more effective than what is currently in play. This strategy is necessary, however, because the nature of the project is to call into question the notion of effective math education altogether.

6.1 Radicality: Reprise

Challenging the problem solving doctrine in contemporary mathematics education is necessarily a radical undertaking, as it aims at the most fundamental underpinnings of what we are doing. I have not called these foundations into question for the sake of proving the whole wrong, or for pushing my favorite philosophers into the education sphere. Instead, my hope is that the theoretical work I have done can help us see the work being done in classrooms more clearly. The lens that I will now turn onto particular teachers and their classrooms, will, I believe, cast their practice into a certain relief, allowing us to see what was not observable before. As we will see in the forthcoming examples, these teachers are enacting pedagogy that cultivates intellectual humility, courage, and judgment, within the context of an authentically and rigorously mathematical curriculum.

6.2 Four Classrooms

I will offer four examples of mathematics education that promote intellectual humility and courage. The first is an example of pure mathematics, offered by Paul

Lockhart (2009). The second is a narrative of teaching practice offered by Magdalene

Lampert (1990a, 1990b). The third example is a concrete piece of elementary mathematics

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curriculum written by Sherry Parrish (2010). Finally, I will recount and offer comments on a lesson from my own teaching at the high school level. Following these illustrations, I will outline the ways in which they each exemplify pedagogy called for in chapters 3, 4, and 5, and argue that the unifying theme is their underlying philosophies of mathematics. This is to say, what makes these examples unique is the ideology of mathematics that the teachers hold, which manifests in the curriculum and teaching methodology. Following these observations, I will summarize pertinent research concerning teachers’ beliefs about mathematics, and contextualize the present project in relation to two trends in teacher education.

6.2.1 Lockhart

The first example is adapted from Paul Lockhart, a research mathematician turned high school teacher. Lockhart’s piece is interesting because in the first place, numbers are to be understood in terms of what they do, rather than what they are. That is, there need not be any external justification or grounding for numbers, rather we learn more about them by exploring how they interact with one another. Doing so is tricky, because they don’t have an external referent by which to define them,30 and which thus constrains them, but they are subject to rules which we do not define, which means we can discover things about them and patterns and different types start to emerge. For example, we can represent numbers with groups of objects, like this:

30 There need not be an ideal 5 in the universe somewhere, like there is an ideal meter stick by which all other meters are measured. There is no essence of “fiveness” that tells us any more about it than we already know.

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! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Figure 12: Natural numbers as groups

A distinction presents itself between two types of groups: those that divide evenly into

rows and those that do not, these types being the even and the odd numbers. By looking at

the diagrams, we can start to think about how the numbers interact with one another, for

example:

! ! ! ! ! ! ! ! ! !

! ! ! & ! ! ! = ! ! ! ! ! !

An even an odd makes An odd

Figure 13: Combining evens and odds The diagram should intuitively show that this is true no matter which even and odd we

pick, and thus implies a general proposition about the sum of an even number and an odd

number. We can make the numbers interact in any combinations we can think of, for

example, what if we put together the first few odd numbers?

1+3=4

1+3+5=9

1+3+5+7=16

1+3+5+7+9=25

This sequence of sums is not just any random sequence of numbers, they are the early

members of their own set—these are the numbers which can be arranged into perfect

squares, in the same way we arranged into rows before. But there is no apparent reason

why that would be the case, and there is no outward property of either group that implies

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this relationship. This lack of an obvious justification is what makes it an interesting pattern—if I cannot see why it is the case, perhaps it is just a pretty coincidence? If it is always the case, perhaps it is still just happenstance? This seems unlikely, but we have no way of knowing if it is always the case, even if we have the patience to carry it out to the millionth iteration, as it might break down on the 1,000,001th term.

Recall that Wittgenstein helped distinguish between a mathematical problem (for which there is an accepted method for finding an answer) and an open question (for which answering the question entails finding a method). From the same observed pattern above, we can ask either type of question. “Is the sum of the first 10 odds a perfect square?”

“Show that the pattern holds for the first 15 sets of odd numbers” and “Identify the rule illustrated by these four examples and state in general terms” are all mathematical problems that could be posed about this pattern, each one of which requires an easily assessable mechanical solution. These are not the only questions that can be asked, though.

We could ask for a visual representation of the pattern, or we could ask about the

1,000,000th sum in the series, which would be so far beyond the bounds of plausible calculations, it would require a different approach.

Returning to the pattern, the mathematician attempts to answer both questions by seeing that the pattern is saying that each square number is made up of a collection of odds.

We should then be able to visually break down the squares into the requisite pieces, and after trying a number of different patterns, the following presents itself:

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" " " "

" " " " " " " " " " " " " " " " " " " " " " " " " "

Figure 14: Sequence of odds The picture is not a proof, nor is it an explanation, but just by adding the right segments to the diagram, it makes visual sense. Each square is made up of little L shaped pieces, each one of which has an odd number of units. To add the next odd number to any square just gives you the next square up. But why should each L shape have an odd number of units?

And how do we know that they fit together like that? And still, how do we know that it will always be like that? These questions are the work of proofs. At some point the visual language we are working in ceases to be descriptive and precise enough, so we want to turn to something more discerning, that will let us talk about the ‘n-th’ iteration. We want to build ourselves some tools for thinking about that 1,000,000th square, without having to draw them all out. This more formalized version comes only after we have an idea of what we are trying to do, though, and is a means of getting more clarity.

The above example is easily characterized as a problem-based approach, as questions are what move from one idea to the next. Not all questions are generative, but it is quite possible to pose open questions within the scope of elementary mathematics. One does not need a higher degree before inventing ideas in mathematics.

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6.2.2 Lampert

The second example comes from Magdalene Lampert, who provides a contrast between a standard problem solving pedagogy and what she calls teaching with problems.

In her widely read 2001 book Teaching Problems and the Problems of Teaching, Lampert gives a holistic account of a year in a fifth grade classroom with the intent to illustrate the practice of teaching, in its complexity. She works with a three part model that highlights the interactions between the students, the teacher, and the content. Each day when the students enter the classroom, there is a question written on the board, which they proceed to copy down into their notebooks. The students are seated in groups (mostly of 4), and they then begin to work on solving the problem. Lampert has the students divide their notebook page into two sections, experiments and reasoning, so that they are encouraged to take space to try out different approaches on their own, and then provide justification for their approach, as they would explain to someone else. After a significant amount of time spent working individually and in groups, attention turns to full group discussion, which begins with students’ approaches being copied onto the chalkboard.

Unlike Lockhart, Lampert works largely with problems and questions taken from standard curriculum materials. For example, one day the students work on the following problem:

Groups Groups a. 12 = 10 6 of of Groups Groups b. 30 2 = 4 of of Groups Groups c. 7 = 21 of of

Figure 15: Making Groups (Lampert, 2003, p. 143)

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After students work independently to fill in the boxes, Lampert coordinates a full group discussion. She begins by asking “who has something to say about a.?” and calls on a student she knows to have a solution that does not work. Lampert fills in the student’s response:

a. Groups of 12 = 10 Groups of 6 Figure 16: Richard's Response (Lampert, 2003, p. 147) and asks him to explain. “By asking Richard to explain his reasoning” Lampert writes, “I initiate a discussion of why he may have done what he did…. I conduct the discussion as if there was a shared assumption that there would be reasoning behind any assertion that would explain why it would make sense” (Lampert, 2003, p. 148). As the teacher facilitates discussion and brings in more students, she supplements their ideas with precise language and notations and at times gives suggestions for streamlining processes. After hearing from a few other students, Richard is asked if he would like to revise his initial assertion, in light of other children’s ideas, and is invited to give an alternate explanation. Lampert is clear that this is a fragile situation. “In terms of ordinary classroom norms, Richard is being asked to be extraordinarily courageous here…. Because I am teaching him in such a public way, I need to manage the problem of helping him save face with the rest of the class, even though he has publicly admitted that he was ‘wrong’” (2003, p. 158). Even though

Richard’s initial answer was wrong, and he displayed some basic confusion with the operations (appearing to add the 10 and 12), he is modeling for the class an important mathematical habit—the willingness to publicly talk about his reasoning, and listen to suggestions that might point out his flaws.

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6.2.3 Parrish

A second example is taken from the work of Sherry Parrish, who is concerned about students who do not make sense of the mathematics they do in the classroom. When students think of mathematics as a set of rules and procedures to be memorized, they do not grasp the underlying mathematical logic in play, and thus cannot engage with it.

Parrish argues that this gap, between the process and the logic, is why many students are unable to evaluate their own work, judging whether an answer is reasonable or not.

Parrish wants to develop students’ number sense, and help them to compute accurately, efficiently, and flexibly (Parrish, 2010, p. 5). She has developed a curricular unit called

Number Talks to help facilitate this sense making with numbers.

Number Talks are activities to be done at the beginning of each class through the elementary grades and possibly into middle school. The author recommends spending 15 minutes each day on them. A computation problem is put on the board, and students begin to solve it mentally. When students have a solution, they give a silent indication to the teacher, and try to find additional methods while the other students are thinking. When a majority of students have solutions, the teacher calls for answers, recording all solutions on the board. Once a number of answers are in play, students present and defend their methods, and judge which answers and methods are reasonable. For example, the question below is given as a third grade problem:

Michelle baked 3 pans of cookies. Each pan has 12 cookies.

How many cookies did Michelle bake?

Figure 17: Parrish 2010, p 238

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Most children are familiar with the context, which gives them an opportunity to come up with a strategy to answer. Four common strategies are given:

Repeated addition: The student adds 12+12+12. The teacher asks whether they could also represent this method with multiplication as 3 x 12. Decomposition: The student adds 10+10+10+2+2+2 The teacher asks whether they could represent this strategy with multiplication as (3 x 10) + (3 x 2) Skip Counting: The student counts by three’s twelve times, counting 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. The teacher asks whether there are any groups that might make this counting more efficient (count by 9’s, or 15’s?) Using friendlier numbers: The student knows that 10 x 3 = 30, and adds two more threes to get 36. Figure 18: 12 x 3 With the teacher’s help, by putting their reasoning up for public scrutiny, the class learns what count as good reasons in mathematics, and what it means to engage in mathematical conversation. While the teacher is offering steps to help develop their thinking, she is not telling them how to solve the problem nor introducing strategies of her own. She begins by recording their work and their thinking, and contributes by drawing connections to other ideas they have seen. Here, one of the main themes is the relationship between addition and multiplication. A problem for more advanced students is 8 x 25, for which parallel methods will likely be offered:

Repeated addition: 25 + 25 + 25 + 25 + 25 + 25 + 25 + 25, which might be simplified to counting by 50’s or 100’s. Decomposition: 8 x 20 + 8 x 5, or 8 x 10 + 8 x 10 + 8 x 5, Using friendlier numbers: 25 x 10 – 25 x 2 Figure 19: 8 x 25

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6.2.4 Brewster

A final example is recalled from my own teaching experience, when I first started toying with the form and function of problems in my mathematics classroom. As an element of ninth grade Algebra I, I was to cover the fundamentals of complex numbers, including establishing a solid understanding of i, and combining complex numbers with the four major operations. These items are now indexed in the Number and Quantity section of the CCSS for High School Mathematics. Typically, a unit on complex numbers begins with the introduction of i, and a lot of confusion. Khan Academy, for example, begins a five part series with the equivalent definitions �! = −1 and � = −1. From there the video considers raising i to integer powers (“Khan Academy,” 2014).

Instead of beginning with a new definition I established a need for a new unit first.

Earlier in the term, we had studied quadratic equations, for which the Quadratic Formula had left us with a surplus of unsolvable equations. In the students’ recent memory, they had gotten stuck, repeatedly, because they ended up with negative numbers below a radical sign, which means that there is no real number solution to the equation. We gathered up a bunch of these unsolvable problems, and talked about other times when people had run up against limits in the number system, and what they had done. How did people adapt when they wanted to divide four items among five people? They needed to make fractions. What did people do when they wanted to subtract 12 from 7? They needed to extend their number system to include some negative numbers. So, here we were with a bunch of unsolvable problems that all are all the same type, and I asked them what it would mean to invent a solution for some of the problems. In groups they worked through a set of open

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ended leading questions that asked them to invent some new numbers and see what they could discover.

Part invention and part discovery, the students had an experience of working with mathematical concepts in order to solve a problem they wanted to solve. After the students had a chance to work with the units they had created, I offered a short historical context for the imaginary numbers, explaining that others had done similar investigations, and presented detailed standardized terminology and symbols. Time was taken to undertake this meandering exploration at the beginning of the unit, but in all four of the classes that I tried it with, I had much better results at the end.

6.3 Criteria

These four examples of classroom practice have important commonalities, which also differentiate them from the traditional model of mathematics teaching.

6.3.1 The end in sight

In each of the cases above, the end-in-sight during the lesson is not the mastery of an isolatable skill. Appropriate mathematical concepts imbue each of the lessons, but none of them take as their objective an atom of mathematics that students are to encounter and learn that day. The role of the problems is thus more open ended, and they are used to stimulate discussion and work and not just for practicing what has been observed. Even in

Lampert’s and Parrish’s classrooms, where the problems at issue are more mechanically oriented, students learn to use the problems as discussion starters, with the learning happening throughout the discussion process.

Lampert conceptualizes her lessons as travelling in certain domains of mathematics, rather than covering certain topics, which is a unique approach. While in a traditional

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approach the teacher lists out the ideas to be covered, and then schedules them into the course according to what needs to be covered and how much time is allotted, Lampert understands herself to be moving within different conceptual fields. She divides the school year into large units, and each unit is dedicated to a different set of problems. The grouping problems shown earlier represent one type of problem, or one piece of the terrain.

Another unit is dedicated to time, speed, and distance problems, many of which are related to the work the students are doing concurrently in their other classes. Given the context of a ship travelling at sea, Lampert crafts mathematical questions about time, speed, and distance which are inherently mathematical questions that rely on multiplication and division. Another unit focuses on delivering a fixed number of cakes equally to a number of bakeries. Again, Lampert crafts a number of questions about the same context. Here, the questions lead them into fractions and mixed numbers, necessary to solve their division problems. While the questions are different, in a general sense, they are still on the terrain of multiplication and division, which are at the heart of fractions.

6.3.2 Growth is not additive

As Lampert explains, however, it is not that the students are studying the same thing over and over, but neither are they travelling along a linear path (nor moving upwards in a spiraling pattern). “While my students were doing multiplication and division in different problem contexts, they were studying aspects of these operations. But they were also studying something new in each context, namely, how those operations need to be tailored to the different constraints of different problem domains” (2003, p. 220). Division and remainders come up in the time, speed and distance problems as well as in the delivering cakes problems. The different contexts require that they are handled differently (a leftover

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cake is handled differently than a leftover hour). In moving through the content as she does, Lampert gives the students the chance to develop nuanced understandings of the operations, and also to experience the particularities of the different contexts.

Parrish exhibits a similar commitment in the material she chooses at each grade level, though it is expressed differently. In each case, the computations students are asked to complete are below their grade level. Given pencils and paper, the students could work through the standard algorithms for all of the calculations. In one sense, there is nothing new being introduced, rather, it is a deepening of what the students already know, and developing connections between numbers and operations. She finds that when the algorithm is students’ only mode of finding the sum, they do not develop mathematical reasoning or judgment skills, and so wants to encourage them to use other means to find it.

Parrish asks the students to do each calculation mentally. She writes the problems horizontally on the board, which does not lend itself to the standard computational algorithms, so that students cannot simply follow memorized procedures, and offers calculations for which there are too many steps to hold mentally at once. The students are thus forced to come up with a method to organize the steps, and to think more abstractly about the numbers in play. For a problem like 1999 + 1999, the standard algorithm would be to line the numbers up vertically, and add by place value. Without a pencil and paper, however, students are unable to keep track of the digits and steps necessary to use the standard algorithm mentally and have to look for an alternate method. These numbers lend themselves to what Parrish calls “using friendlier numbers,” as students see that the sum will be 2 less than 2000+2000, so 3998. In algebraic terms, this is an application of the associative property of addition. The use of mental math provides a way to encourage

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students to think in abstract terms about numbers in contexts that are accessible to all of them.

6.3.2 Wrong answers and courage

The spaces created by each of these approaches to problems in the classroom are conducive to developing the intellectual virtues discussed earlier. I hesitate to write too extensively here because I am not in a position to make any causal claims about the methodologies presented. However, even superficially, we can see how the instructors cultivate humility and courage in intellectual work.

Lampert and Parrish both cultivate courage by bringing children’s reasoning into the public sphere. In creating the safe spaces that they do, and in not only inviting all contributions, but incorporating them into the discussion, both of these teachers are habituating students to examining their thinking, exposing it to the world, and committing to changing it if it is shown to be wrong. Parrish writes that “In number talks, wrong answers are used as opportunities to unearth misconceptions and for students to investigate their thinking and learn from their mistakes. In number talks classrooms, mistakes play an important role in learning. They provide opportunities to question and analyze thinking, bring misconceptions to the forefront, and solidify understanding” (Parrish, 2010, pp. 11–12). If students can learn in mathematics class that mistakes are opportunities to know better, they may be able to do so in other aspects of their lives, as well.

In Lockhart’s and my classrooms, the courage required to participate is less obvious, because it is more purely intellectual. In both of these lessons, the student is not risking being wrong in front of his peers, he is being asked to venture into the unknown and see what happens. In my imaginary numbers exploration in particular, students are asked to

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create something outside of the bounds of what they know is certain and correct, and arguing on its behalf, showing that it is coherent within the greater set of numbers. In a realm where students have generally always been given perfectly stated boundaries within which to operate, striking into the unknown is intellectually frightening.

6.3.3 Authority and humility

Recall from chapter 4 that intellectual humility was defined a subservience to reason and fact. Or, in Murdoch’s more graceful phrasing, it is a “selfless respect for reality”

(2001, p. 93). Closely bound up with courage, it was the disposition to change one’s position given new information, and it also entailed a disregard for vanity and social status in evaluating a position. In all four teaching examples, teachers demonstrate both of these aspects of humility. Parrish and Lampert, in putting discussion at the center of the activity, give priority to the reasoning expressed by students over their own authority. This is not to say that the teachers abdicate their positions, but rather that positions are evaluated and ideas are discussed based on the reasoning that accompanies them. Because of its structure, mathematics is uniquely suited to this sort of inquiry and teacherly posture; I am reminded of Socrates’ conversation with the Slave Boy in Meno.

But while mathematics is uniquely able to support, and even demands, this kind of intellectual humility, too often teachers in the United States are ill-equipped to engage in it, and so fall back on traditional, authoritative methods. In a qualitative study of United

States elementary mathematics teachers, Liping Ma (2010) asked teachers to suppose that a student came to them with “a new mathematical theory” asserting that when the perimeter of a rectangle increases, so too must its area. Out of 23 teachers, Ma reports that only three attempted to investigate the claim mathematically. Of the three that

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investigated, only one arrived at a correct proposition (that the student’s claim was false).

Thirteen teachers did look for examples of the hypothesis’s alleged truth, but this would be only the very first step in investigating the claim mathematically. Sixteen teachers expressed a desire to find out more along with the inquiring student, but seemingly without a sense of what mathematical investigation even looks like.

Mathematics is an ideal milieu for developing the sorts of intellectual courage and humility that can contribute to socially responsive thinking outside of the classroom. What is needed, however, are teachers who can undertake genuine mathematical inquiry, even at the level of elementary mathematics.

6.4 Teacher beliefs

In 1902, John Dewey wrote that “every study or subject thus has two aspects: one for the scientist as a scientist; the other for the teacher as teacher. These two aspects are in no sense opposed or conflicting. But neither are they immediately identical.” (Dewey,

1902). Research into mathematics teachers’ knowledge of the subject, mathematical knowledge for teaching, did not start in earnest until the 1960’s and 70’s. The first wave of research (e.g. Begle, 1979), was primarily focused on substantiating the seemingly obvious claim that the more mathematical knowledge a teacher has, the more effective he will be as an instructor. No convincing causal link was identified between teachers knowledge,

(measured via number of mathematics courses taken, major or minor in mathematics, and grade point average) and student achievement . The absence of significant results gave rise to a second phase of research, which continued through the 1980’s and sought to more finely asses teachers’ mathematical knowledge of specific content areas (e.g. Baturo &

Nason, 1996; Graeber, Tirosh, & Glover, 1989; Simon & Blume, 1996; Simon, 1993). The

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majority of this research has been done with pre-service teachers, and reveals while the majority are proficient with the algorithms and procedures of elementary mathematics, they lack more general conceptual understandings. These results echo Ma’s findings, above.

A further development in the field brought a number of comparative studies, both comparing elementary teachers to secondary teachers and comparing United States teachers to their counterparts around the world (e.g. Ma, 2010)

Through the late 1990’s, research on teachers’ mathematical beliefs increased, ultimately showing that teachers’ meta-understanding of the discipline shapes the ways that curriculum is implemented. Beliefs have generally been partitioned into: understanding and beliefs about what mathematics is, how mathematics learning occurs in practice, and how mathematics learning ought ideally to occur (Handal, 2003). Under these broad categories, beliefs about personal autonomy, assessment, calculators, instructional strategies, the overarching aim of the school, and attitudes toward available curriculum and school policy all interact. Teachers’ mathematical beliefs tend to cluster around one of two poles. Progressive beliefs are associated with socio-constructivist approaches, which advocate learning mathematics by personally and discursively constructing knowledge.

Problem solving, exploration, and real world applications are emphasized. Traditional instruction is associated with behaviorist approaches, which prioritize knowledge transmission, and figure learning as the assimilation of given content. Traditional approaches value conformity to established ways of thinking and abstract mathematics.

Some studies indicate teachers’ beliefs are well articulated and consciously held, shaping their practice and effect their students’ outcomes (e.g. Stipek, Givvin, Salmon, & MacGyvers,

2001).

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Beliefs about mathematics, curriculum, learning, and teaching all act as filters though which teachers understand their practice and make decisions in the classroom.

This is to say, curricular reforms and instructional materials are always mediated by the teachers’ mathematical beliefs. For this reason, research on efficacy of curricular reforms is often concerned with the extent to which teachers’ beliefs are facilitating or hindering the experimental project (Stemhagen, 2011). A consistent theme is that while a given curricular element is progressive in nature, teachers’ traditional beliefs hinder its implementation (Handal & Herrington, 2003).

For the most part, attention in teacher education programs is so heavily focused on pedagogical technique, that little attention is paid to altering, examining, or developing beliefs about the subject matter (Tillema, 1995). As a result, teachers’ beliefs come, from their own educations and time as students in classrooms. Teacher training programs, by which I mean undergraduate courses in pedagogy and education taken in advance of certification for teaching, are unique among vocational courses, in that only in teacher training courses do students arrive with 12 or more years of observation of the profession they are entering. In the classrooms they have been brought up in, students develop proficiency in the disciplines as well as beliefs about them. When left unexamined, the beliefs that new teachers developed as independent observers persist, becoming influential in their teaching practice (Tillema, 1995).

Stemhagen (2011) offers a recent investigation into teachers beliefs about mathematics that both corroborates many of the earlier findings and sheds light on the particular pressures put on teachers and administrators by the current assessment climate.

He conducts an empirical study of fourth through eighth grade teachers, and finds that:

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“Teachers’ philosophies of mathematics— that is, how they view the nature of their subject matter— are highly correlated with their pedagogical orientation. Philosophy of mathematics is also relatively highly correlated with reported practices” ( Stemhagen,

2011, p. 9, see also Ernest, 1991;). That is to say, teachers’ understanding of mathematics matter. Of particular importance, though, are comments made in the discussion section of the piece, where the author reflects on his interactions with the subjects of the study. The author was disappointed by “the expression of consternation that teachers were being introduced to ideas (the democratic education questions) that would somehow be damaging or distracting from their jobs. One study-team member wondered whether we would disturb/upset the teachers with such aims-related and politically oriented material… [and] one school-district central office employee, after reading the survey, asked why we needed to ask such questions during this time of year when teachers were busy with the requirements of year-end testing” (Stemhagen, 2011, p. 11). Because the administrators’ priority was impact on students’ achievement scores, they were in many ways unable to engage with the questions that the researchers were asking. For these administrators, and for many teachers, there is an experienced gulf between the priorities of mathematics education and the priorities of democratic education.

6.5 Changing the paradigm: Teacher as Mathematician

6.5.1 Math circles

There is a slowly growing trend in North America of Math Circles, which are modeled after similar programs in Eastern Europe. Math circles are organized extra- curricular spaces for students (generally talented middle school and high school students) to explore mathematics in an informal environment. In these spaces, students approach

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mathematical concepts as mathematicians do. The website of one such math circle located in Boston, Massachusetts announces:

Our classes begin with a free discussion of ideas and play of invention around a developing problem; then - once insight blossoms - we link this insight formally to axioms, aiming for elegance and clarity. While the courses are mathematically rigorous, the atmosphere is friendly and relaxed. We want our students to feel free to express their ideas, to suggest their own approaches, and to make mistakes. We work in a spirit of friendship, cooperation, and enjoyment of one another. (“The Math Circle,” n.d.)

One listing shows 23 active math circles in North America as of early 2014. While many of these math circles cite improved performance on academic measures, the math circle model is not directed toward external performance measures, it is simply a space in which to do mathematics.

In addition to their programming for adolescents, The New York Math Circle offers programming for teachers, as well. The format is similar to the student series, in that a space is created for teachers to work on and develop mathematical problems and puzzles, and engage in mathematical discussion outside of their classroom contexts. One outcome of these sessions is that teachers are exposed to interesting problems, many or which are suitable to use as classroom supplements. The immediate end in sight, however, is to provide a space for teachers to develop their own mathematical thinking, by exploring, inventing, and discussing.

6.5.3 Concluding thoughts and open questions: The teacher as

mathematician

This dissertation has made the case that the intellectual virtues, courage and humility in particular, are integral to mathematical practice as well as to democratic social engagement. Mathematics is commonly understood to be apolitical and asocial, and by

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extension the teaching of mathematics is seen as transparently unrelated to social and moral concerns. As I have argued, however, much of contemporary mathematics teaching promotes ways of thinking that are rigid and systematized, and which habituate students to uncritically internalizing received knowledge. These habits are problematic because we exist in a social context where much of the knowledge that we receive bears traces of oppressive social orders. Scholars of education, of gender, and of race have all shown that disrupting oppressive legacies requires changing the way we think about the world. It requires of us as individuals the epistemic courage and humility to call into question the frameworks through which we understand ourselves and the world. Mathematics education, because of the discipline’s social and cultural cache and identification with good clear thinking, plays an important role in shaping the thinking habits we develop. I have made the case that the intellectual virtues called for by social and educational thinkers are in fact characteristic of authentic mathematical practice.

Through the series of vignettes in this final chapter, I have shown that intellectual courage and humility can be cultivated within elementary mathematics, when the teacher has a rich understanding of the role and nature of problem solving and mathematical thinking. It is not the case that genuine mathematical inquiry is suited only for higher mathematics, rather, teachers like Lampert and Parrish facilitate them in their grade school classrooms. Too often, administrative and curricular requirements preclude the sort of work that these teachers are doing. Strict requirements on content sequencing and lesson planning, for example, pressure teachers to conform to traditional skills-oriented practices.

In many districts and schools, teachers are asked to articulate learning objectives for each class period, and some are required to post the day’s objective for the students to see. All

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of the activities during the class period are expected to be in service of learning the day’s objective. Within this requirement, it is difficult if not impossible for a teacher to carve out space for open-ended inquiry or contemplation. There is no room in such a system for genuine (as it has been characterized here) mathematical thinking. Improving mathematics education in the ways recommended here requires problematizing and resisting the paradigm that casts mathematics as the sort of knowledge that can be atomized and prescribed in these ways.

An important lingering question that is raised by the forgoing analysis is whether there can be a causal relationship between thinking well in mathematics and thinking well socially. From an educationalist’s perspective, more research is needed before the claim can be made that promoting flexible and reflective thinking within the walls of the mathematics classroom can bring about flexible and reflective thinking about complex social issues. I have a weak claim that there is an isomorphism between the two here, but the stronger claim that the former can bring about the latter remains open.

A related question, and one that might be instructive in investigating that causal question is whether it is possible to be a good mathematician, and to exemplify mathematical thinking within the discipline, and still be a racist, a misogynist, or to otherwise espouse anti-democratic (in the Deweyan sense) commitments. The mathematician R.L. Moore is an instructive example here, as a successful mathematician who pioneered what is known as “The Moore Method” for teaching higher mathematics through inquiry, and who was known to hold a variety of bigoted beliefs (“Moore_Robert biography,” 2001). The intellectual virtues framework is promising for investigating this question.

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Mathematics education plays an important role in teaching students to think well.

In mathematics class we learn to reason, to distill information, and to leverage the information we have to find out about what we do not yet know. Unfortunately, the current paradigm in mathematics education relies on a deficient understanding of mathematics, and rewards rigid and mechanistic thinking. The paradigm is resilient to attack, however, as paradigms tend to be, because it shrinks from view and lets us believe that it is no more than a faithful representation of the way the world is. If teachers begin to think differently about mathematics, beginning with a new understanding of the role the problems and questions, this hugely influential space may begin to cultivate intellectual courage and humility.

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References

Adler, J. E. (2008). Surprise. Educational Theory, 58(2), 149–174.

Ahmed, S. (2007). A phenomenology of whiteness. Feminist Theory, 8(2), 149–168. doi:10.1177/1464700107078139

Alcoff, L. M. (2010). Book Symposium: Miranda Fricker’s Epistemic Injustice" Power and the Ethics of Knowing. Episteme, 7, 128–137. doi:10.3366/E1742360010000869

Alston, K. (2001). Re / Thinking Critical Thinking : The Seductions of Everyday Life. Studies in Philosophy and Education, 20, 27–40.

Appelbaum, P. M. (1995). Popular Culture, Educational Discourse, and Mathematics. Albany, NY: SUNY Press.

Applebaum, B. (2010). Being White, Being Good: White Complicity, White Moral Responsibility, and Social Justice Pedagogy. Lanham, MD: Lexington Books.

Argyle, S. F. (2012). Mathematical Thinking: From Cacaphony to Consensus. Kent State University.

Aristotle. (1998). The Nichomachean Ethics. (D. Ross, J. . rev. Urmson, & J. L. Ackrill, Trans., D. Ross, Ed.). Oxford, UK: Oxford University Press.

Auden, W. H. (1976). W.H. Auden: Collected Poems. (E. Mendelson, Ed.). New York, NY: Random House.

Aufmann, R. N., & Lockwood, J. (2010). Basic College Mathematics: An Applied Approach (9th editio.). Belmont, CA: Cengage Learning.

Baehr, J. S. (2004). Virtue Epistemology. Internet Encyclopedia of Philosophy. Retrieved from http://www.iep.utm.edu/virtueep/

Bailin, S. (2003). The Virtue of Critical Thinking. Philosophy of Education, 327–330.

Bailin, S., & Siegel, H. (2003). Critical Thinking. In N. Blake, P. Smeyers, R. Smith, & P. Standish (Eds.), The Blackwell Guide to the Philosophy of Education. Oxford, UK: Blackwell Publishing Ltd.

Baturo, A., & Nason, R. (1996). Student teachers’ subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31(3), 235–268. doi:10.1007/BF00376322

169

Bauerlein, M. (2008). The Dumbest Generation: How the Digital Age Stupefies Young Americans and Jeopardizes Our Future(Or, Don’t Trust Anyone Under 30). New York, NY: Penguin Books.

Beckmann, P. (1976). A History of Pi. New York, NY: St. Martin’s Press.

Beckmann, S. (2012). Mathematics for Elementary Teachers with Activities (4th ed.). Pearson.

Begle, E. G. (1979). Critical Variables in Mathematics Education: Findings from a Survey of the Empirical Literature. Washington, D.C.: Mathematical Association of America, National Council of Teachers of Mathematics.

Benacerraf. (1965). What Numbers Could Not Be. The Philosophical Review, 74(1), 47–73.

Biesta, G. (2006). Beyond Learning: Democratic Education for a Human Future. Paradigm Publishers. Boulder, CO: Paradigm Publishers.

Blum, L. (2008). `White privilege’: A mild critique1. Theory and Research in Education, 6(3), 309–321. doi:10.1177/1477878508095586

Boaler, J. (2002). Paying the Price for “Sugar and Spice”: Shifting the Analytical Lens in Equity Research. Mathematical Thinking and Learning, 4(2 & 3), 127–144. doi:10.1207/S15327833MTL04023

Bonilla-Silva, E. (2003). Racism Without Racists: Color-Blind Racism and the Persistence of Racial Inequality in the United States. New York, NY: Rowman and Littlefield.

Boyer, C. B., & Merzbach, U. C. (1968). A History of Mathematics (2nd ed.). USA: John Wiley & Sons.

Brantlinger, a. (2013). Between Politics and Equations: Teaching Critical Mathematics in a Remedial Secondary Classroom. American Educational Research Journal, 50(5), 59–70. doi:10.3102/0002831213487195

Buerk, D. (1982). An Experience with Some Able Women Who Avoid Mathematics. For the Learning of Mathematics, 3(2), 19–24.

Buerk, D. (1985). The voices of women making meaning in mathematics. Journal of Education, 167(3), 59–70.

Castelvecchi, D. (2012). In Their Prime: Mathematicians Come Closer to Solving Goldbach’s Weak Conjecture. The Scientific American. Retrieved from http://www.scientificamerican.com/article.cfm?id=goldbachs-prime-numbers

170

Code, L. (1987). Epistemic Responsibility. Hanover, NH: Brown University Press.

D’Ambrosio, U. (1984). Socio-cultural bases for mathematical education. In Proceeding of the fifth international congress on mathematical education (pp. 1–6). Boston, MA: Birkhauser.

D’Ambrosio, U. (1985). Ethnomathematics and its Place in the History and Pedagogy of Mathematics. For the Learning of Mathematics, 5(1), 44–48.

Davis, P. J. (1989). Applied Mathematics as a Social Contract. In C. Keitel (Ed.), Mathematics, Education, and Society. Paris: UNESCO Document Series no. 35, Science and Technology Education.

Devine, P. G. (1989). Stereotypes and Prejudice: Their Automatic and Controlled Components. Journal of Personality and Social Psychology, 56(1), 5–18.

Dewey, J. (1902). The Child and the Curriculum. Chicago, IL: University of Chicago Press. Dewey, J. (1909). Moral Principles in Education. Boston, MA: Houghton Mifflin.

Dewey, J. (1916). Democracy And Education: An Introduction to the Philosophy of Education. New York, NY: Free Press.

Dewey, J. (1934). The Need for a Philosophy of Education. Schools: Studies in Education, 7(2), 244–245.

Dewey, J. (1938). Experience And Education. New York, NY: Touchstone.

Dewey, J. (2009). Moral Principles In Education (p. 72). Merchant Books.

Eberhardt, J. L., Goff, P. A., Purdie, V. J., & Davies, P. G. (2004). Seeing Black: Race, Crime, and Visual Processing. Journal of Personality and Social Psychology, 87(6), 876–893.

Ernest, P. (1991). The Philosophy of Mathematics Education (Studies in Mathematics Education). Oxon, UK: Routledge.

Ernest, P. (1996). Popularization: Myths, Mass Media, and Modernism. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International Handbook of Mathematics Education, Part 2 (Vol. 64, pp. 785–820). Dordecht, The Netherlands: Springer.

Eves, H. W. (1964). An introduction to the history of mathematics (Revised.). USA: Holt, Rinehart, and Winston.

Fennema, E. H. (2000). Gender and Mathematics: What is Known and What do I Wish Was Known? In Fifth Annual Forum of the National Institute for Science Education. Detroit,

171

MI. Retrieved from http://www.wcer.wisc.edu/archive/nise/news_Activities/Forums/Fennemapaper.ht m

Fennema, E. H., & Sherman, J. A. (1977). Sex-Related Differences in Mathematics Achievement, Spatial Visualization and Affective Factors. American Educational Research Journal, 14(1), 51–71. doi:10.3102/00028312014001051

Fennema, E. H., & Sherman, J. A. (1978). Sex-Related Differences in Mathematics Achievement and Related Factors: A Further Study. Journal for Research in Mathematics Education, 9(3), 189–203.

Feynman, R. P., & Leighton, R. (1997). Surely You’re Joking, Mr. Feynman! (Adventures of a Curious Character). (E. Hutchings, Ed.). New York, NY: W. W. Norton & Company.

Fleener, M. J., Carter, A., & Reeder, S. (2004). Language games in the mathematics classroom: teaching a way of life. Journal of Curriculum Studies, 36(4), 445–468. doi:10.1080/0022027032000150411

Floden, E., & Clark, M. (1988). Teachers for Uncertainty. Teachers College Record, 89(4), 505–524.

Frank, T. (2004). What’s the Matter with Kansas?: How Conservatives Won the Heart of America. New York, NY: Holt Paperbacks.

Frankenstein, M. (1990). Relearning Mathematics: A Different Third R - Radical Maths. London, UK: Free Assn Books.

Frankenstein, M. (1997). Ethnomathematics: Challenging Eurocentrism in Mathematics Education. Albany, NY: State University of New York Press.

Freeman, A. D. (1996). Legitimizing Racial Discrimination through Antidiscrimination Law: A Critical Review of Supreme Court Doctrine. In K. Crenshaw, N. Gotanda, G. Peller, & K. Thomas (Eds.), Critical Race Theory: The Key Writings That Formed the Movement. New York, NY: The New Press.

Freire, P. (1985). Reading the World and Reading the Word: An Interview with Paulo Freire. Language Arts, 62(1).

Freire, P. (2000). Pedagogy of the Oppressed (30th Anniv.). New York, NY: Continuum.

Frye, M. (1983). In and Out of Harm’s Way: Arrogance and Love. In The Politics of Reality: Essays in Feminist Theory (pp. 52–83). Trumansburg, NY: Crossing Press.

Frye, M. (1992). Willful Virgin: Essays in Feminism 1976-1992. Freedon, CA: Crossing Press.

172

Gadamer, H.-G. (1989). Truth and Method. (J. Weinsheimer & D. Marshall, Eds.) (2nd Revise.). New York, NY: Crossroad.

Garrison, J. (2009). Teacher As Prophetic Trickster. Educational Theory, 59(1), 67–83. doi:10.1111/j.1741-5446.2009.00307.x

Gill, P. (1999a). Aspects of Undergraduate Engineering Students’ Understanding of Mathematics. International Journal of Mathematics Education in Science and Technology, 30(4), 557–563.

Gill, P. (1999b). The physics/maths problem again. Physics Education, 34(2), 83–87.

Gilligan, C. (1982). In a Different Voice: Psychological Theory and Women’s Development. Cambridge, MA: Harvard University Press.

Gilligan, C. (2011). Looking Back to Look Forward: Revisiting “In a Different Voice.” Classics@, (9, “Defense Mechanisms”).

Giroux, H. (1988). Schooling and the Struggle for Public Life: Critical Pedagogy in the Modern Age. Minneapolis, MN: University of Minnesota Press.

Giroux, H. (1997). Pedagogy and the Politics of Hope: Theory, Culture, and Schooling, A Critical Reader. Boulder, CO: Westview Press.

Gore, A. (2007). The Assault on Reason. New York, NY: Penguin Books.

Gowers, W. T. (2000). The Two Cultures of Mathematics. In V. Arnold, M. Atiyah, P. Lax, & B. Mazur (Eds.), Mathematics: Frontiers and Perspectives (pp. 65–78). USA: International Mathematical Union. Retrieved from https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

Graeber, A. O., Tirosh, D., & Glover, R. (1989). Preservie Teachers’ Misconceptions in Solving Verbal Problems in Multiplication and Division. Journal for Research in Mathematics Education, 20(1), 95–102.

Graham, R. (2012, June). Carol Gilligan’s Persistent “Voice.” The Boston Globe. Boston, MA. Retrieved from http://www.bostonglobe.com/ideas/2012/06/23/carol- gilligan/toGqkSSmZQC3v4KhFyQ5bK/story.html

Greco, J. (1993). Virtues and Vices of Virtue Epistemology. Canadian Journal of Philosophy, 23, 413–432.

Greene, M. (1988). The Dialectic of Freedom. New York, NY: Teachers College Press.

173

Greenwald, A. G., McGhee, D. E., & Schwartz, J. L. K. (1998). Measuring Individual Differences in Implicit Cognition: The Implicit Association Test. Journal of Personality and Social Psychology, 74(6), 1464–1480.

Grunbaum, B. (2009). An Enduring Error. Elemente Der Mathematik, 89–101.

Gutierrez, R. (2002). Enabling the Practice of Mathematics Teachers in Context : Toward a New Equity Research Agenda. Mathematical Thinking and Learning, 4(2 & 3), 145–187. doi:10.1207/S15327833MTL04023

Gutiérrez, R. (2010). The Sociopolitical Turn in Mathematics Education. Journal for Research in Mathematics Education, 44(1), 37–68.

Gutstein, E. (2003). Teaching and Learning Mathematics for Social Justice in an Urban , Latino School. Journal for Research in Mathematics Education, 34(1), 37–73.

Gutstein, E. (2006). Reading and Writing the World with Mathematics: Toward a Pedagogy for Social Justice. New York, NY: Taylor & Francis.

Halliday, J., & Hager, P. (2002). Context, Judgment, and Learning. Educational Theory, 52(4), 429–443.

Handal, B. (2003). Teachers ’ Mathematical Beliefs : A Review. The Mathematics Educator, 13(2), 47–57.

Handal, B., & Herrington, A. (2003). Mathematics teachers’ beliefs and curriculum reform. Mathematics Education Research Journal, 15(1), 59–69. doi:10.1007/BF03217369

Hanna, G., & de Villiers, M. (2012). Proof and Proving in Mathematics Education: The 19th ICMI Study. New York, NY: Springer.

Hanna, G., & Jahnke, H. N. (1997). Proof and Proving. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International Handbook of Mathematics Education. Dordrecht: Springer Netherlands. doi:10.1007/978-94-009-1465-0

Hansen, D. T. (2001). Exploring the Moral Heart of Teaching: Toward a Teacher’s Creed. New York, NY: Teachers College Press.

Hansen, D. T. (2002). Montaigne and the Values in Educating Judgment. In Philosophy of Education 2002. Urbana, Il.

Hansen, D. T. (2011). The Teacher and the World: A Study of Cosmopolitanism as Education. New York, NY: Routledge.

174

Hare, W. (1979). Open Mindedness and Education. Montreal, Quebec, Canada: Mcgill Queens University Press.

Hare, W. (1985). In Defence of Open-Mindedness. Montreal, Quebec, Canada: Mcgill Queens University Press.

Hare, W. (1992). Humility as a virtue in teaching. Journal of Philosophy of Education, 26(2), 227–236. Retrieved from http://www.williamhare.org/assets/hare_humilityasvirtue.pdf

Hare, W. (2009). Socratic Open-mindedness. Paideusis, 18(1), 5–16.

Haslanger, S. (1993). On Being Objective and Being Objectified. In L. M. Antony & C. E. Witt (Eds.), A Mind of One’s Own: Feminist Essays on Reason and Objectivity (pp. 85–125). Boulder, CO: Westview Press.

Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399. doi:10.1007/BF01273372

Hersh, R. (1998). What is Mathematics, Really? (p. 343). Vintage.

Hersh, R., & John-Steiner, V. (2011). Loving and Hating Mathematics: Challenging the Myths of Mathematical Life. Princeton, NJ: Princeton University Press.

Hofstadter, D. R. (1999). Gödel, Escher, Bach: An Eternal Golden Braid (20th Anniv.). New York, NY: Basic Books.

Hofstadter, R. (1966). Anti-Intellectualism in American Life. New York, NY: Vintage.

Hyslop-Margison, E. J. (2003). The Failure of Critical Thinking : Considering Virtue Epistemology as a Pedagogical Alternative. In Philosophy of Education 2003. Urbana, Il: Philosophy of Education Society.

Hytten, K., & Warren, J. (2003). Engaging Whiteness: How racial power gets reified in education. Qualitative Studies in Education, 16(1), 65–89.

Jacman, S., Goldfinch, J., & Searl, J. (2001). The eﬀectiveness of coursework assessment in mathematics service courses--Studies at two universities. International Journal of Mathematics Education in Science and Technology, 32(2), 175–187.

Jacoby, S. (2008). The Age of American Unreason. New York, NY: Vintage.

Johnson, S. (2010). Teaching Thinking Skills. In C. Winch (Ed.), Teaching Thinking Skills (2nd Editio., pp. 1–50). New York, NY: Continuum.

175

Kant, I. (1999). An Answer to the Question: What is Enlightenment? (1784). In M. J. Gregor (Ed.), M. J. Gregor (Trans.), Practical Philosophy (The Cambridge Edition of the Works of Immanuel Kant) (pp. 16–22). New York, NY: Cambridge University Press.

Kerdeman, D. (2003). Pulled Up Short: Challenging Self-Understanding as a Focus of Teaching and Learning. Journal of Philosophy of Education, 37(2), 293–308. doi:10.1111/1467-9752.00327

Khan Academy. (2014). Retrieved March 25, 2014, from https://www.khanacademy.org/

Khan, S. (2010). YouTube U. Beats YouSnooze Through. The Chronicle of Higher Educaion. Retrieved March 25, 2014, from http://chronicle.com/article/YouTube-U-Beats- YouSnooze/125105/

Kitcher, P. (1977). On the Uses of Rigorous Proof. Science, 196(4291), 782–783.

Kitcher, P. (1984). The Nature of Mathematical Knowledge. New York, NY: Oxford University Press.

Kitcher, P. (1988). Mathematical Naturalism. In P. Kitcher & W. Aspray (Eds.), History and Philosophy of Modern Mathematics, XI. Minneapolis, MN: University of Minnesota Press.

Kitcher, P., & Aspray, W. (1988). An Opinionated Introduction. In P. Kitcher & W. Aspray (Eds.), History and Philosophy of Modern Mathematics, XI. Minneapolis, MN: University of Minnesota Press.

Kline, M. (1974). Why Johnny Can’t Add: The Failure of the New Math. New York, NY: Vintage.

Kline, M. (1980). Mathematics: The Loss of Certainty. New York, NY: Oxford University Press.

Kohlberg, L. (1958). The Development of Modes of Thinking and Choices in Years 10 to 16. University of Chicago.

Kohlberg, L. (1981). The Philosophy of Moral Development. San Francisco, CA: Harper and Row.

Kontorovich, A. (2007). Transgressing the Boundaries: Toward a Non-Hegemonic Feminist Mathematical Theory. Absolutely Regular. Retrieved March 18, 2014, from http://absolutely-regular.blogspot.com/2007/01/transgressing-boundaries-toward- non.html

Kuhn, T. S. (1996). The Structure of Scientific Revolutions (3rd ed.). Chicago: University of Chicago Press.

Lakatos, I. (1961). Essays in the Logic of Mathematical Discovery. University of Cambridge.

176

Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery. (J. Worrall & E. Zahar, Eds.). Cambridge, UK: Cambridge University Press.

Lampert, M. (1990a). When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching. American Educational Research Journal, 27(1), 29. doi:10.2307/1163068

Lampert, M. (1990b). When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching. American Educational Research Journal, 27(1), 29. doi:10.2307/1163068

Lampert, M. (2003). Teaching Problems and the Problems of Teaching. New Haven, CT: Yale University Press.

Langton, R. (2000). Feminism in Epistemology: Exclusion and Objectification. In M. Fricker & J. Hornsby (Eds.), The Cambridge Companion to Feminism in Philosophy (pp. 127– 145). Cambridge, UK: Cambridge University Press.

Larvor, B. (1998). Lakatos: An Introduction. New York, NY: Routledge.

Lerman, N. (2000). The Social Turn in Mathematics Education Research. In J. Boaler (Ed.), Multiple Perspectives on Mathematics Teaching and Learning (pp. 19–44). Westport, CT: Ablex.

Lindberg, S. M., Hyde, J. S., Petersen, J. L., & Linn, M. C. (2010). New trends in gender and mathematics performance: A meta-analysis. Psychological Bulletin, 136(6), 1123–1135. doi:10.1037/a0021276

Linnebo, Ø. (2011). Platonism in the Philosophy of Mathematics. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Fall 2011 edition). Retrieved from http://plato.stanford.edu/entries/platonism-mathematics/

Livingston, C. V. (2009). The Privilege of Pedagogical Capital: A Framework for Understanding Scholastic Success in Mathematics. In P. Ernest, B. Greer, & B. Sriraman (Eds.), Critical Issues in Mathematics Education (p. 502). Missoula, MT: Information Age Publishing.

Lloyd, E. A. (1993). Pre-theoretical assumptions in evolutionary explanations of female sexuality. Philosophical Studies, 69(2-3), 139–153. doi:10.1007/BF00990080

Lockhart, P. (2009). A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form (p. 140). New York, NY: Bellevue Literary Press.

177

Lockhart, P. (2012). Measurement. Cambridge, MA: Belknap Press of Harvard University Press.

Long, C. T., DeTemple, D. W., & Millman, R. (2008). Mathematical Reasoning for Elementary Teachers (5th ed.). Pearson.

Ma, L. (2010). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series) (p. 232). Routledge.

Macintyre, A. (2007). After Virtue: A Study in Moral Theory, Third Edition [Paperback] (3rd ed.). Notre Dame, IN: University of Notre Dame Press.

Masterman, M. (1970). The Nature of a Paradigm. In I. Lakatos & A. Musgrave (Eds.), Criticism and the Growth of Knowledge (pp. 59–89). Cambridge, UK: Cambridge University Press.

Maxwell, J. (1989). Mathephobia. In P. Ernest (Ed.), Mathematics Teaching: The State of the Art (pp. 221–226). London, UK: Falmer Press.

Mayer, R. E., & Wittrock, M. C. (1996). Problem Solving Transfer. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of Educational Psychology (pp. 47–62). Mahwah, NJ: Lawrence Erlbaum Associates.

Mayo, C. (2004). Certain Privilege : Rethinking White Agency. In C. Higgins (Ed.), Philosophy of Education 2004 (pp. 308–316). Urbana, IL: Philosophy of Education Society.

McClintock, R. (2004). Homeless in the House of Intellect. New York: Laboratory for Liberal Learning.

McClintock, R. (2012). Enough: A Pedagogic Speculation. New York, NY: Collaboratory for Liberal Learning.

McIntosh, P. (1997). White Privilege and Male Privilege: A Personal Account of Coming to See Correspondences Through Work in Women’s Studies. In R. Delgado & J. Stefancic (Eds.), Critical White Studies: Looking behind the mirror (pp. 291–199). Philadelphia, PA: Temple University Press.

McKinnon, C. A. (1987). Feminism Unmodified: Discourses on Life and Law. Boston, MA: Harvard University Press.

Mehta, J. (2013). The Allure of Order: High Stakes, Dashed Expectations, and the Quest to Remake American Education. New York, NY: Oxford University Press.

178

Melhuish, E. C., Sylva, K., Sammons, P., Siraj-Blatchford, I., Taggart, B., Phan, M. B., & Malin, A. (2008). The Early Years: Preschool Influence on Mathematics Achievement. Science, 321(5893), 1161–2.

Mellin-Olsen, S. (1977). Indlæring som social proces: Fra Piaget til Marx. Copenhagen, Denmark: Rhodos.

Mellin-Olsen, S. (1987). The Politics of Mathematics Education. Dordecht, The Netherlands: Springer.

Mendick, H. (2006). Masculinities in Mathematics. McGraw Hill International.

Mendick, H., Moreau, M.-P., & Hollinworth, S. (2008). Mathematical Images and Gender Identities: A report on the gendering of representations of matheamtics and mathematicians in popular culture and their influences on learners. Project Report. Bradford, UK.

Montmarquet, J. (1987). Epistemic Virtue. Mind, 96, 482–497.

Moore_Robert biography. (2001). MacTutor History of Mathematics. Retrieved May 12, 2014, from http://www-groups.dcs.st- and.ac.uk/~history/Biographies/Moore_Robert.html

Moses, R. P., & Cobb, C. (2002). Radical Equations: Math Literacy and Civil Rights. Boston, MA: Beacon Press.

Mulryan, S. (2009). The Courage of Dialogue. (D. Kerdeman, Ed.) (pp. 141–149). Urbana, IL: Philosophy of Education Society.

Munroe, R. (2008). Purity. xkcd.com.

Murdoch, I. (2001). The Sovereignty of Good (2nd ed.). New York, NY: Routledge.

Murphy, R., Gallagher, L., Krumm, A. E., Mislevy, J., & Hafter, A. (2014). Research on the Use of Khan Academy in Schools.

National Council of Teachers of Mathematics. (1980). An Agenda for Action. Reston, VA.

National Council of Teachers of Mathematics. (2000). Professional Standards for School Mathematics. Reston, VA.

National Governors Association Center for Best Practices & The Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, D.C.

179

National Science Foundation. (2009). Women, Minorities, and Persons with Disabilities in Science and Engineering: 2009. Arlington, VA.

Parrish, S. (2010). Number Talks: Helping Children Build Mental Math and Computation Strategies. Sausalito, CA: Math Solutions.

Picker, S. H., & Berry, J. (2000). Investigating pupils’ Images of Mathematicians. Educational Studies in Mathematics, 43(1), 65–94.

Polya, G. (1957). How to Solve It: A New Aspect of Mathematical Method (2nd ed.). Princeton, NJ: Princeton University Press.

Powell, A. B., & Frankenstein, M. (1997). Ethnomathematics: Challenging Eurocentrism in Mathematics Education. Albany, NY: SUNY Press.

Rhee, R. J. (2007). The Socratic Method and the Mathematical heuristic of George Polya. St John’s Law Review, 81(4), 881–898. Retrieved from http://scholarship.law.stjohns.edu/lawreview/vol81/iss4/5

Roberts, R. C., & Wood, W. J. (2007). Intellectual Virtues: An Essay in Regulative Epistemology. Oxford, UK: Oxford University Press.

Rota, G. C. (1997). Indiscrete Thoughts. (F. Palombi, Ed.). Boston, MA: Birkhauser.

Russell, B. (1918). Mysticism and Logic. Mineola, NY: Dover.

Russell, B., & Griffin, N. (1992). The selected letters of Bertrand Russell: The Public Years, 1914-1970. New York, NY: Allen Lane.

Säätelä, S. (2011). From Logical Method to “Messing About”: Wittgenstein on Open Problems. In O. Kuusela & M. McGinn (Eds.), The Oxford Handbook of Wittgenstein. New York, NY: Oxford University Press.

Schoenfeld, A. H. (1985). Students’ Beliefs about Mathematics and their Effects on Mathematical Performance: A questionnaire analysis. In Annual Meeting of the Americal Educational Research Association. Chicago, IL.

Schoenfeld, A. H. (1987). Polya, Problem Solving, and Education. Mathematics Magazine, 60(5), 283–291. doi:10.2307/2690409

Schoenfeld, A. H. (1988). When Good Teaching Leads to Bad Results: The Disasters of “Well-Taught” Mathematics Courses. Educational Psychologist, 23(2), 145–166. doi:10.1207/s15326985ep2302_5

180

Schoenfeld, A. H. (1989). Exploration of Students’ Mathematical Beliefs ad Behavior. Journal for Research in Mathematics Education, 20(4), 338–355.

Schroeder, T., & Lester, F. (1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), New directions for elementary school mathematics (pp. 31–42). Reston, VA: NCTM.

Schulz, K. (2010). Being Wrong: Adventures in the Margin of Error. New York, NY: Harper Collins.

Shenkman, R. (2008). Just How Stupid Are We?: Facing the Truth About the American Voter. New York, NY: Basic Books.

Sherman, J. A., & Fennema, E. H. (1977). The Study of Mathematics By High School Girls and Boys: Related Variables. American Educational Research Journal, 14(2), 159–168. doi:10.3102/00028312014002159

Simon, M. A. (1993). Prospevtive Elementary Teachers’ Knowledge of Division. Journal for Research in Mathematics Education, 24(3), 233–254.

Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15(1), 3–31.

Simons, M., & Masschelein, J. (2008). The Governmentalization of Learning and the Assemblage of a Learning Appartus. Educational Theory, 58(4), 391–415.

Singh, S. (1996). Fermat’s Last Theorem. U.K.: BBC TV/WGBH Boston.

Skovsmose, O. (1980). Forandringer i matematikundervisningen. Copenhagen, Denmark: Gyldendal.

Skovsmose, O. (1981). Matematikundervisning og kritisk pædagogik. Copenhagen, Denmark: Gyldendal.

Skovsmose, O. (1994). Towards a Philosophy of Critical Mathematics Education. Dordecht, The Netherlands: Kluwer.

Skovsmose, O., & Greer, B. (2012). Opening the Cage: Critique and Politics of Mathematics Education. Rotterdam, The Netherlands: Springer.

Skovsmose, O., & Nielsen, L. (1997). Critical Mathematics Education. In A. J. Bishop, M. A. (Ken) Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International Handbook of Mathematics Education, Part 2 (4th ed., Vol. 64, pp. 1257–1288). Dordecht, The Netherlands: Kluwer.

181

Sockett, H. (2012). Knowledge and Virtue in Teaching and Learning: The Primacy of Dispositions. New York, NY: Routledge.

Sommers, C. H. (2001). The War Against Boys: How Misguided Feminism Is Harming Our Young Men. New York, NY: Simon & Schuster.

Sosa, E. (1993). Proper Functionalism and Virtue Epistemology. Nous, 27(1), 51–65.

Sosa, E. (1999). How must Knowledge be Modally Related to What is Known? Philosophical Topics, 26(1-2), 373–384.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building Student Capacity for Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform Classrooms. American Educational Research Journal, 33(2), 455–488. doi:10.3102/00028312033002455

Stemhagen, K. (2006). Social Justice Mathematics: Rethinking the Nature and Purposes of School Mathematics. Philosophy of Mathematics Education, 19.

Stemhagen, K. (2011). Democracy and School Math: Democracy and Education, 19(2), 1–13.

Stigler, J. W., & Hiebert, J. (2009). The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom (Updated.). New York, NY: Free Press.

Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17(2), 213–226.

Sullivan, S. (2006). Revealing Whiteness: The Unconscious Habits of Racial Privilege (American Philosophy). Bloomington, IL: Indiana University Press.

Swalwell, K. M. (2013). Educating Activist Allies: Social Justice Pedagogy with the Suburban and Urban Elite (Critical Social Thought). New York, NY: Routledge.

Tatum, B. D. (2003). “Why Are All the Black Kids Sitting Together in the Cafeteria?” and Other Conversations About Race. New York, NY: Basic Books.

The Center for Education Reform. (2014). K-12 Facts. Retrieved March 18, 2014, from http://www.edreform.com/2012/04/k-12-facts/

The Math Circle. (n.d.). Retrieved March 19, 2014, from http://www.themathcircle.org/

Tillema, H. H. (1995). Changing the professional knowledge and beliefs of teachers: A training study. Learning and Instruction, 5(4), 291–318.

182

Walle, J. Van de, Karp, K. S., & Bay-Williams, J. M. (2012). Elementary and Middle School Mathematics: Teaching Developmentally (8th Edition) (Teaching Student-Centered Mathematics Series). Boston, MA: Pearson.

Warnick, B. R., & Stemhagen, K. (2007). Mathematics teachers as moral educators: the implications of conceiving of mathematics as a technology. Journal of Curriculum Studies, 39(3), 303–316. doi:10.1080/00220270600977683

Weisstein, N. (1993). Power, Resistance and Science: A call for a revitalized feminist psychology. Feminism and Psychology, 3, 239–245.

Willis, S. (1995). Mathematics: From Constructing Privilege to Deconstructing Myths. In J. Gaskell & J. Willinsky (Eds.), Gender In/Forms Curriculum: From Enrichment to Transformation (Critical Issues in Curriculum Series) (pp. 262–284). New York, NY: Teachers College Press.

Willis, S. (1996). Gender Justice and the Mathematics Curriculum: Four Perspectives. In L. H. Parker, L. J. Rennie, & B. J. Fraser (Eds.), Gender, Science and Mathematics: Shortening the Shadow (pp. 41–52). Dordecht, The Netherlands: Springer.

Wilson, T. (2003). Maxine’s Table: Connecting Action with Imagination in the thought of Maxine Greene and Hannah Arendt. Educational Theory, 53(2), 203–220.

Wittgenstein, L. (1983). Remarks on the Foundations of Mathematics. (G. E. M. Anscombe, Trans., G. H. von Wright & R. Rhees, Eds.) (Revised Ed.). Boston, MA: MIT Press.

Yancy, G. (2008). Black Bodies, White Gazes: The Continuing Significance of Race. Lanham, MD: Rowman and Littlefield.

Zagzebski, L. (1996). Virtues of the Mind. Cambridge, UK: Cambridge University Press.

183