Critical Postmodern Theory Critical Postmodern Theory in ...

RUNNING HEAD: Critical Postmodern Theory

Critical Postmodern Theory in Mathematics Education Research: A Praxis of Uncertainty Submitted by David W. Stinson, Ph.D. Georgia State University [email protected] Erika C. Bullock Georgia State University [email protected]

Submitted to Educational Studies in Mathematics Special Issue Mathematics Education and Contemporary Theory Editors: Tony Brown, Ph.D. Manchester Metropolitan University Margaret Walshaw, Ph.D. Massey University

Critical Postmodern Theory

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Abstract In this article, the authors provide an overview of mathematics education as a research domain, identifying and briefly discussing four shifts or historical moments in mathematics education research. Using the Instructional Triangle as a point of reference for the dynamics of mathematics instruction, they illustrate how mathematics education researchers working in various moments reconceptualize not only the interactions among teachers, students, and mathematics but also teachers, students, and mathematics as subjects of inquiry as they ask different questions made possible by different theoretical perspectives. The authors next provide brief descriptions of critical theory and postmodern theory, and suggest critical postmodern theory (CPT) as a composite theory that offers a praxis of uncertainty for reconceptualizing and conducting mathematics education research. To conclude, they return to the Instructional Triangle to provide research exemplars of how teachers, students, and mathematics might indeed be reconceptualized within a critical postmodern theoretical perspective.

Keywords: critical theory, critical postmodern theory, mathematics education research, postmodern theory


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Even after 3 decades of the Mathematics for all rhetoric in the United States (Martin, 2003), the discipline of mathematics continues to be used for social stratification. Beginning as early as third grade (or sooner), children are often tracked or ―sorted‖ according to their perceived mathematics ―abilities‖ (Oates, Ormseth, Bell, & Camp, 1990). This (perceived) ability tracking, specifically in mathematics, continues to be a fundamental aspect of U.S. education even after research has shown the positive possibilities of de-tracked mathematics classrooms (Boaler & Staples, 2008). In the United States, as in most Western countries, it appears that we are stuck in a discursive binary (cf. Derrida, 1974/1997) derived from a Platonic Ideal (cf. Plato, trans. 1996): either a child possesses the aristocratic characteristic of intelligence or not, and those who do ―should persist in their studies until they reach the level of pure thought, where they will be able to contemplate the very nature of number‖ (Plato, trans. 1996, p. 219). This discursive practice (cf. Foucault, 1969/1972) of linking (perceived) mathematics ability to the aristocratic characteristic of intelligence, and thus to power and privilege, continues to permeate Western ideology. Bourdieu (1989/1998) referred to this linkage as ―psychological brutality,‖ highlighting the reproductive stratifying strategies of schools that lay down final judgments and verdicts with no appeal, ―ranking all students in a unique hierarchy of forms of excellence, nowadays dominated by a single discipline, mathematics‖ (p. 28). For those who argue that schools evolved not in the pursuit of equity but to (re)produce the social stratification needs of capitalism (Bowles, 1977; Bowles & Gintis, 2002), this strategic hierarchical sorting mechanism of school mathematics although problematic is, in fact, somewhat welcomed. Indeed, the often-repeated concepts of capitalism such as individualism, innovation, entrepreneurship, competitiveness, and preeminence permeate U.S. education policies; with the latter two clearly entrenched in


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mathematics education policy. Gutstein (2009, 2010) recently drew explicit parallels between U.S. education policies, capitalism, income inequalities, and persistent inequities of educational opportunities in general and mathematics education in particular. In this article, to critical examine and deconstruct the persistent inequities of mathematics education or, more specifically, to open up the ―fictions, fantasies and plays of power inherent in mathematics education‖ (Walkerdine, 2004, p. viii), we make a case for considering critical postmodern theory (CPT) (Kincheloe & McLaren, 1994) in mathematics education research. We believe that CPT provides a means to make visible the Trojan Horse of the mathematics for all discourse by critically deconstructing the implicit and explicit plays of power working within the discourse by motivating different questions for mathematics education researchers to explore. We structure the article into three sections. In the first section, we provide an overview of mathematics education as a research domain, identifying and briefly discussing four shifts or moments in mathematics education research. We demonstrate how different theoretical perspectives permit researchers working in various moments to reconceptualize the interactions among teachers, students, and mathematics while simultaneously reconceptualizing teachers, students, and mathematics as subjects of inquiry. In the second section, we provide brief descriptions of critical theory and postmodern theory, and make a case for CPT as a composite theory that offers a praxis of uncertainty. In the concluding third section, we provide exemplars of how teachers, students, and mathematics might indeed be reconceptualized within a critical postmodern theoretical perspective. Zooming Out and In Yet Again In this section, we do not attempt to provide a comprehensive historical review of mathematics education as a research domain; that has been done elsewhere (Kilpatrick, 1992).


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Nor do we intend to suggest that the four shifts or historical moments of mathematics education research that we identify and discuss here—the process-product moment (1970s–), the interpretivist-constructivist moment (1980s–), the social-turn moment (mid 1980s–), and the sociopolitical-turn moment (2000s–)—have developed in some discrete, linear fashion, moving toward some ―ideal‖ of conceptualizing and conducting mathematics education research. Rather, we understand the four moments as distinct yet overlapping and simultaneously operating moments; therefore, we do not identify end dates. Furthermore, we understand that our attempt to mark the beginning of a moment within a specific decade is somewhat misleading, given that there have been researchers who began developing different possibilities for mathematics education research long before the decades that we identify.1 Nonetheless, because mathematics education as a research domain prior to the 1970s is rooted largely in a combination of two disciplines: mathematics and cognitive psychology (Kilpatrick, 1992), we start our discussion with the 1970s, identifying this decade as the beginning of the process-product moment. Most of the research in this moment attempts to quantify ―effective‖ mathematics teaching; quantitative statistical inference is the primary methodology. Here mathematics teachers‘ classroom practices are described (process) and linked to student outcomes (product) (see, e.g., Good & Grouws, 1979). Securely embedded in the Enlightenment, this moment is theoretical grounded in positivism; its aim is to predict social phenomena by ―objectively‖ observing and measuring a ―reasonable‖ universe. In the late 1970s 1

For example, Brownell (1947/2004) began writing about children making meaning of mathematics

decades before the interpretivist-constructivist moment of the 1980s. Likewise, Frankenstein (1983/1987) and Skovsmose (1985) began examining the sociopolitical implications of critical mathematics education several years before the sociopolitical-turn moment of the 2000s.


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and early 1980s, however, mathematics education researchers began transitioning away from quantitative statistical inference and toward qualitative methodologies adapted from the disciplines of anthropology, cultural/social psychology, history, philosophy, and sociology (Lester & Lambdin, 2003).2 Because research methodologies are inextricably linked to theoretical perspectives, the favoring of qualitative methodologies transitioned many mathematics education researchers into new theoretical perspectives such as interpretivism and constructivism. Although embedded in the Enlightenment, within the interpretivist-constructivist moment (1980s–), the aim of research is no longer to predict social phenomena, but rather to understand it. For example, researchers explore how mathematics teachers come to understand or make meaning of their pedagogical practices (Thompson, 1984), or how students come to understanding or making meaning of the very mathematics they are taught (Steffe & Tzur, 1994). Here mathematics teaching and learning are examined within the dynamic interactions among teachers-and-students and students-andstudents as they engage with mathematics in the classroom. Figure 1, The Instructional Triangle is a familiar and often-referenced model within U.S. mathematics education research intended to 2

It is important to note that statistical inferential research has experienced a strong resurgence in

the United States as randomized- and quasi-experimental research have been established as the ―gold standard‖ in education research generally and mathematics education research specifically, based largely on two highly influential reports: Scientific Research in Education (National Research Council, 2002) and Foundations for Success: The Final Report of the National Mathematics Advisory Panel (NMAP, 2008; for a critique of the final report and of the resurgence of statistical inferential research, see Kelly, 2008).


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represent these dynamics. [Figure 1 about here] Figure 1. The Instructional Triangle.3 Although it has been argued that the Instructional Triangle is a useful model to conduct ―research in education‖ (Ball & Forzani, 2007, p. 529, emphasis in original), as mathematics education researchers continue to explore the complexities of mathematics teaching and learning, adapting methodologies and theoretical perspectives from other disciplines, some have begun to understand the indispensable requirement of contextualizing not only the triangle but also the interactions among teachers, students, and mathematics. In so doing, they make the social turn in mathematics education research as they acknowledge that meaning, thinking, and reasoning are products of social activity (Lerman, 2000). For example, in the social-turn moment (mid1980s–), mathematics teaching and learning is understood within the sociocultural contexts in which it is learned (Carraher, Carraher, & Schliemann, 1987), the situated contexts in which it is practiced (Boaler, 1998), and the taken-as-shared meanings constructed in the mathematics classroom (Cobb, Perlwitz, & Underwood, 1996). This further expansion of mathematics education research from its roots of mathematics and cognitive psychology, however, presents researchers 3

The Instructional Triangle: Instruction as the interaction among teachers, students, and

mathematics, in contexts. From NRC. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press, p. 314. Copyright 2001 by the National Academy of Sciences. Reprinted by permission.


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with unique challenges as they attempt to bring together individual agency and trajectories and the socio-cultural and -historical origins of the ways people think, act, and understand the world (Lerman, 2000). Research in this moment in general does not abandon psychology altogether but rather calls for a sociocultural, discursive psychology in which mathematics teaching and learning might be understood as a particular moment in the zoom of a lens (Lerman, 2001). By zooming out, mathematics education researchers explore not only the complexities of the concentric contexts in which the Instructional Triangle is embedded (e.g., classroom, school, district, community, society) but also the multiplicities of the socio-cultural and -historical discourses that construct and continuously shape those contexts (Weissglass, 2002). By zooming in, researchers explore the dynamic complexities of how socio-cultural and -historical discourses have constructed and continuously shape teachers, students, and mathematics—thus, the possibility of the very existence of the triangle. This back-and-forth zooming has compelled a small (but growing) number of mathematics education researchers to abandon theoretical perspectives that merely explore understanding social phenomena and to embrace theoretical perspectives that explore emancipation from or deconstruction of social phenomena. These researchers, in many ways, have adopted ―a degree of social consciousness and responsibility in seeing the wider social and political picture‖ (Gates & Vistro-Yu, 2003, p. 63) of mathematics education research. Seeing the wider social and political picture characterizes the sociopolitical-turn moment (2000s–) in mathematics education research; it signals a shift toward ―theoretical perspectives that see knowledge, power, and identity as interwoven and arising from (and constituted within) social discourses‖ (Gutiérrez, 2010, p. 4). Researchers who position their work within this moment recognize that their work is ―profoundly cultural and political‖ (Hardy, 2004, p. 105) as


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they use old theoretical perspectives in novel and unexpected ways and/or embrace contemporary theoretical perspectives. For example, social theory is employed to deconstruct mathematics education policy, demonstrating how it reifies mathematics as an institutional space of whiteness (Martin, 2010); critical theory is used as a foundation for mathematics curriculum, illustrating the empowering possibilities of mathematics teaching and learning for students and teachers alike (Gutstein, 2003); critical race theory is applied to the schooling experiences of mathematically successful middle school boys, highlighting the endemic nature of race and racism in schools generally and school mathematics specifically (Berry, 2008); and postmodern theory is employed to reexamine girls‘ participation in the mathematics classroom, turning earlier research upside down (Walshaw, 2001). The sociopolitical-turn moment, as we envision it, permits mathematics education researchers to trouble the Instructional Triangle—its existence, its assumptions, and its implications—by maintaining the exhausting process of concurrently zooming out and zooming in on the triangle only to zoom out and in yet again. This simultaneous zooming out/in steals the innocence of the triangle, deconstructing it, as the discursive binaries used to name the vertices, and thus the triangle, are put under erasure (cf. Derrida, 1974/1997). Here teachers, students, and mathematics are understood as discursive formations (cf. Foucault, 1969/1972), named and renamed (but not determined) within hegemonic socio-cultural, -historical, and -political assumptions, conditions, and power relations. With this simultaneously zooming out/in, the vertices are no longer brought into focus, but become monsters, no longer intelligible, as they resist the surveilling and disciplining gazes of normalization (cf. Foucault, 1977/1995). We believe that the sociopolitical-turn moment has the potential to move mathematics education researchers away from an agenda that primarily explores questions of how to improve


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mathematics teaching and learning toward an agenda strongly concerned with the question of why mathematics education (Pais, Stentoft, & Valero, 2010). In exploring this—in many ways, forbidden—why question, mathematics education as a research domain is cracked wide open, revealing its fictions, fantasies, and plays of power (Walkerdine, 2004), its inclusions and exclusions (Skovsmose, 2005). Within the sociopolitical-turn moment, we believe that CPT provides a means to not only ask this forbidden why question but also other why and how questions, opening up a praxis of uncertainty within mathematics education research. A Praxis of Uncertainty In this section, we do not provide accounts of what mathematics education research ―looks like‖ within critical theory or postmodern theory; that has been done elsewhere (see, e.g., Burton, 2003; Walshaw, 2004, 2010). Nor do we discuss the complexities and often inherent contradictions of pulling concepts from different theories while conducting research; that too, has been done elsewhere (see, e.g., Brown & Jones, 2001; Koro-Ljungberg, 2004; Lather, 1991). Our intent here is to briefly describe critical theory and postmodern theory from our current understandings of these complex and far-reaching (and somewhat contradictory) theories, and suggest that concepts from both theoretical perspectives might be used side by side—like tools pulled from a tool box—to short-circuit systems of power (Foucault, 1975/1996). Consequently, we believe for a researcher to assume a fixed theoretical perspective has the same effect as choosing a hammer to tackle every household task: some things may be repaired quickly while others may remain unrepaired or, worse yet, irreparably broken. As a single household task may require multiple tools for repair, a single question or social phenomenon may require multiple concepts derived from different theories for analysis, which demands a break from theoretical fundamentalism (Lather, 2006) in favor of a more eclectic


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theoretical approach (Stinson, 2009). Lerman (2000) suggested theoretical flexibility as a ―space for the play of ideologies in the process‖ (p. 20) of adapting and applying theory. We see theoretical flexibility as sifting data through one theoretical sieve, analyzing what is captured, and then catching that which remains with the next sieve of theory. Effective use of theory therefore requires that the researcher assume the responsibility of scholarly work: the difficult intellectual work of studying the strengths and weaknesses and the convergences and divergences of different theoretical concepts pulled from (at times) contradictory theoretical perspectives (Paul & Marfo, 2001). Our purpose here is to do this difficult—yet rewarding—work as it relates to critical theory and postmodern theory. CPT, as a composite theory, we believe, holds fruitful possibilities for reconceptualizing and conducting mathematics education research. We next turn to our brief descriptions of critical theory and postmodern theory, followed by a description of what we call a praxis of uncertainty of critical postmodern theory. Critical Theory The origin of critical theory is associated with the Frankfurt School (circa 1920), which holds a Marxist theoretical perspective: to critique and subvert domination in all its forms (Bottomore, 2001). In the most general sense, critical theory maintains sociopolitical critiques on social practices and ideology that mask ―systematically distorted accounts of reality which attempt to conceal and legitimate asymmetrical power relations‖ (Bottomore, 2001, p. 209). As an activist and emancipatory project, critical theory calls its claimant to question the structures that are developed and maintained by ―constructors‖ (Skovsmose, 2005, p. 140) and manifested as false consciousness for those who are constructed within hegemonic power. Hegemony constructs people as objects—those who are acted upon, rather than Subjects, those who act—


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who become so entrenched in their own oppressive condition that they do not realize their own subjugation or their complicity in the perpetuation of unjust social and economic systems (Freire, 1970/2000). Although emancipatory (at times), employing critical theory creates a tremendous burden of responsibility for the researcher because she or he must reflect upon and touch the problem in order to change it, instead of simply talking about it (Crotty, 1998); she or he must respond to Marx‘s chastening: ―The philosophers have only interpreted the world in different ways; the point is to change it‖ (as cited in Rush, 2004, p. 10, emphasis in original). Employing critical theory therefore generally requires the researcher to use her or his scholarship to dismantle the constructors‘ hegemonic power and the reproduction and execution of that power through institutions such as media and schools (Slott, 2002). The critical theorist questions the production, validation, dissemination, and reproduction of knowledge through these institutions; there is no innocence in even those efforts to include those who have hitherto been alienated (Marx & Engels, 1848/1978) in the conversations: that which appears to be neutral is too often rife with hegemonic power. Critical theorists therefore call for all efforts to disseminate knowledge to be accompanied by an investigation of not only its relation to ideology and power but also the subjectivities of those who (re)produce knowledge (Leistyna & Woodrum, 1996). As a modernist project, embedded in the Enlightenment, critical theorists believe that as marginalized individuals and/or groups become critically aware of their ―true‖ situation, intervene in its reality, thus take charge of their destiny, they will exercise their right to participate with a critical consciousness in the socio-cultural and -historical transformation of their society (Crotty, 1998). Accordingly, critical mathematics (accredited to critical theory) acknowledges students (and teachers) as members of a society rife with hegemonic power and builds mathematics


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around the cultural identities of students in such a way that doing mathematics necessarily takes up social and political issues as it aims to provide access to dominant mathematics (i.e., school mathematics) for those who have traditionally been excluded (Gutiérrez, 2002). Skovsmose (1994) named this critical theoretical perspective mathemacy: using mathematics as a sociopolitical tool ―to organize and reorganize interpretations of social institutions, traditions and proposals for political reforms‖ (p. 39). Therefore, a challenge for critical mathematics education ―is to retrieve and reshape school mathematics so that it is empowering for all peoples and also edifying for the human spirit of all‖ (Ernest, 2010, p. 82). In short, critical mathematics education researchers attempt to broaden the accessibility and purposes of school mathematics in hopes of transforming mathematics teaching and learning into ways that are more equitable and just. Postmodern Theory Postmodern theory is a critique of the Enlightenment that rejects any static foundational systems of logic, resulting in truth—and thus, knowledge—becoming fluid and avoiding any claims of certainty (Ernest, 1997). Here thought (or knowledge) is understood not as a denial of the existence of Truth, but rather as an acceptance of multiple forms of truth, made and re-made with/in socio-cultural, -historical, and -political discourses (Foucault, 1984/1996). Discourses, however, are no longer mere intersections of things and words that might be heard or read but rather discursive practices that systematically form the possibilities (and impossibilities) of knowledge discourses, too often (re)producing régimes of truth (Foucault, 1969/1972, 1977/1980). Postmodern theory then is ―a movement of ‗unmaking‘‖ (R. Wolin, cited in Crotty, 1998, p. 192) these régimes, pulling apart reductionistic discursive binaries (e.g., White/Black, man/woman, rational/irrational, etc.) that undergird notions of universal Truth. This pulling


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apart—or deconstruction (cf. Derrida, 1974/1997)—of discursive binaries aims to unsettle and displace binary hierarchies, to uncover their historically contingent origin and politically charged roles, not to provide a ―better‖ foundation for knowledge and society, but to dislodge their dominance, creating a social space that demonstrates its tolerance of difference, ambiguity, and playful innovations (Seidman, 1994). In embracing difference and ambiguity, the postmodern theorist rejects the single story or grand meta-narrative (cf. Lyotard, 1979/1984), which attempts to bring order and closure to a world spinning out of control (Usher & Edwards, 1994). Central to the making of the metanarrative is the subject (i.e., the individual) who has been wholly constituted by and with/in discursive practices. In postmodern research, however, the subject does not exist as a unified or fixed being but rather as an infinitely dynamic—neither one nor multiple—fragmented self (Stinson, 2010). Here the subject, no longer defined once and for all, is continually created and re-created through socio-cultural, -historical, and -political discourses. But while the subject is limited by these discourses in ways that she or he may not comprehend, they do not define her or him. The subject is always already open to the possibilities of subversive repetition: subverting (or not) the very discourses that attempt to construct or constitute her or him (Butler, 1990/1999). In the end, within postmodern theory, both knowledge and identity are infinitely dynamic rhizomatic processes always already open to different lines of flight (Deleuze & Guattari, 1980/1987). Thus mathematics, within the context of postmodern theory, might be understood as a discursively constituted ―language game‖4 of deeply entrenched rules and patterns that are not 4

See Ernest (1998) for a discussion of Wittgenstein‘s philosophy of mathematics as ―a whole

language-game with questions and answers‖ (L. Wittgenstein, as cited in Ernest, 1998, p. 76).


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only stable and enduring but also always already open to the possibility of change—and, in fact, do change (Ernest 2004). This infinitely dynamic postmodern approach to knowledge, including knowledge created by mathematics education researchers, takes critique a step further by examining mathematics education research itself, investigating how researchers are complicit in the (re)production of the power relations inherent in the discursive practices of mathematics and mathematics education research (Valero, 2004). In short, postmodern theory provides a vehicle through which mathematics education researchers can unmask both the apparent neutrality of mathematics education reform (Hardy, 2004) and their own complicity in (re)producing the régime of truth ―that the teaching and learning of mathematics deserves a privileged place in the education of all‖—a privileging that, in the end, is complicit in ―the degradation of the human spirit‖ (Ernest, 2010, pp. 72–73). Critical Postmodern Theory Employing concepts from critical theory and postmodern theory—or any other theoretical combination—side by side is messy work that is ―necessary and fruitful in ‗the search for meaning‘‖ (Cook, as cited in Lather, 2010, p. 9). Working against theoretical fundamentalism (Lather, 2006), CPT operates as a differential consciousness, which Sandoval (2004) compared to the clutch of an automobile: ―the mechanism that permits the driver to select, engage, and disengage gears in a system for the transmission of power‖ (p. 203). She further described differential consciousness as representative of the variance that emerges out of correlations, intensities, junctures, and crises. As we consider critical theory and postmodern theory independently, we encounter such variance from which CPT—the synergy of the two—emerges (Kincheloe & McLaren, 1994). To illustrate this synergy, we provide an example of how


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marginalization and resistance might be reconceptualized when considering the both-and theoretical perspective of critical postmodern theory rather than the either-or perspective. While both critical and postmodern theorists are concerned with marginalization and resistance, their approaches are indeed significantly different. Consider, for example, the language game of mathematics. The critical theorist sees the game on two distinct playing fields: dominant mathematics (or as previously noted, school mathematics) and critical mathematics (also consider ethnomathematics and/or liberatory mathematics) (Gutiérrez, 2002). Dominant mathematics is a system established as right and True by the White, middle-class men who have historically controlled and constructed the game; controlling not only the rules of the game but also those who might even gain access. Critical mathematics, however, is an oppositional system that exposes the power dynamic between the oppressor—White, middle-class, male mathematicians—and the oppressed—the marginalized Other, with the hope of opening the field of dominant mathematics to new players. The challenge here, however, is in continuing the ascribed privilege granted to the field of dominant mathematics; unfortunately, critical mathematics is too often reduced to a mere ―bridge‖ that only leads students to the delimiting possibilities of ―real‖ dominant mathematics. In this context, the possibilities of mathematics teaching and learning remain limited and oppressive as dominant mathematics maintains a régime of truth, yielding no real sense of liberation. The postmodern theorist, on the other hand, troubles not only the game and both playing fields but also any emancipatory project intended to transform the game. In other words, when considering the game of mathematics, the postmodern theorist questions not only the rules and the fields but also the very existence of the game. Why play? Through continual de-construction and re-construction, the game of mathematics itself exists in a perpetual state of becoming


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(Fleener, 2004) as inner forces and external circumstances intertwine and cross over the shifting boundaries of the game, flowing one way and ebbing the other (Ernest, 2004). In the end, the issue for the postmodernist is not emancipatory transformation of the game but rather purpose: forcing mathematics education researchers to think about that which we do not think about, but yet, cannot, not think about (Fleener, 2004). Or, said in another way, how might we come to know ―in not-ever-knowing,‖ but yet, continue to learn ―something of that ‗not-ever‘ of knowing (I. Stronach, as cited in Brown & Jones, 2001, p. 104). This knowing through not-ever knowing leads the postmodern theorist to advise extreme caution with the hopeful emancipation of critical theory because any emancipatory project presupposes values that cannot be known universally or once and for all (Brown & Jones, 2001). This cautious stance, however, results in critics of postmodern theory to be rather skeptical of the possibilities of any practical contribution postmodern theory might make to mathematics teaching and learning (Klein, 2002). So in returning to the game of mathematics, we are confronted on both sides by the question ―Why play?‖ The postmodern theorist questions the very existence of the game, while, in response, the critical theorist questions her or his purpose if the game is altogether dissolved. But what if we hope for emancipation while simultaneously placing emancipation under erasure? What if we do act while simultaneously rethinking our rethinking of action? What if we put our knowing to work while simultaneously acknowledging not-ever knowing? In short, what if we engage in praxis while simultaneously opening praxis to uncertainty? But then, what is a praxis of uncertainty? Borrowing from Freire (1970/2000), we see praxis as a continuous cycle of action and reflection in which sacrificing action equates to empty verbalism (i.e., an alienated and alienating ―blah, blah‖) while sacrificing reflection equates to


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mere activism (i.e., action for action‘s sake). Freire believed: ―There is no true word that is not at the same time praxis. Thus, to speak a true word is to transform the world‖ (p. 87). But here, within a praxis of uncertainty, speaking a true word and transforming the world are both left open to multiplicitous possibilities; we can only know what these phrases might mean in our notever-knowing as we continue learning in and through that not-ever of knowing. Indeed, we concede that this ―theoretical border crossing‖ (Walshaw, 2004, p. 11) is messy and not without its own challenges. But in spite of these challenges, we argue that CPT is a valid approach, believing that both the seductions of and resistance to postmodern theory can assist us getting smart about the limits of critical theory (Lather, 2007). Or, said in another way, the synergy between critical theory and postmodern theory is found in the ―interplay between the praxis of the critical and the radical uncertainty of the postmodern‖ (Kincheloe & McLaren, 1994, p, 144). Therefore, the critical postmodern mathematics education researcher is concerned with a continual reconceptualization of the game of mathematics—leaving teachers, students, and mathematics always already open to uncertain possibilities. The Instructional Triangle and Critical Postmodern Theory In this concluding section, we return to the Instructional Triangle to view it from the perspective of CPT. Previously, we argued that within the sociopolitical-turn moment (2000s–) the vertices and, in turn, the triangle itself, become unintelligible with the simultaneous zooming out/in, opening the triangle—thus, teachers, students, and mathematics—to uncertain possibilities. Our purpose in troubling the Instructional Triangle was not to claim that it is a ―bad‖ model or representation of the complexities of mathematics teaching and learning but only to suggest that it is ―dangerous.‖ Being dangerous, however, means that there is more work to do, and leads not to apathy but to hyper- and pessimistic activism (Foucault, 1983/1997). That


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activism, for us, is to consider CPT as a composite theory that offers a praxis of uncertainty in reconceptualizing not only the triangle but also each vertex. Here we briefly summarize three research articles, each primarily focused, in turn, on a single vertex, that we believe exemplify this praxis of uncertainty. That is to say, research articles in which the respective authors, we claim, are working in the synergistic space of the praxis of the critical and the radical uncertainty of the postmodern. Teachers. In ―Plotting Intersections Along the Political Axis: The Interior Voice of Dissenting Mathematics Teachers,‖ de Freitas (2004) uses ―fiction-as-research‖ to access inner dissenting voices to illustrate how the discursive practices of mathematics instruction are determined by the regulative and normative discourses that frame society. Here de Freitas is compelled to use fiction, as only through fiction can these dissenting voices of mathematics teachers be explicitly heard, because ―fiction, as a methodology, has the potential to defamiliarize, to cross boundaries, to transgress cultural norms‖ (p. 272). She argues that the mathematics teacher is in a marginalized space relative to the dominant discourse of mathematics education and positions her work as an effort to ―re-write the mainstream from a marginal post‖ (p. 262). Agnes, the fictional teacher of de Freitas‘s inquiry, reflects upon her experiences as both a student and teacher of mathematics. She recalls times when, as an exemplary mathematics student, she questioned the purpose of the mathematics tasks that she encountered, surmising that the only one who stood to benefit was the teacher. As the student, Agnes believed her spoken voice was mere disruptive interference. Agnes then laments that now as the mathematics teacher she is ―part of the fraudulence that torments youth‖ (p. 268) and expresses remorse for the students for whom she continues to surrender to normative expectations due to their exhaustion


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produced by resistance. Agnes emerges resolutely from her guilt and confusion determined to expose the scandalous foundation of mathematics to right a terrible wrong. Students. In ―Discourse Positioning and Emotion in School Mathematics Practices,‖ Evans, Morgan, and Tsatsaroni (2006) address the often-ignored affective dimension of mathematics teaching and learning. By adopting an interdisciplinary critical theoretical approach, which draws on discourse theory, social theory, semiotics, and psychoanalysis, they avoid the binaries of individual/social and cognitive/affective. They examine pedagogic discourse using both structural and textual analysis to highlight the positions, power relations, and emotional opportunities within the official and everyday discourses that surround three eighth-grade mathematics students in Lisbon, Portugal. Evans and colleagues use official policy and student self-evaluation for structural analysis to inform the positions available for students and the power relationships contained therein. This form of analysis also reveals how ―often there is more than one available position for an individual, either within one discourse or several competing discourses‖ (p. 213). They combine the structural analysis with textual analysis of classroom video transcripts that include emotional observations to reveal how the students reproduce discursive positionings and how those discourses conflict. Perhaps the most significant observation in the study is the effect of the teacher‘s intervention. As students combine local and academic discourses to struggle through a mathematics problem, the teacher‘s intervention, despite helpful intentions, ―re-establishes traditional pedagogic relations…elicits obedience from the students, and acceptance of positioning as followers, rather than as directors of their own learning‖ (p. 223). This intervention also inadvertently dictates the acceptable student emotions in the classroom. In the end, Evans and colleagues suggest that considering the limitations and


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conflicts both within and among discourses operating in the classroom can open up possibilities for developing student emotional competencies along with mathematical skills. Mathematics. In ―Criticisms and Contradictions of Ethnomathematics,‖ Pais (2011) proposes the expansion of the common definition of ethnomathematics as indigenous—or nonschool—mathematics to include ―a social, historical, political, and economical analysis of how mathematics has become what it is today‖ (p. 210). He argues that, to date, ethnomathematics has been a project focused on defining non-Western and non-European mathematical practice in Western and European terms. Students‘ local or indigenous mathematical knowledge, then, becomes an actor in a foreign classroom, a process that only reifies its marginalization. In this context, emancipation becomes ―access to an participation in a world where mathematical knowledge is central‖ (p. 214) and multicultural mathematics education values what the culture can contribute to Western mathematical understanding rather than the culture itself. Ethnomathematics, then, exists only as a counternarrative within the existing structure of dominant mathematics rather than a voice calling for a reconceptualization of the very structure and existence of mathematics. Pais posits that mathematics‘ emancipatory and empowering value is not a result of the knowledge that it offers to its possessor, but rather as a result of the social capital that it affords her or him. Pais calls for mathematics education researchers who employ ethnomathematics to consider how they too are complicit in perpetuating ―Otherness‖ through ―the benevolent multicultural ideas [they] want to enforce‖ (p. 227). These studies demonstrate how the vertices of the Instructional Triangle are no longer reduced to static objects of inquiry but rather are understood as dynamic subjects of rhizomatic possibilities named and re-named with/in discursive practices. de Freitas (2004) has opened not only the teacher to new possibilities but also the mathematics education research text (de Freitas


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& Nolan, 2007) by using fiction to amplify the dissenting voice of the mathematics teacher transgressing normative discourses. By avoiding the individual/social and cognitive/affective discursive binaries, Evans, Morgan, and Tsatsaroni (2006) reconceptualize the relationship between the student and mathematics when the students‘ mathematical experience is opened to the uncertainty of the multiplicity of discursive positioning(s). And Pais (2011) folds the emancipatory project of ethnomathematics onto itself opening ethnomathematics and the entire emancipatory project of critical mathematics to radical uncertain possibilities. We have attempted here to illustrate the value and the possibilities of this synergistic space of critical theory and postmodern theory based upon our current understanding of these complex and ever-evolving theories. Theoretical perspective—claimed or not, articulated or not—determines not only the questions a researcher might explore but also, and perhaps more importantly, the findings. All research is value-laden (Lather, 1991): but too often mathematics—positioned as an asocial, ahistorical, and apolitical discipline (de Freitas, 2004)— has manifested mathematics education research that posits itself to be atheoretical (Lerman, Xu, & Tsatsaroni, 2002). Removing mention of theory from mathematics education research, however, does not absolve the researcher of theoretical responsibilities or the subsequent consequences of her or his research. Mathematics education research is not an innocent science that exists outside of socio-cultural, -historical, and -political discourses (Gutiérrez, 2010). Therefore, we challenge mathematics education researchers to determine their own synergistic theoretical space that might offer a praxis of uncertainty within their research passions—be it teachers, students, mathematics, or the complex multiplicity of interactions therein.


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Critical Postmodern Theory Figure 1

The Instructional Triangle: Instruction as the Interaction Among Teachers, Students, and Mathematics, in Contexts

SOURCE: Adapted from Cohen and Ball, 1999, 2000, in press.

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