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Signs of Power: A Critical Approach to the Study of Mathematics Cognition and Instruction

  • Author(s): Gutierrez, Jose Francisco;
  • Advisor(s): Abrahamson, Dor;
  • et al.

My dissertation investigates the social power dynamics inherent in a single mathematics classroom and their effects on the quality of students’ engagement and therefore learning. I examine the ways in which individual learning and relations of power are mutually constituted through discourse. That is, I view discourse not only as a lens for investigating these phenomena, but as the medium through which mathematical knowledge and power relations simultaneously become objectified, stabilized, and reproduced. To trace this “learning–power imbrication” and its impact on students’ identity, agency, and conceptual understanding of mathematics, I conducted a year-long participant ethnography where I immersed myself within a high school mathematics community.

I present a theoretical model of mathematics teaching and learning that captures the fundamentally dialectical relationship between objectification and subjectification, which I refer to as the “obj–subj dialectic.” Luis Radford’s semiotic–cultural approach to the study of mathematics learning is central to my research on the “obj” side of the dialectic. I focus particularly on his theory of knowledge objectification, which was developed in the context of algebraic-generalization activity. To research the “-subj” side, I combine theoretical perspectives set forth by Anna Sfard and collaborators with those of Rom Harré to analyze the positional identities that emerge during social interaction. I used this theoretical model to conduct a series of qualitative microgenetic analyses of the emergence of the learning–power imbrication during one particular instructional lesson centered on an algebraic-generalization activity.

As students attempted to construct mathematical generalizations, their discursive actions created hierarchical positional identities that they then, in turn, took up, accepted, contested, negotiated, or rejected. Thus some students gain “mathematical ascendency” over others, which is a construct I propose to describe the co-construction of hierarchical subject positions emerging from the obj–subj dialectic during multimodal interactions. Also, whereas the obj–subj dialectic accounted for emergent forms of power, other aspects of the interaction were better explained through the lens of systemic dimensions of power, such as teachers’ orientations and the enactment of these orientations. In particular, my data analysis shows that a teacher’s “cognitive–conceptual script” resulted in differential learning opportunities.

The research contributes theory refinement and novel methodological techniques for both the learning sciences and critical educational studies. These contributions include: (1) a qualification of Luis Radford’s theory of knowledge objectification, specifically as it relates to students’ mathematical generalizations; (2) empirical evidence supporting the general hypothesis that “learning” and “relations of power” are intrinsically reciprocal and mutually constitutive; and (3) analytic techniques for identifying and describing specific mechanisms of power inherent to mathematics instruction and their consequences on learning.

Ultimately, the research presented in this dissertation should inform educational practice. In particular, the signs of power articulated in this work should illuminate and complexify professional discourse and design efforts toward better serving students who have been historically underrepresented in the field of mathematics to navigate both its conceptual and systemic challenges.

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