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Marta Kobiela and Richard Lehrer

We examined the codevelopment of mathematical concepts and the mathematical practice of defining within a sixth-grade class investigating space and geometry. Drawing upon existing literature, we present a framework for describing forms of participation in defining, what we term aspects of definitional practice. Analysis of classroom interactions during 16 episodes spanning earlier and later phases of instruction illustrate how student participation in aspects of definitional practice influenced their emerging conceptions of the geometry of shape and form and how emerging conceptions of shape and form provided opportunities to develop and elaborate aspects of definitional practice. Several forms of teacher discourse appeared to support students' participation and students' increasing agency over time. These included: (a) requesting that members of the class participate in various aspects of practice, (b) asking questions that serve to expand the mathematical system, (c) modeling participation in aspects of practice, (d) proposing examples that create contest (i.e., monsters), and (e) explicitly stating expectations of and purposes for participating in the practice.

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Marta Kobiela, Kara J. Jackson, Annie Savard, and Emily Shahan

All too often, definitions of mathematical ideas and objects are presented as facts to memorize in mathematics classrooms (de Villiers 1998; Keiser 2000). This is unfortunate, as it means that students are not provided opportunities to engage in a form of reasoning that is arguably at the heart of mathematics—definitional reasoning. Making sense of, constructing, and using definitions to determine what counts as an object (e.g., an odd number, a triangle) develop students' ability to communicate about mathematical ideas and their conceptual understanding of properties and relations (Keiser 2000).