Abstract
This paper aims to illustrate how a teacher instilled norms that regulate the theorem construction process in a three-dimensional geometry course. The course was part of a preservice mathematics teacher program, and it was characterized by promoting inquiry and argumentation. We analyze class excerpts in which students address tasks that require formulating conjectures, that emerge as a solution to a problem and proving such conjectures, and the teacher leads whole-class activities where students’ productions are exposed. For this, we used elements of the didactical analysis proposed by the onto-semiotic approach and Toulmin’s model for argumentation. The teacher’s professional actions that promoted reiterative actions in students’ mathematical practices were identified; we illustrate how these professional actions impelled students’ actions to become norms concerning issues about the legitimacy of different types of arguments (e.g., analogical and abductive) in the theorem construction process.
Highlights
In an inquiry classroom, an atmosphere of intellectual challenge is generated in which students are expected to: (i) propose and defend mathematical ideas and conjectures and (ii) respond thoughtfully to the mathematical arguments of their peers
Mathematical inquiry begins when a task is proposed that requires solving an openended problem using a Dynamic Geometry Software (DGS), formulating a conjecture that encapsulates the solution of the problem, and proving the conjecture
In attempting to answer this question, we focus on some specificities or concretizations of teachers’ professional actions described in the literature, and we illustrate how, in the course, the students’ and teacher’s actions feed upon each other
Summary
An atmosphere of intellectual challenge is generated in which students are expected to: (i) propose and defend mathematical ideas and conjectures and (ii) respond thoughtfully to the mathematical arguments of their peers. The mathematical practice of a classroom with these characteristics requires focusing on students’ production and teacher and students collectively building norms (social and socio-mathematical) that regulate and support these practices [1]. Examples of socio-mathematical norms are: research in mathematics involves creatively solving problems; valid arguments should be based on properties of mathematical objects [1,4,5]; open-ended problems require exploration, formulation of conjecture, and argumentation of conjecture [3]. Conner et al [6]
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