Geometry Teachers Perspectives on Convincing and Proving When Installing a Theorem in Class

Recent research on mathematics learning has called attention to the nature of the situation that serves as context for that learning (Brousseau, 1997; Lave & Wenger, 1991; Schoenfeld, 1998). In our research, we conceive of classroom life as organized by recurring instructional situations: frames that allow teacher and students to exchange the work they do for claims on the stakes of teaching and learning. Decisions and actions made by teacher or students not only result from individual thinking or belief, but also respond to the norms of the instructional situation in which those decisions and actions are made. Depending on the instructional situation, different actions may be normative. For example, in the situation we’ve called ‘doing proofs’ in high school geometry classrooms, it is normative for the teacher to state the proposition to be proved in terms of a specific diagram (Herbst & Brach, 2006), but in the situation of ‘installing’ a new theorem, it is normative for the teacher to state a theorem in terms of abstract concepts (Herbst & Nachlieli, 2007). In this paper, we advance understanding of instructional phenomena by focusing on the norms of those instructional situations as they relate to two different activities which have traditionally been studied in their cognitive and epistemological dimensions: convincing (or bringing someone to a state of belief) on the truth of a statement (Harel and Sowder, 1998) and proving (or establishing the truth of a statement for a given community of knowers; Balacheff, 1987). We argue that a teacher’s management of those activities requires them to respond to norms of the instructional situations where those activities occur. A teacher’s response to those norms is constructed by decisions and actions that articulate various dispositions that might contradict each other. We illustrate this point examining further the situation of installing theorems in the high school geometry class and drawing from teachers’ responses to an animated representation of the teaching of a theorem about medians.