Proof, Proving, and Teacher-Student Interaction: Theories and Contexts
New ICMI Studies Series
This chapter focuses on the teachers' role in teaching proof and proving in the mathematics classroom. Within an over-arching theme of diversity (of countries, curricula, student age-levels, teachers' knowledge), the chapter presents a review of three carefully-selected theories: the theory of socio-mathematical norms, the theory of teaching with variation, and the theory of instructional exchanges. We argue that each theory starts by abstracting from observations of school mathematics
... s. Each then uses those observations to probe into the teachers' rationality in order to understand what sustains those classroom contexts and how these might be changed. Here, we relate each theory to relevant research on the role of the teacher in the teaching and learning of proof and proving. Our review offers evidence and support for mathematics educators meeting the challenge of theorising about proof and proving in mathematics classrooms across diverse contexts worldwide. 262 didactics in mainland Europe (c.f., Best, 1988; Chevallard, 1999a; Murphy, 2008) . The word instruction, as used by Cohen, Raudenbush and Ball (2003) to refer to the interactions among teacher-students-content in classroom environments, is probably a better word to designate the locus of the phenomena we target. In focusing on teacher-student interaction, we acknowledge that what learners bring to the classroom (from developmental experiences prior to schooling, to ongoing experiences across varied out-of-school contexts) impacts on such interactions, just as, most certainly, can the diversity of countries, of instructional courses, of student ages, of levels of teacher knowledge, and so on, around the world. Whatever the terminology, our over-arching focus is on the teacher -and, in particular, on the teacher's part in the teacher-student interactions that occur day-to-day in mathematics classrooms. In theorising about proof, proving, and teacher-student interaction, we are aware that theories can appear in different guises and operate at different levels and grain sizes. As Silver and Herbst (2007) identify in their analysis, there can be "grand theories", "middlerange theories", and "local theories": where "grand theories" aim at the entirety of phenomena within, say, mathematics education; "middle-range theories" focus on subfields of study; and "local theories" apply to specific phenomena within the field. We also note Kilpatrick's (2010, p. 4) observation: "To call something a theory ... is an exceedingly strong claim". It is not our intention to consider whether or not some proposed approach is, or is not, a "theory"; rather, we use the term "theory" as short-hand for 'theoretical framework', 'theoretical perspective', 'theoretical model', or other equivalent terms. Across all these considerations, we take proof and proving to be "an activity with a social character" (Alibert & Thomas, 1991, p. 216). As such, mathematics classroom communities involve students in communicating their reasoning and in building norms and representations that provide the necessary structures for mathematical proof to have a central presence. Hence, our focus on the role of the teacher in teaching proof and proving in mathematics encompasses the teacher managing the work of proving in the classroom even when proof itself is not the main object of teaching. Clearly, in such situations proofs may be requested, and offered, even when proof itself is not the object of study; such possibilities hinge on customary practices (including matters of language) that the teacher has the responsibility to establish and sustain. Balacheff (1999) , Balacheff (2009) and Sekiguchi (2006) , for examples, have studied these forms of classroom practices, and the role of the teacher in establishing and sustaining the practices. As Balacheff (2010, p. 116-117) shows, basing classroom practices on "grand theories" such as those of Piaget or Vygotsky has not worked very well. Balacheff argues "The responsibility for all these failures does not belong to the theories which supposedly underlie the educational designs, but to naive or simplifying readers who have assumed that concepts and models from psychology can be freely transferred to education". Balacheff goes on to consider the didactical complexity of learning and teaching mathematical proof by analyzing the gap between knowing mathematics and proving in mathematics. In contrast, our approach in selecting relevant theories to review is to choose ones that represent ongoing and current foci for classroom-based research and, importantly, that start from the abstraction of observations in existing school mathematics classrooms. Using these criteria, we review the theory of socio-mathematical norms, the theory of teaching with variation, 263 and the theory of instructional exchanges. We conclude by giving pointers to future research -both empirical and theoretical -that we hope can advance the field.