Transforming Perceptions of Proof: A Four-Part Instructional Sequence

Abstract

Mathematics teachers are expected to engage their students in critiquing and constructing viable arguments. These classroom expectations are necessary, given that proof is a central mathematical activity. However, mathematics teachers have been provided limited opportunities as learners to construct arguments and critique the reasoning of others, and hence have developed perceptions of proof as an object that must follow a strict format. In this article, we describe a four-part instructional sequence designed to broaden and deepen teachers' perception of the nature of proof. We analyzed participants' reflections on the instructional sequence in order to gain insight into (a) the differences between this instructional sequence and participants' previous proof learning opportunities and (b) the ways this activity was influential in transforming participants' perceptions of proof. Participants' previous learning experiences were focused on memorizing and reproducing textbook or instructor proofs, and our sequence was different because it actively and collaboratively engaged participants in constructing their own arguments, critiquing others' reasoning, and creating criteria for what counts as proof. Participants found these activities transformative as they became more clear about what counts as proof, began to view proof as socially negotiated, and expanded their conception of proof beyond a rigid structure or format.

Who Should Learn Proving and Why: An Examination of Secondary Mathematics Teachers’ Perspectives

“…In this way, an emphasis on disciplinary obligations, even if intended to motivate all students to experience this important mathematical practice, may be perceived by teachers as a rationale for emphasizing proof only with those students who are pursuing higher mathematics. Further research is needed on these dynamics, and we think it would be particularly wise to look further at the role of teachers' own conceptions of proof (e.g., Boyle et al, 2015;Knuth, 2002) and how those perceptions are linked to their perspectives on who should learn proof.…”

Exploring Prospective Teachers’ Ability to Generate and Analyze Evidence-based Explanatory Arguments

“…Prior research in the area of mathematical argumentation includes studies conducted primarily within the proofrelated research paradigm. This body of research largely explored PSTs' conceptions of mathematical argumentation and their argumentation skills from the perspective of proof (e.g., Boyle, Bleiler, Yee, & Ko, 2015;Felton, 2007;Martin & Harel, 1989;Morris, 2007;. Some researchers studied PSTs' proof-related knowledge with a focus on specific proof methods (e.g., Stylianides, Stylianides, & Phillippou, 2007).…”

A multi-faceted framework for identifying students’ understanding of the generality requirement of proof

“…Individual written work was used to assess students' current understanding of proof and ability to construct proofs. Task was posed in this manner to minimize the mathematical vocabulary and evoke uncertainty about the claim's validity (Buchbinder & Zaslavsky, 2008) 14 Students developed shared criteria for features of "good proofs"; based on Boyle et al (2015).…”