Mathematicians’ perspectives on the teaching and learning of proof

Abstract

Abstract. This paper reports on an exploratory study of mathematicians' views on the teaching and learning that occurs in a course designed to introduce students to mathematical reasoning and proof. Based on a sequence of interviews with five mathematicians experienced in teaching the course, I identify four modes of thinking that these professors indicate are used by successful provers. I term these instantiation, structural thinking, creative thinking and critical thinking. Through the mathematicians' comments, I explain these modes and highlight ways in which students sometimes fail to use them effectively. I then discuss teaching strategies described by the participants, relating these to the four modes of thinking. I argue that teaching aimed at improving structural thinking tends to dominate, and that courses that introduce proof, regardless of classroom organization, should address all four modes in a balanced and integrated way.

Undergraduates’ Example Use in Proof Construction: Purposes and Effectiveness

“…Alcock (2004Alcock ( , 2009 found that mathematicians taking part in a research study indicated that examples were useful to them for three specific purposes: understanding the meaning of mathematical statements, generating ideas for how a statement might be proven, and checking that inferences drawn within a proof are not invalid. Alcock and Inglis (2009) Proof strategies are not explicitly listed as a topic of discussion.…”

“…Housman and Porter (2003) found that when introduced to a new concept or asked to evaluate conjectures, many undergraduates did not generate examples unless they were explicitly prompted to do so. Similarly, the mathematicians interviewed by Alcock (2004Alcock ( , 2009 lamented that the undergraduates in their transition-to-proof courses rarely seemed to use examples when working on proofs.…”

Referential and syntactic approaches to proving: Case studies from a transition-to-proof course

“…The issue of how referential and syntactic strategies contribute to effective mathematical reasoning is both interesting and complex. Certainly a successful mathematician will be able to use both, switching from one to the other in response to the perceived demands of the situation (Alcock, 2008). For instance, a mathematician who is uncertain about the truth of a statement is likely to examine examples in a more or less systematic counterexample search, whereas one who is certain of a result may proceed directly to working syntactically with appropriate definitions (Inglis, Mejia-Ramos, & Simpson, 2007).…”

Does Content Matter in an Introduction-to-Proof Course?

“…For instance, in one study, Alcock [2] interviewed five mathematicians experienced in teaching an introduction-to-proof course. She identified four modes of thinking (instantiation, creative thinking, critical thinking, and structural thinking) considered important by the mathematicians for successful proving.…”

“…Her conclusion was that "it certainly seems reasonable to claim that collaborative classroom environments, in which students investigate, refine, and prove mathematical conjectures" [2, page 94] foster the flexible use of all four modes. Although researchers (e.g., [2,8]) have discussed pedagogical strategies, implications, and suggestions for teaching these four modes of thinking, they did not specify which mathematical content would be useful in a transition-to-proof course for developing these four modes of thinking, nor did they address the question of whether transition-to-proof courses adequately prepare students for more advanced mathematics courses.…”