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

Abstract

Abstract. The goal of this paper is to increase our understanding of different approaches to proving in advanced mathematics. We present two case studies from an interview-based investigation in which students were asked to complete proof-related tasks. The first student consistently took what we call a referential approach toward these tasks, examining examples of the objects to which the mathematical statements referred, and using these to guide reasoning. The second consistently took what we call a syntactic approach toward these tasks, working logically with definitions and proof structures without reference to examples. Both students made substantial progress on each of the tasks, but they exhibited different strengths and experienced different difficulties. In this paper we: demonstrate consistency in these students' approaches across a range of tasks, examine the different strengths and difficulties associated with their approaches to proving, and consider the pedagogical issues raised by these apparent student preferences for reasoning in certain ways.

Undergraduates’ Example Use in Proof Construction: Purposes and Effectiveness

“…Each of these tasks was chosen so that they could be sensibly approached with or without the use of examples. For the relations task, for instance, a participant could examine particular functions that satisfy the relation or they could immediately try to prove from the definition that the relation was reflexive, symmetric, and transitive (see Alcock & Weber, 2005, 2009, for case studies of students who did each of these two things). We note that the two tasks are of different types: one says "prove or disprove" and the other says "prove."…”

“…Brad, Karen, and Lisa all constructed examples of increasing functions and generated explanations based on these examples that were similar to Hannah's. (Brad's argument is studied in detail in Alcock & Weber, 2005, 2009). However, none came close to formalizing their arguments as proofs.…”

Factors Influencing Students’ Propensity for Semantic and Syntactic Reasoning in Proof Writing: a Case Study

“…Finally, as Table 1 illustrates, when researchers claim that students have a semantic or syntactic reasoning style in advanced mathematics, they often base these findings as much on interview data as on performance on mathematical tasks (e.g., Alcock and Weber 2010;Duffin and Simpson 2006;Pinto and Tall 2002;Raman 2003;Weber 2009). For Caleb, we document patterns of reasoning he used in his calculus and linear algebra tasks and report on interview data when he was asked how he commonly approached these proofs.…”

“…In mathematics education, researchers have also hypothesized that, like mathematicians, students may have cognitive styles in advanced mathematics (e.g., Alcock and Inglis 2008;Simpson 2004, 2005;Alcock and Weber 2010;Duffin and Simpson 2006;Moutsios-Rentzos 2009;Pinto and Tall 2002;Raman 2003;Weber 2009), and have begun to explore what makes students with either cognitive style successful or unsuccessful.…”

“…Alternatively, an individual can understand and reason about mathematical concepts using informal representations of these concepts, such as thinking about these concepts in terms of graphs, diagrams, gestures, or examples Tall 1999, 2001;Raman 2003;Vinner 1991;Alcock 2004, 2009). Recently a number of research reports have identified individual students or groups of students who predominantly engage in one of these two types of reasoning while rarely engaging in the other (e.g., Alcock and Inglis 2008;Simpson 2004, 2005;Alcock and Weber 2010;Duffin and Simpson 2006;MoutsiosRentzos 2009;Tall 1999, 2002;Weber 2009). However, in these studies, students' reasoning styles were often identified by their performance on a small number of tasks and these tasks were nearly always situated in a single mathematical domain.…”