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

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

In this paper, we present data from an exploratory study that aimed to investigate the ways in which, and the extent to which, undergraduates enrolled in a transition-to-proof course considered examples in their attempted proof constructions. We illustrate how some undergraduates can and do use examples for specific purposes while successfully constructing proofs, and that these purposes are consistent with those described by mathematicians. We then examine other cases in which students used examples ineffectively. We note that in these cases, the purposes for which the students attempted to use examples are again appropriate, but the implementation of their strategies is inadequate in one of two specific ways. On this basis we identify points that should be borne in mind by a university teacher who wishes to teach students to use examples effectively in proof-based mathematics courses

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

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

How mathematicians obtain conviction: implications for mathematics instruction and research on epistemic cognition

The received view of mathematical practice is that mathematicians gain certainty in mathematical assertions by deductive evidence rather than empirical or authoritarian evidence. This assumption has influenced mathematics instruction where students are expected to justify assertions with deductive arguments rather than by checking the assertion with specific examples or appealing to authorities. In this paper, we argue that the received view about mathematical practice is too simplistic; some mathematicians sometimes gain high levels of conviction with empirical or authoritarian evidence and sometimes do not gain full conviction from the proofs that they read. We discuss what implications this might have, both for for mathematics instruction and theories of epistemic cognition

Challenges in mathematical cognition: a collaboratively-derived research agenda

This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education, psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into six broad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in mathematical cognition

Mathematics education research on mathematical practice

In the mathematics education research literature, there is a growing body of scholarship on how mathematicians practice their craft. The purpose of this chapter is to survey some of this literature and explain how it can contribute to the philosophy of mathematical practice. We first describe how mathematics educators use empirical methodologies to investigate the behaviors of mathematicians and argue that findings from these studies can inform the philosophy of mathematical practice. We then illustrate this by summarizing research on mathematicians’ understandings and mathematicians’ proof reading

Proof in advanced mathematics classes: semantic and syntactic reasoning in the representation system of proof

Proof in advanced mathematics classes: semantic and syntactic reasoning in the representation system of proo

On mathematicians' different standards when evaluating elementary proofs

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians