Many mathematics departments have transition to proof (TTP) courses, which prepare undergraduate students for proof-oriented mathematics. Here we discuss how common TTP textbooks connect three topics ubiquitous to such courses: logic, proof techniques, and sets. In particular, we were motivated by recent research showing that focusing on sets is…

Many tertiary institutions with mathematics programmes offer introduction to proof courses to ease mathematics students' transition from primarily calculation-based courses like Calculus and differential equations to proof-centred courses like real analysis and number theory. However, unlike most tertiary mathematics courses, whose mathematical…

This article documents differences between novice and experienced undergraduate students' processes of reading mathematical proofs as revealed by moment-by-moment, think-aloud protocols. We found three key reading behaviors that describe how novices' reading differed from that of their experienced peers: alternative task models, accrual of…

This paper presents selected findings from an assessment of university students' moment-by-moment reading of mathematical proof. This method, adapted from an assessment of narrative reading validated by psychologists, yields novel insights into the strategies students use to construct meaning for the equations in a proof text. In particular, we…

We examine a commonly suggested proof construction strategy from the mathematics education literature--that students first produce a graphical argument and then work to construct a verbal-symbolic proof based on that graphical argument. The work of students who produce such graphical arguments when solving proof construction tasks was analyzed to…

Translating an informal mathematical argument into a proof which conforms to the norms of the mathematical community in which it is situated is a non-trivial task. Here we discuss several types of products, other than the initial informal argument and its direct formalisation, which we observed students generating in a master's level analysis…

In this paper we explore how students construe what it means for an informal argument to be the basis of a proof. We use students' attention as a proxy for their conceptions and document what they pay attention to when assessing whether a proof is based on an informal argument. The data illustrate that some undergraduate mathematics students pay…

A significant body of research literature in mathematics education attends to mathematical proofs. However, scant research attends to proof comprehension, which is the focus of this study. We examine perspective secondary teachers' conceptions of what constitutes comprehension of a given proof and their ideas of how students' comprehension can be…

This article argues for a shift in how researchers discuss and examine students' uses and understandings of multiple representations within a calculus context. An extension of Zazkis, Dubinsky, and Dautermann's (1996) visualization/analysis framework to include contextual reasoning is proposed. Several examples that detail transitions between…

In this paper we explore how students construe what it means for an informal argument to be the basis of a formal proof and what students pay attention to when assessing whether a proof is based on an informal argument. The data point to some undergraduate mathematics students having underdeveloped conceptions of what it means for a proof to be…

Script writing by learners has been used as a valuable pedagogical strategy and a research tool in several contexts. We adopted this strategy in the context of a mathematics course for prospective teachers. Participants were presented with opposing viewpoints with respect to a mathematical claim, and were asked to write a dialogue in which the…

In this paper we examine a commonly suggested proof construction strategy from the mathematics education literature--that students first produce an informal argument and then use this as a basis for constructing a formal proof. The work of students who produce such informal arguments during proving activities was analyzed to distill three…

This article argues for a shift in how researchers discuss and examine students' uses of representations during their calculus problem solving. An extension of Zazkis, Dubinsky, and Dautermann's (1996) Visualization/Analysis-framework to include physical modes of reasoning is proposed. An example that details how transitions between visual,…