Why and how mathematicians read proofs

An important goal in undergraduate mathematics education is to lead math majors to think and behave more like mathematicians with respect to proof. In order to address this goal we first need to have an accurate understanding of the different proof-related activities that mathematicians engage in, the ways in which they perform them, and what their beliefs about proof actually are. Recently, we have conducted studies on the ways in which, and reasons why, mathematicians read proofs (see Weber & Mejía Ramos, 2011; Mejía Ramos & Weber, 2014).

Assessment of proof comprehension in undergraduate mathematics

Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. We addressed these issues by developing (i) a model for assessing proof comprehension in advanced mathematics and (ii) a method for designing and validating proof comprehension tests. We describe ways in which teachers and researchers can generate tests to evaluate students' understanding of a proof in advanced mathematics (see Mejía Ramos et al. 2012; in press).

Comprehending different methods of proof presentation

In order to address students’ difficulties understanding proof in undergraduate mathematics, some researchers have proposed methods for presenting proofs that differ from the linear, abstract format in which they are traditionally presented. In a series of both qualitative and quantitative studies, we have started to address the assessment of novel proof presentation formats, focusing on the effect of structured proofs and generic proofs on student comprehension (see Lai et al, 2014; Fuller et al. 2014).

Students' comprehension of lectures in advanced mathematics

Our research interests in this area concern: (i) how do mathematics professors present proofs in advanced mathematics, (ii) what are professors' rationales for the pedagogical choices that they make, (iii) how do students interpret the proofs that their professors present, and (iv) why do students frequently interpret mathematics lectures in a different way than their professors intend? Currently, we have proposed two broad accounts that answer these questions (see Lew et al. 2016; Weber et al., 2016).