Undergraduate students’ conceptions of intuition and rigour and their awareness of the roles of proof

In this thesis, I address the constructs of mathematical intuition and rigour, which lack a single definition in the research literature, and for these constructs, there is little empirical research in undergraduate mathematics (e.g., Suwarto et al., 2023). This research fills a gap in the literature, which is mostly theoretical, by addressing the understanding of these constructs through the learner’s perspective.

I first conducted two studies investigating undergraduate students’ conceptions of intuition and rigour. I asked undergraduate students to express their conceptions of each construct through an online survey and analysed their responses using comparative judgment as a research method. This entailed asking mathematics PhD students to evaluate responses provided by undergraduate students through making pairwise comparisons between the responses. I then used these judgements to generate a scaled rank order for responses from best to worst. Moreover, through content-coding the responses given by undergraduate students, I generated themes inductively (in the first study), which were applied deductively in the second study. Regression analyses assessed which themes predicted higher CJ scores as judged by mathematics PhD students. The findings for intuition suggested undergraduate students associated the themes of ‘previous experience’, ‘instinct’, ‘logical thinking’ and ‘understanding’ with intuition, yielding a cluster conception for this construct. The findings for rigour implied undergraduate students associated the themes of ‘logic’, ‘clarity’, ‘proof’, ‘per- sonal traits’, and ‘incontrovertibility’ with rigour, aligning with a formal under- standing in the educational sense. Moreover, third/fourth-year undergraduate students tended to give better responses than first-year undergraduate students (as judged by mathematics PhD students), suggesting exposure to more mathematics may positively impact the understanding of these constructs.

To further investigate undergraduate students’ understanding of rigour, I conducted a third study employing task-based semi-structured interviews. I selected five proofs from Ording (2019) with different characteristics to highlight the roles of proof: verification, explanation, communication, and discovery (cf., de Villiers, 1990). In the existing literature, there is a lack of empirical studies into the roles of communication and discovery, and this study aimed to address this gap. The role-of-proof framework was used as a theoretical tool to code undergraduate student utterances. I initially coded undergraduate student utterances through ‘in vivo’ coding using the participants’ own language to preserve meaning. In the next stage, I allocated these in vivo codes to the respective role of proof deductively. I then generated axial codes by grouping similar in vivo codes into distinct thematic categories. The findings from this third study implied that first-year undergraduate students were unaware of the explanation role of proof (unlike third/fourth-year undergraduate students). Furthermore, undergraduate students did not demonstrate an awareness of the discovery role of proof, however some were able to make links and display positive learning, while others found this difficult suggesting a lack of secure background knowl- edge. All participants were aware of the roles of verification and communication.

Furthermore, findings from this study were nuanced and detailed regarding the similarities and differences between both groups of students in terms of their awareness (or not) of the roles of proof.

The findings from all studies combined suggest some overlap in undergraduate students’ conceptions of these constructs, where the themes of ‘previous experience’, ‘logical thinking’ and ‘understanding’, in addition to composing a definition of intuition, are also relevant to a conception of rigour. This suggests that these constructs are not necessarily distinct in the minds of students, although they are in theoretical research.