Improving proof comprehension in undergraduate mathematics

When studying for a mathematics degree, it has been shown students have great difficulty working with proof (Moore, 1994). Yet, to date, there has been surprisingly little research into how we could improve the way students study mathematical proofs. Furthermore, there is relatively less research on students' proof comprehension skills when compared with that of their proof construction skills (Ramos and Inglis, 2009). The aim of this thesis was therefore to build upon the existing proof comprehension literature to determine methods of improving undergraduate proof comprehension. Previously, text based manipulations (e.g. Leron, 1985; Rowland, 2001; Alcock, 2009a) have been tested as a way of improving proof comprehension but these have often not been as successful as we would have liked. However, an alternative method, called self-explanation training, has been shown to be successful at improving comprehension of texts in other fields (Chi et al., 1989; Wong et al., 2002; Rittle-Johnson, 2006; Ainsworth and Burcham, 2007). This thesis reports three studies that investigate the effects of self-explanation training on proof comprehension. The first study confirmed the findings of previous self-explanation training research in other fields. Students in the study who received the self-explanation training showed a significantly greater understanding of the proof text compared to that of a control group. Study 2 used eye-tracking analysis to show that self-explanation training actually changed the way students in the study read proofs; they concentrated harder on the proof (as measured by mean fixation durations), and made more between-line transitions. The final study revealed that self-explanation training can be implemented into a genuine pedagogical setting with relative ease and also showed the positive effects on proof comprehension last for a longer term of three weeks. From the findings of the research reported in this thesis it can be concluded that many students who participated in these studies appeared to have the knowledge required to understand proofs, it is perhaps they just needed some guidance on how to apply their knowledge. Self-explanation training appears to do this as it significantly improved proof comprehension in the short-term as well as offering longer-term benefits. More research will be needed to confirm these findings, given that the studies here involved participants from only one UK university on what would be considered as typical mathematics degree courses for the UK. However, these findings are promising and provide the foundation for improvements in undergraduate proof comprehension

Investigating and improving undergraduate proof comprehension

Undergraduate mathematics students see a lot of written proofs. But how much do they learn from them? Perhaps not as much as we would like – every professor knows that students struggle to make sense of the proofs presented in lectures and textbooks. Of course, written proofs are only one resource for learning; students also attend lectures and work, independently or with support, on problems. But, because mathematics majors are expected to learn much of their mathematics by studying proofs, it is important that we understand how to support them in reading and understanding mathematical arguments. This observation was the starting point for the research reported in this article. Our work uses psychological research methods to generate and analyse empirical evidence on mathematical thinking, in this case via experimental studies of teaching interventions and quantitative analyses of eye-­‐movement data. What follows is a chronological account of three stages in our attempts to better understand students’ mathematical reading processes and to support students in learning to read effectively