Mathematical proof skills have traditionally been studied through proof construction and proof comprehension. Mejía-Ramos and Inglis (2009a) proposed a third activity—presenting a given proof to demonstrate understanding—but little is known about how this activity can be assessed or how it relates to students’ broader proof competencies. This study investigates proof presentation in a sample of 143 undergraduate mathematics students. Students completed a video-based assignment in which they explained, line-by-line, a lecturer-provided proof that √(6) is irrational. The task required students to articulate the mathematical warrants and logical connections within the argument, making their mathematical reasoning observable. Videos were assessed by an experienced GTA and peers. A delayed near-transfer measure of proof construction was obtained from a final examination task requiring students to prove that √(10) is irrational. Although students generally performed well on the proof-presentation task, performance was only moderately associated with subsequent proof-construction performance. Peer and GTA ratings showed moderate agreement overall, although the analysis identified particular proof steps at which ratings diverged. Moreover, peer ratings explained additional variance in final examination performance beyond that accounted for by GTA ratings alone. One plausible interpretation, to be tested in future research, is that peers were sensitive to motivational, affective, and communicative dispositions reflected in students’ presentations that later influenced learning. The findings provide initial evidence that video-based proof-presentation tasks, supported by peer assessment, can be implemented at scale to make students’ mathematical reasoning more observable. This is particularly relevant as generative AI challenges the use of unsupervised written work as evidence of independent competence.
This research takes a qualitative approach to examine how the transition from high school to university (the collegiate mathematics transition or the secondary-tertiary transition) can affect mathematics self-efficacy (MSE) and mindset for undergraduate students. Thirteen first-year and nine non-first year students from three different mathematics courses (Intermediate Algebra, Calculus 1, and Discrete Mathematics) were interviewed. Results show that the collegiate mathematics transition provided an opportunity for increases in MSE for first-year students who entered university with low MSE; some of these students also experienced shifts towards growth mindset. Conversely, most first-year students with high MSE experienced no change in MSE or mindset, with one student experiencing a decline in MSE. Changes to MSE appear tied to students’ attributions. In addition, we compare these results with students not undergoing the collegiate mathematics transition to highlight the specific effects of the transition. Results have implications for our understanding of how MSE can change as well as for our understanding of how to better understand and support students during their transition to university mathematics.
Two-stage collaborative examinations have gained increasing attention in undergraduate mathematics education as a means of integrating individual accountability with peer discussion during assessment. Prior research has highlighted the value of such approaches for student performance and engagement, while also raising questions about their role in supporting mathematical learning. This study extends previous work on two-stage collaborative examinations by examining their implementation in an undergraduate engineering mathematics module in which the group stage used contextualised real-world problems. Drawing on 240 student-assessment observations from four consecutive cohorts, the study employed a multi-method design combining quantitative analysis of assessment performance, structured tutor observations of group interaction, and a post-assessment student survey. Across the pooled dataset, mean group-stage performance exceeded mean individual-stage performance by 12.32
We implemented a semester-long professional development workshop series to help Calculus I and II instructors implement active learning and equitable teaching practices. The workshops actively engaged instructors in approximations of practice (Grossman et al., 2009) in which they learned to enact various teaching practices like designing tasks; eliciting, noticing, and responding to evidence of student thinking; orchestrating class discussions; fostering inclusive social norms; and equitably managing student participation. After they participated in the workshops, we interviewed the instructors about their decisions to implement the teaching practices from the approximations of practice in their teaching. We analyzed the instructors’ resources, orientations, and goals that informed their decisions to enact those practices. We then identified the aspects of the approximations of practice that the instructors viewed as supporting their implementation of those teaching practices.
