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 $$\:\sqrt{6}$$ 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 $$\:\sqrt{10}$$ 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.

Teacher self-efficacy (TSE) significantly influences teacher well-being and pedagogical quality, thereby impacting student learning outcomes. Despite the recognized effectiveness of Lesson Study (LS) in shaping teachers' practices and beliefs, there is a research gap in TSE, particularly in higher education. To address this, we conducted a longitudinal study at a northeastern Chinese university, tracking TSE evolution in three English as a Foreign Language (EFL) teachers, encompassing both novices and experienced educators, during a six-month LS program. We collected both quantitative and qualitative data, encompassing interviews, surveys, and observations. Underpinned by Bandura and Wyatt's self-efficacy theory, the impact of LS on teachers' TSE development was explored and interpreted. Our findings shed light on the many ways in which LS effects individual teachers' TSE. Notably, both novice and experienced teachers improved significantly in TSE, especially in the domains of instructional techniques and student engagement. However, in the context of classroom management, there was considerably less growth. Furthermore, our research identifies a number of challenges that can have a negative impact on teachers' TSE, such as adapting to students' changing needs, dealing with limited resources, managing time constraints, navigating the school examination system, and overcoming initial resistance to the LS approach.