Laura Wirth ^1* Katharina Kirsten ¹ Christian Serpé ² Gilbert Greefrath ¹

• ¹ Institute of Mathematics Education and Computer Science Education, University of Münster, Münster, Germany
• ² Mathematical Institute, University of Münster, Münster, Germany

Background: Proofs are a key component in undergraduate mathematics, but understanding presented proofs and constructing proofs is a challenge for many students. Flipped undergraduate mathematics classrooms often employ instructional videos, yet little is known about their potential to help students understand and construct proofs. Objective: This study investigates the potential of video-based proof presentations on student learning. We compared a video that presented the proof construction process (proof video); a video that heuristically presented the proof construction process, which modeled key decisions and named the phases of proof construction and activities (heuristic proof video); and a video that offered prompts during the proof construction process, where self-explanation prompts guided students through these phases and activities (prompted proof video). Methods: A between-subjects design was employed, involving 177 mathematics (teacher) students in a first-semester proof-based linear algebra course. Data were collected on students' comprehension of the presented proof, their knowledge for proof construction, and their evaluative perceptions. Statistical analyses were performed using ANOVA (proof comprehension) and MANOVA (evaluative perceptions) to compare the groups. Qualitative content analysis was employed to identify different facets of knowledge for proof construction and the groups were contrasted using 𝜒^2 -tests. Results: We found that independent of the video they watched, students achieved a rather local comprehension of the presented proof. The heuristic proof video showed potential for offering metaknowledge of how to approach proof construction and knowledge on process-related activities that support individual phases of proof construction but required more time. Yet, while students perceived all videos positively, they liked the heuristic proof video best. Conclusion: The results provide insights into the design of instructional videos, suggesting that, in the early stages of learning about proofs, a heuristic proof video may help address the challenges students face.