Many college students have difficulty developing and evaluating mathematical arguments and constructing proofs. As a result, a number of colleges and universities have instituted introduction-to-proof courses to help prepare these students for more theoretical upper level math courses. We have developed a library of short edited videos (3 to 10 minutes in length) showing pairs of students as they work to construct proofs of theorems that are new to them. These authentic videos reveal student thinking, where and why they get stuck and how they get themselves unstuck; they can be used in and out of class to help students focus on specific issues and develop their critical thinking skills. The videos are relevant for use in introduction-to-proof courses and other courses in which students learn to develop their own proofs.
The videos we have created focus on students as they discuss how to write a proof. The key benefit of these videos is that, when used in the classroom together with a well-guided discussion, they allow students to see peers address obstacles and articulate reasoning as they move toward the proof of a statement (or away from one). This interactive experience gives the viewer a chance to think about how the students in the video are approaching a problem and challenges them to articulate why a particular approach may or may not be working. We have observed that engaging in such a critique facilitates student understanding of the problem solving process, especially for the majority of students who are mid-level performers in our courses. It also helps students develop the metacognitive skills necessary to succeed in higher mathematics.
Even if not used in the classroom, our videos are helpful for faculty teaching introduction to proof courses. In our work with colleagues in our project and with giving workshops and presentations, faculty who have watched these videos have reported gaining a greater insight into their students’ thinking processes and a greater understanding of where their students’ difficulties lie.
This workshop introduces faculty to a resource that will enhance their teaching and students’ learning of proof writing. Participants will explore our online library of authentic videos of students attempting to construct proofs of theorems that are new to them, use the videos to better understand struggles students encounter in learning to construct proofs, learn various ways these videos can be used to help students overcome difficulties, and develop plans to use these videos in their own learning environment. Before the workshop, participants will access a video and engage in discussions of their experiences teaching introductory proof writing. The on-site component will involve watching videos, break-out and whole group discussions of innovative ways to use this resource and what it teaches us, and development of plans for participants’ use of videos. Participants will have access to the video library, and on-line discussions will continue so that participants can share experiences.