Mathematical Sciences (21) This project is considering the problem of helping undergraduates learn to produce mathematical proofs. Difficulties in producing proofs often occurs at the collegiate level, for both students and their teachers, as students try to make the transition from more skills oriented courses such as Calculus and Differential Equations to more theoretical upper level mathematics courses such as Abstract Algebra and Analysis. This project builds upon the experience of the PI and Co-PI in developing videos of students solving problems that are typically found to be difficult in such courses. The students being videotaped are somewhat more experienced problem solvers than students typically taking such a course. The videos show where the students had difficulty, what methods they used to get unstuck and why they chose those methods. Preliminary research indicates that appropriate discussion of such videos helps students unpack and develop the thinking processes that are required to be successful problem solvers, such as using examples to help understand the reasoning behind the proof, trying a variety of approaches, and developing informal, heuristic arguments into acceptable formal proofs.
The project is research-designed, meaning iterative cycles are being used to refine the videos based on what is learned about how the teachers and students interact with the media. In addition to producing videos, the project is also generating research findings regarding how students learn (or don't learn) from video. Since this has not yet been studied in any depth at the university level, this will be an important contribution to the literature.
These carefully researched video-media provide faculty throughout the country both mathematical and pedagogical support in their effort to help students read and write mathematical proofs. The videos expose what is normally invisible about mathematical reasoning, and provide a platform for making this visible to the student. These videos can be used in a variety of ways in different courses, both for in-class presentation and discussion, homework, and within supplemental skills laboratories. The videos are being developed and pilot tested by a consortium of faculty and math educators who have many years experience teaching undergraduates how to do mathematical proofs.
