Research-Based Video for Teaching Undergraduate Proof
This design-research project aims to improve the teaching of proof at the undergraduate level by using videos of students working on proof-related problems. These videos can be used for class discussion or for professional development. The research component of the project looks at what makes proving difficult for students, and unpacking the knowledge that mathematicians have about proof which might be difficult for them to access and/or communicate.
This project is a cooperation between several mathematicians and math educators, to help develop classroom materials to help university students learn to construct proofs. After three years of piloting, we have created videos of "advanced" students working on proofs known to be hard for most students. These "advanced" students do not know exactly how to proceed, but in the end are able to construct a reasonable proof. These videos are then watched by students in "Introduction to Mathematical Reasoning" (or some such "bridge" course that prepares students for upper level mathematics) at the university level, allowing the usually "invisible" aspects of mathematical thinking--including getting stuck and unstuck, how to choose appropriate methods, how to coordinate informal and formal types of reasoning-not only to become visible, but also to be the center of classroom discussion.
In recent years there has been increased attention to trying to improve the teaching of university level mathematics. One problem is that many students, who have been exposed primarily to computational aspects of mathematics are unprepared for the more theory and rigor of university level mathematics.
One solution to this problem has been to create a course called at many institutions something like "Introduction to Proof" which serves as a bridge from the more computational courses of earlier mathematical study to the more theoretical courses at the university. These courses, however, are known to be difficult to teach, in part because mathematical knowledge is so difficult to unpack and articulate. This project aims to help improve the teaching and learning in this type of course.
The idea is to use videos of students who struggle and eventually succeed on proof problems as a basis for discussing the mathematical processes involved in proving. These videos, edited to highlight common difficulties that students encounter with proof, help to make visible some of the mathematical thinking processes involved in proving that are often invisible in a typical mathematical classroom.
This project has both a research and a development component. The development component, directed by Jim Sandefur at Georgetown University, focuses on making good quality videos and exploring different types of pedagogy for using the videos in classrooms. The research component, directed by Manya Raman Sundström at Umeå University, focuses on the mathematical thinking processes of both students and teachers to better understand how people learn to produce mathematical proof.
The development part of the project is funded by the National Science Foundation in USA and the research part is funded by a Forskarassistent position, sponsored by School of Education at Umeå University.