Writing Proofs: How Do We Teach Students What Is Second Nature To Us?

James Sandefur, Georgetown University

I have found it difficult to teach students how to write proofs. A large part of this is that I, like most of my colleagues, can usually look at a problem and immediately know what the most likely approach should be. I have been doing proofs for so long that I don’t even have to think about it. Thus, how can I get students to learn how to approach problems if I don’t know how I approach them?

To help overcome this problem, for the past 2 years I have worked with Georgetown University’s Center for New Designs in Learning and Scholarship on developing better strategies to help students learn to write proofs and give oral presentations. Through this work, I have developed some understanding about how students approach problems. I have also learned there is no magic bullet for teaching students to write. Learning to write and problem solve is difficult and time consuming. In this presentation, I will discuss some of what I have learned through this collaboration, and what questions I still have?

The students we are working with are generally sophomore math majors and upper level math minors taking Foundations of Mathematics. These students are given a variety of written and oral assignments throughout the course. We have studied the effectiveness of the assignments and teaching strategies using periodic questionnaires and interviews, both during the course and after students have taken later courses. One particularly effective method has been Think-Alouds, in which we videotape students doing their homework. A group of students is given a moderately difficult problem to turn in. Each student is videotaped working on the problem individually and as a group. While they are videotaped, they are encouraged to continue talking about what they are thinking. When students get stuck for a prolonged period of time, they are given prompts to help direct them in a better direction. These sessions have helped me understand when students get stuck, how they get stuck, and what type of prompts can help the get over the impasse.

One strategy we have found effective is classroom presentations. The presentations are required to be typed and on transparencies. A second group prepares a critique of the logistics of the presentation, and a third group critiques the content of the proof. Not only does the group giving the presentation learn from this, the groups giving the feedback learn to incorporate effective techniques into their presentation and to avoid pitfalls others have made.

The problems used are typical for this course and can be found in a variety of texts on Foundations of Mathematics. Solutions are to be problems are to be typed and revised. By typing solutions, students learn how to revise and edit proofs, just as they would for papers submitted to other courses. Email makes it easy for group members to edit and revise each other’s work as long as they plan far enough ahead. I oversee that the emailing is being done.