The case of Brandon: the dual nature of key ideas in the classroom

Abstract

This paper looks at proof production in the midst of classroom interaction. The setting is a college level geometry course in which students are working on the following task: Prove that two parallel transported lines in the plane are parallel in the sense that they do not intersect. A proof of this statement is traced from a student’s idea, through a small group discussion, to a large class discussion moderated by a teacher. As the proof emerges through a series of increasingly public settings we see ways in which the key idea of the proof serves to both open and close class discussion. We look at several examples of opening and closing, showing how not only the key idea, but also the warrants and justifications connected to it, play an important role in the proof development.