Knowledge Needed by a Teacher to Provide Analytic Scaffolding during Undergraduate Mathematics Classroom Discussions

Using case study analysis and a cognitive theoretical orientation, we examine elements of knowledge for teaching needed by a mathematician to orchestrate whole-class discussions in an undergraduate mathematics classroom. The instructor, an experienced teacher and mathematics researcher, used an inquiry-oriented curriculum to teach a differential equations course for the first time after teaching it with traditional lecture methods for many years. Examples of classroom teaching and interview data demonstrate that, despite having extensive teaching experience and possessing strong content knowledge, some instructors may still face challenges when trying to provide analytic scaffolding to move whole-class discussions toward a lesson's mathematical goals. We also hypothesize several component practices necessary for the successful use of analytic scaffolding. Our analysis focuses on the relationship between the instructor's pedagogical content knowledge and specialized content knowledge and his capacity to enact these component practices during whole-class discussions.