Case studies of instructional practices in proof-based mathematics lectures

Abstract

This multi-case study investigates the teaching practices of four instructors who were teaching undergraduate level proof-based mathematics courses using lecture methods. Both interview data with the instructors and several video observations of each classroom were gathered throughout the course of a semester. The data analysis techniques were primarily qualitative, but included some quantitative methods such as frequency counts and percentages to give an overall picture of the instructors' teaching. Analysis occurred in several phases, and used multiple units of analysis including the proof presentations, examples used in proof presentations, the class period, and the individual instructor questions.

The first and second research questions addressed the pedagogical moves that the instructor makes during proof presentations, and the instructors' allocation of class time. The instructors spent between 35% and 70% of their class time presenting proofs. The proof presentation techniques that were identified in the interviews were outline, examples, logical structure, and context. At least one of these strategies was observed in 94% of the proof presentations in the video data. Three of the four instructors expected active engagement in 95% of their proof presentations, while the fourth expected active engagement in 50% of his proof presentations. The proportion of class time spent on interactive lecture ranged from 26% to 62%.

The third and fourth research questions addressed the uses, types, and timing of examples during the instructors' proof presentations. Examples used during the observed proof presentations were used to create a framework that describes the uses of examples, types of examples, and chronological placement of examples in their proof presentations.

The fifth and sixth research questions addressed the questions posed by the instructors. The question rates ranged from 0.69 to 1.81 questions per minute, and that the percentage of higher order questions ranged from 30.1% to 54.2%. The percentage of questions to which students responded ranged from 35% to 52%. When restricting to only questions which were answered by students, if was found that in all four cases the percentage of answered questions that were higher-order matched the percentage of asked questions that were higher order. Thus, students were answering a variety of types of questions.