Supporting Proof Comprehension in University Mathematics Studies-Comparison of Different Options for the Strategy "Illustrating with Examples"

Reading and understanding proofs is an essential activity in the scientific discipline of mathematics. In the introductory phase of a university mathematics program, it is a major challenge for students to engage with mathematical proofs in a way that supports learning. Understanding proofs does not only mean to comprehend individual steps of the proof, but it also means, for instance, to be able to identify the main ideas of the proof. The present study focuses on the question to which degree individual factors predict proof comprehension, and how students' proof comprehension processes can be supported through the specific strategy of using examples. The proof of the Monotone Subsequence Theorem was presented to 166 students from several Real Analysis courses. The students were prompted to read the proof and to use examples for illustrating every step of the proof. Groups of students were formed according to the way in which the use of examples is specified (2 x 2-design): The first condition ('learning activity') differentiates between actively constructing an example to a given proof and passively retracing a given example. The second condition ('representation format') differentiates between symbolic vs. graphical presentation of the example. Our results show that final grades from high school and previous subject-matter knowledge strongly predict proof comprehension, as expected. By contrast, we could not give evidence for any advantages of the various conditions in the proof comprehension process, but we get illuminating insights into students' understanding processes. We discuss the results with an eye toward theoretical and practical implications.