Instructor Perceptions of Student Example Use and Understanding in an Introduction to Proof Course

Abstract

As students transition from the calculus sequence to upper-level mathematics classes, many struggle to understand the purpose of proofs and how to produce them on their own. One tool students and mathematicians alike frequently use is examples to aid their understanding and production of proof. Using examples productively has been argued to be beneficial to the development of deductive reasoning used in proving mathematical claims. Although much research exists on how students use examples and why they choose the examples to be used, there is little research on how to help instructors aid their students in the productive use of examples. This study seeks to answer this lack of research by asking the research question: How do instructors perceive the example-use of their students in an introduction to proof course? Instructors of introduction to proof courses from across the southeastern United States were interviewed and shown samples of student work with varying degrees of effectiveness in using examples to answer two different questions. Participants of the interview later categorized the sample student work and finally ranked the student responses for how well they believed the students gave a proof of the problem presented. Qualitative analysis of the interviews indicated instructors commonly ranked and categorized the student responses in alignment with mathematics education research related to example-use. Although most participants recommended students perceived as struggling with the task use examples to help build intuition to solve the problems, participants were split on whether students commonly ranked at the top should include examples in the formal written work presented in class. Other findings indicated participants typically suggested their students use examples along a continuum of example-use rather than jump straight to formal proof excluding examples. Common instructional responses to students at varying levels of example-use were again aligned with mathematics education research and aimed to help students advance in understanding along a continuum of responses. Finally, a lesson was developed to implement the research findings discussed in the first part of this dissertation in an introduction to proof course with the goal of highlighting the benefits of examples to both students and instructors.