Transitioning to Proof with Worked Examples

Proof is an essential part of mathematical practice both for mathematicians and for students at the undergraduate and graduate levels. In transitioning from computation-based courses to proof-based courses, the literature suggests that undergraduate students have difficulty with understanding and constructing mathematical proof. In the present study, I explored the use of a worked-examples-based proof-writing framework as a pedagogical tool to improve undergraduate students' ability to construct proofs. Over the course of three months, I ran three workshops with five undergraduate students who had no experience with formal mathematical proof. First, students completed a pretest which prompted them to write proofs for four claims from elementary number theory, with no instruction or statement of expectations. In each workshop, I provided students with worksheets containing completed worked examples of mathematical proofs, followed by partially completed worked examples of proofs (to be completed by the participants), and, lastly, exercises. In analyzing students' work, written reflections, and group conversations, several key findings emerged. First, a worked-examples-based proof-writing framework, such as the one used in this study, is most beneficial to the novice proof-writer. Where most students' attempts at proof on the pretest were non-deductive, colloquial arguments, on the partial examples and exercises immediately following the introduction of the worked examples students' proof-writing improved dramatically, both in terms of mathematical validity and formal-rhetorical quality. In later sessions, there was a decline in the formal-rhetorical aspect of student proof-writing. More significantly, in the last two sessions, students began to exhibit strategic approaches to proof, experimenting with different proof methods and approaches. Finally, regarding student engagement with the proof-writing framework, their reflections and comments most frequently referenced brainstorming and experimenting as being the most valuable components of the framework. These findings have implications for instruction in transition to proofs courses, as well as future research in the use of worked examples in proof-based courses.Ed.D., Educational Leadership and Management -- Drexel University, 201