Examining Undergraduate Students' Proof Schemes in a Transition Proofs Course

This study uses narrative analysis to track undergraduates’ proof schemes throughout a transition proofs course at a research 1 university in the Midwest, with a focus on student example use and how it influences their schemes. Assessment data and cognitive interviews were conducted at four points throughout the semester in order to track the cognitive shift students go through as they transition from novice, naïve proovers to expert ones. Two predominant trends were identified: some students’ schemes went from naïve empirical schemes to correct proof schemes during the semester, and some students’ schemes went from naïve empirical schemes to correct proof schemes and then reverted back to naïve schemes by the end of the semester. Case studies that exemplified these two trends were then analyzed using linguistic markers and a type 1 / type 2 narrative analysis framework. A correlation between use of authentic mathematical language and correct proofs was identified in many of the narratives. This study begins to answer the questions about the cognitive shift that undergraduate mathematics students go through as they transition from novices to experts and will help to inform instruction and further research. The instructional implications of the study indicate that transition proof course instruction should include instructor observation of students proving, rather than examination of their final proofs. Further, the successful application of narrative analysis to undergraduate proving schemes indicates that this type of analysis can be more broadly used in order to gain insight into student cognition about proving.