Abstract:

In recent years, a significant emphasis has been placed on the teaching and learning of proof across grade levels and mathematical content areas. However, undergraduate students’ difficulties with proof are well-documented. To better support undergraduate students’ learning of proof, the use of student-centered instruction in teaching and learning proof has been found to promote their understanding of what constitutes proof and their ability to construct proofs. Yet, to date, we do not know how implementing student-centered instruction affects undergraduate students’ perceptions on proof throughout the semester. In this study, we investigated 11 undergraduate students’ written responses to a question, “What is proof in mathematics?” in a proof-focused course entitled “Discrete Mathematics” at the beginning and the end of the semester. Prior to engaging in student-centered proof instruction, the two most frequently cited roles of proof at the beginning of the semester were verification and explanation. After experiencing student-centered proof instruction throughout the semester, the three most vital roles of proof in mathematics categorized by undergraduate students’ responses were the verification role, the explanation role, and the communication role. This result suggest that the undergraduate students in this student-centered course acknowledge the communication role of proof as involving the processes for individuals to share ideas understood by people in their mathematics community. Regarding the five roles of proof suggested by de Villiers (1990), the discovery role of proof was never cited by the participants at the beginning of the semester. Although all the five roles of proof suggested by de Villiers (1990) were categorized by our students’ descriptions at the end of the semester, our result showed that the two least frequently cited roles of proof were discovery and systemization. Future research could incorporate more unfamiliar questions and statements into a student-centered proof course, allowing undergraduate students to use previously proven theorems and accepted definitions to structure their constructions for new results and to explore unexpected findings through the completion of their proofs.

Keywords:

Advanced Mathematical Thinking, Reasoning and Proof, Student-Centered Instruction, Undergraduate Education.