A Sense-making Approach to Proof: Initial Strategies of Students in Traditional and Problem-based Number Theory Courses

This paper reports the results of an exploratory study of the perceptions of and approaches to mathematical proof of undergraduates enrolled in lecture-based and problem-based transition to proof courses. While the students in the lecture-based course demonstrated conceptions of proof that reJect those reported in the research literature as insuff~cient and typical of undergraduates, the students in the problem-based course were found to approach the construction of proofs in ways that demonstrated efforts to make sense of mathematical ideas. This sense-making manfested itself in the ways in which students employed initial strategies, notation, prior knowledge and experiences, and concrete examples in the proof construction process. These results suggest that such a problem-based course may provide opportunities for students to develop conceptions of proof that are more meanincJu1 and robust than does a traditional lecture- based course.