Abstract
This qualitative case study examined the prospective mathematics teachers’ conception of mathematics, proof, and mathematical definitions. Moreover, it also investigated their proof writing approaches, using mathematical definitions and validation assessment practice. Participants of the study were four prospective mathematics teachers in a large southern research university who had taken a proof transition course as well as Linear Algebra. Four semi-structured interviews were conducted to collect data. The first interview protocol was designed to determine participants’ conception of mathematics, proof, and mathematical definitions. The other three were task-based interviews that were designed to investigate participants’ use of mathematical definitions in simple exercises, proof production, and proof validation in three different content areas: Geometry, Set Theory, and Linear Algebra. Ernest’s (1989) framework was used to identify the mathematical beliefs of participants, while Raman’s (2002) framework guided the analysis of students’ proof production and validity assessment practices. Results of the two cases were presented in this study. They were chosen based on their conception of mathematics: one held an instrumentalist view of mathematics and the other held a Platonist perspective of mathematics. The study intended to create a clear picture of the practices of students with different perspectives of mathematics.
The results of the study suggested that students’ mathematical beliefs might inform their proof production approaches. It was found that the student with an instrumentalist view tended to use heuristic approaches; on the other hand, the student with a Platonist perspective was inclined to use a procedural approach in proof production. Moreover, the study addressed that students’ conception of proof was framed within the justification, verification, and occasionally the explanation role of proof. This limited conception of proof constructed their criteria to assess the validity of a given proof. Another finding of the study was that students tend to bypass the concept definitions as long as they can reach a conclusion with their concept images. Lastly, students’ experiences within a mathematical context were distinctive in their comfort to make comments and develop connections between mathematical concepts to make logical deductions.
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