Constructing-Evaluating-Refining Mathematical Conjectures and Proofs
This study focused on investigating the ability of 58 pre-service mathematics teachers' (PSMTs) to construct-evaluate-refine mathematical conjectures and proofs. The PSMTs enrolled in a three-credit mathematics course that offered various opportunities for them to engage with mathematical activities including constructing-evaluating-refining proofs in various topics. The PSMTs' proof constructions were coded in three categories as: Type P1, Type P2 and Type P3 in decreasing levels of sophistication (from a mathematical stand point) and the constructions of conjectures were coded in two categories as: Type C1: correct conjectures and Type C2: incorrect conjectures. In addition to classifying the PSMTs' proof and conjecture constructions, how they reacted when they needed to refine conjectures and proofs were also classified. Samples of classroom episodes were provided to exemplify these different proof-conjecture constructions-evaluations-refining processes. The results of the study demonstrated that the combined construction-evaluation-refining activities of conjectures and proofs were not only helpful to better illuminate the PSMTs' understanding of mathematical proofs, but they were also an essential instructional tool to help PSMTs comprehend mathematical ideas and relations.