A Study of Prospective Secondary Mathematics Teachers’ Evolving Understanding of Reasoning-and-Proving

\ud Proof is a foundational mathematical activity that has been underrepresented in school mathematics. The recently adopted Common Core State Standards in Mathematics includes eight process standards, several of which promote the inclusion of reasoning and proof across all grades, courses, and students. If students are to reach the expectations recommended by mathematics researchers and explicitly identified in the Common Core State Standards, then students will need opportunities to construct and validate proof arguments. However, secondary students find it challenging to validate arguments and produce proofs and do not know what a mathematical proof is. Furthermore, those preparing to be secondary mathematics teachers in undergraduate mathematics courses are unable to construct proofs on a consistent basis, and practicing secondary teachers possess a limited conception of proof. \ud A six-week graduate-level course was taught with the purpose of increasing practicing mathematics teachers’ knowledge, expanding their conceptions of reasoning and proof, and preparing them to create similar experiences for their students. Research was conducted on the course to study the participants’ evolving understanding of reasoning-and-proving. The results suggest that: 1) the course was successful at expanding the participants conception of proof; 2) the prospective teachers encountered five challenges when asked to write proofs that are at the secondary mathematics level; 3) specific types of arguments were challenging for participants to classify as proofs or non-proofs; and 4) even though the participants were skillful in selecting high-level tasks that they could modify to include reasoning-and-proving opportunities, more work is needed to integrate such task across any secondary curricula. \u