Enacting Reasoning-and-Proving in Secondary Mathematics Classrooms through Tasks

摘要

Proof is the mathematical way of convincing oneself and others of the truth of a claim for all cases in the domain under consideration. As such, reasoning-and-proving is a crucial, formative practice for all students in kindergarten through twelfth grade, which is reflected in the Common Core State Standards in Mathematics. However, students and teachers exhibit many difficulties employing, writing, and understanding reasoning-and-proving. In particular, teachers are challenged by their knowledge base, insufficient resources, and unsupportive pedagogy.udThe Cases of Reasoning and Proving (CORP) materials were designed to offer teachers opportunities to engage in reasoning-and-proving tasks, discuss samples of authentic practice, examine research-based frameworks, and develop criteria for evaluating reasoning-and-proving products based on the core elements of proof. A six-week graduate level course was taught with the CORP materials with the goal of developing teachers’ understanding of what constitutes reasoning-and-proving, how secondary students benefit from reasoning-and-proving, and how they can support the development of students’ capacities to reason-and-prove. Research was conducted on four participants of the course during either their first or second year of teaching. The purpose of the research was to study the extent to which the participants selected, implemented, and evaluated students’ work on reasoning-and-proving tasks. The participants’ abilities were examined through an analysis of answers to interview questions, tasks used in class, and samples of student work, and scoring criteria. The results suggest that: 1.) participants were able to overcome some of the limitations of their insufficient resource by modifying and creating some reasoning-and-proving exercises; 2.) participants were able to maintain the level of cognitive demand of proof tasks during implementation; and 3) participants included some if not all of the core elements of proof in their definition of proof and in their evaluation criteria for student products of reasoning-and-proving products.ud