Wading Toward the Deep End of Proof: Tools That Can Support Mathematics Teachers

• In Event: Tools and Routines for Preparing STEM Teachers

Abstract

Purpose: Learning to think and reason both formally and informally is an important goal in the mathematics classroom (NCTM, 2009). In the recently published Reasoning and Sense-Making document (NCTM, 2009), formal reasoning (i.e., proof) was situated as the final of three stages in the reasoning progression required for increasing levels of understanding in the high school mathematics classroom. The authors pointed out that the effort to help students progress from less formal to more formal reasoning requires that “teachers play an essential role in encouraging students to explore more sophisticated levels of reasoning” (p. 11). One might wonder, however, how and how well are teachers being prepared to play this essential role.
Perspective: Research conducted by Herbst and colleagues (2009) indicates that students’ roles in the activity of “doing proofs” in the classroom continue to be minimal. Other research (Knuth, 2002a, 2002b) has suggested that perhaps the reason that teachers have not moved their students beyond the traditional two-column approach to proof is related to their beliefs about the purpose of proof and their students’ abilities to complete a proof. Additionally, teachers may not have had opportunities to consider alternative ways of teaching proof that fall outside of the “apprenticeship of observation” (Lortie, 1975) experienced in their own mathematics backgrounds.
Methods: To learn more about the challenges that teachers face when cultivating formal proof in their classrooms, the Geometry Proof Project, a two-year study that makes use of qualitative methods of inquiry, was designed.
Data sources: Six teachers were recruited for this study. Baseline data, collected in Fall, 2010, included two weeks of classroom observations in one target classroom of each teacher. Interviews designed to help the researcher better understand the data and the teachers’ beliefs about teaching proof were conducted. In Spring, 2011, professional development sessions were designed to assist the project teachers in: examining their teaching more carefully; attending to the ways in which they introduced proof; and re-considering the discourse in their classrooms as they were teaching proof.
Results: Results from the first year of the study suggest that the teachers believed that the only way to teach proof is to have students observe them doing proofs. For example, one teacher told his students, “There’s no shallow end to teaching proof. I just have to throw you into the deep end”. Through this collaboration with beginning and experienced teachers, tools intended to support teachers in navigating the “shallow end” of proof (i.e., scaffolding proof as a mathematical practice) were developed. The data suggests that these tools can be used to support teachers in inducting their students into formal reasoning. The tools were used with preservice teachers this past spring and will be piloted with the high school students this fall.
Scholarly significance: Understanding more about how a teacher learns to teach proof to students who are just learning how to develop a formal proof is an important implication of this work. This research has important implications, not only for teacher preparation, but also for teacher professional development.

Author

• Michelle Cirillo, University of Delaware