Through lesson study, the PISC Project explores the effect of an intervention to support the teaching and learning of proof in secondary geometry. PISC takes as its premise that if we scaffold proof, by first teaching particular sub-goals of proof, then students will be more successful with proof later on.
Despite that fact that proof is considered a central mathematical process, and policy documents have consistently recommended that proof be taught in school mathematics, success with proof remains elusive. A preponderance of evidence suggests that proof is challenging for teachers to teach (e.g., Cirillo, 2011; Knuth, 2002) and for students to learn (e.g., Chazan, 1993; Senk, 1985). Factors identified as contributing to these challenges include: impoverished curricula (Otten et al., 2014); teachers’ content and pedagogical knowledge (Knuth, 2002); and the lack of recommendations about how to scaffold proof so that students can be successful (Cirillo et al, 2017).
PISC draws on pilot study data and findings that suggest a promising approach to scaffolding the introduction to proof in geometry. Based on these findings, we developed the Geometry Proof Scaffold (GPS)—a pedagogical framework that outlines eight sub-goals and corresponding competencies that can be taught one at a time. For example, prior to being asked to work on a proof, students learn to draw valid conclusions from given information or assumptions. The eight sub-goals in the GPS are: Understanding Geometric Concepts, Defining, Coordinating Geometric Modalities, Conjecturing, Drawing Conclusions, Using Common Sub-Arguments, Understanding Theorems, and Understanding the Nature of Proof.
A set of 16 detailed lessons plans and corresponding student investigations, focused on the sub-goals of proof, served as the study intervention. Using a mixed-methods approach, data were collected from control and experimental groups to test the effect of the intervention. Comparing student interviews and written assessments from these groups provided compelling evidence that the PISC lessons had a positive impact on student learning. Statistical analyses demonstrate that gains made by students were significantly larger under the PISC curriculum. Clinical interviews conducted with students in control and experimental groups also provided compelling qualitative evidence about the effect of the intervention.