Investigating Students’ Proof Reasoning: Analyzing Students’ Oral Proof Explanations and their Written Proofs in High School Geometry

Constructing formal geometry proofs in is an important topic in the mathematics curriculum. But students’ difficulties with proof are well documented. This article focuses on proofs that use triangle congruence postulates in US high school geometry. Examining students’ proof reasoning in one-on-one task-based interviews, we analyzed students’ oral planning/explanations and two-column written proofs. We found that much of the reasoning students conveyed in their oral planning/explanations was not incorporated into their written proofs, resulting in formalization and fatal logical reasoning gaps. We found that the majority of the students utilized sound reasoning in their oral explanations, but struggled to capture that reasoning in their written two-column proofs. We link our findings to learning progression research, further elaborating the van Hiele levels. Our research contributes to understanding the learning and teaching of proof and to the ongoing controversy over the use of two-column proofs in the high school geometry curriculum.