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

Abstract

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.

Students' Mathematical Argumentation with the Visualizer Cognitive Style in Proving the Congruence of Triangles

“…Some of these include difficulties in drawing or interpreting geometric images and applying more than one theorem to solve certain geometric problems (Casanova & Lapinid, 2021;Dinata, 2019;Dirgantoro, 2019;Juman et al, 2022 ;Ramdjid et al, 2022) Regarding the difficulty of geometry material, it cannot be separated from the discussion regarding reasoning and proof, which is a challenge that is not easy. Researchers also agree that proof and reasoning are very important in supporting students' mathematical abilities and are important topics in the mathematics curriculum (Beni, 2015;Winer & Battista, 2022). Difficulty in translating mathematical language into geometric models is a problem that is often faced.…”

“…The author argues that the geometric proof related to the congruence of triangles is relevant to use as a test question for the title. With the reason that this question can explore the ability to carry out the process of formulating, applying theorems, and interpreting geometric images, where basic mathematical abilities such as reasoning and argumentation are needed in these three processes (Winer & Battista, 2022). A few misconceptions on class inclusion, especially when considering isosceles right triangles and obtuse triangles.…”