The role of theory building in the teaching of secondary geometry

Abstract

Although mathematical practice has traditionally valued two distinct kinds of mathematical work-referred to by Gowers (2000) as theory building and problem solving-activity in classrooms appears to be organized largely around the latter, rather than the former. This study takes up the question of whether there is a customary role for theory building in the secondary geometry course and to what extent teachers of geometry hold students accountable for that disposition. We analyze records from study groups composed of experienced geometry teachers in the USA, and ask to what extent the disposition towards theory building is visible in those records, and how teachers react to the suggestion that it could play a larger role in their practice. Among our findings, we report that participants consistently attached a higher value to work organized around problem solving, rejecting the idea that they might hold their students accountable for theory building.

“Is It Valid or Not?”: Pre-Service Teachers Judge the Validity of Mathematical Statements and Student Arguments

“…This shows that the PSMTs need not only an understanding of what counts as valid argument, but also an adequate knowledge of choosing accepted definitions, axioms, and facts. Various properties and postulates that underlie in an argument made in the proof are usually not spelled out, but rather are assumed to have been already learned and internalized by students (Schleppegrell, 2007;Weiss & Herbst, 2015). Therefore, it might not be surprising to see students have difficulties interpreting or using theorems on their own.…”

“…Research has demonstrated that various properties and postulates that underlie in an argument made in the proof are not spelled out, but rather are assumed to have been already learned and internalized by students (Schleppegrell, 2007;Weiss & Herbst, 2015). As a result, students have difficulties interpreting, or using theorems on their own (Zeybek Simsek, 2020).…”