Ways secondary mathematics teachers apply definitions in Taxicab geometry for a real-life situation: Midset

Abstract

Definitions in mathematics are an integral part of understanding concepts and are often not used correctly by students in mathematical proofs and problem-solving situations. Research shows that by observing properties and making conjectures in non-Euclidean geometry, students can better develop their understanding of concepts in Euclidean geometry. For this study, Taxicab geometry (defined by Taxicab distance, or the L1 norm) was introduced to students enrolled in a College Geometry course at a university. Action-Process-Object-Schema (APOS) Theory was used as a guiding framework in the data analysis of responses from eleven secondary mathematics teachers to a real-life problem situated in Taxicab geometry. This report provides illustrations of the conceptual understanding of midset (known as a perpendicular bisector in Euclidean geometry) found among participants and suggestions for teaching material to help facilitate development of a deeper understanding of definitions in geometry.