An examination of a pre-service mathematics teacher's mental constructions of relationships in a right triangle

Abstract

Students need to construct strong knowledge of angles as well as relationships between angles and side lengths in a triangle to succeed in geometry. Although many researchers pointed out the importance of angles and angle-related concepts, students and teachers have had limited knowledge of these concepts. This study is a part of a larger study, and examines pre-service secondary mathematics teachers‘ (PSMTs) mental constructions of relationships between angles and side lengths in a right triangle (RASR). The Action-Process-Object-Schema (APOS) learning theory was used as the theoretical lens and clinical interview methodology was used as the methodology in the study. The study was conducted with four PSMTs, but it focuses on one of PSMTs, Linda, who revealed evidence of schema for RASR. As a result of fine-grained analysis of Linda‘s responses to the tasks, this article reports the mental constructions enough to develop a schema for RASR. The model describes that schema for 2-line angles, right triangles and relationships between opposite angles and side lengths and process level for some relationships including ‗Pythagorean Theorem‘, ‗The hypotenuse is always the longest side in a right triangle‘, ‗Special right triangles‘, ‗Complementary Angles‘, and ‗Triangle inequalities‘ are enough to construct a schema for RASR.