# Learning of trigonometry: An examination of pre-service secondary mathematics teachers' trigonometric ratios schema

#### Abstract

Mathematics education research has emphasized the importance of trigonometry in advanced mathematical learning and highlighted students' difficulties with trigonometry, which stem from underdeveloped foundational concepts of angle and angle measurement. The purpose of this dissertation was to investigate pre-service secondary mathematics teachers' (PSMTs) constructions of relationships between angles and side lengths in a right triangle (RASR) and their relationship to constructions of trigonometric ratios. Dubinsky and his colleagues' (Arnon et al., 2014; Asiala et al., 1996; Dubinsky, 1991, Dubinsky & McDonald, 2001) APOS theory was used to define knowledge as a collection of physical and mental constructions and operations with them (action, process, object, and schema). To explore constructions of RASR and relationships to constructions of trigonometric ratios, evidence of types of processing was sought. Using Clements' (2000) clinical interview methodology, this study utilized a series of controlled interviews to gather evidence of four PSMTs' constructions of angles, angle measurement, RASR, trigonometric expressions and trigonometric ratios. The analysis consisted of case study and cross-case study analysis. The case study analysis focused on constructing a model for each individual PSMT from observations during task-based interviews. Inferences about constructions of the APOS levels associated with each concept were developed. After each model was created for the PSMT, similarities and differences in actions, processes, objects, and schema were noted. This cross-case analysis allowed the construction of a cognitive model which describes how the PSMTs' mental constructions were related to their constructions trigonometric ratios schema. PSMTs participating in the study all had a schema related to 2-line angles and angle measurement. However, constructions of 1-line and 0-line angles and angle measurement were at lower levels. Yet, constructions of 1-line and 0-line angles and angle measurement were not required to operate with right triangles. The schema level for 2-line angles and angle measurement was sufficient for constructions of schema for RASR and trigonometric ratios in a right triangle context. The findings also support the claim that a schema of right triangles and RASR is necessary to construct trigonometric ratios schema. Particularly, having a schema of RASR provided PSMTs opportunities to act on dynamic right triangles presented in dynamic geometric software (DGS), specifically in GeoGebra, and reason about RASR. If students reached the schema level for RASR, they could then flexibly act on the trigonometric ratios considering the RASR. A cognitive model which explains relationships in PSMTs' constructions that are related to their constructions of trigonometric ratios schema is shared. Findings also suggest that reaching the schema level for 2-line angles, right triangle, RASR, and trigonometric expressions as well as reaching the object level for ratios was sufficient to construct a trigonometric ratios schema in a right triangle context. In addition, it was found that using the mnemonic, SOH-CAH-TOA, and the special right triangles [45°-45° -90° and 30°-60°-90°] supported PSMTs' construction of trigonometric ratios schema.

#### Degree

Ph.D.

Kastberg, Purdue University.

#### Subject Area

Mathematics education|Secondary education

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