## Graduate teaching assistants' mathematical understanding for teaching trigonometry

##### Abstract

This study described the mathematical understanding, exhibited by graduate teaching assistants in a Department of Mathematics (GTA-Ms), that is useful for teaching trigonometry. The following two research questions guided this study: (1) To what extent do GTA-Ms exhibit an understanding of trigonometric concepts when solving and explaining trigonometry problems? (2) What understanding of trigonometry do GTA-Ms use in analyzing and responding to students’ mathematical thinking about concepts of trigonometry in hypothetical teaching contexts? I used the framework of Mathematical Understanding for Secondary Teaching (MUST) developed by the Situations Project of the Mid-Atlantic Center for Mathematics Teaching and Learning (MAC-MTL) at Pennsylvania State University and the Center for Proficiency in Teaching Mathematics (CPTM) at the University of Georgia. This framework was a useful guide for designing task items and a good tool for analysis of the data collected from three task-based interviews with each participant because it helped me systemically organize, categorize, and describe the mathematical understanding that emerged from the participants’ mathematical work.
In this study, I considered fundamental concepts of trigonometry to mean the basic core concepts that underlie teaching and learning trigonometry. The findings from this study showed that the participants exhibited a mathematical understanding characterized in most of the strands of the MUST framework. Although GTA-Ms exhibited proficiency with advanced mathematical concepts, they showed a lack of conceptual understanding of some fundamental concepts useful for teaching mathematics when solving and explaining trigonometry problems. Given a hypothetical teaching context describing students’ mathematical thinking, the participants tended to use their mathematical understanding to respond in formal ways, such as providing rigid definitions, deductive reasoning, and conventional manipulations of mathematical symbols. In particular, their explanations of both advanced and fundamental concepts were more procedural than conceptual, equation-oriented, and definition-based. The findings from this study suggested that mathematical concepts–fundamental as well as advanced concepts–within courses that GTA-Ms teach should be revisited and conceptually developed as part of their preparation for teaching.