Student Understanding of Trigonometric Functions: Exploring Understanding Through the Use of Contextual Tasks

Researchers in mathematics education have studied many facets of teaching and learning mathematical functions, however, research focusing on trigonometric functions is sparse. Mathematics education researchers have also studied the importance of and impact of mathematical tasks for learning. For example, the framework of Realistic Mathematics Education (RME) emphasizes learning mathematics situated in reality and using rich contexts that provide learners with opportunities to reinvent mathematics in a way that is meaningful to them. Despite this valuable research, there exist gaps between the findings from these studies and their enactment in classrooms. As such, this qualitative study sought to extend frameworks for student understanding of functions and of realistic tasks to the area of trigonometric functions in a high school precalculus course. I studied 14 students during their time in a precalculus course in a United States public high school. As a teacher-researcher, I employed tasks consistent with the principles of RME, and I captured students’ words and written work as they engaged in the tasks and other classroom activities. Aligning with the methods of a teaching experiment, I developed learning goals for three concepts in trigonometry — radian measure, evaluating trigonometric expressions, and graphs and periodicity of trigonometric functions. Through analysis of students’ responses on assessments, their written work, and transcripts of class audio, I found that all students made progress toward all three learning goals, especially in the area of radian measure. Most students referred to radians as an arc length measured using the radius. I also found that all of the students understood π as a real number with some preferring to use 6.28 rather than 2π when talking or writing about radians or arc lengths, which came from the realistic tasks used in this study. Also, all students could explain the definition of a period of a trigonometric function, but some had trouble applying this definition to modeling contexts such as modeling hours of daylight throughout the year. Benefits of the realistic tasks used in this study were valuable discussions that provided students with opportunities to articulate their thinking using their own, informal language. Their articulations helped me as their teacher to address misconceptions and to use students’ language when talking about the concepts addressed in the learning goals. Students were engaged in all of the tasks and, for one student especially, the tasks proved to be valuable tools for expression and development of thinking that traditional assessments did not capture. The findings from this study benefit researchers and teachers of mathematics as they work to understand student thinking or implement classroom activities to improve student understanding of trigonometric functions. More broadly, this work contributes to the larger body of research on realistic tasks and their importance in mathematics classrooms for supporting conceptual understanding and student engagement.