An Analysis of Conceptions of Inverse Trigonometric Functions

• In Event: 75.031 - Deepening Our Understanding of Student Mathematics Knowledge in Community Colleges

Abstract

Trigonometry is offered in post-secondary institutions as part of a sequence of preparatory courses that lead to a calculus sequence. Nearly 50,000 students take trigonometry at two-year colleges (Lutzer, Rodi, Kirkman, & Maxwell, 2007), yet we know very little about how this topic is taught. Scholarship on this area is limited to students’ understanding of radian measure (Moore, 2010), advantages and disadvantages of teaching a ratio or a functional approach to the trigonometric relationships (Kendal & Stacey, 1997; Weber, 2005), and future teachers’ knowledge of trigonometry (Fi, 2003). As part of a larger project we studied how instructors explained inverse trigonometric functions to their students and also how students interpreted those explanations. We found discrepancies between what teachers did in classroom and what their students revealed during interviews. In search for understanding this phenomenon, we further investigated how textbooks present ideas associated with this topic. Our main goal was to describe the conceptions of inverse trigonometric functions present in explanations in textbooks and given by instructors and students, and use the discrepancies between these as a way to explain learning difficulties associated with these notions.

We use Balacheff’s model of mathematical conceptions (Balacheff & Gaudin, 2010) to frame the study. In his model a conception results from the interaction between the cognizant subject and the milieu. Conceptions of mathematical notions are strongly tied to particular mathematical problems, which frame the notions, representations, operations, and control strategies.

The data was collected in the context of three courses, trigonometry, pre-calculus, and calculus taught by two community college instructors. We collected students’ responses to a questionnaire on inverse trigonometric functions (n=45), interviews with students on various trigonometric ideas (n=15), interviews with faculty on those results (2 faculty 6 interviews total), video-recordings of classroom sessions (n=9), and sections from the textbooks teachers used. Analytically, we identified the conceptions on trigonometry functions on each of the sources collected.

Overall we found a high preponderance of mathematical problems that gave way to two distinct conceptions about angles. The repertoire of operations was small and directly tied to the dominant representation, numeric, based either on triangles or on unit circles. There was no smooth transition from these to a Cartesian representation of the functions and a limited set of control strategies. The analysis of the student interview data revealed difficulties interpreting the meaning of intervals, composition of functions, inverse functions, injective (one to one) functions, domain, range, and angle measures. We traced some of these difficulties to instructors’ and textbooks’ explanations. Our analysis of textbooks revealed two competing conceptions of angles, a static one based on triangle trigonometry, and a dynamic one based on the unit circle. These representations generate operations and means of control that are very specific to each conception, yet faculty and students use them interchangeably.

These discrepancies contribute to the difficulty of these topics and suggest the need for faculty development that can help faculty realize the difference between these notions and to create explanatory paths for students as they teach.

Author

• Vilma M. Mesa, University of Michigan