Trigonometry is a rich mathematical content area that blends geometric, graphical, and algebraic reasoning. Modern instruction is typically conducted in a piecemeal fashion over a period of years with a focus on computation and procedures. Although students learn a variety of techniques for solving triangles, manipulating trigonometric expressions, and graphing functions, many rich opportunities for reasoning and sense making are missed. A number of authors have called for more meaningful connections in the teaching and learning of trigonometry (e.g., Bressoud, 2010; Thompson, 2008; Weber, 2005). This symposium aims to contribute to the conversation by sharing recent research in the area. Speakers will discuss three studies – understanding trigonometric functions through directed length, angle measure as a foundation for trigonometric functions, and the effects of visualization on student understanding of trigonometry – with attention to implications for research and practice. Presentations will focus around three central questions:

1. What are the critical ways of reasoning involved in constructing robust understandings of trigonometric functions?

2. What instructional approaches and tools (e.g., technology) support student engagement in the critical ways of reasoning articulated in response to question 1?

3. What are the implications of the studies' findings for future research into the teaching and learning of trigonometry?

Session Overview (3-5 minutes)

Presentation 1: Understanding Trigonometric Functions Through Directed Length (15 minutes)

This study investigated how an instructional sequence using a directed length definition of the trigonometric functions could influence participants reasoning about trigonometry. APOS Theory (Asiala et al., 1996) guided our conceptual framework and influenced the design, implementation, and analysis. We designed three lessons to engage participants with a directed length definition of the trigonometric functions within a dynamic geometry environment. The lessons were taught sequentially to a group of preservice teachers (n=23) enrolled in a course that focused on the use of technology tools for the secondary mathematics classroom. A pre/post assessment was administered and written work was analyzed using a constant comparative method (Merriam, 1998) to identify and categorize solution strategies.

Results of the two-tailed dependent t-test indicated a statistically significant change in scores t(22) = 7.19, p < .001 with calculated effect size d = 1.04. Analysis of written work resulted in the identification of three different solution strategies: ratio, directed length, and graph. We identified changes in student strategy use and an increase in effectiveness across the three identified strategies. Based on these results, we believe that students understood the directed length definition and integrated it into their trigonometric schema thereby developing new ways to reason about the trigonometric functions.

Presentation 2: Using Angle Measure as a Foundation for Trigonometric Functions (15 minutes)

This study used theories of quantitative and covariational reasoning (Carlson et al, 2002; Smith III & Thompson, 2008) to investigate students’ notions of angle measure and trigonometric functions. Specifically, the study explored how students’ understandings of angle measure influence their conceptions of the sine and cosine functions.

The data (video, audio, and written) of the study were generated during a teaching experiment (Steffe & Thompson, 2002) with three precalculus students from a university in the southwest US. The students attended group and individual teaching sessions taught by the researcher.

All teaching sessions were videotaped, all student work was collected, and a conceptual analysis (Thompson, 2008) was performed in order to characterize the students’ thinking.

Findings revealed that students who understood angle measure as a process of measuring a subtended arc in a unit length (e.g., the radius) were supported in reasoning about the sine and cosine functions as formalizing covariational relationships between two quantities’ values. Also, understanding the radius as a unit of measure provided the students a reasoning tool to construct connections between circle and right triangle contexts. In contrast, a student that conceived of angle measures as labels of geometric objects (e.g., not the result of a measurement process) encountered difficulties in using the sine and cosine functions to relate two quantities’ values in circle and triangle contexts.

Presentation 3: Effects of Visualization on Student Understanding of Trigonometry (15 minutes)

Since representation is an essential tool for mathematical thinking (Greeno & Hall, 1997), particularly regarding the “big three:” algebraic, numerical, and graphical (Nemirovsky, Tierney, & Wright, 1998), this study was designed to investigate the extent to which using technology for visualization (Arcavi, 2003) would affect students’ understanding of the trigonometric concepts of radian, reference angle, and the Unit Circle. A mixed-method approach (Johnson & Onwuegbuzie, 2004) consisting of a modified experimental design and a qualitative component was used with two high school precalculus classes. One class was taught by an experienced teacher using a traditional approach, and the other was taught by a student teacher employing a technology-enhanced approach.

Student interviews, classroom observations, and work samples were collected over a six-week period, with item analysis serving as a basis for between-group comparison. The main finding was that students who received technology-enhanced whole-class instruction emphasizing visualization demonstrated a higher degree of “trig sense” than students receiving traditional instruction. Trig sense is defined as understanding the characteristics of and relationship among right triangle trigonometry, the Unit Circle, and circular functions. More specifically, this study also found that technology use for animation and dynamic image manipulation had a positive effect upon student learning.

Disscussant (10-12 minutes)

The discussant will address the ways that the three presentations tie into the overarching questions and implications for future research.

Audience Discussion (30 minutes)

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References

Arcavi, A. (2003). The role of visualization in the learning of mathematics. Educational Studies in Mathematic, 52: 215-241.

Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D. & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education, 2, 1-32.

Bressoud, D. M. (2010). Historical reflections on teaching trigonometry. Mathematics Teacher, 104(2), 106-112.

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352-378.

Greeno, J. G., & Hall, R. P. (1997). Practicing representation. Phi Delta Kappan, 78(5), 361-367.

Johnson, R., & Onwuegbuzie, A. (2004). Mixed methods research: A research paradigm whose time has come. Educational Researcher, 33(7), 14-26.

Merriam, S. B. (1998). Qualitative research and case study applications in education. San Fransisco, CA: Jossey-Bass.

Nemirovsky, R., Tierney, C. & Wright, T. (1998). Body motion and graphing. Cognition and Instruction, 16(2), 119-172.

Smith III, J., & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the Early Grades (pp. 95-132). New York, NY: Lawrence Erlbaum Associates.

Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267-307). Hillside, NJ: Erlbaum.

Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Sépulveda (Eds.), Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 45-64). Morélia, Mexico: PME.

Weber, K. (2005). Students' understanding of trigonometric functions. Mathematics Education Research Journal, 17(3), 91-112.