FRAMING STUDENTS\u27 TRIGONOMETRIC THINKING

This study took place in a course focused on the roles of technology in the teaching and learning of mathematics (Cullen, Hertel, & Nickels, in press) populated by preservice secondary mathematics teachers. The focus was on students working in different groups trying to identify the tangent function in a directed length model (Cullen & Martin, 2018; Hertel & Cullen, 2011). Students were interacting with a dynamic geometry environment relating arc lengths and vectors as an alternative to traditional definitions for trigonometric functions. We used three different theoretical frameworks to analyze student thinking and then examined the limitations and affordances of each one. We identified one episode that contained student argumentation (Connor, Singletary, Smith, Wagner, Francisco, 2014) one episode that exposed student concept images and definitions (Tall & Vinner, 1981), and one episode that showed creative and imitative reasoning (Lithner, 2008). We will highlight the types of conclusions that can be drawn from using each framework. We will provide recommendations about the utility of each of the frameworks. Conner, A. M., Singletary, L. M., Smith R. C., Wagner, P. A., & Francisco, R. T. (2014). Educational Studies in Mathematics, 86, pp. 401-429. doi:10.1007/s10649-014-9532-8 Cullen, C. J., Hertel, J. T., & Nickels, M. (in Press). The roles of technology in mathematics education. The Educational Forum. Cullen, C. J., & Martin, T. S. (2018). Exploring trigonometric relationships: Is it a function? In Editors, Proceedings of the 40th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 259-262). Greenville, SC: Publisher. Cullen, C. J., & Martin, T. S. (2018). Discovering trig identities in geometric representations. The Mathematics Teacher, 112(3), p. 260. Hertel, J., & Cullen, C. (2011). Teaching trigonometry: A directed length approach. In L. R., Wiest, & T. Lamberg (Eds.), Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1400-1407). Reno, NV: University of Nevada, Reno. Lithner, J. (2008) A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67, pp. 255-276. Available at https://doi.org/10.1007/s10649-007-9104-2 Tall, D., & Vinner, S. (1981). Concept images and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, pp. 151-169. doi:10.1007/BF0030561