Session Information
24 SES 05, The Role of Mathematical Tasks in Promoting Domain-Specific and Domain-Transcendent Mathematical Reasoning - Mathematically-Situated Reasoning (Part 1)
Symposium to be continued in 24 SES 06
Contribution
Forming and interpreting relationships between changing quantities (covariational reasoning) is essential for secondary students, yet little is known about its development. If students were to consistently engage in covariational reasoning, they likely would develop more robust conceptions of rate of change (e.g., Carlson et al., 2002, Thompson, 1994). However, secondary students may envision only one changing quantity (variational reasoning) when working on tasks involving multiple changing quantities (Johnson, 2013). In this research I investigated the question: What design aspects of mathematical tasks might foster students’ shifts from variational to covariational reasoning? Using design experiment methodology (Cobb et al., 2003), I developed a dynamic computer environment (DCE) involving a turning Ferris wheel to provide students with opportunities to form and interpret relationships between non-temporal quantities of distance, height and width. I drew on variation theory (Marton & Booth, 1997) to design and sequence a set of tasks that could provide opportunities for students to experience differences (e.g., relating different pairs of quantities; varying graphical representations of related quantities) that thereby could foster shifts in reasoning. Implementing tasks with five ninth grade students (~15 years old), I conducted a series of six clinical interviews with pairs, a trio, or individual students. Using comparative analysis (Corbin & Strauss, 2008), I examined portions of data when shifts in reasoning seemed likely to occur, looking across the set of tasks to trace shifts in students’ reasoning. These tasks provided the opportunity for a student to shift from variational to covariational reasoning, a key finding from the study. Implications for task design include: (1) incorporating DCEs with innovative graphs containing dynamic segments linked to changing quantities, (2) incorporating changing quantities measured with the same type of unit (e.g., distance and height), and (3) pairing a student engaging in variational reasoning with a student engaging in covariational reasoning.
References
Carlson, M. P., 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. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13. Corbin, J., & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory (3rd ed.). London: Sage Publications. Johnson, H. L. (2013). Designing covariation tasks to support students reasoning about quantities involved in rate of change. In C. Margolinas (Ed.), Task design in Mathematics Education. Proceedings of ICMI Study 22 (Vol. 1, pp. 213-222). Oxford. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Lawrence Erlbaum Associates. Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany, NY: State University of New York Press
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