On Transitions Between Representations: The Role of Contextual Reasoning in Calculus Problem Solving

References

• Aspinwall, L., & Shaw, K. (2002). Report on calculus: Two contrasting cases. Mathematics Teacher, 95(6), 434–439.
   Google Scholar
• Aspinwall, L., Shaw, K., & Presmeg, N. (1997). Uncontrollable mental imagery: visual connections between a function and its derivative. Educational Studies in Mathematics, 33, 301–317.
   Google Scholar
• Bressoud, D. (2011). The Calculus I instructor. Washington, DC: Mathematical Association of America.
   Google Scholar
• 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.
   Google Scholar
• Carlson, M., Madison, B., & West, R. (2010). The Calculus Concept Readiness (CCR) instrument: Assessing student readiness for calculus. Preprint, http://arxiv.org/abs/1010.2719.
   Google Scholar
• Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. In Proceedings of the 21st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Cuernavaca, Morelos, Mexico.
   Google Scholar
• Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.
   Google Scholar
• Geometer's Sketchpad. (2011). Retrieved from http://www.dynamicgeometry.com/.
   Google Scholar
• Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1/3), 111–129.
   Google Scholar
• Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. C. (2010). Contrasting cases of calculus students' understanding of derivative graphs. Mathematical Thinking and Learning, 12(2), 152–176.
   Google Scholar
• Harel, G. (2008a). DNR perspective on mathematics curriculum and instruction, Part I: focus on proving. ZDM, 40(3), 487–500.
   Google Scholar
• Harel, G. (2008b). A DNR perspective on mathematics curriculum and instruction. Part II: with reference to teacher's knowledge base. ZDM, 40(5), 893–907.
   Google Scholar
• Hughes-Hallett, D. H., Gleason, A., Flath, D., Gordon, S., Lomen, D., Lovelock, D., … Thrash, K. (1994). Calculus. New York, NY: John Wiley & Sons.
   Google Scholar
• Kaput, J. J. (1987). Representation and mathematics. In C. Janvier (Ed.), Problems of representation in mathematics learning and problem-solving (pp. 19–26). Hillsdale, NJ: Lawrence Erlbaum Associates.
   Google Scholar
• Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500–507.
   Google Scholar
• Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago, IL: University of Chicago Press.
   Google Scholar
• Leinhardt, G., Zaslavsky, O., & Stein, M. M. (1990). Functions, graphs, and graphing: Tasks, learning and teaching. Review of Educational Research, 60, 1–64.
   Google Scholar
• Mejia-Ramos, J. P., & Weber, K. (2015). Mathematics majors' example and diagram usage when writing calculus proofs. In Proceedings of the Conference for Research in Undergraduate Mathematics Education. Pittsburgh, PA, USA.
   Google Scholar
• Nemirovsky, R., Tierney, C., & Wright, T. (1998). Body motion and graphing. Cognition and Instruction, 16(2), 119–172.
   Google Scholar
• Presmeg, N. C. (1986). Visualization and mathematical giftedness. Educational Studies in Mathematics, 17, 297–311.
   Google Scholar
• Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics: Emergence from psychology. In A. Gutierrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 205–235). Rotterdam, The Netherlands: Sense.
   Google Scholar
• Presmeg, N. C., & Bergsten, C. (1995). Preference for visual methods: An international study. In L. Meira, & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 3, 58–65.
   Google Scholar
• Roth, W. M., & Bowen, G. M. (2001). Professionals read graphs: A semiotic analysis. Journal for Research in Mathematics Education, 32, 159–194.
   Google Scholar
• Roth, W. M., & McGinn, M. K. (1998). Inscriptions: Toward a theory of representing as social practice. Review of Educational Research, 68(1), 35–59.
   Google Scholar
• Samkoff, A., Lai, Y., & Weber, K. (2012). On the different ways that mathematicians use diagrams in proof construction. Research in Mathematics Education, 14(1), 49–67.
   Google Scholar
• Schnotz, W. (2002). Commentary: Towards an integrated view of learning from text and visual displays. Educational Psychology Review, 14(1), 101–120.
   Google Scholar
• Vinner, S. (1989). The avoidance of visual considerations in calculus students. Focus on Learning Problems in Mathematics, 11(2), 149–156.
   Google Scholar
• Wawro, M., Rasmussen, C., Zandieh, M., Larson, C., & Sweeney, G. (2012). An inquiry-oriented approach to span and linear independence: The case of the magic carpet ride sequence. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 22(8), 577–599. doi:10.1080/10511970.2012.667516.
   Google Scholar
• Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2–3), 209–234.
   Google Scholar
• Wittmann, M. C., Flood, V. J., & Black, K. E. (2013) Algebraic manipulation as motion within a landscape. Educational Studies in Mathematics, 82(2), 169–181.
   Google Scholar
• Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education IV (Vol. 8, pp. 103–126). Providence, RI: American Mathematical Society.
   Google Scholar
• Zandieh, M. J., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. The Journal of Mathematical Behavior, 25(1), 1–17.
   Google Scholar
• Zazkis, R., Dubinsky, E., & Dautermann, J. (1996). Coordinating visual and analytic strategies: A study of students' understanding of the group D4. Journal for Research in Mathematics Education, 27(4), 435–457.
   Google Scholar