Abstract
Too often powerful and beautiful mathematical ideas are learned (and taught) in a procedural manner, thus depriving students of an experience in which they create and refine ideas for themselves. As a first step toward improving the current undesirable situation in undergraduate mathematics education, this chapter describes several different modeling perspectives and their implications for teaching and learning.
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Notes
- 1.
DNR stands for duality, necessity, and repeated reasoning (Harel and Sowder, 2007).
References
Black, M. (1962). Models and Metaphors. Ithaca, NY: Cornell University Press.
Black, M. (1977). More about Metaphor. Dialectica, 31, 433–57.
Black, M. (1979). How Metaphors Work: A Reply to Donald Davidson. Critical Inquiry, 6, 131–143.
Cobb, P., and Gravemeijer, K. (in press). Experimenting to support and understand learning processes. In A. E. Kelly, and R. Lesh (Eds.), Design Research in Education. Mawah, NJ: Erlbaum.
Dewey, J. (1896). The reflex arc concept in psychology. Psychological Review, 3, 357–370.
Fauconnier, G., and Turner, M. (2002). The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities. New York: Basic Books.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1, 155–177.
Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6, 105–128.
Harel, G., and Sowder, L. (1998). Students’ proof schemes. In E. Dubinsky, A. Schoenfeld, and J. Kaput (Eds.), Research on Collegiate Mathematics Education, Vol. III (pp. 234–283), AMS.
Harel, G., and Sowder, L. (2007). Toward a comprehensive perspective on proof. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning, National Council of Teachers of Mathematics (pp. 805–842).
Hughes-Hallett, D., Gleason, A., et al. (2002). Calculus (3rd ed.). New York: John Wiley.
Lakoff, G. (1987). Women, Fire, and Dangerous Things: What Categories Reveal About the Mind. Chicago: University of Chicago Press.
Lakoff, G., and Johnson, M. (1980). Metaphors We Live By. Chicago: The University of Chicago Press.
Lakoff, G., and Núñez, R. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
Lesh, R., and Doerr, H.M. (2003). Foundations of a models and modeling perspective on mathe-matics teaching, learning, and problem solving. In R. Lesh, and H.M. Doerr (Eds.), Beyond Constructivism: Models and Modeling Perspective on Mathematics Teaching, Learning, and Problem Solving (pp. 3–33). Mahwah, NJ: Lawrence Erlbaum Associates.
Lesh, R., Hoover, M., Hole, B., Kelly, A., and Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. Kelly, and R. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education (pp. 665–708). Mahwah, NJ: Lawrence Erlbaum Associates.
Oehrtman, M. (in press). Layers of abstraction: Theory and design for the instruction of limit concepts. To appear in M. Carlson, and C. Rasmussen (Eds.), Making the Connection: Research and Practice in Undergraduate Mathematics, MAA Notes Volume.
Oehrtman, M. (2007). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Manuscript submitted for publication.
Schoenfeld, A. (1985). Mathematical Problem Solving. Orlando, FL: Academic Press.
Sealey, V., and Oehrtman, M. (2005). Student understanding of accumulation and Riemann Sums. In G.M. Lloyd, M.R. Wilson, J.L.M. Wilkins, and S.L. Behm (Eds.), Proceedings of the 27th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education [CD-ROM]. Eugene, OR: All Academic.
Sealey, V., and Oehrtman, M. (2007). Calculus students’ assimilation of the Riemann integral. Manuscript submitted for publication.
Silver, E. (1985). Teaching and Learning Mathematical Problem Solving: Multiple Research Perspective. Hillsdale, NJ: Lawrence Erlbaum.
Speiser, B., and Walter, C. (2004). Placenticeras: Evolution of a task. Journal of Mathematical Behavior, 23(3), 259–269.
Speiser, R., and Walter, C. (2007). Rethinking change. In M. Carlson, and C. Rasmussen (Eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics. Washington, DC: Mathematical Association of America (in press).
Taylor, P. D. (1992). Calculus: The Analysis of Functions. Toronto: Wall & Emerson.
Zandieh, M., and Rasmussen, C. (2007). A case study of defining: Creating a new mathematical reality. Manuscript submitted for publication.
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Larson, C. et al. (2010). Modeling Perspectives in Math Education Research. In: Lesh, R., Galbraith, P., Haines, C., Hurford, A. (eds) Modeling Students' Mathematical Modeling Competencies. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0561-1_5
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