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Dynamic mathematics and the blending of knowledge structures in the calculus

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Abstract

This paper considers the role of dynamic aspects of mathematics specifically focusing on the calculus, including computer software that responds to physical action to produce dynamic visual effects. The development builds from dynamic human embodiment, uses arithmetic calculations in computer software to calculate ‘good enough’ values of required quantities and algebraic manipulation to develop precise symbolic values. The approach is based on a developmental framework blending human embodiment, with the symbolism of arithmetic and algebra leading to the formalism of real numbers and limits. It builds from dynamic actions on embodied objects to see the effect of those actions as a new embodiment that needs to be calculated accurately and symbolised precisely. The framework relates the growth of meaning in history to the mental conceptions of today’s students, focusing on the relationship between potentially infinite processes and their consequent embodiment as mental concepts. It broadens the strategy of process-object encapsulation by blending embodiment and symbolism.

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References

  • Artigue, M. (1991). Analysis. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 167–198). Dordrecht: Kluwer.

    Google Scholar 

  • Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II. CBMS Issues in Mathematics Education, 6, 1–32.

    Google Scholar 

  • Blokland, P., Giessen, C., & Tall, D. O. (2000). Graphic calculus for windows. Retrieved September 24, 2008 from http://www.vusoft2.nl.

  • Chae, S. D. (2002). Imagery and construction of conceptual knowledge in computer experiments with period doubling. Unpublished PhD, University of Warwick.

  • Cornu, B. (1991). Limits. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 153–166). Dordrecht: Kluwer.

    Google Scholar 

  • Donald, M. (2001). A mind so rare. New York: Norton.

    Google Scholar 

  • Gray, E. M., & Tall, D. O. (2001). Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 65–72). Utrecht, The Netherlands.

  • Heath, T. L. (1921). History of Greek mathematics volume 1. Oxford: Oxford University Press. (Reprinted Dover Publications 1963).

  • Hughes-Hallett, D., Gleason, A. M., McCallum, W. G., et al. (2007). Calculus: Single and multivariable. New York: Wiley.

    Google Scholar 

  • Keisler, H. J. (1976). Elementary calculus: An infinitesimal approach. Boston: Prindle, Weber and Schmidt.

    Google Scholar 

  • Kepler, J. (1858–1871). Opera Omnia (Vol. 8) (ed. C. von Frisch). Frankfurt: Heyder & Zimmer.

  • Leibniz, G. W. (1849–1860). Mathematische Schriften (7 vols) (ed. C. I. Gerhardt, Berlin and Halle).

  • Robinson, A. (1966). Non-standard analysis. Amsterdam: North Holland.

    Google Scholar 

  • Stewart, J. (2003). Calculus (5th ed.). Belmont: Thomson.

    Google Scholar 

  • Stewart, I. N., & Tall, D. O. (1977). Foundations of mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Tall, D. O. (1980a). Intuitive infinitesimals in the calculus. Abstracts of short communications, Fourth International Congress on Mathematical Education, Berkeley. Retrieved September 24, 2008 from http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1980c-intuitive-infls.pdf.

  • Tall, D. O. (1980b). Looking at graphs through infinitesimal microscopes, windows and telescopes. Mathematical Gazette, 64, 22–49. doi:10.2307/3615886.

    Article  Google Scholar 

  • Tall, D. O. (1981). Comments on the difficulty and validity of various approaches to the calculus. For the Learning of Mathematics, 2(2), 16–21.

    Google Scholar 

  • Tall, D. O. (1985). Understanding the calculus. Mathematics Teaching, 110, 49–53.

    Google Scholar 

  • Tall, D. O. (1986). Building and testing a cognitive approach to the calculus using interactive computer graphics. Unpublished PhD, University of Warwick, Coventry.

  • Tall, D. O. (1991a). The psychology of advanced mathematical thinking. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 3–21). Dordrecht: Kluwer.

    Google Scholar 

  • Tall, D. O. (1991b). Real functions and graphs for the BBC computer. Cambridge: Cambridge University Press.

    Google Scholar 

  • Tall, D. O. (2004). The three worlds of mathematics. For the Learning of Mathematics, 23(3), 29–33.

    Google Scholar 

  • Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20, 5–24.

    Google Scholar 

  • Tall, D. O., & Schwarzenberger, R. L. E. (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44–49.

    Google Scholar 

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Correspondence to David O. Tall.

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Tall, D.O. Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM Mathematics Education 41, 481–492 (2009). https://doi.org/10.1007/s11858-009-0192-6

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