Dynamic mathematics and the blending of knowledge structures in the calculus
 David O. Tall
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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 processobject encapsulation by blending embodiment and symbolism.
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Within this Article
 Introduction
 A framework for the development of mathematical thinking
 A locally straight approach to calculus
 Cultural aspects
 Reflections
 References
 References
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 Artigue, M. (1991). Analysis. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 167–198). Dordrecht: Kluwer.
 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.
 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 Unpublished PhD, University of Warwick.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.
 Donald, M. (2001). A mind so rare. New York: Norton.
 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 HeuvelPanhuizen (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).
 HughesHallett, D., Gleason, A. M., McCallum, W. G., et al. (2007). Calculus: Single and multivariable. New York: Wiley.
 Keisler, H. J. (1976). Elementary calculus: An infinitesimal approach. Boston: Prindle, Weber and Schmidt.
 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). Nonstandard analysis. Amsterdam: North Holland.
 Stewart, J. (2003). Calculus (5th ed.). Belmont: Thomson.
 Stewart, I. N., & Tall, D. O. (1977). Foundations of mathematics. Oxford: Oxford University Press.
 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/dot1980cintuitiveinfls.pdf.
 Tall, D. O. (1980b). Looking at graphs through infinitesimal microscopes, windows and telescopes. Mathematical Gazette, 64, 22–49. doi:10.2307/3615886. CrossRef
 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.
 Tall, D. O. (1985). Understanding the calculus. Mathematics Teaching, 110, 49–53.
 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.
 Tall, D. O. (1991b). Real functions and graphs for the BBC computer. Cambridge: Cambridge University Press.
 Tall, D. O. (2004). The three worlds of mathematics. For the Learning of Mathematics, 23(3), 29–33.
 Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20, 5–24.
 Tall, D. O., & Schwarzenberger, R. L. E. (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44–49.
 Title
 Dynamic mathematics and the blending of knowledge structures in the calculus
 Journal

ZDM
Volume 41, Issue 4 , pp 481492
 Cover Date
 20090801
 DOI
 10.1007/s1185800901926
 Print ISSN
 18639690
 Online ISSN
 18639704
 Publisher
 SpringerVerlag
 Additional Links
 Authors

 David O. Tall ^{(1)}
 Author Affiliations

 1. Institute of Education, University of Warwick, Coventry, CV4 7AL, UK