# An essential tension in mathematics education

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## Abstract

There is a problem that goes through the history of calculus: the tension between the intuitive and the formal. Calculus continues to be taught as if it were natural to introduce the study of change and accumulation by means of the formalized ideas and concepts known as the mathematics of ε and δ. It is frequently considered as a failure that “students still seem to conceptualize limits via the imagination of motion.” These kinds of assertions show the tension, the rift created by traditional education between students’ intuitions and a misdirected formalization. In fact, I believe that the internal connections of the intuition of change and accumulation are not correctly translated into that arithmetical approach of ε and δ. There are other routes to formalization which cohere with these intuitions, and those are the ones discussed in this paper. My departing point is epistemic and once this discussion is put forward, I produce a narrative of classroom work, giving a special place to *local* conceptual organizations.

## Keywords

Variation Infinitesimal Infinite Archimedean principle Symbol Intuition Formalization Continuum Limit Derivative## References

- Biza, I. (2011). Students’ evolving meaning about tangent line with the mediation of a dynamic geometry environment and an instructional example space.
*Technology, Knowledge and Learning,**16*(2), 125–151.Google Scholar - Borba, M., & Villarreal, M. (2005).
*Humans-with-media and the reorganization of mathematical thinking: Information and communication technologies, modeling, visualization, and experimentation*. New York: Springer.Google Scholar - Bottazzini, U. (1986).
*The higher calculus: A history of real and complex analysis from Euler to Weierstrass*. New York: Springer.Google Scholar - Bradley, R. E., & Sandifer, C. E. (2009).
*Cauchy’s cours d’analyse: An annotated translation*. New York: Springer.CrossRefGoogle Scholar - Demidov, S. S., & Shenitzer, A. (2000). Two letters by N. N. Luzin to M. Ya. Vygodskii.
*The American Mathematical Monthly,**107*(1), 64–82.CrossRefGoogle Scholar - Donald, M. (2001).
*A mind so rare: The evolution of human consciousness*. New York: Norton.Google Scholar - Edwards, C. H. (1979).
*The historical developments of the calculus*. New York: Springer.CrossRefGoogle Scholar - Euler, L. (1988).
*Introduction to analysis of the infinite*(trans.: Blanton, J. D.). New York: Springer.Google Scholar - Forinash, K., Ramsey, W., & Lang, C. (2000). Galileo’s mathematical language of nature.
*Science & Education,**9*(5), 449–456.CrossRefGoogle Scholar - Gordon, E., Kusraev, A. G., & Kutateladze, S. S. (2002).
*Infinitesimal analysis*. Dordrecht: Kluwer.CrossRefGoogle Scholar - Grabiner, J. (1983). Who gave you the epsilon? Cauchy and the origins of rigorous calculus.
*The American Mathematical Monthly,**90*(3), 185–194.CrossRefGoogle Scholar - Klein, F. (1896). The arithmetizing of mathematics.
*Bulletin of the American Mathematical Society,**2*(8), 241–249.CrossRefGoogle Scholar - Klein, F. (2004).
*Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis*(trans.: Hedrick, E., & Noble, C.). New York: Dover (Translated from the Third German Edition, 1932, New York: Macmillan.).Google Scholar - Kleiner, I. (2001). History of the infinitely small and the infinitely large in calculus.
*Educational Studies in Mathematics,**48*(2–3), 137–174.CrossRefGoogle Scholar - Maschietto, M. (2008). Graphic calculators and micro-straightness: Analysis of a didactic engineering.
*International Journal of Computers for Mathematical Learning,**13*(3), 207–230.CrossRefGoogle Scholar - Moreno-Armella, L. (1996). Mathematics: A historical and didactic perspective.
*International Journal of Mathematics Education in Science and Technology,**27*(5), 633–639.CrossRefGoogle Scholar - Moreno-Armella, L., & Hegedus, S. (2009). Co-action with digital technologies. ZDM—The International Journal on Mathematics Education,
*41*, 505–519.Google Scholar - Moreno-Armella, L., & Hegedus, S. (2013). From static to dynamic mathematics: Historical and representational perspectives. In S. Hegedus & J. Roschelle (Eds.),
*The SimCalc vision and contributions*(pp. 27–45). Dordrecht: Springer. doi: 10.1007/978-94-007-5696-0_3. - Moreno-Armella, L., & Sriraman, B. (2010). Symbols and mediation in mathematics education. In B. Sriraman & L. English (Eds.),
*Theories of mathematics education: Seeking new frontiers*. Berlin Heidelberg: Springer.Google Scholar - Morgan, F. (2005).
*Real analysis*. Providence: American Mathematical Society.Google Scholar - Otte, M. (2006). Mathematical epistemology from a Peircean point of view.
*Educational Studies of Mathematics,**61*(1–2), 11–38.CrossRefGoogle Scholar - Robinson, A. (1966).
*Non-standard analysis*. Amsterdam: North-Holland.Google Scholar - Roh, K. H. (2008). Students’ images and their understanding of definitions of the limit of a sequence.
*Educational Studies in Mathematics,**69*(3), 217–233.CrossRefGoogle Scholar - Struik, D. J. (Ed.). (1969).
*A source book in mathematics, 1200–1800*. Cambridge: Harvard University Press.Google Scholar - Tall, D. (2003). Using technology to support an embodied approach to learning concepts in mathematics. In L. Carvalho & L. Guimarães (Eds.),
*História e tecnologia no ensino da matemática*(Vol. 1, pp. 1–28).Google Scholar - Tall, D. (2013). A sensible approach to the calculus. F. Pluvinage & A. Cuevas (Eds.),
*Handbook on calculus and its teaching*. México: Pearson.Google Scholar - Thom, R. (1973). Modern mathematics: does it exist? In A. G. Howson (Ed.),
*Development in mathematical education. Proceedings of the Second International Congress on Mathematical Education*(pp. 159–209). Cambridge: Cambridge University Press.Google Scholar - Toeplitz, O. (2007).
*The calculus, a genetic approach*. Chicago: University of Chicago Press.CrossRefGoogle Scholar