# Bridging History of the Concept of Function with Learning of Mathematics: Students’ Meta-Discursive Rules, Concept Formation and Historical Awareness

- 529 Downloads
- 3 Citations

## Abstract

In this paper we present a matrix-organised implementation of an experimental course in the history of the concept of a function. The course was implemented in a Danish high school. One of the aims was to bridge history of mathematics with the teaching and learning of mathematics. The course was designed using the theoretical frameworks of a multiple perspective approach to history, Sfard’s theory of thinking as communicating, and theories from mathematics education about concept image, concept definition and concept formation. It will be explained how history and extracts of original sources by Euler from 1748 and Dirichlet from 1837 were used to (1) reveal students’ meta-discursive rules in mathematics and make them objects of students’ reflections, (2) support students’ learning of the concept of a function, and (3) develop students’ historical awareness. The results show that it is possible to diagnose (some) of students’ meta-discursive rules, that some of the students acted according to meta-discursive rules that coincide with Euler’s from the 1700s, and that reading a part of a text by Dirichlet from 1837 created obstacles for the students that can be referenced to differences in meta-discursive rules. The experiment revealed that many of the students have a concept image that was in accordance with Euler’s rather than with our modern concept definition and that they have process oriented thinking about functions. The students’ historical awareness was developed through the course with respect to actors’ influence on the formation of mathematical concepts and the notions of internal and external driving forces in the historical development of mathematics.

## Keywords

Mathematics Education Expert Group Mathematical Concept Basis Group Discontinuous Function## References

- Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function.
*Educational Studies in Mathematics,**23*, 247–285.CrossRefGoogle Scholar - Dubinsky, E., & Harel, G. (1992). The process conception of function. In: Harel, G., & Dubinsky, E., (Eds.), The Concept of function: Aspects of epistemology and pedagogy, MAA Notes, 28, 85–106.Google Scholar
- Jensen, B. E. (2003).
*Historie—livsverden og fag*. Copenhagen: Gyldendal.Google Scholar - Jankvist, U. T., & Kjeldsen, T. H. (2011). New avenues for history in mathematics education: Mathematical competencies and anchoring.
*Science & Education,**20*, 831–862.CrossRefGoogle Scholar - Jensen, B. E. (2010).
*Hvad er historie*. Copenhagen: Akademisk Forlag.Google Scholar - Katz, V. J. (2009).
*A history of mathematics. An introduction*. Boston: Addison-Wesley.Google Scholar - Kjeldsen, T. H. (2011a). History in a competency based mathematics education: A means for the learning of differential equations. In V. Katz & C. Tzanakis (Eds.),
*Recent developments on introducing a historical dimension in mathematics education*(pp. 165–173). Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar - Kjeldsen, T. H. (2011b). Does History have a significant role to play for the learning of mathematics? Multiple perspective approach to history, and the learning of meta level rules of mathematical discourse. In E. Barbin, M. Kronfellner, & C. Tzanakis (Eds.),
*History and epistemology in mathematics education proceedings of the sixth European summer university ESU 6*(pp. 51–62). Vienna: Verlag Holzhausen GmbH.Google Scholar - Kjeldsen, T.H. (2012). Uses of history for the learning of and about mathematics: Towards a theoretical framework for integrating history of mathematics in mathematics education. Plenary address in The proceedings for the HPM 2012 satellite meeting of ICME-12, Daejeon, Korea, 2012, 1–21.Google Scholar
- Kjeldsen, T. H., & Blomhøj, M. (2012). Beyond motivation: History as a method for the learning of meta-discursive rules in mathematics.
*Educational Studies in Mathematics,**80*, 327–349.CrossRefGoogle Scholar - Lützen, J. (1978). Funktionsbegrebets udvikling fra Euler til Dirichlet.
*Nordisk Matematisk Tidsskrift,**25*(26), 5–32.Google Scholar - Lützen, J. (1983). Euler’s vision of a general partial differential calculus for a generalized kind of function.
*Mathematics Magazine,**56*(5), 299–306.CrossRefGoogle Scholar - Petersen, P.H. (2011).
*Potentielle vindinger ved inddragelse af matematikhistorie i matematikundervisningen.*Master Thesis in mathematics, Roskilde University, Denmark. http://milne.ruc.dk/ImfufaTekster/pdf/483web.pdf. - Sfard, A. (1991). On the dual nature of mathematical conception: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics,**22*, 1–36.CrossRefGoogle Scholar - Sfard, A. (2000). On reform movement and the limits of mathematical discourse.
*Mathematical Thinking and Learning,**2*(3), 157–189.CrossRefGoogle Scholar - Sfard, A. (2008).
*Thinking as communicating*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.
*Educational Studies in Mathematics,**12*, 151–169.CrossRefGoogle Scholar