Abstract
In this paper we explore the possibility of using the historical development of cognitive artifacts as a resource in mathematics education. We present three examples where the introduction of new artifacts has played a role in the development of a mathematical theory. Furthermore, we present a methodological approach for using original sources in the classroom. The creation of an inquiry-reflective learning environment in mathematics is a significant element of this methodology. It functions as a mediating link between the theoretical analysis of sources from the past and a classroom practice where the students are invited into the workplace of past mathematicians through history. We illustrate our methodology by applying it to the use of artifacts in original sources, hereby introducing a first version of such an inquiry-reflective learning environment in mathematics through history.
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Notes
- 1.
For an account of the development since 2000, see Clark et al. (2016).
- 2.
We are indebted to Professor Jesper Lützen, University of Copenhagen, for bringing this example to our attention in his talk at the Second Joint International Meeting of the Israel Mathematical Union and the American Mathematical Society, IMU-AMS in Tel Aviv, Israel, June 16–19, 2014.
- 3.
Or, in modern terms, that the discriminant of the resulting quadratic equation is non-negative.
- 4.
The table comes from al-Samaw’al’s work Al-Bahir fi’l-Hisab (The Shining Book on Calculation). As far as we know the book is not translated into English in its entirety. We here consider the original source to consist of the table and of al-Samaw’al’s explanation of the law of exponentials given below.
- 5.
The relevant part of the text is also available in Smith (1959), pp. 46–54.
- 6.
Caspar Wessel’s work On the analytic representation of direction is another source that could be used in this context.
References
Abd-El-Khalick, F. (2013). Teaching with and about nature of science, and science teacher knowledge domains. Science & Education, 22(9), 2087–2107.
Bagni, G. (2011). Equations and imaginary numbers: A contribution from Renaissance Algebra. In V. Katz & C. Tzanakis (Eds.), Recent developments on introducing a historical dimension in mathematics education. MAA Notes (Vol. 78, pp. 45–56). Washington, DC: The Mathematical Association of America.
Barbin, E. (2011). Dialogism in mathematical writing: Historical, philosophical and pedagogical issues. In V. Katz & C. Tzanakis (Eds.), Recent developments on introducing a historical dimension in mathematics education. MAA Notes (Vol. 78, pp. 9–16). Washington, DC: The Mathematical Association of America.
Barnett, J., Pengelley, D., & Lodder, J. (2014). The pedagogy of primary historical sources in mathematics: Classroom practice meets theoretical frameworks. Science & Education, Special Issue on History and Philosophy of Mathematics in Mathematics Education, 23(1), 7–27.
Berggren, J. L. (1986). Episodes in the mathematics of medieval Islam. New York: Springer.
Cardano, G. (2007). The rules of algebra (Ars Magna) (T. R. Witmer, Trans.). New York: Dover (Original work published 1545).
Carter, J. (2010). Diagrams and proofs in analysis. International Studies in The Philosophy of Science, 24(1), 1–14.
Clark, K., Kjeldsen, T. H., Schorcht, S., Tzanakis, C., & Wang, X. (2016). History of mathematics in mathematics education: Recent developments. In L. Radford, F. Furinghetti, & T. Hausberger (Eds.), Proceedings of the 2016 ICME Satellite Meeting—HPM 2016 (pp. 135–179). Montpellier: IREM de Montpellier.
Gadamer, H. G. (1975). Truth and method (2nd ed.). (J. Weisenheimer & D. G. Marshall, Trans.). London: Sheed & Ward.
Heath, T. (1921). A history of Greek mathematics. Vol. II: From Aristarchus to Diophantus. Oxford: The Clarendon Press.
Hutchins, E. (1995). Cognition in the wild. Cambridge, MA: MIT Press.
Jankvist, U. (2010). An empirical study of using history as a ‘goal’. Educational Studies in Mathematics, 74(1), 53–74.
Johansen, M. W., & Kjeldsen, T. H. (2015). Mathematics as a tool driven practice. The use of material and conceptual artefacts in mathematics. In E. Barbin, U. T. Jankvist, & T. H. Kjeldsen (Eds.), History and Epistemology in Mathematics Education. Proceedings of the 7th ESU. (pp. 79–95). Copenhagen: Danish School of Education, Aarhus University.
Johansen, M. W., & Misfeldt, M. (2015). Semiotic scaffolding in mathematics. Biosemiotics, 8, 325–340.
Katz, V. J. (1998). A history of mathematics: An introduction (2nd ed.). Reading, MA: Addison-Wesley.
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. In E. Barbin, C. Tzanakis, & S. Hwang (Eds.), Proceedings of HPM 2012 – The HPM Satellite Meeting of ICME-12. (pp. 1–21). Daejeon: Korean Society of Mathematical Education & Korean Society for History of Mathematics.
Kjeldsen, T. H. (2016). Enacting inquiry learning in mathematics through history. In L. Radford, F. Furinghetti, & T. Hausberger (Eds.), Proceedings of the 2016 ICME Satellite Meeting—HPM 2016 (pp. 453–464). Montpellier: IREM de Montpellier.
Kjeldsen, T. H., & Petersen, P. H. (2014). Bridging history of the concept of a function with learning of mathematics: Students’ meta-discursive rules, concept formation and historical awareness. Science & Education, 23(1), 29–45.
Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.
Smith, D. E. (Ed.). (1959). A source book in mathematics. New York: Dover.
Steensen, A. K., & Johansen, M. W. (2016). The role of diagram materiality in mathematics. Cognitive Semiotics, 9(2), 183–201.
Struik, D. J. (1969). A source book in mathematics 1200–1800. Cambridge, MA: Harvard University Press.
Svensson, R. O. (2016). Komparativ undersøgelse af deduktiv og induktiv matematikundervisning. IND’s studenterserie nr. 45. Copenhagen: Institut for Naturfagenes Didaktik. Master’s thesis. http://www.ind.ku.dk/publikationer/studenterserien/ros/. Accessed August 7, 2017.
Thomaidis, Y. (2005). A framework for defining the generality of Diophantos’ methods in ‘Arithmetica’. Archive for History of Exact Sciences, 59(6), 591–640.
Wallis, J. (1685). A treatise of algebra, both historical and practical, shewing the original, progress, and advancement thereof, from time to time, and by what steps it hath attained to the height at which it now is, with some additional treatises. London: Printed by John Playford for Richard Davis.
Zhang, J., & Patel, V. L. (2006). Distributed cognition, representation, and affordance. Pragmatics & Cognition, 14(2), 333–341.
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Johansen, M.W., Kjeldsen, T.H. (2018). Inquiry-Reflective Learning Environments and the Use of the History of Artifacts as a Resource in Mathematics Education. In: Clark, K., Kjeldsen, T., Schorcht, S., Tzanakis, C. (eds) Mathematics, Education and History . ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-73924-3_2
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