Inquiry-Reflective Learning Environments and the Use of the History of Artifacts as a Resource in Mathematics Education

  • Mikkel Willum Johansen
  • Tinne Hoff Kjeldsen
Part of the ICME-13 Monographs book series (ICME13Mo)


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.


Cognitive artifacts Inquiry-based learning Original sources Mathematics education History of mathematics Reflective learning environment 


  1. Abd-El-Khalick, F. (2013). Teaching with and about nature of science, and science teacher knowledge domains. Science & Education, 22(9), 2087–2107.CrossRefGoogle Scholar
  2. 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.Google Scholar
  3. 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.Google Scholar
  4. 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.Google Scholar
  5. Berggren, J. L. (1986). Episodes in the mathematics of medieval Islam. New York: Springer.Google Scholar
  6. Cardano, G. (2007). The rules of algebra (Ars Magna) (T. R. Witmer, Trans.). New York: Dover (Original work published 1545).Google Scholar
  7. Carter, J. (2010). Diagrams and proofs in analysis. International Studies in The Philosophy of Science, 24(1), 1–14.CrossRefGoogle Scholar
  8. 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.Google Scholar
  9. Gadamer, H. G. (1975). Truth and method (2nd ed.). (J. Weisenheimer & D. G. Marshall, Trans.). London: Sheed & Ward.Google Scholar
  10. Heath, T. (1921). A history of Greek mathematics. Vol. II: From Aristarchus to Diophantus. Oxford: The Clarendon Press.Google Scholar
  11. Hutchins, E. (1995). Cognition in the wild. Cambridge, MA: MIT Press.Google Scholar
  12. Jankvist, U. (2010). An empirical study of using history as a ‘goal’. Educational Studies in Mathematics, 74(1), 53–74.CrossRefGoogle Scholar
  13. 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.Google Scholar
  14. Johansen, M. W., & Misfeldt, M. (2015). Semiotic scaffolding in mathematics. Biosemiotics, 8, 325–340.CrossRefGoogle Scholar
  15. Katz, V. J. (1998). A history of mathematics: An introduction (2nd ed.). Reading, MA: Addison-Wesley.Google Scholar
  16. 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.Google Scholar
  17. 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.Google Scholar
  18. 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.CrossRefGoogle Scholar
  19. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  20. Smith, D. E. (Ed.). (1959). A source book in mathematics. New York: Dover.Google Scholar
  21. Steensen, A. K., & Johansen, M. W. (2016). The role of diagram materiality in mathematics. Cognitive Semiotics, 9(2), 183–201.Google Scholar
  22. Struik, D. J. (1969). A source book in mathematics 1200–1800. Cambridge, MA: Harvard University Press.Google Scholar
  23. Svensson, R. O. (2016). Komparativ undersøgelse af deduktiv og induktiv matematikundervisning. IND’s studenterserie nr. 45. Copenhagen: Institut for Naturfagenes Didaktik. Master’s thesis. Accessed August 7, 2017.
  24. Thomaidis, Y. (2005). A framework for defining the generality of Diophantos’ methods in ‘Arithmetica’. Archive for History of Exact Sciences, 59(6), 591–640.CrossRefGoogle Scholar
  25. 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.Google Scholar
  26. Zhang, J., & Patel, V. L. (2006). Distributed cognition, representation, and affordance. Pragmatics & Cognition, 14(2), 333–341.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Science EducationUniversity of CopenhagenCopenhagen CDenmark
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark

Personalised recommendations