Educational Studies in Mathematics

, Volume 66, Issue 2, pp 165–183 | Cite as

The notion of historical “parallelism” revisited: historical evolution and students’ conception of the order relation on the number line



This paper associates the findings of a historical study with those of an empirical one with 16 years-old students (1st year of the Greek Lyceum). It aims at examining critically the much-discussed and controversial relation between the historical evolution of mathematical concepts and the process of their teaching and learning. The paper deals with the order relation on the number line and the algebra of inequalities, trying to elucidate the development and functioning of this knowledge both in the world of scholarly mathematical activity and the world of teaching and learning mathematics in secondary education. This twofold analysis reveals that the old idea of a “parallelism” between history and pedagogy of mathematics has a subtle nature with at least two different aspects (metaphorically named “positive” and “negative”), which are worth further exploration.


Order relation Number line History in mathematics education Phylogeny Ontogeny Historical parallelism Students’ conceptions 


  1. Arcavi, A. (2004). Solving linear equations – why, how and when? For the Learning of Mathematics, 24(3), 25–28.Google Scholar
  2. Arcavi, A., & Bruckheimer, M. (2000). Didactical uses of primary sources. Themes in Education, 1(1), 55–74.Google Scholar
  3. Artigue, M. (1992). Functions from an algebraic and graphic point of view: Cognitive difficulties and teaching practices. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 109–132). Washington, DC: M.A.A.Google Scholar
  4. Bednarz, N., Kieran, C., & Lee, L. (Eds.) (1996). Approaches to algebra: Perspectives for research and teaching. Dordrecht: Kluwer.Google Scholar
  5. Bottazzini, U. (1986). The higher calculus: A history of real and complex analysis from Euler to Weierstrass. New York: Berlin Heidelberg Springer.Google Scholar
  6. Brousseau, G. (1983). Les obstacles épistémologiques et les problèmes en mathématiques. Recherches en Didactique des Mathématiques, 4, 164–198.Google Scholar
  7. Cauchy, A. L. (1821/1897). Cours d’Analyse de l’École Royale Polytechnique, 1ère Partie: Analyse Algébrique. In Oeuvres Complètes D’A. Cauchy. Paris: IIème Serie, tome III, Gauthier-Villars.Google Scholar
  8. Descartes, R. (1637/1954). The geometry of René Descartes. D. E. Smith & M. L. Latham (trans.). Dover, New York.Google Scholar
  9. Euler, L. (1770/1984). Elements of algebra. J. Hewlett (trans.), Springer, New York (French translation 1774; English translation 1840).Google Scholar
  10. Euler, L. (1748/1990). Introduction to analysis of the infinite, Book II. J.D. Blanton (trans.). Berlin Heidelberg New York: Springer.Google Scholar
  11. Fauvel, J. (1991). Using history in mathematics education. For the Learning of Mathematics, 11(2), 3–6.Google Scholar
  12. Fauvel, J., & van Maanen, J. (Eds.) (2000). History in mathematics education: The ICMI study. Dordrecht: Kluwer.Google Scholar
  13. Furinghetti, F., & Radford, L. (2002). Historical conceptual developments and the teaching of mathematics: From phylogenesis and ontogenesis theory to classroom practice. In L. English (Ed.), Handbook of international research in mathematics education (pp. 631–654). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  14. Gagatsis, A., & Thomaidis, Y. (1995). Eine Studie zur historischen Entwicklung und didaktischen Transposition des Begriffs absoluter Betrag. Journal für Mathematik-Didaktik, 16, 3–46.Google Scholar
  15. Gericke, H. (1970). Geschichte des Zahlbegriffs. Mannheim: Bibliographisches Institute.Google Scholar
  16. Glaeser, G. (1981). Epistémologie des nombres relatifs. Recherches en Didactique des Mathématiques, 2, 303–346.Google Scholar
  17. Grabiner, J. (1981). The origins of Cauchy’s rigorous calcula. Cambridge, Mass: The MIT Press.Google Scholar
  18. Harper, E. (1987). Ghosts of Diophantus. Educational Studies in Mathematics, 18, 75–90.CrossRefGoogle Scholar
  19. Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 60–86). Reston, VA: N. C. T. M.Google Scholar
  20. Klein, F. (1908/1939). Elementary mathematics from an advanced standpoint: Part 1. Arithmetic-Algebra-Analysis. E.R. Hedrick & C.A. Noble (trans.). New York: Dover.Google Scholar
  21. Lagrange, J. L. (1770/1868). Addition au mémoire sur la résolution des équations numériques. In Ouevres (pp. 581–652). Paris: Tome 2. Gauthier-Villars.Google Scholar
  22. Moreno, L. E., & Waldegg, G. (1991). The conceptual evolution of actual mathematical infinity. Educational Studies in Mathematics, 22, 211–231.CrossRefGoogle Scholar
  23. Mosvold, R. (2003). Genesis principles in mathematics education. In O. Bekken & R. Mosvold (Eds.), Study the masters (pp. 85–96). Göteborg: Nationel Centrum för Matematikutbildning.Google Scholar
  24. Napier, J. (1616/1969). A description of the admirable table of logarithms. Amsterdam: Da Capo Press.Google Scholar
  25. Pycior, H. (1987). British abstract algebra: development and early reception. Cahiers d’Histoire et de Philosophie des Sciences, 21, 152–168.Google Scholar
  26. Russ, S. (1980). A translation of Bolzano’s paper on the intermediate value theorem. Historia Mathematica, 7, 156–185.CrossRefGoogle Scholar
  27. Schubring, G. (1986). Ruptures dans le statut mathématique des nombres négatifs. Petit x, 12, 5–32.Google Scholar
  28. Sfard, A. (1994). What history of mathematics has to offer to psychology of mathematics learning? In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the eighteenth international conference for the psychology of mathematics education, volume I (pp. 129–132). Lisbon, Portugal: University of Lisbon.Google Scholar
  29. Sierpinska, A. (1994). Understanding in mathematics. London: The Falmer Press.Google Scholar
  30. Sip, J. (1990). But everybody accepts this explanation: operations on signed numbers. In J. Fauvel (Ed.), History in the mathematics classroom. The I.R E.M. Papers (pp. 73–84). Leicester, UK: The Mathematical Association.Google Scholar
  31. Stacey, K., Chick, H., & Kendal, M. (Eds.) (2004). The future of the teaching and learning of algebra. Dordrecht: Kluwer.Google Scholar
  32. Stein, S. (1987). Gresham’s law: Algorithm drives out thought. For the Learning of Mathematics, 7(2), 2–4.Google Scholar
  33. Sutherland, R., Rojano, T., Bell, A., & Lins, R. (Eds.) (2001). Perspectives on school algebra. Dordrecht: Kluwer.Google Scholar
  34. Thomaidis, Y. (1993). Aspects of negative numbers in the early 17th century: An approach for didactic reasons. Science & Education, 2, 69–86.Google Scholar
  35. Thomaidis, Y. (1995). Didactic transposition of mathematical concepts and learning obstacles (The case of absolute value), Doctoral thesis. Greece: Aristotle University of Thessaloniki (in Greek).Google Scholar
  36. Tzanakis, C., & Kourkoulos, M. (2004). May history and physics provide a useful aid for introducing basic statistical concepts? In F. Furinghetti, S. Kaisjer, & A. Vretblad (Eds.), Proceedings of HPM 2004 (The HPM satellite meeting of ICME-10 & the 4th Summer University on the history and epistemology in mathematics education) (pp. 425–437). Sweden: Uppsala.Google Scholar
  37. Whiteside, D. T. (Ed.) (1972). The mathematical papers of Isaac Newton, Volume V. Cambridge: Cambridge University Press.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Experimental SchoolUniversity of MacedoniaThessalonikiGreece
  2. 2.Department of EducationUniversity of CreteRethymnonGreece

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