Educational Studies in Mathematics

, Volume 62, Issue 2, pp 149–168 | Cite as

An Historical Phenomenology of Mean and Median

  • Arthur BakkerEmail author
  • Koeno P. E. Gravemeijer


Using Freudenthal’s method of historical phenomenology, the history of statistics was investigated as a source of inspiration for instructional design. Based on systematically selected historical examples, hypotheses were formulated about how students could be supported in learning to reason with particular statistical concepts and graphs. Such a historical study helped to distinguish different aspects and levels of understanding of concepts and helped us as instructional designers to look through the eyes of students. In this paper, we focus on an historical phenomenology of mean and median, and give examples of how hypotheses stemming from the historical phenomenology led to the design of instructional activities used for teaching experiments in grades 7 and 8 (12–14-years old).

Key Words

history of statistics inspiration for instructional activities preparation phase of design research 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bakker, A.: 2004, Design Research in Statistics Education: On Symbolizing and Computer Tools, CD Beta Press, Utrecht, The Netherlands.Google Scholar
  2. Bakker, A. and Gravemeijer, K.P.E.: 2004, ‘Learning to reason about distribution’, in D. Ben-Zvi and J. Garfield (eds.), The Challenge of Developing Statistical Literacy, Reasoning, and Thinking, Kluwer, Dordrecht, The Netherlands, pp. 147–168.Google Scholar
  3. Brousseau, G.: 1997, ‘Theory of didactical situations in mathematics’, in N. Balacheff, M. Cooper, R. Sutherland, and V. Warfield (eds. and trans.), Kluwer, Dordrecht, The Netherlands.Google Scholar
  4. Cobb, P., McClain, K. and Gravemeijer, K.P.E.: 2003, ‘Learning about statistical covariation’, Cognition and Instruction 21, 1–78.CrossRefGoogle Scholar
  5. Cortina, J.L.: 2002, ‘Developing instructional conjectures about how to support students' understanding of the arithmetic mean as a ratio’, in B. Phillips (ed.), Proceedings of the Sixth International Conference of Teaching Statistics [CD-ROM], International Statistical Institute, Voorburg, The Netherlands.Google Scholar
  6. Cournot, A.A.: 1843/1984, Exposition de la théorie des chances et des probabilités Treatise of the theory of chances and probabilities, Librairie philosophique J. Vrin, Paris.Google Scholar
  7. David, H.A.: 1995, ‘First (?) occurences of common terms in mathematical statistics’, The American Statistician 49, 121–133.CrossRefGoogle Scholar
  8. David, H.A.: 1998, ‘First (?) occurences of common terms in probability and statistics – A second list, with corrections’, The American Statistician 52, 36–40.CrossRefGoogle Scholar
  9. Eisenhart, C.: 1974, ‘The development of the concept of the best mean of a set of measurements from antiquity to the present day’, 1971 ASA Presidential Address. Unpublished manuscript.Google Scholar
  10. Eisenhart, C.: 1977, ‘Boscovich and the combination of observations’, in M.G. Kendall and R.L. Plackett (eds.), Studies in the History of Statistics and Probability, Vol. 2, Charles Griffin, London.Google Scholar
  11. Euclid: 1956, The thirteen books of The Elements. Translation with introduction and commentary by Sir Thomas L. Heath (T. H. Heath, trans.). New York: Dover.Google Scholar
  12. Fauvel, J. and Van Maanen, J. (Eds.).: 2000, History in Mathematics Education. Kluwer Academic Publishers, Dordrecht, the Netherlands.Google Scholar
  13. Freudenthal, H.: 1983a, Didactical Phenomenology of Mathematical Structures. Dordrecht, the Netherlands: Reidel.Google Scholar
  14. Freudenthal, H.: 1983b, ‘The implicit philosophy of mathematics: History and education’, in Proceedings of the International Congress of Mathematicians. Warsaw, pp. 1695–1709.Google Scholar
  15. Freudenthal, H.: 1991, Revisiting Mathematics Education: China Lectures. Dordrecht, the Netherlands: Kluwer Academic Publishers.Google Scholar
  16. Galton, F.: 1889, Natural Inheritance. Macmillan, London.Google Scholar
  17. Godard, R. and Crépel, P.: 1999, ‘An historical study of the median’, in Proceedings of the Canadian Society for History and Philosophy of Mathematics, Canadian Mathematical Society: Ottawa, pp. 207–218.Google Scholar
  18. Gravemeijer, K.P.E.: 2002, ‘Emergent modeling as the basis for an instructional sequence on data analysis’, in B. Phillips (ed.), Developing a Statistically Literate Society. Proceedings of the International Conference on Teaching Statistics [CD-ROM]. Cape Town, South-Africa, July 7–12, 2002.Google Scholar
  19. Gulikers, I. and Blom, K.: 2001, ‘‘An historical angle’, a survey of recent literature on the use and value of history in geometrical education’, Educational Studies in Mathematics 47, 223–258.CrossRefGoogle Scholar
  20. Hacking, I.: 1975, The Emergence of Probability. A Philosophical Study of Early Ideas About Probability, Induction and Statistical Inference. Cambridge University Press,London.Google Scholar
  21. Hald, A.: 1990, A History of Probability and Statistics and their Applications before 1750, John Wiley and Sons, New York.Google Scholar
  22. Harter, H.L.: 1977, A Chronological Annotated Bibliography on Order Statistics (Vol. 1): US Air Force.Google Scholar
  23. Heath, T.H.: 1981, A History of Greek Mathematics. Dover, New York.Google Scholar
  24. Huygens, C.: 1895, Oeuvres Completes de Christiaan Huygens (Vol. VI). Den Haag, the Netherlands: Nijhoff.Google Scholar
  25. Konold, C. and Pollatsek, A.: 2002, ‘Data analysis as the search for signals in noisy processes’, Journal for Research in Mathematics Education 33, 259–289.CrossRefGoogle Scholar
  26. Kotz, S. and Johnson, N.L. (eds.): 1981, Encyclopedia of Statistical Sciences, Wiley, New York.Google Scholar
  27. Laplace, P.S.: 1812/1891, Thé orie analytique des probabilités. Oeuvres completes Vol. 7. Gauthier-Villars, Paris.Google Scholar
  28. Legendre, A.M.: 1805, Nouvelles mé thodes pour la détermination des orbites des cometes, Courcier, Paris.Google Scholar
  29. Mokros, J. and Russell, S.J.: 1995, ‘Children's concepts of average and representativeness’, Journal for Research in Mathematics Education 26, 20–39.CrossRefGoogle Scholar
  30. Monjardet, B.: 1991, ‘élements pour une histoire de la médiane métrique’, in J. Feldman, G. Lagneau and B. Matalon (eds.), Moyenne, Milieu, Centre: Histoire et Usages, Ecole des Hautes études en Sciences Sociales, Paris, pp. 45–62.Google Scholar
  31. Pannekoek, A.: 1961, A History of Astronomy, Allen and Unwin, London.Google Scholar
  32. Peirce, C.S.: 1958, CP, Collected Papers of Charles Sanders Peirce. Harvard University Press, Cambridge, MA.Google Scholar
  33. Perry, M. and Kader, G.: 1998, ‘Counting penguins’, Mathematics Teacher 91(2), 110–116.Google Scholar
  34. Petrosino, A.J., Lehrer, R. and Schauble, L.: 2003, ‘Structuring error and experimental variation as distribution in the fourth grade’, Mathematical Thinking and Learning 5, 131–156.CrossRefGoogle Scholar
  35. Plackett, R.L.: 1970, ‘The principle of the arithmetic mean’, in E. Pearson and M.G. Kendall (eds.), Studies in the History of Statistics and Probability, Vol. 1, Griffin, London.Google Scholar
  36. Porter, T.M.: 1986, The Rise of Statistical Thinking, 1820–1900. Princeton University Press, Princeton.Google Scholar
  37. Portney, S. and Koenker, R.: 1997, ‘The Gaussian hare and the Laplacian tortoise: Computability of squared-error versus absolute-error estimators’, Statistical Science 12, 279–300.CrossRefGoogle Scholar
  38. Radford, L.: 2000, ‘Historical formation and student understanding of mathematics’, in J. Fauvel and J. van Maanen (eds.), History in Mathematics Education: The ICMI Study, Kluwer Academic Publishers, Dordrecht, the Netherlands.Google Scholar
  39. Rubin, E.: 1971, ‘Quantitative commentary on Thucydides’, The American Statistician 1971 (December), 52–54.Google Scholar
  40. Sheynin, O.: 1996, The History of the Theory of Errors, Verlag Dr. Markus Hänsel-Hohenhausen, Egelsbach, Germany.Google Scholar
  41. Simon, M.A.: 1995, ‘Reconstructing mathematics pedagogy from a constructivist perspective’, Journal for Research in Mathematics Education 26, 114–145.CrossRefGoogle Scholar
  42. Simpson, J.A. and Weiner, E.S.C. (eds.).: 1989, The Oxford English Dictionary, 2nd edn., Vol. 1, Clarendon Press, Oxford.Google Scholar
  43. Stamhuis, I.H.: 1996, ‘Christiaan Huygens correspondeert met zijn broer over levensduur Christiaan Huygens corresponds with his brother about life span’, De Zeventiende Eeuw 12(1), 161–170.Google Scholar
  44. Steinbring, H.: 1980, Zur Entwicklung des Wahrscheinlichkeitsbegriffs – Das Anwendungsproblem in der Wahrscheinlichkeitstheorie aus didaktischer Sicht. Bielefeld: Institut fur Didaktik der Mathematik der Universität Bielefeld.Google Scholar
  45. Stigler, S.M.: 1984, ‘Boscovich, Simpson and a 1760 manuscript on fitting a linear relation’, Biometrika 71(3), 615–620.CrossRefGoogle Scholar
  46. Stigler, S.M.: 1986, The History of Statistics. The Measurement of Uncertainty Before 1900, Harvard University Press, Cambridge, MA.Google Scholar
  47. Strauss, S. and Bichler, E.: 1988, ‘The development of children's concepts of the arithmetic mean’, Journal for Research in Mathematics Education 19, 64–80.CrossRefGoogle Scholar
  48. Székely, G.: 1997, ‘Problem corner’, Chance 10(4), 25.Google Scholar
  49. Véron, J. and Rohrbasser, J.-M.: 2000, ‘Lodewijk et Christiaan Huygens: La distinction entre vie moyenne et vie probable’, Mathé matiques et Sciences Humaines 38(149), 7–21.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institute of EducationUniversity of LondonLondonUnited Kingdom
  2. 2.Freudenthal InstituteUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations