ZDM

, Volume 39, Issue 1–2, pp 51–61 | Cite as

Game and decision theory in mathematics education: epistemological, cognitive and didactical perspectives

Original article
First Online:
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Abstract

In the 1950s, game and decision theoretic modeling emerged—based on applications in the social sciences—both as a domain of mathematics and interdisciplinary fields. Mathematics educators, such as Hans Georg Steiner, utilized game theoretical modeling to demonstrate processes of mathematization of real world situations that required only elementary intuitive understanding of sets and operations. When dealing with n-person games or voting bodies, even students of the 11th and 12th grade became involved in what Steiner called the evolution of mathematics from situations, building of mathematical models of given realities, mathematization, local organization and axiomatization. Thus, the students could participate in processes of epistemological evolutions in the small scale. This paper introduces and discusses the epistemological, cognitive and didactical aspects of the process and the roles these activities can play in the learning and understanding of mathematics and mathematical modeling. It is suggested that a project oriented study of game and decision theory can develop situational literacy, which can be of interest for both mathematics education and general education.

Keywords

Mathematics Education Decision Theory Winning Coalition United Nations Security Council Vote Situation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgments

The author wants to thank Rolf Biehler and Sören Scholz for their comments on an earlier version, Stephanie Keller for editorial support and Erika Louise Steiner for their recommendations on literature.

References

  1. Arrow, K. J. (1951). Social choice and individual values. New York: WileyGoogle Scholar
  2. Aumann, R. J. (2005). War and peace. Nobel Prize Lecture 8 December 2005. Retrieved 27 December 2006, from http://www.nobelprize.org/nobel_prizes/economics/laureates/2005/aumann- lecture.pdf
  3. Aumann, R. J. & Maschler, M. (1964). The bargaining set for cooperative games. In M. Drescher, L. S. Shapley & A. W. Tucker (Eds.), Advances in Game Theory. Annals of Mathematical Studies, 52, (pp. 443–476)Google Scholar
  4. Banzhaf, J. F. (1965). Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Review, 19, 31–343Google Scholar
  5. Ben-Zvi, D. & Garfield, J. (Eds.) (2004). The challenge of developing statistical literacy, reasoning and thinking. Dordrecht: KluwerGoogle Scholar
  6. Berger, P. L & Luckmann, T. (1966). The social construction of reality: a treatise in the sociology of knowledge. Garden City, NY: AnchorGoogle Scholar
  7. Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: KluwerGoogle Scholar
  8. Brousseau, G. (2004). Introduction to the study of the teaching of reasoning and proof: paradoxes. International newsletter on the teaching and learning of mathematical proof, 04 070809. Retrieved 25 December 2006, from http://www.lettredelapreuve.it/Newsletter/04Ete/04EteThemeUK.html
  9. Bruner, J. (1966). Towards a theory of instruction. Cambridge, MA: Harvard University PressGoogle Scholar
  10. Bruner, J. (1990). Acts of meaning. Cambridge, MA: Harvard University PressGoogle Scholar
  11. Brunswik, E. (1952). The conceptual framework of psychology. Chicago: University of Chicago PressGoogle Scholar
  12. Bush, R. R. & Estes, W. K. (Eds.) (1959). Studies in mathematical learning theory. Stanford, CA: Stanford University PressGoogle Scholar
  13. Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2–33CrossRefGoogle Scholar
  14. Collins, A. & Ferguson, W. (1993). Epistemic forms and games. Educational Psychologist, 28(1), 25–42CrossRefGoogle Scholar
  15. Damerow, P. (1977). Die Reform des Mathematikunterrichts in der Sekundarstufe I. Stuttgart: Klett-CottaGoogle Scholar
  16. Frey, K. (1982). Die Projektmethode. Beltz: WeinheimGoogle Scholar
  17. Heid, M. K. (1997). The technological revolution and the reform of school mathematics. American Journal of Education, 106(1), 5–61CrossRefGoogle Scholar
  18. Hicks, J. R. (1937). Mr. Keynes and the “Classics”. A suggested interpretation. Econometrica, 5, 147–159CrossRefGoogle Scholar
  19. Hogarth, R. M. (2001). Educating intuition. Chicago: University of Chicago PressGoogle Scholar
  20. Holler, M. J. & Illing, G. (2003). Einführung in die Spieltheorie. Berlin: SpringerGoogle Scholar
  21. Jahnke, H. N. (1978). Zum Verhältnis von Wissensentwicklung und Begründung in der Mathematik—Beweisen als didaktisches Problem. Materialien und Studien des IDM, Bd.10. Bielefeld: Universität Bielefeld, IDMGoogle Scholar
  22. Kahane, J. P. (1988). ICMI and recent developments in mathematical education. In M. Otte et al. (Eds.), Wissenschaftliches Kolloquium Hans-Georg Steiner zu Ehren. IDM Occasional Paper 116. pp. 13–24. Bielefeld: Universität Bielefeld, IDMGoogle Scholar
  23. Kemeny, J. G. & Snell, J. L. (1962). Mathematical models in the social sciences. New York: BlaisdellGoogle Scholar
  24. Laan, G. v.d., Brink, R. v.d. (1998). Axiomatizations of the normalized Banzhaf value and the Shapley value. Social Choice and Welfare, 15(4), 567–582CrossRefGoogle Scholar
  25. Larzarsfeld, P. F. (1954). Mathematical thinking in the social sciences. Glencoe, IL: The Free PressGoogle Scholar
  26. Lesh, R. (1985). Processes, skills, and abilities needed to use mathematics in everyday situations. Education and Urban Society, 17(4), 439–446Google Scholar
  27. Luce, R. C. & Raiffa, H. (1989). Games and decisions. New York: Dover PublicationsGoogle Scholar
  28. Lucas, W. F. & Billera, L. J. (1982). Modeling coalitional values. In S. J. Brams, W. F. Lucas & P. D. Straffin (Eds.) Political and related models (pp. 66–97). New York: SpringerGoogle Scholar
  29. Maschler, M. (ed.) (1962). Recent advances in game theory. Princeton: Princeton UniversityGoogle Scholar
  30. Morgenstern, O. (1951). Abraham Wald, 1902–1950. Econometrica, 19, 361–367CrossRefGoogle Scholar
  31. Nash, J. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United States of America, 36, 48–49CrossRefGoogle Scholar
  32. Negrete, M. A. M. (2000). Probability, an epistemological focus. Revista Mexicana de Fisica, 46(5), 490–495Google Scholar
  33. Neisser, U. (1967). Cognitive Psychology. Englewood Cliffs, NJ: Prentice-HallGoogle Scholar
  34. Neumann, J. & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University PressGoogle Scholar
  35. Olivier, A. (1999). Constructivist learning theory. In P. Human & A. Olivier (Eds.), Advanced numeracy course. Facilitator’s guide (pp. 24–30). Parow-East: Ebony Books. Retrieved 17 December 2006, from http://www.academic.sun.ac.za/mathed/MALATI/Files/Constructivism.pdf
  36. Osborne, M. (2003). An introduction to game theory. Oxford: Oxford University PressGoogle Scholar
  37. Ostmann, A. (1988). Theorie und Anwendung einfacher Spiele. Discussion Paper B-101, Sonderforschungsbereich 303, Bonn: Universität BonnGoogle Scholar
  38. Petrosjan, L. & Zaccour, G. (2003). Time-consistent Shapley value allocation of pollution cost reduction. Journal of Economic Dynamics & Control, 27(3), 381–398CrossRefGoogle Scholar
  39. Piaget, J. (1973). Comments on mathematical education. In A. G. Howson (Ed.), Developments in mathematical education: proceedings of the 2nd International congress on mathematical education (pp. 79–87). London: Cambridge University PressGoogle Scholar
  40. Pierce, R. & Stacey, K.(2006). Enhancing the image of mathematics by association with simple pleasures from real world contexts. Zentralblatt für Didaktik für Mathematik, 38(3), 214–225Google Scholar
  41. Raiffa, H. (2002). Decision analysis: a personal account of how it got started and evolved. Operations Research 50(1), 179–185Google Scholar
  42. Resnick, L. & Ford, W. W. (1981). The psychology of mathematics for instruction. Erlbaum: LondonGoogle Scholar
  43. Reumann, K. (1972). Polizei gegen die Professoren rufen? Ein Marburger Kapitel aus dem Kampf um die Abschaffung von Prüfungen. Frankfurter Allgemeine Zeitung vom 16.05.1972, S. 12Google Scholar
  44. Schoenfield, A. H. (2000). Purposes and methods of research in mathematics education. Notices of the AMS, June/July, 2000, 641–649. Retrieved December 17, 2006, from http://www.ams.org/notices/200006/fea-schoenfeld.pdf
  45. Scholz, R. W. (1987). Cognitive strategies in stochastic thinking. Dordrecht: ReidelGoogle Scholar
  46. Scholz R. W. & Tietje, O. (2002). Embedded case study methods: integrating quantitative and qualitative knowledge. Thousand Oaks: SageGoogle Scholar
  47. Scholz, R. W., Steiner, R. & Hansmann, R. (2004). Role of internship in higher education in environmental sciences. Journal of Research in Science Teaching, 41(1), 24–46CrossRefGoogle Scholar
  48. Scholz, R. W., Lang, D. J., Wiek, A., Walter, A. I., & Stauffacher, M. (2006). Transdisciplinary case studies as a means of sustainability learning: historical framework and theory. International Journal of Sustainability in Higher Education, 7(3), 226–251CrossRefGoogle Scholar
  49. Shaffer, D. W. (2006). Epistemic frames for epistemic games. Computers and Education, 46(3), 223–234CrossRefGoogle Scholar
  50. Shulman, L. S. & Carey, N. B. (1984). Psychology and the limitations of individual rationality: Implications for the study of reasoning and civility. Review of Educational Research, 54, 501–524CrossRefGoogle Scholar
  51. Simon, H. A. (1982). Models of bounded rationality. Cambridge, MA: MIT PressGoogle Scholar
  52. Skemp, R. R. (1987). The psychology of learning mathematics, 2nd Edn. London: ErlbaumGoogle Scholar
  53. Sneed, J. D. (1971). The logical structure of mathematical physics. Dordrecht: ReidelGoogle Scholar
  54. Stake, R. E. (1995). The art of case study research. Thousand Oaks, CA: SageGoogle Scholar
  55. Stauffacher, M., Walter, A. I., Lang, D. J., Wiek, A. & Scholz, R. W. (2006). Learning to research environmental problems from a functional socio-cultural constructivism perspective: The transdisciplinary case study approach. International Journal of Sustainability in Higher Education, 7(3), 252–275CrossRefGoogle Scholar
  56. Steiner, H. G. (1965). Menge, Struktur, Abbildung als Leitbegriffe für den mathematischen Unterricht. Der Mathematikunterricht, 11(1), 5–19Google Scholar
  57. Steiner, H. G. (1966). Mathematisierung und Axiomatisierung einer politischen Struktur. Der Mathematikunterricht 12(3), 66–86Google Scholar
  58. Steiner, H. G. (1967) Eine mathematische Theorie der Rangordnungen und Konsensus-Rangordnungen. Material zum Unterrichtsthema: Mathematische Modelle der Wirklichkeit. Mathematisch-Physikalische Semesterberichte XIV(2), 212–245Google Scholar
  59. Steiner, H. G. (1968). Examples of exercises in mathematization on secondary school level. Educational Studies in Mathematics, 1(1/2), 181–201CrossRefGoogle Scholar
  60. Steiner, H. G. (1969). Eine mathematische Theorie der Abstimmungsgremien. Technische Universität Darmstadt, DissertationGoogle Scholar
  61. Steiner, H.G. (1976). Zur Methodik des mathematisierenden Unterrichts. In W. Dörfler, R. Fischer (Eds) Anwendungsorientierte Mathematik in der Sekundarstufe II (pp. 211–246). Klagenfurt: HeynGoogle Scholar
  62. Steiner, H. G. (1982) Eine gemeinsame Charakterisierung der Additivität von Rangordnung auf endlichen Booleschen Verbänden und der Messbarkeit von Abstimmungsgebilden. Mathematische Semesterberichte XXIX(1), 21–50Google Scholar
  63. Steiner, H. G. (1984). The process of mathematization and the social dimension of mathematics—epistemological and didactical considerations. Occasional Paper 41. Bielefeld: Universität Bielefeld, IDMGoogle Scholar
  64. Steiner, H. G. (1988). Mathematization of voting systems: some classroom experiences. International Journal of Mathematical Education in Science and Technology, 19(2), 199–213Google Scholar
  65. Straf, M. L. (2003). Statistics: the next generation. Journal of the American Statistical Association, 98(461), 1–6CrossRefGoogle Scholar
  66. Törner, G. & Sriraman, B. (2007). A contemporary analysis of the six “theories of mathematics education”—Theses of Hans Georg Steiner. ZDM, 39(1), in this issueGoogle Scholar
  67. Vollrath, H. J. (2007). Towards an authentic teaching of mathematics – Hans Georg Steiner’s contribution to the reform of mathematics teaching. ZDM, 39(1) (in this issue)Google Scholar
  68. Walras, L. (1874). Elements of pure economics. Oxford: RoutledgeGoogle Scholar
  69. Wittenberg, A. I. (1963). Bildung und Mathematik, (1990, 2nd Edn) Stuttgart: KlettGoogle Scholar
  70. Zint, M. T. (2001). Advancing environmental risk education. Risk Analysis, 21(3), 417–426Google Scholar

Copyright information

© FIZ Karlsruhe 2007

Authors and Affiliations

  1. 1.ETH Zürich, Institute for Environmental Decisions (IED), Natural and Social Science Interface (NSSI)ZürichSwitzerland

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