Advertisement

Technology, Knowledge and Learning

, Volume 17, Issue 1–2, pp 1–21 | Cite as

Abstraction Through Game Play

  • Antri Avraamidou
  • John Monaghan
  • Aisha Walker
Article
First Online:

Abstract

This paper examines the computer game play of an 11-year-old boy. In the course of building a virtual house he developed and used, without assistance, an artefact and an accompanying strategy to ensure that his house was symmetric. We argue that the creation and use of this artefact-strategy is a mathematical abstraction. The discussion contributes to knowledge on mathematical abstraction: of non-traditional knowledge; without teacher mediation; through game play. The paper also considers learning without instruction/instructional design and questions received distinctions between scientific and everyday knowledge.

Keywords

Mathematical abstraction Game play Computers 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

An early version of this paper appeared in PME, Avraamidou and Monaghan (2009). We would like to thank three reviewers who made very useful comments on the original submission. We dedicate this paper to the memory of Phil Scott (1953–2011), a lovely man who did so much to develop scholarship on Vygotsky’s distinction between everyday and scientific concepts and who provided personal comment on a draft of this paper.

References

  1. Avraamidou, A., & Monaghan, J. (2009). Abstraction through game play. In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education. (Vol. 2, pp.73–30), Thessaloniki, Greece.Google Scholar
  2. Bragg, L. (2006). Students’ impressions of the value of games for the learning of mathematics. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 217–224) Prague.Google Scholar
  3. Bright, G., Harvey, J. & Wheeler, M. (1985). Learning mathematics and games. Journal for Research in Mathematics Education. Monograph, Vol 1. Google Scholar
  4. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.CrossRefGoogle Scholar
  5. Davydov, V. V. (1990). Soviet studies in mathematics education: Vol. 2. Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula (J. Kilpatrick, ed. & J. Teller, trans.). Reston, VA: National Council of Teachers of Mathematics (Original work published in 1972).Google Scholar
  6. De Corte, E., Verschaffel, L. & Greer, B. (2000). Connecting mathematical problem solving to the real world. In Proceedings of the International Conference on Mathematics Education into the 21st Century: Mathematics for living. Amman, Jordan: The National Center for Human Resource Development (pp. 66–73).Google Scholar
  7. Dreyfus, T., & Kidron, I. (2006). Interacting parallel constructions: A solitary student and the bifurcation diagram. Recherches en didactique des mathématiques, 26(3), 295–336.Google Scholar
  8. Dreyfus, T., & Monaghan, J. (2009). Abstraction beyond a delicate shift of attention. In S. Lerman & B. Davies (Eds.), Mathematical action & structures of noticing (pp. 101–110). Sense: Rotterdam.Google Scholar
  9. Dreyfus, T., & Tsamir, P. (2004). Ben’s consolidation of knowledge structures about infinite sets. Journal of Mathematical Behavior, 23, 271–300.CrossRefGoogle Scholar
  10. Engeström, Y. (2001). Expansive learning at work: Towards an activity theoretical reconceptualization. Journal of Education and Work, 14(1), 133–156.Google Scholar
  11. Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer.Google Scholar
  12. Gee, J. P., & Hayes, E. R. (2010). Women and gaming: The Sims and 21st century learning. New York: Palgrave Macmillan.Google Scholar
  13. Gresalfi, M., Martin, T., Hand, V., & Greeno, J. (2009). Constructing competence: An analysis of student participation in the activity systems of mathematics classrooms. Educational Studies in Mathematics, 70(1), 49–70.CrossRefGoogle Scholar
  14. Hershkowitz, R., Schwarz, B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32(2), 195–222.CrossRefGoogle Scholar
  15. Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  16. Leont’ev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs: Prentice-Hall.Google Scholar
  17. Mason, J. (1989). Mathematical abstraction as a result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2–8.Google Scholar
  18. Mitchelmore, M., & White, P. (2007). Editorial: Abstraction in mathematics learning. Mathematics Education Research Journal, 19(2), 1–9.CrossRefGoogle Scholar
  19. Monaghan, J. (2007). Linking school mathematics to out-of-school mathematical activities: student interpretation of task, understandings and goals. International Electronic Journal of Mathematics Education, 2(2), 50–71. Available, http://www.iejme.com.
  20. Monaghan, J., & Ozmantar, M. F. (2006a). Abstraction and consolidation. Educational Studies in Mathematics, 62(3), 233–258.CrossRefGoogle Scholar
  21. Monaghan, J., & Ozmantar, M. F. (2006b). Comments on the co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection. International Journal of Computers for Mathematical Learning, 11(3), 351–360.CrossRefGoogle Scholar
  22. NCTM. (2004). Communicating about mathematics using games. Available http://standards.nctm.org/document/eexamples/chap5/5.1/.
  23. Noss, R., & Hoyles, C. (1996a). Windows on mathematical meanings: Learning cultures and computers. Dordrecht: Kluwer.CrossRefGoogle Scholar
  24. Noss, R., & Hoyles, C. (1996b). The visibility of meanings: Modelling the mathematics of banking. International Journal of Computers for Mathematical Learning, 1(1), 3–31.CrossRefGoogle Scholar
  25. Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street mathematics and school mathematics. Cambridge: Cambridge University Press.Google Scholar
  26. Ohlsson, S., & Lehtinen, E. (1997). Abstraction and the acquisition of complex ideas. International Journal of Educational Research, 27, 37–48.CrossRefGoogle Scholar
  27. Ozmantar, M. F., & Monaghan, J. (2007). A dialectical approach to the formation of mathematical abstractions. Mathematics Education Research Journal, 19(2), 89–112.CrossRefGoogle Scholar
  28. Pozzi, S., Noss, R., & Hoyles, C. (1998). Tools in practice, mathematics in use. Educational Studies in Mathematics, 36(2), 105–122.CrossRefGoogle Scholar
  29. Pratt, D., & Noss, R. (2002). The microevolution of mathematical knowledge: the case of randomness. The Journal of the Learning Sciences, 11(4), 453–488.CrossRefGoogle Scholar
  30. Prensky, M. (2006). Don’t bother me mom—I’m learning!. USA: Paragon House.Google Scholar
  31. Roth, W.-M. (2009). Learning in schools: A dialectical materialistic, cultural-historical activity-theoretic prespective. In B. Schwarz, et al. (Eds.), Transformation of knowledge through classroom interaction (pp. 281–301). London: Routledge.Google Scholar
  32. Ruthven, K., Laborde, C., Leach, J., & Tiberghien, A. (2009). Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences. Educational Researcher, 38(5), 329–342.CrossRefGoogle Scholar
  33. Salen, K., & Zimmerman, E. (2003). Rules of play: Game design fundamentals. Cambridge: MIT press.Google Scholar
  34. Saxe, G. B. (1991). Culture and cognitive development: Studies in mathematical understanding. Hillsdale: Laurence Erlbaum Associates.Google Scholar
  35. Schwarz, B., Dreyfus, T., & Hershkowitz, R. (2009). The nested epistemic actions model for abstraction in context. In B. Schwarz, et al. (Eds.), Transformation of knowledge through classroom interaction (pp. 11–41). London: Routledge.Google Scholar
  36. Scott, P., Mortimer, E., & Ametller, J. (2011). Pedagogic link-making: A fundamental aspect of teaching and learning scientific conceptual knowledge. Studies in Science Education, 47(1), 3–36.CrossRefGoogle Scholar
  37. Shaffer, D. (2005). Epistemic games. Innovate, 1(6). http://www.innovateonline.info/index.php?view=article&id=79.
  38. Skemp, R. (1986). The psychology of learning mathematics (2nd ed.). Harmondsworth: Penguin.Google Scholar
  39. Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks: Sage.Google Scholar
  40. The Sims 2. (2006). PC DVD-ROM. USA: Electronics Arts Inc.Google Scholar
  41. Tiberghien, A., & Malkoun, L. (2009). The construction of physics knowledge in the classroom from different perspectives: The classroom as a community and the students as individuals. In B. Schwarz, et al. (Eds.), Transformation of knowledge through classroom interaction (pp. 42–55). London: Routledge.Google Scholar
  42. van Oers, B. (2001). Contextualisation for abstraction. Cognitive Science Quarterly, 1(3), 279–305.Google Scholar
  43. Wei, F.-Y. F., & Hendrix, K. G. (2009). Gender differences in preschool children’s recall of competitive and noncompetitive computer mathematics games. Learning, Media and Technology, 34(1), 27–43.CrossRefGoogle Scholar
  44. Wells, G. (1993). Reevaluating the IRF sequence: A proposal for the articulation of theories of activity and discourse for the analysis of teaching and learning in the classroom. Linguistics and Education, 5, 1–37.CrossRefGoogle Scholar
  45. Wertsch, J. V. (1991). Voices of the mind. Cambridge: Harvard University Press.Google Scholar
  46. Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press.Google Scholar
  47. Williams, G. (2007). Abstracting in the context of spontaneous learning. Mathematics Education Research Journal, 19(2), 69–88.CrossRefGoogle Scholar
  48. Yin, R. K. (2003). Case study research: design and methods (3rd ed.). Thousand Oaks: Sage.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of EducationUniversity of LeedsLeedsUK

Personalised recommendations

Cite article

Buy options