Abstract
In this chapter we address the question: are mathematical abstractions situated? We first consider empiricist accounts of abstraction which see abstraction as a development process from the concrete to the abstract achieved through the recognition of commonalties isolated in a large number of instances. We discuss difficulties involved in empiricist accounts and propose an alternative approach which we call a dialectical account of abstraction. In this approach, an undeveloped initial idea develops through the use of mediational means and social interaction. This development is not from the concrete to the abstract but, rather, a dialectical to and fro between the concrete and the abstract. Unlike empiricist views, our approach regards context, in the formation of mathematical abstractions, as paramount. Although the construct ‘context’ is difficult to delineate precisely, we focus on the importance of students’ personal mathematical histories, the tools and knowledge artefacts they work with, the people they work with and the tasks they work on. We exemplify the importance of these contextual factors through a study where two teenage girls worked collaboratively, with an interviewer assisting them, in completing tasks designed to generate abstractions in the field of graphs of linear absolute value functions.
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References
Boero, P., Dreyfus, T., Gravemeijer, K., Gray, E., Hershkowitz, R., Schwarz, B., Sierpinska, A., & Tall, D. (2002). Research forum 1, abstraction: Theories about the emergence of knowledge structures. In A. D. Cockburn & E. Nardi (Eds.) Proceedings of the 26th annual conference of the International Group for the Psychology of Mathematics Education (Vol. I, pp. 113-138). Norwich, England: PME.
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221-266.
Chi, M. T. H., Siler, S. A., Jeong, H., Yamauchi, T., & Hausmann, R. G. (2001). Learning from human tutoring. Cognitive Science, 25, 471-533.
Cole, M. (1996). Cultural psychology: A once and future discipline. Cambridge, MA: Harvard University Press.
Davydov, V. V. (1990). Soviet studies in mathematics education (J. Teller, Trans.). In J. Kilpatrick (Ed.), Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula (Vol. 2). Reston, VA: National Council of Teachers of Mathematics.
Dienes, Z. P. (1963). An experimental study of mathematics learning. London: Hutchinson.
Engeström, Y., & Cole, M. (1997). Situated cognition in search of an agenda. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition: Social, semiotic and psychological perspectives (pp. 301-310). New Jersey: Lawrence Erlbaum Associates.
Forman, E. A. (1989). The role of peer interaction in the social construction of mathematical knowledge. International Journal of Educational Research, 13, 55-70.
Gee, J. P. (1997). Thinking, learning, and reading: The situated sociocultural mind. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition: Social, semiotic and psychological perspectives (pp. 235-260). New Jersey: Lawrence Erlbaum Associates.
Goldstein, L. S. (1999). The relational zone: The role of caring relationships in the co-construction of mind. American Educational Research Journal, 36(3), 647-673.
Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32(2), 195-222.
Hoyles, C. (2001). From describing to designing mathematical activity: The next step in developing a social approach to research in mathematics education? Educational Studies in Mathematics, 46(1), 273-286.
Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life. Cambridge: Cambridge University Press.
Leontiev, A. N. (1981). The problem of activity in psychology (J. V. Wertsch, Trans.). In J. V. Wertsch (Ed.), The concept of activity in soviet psychology (pp. 37-71). Armonk, NY: M.E. Sharpe. Inc.
Locke, J. (1689). An essay concerning human understanding. London: Fontana.
Monaghan, J., & Ozmantar, M. F. (2006). Abstraction and consolidation. Educational Studies in Mathematics, 62(3), 233-258.
Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers. Dordrecht: Kluwer.
Ohlsson, S., & Lehtinen, E. (1997). Abstraction and the acquisition of complex ideas. International Journal of Educational Research, 27, 37-48.
Ozmantar, M. F. (2005). An investigation of the formation of mathematical abstractions through scaffolding: Unpublished PhD Thesis, University of Leeds.
Ozmantar, M. F., & Monaghan, J. (2005). Voices in scaffolding mathematical constructions, Fourth Congress of the European Society for Research in Mathematics Education. Available through http://cerme4.crm.es/, see Working Group 8. Accessed 13 March, 2006.
Piaget, J. (1970). Genetic epistemology. New York: W.W. Norton.
Piaget, J. (2001). Studies in reflecting abstraction (R. L. Campbell, Trans.). Philadelphia, PA: Psychology Press.
Rogoff, B., Ellis, S., & Gardner, W. (1984). The adjustment of adult-child instruction according to child’s age and task. Developmental Psychology, 20(2), 193-199.
Sahlberg, P., & Berry, J. (2003). Small group learning in mathematics: Teachers’ and pupils’ ideas about groupwork in school. Painosalama Oy: Finnish Educational Research Association.
Skemp, R. (1986). The psychology of learning mathematics (2nd ed.). Harmondsworth: Penguin.
van Oers, B. (2001). Contextualisation for abstraction. Cognitive Science Quarterly, 1(3), 279-305.
Wertsch, J. V. (1998). Mind as action. New York/Oxford: Oxford University Press.
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Ozmantar, M., Monaghan, J. (2008). Are Mathematical Abstractions Situated?. In: Watson, A., Winbourne, P. (eds) New Directions for Situated Cognition in Mathematics Education. Mathematics Education Library, vol 45. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-71579-7_6
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DOI: https://doi.org/10.1007/978-0-387-71579-7_6
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