Mathematics Education Research Journal

, Volume 19, Issue 2, pp 23–40

Abstraction as a natural process of mental compression

  • Eddie Gray
  • David Tall
Articles

Abstract

This paper considers mathematical abstraction as arising through a natural mechanism of the biological brain in which complicated phenomena are compressed into thinkable concepts. The neurons in the brain continually fire in parallel and the brain copes with the saturation of information by the simple expedient of suppressing irrelevant data and focusing only on a few important aspects at any given time. Language enables important phenomena to be named as thinkable concepts that can then be refined in meaning and connected together into coherent frameworks. Gray and Tall (1994) noted how this happened with the symbols of arithmetic, yielding a spectrum of performance between the more successful who used the symbols as thinkable concepts operating dually as process and concept (procept) and those who focused more on the step-by-step procedures and could perform simple arithmetic but failed to cope with more sophisticated problems. In this paper, we broaden the discussion to the full range of mathematics from the young child to the mature mathematician, and we support our analysis by reviewing a range of recent research studies carried out internationally by research students at the University of Warwick.

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References

  1. Baturo, A. R., & Cooper, T. J. (1999). Fractions, reunitisation and the number-line representation. In O. Zaslavsky (Ed),Proceedings of the 23rd annual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 81–88). Haifa, Israel: PME.Google Scholar
  2. Bayazit, I., & Gray, E. (2006). Acontradiction between pedagogical content knowledge and teaching indications. In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková (Eds),Proceedings of 31st international conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 121–128). Prague, Czech Republic: PME.Google Scholar
  3. Biggs, J., & Collis, K. (1982).Evaluating the quality of learning: the SOLO Taxonomy. New York: Academic Press.Google Scholar
  4. Bruner, J. S. (1966).Towards a theory of instruction. Cambridge: Harvard University Press.Google Scholar
  5. Denvir, H., & Askew, M. (2001). Pupils’ participation in the classroom examined in relation to “interactive whole class teaching”. In T. Rowland (Ed.),Proceedings of the British Society for Research into Learning Mathematics day conference held at Manchester Metropolitan University, March 2001 (Vol. 2, pp. 25–30). London, UK: BSRLM.Google Scholar
  6. 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, 12–33.CrossRefGoogle Scholar
  7. Crick, F. (1994).The astonishing hypothesis. London: Simon & Schuster.Google Scholar
  8. Department for Education and Employment (DfEE). (1999).The National Numeracy Strategy: Framework for teaching mathematics from reception to year 6. London: DfEE.Google Scholar
  9. Doritou, M. (2006).Understanding the number line: Conception and practice. Unpublished PhD thesis: University of Warwick.Google Scholar
  10. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.),Advanced Mathematical Thinking (pp. 95–123). Dordrecht: Kluwer.Google Scholar
  11. Edelman, G. M. (1992).Bright air, brilliant fire. New York: Basic Books.Google Scholar
  12. Escudero, I., & Sanchez, V. (2002). Integration of domains of knowledge in mathematics teachers’ practice. In A. D. Cockburn & E. Nardi (Eds),Proceedings of the 26th annual conference of the International Group for the Psychology of Mathematics Education (Vol.3, pp. 177–184). Norwich, UK: PME.Google Scholar
  13. Fischbein, E. (1978). Intuition and mathematical education.Osnabrücker Schriften zür Mathematik, 1, 148–176.Google Scholar
  14. Gray, E. M., & Pitta, D. (1996). Number processing: Qualitative differences in thinking and the role of imagery. In L. Puig & A Guitiérrez (Eds),Proceedings of 20th annual conference of the International Group for the Psychology of Mathematics Education (Vol.4, pp. 155–162).Valencia, Spain: PME.Google Scholar
  15. Gray, E. M., & Pitta, D. (1997). Emily and the supercalculator. In E. Pehkonen (Ed.),Proceedings of 21st International Conference for the Psychology of Mathematics Education (Vol.4, pp. 17–25). Lahti, Finland: PME.Google Scholar
  16. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic.Journal for Research in Mathematics Education, 26, 115–141.Google Scholar
  17. Gray, E. M., & Tall, D. O. (2001). Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. In M. van den Heuvel-Panhuizen (Ed.),Proceedings of 25th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 65–72). Utrecht, The Netherlands: PME.Google Scholar
  18. Gray, E. M., & Tall, D. O. (2002). Abstraction as a natural process of mental construction. In A. D. Cockburn & E. Nardi (Eds.),Proceedings of 26th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 115–119). Norwich, UK: PME.Google Scholar
  19. Hebb, D. O. (1949).Organization of behavior: A neuropsychological theory. New York: John Wiley.Google Scholar
  20. Howatt, H. (2005).Participation in elementary mathematics: An analysis of engagement, attainment and intervention. Unpublished PhD thesis: University of WarwickGoogle Scholar
  21. Krutetskii, V. A. (1976).The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.Google Scholar
  22. Lima, R. N. de, & Tall, D. O. (2006). The concept of equation: What have students met before? In J. Novotná, H. Moraová, M. Krátká, & N. Steliková (Eds),Proceedings of the 30th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 233–241). Prague, Czech Republic: PME.Google Scholar
  23. Md Ali, R. (2006).Teachers’ indications and pupils’ construal and knowledge of fractions: The case of Malaysia. Unpublished PhD thesis: University of Warwick.Google Scholar
  24. Mejia-Ramos, P., & Tall, D. O. (2006, November).The long-term cognitive development of different types of reasoning and proof. Paper presented to the conference on Explanation and Proof in Mathematics: Philosophical and Educational Perspectives, Essen, Germany.Google Scholar
  25. Office for Standards in Education (OfSTED). (2006).Evaluating mathematics provision for 14–19 year-olds. London: OfSTED.Google Scholar
  26. Pegg, J. (1992). Assessing students’ understanding at the primary and secondary level in the mathematical sciences. In J. Izard & M. Stephens (Eds.),Reshaping assessment practice: Assessment in the mathematical sciences under challenge (pp. 368–385). Melbourne: Australian Council of Educational Research.Google Scholar
  27. Pegg, J., & Tall, D. O., (2002). Fundamental cycles of cognitive growth. In A. D. Cockburn & E. Nardi (Eds.),Proceedings of the 26th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 41–48). Norwich, UK: PME.Google Scholar
  28. Piaget, J. (1972).The principles of genetic epistemology (W. Mays, Trans). London: Routledge & Kegan Paul.Google Scholar
  29. Pitta, D. (1998).Beyond the obvious: Mental representations and elementary arithmetic. Unpublished PhD thesis: University of WarwickGoogle Scholar
  30. Rosch, E. (1978). Principles of categorization. In E. Rosch & B. B. Lloyd (Eds.),Cognition and categorization. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  31. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.Educational Studies in Mathematics, 22, 1–36.CrossRefGoogle Scholar
  32. Skemp, R. R. (1971).The psychology of learning mathematics. London: Penguin.Google Scholar
  33. Skemp, R. R. (1979).Intelligence, learning, and action. London: Wiley.Google Scholar
  34. Tall, D. O. (1999). The cognitive development of proof: Is mathematical proof for all or for some? In Z. Usiskin (Ed.),Developments in school mathematics education around the world (Vol, 4, pp. 117–136). Reston, VA: NCTM.Google Scholar
  35. Tall, D. O. (2002). Differing modes of proof and belief in mathematics.Proceedings of the international conference on Understanding Proving and Proving to Understand (pp. 91–107). Taiwan: National Taiwan Normal University.Google Scholar
  36. Tall, D. O. (2004). The three worlds of mathematics.For the Learning of Mathematics, 23 (3), 29–33.Google Scholar
  37. Tall, D. O., Gray, E M., Ali, M. b., Crowley, L. R. F., DeMarois, P., McGowen, M. C., Pitta, D., Pinto, M. M. F., Thomas, M., & Yusof, Y. b. (2001). Symbols and the bifurcation between procedural and conceptual thinking.Canadian Journal of Science, Mathematics and Technology Education, 1, 81–104.CrossRefGoogle Scholar
  38. Thurston, W. P. (1990). Mathematical education.Notices of the American Mathematical Society, 37, 844–850.Google Scholar
  39. Tirosh, D., Even, R., & Robinson, N. (1998), Simplifying algebraic expressions: Teacher awareness and teaching approaches.Educational Studies in Mathematics, 35, 51–64.CrossRefGoogle Scholar
  40. White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus,Journal for Research in Mathematics Education, 27, 79–95.CrossRefGoogle Scholar

Copyright information

© Mathematics Educations Research Group of Australasia Inc. 2007

Authors and Affiliations

  • Eddie Gray
    • 1
  • David Tall
    • 1
  1. 1.Institute of EducationThe University of WarwickCoventryEngland

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