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Knowledge construction and diverging thinking in elementary & advanced mathematics

By Edward Martin Gray, Marcia Pinto, Demetra Pitta and David Tall


This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action upon it and reflection on both perception and action. We see an emphasis on one or more of these activities leading not only to different kinds of mathematics, but also to a spectrum of success and failure depending on the nature of the focus in the individual activity. For instance, geometry builds from the fundamental perception of figures and their shape, supported by action and reflection to move from practical measurement to theoretical deduction and euclidean proof. Arithmetic, on the other hand, initially focuses on the action of counting and later changes focus to the use of symbols for both the process of counting and the concept of number. The evidence that we draw together from a number of studies on children's arithmetic shows a divergence in performance. The less successful seem to focus more on perceptions of their physical activities than on the flexible use of symbol as process and concept appropriate for a conceptual development in arithmetic and algebra. Advanced mathematical thinking introduces a new feature in which concept definitions are formulated and formal concepts are constructed by deduction. We show how students cope with the transition to advanced mathematical thinking in different ways leading once more to a diverging spectrum of success.\ud This revised version was published online in September 2005 with corrections to the Cover Date.\u

Topics: L1
Publisher: Springer Netherlands
Year: 1999
OAI identifier: oai:wrap.warwick.ac.uk:470

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  1. 1993a, ‘Mathematicians Thinking about Students Thinking about Mathematics’,
  2. 1993b, ‘Real Mathematics, Rational Computers and Complex People’,
  3. (1992). A constructivist alternative to the representational view of mind in mathematics education’, doi
  4. (1975). An experimental test of five process models for subtraction’, doi
  5. (1991). Assessment of mathematical performance: An analysis of open-ended test items’
  6. (1997). Changing Emily’s Images’,
  7. (1983). Children’s Counting Types: Philosophy, Theory and Applications, Preagar,
  8. (1995). Cognitive growth in elementary and advanced mathematical thinking’, in doi
  9. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity’, doi
  10. (1992). Conceptual entitities in advanced mathematical thinking: The role of notations in their formation and use’,
  11. (1993). Count-on: The parting of the ways in simple arithmetic’. In
  12. (1995). Difficulties teaching mathematical analysis to nonspecialists’, in D. Carraher and L.Miera (Eds.),
  13. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic’, doi
  14. (1997). Emily and the supercalculator’, in E. Pehkonen (Ed.),
  15. (1998). In the mind. Internal representations and elementary arithmetic’, Unpublished Doctoral Thesis,
  16. (1997). In the Mind. What can imagery tell us about success and failure in arithmetic?’,
  17. (1984). Learning mathematics: the cognitive science approach to mathematics education. doi
  18. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics’,
  19. (1971). Mental imagery in the child, doi
  20. (1993). Natural, conflicting, and alien’,
  21. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin’, doi
  22. (1935). Psychological considerations in the learning and teaching of arithmetic’.
  23. (1991). Reflective Abstraction’, doi
  24. (1976). Relational understanding and instrumental understanding’, doi
  25. (1986). Structure and Insight.
  26. (1998). Students’ Understanding of Real Analysis, Unpublished Doctoral Thesis,
  27. (1994). The Astonishing Hypothesis, doi
  28. (1959). The child’s thought and geometry. Reprinted
  29. (1985). The Equilibrium of Cognitive Structures, Cambridge Massechusetts:
  30. (1994). The gains and pitfalls of reification–the case of algebra’, doi
  31. (1972). The Principles of Genetic Epistemology, doi
  32. (1922). The Psychology of Arithmetic, doi
  33. (1976). The Psychology of Mathematical Abilities in doi
  34. (1986). The relationship between initial meaningful and mechanical knowledge of arithmetic’. In
  35. (1988). The students construction of quantification’,
  36. (1989). Transition from operational to structural conception: The notion of function revisited’,
  37. (1996). Understanding the limit concept: Beginning with a co-ordinated process schema’, doi

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