Gray, Edward Martin, Pinto, Marcia, Pitta, Demetra and Tall, David. (1999) Knowledge construction and diverging thinking in elementary & advanced mathematics. Educational Studies in Mathematics, Vol.38 (No.1-3). pp. 111-133. ISSN 0013-1954

Preview

PDF WRAP_Tall_dot1999k-ed-dem-marcia-esm.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (423Kb)

Abstract

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. This revised version was published online in September 2005 with corrections to the Cover Date.

Item Type:	Journal Article
Subjects:	L Education > L Education (General)
Divisions:	Faculty of Social Sciences > Institute of Education
Library of Congress Subject Headings (LCSH):	Mathematics -- Study and teaching, Cognitive learning
Journal or Publication Title:	Educational Studies in Mathematics
Publisher:	Springer Netherlands
ISSN:	0013-1954
Date:	March 1999
Volume:	Vol.38
Number:	No.1-3
Page Range:	pp. 111-133
Identification Number:	10.1023/A:1003640204118
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	Baroody, A.J. & Ginsburg, H.P.: 1986, ‘The relationship between initial meaningful and mechanical knowledge of arithmetic’. In J. Hiebert (Ed.), Conceptual and Procedural Knowledge: The Case for Mathematics, (pp. 75–112), Hillsdale, N.J., Lawrence Erlbaum Associates. Beth, E.W. & Piaget, J. :1966, Mathematical Epistemology and Psychology, (W. Mays, trans.), Reidel. Dordrecht (originally published 1965). Brownell, W. A.: 1935, ‘Psychological considerations in the learning and teaching of arithmetic’. In W. D. Reeve (Ed.), Teaching of Arithmetic, The Tenth Yearbook of the National Council of Teacher’s of Mathematics, Bureau of Publication, Teachers College, Columbia University. 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, 2–23. Collis., K. & Romberg, T.: 1991, ‘Assessment of mathematical performance: An analysis of open-ended test items’ (pp. 82–116). In C. Wittrock & E. Baker (Eds.), Testing and Cognition, New Jersey, Prentice-Hall. Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D.: 1996, ‘Understanding the limit concept: Beginning with a co-ordinated process schema’, Journal of Mathematical Behavior, 15, 167–192. Crick, F.: 1994, The Astonishing Hypothesis, London: Simon & Schuster. Davis, R. B.: 1984, Learning mathematics: the cognitive science approach to mathematics education. Norwood, NJ: Ablex. Dienes, Z. P.: 1960, Building up Mathematics, London: Hutchinson Educational. Dubinsky E.: 1991, ‘Reflective Abstraction’, in D.O.Tall (Ed), Advanced Mathematical Thinking, (pp. 95–123), Dordrecht: Kluwer Academic Publishers. Dubinsky, E., Elterman, F. & Gong, C.: 1988, ‘The students construction of quantification’, For the Learning of Mathematics, 8, 44–51. Duffin, J. M. & Simpson. A. P.: 1993, ‘Natural, conflicting, and alien’, Journal of Mathematical Behaviour, 12, 313–328. Gray, E.M.: 1993, ‘Count-on: The parting of the ways in simple arithmetic’. In I. Hirabayashi, N. Hohda, K Shigematsu and Fou-Lai Lin (Eds.), Proceedings of XVII International Conference for the Psychology of Mathematics Education, Vol. I, 204–211, Tsukuba, Japan. Gray, E.M.& Tall, D.O.: 1994, ‘Duality, ambiguity and flexibility: A proceptual view of simple arithmetic’, Journal for Research in Mathematics Education, 25, 2, 115–141. Gray, E.M. & Pitta, D.: 1997, ‘Changing Emily’s Images’, Mathematics Teaching, 161, 38–51 Greeno, J. (1983). Conceptual Entities. In D. Genter & A. L. Stevens (Eds.), Mental Models (pp. 227–252), Hillsdale, NJ: Lawrence Erlbaum. Harel, G., & Kaput, J.: 1992, ‘Conceptual entitities in advanced mathematical thinking: The role of notations in their formation and use’, in D.O.Tall (Ed), Advanced Mathematical Thinking, (pp. 82–94), Dordrecht, Kluwer Academic Publishers. Krutetskii, V.A.: 1976, The Psychology of Mathematical Abilities in Schoolchildren, (Trans, J. Teller; Eds., J. Kilpatrick & I. Wirzup). Chicago: University of Chicago. Piaget, J.: 1950, The Psychology of Intelligence, (M. Piercy trans.), London, Routledge & Kegan Paul. Piaget, J.: 1972, The Principles of Genetic Epistemology, (W. Mays trans.), London, Routledge & Kegan Paul. Piaget, J.: 1985, The Equilibrium of Cognitive Structures, Cambridge Massechusetts: Harvard University Press. Piaget, J. & Inhelder, B.: 1971, Mental imagery in the child, New York, Basic. Pinto, M. M. F.: 1998, Students’ Understanding of Real Analysis, Unpublished Doctoral Thesis, Mathematics Education Research Centre, University of Warwick, UK. Pinto, M. M. F. & Gray, E.: 1995, ‘Difficulties teaching mathematical analysis to nonspecialists’, in D. Carraher and L.Miera (Eds.), Proceedings of X1X International Conference for the Psychology of Mathematics Education, Recife, Brazil, 2, 18–25. Pinto, M. M. F. & Tall, D. O.: 1996, ‘Student teachers’ conceptions of the rational numbers’, in L. Puig and A Guitiérrez (Eds.), Proceedings of XX International Conference for the Psychology of Mathematics Education, Valencia, 4, 139–146. Pitta, D. & Gray, E.: 1997,. ‘In the Mind. What can imagery tell us about success and failure in arithmetic?’, In G. A. Makrides (Ed.), Proceedings of the First Mediterranean Conference on Mathematics, Nicosia: Cyprus, pp. 29–41. Pitta, D. & Gray, E.: 1997, ‘Emily and the supercalculator’, in E. Pehkonen (Ed.), Proceedings of XXI International Conference for the Psychology of Mathematics Education, Lahti: Finland, 4, 17–25. Pitta, D.: 1998, ‘In the mind. Internal representations and elementary arithmetic’, Unpublished Doctoral Thesis, Mathematics Education Research Centre, University of Warwick, UK. Schoenfeld, A.H.: 1992, ‘Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics’, in D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics, (pp. 334–370), New York: MacMillan. Sfard, A.: 1989, ‘Transition from operational to structural conception: The notion of function revisited’, in G. Vergnaud, J. Rogalski, M. Artigue, (Eds.), Proceedings of X111 International Conference for the Psychology of Mathematics Education, Paris, France, Vol. 3, 151–158. 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. Sfard, A. & Linchevski, L.: 1994, ‘The gains and pitfalls of reification–the case of algebra’, Educational Studies in Mathematics, 26, 191–228. Skemp, R.R.: 1976, ‘Relational understanding and instrumental understanding’, Mathematics Teaching, 77, 20-26. Skemp, R.R.:1979, Intelligence, Learning and Action, John Wiley & Sons, Chichester, U.K. Steffe, L., von Glaserfeld, E., Richards, J. & Cobb, P.: 1983, Children’s Counting Types: Philosophy, Theory and Applications, Preagar, New York. Tall, D. O.: 1991, Advanced Mathematical Thinking, Dordrecht: Kluwer. Tall, D.O.: 1993a, ‘Mathematicians Thinking about Students Thinking about Mathematics’, Newsletter of the London Mathematical Society, 202, 12–13. Tall, D.O.: 1993b, ‘Real Mathematics, Rational Computers and Complex People’, Proceedings of the Fifth Annual International Conference on Technology in College Mathematics Teaching, 243–258. Tall, D. O.: 1995, ‘Cognitive growth in elementary and advanced mathematical thinking’, in D. Carraher and L. Miera (Eds.), Proceedings of X1X International Conference for the Psychology of Mathematics Education, Recife: Brazil. Vol. 1, 61–75. Tall, D. O. & Vinner, S.: 1981, ‘Concept image and concept definition in mathematics with particular reference to limits and continuity’, Educational Studies in Mathematics, 12, 151-169. Thorndike, E. L.: 1922, The Psychology of Arithmetic, New York: Macmillan. Van Hiele, P. & D.: 1959, The child’s thought and geometry. Reprinted (1984) in D. Fuys, D. Geddes & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele, (pp.1–214), Brooklyn NY: Brooklyn College. Van Hiele, P.: 1986, Structure and Insight. Orlando: Academic Press. Woods, S. S., Resnick, L. B. & Groen, G. J.: 1975, ‘An experimental test of five process models for subtraction’, Journal of Educational Psychology, 67, 17-21.
URI:	http://wrap.warwick.ac.uk/id/eprint/470

Request changes to a record

Actions (login required)

View Item