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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baroody, A.J. and 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, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 75–112.
Beth, E.W. and Piaget, J.: 1966, Mathematical Epistemology and Psychology, (translated by W. Mays), D. Reidel. Dordrecht, The Netherlands (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. and 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. and Romberg, T.: 1991, ‘Assessment of mathematical performance: An analysis of open-ended test items’, in C. Wittrock and E. Baker (eds.), Testing and Cognition, Prentice-Hall, New Jersey, pp. 82–116.
Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K. and Vidakovic, D.: 1996, ‘Understanding the limit concept: Beginning with a co-ordinated process schema’, Journal of Mathematical Behaviour 15, 167–192.
Crick, F.: 1994, The Astonishing Hypothesis, Simon 0026 Schuster, London.
Davis, R.B.: 1984, Learning mathematics: the cognitive science approach to mathematics education, Ablex. Publishing Co., Norwood, NJ.
Dienes, Z.P.: 1960, Building up Mathematics, Hutchinson Educational, London.
Dubinsky, E.: 1991, ‘Reflective abstraction’, in D.O. Tall (ed.), Advanced Mathematical Thinking, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 95–123.
Dubinsky, E., Elterman, F. and Gong, C.: 1988, ‘The students construction of quantification’, For the Learning of Mathematics 8, 44–51.
Duffin, J.M. and 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 on the Psychology of Mathematics Education, Tsukuba, Japan, Vol. I, pp. 204–211.
Gray, E.M. and 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. and Pitta, D.: 1997, ‘Changing Emily’s Images’, Mathematics Teaching 161, 38–51.
Greeno, J.: 1983, ‘Conceptual entities’, in D. Genter and A.L. Stevens (eds.), Mental Models, Lawrence Erlbaum, Hillsdale, NJ, pp. 227–252.
Harel, G. and 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, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 82–94.
Krutetskii, V.A.: 1976, The Psychology of Mathematical Abilities in Schoolchildren, (translated by J. Teller), University of Chicago, Chicago.
Piaget, J.: 1950, The Psychology of Intelligence, (translated by M. Piercy ), Routledge and Kegan Paul, London.
Piaget, J.: 1972, The Principles of Genetic Epistemology, (translated by W. Mays), Routledge 0026 Kegan Paul, London.
Piaget, J.: 1985, The Equilibrium of Cognitive Structures, Harvard University Press, Cambridge Massechusetts.
Piaget, J. and Inhelder, B.: 1971, Mental Imagery in the Child, Basic, New York.
Pinto, M.M.F.: 1996, Students’ Use of Quantifiers,Paper presented to the Advanced Mathematical Thinking Working Group at The Twentieth Conference of the International
Group for the Psychology of Mathematics Education, Valencia, Spain.
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. and Gray, E.: 1995, ‘Difficulties teaching mathematical analysis to nonspecialists’, in D. Carraher and L. Miera (eds.), Proceedings of XIX International Conference for the Psychology of Mathematics Education, Recife, Brazil, 2, 18–25.
Pinto, M.M.F. and 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, pp. 139–146.
Pitta, D. and Gray, E.: 1997a, 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. and Gray, E.: 1997b, ‘Emily and the supercalculator’, in E. Pehkonen (ed.), Proceedings of XXI International Conference for the Psychology of Mathematics Education, Lahti, Finland, 4, pp. 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, MacMillan, New York, pp. 334–370.
Sfard, A.: 1989, ‘Transition from operational to structural conception: The notion of function revisited’, in G. Vergnaud, J. Rogalski, M. Arigue (eds.), Proceedings of XIII International Conference for the Psychology of Mathematics Education, Paris, France, Vol. 3, pp. 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. and 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,Chichester, U.K., John Wiley 0026 Sons.
Steife, L., Von Glaserfeld, E., Richards, J. and Cobb, P.: 1983, Children’s Counting Types: Philosophy, Theory and Applications, Preagar, New York.
Tall, D.O.: 1991, Advanced Mathematical Thinking, Kluwer Academic Publishers, Dordrecht, The Netherlands.
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, pp. 243–258.
Tall, D. 0.: 1995, ‘Cognitive growth in elementary and advanced mathematical thinking’, in D. Carraher and L. Miera (eds.), Proceedings of XIX International Conference for the Psychology of Mathematics Education, Recife, Brazil. Vol. 1, pp. 61–75.
Tall, D.O. and Vinner, S.: 1981, ‘Concept image and concept definition in mathematics with particular reference to limits and continuity’, Educational Studies in Mathematics 12, 151–169.
Thomdike, E.L.: 1922, The Psychology of Arithmetic, MacMillan, New York.
Van Hiele, P. and D.: 1959, The child’s thought and geometry, Reprinted (1984), in D. Fuys, D. Geddes and R. Tischler (eds.), English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele, Brooklyn College, Brooklyn, NY, pp. 1–214.
Van Hiele, P.: 1986, Structure and Insight, Academic Press, Orlando.
Woods, S.S., Resnick, L.B. and Groen, G.J.: 1975, ‘An experimental test of five process models for subtraction’, Journal of Educational Psychology 67, 17–21.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Gray, E., Pinto, M., Pitta, D., Tall, D. (1999). Knowledge Construction and Diverging Thinking in Elementary & Advanced Mathematics. In: Tirosh, D. (eds) Forms of Mathematical Knowledge. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1584-3_6
Download citation
DOI: https://doi.org/10.1007/978-94-017-1584-3_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5330-5
Online ISBN: 978-94-017-1584-3
eBook Packages: Springer Book Archive