Mathematics Education Research Journal

, Volume 12, Issue 3, pp 196–218 | Cite as

Cognitive development in advanced mathematics using technology

  • David Tall


This paper considers cognitive development in mathematics and its relationship with computer technology, with special emphasis on the use of visual imagery and symbols and the later shift to formal axiomatic theories. At each stage, empirical evidence is presented to show how these forms of thinking are enhanced, changed, or impeded by the use of technology.


Cognitive Development Mathematical Idea Mathematical Thinking Computer Algebra System Visual Imagery 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Blokland, P., Giessen, C, & Tall, D. O. (2000). Graphic Calculus for Windows [Computer software). Available on the World Wide Web: Google Scholar
  2. Chae, S. D., & Tall, D. O. (in press). Aspects of the construction of conceptual knowledge in the case of computer aided exploration of period doubling. In C. Morgan & T. Rowlands (Eds.),Research in Mathematics Education: Vol. 3. Bristol, England: British Society for Research in the Learning of Mathematics, Faculty of Education.Google Scholar
  3. Collis, K. F. (1972).A study of the relationship between formal thinking and combinations of operations. Newcastle, NSW: University of Newcastle.Google Scholar
  4. Carmi, B. (1991). Limits. In D. O. Tall (Ed.),Advanced Mathematical Thinking (pp.153–166). Dordrecht, The Netherlands: Kluwer.Google Scholar
  5. Crick, F. (1994).The astonishing hypothesis: The scientific search for the soul. London, England: Simon & Schuster.Google Scholar
  6. Crowley, L. R. F. (2000).Cognitive structures in college algebra. Unpublished Ph.D. thesis, University of Warwick, England.Google Scholar
  7. Douady, R. (1986), Jeu de cadres et dialectique outil-objet.Recherches en Didactique des Mathematiques, 7(2), 5–32.Google Scholar
  8. Davis, R. B., Jockusch, E., & McKnight, C. (1978). Cognitive processes in learning algebra.Journal of Children’s Mathematical Behavior, 2(1), 10–320.Google Scholar
  9. Dehaene, S. (1997).The number sense: How the mind creates mathematics. New York, NY: Oxford University Press.Google Scholar
  10. DeMarois, P. (1998).Aspects and layers of the function concept. Unpublished doctoral thesis, University of Warwick, England.Google Scholar
  11. Gleick, J. (1987).Chaos. New York, NY: Penguin.Google Scholar
  12. Gray, E. M., & Pitta, D. (1997). Changing Emily’s images.Mathematics Teaching, 161, 38–51.Google Scholar
  13. Gray, E., M, Pitta, D., Pinto, M. M. F., & Tall, D. O. (1999). Knowledge construction and diverging thinking in elementary and advanced mathematics.Educational Studies in Mathematics, 38, 111–133.CrossRefGoogle Scholar
  14. Gray, E. M. & Tall, D. O. (1991). Duality, ambiguity and flexibility in successful mathematical thinking. In F. Furinghetti (Ed.),Proceedings of the 15th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 72–79). Assisi, Italy: Program Committee.Google Scholar
  15. 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
  16. Htillter, M., Monaghan, J. D., & Roper, T. (1993). The effect of computer algebra use on students’ algebraic thinking. In R. Sutherland (Ed.),Working Papers for ESRC Algebra Seminar. London, England: London University, Institute of Education.Google Scholar
  17. Kieran C. (1981). Pre-algebraic notions among 12 and 13 year olds. In Equipe de Recherche Pedagogique Laboratoire I.M.A.G. (Ed.),Proceedings of the 5th annual conference of the International Group for the Psychology of Mathematics Education (pp. 158–164). Grenoble, France: Program Committee.Google Scholar
  18. Lanford, O. E. (1982). A computer-assisted proof of the Feigenbaum conjectures.Bulletin of the American Mathematical Society, 6, 427.CrossRefGoogle Scholar
  19. Li, L., & Tall, D. O. (1993). Constructing different concept images of sequences and limits by programming. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F.-L. Lin (Eds.),Proceedings of the 17th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 41–48). Tsukuba, Japan: Program Committee.Google Scholar
  20. Matz, M. (1980). Towards a computational theory of algebraic competence,Journal of Mathematical Behavior, 3 (1), 93–66.Google Scholar
  21. Monaghan, J. D. (1986).Adolescents’ understanding of limits and infinity. Unpublished doctoral thesis, University of Warwick, England.Google Scholar
  22. Monaghan, J., Sun, S., & Tall, D. O. (1994), Construction of the limit concept with a computer algebra system. In J. P. da Ponte & J. F. Matos (Eds.),Proceedings of the 18th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 279–286). Lisbon, Portugal: Program Committee.Google Scholar
  23. Pinto, M. M. F., (1998).Students’ understanding of real analysis. Unpublished doctoral thesis, University of Warwick, England.Google Scholar
  24. School Mathematics Project (1991).Introductory calculus. Cambridge, England: Cambridge University Press.Google Scholar
  25. Skemp, R. R. (1971).The psychology of learning mathematics. London, England: Penguin.Google Scholar
  26. Still, S. (1993).Students’ understanding of limits and the effect of computer algebra systems. Unpublished M.Ed. thesis, Leeds University, England.Google Scholar
  27. Tall, D. O. (1982). The blancmange flmction, continuous everywhere but differentiable nowhere.Mathematical Gazette, 66, 11–22.CrossRefGoogle Scholar
  28. Tall, D. O. (1985). Understanding the calculus.Mathematics Teaching, 110, 49–53.Google Scholar
  29. Tall, D. O. (1993a). Interrelationships between mind and computer: Processes, images, symbols In D. L. Ferguson (Ed.),Advanced technol,ogies in the teaching of mathematics and science (pp. 385–413). New York, NY: Springer-Verlag.Google Scholar
  30. Tall, D. A. (1993b). Real mathematics, rational computers and complex people. In Lum, L. (Ed.),Proceedings of the 5th annual international conference on Technology in College.Mathematics Teaching (pp. 243–258). Reading, MA: Addison-Wesley.Google Scholar
  31. Tall, D. A. (1998). Information technology and mathematics education: Enthusiasms, possibilities & realities. In C. Alsina, J. M. Alvarez, M. Niss, A. Perez, L. Rico, & A. Sfard (Eds.),Proceedings of the 8th International Congress on Mathematical Education. (pp. 65–82). Seville: SAEM Thales.Google Scholar
  32. Tall, D. A., Blokland, P., & Kok, D. (1990).A graphic approach to the calculus. Pleasantville, NY: Sunburst.Google Scholar
  33. Tall, D. A. & Thomas, M. A. J. (1991). Encouraging versatile thinking in algebra using the computer.Educational Studies in Mathematics, 22, 125–147.CrossRefGoogle Scholar
  34. Tall, D. A., Thomas, M. A. J., Davis, G., Gray, E. M., & Simpson, A. (2000). What is the object of the encapsulation of a process.Journal ofMathematical Behavior, 18(2), 1–19.Google Scholar
  35. Thomas, M. A. J. (1988).A conceptual approach to the early learning of algebra using a computer. Unpublished doctoral thesis, University of Warwick, England.Google Scholar
  36. Williams, S. R. (1991). Models of limit held by college calculus students.Journal for Research in Mathematics Education, 22, 237–251.CrossRefGoogle Scholar

Copyright information

© Mathematics Education Research Group of Australasia Inc. 2000

Authors and Affiliations

  • David Tall
    • 1
  1. 1.Mathematics Education Research Centre, Institute of EducationUniversity of WarwickCoventryEngland

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