# Cognitive development in advanced mathematics using technology

Articles

- 16 Citations
- 250 Downloads

## Abstract

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.

## Preview

Unable to display preview. Download preview PDF.

### References

- Blokland, P., Giessen, C, & Tall, D. O. (2000). Graphic Calculus for Windows [Computer software). Available on the World Wide Web:http://www.vusoft.nl Google Scholar
- 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 - Collis, K. F. (1972).
*A study of the relationship between formal thinking and combinations of operations*. Newcastle, NSW: University of Newcastle.Google Scholar - Carmi, B. (1991). Limits. In D. O. Tall (Ed.),
*Advanced Mathematical Thinking*(pp.153–166). Dordrecht, The Netherlands: Kluwer.Google Scholar - Crick, F. (1994).
*The astonishing hypothesis: The scientific search for the soul*. London, England: Simon & Schuster.Google Scholar - Crowley, L. R. F. (2000).
*Cognitive structures in college algebra*. Unpublished Ph.D. thesis, University of Warwick, England.Google Scholar - Douady, R. (1986), Jeu de cadres et dialectique outil-objet.
*Recherches en Didactique des Mathematiques, 7*(2), 5–32.Google Scholar - 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 - Dehaene, S. (1997).
*The number sense: How the mind creates mathematics*. New York, NY: Oxford University Press.Google Scholar - DeMarois, P. (1998).
*Aspects and layers of the function concept*. Unpublished doctoral thesis, University of Warwick, England.Google Scholar - Gleick, J. (1987).
*Chaos*. New York, NY: Penguin.Google Scholar - Gray, E. M., & Pitta, D. (1997). Changing Emily’s images.
*Mathematics Teaching, 161*, 38–51.Google Scholar - 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 - 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 - 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 - 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 - 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 - Lanford, O. E. (1982). A computer-assisted proof of the Feigenbaum conjectures.
*Bulletin of the American Mathematical Society*, 6, 427.CrossRefGoogle Scholar - 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 - Matz, M. (1980). Towards a computational theory of algebraic competence,
*Journal of Mathematical Behavior, 3*(1), 93–66.Google Scholar - Monaghan, J. D. (1986).
*Adolescents’ understanding of limits and infinity*. Unpublished doctoral thesis, University of Warwick, England.Google Scholar - 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 - Pinto, M. M. F., (1998).
*Students’ understanding of real analysis*. Unpublished doctoral thesis, University of Warwick, England.Google Scholar - School Mathematics Project (1991).
*Introductory calculus*. Cambridge, England: Cambridge University Press.Google Scholar - Skemp, R. R. (1971).
*The psychology of learning mathematics*. London, England: Penguin.Google Scholar - Still, S. (1993).
*Students’ understanding of limits and the effect of computer algebra systems*. Unpublished M.Ed. thesis, Leeds University, England.Google Scholar - Tall, D. O. (1982). The blancmange flmction, continuous everywhere but differentiable nowhere.
*Mathematical Gazette*, 66, 11–22.CrossRefGoogle Scholar - Tall, D. O. (1985). Understanding the calculus.
*Mathematics Teaching, 110*, 49–53.Google Scholar - 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 - 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 - 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 - Tall, D. A., Blokland, P., & Kok, D. (1990).
*A graphic approach to the calculus*. Pleasantville, NY: Sunburst.Google Scholar - 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 - 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 - 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 - 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