Skip to main content

Advertisement

Springer Nature Link
Log in

A critique of the selection of “Mathematical objects” as a central metaphor for advanced mathematical thinking

  • Published:
International Journal of Computers for Mathematical Learning Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Amol'd, V. I. (1990).Huygens & Barrow and Newton & Hooke (E. J. F. Primose, Trans.). Basel: Bierkhäuser Verlag.

    Google Scholar 

  • Artigue, M. (1992). The importance and limits of epistemological work in didactics. In W. Geeslin and K. Graham (Eds.),Sixteenth Psychology of Mathematics Education, vol. 3 (pp. 195–216). Durham, NH.

  • Bachelard, G. (1938).La formation de l'esprit scientifique (reprinted 1983 ed.). Paris: J. Vrin.

    Google Scholar 

  • Brousseau, G. (1983). Les obstacles epistemolgiques et les problemes en mathematiques.Recherche in Didactiques des Matematiques 4(2): 164–198.

    Google Scholar 

  • Chevallard, Y (1991). Concepts fondamentaux de didactique: Perspectives apportees par une perspective anthropologique. InLecture at the Vleme Ecole d'Ete de didactique des mathematiques. Plestin les Greves, France.

    Google Scholar 

  • Confrey, J. (1980).Conceptual change, number concepts and the introduction to calculus. Doctorial dissertation, Cornell University, Ithaca, NY.

    Google Scholar 

  • Confrey, J. (1990). A review of the research on student conceptions in mathematics, science and programming. In C. Cazden (Ed.),Review of Research in Education (pp. 3–56). Washington, DC: American Education Research Association.

    Google Scholar 

  • Confrey, J. (1994a). Splitting, similarity, and rate of change: New approaches to multiplication and exponential functions. In G. Harel and J. Confrey (Eds.),The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 293–332). Albany, NY. State University of New York Press.

    Google Scholar 

  • Confrey, J. (1994b). A theory of intellectual development, Part I.For the Learning of Mathematics 14(3, November): 2–8.

    Google Scholar 

  • Confrey, J. (1995). Student voice in examining “splitting” as an approach to ratio, proportion, and fractions. In L. Meira and D. Carraher (Eds.),Proceedings of the Nineteenth International Conference for the Psychology of mathematics Education, vol. 1 (pp. 3–29). Recife, Brazil: Universidade Federal de Pernambuco.

    Google Scholar 

  • Confrey, J. and Scarano, G. H. (1995). Splitting reexamined: Results from a three-year longitudinal study of children in grades three to five. In D. T. Owens, M. K. Reed and G. M. Millsaps (Eds.),Proceedings of the Seventeenth Annual Meeting for the Psychology of Mathematics Education-NA, vol. 1 (pp. 421–426). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

    Google Scholar 

  • Cornu, B. (1991). Limits. In D. Tall (Ed.),Advanced Mathematical Thinking (pp. 153–166). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Costa, S. (1995). “Our” notation from their quarrel: The Leibniz-Newton controversy as embodied in calculus textbooks. Paper presented at the History of Science Society Annual Meeting, October. Minneapolis, MN.

  • Dennis, D. (1995).Historical perspectives for the reform ofmathematics curriculum: Geometric curve drawing devices and their role in the transition to an algebraic description of functions. Doctoral dissertation, Cornell University, Ithaca, NY.

    Google Scholar 

  • Dennis, D. and Confrey, J. (1995). Functions of a curve: Leibniz's original notion of functions and its meaning for the parabola.The College Mathematics Journal 26(3, May): 124–131.

    Google Scholar 

  • Dennis, D. and Confrey, J. (in press). The creation of continuous exponents: A study of the methods and epistemology of Alhazen and Wallis. In J. Kaput and E. Dubinsky (Eds.),Research in Collegiate Mathematics Education II Providence, RI: American Mathematical Society.

  • Dennis, D., Smith, E. and Confrey, J. (1992). Rethinking functions: Cartesian constructions. In S. Hills (Ed.),Second International Conference on the History and Philosophy of Science in Science Education, vol. 2 (pp. 449–466). Queens University: Kingston Ontario: The Mathematics, Science, Technology and Teacher Education Group.

    Google Scholar 

  • Doerr, H. (1995). An integrated approach to mathematical modeling: A classroom study. InAnnual Meeting of the American Educational Research Association, April. San Francisco, CA.

  • Doerr, H. and Confrey, J. (1994). A modeling approach to understanding the trigonometry of forces: A classroom study. InProceedings of the Eighteenth Annual Meeting of Psychology of Mathematics Education, vol. II (pp. 264–271). Lisbon, Portugal.

  • Dorfler, W. (1993). Computer use and views of the mind. In C. Keitel and K. Ruthven (Eds.),Learning from Computers: Mathematics Education and Technology (pp. 159–186). New York: Springer-Verlag.

    Google Scholar 

  • Douady, R. (1991). Tool, object, setting, window: Elements for analysing and constructing didactical situations in mathematics. In A. J. Bishop and S. Mellin-Olsen (Eds.),Mathematical Knowledge: Its Growth Through Teaching Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.),Advanced Mathematical Thinking (pp. 95–126). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Dubinsky, E. and Harel, G. (1992). The nature of the process conception of function. In G. Harel and E. Dubinsky (Eds.), The Concept of Function Washington, DC: Mathematical Association of America.

    Google Scholar 

  • Frye, M. (1983).The Politics of Reality: Essays in Feminist Theory. Trumansburg, NY: Trumansburg Press.

    Google Scholar 

  • Harel, G. and Dubinsky, E. (Eds.). (1992).The Concept of Function. Washington, DC: Mathematical Assocation of America.

    Google Scholar 

  • Hoyles, C. (1993). Microworlds/schoolworlds: The transformation of an innovation. In C. Keitel and K. Ruthven (Eds.),Learning from Computers: Mathematics Education and Technology New York: Springer-Verlag.

    Google Scholar 

  • Hoyles, C. and Noss, R. (1993). Out of the cul-de-sac? In J. R. Becker and B. J. Pence (Eds.),Proceedings of the Fifteenth Annual Meeting of the Psychology of Mathematics Education-NA, vol. 1 (pp. 83–90). Pacific Grove, CA: Center for Mathematics & Computer Science Education, San Jose State University.

    Google Scholar 

  • Jahnke, H. N. (1994). The historical dimension of mathematical understanding — objectifying the subjective. In J. P. daPonte and J. F. Matos (Eds.),Proceedings of the Eighteenth International Conference of the Psychology of Mathematics Education, vol. 1 (pp. 139–156). Lisbon, Portugal: University of Lisbon.

    Google Scholar 

  • Lakatos, I. (1984). A renaissance of Empiricism in the recent philosophy of mathematics? In T. Tymockzko (Ed.),New Directions on the Philosophy of Mathematics (pp.29–48).

  • Lakatos, I. and Musgrave, A. (Eds.). (1970).Criticism and the Growth of Knowledge. London: Cambridge University Press.

    Google Scholar 

  • Otte, M. (1993). Towards a social theory of mathematical knowledge. In C. Keitel and K. Ruthven (Eds.),Learning from Computers: Mathematics, Education, and Technology (pp. 280–327). New York: Springer-Verlag.

    Google Scholar 

  • Papert, S. (1980).Mindstorms. New York: Basic Books, Inc.

    Google Scholar 

  • Polanyi, M. (1958).Personal Knowledge: Towards a Post-Critical Philosophy. Chicago: University of Chicago Press.

    Google Scholar 

  • Rizzuti, J. (1991).Students' Conceptualizations of Mathematical Functions: The Effects of a Pedagogical Approach Involving Multiple Representations. Doctoral dissertation, Cornell University, Ithaca, NY

    Google Scholar 

  • Sfard, A. (1992).Operational origins of mathematical objects and the quandary of reification: The case of function. In G. Harel and E. Dubinsky (Eds.),The Concept of Function (pp. 59–84). Washington, DC: Mathematical Association of America.

    Google Scholar 

  • Sfard, A. (1994). Reification as the birth of metaphor.For the Learning of Mathematics 14(1): 44–55.

    Google Scholar 

  • Sfard, A., and Linchevski, L. (1994). The gains and the pitfalls of reification: The case of algebra.Educational Studies in Mathematics 26: 191–228.

    Google Scholar 

  • Sfard, A. and Thompson, P. W. (1994). Problems of reification: Representations and mathematical objects. In D. Kirshner (Ed.),Proceedings of the Sixteenth Annual Conference of the Psychology of Mathematics Education-NA, vol. 1 (pp. 3–32). Baton Rouge, LA: Louisiana State University.

    Google Scholar 

  • Shapin, S. and Schaffer, S. (1985).Leviathan and the Air-pump: Hobbes, Boyle, and the Experimental Life. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Sierpinska, A. (1992). On understanding the notion of function. In G. Harel and E. Dubinsky (Eds.),The Concept of Function (pp. 25–58). Washington, DC: Mathematical Association of America.

    Google Scholar 

  • Tall, D. (Ed.). (1991a).Advanced Mathematical Thinking. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Tall, D. (1991b). The psychology of advanced mathematical thinking. In D. Tall (Ed.),Advanced Mathematical Thinking (pp. 3–21). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Tall, D. (1991c). Reflections. In D. Tall (Ed.),Advanced Mathematical Thinking (pp. 251–260). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Tall, D. (1995). Cognitive growth in elementary and advanced mathematical thinking. In L. Meira and D. Carraher (Eds.),Proceedings of the Nineteenth International Conference for the Psychology of Mathematics Education, vol. 1 (pp. 61–76). Recife, Brazil: Graducate Program in Cognitive Psycholgy, Universidade Federal de Pernambuco.

    Google Scholar 

  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel and J. Confrey (Eds.),The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 181–236). Albany, NY: State University of New York Press.

    Google Scholar 

  • Tymoczko, T (1979). The four-color problem and its philosophical significance.Journal of Philosophy 76(2): 57–82.

    Google Scholar 

  • van Schooten, F. (1657).Exercitationum Mathematicorum, Liber IV, Organica Coniccarum Sectionum in Plano Descriptione. (original edition in the rare books collection of Cornell University, Ithaca, NY).

  • Viete (1591).The Analytic Art (Ars Analytica).

  • Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel and S. Papert (Eds.),Constructionism Norwood, NJ: Ablex Publishing Corp.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Education, Cornell University, 14853, Ithaca, NY, USA

    Jere Confrey & Shelley Costa

Authors
  1. Jere Confrey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shelley Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Confrey, J., Costa, S. A critique of the selection of “Mathematical objects” as a central metaphor for advanced mathematical thinking. Int J Comput Math Learning 1, 139–168 (1996). https://doi.org/10.1007/BF00571076

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00571076

Keywords