Skip to main content
SpringerLink
Log in

Experiencing equivalence but organizing order

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our own conceptions of it. This paper considers various conceptions of equivalence, in history, in mathematics today, and in mathematics education. It reveals critical differences in the notion of equivalence at different points in history and a meaning for equivalence proposed by mathematicians and mathematics educators that is at variance with the ways that learners may think. These differences call into question the most popular view of the subject: that the mathematical notion of equivalence relation is the result of spelling out our experience of equivalence. Moreover, the findings of this study suggest that the standard definition of an equivalence relation is ill-chosen from a pedagogical point of view but well-crafted from a mathematical point of view.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Asghari, A. H. (2004). Organizing with a focus on defining, a phenomenographic approach. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics, vol. 2 (pp. 63–70). Bergen, Norway.

  • Asghari, A. H. (2005a). A mad dictator partitions his country. Research in Mathematics Education, 7, 33–45. doi:10.1080/14794800008520144.

    Article  Google Scholar 

  • Asghari, A. H. (2005b). Equivalence. Unpublished Ph.D. thesis, University of Warwick.

  • Chin, E.-T., & Tall, D. O. (2001). Developing formal mathematical concepts over time. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, vol. 2 (pp. 241–248). Utrecht, Netherlands.

  • Cantor, G. (1895). Contributions to the founding of the theory of transfinite numbers. In Philip, E. B. Jourdain (Tr.). (1915). Translation of Beiträge zur Begründung der transfiniten Mengenlehre. New York: Dover Publications.

  • Dienes, Z. (1971). Building up mathematics (4th ed.). London: Hutchinson Educational.

    Google Scholar 

  • Dienes, Z. (1976). Relations and functions. London: Hodder and Stoughton.

    Google Scholar 

  • Fowler, D. (1999). The mathematics of Plato’s Academy: A new reconstruction (2nd ed.). Oxford: Clarendon.

    Google Scholar 

  • Freudenthal, H. (1966). The language of logic. Amsterdam: Elsevier.

    Google Scholar 

  • Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.

    Google Scholar 

  • Freudenthal, H. (1978). Weeding and sowing: Preface to a science of mathematical education. Dordrecht: Reidel.

    Google Scholar 

  • Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.

    Google Scholar 

  • Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer Academic.

    Google Scholar 

  • Furinghetti, F., & Radford, L. (2002). Historical conceptual developments and the teaching of mathematics: From phylogenesis and ontogenesis theory to classroom practice. In L. English (Ed.), Handbook of International Research in Mathematics Education (pp. 631–654). London: LEA.

    Google Scholar 

  • Gauss, C. F. (1801). Disquisitiones arithmeticae. In A. A. Clarke (Tr.). (1966). New Haven: Yale University Press.

  • Halmos, P. R. (1982). The thrills of abstraction. The Two-Year College Mathematics Journal, 13(4), 243–251, Sep.

    Article  Google Scholar 

  • Hamdan, M. (2006). Equivalent structures on sets: Equivalence classes, partitions and fiber structures of functions. Educational Studies in Mathematics, 62(2), 127–147. doi:10.1007/s10649-006-5798-9.

    Article  Google Scholar 

  • Hausdorff, F. (1914). Set theory (2nd English edn, 1962). J. R. Aumann, et al (Tr.), Translation of 3rd German edition of “Mengenlehre” (1937). New York: Chelsea.

  • Heath, T. L. (1926). The thirteen books of Euclid’s Elements (2nd ed.). New York: Dover.

    Google Scholar 

  • Hilbert, D. (1971). Foundations of geometry, 2nd edn. L. Unger (Tr.), from the 10th German edition of Grundlagen der Geometrie. Open Court: La Salle.

  • Johnson, M. (1987). The body in the mind: The bodily basis of meaning, imagination, and reason. London: University of Chicago Press.

    Google Scholar 

  • Joyce, D. (2008). Euclid’s Elements. from http://aleph0.clarku.edu/~djoyce/java/elements/bookI/cn.html. Accessed 08 November 2008.

  • Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34, 219–248. doi:10.1023/A:1002985109323.

    Article  Google Scholar 

  • Marton, F. (1981). Phenomenography—describing conceptions of the world around us. Instructional Science, 10, 177–200. doi:10.1007/BF00132516.

    Article  Google Scholar 

  • Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah: LEA.

    Google Scholar 

  • Poincaré, H. (1899). La logique et l’intuition dans la science mathématique et dans l’enseignement (Logic and intuition in the science of mathematics and in teaching). L’enseignement Mathématique, 1, 157–162.

    Google Scholar 

  • Russell, B. (1903). Principles of mathematics (2nd ed.). London: George Allen & Unwin (1937).

    Google Scholar 

  • Russell, B. (1919). Introduction to mathematical philosophy. London: George Allen & Unwin.

    Google Scholar 

  • Sinclair, H. (1990). Learning: the interactive recreation of knowledge. In L. Steffe, & T. Wood (Eds.), Transforming children’s mathematics education: International perspectives (pp. 19–29). Hillsdale: Lawrence Erlbaum.

    Google Scholar 

  • Skemp, R. R. (1971). The psychology of learning mathematics. Harmondsworth: Penguin.

    Google Scholar 

  • Stewart, I., & Tall, D. O. (2000). The Foundations of mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Tall, D. O., & Chin, E. T. (2002). University students embodiment of quantifier. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, vol. 4 (pp. 273–280). Norwich, UK.

  • Weyl, H. (1949). Philosophy of mathematics and natural science. Princeton: Princeton University Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shahid Beheshti University, Tehran, Iran

    Amir H. Asghari

Authors
  1. Amir H. Asghari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amir H. Asghari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Asghari, A.H. Experiencing equivalence but organizing order. Educ Stud Math 71, 219–234 (2009). https://doi.org/10.1007/s10649-008-9173-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-008-9173-x

Keywords

Search

Navigation