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The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

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Abstract

Is there a shared meaning of “mathematical proof” among researchers in mathematics education? Almost all researchers may agree on a formal definition of mathematical proof. But beyond this minimal agreement, what is the state of our field? After three decades of activity in this area, being familiar with the most influential pieces of work, I realize that the sharing of keywords hides important differences in the understanding. These differences could be obstacles to scientific progress in this area, if they are not made explicit and addressed as such. In this essay I take a sample of research projects which have impacted the teaching and learning of mathematical proof, in order to describe where the gaps are. Then I suggest a possible scientific programme which aspires to strengthen the research practice in this domain. Eventually, I make the additional claim that this programme could hold for other areas of research in mathematics education.

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Notes

  1. I first raised and discussed these issues on the occasion of the Taipei International Conference “Understanding Proof and Proving to Understand” (16–19 November 2002), on the invitation of Professor Fou-Lai Lin.

  2. This has been the key point of the lecture I gave to the XI° summer school of didactique des mathématiques (Plestin-Les-Grèves 1999) under the title “Preuve, démonstration et écriture mathématique (entre essai et synthèse)” (unpublished, power-point available).

  3. In this text, as we did in the translation of Brousseau work, I use “knowing” as a noun to convey the useful distinction we have in French, as in most roman languages, between “connaissance” and “savoir”—the later being translated by “knowledge” in English.

  4. Habermas makes very clearly this point in his essay about truth and justification (Habermas 1999, especially Chap. 2).

  5. Following what we did for the translation of Brousseau’ theory of didactical situations, I will here use the word “knowing” to keep track of the difference we have in roman languages between connaissance and savoir; the latter being translated by the word “knowledge”. A “knowing” is a personal construct, which becomes a “knowledge” if it is shared by a community under an institutionalised form.

  6. The two-column form precedes Fawcett historically. His three-column proposal was an innovation that never took hold. Fawcett wrote in 1936 and the two columns appeared in the 1910s (in US) apparently coming from UK (P. Herbst, personal communication).

  7. A total of 182 students in the pilot study, 2,459 students queries form 94 classes in 90 schools.

  8. Because this reference points to a document available on the web under un html format, I indicate here the section number instead of the page since the latter may vary from a browser to another.

  9. I must add that it does mean that Mariotti reduces mathematics to a formal game, on the contrary she relates it all along her work to problem solving likely to provide a “concrete meaning” to students activity. Her problématique is that of allowing students passing from a pragmatic to a theoretical conception of proving—hence proving getting its meaning from within mathematics.

  10. Actually, I am here paraphrasing the following quotation: “[to pass] from the need of justifying towards the idea of validating within a geometrical system” (Mariotti 1997, Sect. 4.2).

  11. My free translation.

  12. One may oppose to this statement the case of “proofs without words”; I will not address it here—I did it for the 1991 summer school in didactique—but I can ensure that considering this issue does not change drastically the claim insofar as mathematics is concerned (and not just its popularization).

  13. Balacheff 1987, p. 160.

  14. I take here epistemic in Piaget sense, that is: “the carrier of knowledge” (in either a social or an individual sense as Furth 1969, p. 193, emphasises).

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Acknowledgments

The author expresses his thanks to Patricio Herbst for his editing and thoughtful conversation.

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  1. CNRS, Laboratoire d’Informatique de Grenoble, Grenoble, France

    Nicolas Balacheff

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Balacheff, N. The role of the researcher’s epistemology in mathematics education: an essay on the case of proof. ZDM Mathematics Education 40, 501–512 (2008). https://doi.org/10.1007/s11858-008-0103-2

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