Philosophy of mathematical practice: a primer for mathematics educators

Abstract

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice. In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education.

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Fig. 1

Notes

  1. 1.

    For other overviews of the field aimed at philosophers we refer to Van Bendegem (2014), Giardino (2017b), and Carter (2019).

  2. 2.

    For discussions relevant to the characterization of the notion of mathematical practice, see also Giardino et al. (2012) and Carter (2019, sec. 4).

  3. 3.

    Two themes that will not be addressed in this paper are the role of computers in mathematical inquiry and the issue of ethics in mathematics. For the former, we refer the reader to Avigad (2008a). For the latter, see Rogers and Kaiser (1995), Ernest (2016, 2018), and Rittberg, Tanswell, and Van Bendegem (forthcoming).

  4. 4.

    For a discussion of Rav’s notion of know-how, see Tanswell (2016, chapter 4).

  5. 5.

    See Hamami (2018) for an analysis of these differences at the level of the elementary components of the two types of proofs.

  6. 6.

    See Hamami (forthcoming) for an attempt to meet this challenge which is nonetheless compatible with the standard view.

  7. 7.

    For extensive reviews of visual and diagrammatic thinking in mathematics, see Giaquinto (2016) and Giardino (2017a).

  8. 8.

    For a critical outlook on Brown’s view, see Folina (1999).

  9. 9.

    For another seminal analysis of the role of diagrams in Greek mathematics, see Netz (1999). For attempts to formalize diagrammatic reasoning in the proofs of Euclid’s Elements see Miller (2007), Mumma (2006, 2010), and Avigad, Dean, and Mumma (2009). For an analysis of the relation between geometric objects and the diagrams that represent them, see Panza (2012).

  10. 10.

    Several important volumes have already brought together philosophers and mathematics educators to reflect on the relation between the two fields (François and Van Bendegem 2007; Van Kerkhove and Van Bendegem 2007; Ernest et al. 2016; Ernest 2018).

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Hamami, Y., Morris, R.L. Philosophy of mathematical practice: a primer for mathematics educators. ZDM Mathematics Education 52, 1113–1126 (2020). https://doi.org/10.1007/s11858-020-01159-5

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