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Banality of mathematical expertise

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Abstract

Practices within research mathematics can and do serve as models for mathematics education. However, typically such inspirations impose a devastating narrowness in relation to reflections on mathematics. This narrowness I refer to as the “banality of mathematical expertise”. Reflections on mathematics can be expressed through a philosophy of mathematics that goes beyond the traditional emphasis on ontological and epistemological dimensions, to become four-dimensional by also addressing social and ethical issues. Many working philosophies of mathematics operate within a narrow scope of reflections, seemingly located within an ethical vacuum. The consequence is a cultivation of a banality, manifest in many university studies in mathematics as well as in dominant research paradigms in mathematics. This constitutes a serious limitation in providing models for mathematics education. By contrast, there exist examples of practices of mathematics education that demonstrate a richness of reflections on mathematics. Accordingly, I address the extent to which such practices of critical mathematics education could serve as models for research mathematics and mathematics education at the university level.

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Notes

  1. For a discussion of the diversity of mathematical practices, see Ferreirós (2016).

  2. In this short summary of Burton’s research, I also drawn on conversations I have had with her.

  3. See Bourbaki (1950); Dieudonné (1970); and the discussion in Skovsmose (1990).

  4. Platonism exists in different versions. In Ravn and Skovsmose (2019), we address “Platonism before Plato”, “Platonisms after Plato”, as well as “Plato’s Platonism”. Hacking (2014) makes profound investigations of different versions of Platonism.

  5. More recommendations for such redirecting of the philosophy of mathematics can be found in publications by Ernest (1998); Hacking (2014); Mancosu (2008); and Tymoczko (1986). See Carter (2019) for an overview of the trend.

  6. A detailed discussion of the historicity of mathematics is found in work by Grabiner (1986), raising the question: Is mathematical truth time-dependent?

  7. See, for instance, Restivo (1992); and Restivo et al. (1993).

  8. Many studies in ethnomathematics investigate the social formation of mathematics. A classic is Powell and Frankenstein (1997), while recent contributions are found in Rosa et al. (2016).

  9. A systematics exploration of the ethical dimension of a philosophy of mathematics is found in Part IV in Ravn and Skovsmose (2019).

  10. See, for instance, Bernacerraf and Putnam (1964); Körner (1968); and Shapiro (2000).

  11. See, for instance, Ayer (1959).

  12. Important ideas are expressed by Latour (1987); and Gibbons et al. (1994).

  13. See Ethics in Mathematics https://en.wikipedia.org/wiki/Ethics_in_mathematics. Accessed 17 February 2020.

  14. In Ravn and Skovsmose (2019), the notion of mathematics in action becomes elaborated in terms of fabrications of fictions, fabrication of facts, fabrication of risks, and creation of illusions of objectivity. See also Yasukawa et al. (2012, 2016).

  15. For a discussion of such phenomena see Chapter 15 ‘Symbolic Power, Robotting, and Surveiling’, and Chapter 17 ‘Mathematics as Part of Technology’ in Skovsmose (2014).

  16. That hypothetical reasoning is an important feature of mathematics has been indicated by Peirce (1960), who states: “Mathematics is the study of what is true of hypothetical state of things.” (p. 193). See also Carter (2014).

  17. See Booss-Bavnbek and Høyrup (2003); and Skovsmose and Yasukawa (2009).

  18. See Skovsmose (2019).

  19. Naturally, I am not claiming that the only content that ethical reflections on mathematics could have is rooted in the mathematics in action. The scope of ethical reflections could be much broader.

  20. Carter (2019), Mancosu (2008) and Ferreirós (2016) do not address ethical issues. Hacking (2014), who otherwise provides a new look at the philosophy of mathematics, also ignores ethics.

  21. Naturally, one finds exceptions. Problem orientation and project-based education can be found in university mathematics education (see, for instance, Vithal et al. (1995). In such an educational format, one gets possibilities for reflecting on mathematics in action.

  22. In Philosophy of Mathematics Education Journal, edited by Ernest, one finds an even broader variety of issues addressed. See also Ernest et al. (2016).

  23. See McKenzie (2007).

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Acknowledgements

I want to thank Denner Barros, Arindam Bose, Manuella Carrijo, Ana Carolina Faustino, Peter Gates, Brian Greer, David Kollosche, Amanda Queiroz Moura, Miriam Godoy Penteado, Célia Roncato, Daniela Alves Soares, Guilherme Henrique Gomes da Silva, and Débora Vieira de Souza for helpful comments and suggestions.

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  1. Aalborg University, Ålborg, Denmark

    Ole Skovsmose

  2. Universidade Estadual Paulista (UNESP), São Paulo, Brazil

    Ole Skovsmose

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Skovsmose, O. Banality of mathematical expertise. ZDM Mathematics Education 52, 1187–1197 (2020). https://doi.org/10.1007/s11858-020-01168-4

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