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International Handbook of Research in History, Philosophy and Science Teaching pp 837–871Cite as

The Role of History and Philosophy in University Mathematics Education

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Abstract

University level mathematics is organised differently in different universities. In this paper we consider mathematics programmes leading to a graduate degree in mathematics. We briefly introduce a multiple perspective approach to the history of mathematics from its practices, reflections about uses of history and the research direction in philosophy of mathematics denoted ‘Philosophy of Mathematical Practice’. We link history and philosophy of mathematical practices to recent ideas in mathematics education in order to identify different roles history and philosophy can play in the learning of mathematics at university level. We present, analyse and discuss different examples of inclusions of history and philosophy in university programmes in mathematics. These presentations are divided into courses in history and philosophy, respectively, since this is the main way they are organised at the universities. We shall see, however, that the history courses address philosophical questions and that the philosophy courses employ historical material. The chapter ends with comments on how mathematics educations at university level can benefit from history and philosophy of mathematics.

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Notes

  1. 1.

    For the roles of the history and philosophy of mathematics in liberal arts education, we refer to the previous chapter by Judith Grabiner.

  2. 2.

    Our joint paper (Kjeldsen and Carter 2012) serves as an example of this mutual beneficial relationship.

  3. 3.

    See, e.g. Lützen and Purkert (1989) where the different historiographical views of Cantor and Zeuthen are discussed.

  4. 4.

    In Kjeldsen (2012) Jensen’s terminology is outlined and used as lens through which we can identify and distinguish between different conceptions and uses of history of mathematics.

  5. 5.

    This is our translation of the Danish word ‘lærd’.

  6. 6.

    Their views on mathematics are very different. For a presentation of their positions, see Shapiro (2000). For a nontraditional description of Plato’s philosophy, in line with the perspective of mathematical practice, see McLarty (2005).

  7. 7.

    Another interesting development in mathematics around the turn of the century was the move towards structuralism. What is studied in mathematics is not the objects as such – it is the relations between objects. This is most famously described by Hilbert saying ‘one must be able to say “tables, chairs, beer-mugs” each time in place of “points, lines, planes”’ (Blumenthal 1935, pp. 402–403), expounded mathematically in his Foundations of Geometry (Hilbert 1899). Traces of this conception about mathematics can still be found in today’s philosophies of mathematics; see Benacerraf (1965), Hellman (1996), Resnik (1999), and Shapiro (1997). More recently philosophers have argued that category theory provides a sound basis of a ‘top-down’ structuralist view (Awodey 1996; Landry and Marquis 2005).

  8. 8.

    Since the original foundational schools failed for a variety of reasons, other ways of obtaining a foundation were looked for. It was, for example, proved by Gentzen during 1930s that if sufficiently strong methods (induction over ε0) are used, then it is possible to prove the consistency of arithmetic. A different approach is to find a weaker system than Primitive Recursive Arithmetic where completeness and consistency are provable.

  9. 9.

    See, e.g. Ferreiros and Gray (2006), Mancosu (2008), Van Kerkhove and Van Bendegem (2007), and Van Kerkhove et al. (2010).

  10. 10.

    Philosophy of Mathematics: Sociological Aspects and Mathematical Practice.

  11. 11.

    One misconception is that when Frege started worrying about the foundation of analysis, it had already been settled by the work of Weierstrass in Berlin. The fact of the matter, Tappenden argues, is that problems of real analysis were being solved, but Frege knew of the (revolutionary) work of Riemann from the 1850s integrating geometry and complex analysis, opening up whole new fields of study.

  12. 12.

    The introduction of Riemann surfaces and K-theory.

  13. 13.

    The project was called Competencies and Mathematical Learning. It was initiated by the Danish National Council for Science Education in 2000. For a shortened English version of the original report, see Niss and Højgaard (2011).

  14. 14.

    It is generally held that students’ beliefs influence their learning, for example, that affective beliefs play a major role (Burton 2004). Here we are only interested in philosophical beliefs.

  15. 15.

    It has also been argued that knowledge of philosophy can turn you into a better person. Philosophy teaches rational thinking, and in particular, ethics deals with the good and bad.

  16. 16.

    http://sis.ku.dk/kurser/viskursus.aspx?knr=121117&sprog=2&forrige=57876

  17. 17.

    http://sis.ku.dk/kurser/viskursus.aspx?knr=121117&sprog=2&forrige=57876

  18. 18.

    http://sis.ku.dk/kurser/viskursus.aspx?knr=121117&sprog=2&forrige=57876

  19. 19.

    http://sofia.nmsu.edu/~history/; http://www.cs.nmsu.edu/historical-projects

  20. 20.

    http://sofia.nmsu.edu/~history/

  21. 21.

    http://sofia.nmsu.edu/~history/

  22. 22.

    See http://www.cs.nmsu.edu/historical-projects.

  23. 23.

    Personal e-mail correspondence between David Pengelley and Tinne Hoff Kjeldsen on Monday the 25. of June, 2012.

  24. 24.

    Three to eight students is the common group size, but students are allowed to perform a project on their own.

  25. 25.

    Under the ‘profession’ requirement, the students have a choice between a modelling project, a pure mathematics project, a history and/or philosophy of mathematics project or a project on aspects of mathematics education, according to in what kind of direction, they want their future profession to move.

  26. 26.

    See also Niss’ (2001) narrative on his 25 years of experiences with the RUC model.

  27. 27.

    The students’ project report can be downloaded at the following address: http://milne.ruc.dk/ImfufaTekster/pdf/403.pdf.

  28. 28.

    All quotes from student reports have been translated into English by us.

  29. 29.

    The students’ project report can be downloaded at the following address: http://milne.ruc.dk/ImfufaTekster/pdf/416.pdf.

  30. 30.

    Until 1971 it was compulsory for all university students in Denmark to take an introductory philosophy course, so-called filosofikum. This course was mainly handled by philosophers.

  31. 31.

    In addition ‘videnskabsteori’ is mentioned. It is best translated as ‘theory of science’ and is usually thought of more broadly than philosophy of science, including social science, history of science and ethics.

  32. 32.

    The following is taken – and translated – from slides used in the course.

  33. 33.

    The topic of the role of models in society is also dealt with at USD, and the following description is mainly based on the treatment there.

  34. 34.

    Maddy’s (1990) set theoretic realism and Shapiro’s (2000) ante rem structuralism are examples of what is presented. In addition we discuss Field’s (1980) fictionalism and the empiricism of Mill.

  35. 35.

    A scientific theory is confirmed as a whole. If mathematics is part of a confirmed scientific theory, then the included mathematics is also confirmed.

  36. 36.

    In Aarhus and Copenhagen, it is very common for teachers to write their own textbook material.

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Authors and Affiliations

  1. Department of Science, Systems and Models, IMFUFA, Roskilde University, Roskilde, Denmark

    Tinne Hoff Kjeldsen

  2. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Jessica Carter

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  1. Tinne Hoff Kjeldsen
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Correspondence to Tinne Hoff Kjeldsen .

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  1. School of Education, University of New South Wales, Sydney, New South Wales, Australia

    Michael R. Matthews

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Kjeldsen, T.H., Carter, J. (2014). The Role of History and Philosophy in University Mathematics Education. In: Matthews, M. (eds) International Handbook of Research in History, Philosophy and Science Teaching. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7654-8_26

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