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Interactions Between Mathematics and Physics: The History of the Concept of Function—Teaching with and About Nature of Mathematics

Abstract

In this paper, we discuss the history of the concept of function and emphasize in particular how problems in physics have led to essential changes in its definition and application in mathematical practices. Euler defined a function as an analytic expression, whereas Dirichlet defined it as a variable that depends in an arbitrary manner on another variable. The change was required when mathematicians discovered that analytic expressions were not sufficient to represent physical phenomena such as the vibration of a string (Euler) and heat conduction (Fourier and Dirichlet). The introduction of generalized functions or distributions is shown to stem partly from the development of new theories of physics such as electrical engineering and quantum mechanics that led to the use of improper functions such as the delta function that demanded a proper foundation. We argue that the development of student understanding of mathematics and its nature is enhanced by embedding mathematical concepts and theories, within an explicit–reflective framework, into a rich historical context emphasizing its interaction with other disciplines such as physics. Students recognize and become engaged with meta-discursive rules governing mathematics. Mathematics teachers can thereby teach inquiry in mathematics as it occurs in the sciences, as mathematical practice aimed at obtaining new mathematical knowledge. We illustrate such a historical teaching and learning of mathematics within an explicit and reflective framework by two examples of student-directed, problem-oriented project work following the Roskilde Model, in which the connection to physics is explicit and provides a learning space where the nature of mathematics and mathematical practices are linked to natural science.

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Notes

  1. 1.

    We use the term “explicit-reflective framework” in the sense of Abd-El-Khalick (2013).

  2. 2.

    For a more thorough history of the concept of function and generalized function, see Yuschkevich (1976) and Lützen (1982). A translation into English of a number of definitions of the concept of function from Bernoulli to Bourbaki can be found in Rüthing (1984).

  3. 3.

    A fine pedagogic introduction to Babylonian and ancient Greek astronomy can be found in Aaboe (2001).

  4. 4.

    Baron (1969) gives a fine introduction to the history of the calculus.

  5. 5.

    Here, we have set the wave velocity equal to 1.

  6. 6.

    In other cases, D’Alembert insisted that” the problem will be impossible” (D’Alembert 1747 §8).

  7. 7.

    In most other instances, D’Alembert was highly influenced by physics.

  8. 8.

    See Leçon 1.

  9. 9.

    For an account of the prehistory of the theory of distributions, see Lützen (1982).

  10. 10.

    For a description, analyses and discussion of the study program and the significance of the student-directed, problem-oriented project work for the learning and teaching of mathematics and science, see, e.g. Blomhøj and Kjeldsen (2009), Kjeldsen and Blomhøj (2009), and Andersen and Heilesen (2015).

  11. 11.

    For further details, see Kjeldsen and Blomhøj (2012).

  12. 12.

    For references, see Abd-El-Khalick (2013, p. 2088).

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Correspondence to Tinne Hoff Kjeldsen.

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Kjeldsen, T.H., Lützen, J. Interactions Between Mathematics and Physics: The History of the Concept of Function—Teaching with and About Nature of Mathematics. Sci & Educ 24, 543–559 (2015). https://doi.org/10.1007/s11191-015-9746-x

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Keywords

  • Function Concept
  • Mathematical Knowledge
  • Mathematical Concept
  • Mathematical Practice
  • Epistemological Understanding