Abstract
Mathematics is widely considered to be a prerequisite for learning physics. However, it is quite naive to believe that learning basic math is sufficient to use mathematics as a reasoning tool to think about the physical world. The main reason is that using mathematics in physics is substantially different than in math. In this chapter we show how the way physicists make use of some basic mathematical concepts (e.g., multiplication, division, functions) is specific to physics by identifying their historical genesis and contrasting with the way these concepts are usually taught in math lessons. We argue that the explicit acknowledgment of these differences has important didactical implications.
This is an extended version of a paper published in Karam, R., Uhden, O. & Höttecke, D. (2016). Das habt ihr schon im Matheunterricht gelernt! Stimmt das wirklich? Ein Vergleich zwischen dem Umgang mit mathematischen Konzepten in der Mathematik und in der Physik. Unterricht Physik 153/154, S. 22–27.
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Notes
- 1.
This is not always the case. Consider, for instance, the equation v = λf, expressing the relation between a wave’s velocity, frequency, and wave length. Phenomenologically, v depends on the characteristics of the medium, whereas f is the frequency of the source. Thus, it is usually not correct to say that v increases when f increases. Since v and f are determined by different causes, λ is the factor that is adjusted to maintain this equality. Overall, this exemplifies how physics makes a rather flexible use of mathematics.
- 2.
The first definition of Newton’s Principia is the “quantity of matter” which is “arises from its density and bulk [volume] conjointly.” In Wallis’ work we also find several other examples of such mutual dependence. Furthermore, Huygens’ vis viva, represented by the scalar quantity mv 2, is also among the first examples of the essentially physical use of multiplication to represent a new quantity.
- 3.
There are many other subtleties in the way functions are taught/learned at school (see Ellermeijer and Heck 2002 as well as the chapter from Heck and Buuren in this collection).
- 4.
See also Ellermeijer and Heck 2002, and the chapter from Planinic in this collection.
- 5.
The interested reader can find another episode that illustrates the great influence physics had in the conceptualization of function, namely, the so-called vibrating string controversy (Wheeler and Crummett 1987).
- 6.
See Kjeldsen and Lützen (2015) for an overview on the historical development of the concept of function.
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Karam, R., Uhden, O., Höttecke, D. (2019). The “Math as Prerequisite” Illusion: Historical Considerations and Implications for Physics Teaching. In: Pospiech, G., Michelini, M., Eylon, BS. (eds) Mathematics in Physics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-04627-9_2
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