Abstract
The starting point for this chapter is the assumption that knowing and reflecting about different roles of mathematics in (learning) physics is part of the desirable body of knowledge about the nature of science (NOS). On the one hand, a NOS approach that does not address roles of mathematics is simply not adequate, as mathematics is as important as are experiments for generating and communicating physics knowledge. On the other hand, the understanding of the roles of mathematics may strongly influence other, more general NOS aspects, e.g. to recognize physics as a creative human endeavour. This chapter aims at examplifying potential learning opportunities based on two case studies. The case studies have been chosen to represent an inductive and a deductive approach to teaching the physics around two equations included in most, if not all, secondary physics curricula. To illustrate the inductive approach “uniform motion” has been chosen, “image formation by a thin (converging) lens” is chosen as an example for a deductive approach. Both case studies shed some light on what is taught implicitly and what could be taught explicitly and reflectively about the roles of mathematics in physics, suggesting exemplary fields of reflection for a first encounter.
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Notes
- 1.
- 2.
See Redfors et al. in this book for examples of deductive elements in physics textbooks that are not used by the teachers. Although there are many examples where equations could be deduced from theoretical or mathematical considerations in high school (Snell’s law, centripetal acceleration, etc.), it is rather hard to find many examples for equations that could be dealt with in middle school (e.g. total resistance in series and parallel circuit).
- 3.
Further examples of equations that are likely to be introduced inductively include x(t) = vt (uniform motion), \( x(t)=\frac{1}{2}a{t}^2 \) (accelerated linear motion), \( R=\frac{U}{I}= const. \) (Ohm’s law), \( R=\varrho \frac{l}{A} \) (Pouillet’s law), Q = mcΔT (heat equation), etc.
- 4.
Depending on your perspective, you may wish to emphasize the epistemological upgrade in terms of generality, width of applicability, elegance and simplicity that comes with the mathematical formulation of physical knowledge about the world. However, focusing on the concrete problem (e.g. the motion of an air bubble) from which this exploration started, this specific problem is downplayed and loses its relative importance as it becomes an example of a class of processes with many other processes in this class that are basically the same in that they are just one of many examples for a uniform motion. While none of these perspectives is more correct than the other, I doubt that many young learners can truly appreciate the (surely valid) upgrade at a first encounter.
- 5.
Of course, the magnification equation can also be derived by using the intercept theorem, which shows the deep structural equivalence between geometry and light propagation based on the ray model of light, which allows the mapping between relevant aspects of the real world and geometric entities.
- 6.
It should be clarified that by no means the author intends to argue for a more deductive and against inductive teaching sequences. Both of them have their advantages and disadvantages. They are adequate to situations, outcomes and learners or they are not. For example, an arranged deductive way of introducing a law of physics (even the lens equation the way it was presented) can lead to lots of low-level activities on the learners’ side and simply illustrate bad teaching. Furthermore, both approaches can help teach valuable lessons about the nature of science in general as well as the role of mathematics in (learning) physics more specifically. What the author wishes to say though, is that both approaches to teaching differ in the learning opportunities they can potentially provide with regards to the mathematics-physics-interplay.
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Krey, O. (2019). What Is Learned About the Roles of Mathematics in Physics While Learning Physics Concepts? A Mathematics Sensitive Look at Physics Teaching and Learning. In: Pospiech, G., Michelini, M., Eylon, BS. (eds) Mathematics in Physics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-04627-9_5
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