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Quod erat demonstrandum: Understanding and Explaining Equations in Physics Teacher Education

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Abstract

In physics education, equations are commonly seen as calculation tools to solve problems or as concise descriptions of experimental regularities. In physical science, however, equations often play a much more important role associated with the formulation of theories to provide explanations for physical phenomena. In order to overcome this inconsistency, one crucial step is to improve physics teacher education. In this work, we describe the structure of a course that was given to physics teacher students at the end of their master’s degree in two European universities. The course had two main goals: (1) To investigate the complex interplay between physics and mathematics from a historical and philosophical perspective and (2) To expand students’ repertoire of explanations regarding possible ways to derive certain school-relevant equations. A qualitative analysis on a case study basis was conducted to investigate the learning outcomes of the course. Here, we focus on the comparative analysis of two students who had considerably different views of the math-physics interplay in the beginning of the course. Our general results point to important changes on some of the students’ views on the role of mathematics in physics, an increase in the participants’ awareness of the difficulties faced by learners to understand physics equations and a broadening in the students’ repertoire to answer “Why?” questions formulated to equations. Based on this analysis, further implications for physics teacher education are derived.

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Notes

  1. However, if we seek to understand the reasons why this practice is so deeply penetrated in physics teaching culture, we have to look at the situation from the teachers’ perspective. It is impossible to deny the practicality of this approach for the everyday life of a teacher. Numerical tasks that follow the “given-sought” scheme are very easy to be created and graded. This can be quite handy in the stressful routine of a school. Therefore, a deep epistemological conviction that physics is not about calculating with formulas seems to be a necessary condition for a different (not as handy) approach.

  2. We thank one of the reviewers for raising our attention to this paper.

  3. A similar categorization is proposed by Romer (1993). The previously mentioned “range equation”, for instance, belongs to category 4. We thank one of the reviewers for pointing that out to us.

  4. The line of reasoning that follows could be equally applied for the other three equations approached in the course (y = gt 2/2, a = v 2/r, \( T = 2\pi \sqrt {l/g} \)).

  5. One has to be careful when speaking about the similarities between physics and mathematics because there are also important differences in the types of reasoning and the methods used by physicists and mathematicians. Feynman (1985, pp. 46–58) provides a good discussion about these differences when he distinguishes between the “Babylonian” and the “Greek” traditions of thinking when using mathematics.

  6. One could actually use physical arguments (inertia) to conclude that the velocity is perpendicular to the position. Then, starting from this “physical fact,” it is possible to “prove” the derivation rules for the trigonometric functions sine and cosine [see Chapter 6 in Levi (2009)].

  7. A first trial was conducted at the University of Hamburg (Dec 2012), a pilot version of the course was held at the University of Helsinki (April–May 2013) and the final version was given at the Technical University of Dresden (November–December 2013). In all cases the teacher students were in the end of their master’s studies, i.e. all physics and mathematics courses have already been taken.

  8. Gingras (2001), Bochner (1963), Falk (1990) and Feynman (1985).

  9. The same kind of critical argument can be made for all equations discussed in the course.

  10. See, for instance, a hypothetical dialogue between Galileo and Aristotle at http://hipstwiki.wikifoundry.com/page/moving+bodies.

  11. More details in Schiffer and Bowden (1984), chapter 3.

  12. An alternative geometric derivation—which is far less (mathematically) demanding and was also presented by the students—is found in Feynman Lectures (Feynman et al. 1964, p. 26-3).

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Acknowledgments

We would like to thank all the students that attended the course for their engagement and the reviewers for their valuable suggestions. Many thanks also to Gesche Pospiech (Dresden), Ismo Koponen (Helsinki) and Dietmar Höttecke (Hamburg) for the opportunity of giving the course in their universities. This work was supported by the Alexander von Humboldt Foundation (Postdoctoral Fellowship to RK—BRA 1146348 STP).

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  1. Department of Science Education, University of Copenhagen, Øster Voldgade 3, 1350, Copenhagen, Denmark

    Ricardo Karam

  2. Institut für Physik - Didaktik der Physik, Martin-Luther-Universität Halle-Wittenberg, Hoher Weg 8, 06120, Halle, Germany

    Olaf Krey

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Correspondence to Ricardo Karam.

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Karam, R., Krey, O. Quod erat demonstrandum: Understanding and Explaining Equations in Physics Teacher Education. Sci & Educ 24, 661–698 (2015). https://doi.org/10.1007/s11191-015-9743-0

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