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Modelling Mathematical Reasoning in Physics Education

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Abstract

Many findings from research as well as reports from teachers describe students’ problem solving strategies as manipulation of formulas by rote. The resulting dissatisfaction with quantitative physical textbook problems seems to influence the attitude towards the role of mathematics in physics education in general. Mathematics is often seen as a tool for calculation which hinders a conceptual understanding of physical principles. However, the role of mathematics cannot be reduced to this technical aspect. Hence, instead of putting mathematics away we delve into the nature of physical science to reveal the strong conceptual relationship between mathematics and physics. Moreover, we suggest that, for both prospective teaching and further research, a focus on deeply exploring such interdependency can significantly improve the understanding of physics. To provide a suitable basis, we develop a new model which can be used for analysing different levels of mathematical reasoning within physics. It is also a guideline for shifting the attention from technical to structural mathematical skills while teaching physics. We demonstrate its applicability for analysing physical-mathematical reasoning processes with an example.

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Notes

  1. Poincaré (1958), Wigner (1960), Zahar (1980), Bochner (1981), Gingras (2001), Paty (2003), Boniolo et al. (2005), amongst many others.

  2. Hesse (1966), Bunge (1973), Redhead (1980), Nersessian (1992).

  3. Another possible approach for analysing this interrelation comes from studies on linguistics. For a deeper analysis of the possibility of making meaning with mathematics in physics from this perspective see Redish and Gupta (2010).

  4. According to Hesse (1966), formal analogies occur when the same axiomatic and deductive relations associate both subjects and objects of similar systems, without the necessity of a material similarity between them.

  5. Surely, we are not suggesting that every single mathematical entity should be physically interpreted. For the implications of adopting a realist or a relativist epistemic perspective in physics education see Quale (2011).

  6. Hesse (1953), Bunge (1973), Redhead (1980), Nersessian (1992).

  7. Of course we don’t mean that the role of mathematics should always be the main focus of physics teaching. In several cases, a pure qualitative approach is definitely more appropriate. If the goal of a particular lesson is to give a group of students an introduction to some phenomena and give rise to the students’ curiosity, the mathematical approach will probably not be the best choice. Moreover, it is important to stress that the teaching and learning of physics has many other purposes (learning about scientific inquiry, investigating the world with hands-on experiments, reflecting on social aspects of science and technology, etc.). Considering the students’ level—although our discussion can be applied to physics education in general—the issue of mathematization tends to be more suitable to high school and university levels. Keeping this complexity in mind, we are addressing here an important (though not the sole) aspect of physical knowledge that has been somehow neglected in physics education research.

  8. Notwithstanding, the modelling cycle seems to be used for that purpose according to our own experience. Although it is possible to pass the modelling stations several times, it could not be disclaimed that the numbering of the steps and the direction of the arrows in the modelling cycle might implicate a chronological order. Therefore, a model which is more flexible in following real reasoning processes could be helpful.

  9. It is not always trivial to establish a hierarchical order to the degree of mathematization. Some relevant aspects to establish this hierarchy are conciseness, generality and coverage of the representations. In addition, a historical analysis can usually provide a fruitful guideline.

  10. It is important to stress that our distinction between technical and structural skills is always made in the context of physics. Similar dichotomies (or even dualities) were already proposed in mathematics education, taking into account that mathematical knowledge can also be taught (and learned) procedurally (focusing on rules and techniques) and conceptually (focusing on meaning and demonstrations). See Skemp (1976) for the distinction between relational and instrumental understanding or Sfard (1991) for an analysis of the dual nature of mathematical concepts (structural and operational).

  11. One common problem with this example is related to the difference between the body’s position at a particular time (s(t)) and its displacement (s(t) − s(0)). These two quantities will have the same value if the initial position is set to be zero (s(0) = 0). In the didactic approach we adopt this convention and will speak about s(t) as the body’s displacement. In a real classroom situation this difference should be clearly stated.

  12. The undemanding manipulation of symbols to reach the formula \(s(t) = 1/2 \cdot g\cdot t^2\) at the didactic approach should not be interpreted as technical skills (arrow (c)) because these manipulations are very short and easy, a connection to the physical meaning is needed to perform the steps and thus they are qualitatively different from the technical manipulation in the abstract approach. In that case, the solving of the differential equation consists just out of technical mathematical skills, a connection to physical meaning is not needed; the mathematical “machine” provided the result of the equation. If the model is used to describe students’ reasoning processes, it should be noted that this interpretation can naturally depend on their level and previous knowledge as well as on the desired resolution degree of the different steps.

  13. It is interesting to note that this view is in concordance with Paul Hewitt, the author of the famous book “Conceptual Physics” (Hewitt 2006), who recently published a paper in which he argues for focusing on formulas and using them as a guide to thinking (see Hewitt 2011).

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Acknowledgments

We would like to thank Ulrike Böhm for fruitful discussions, Edward Redish for his valuable remarks, as well as Brian Danielak and Eric Kuo for their help concerning language issues. We also thank the reviewers for their relevant and constructive comments. This research is financially supported by the European Social Fund (ESF), the Free State of Saxony, the Coordination for the Improvement of Higher Level Personnel (CAPES), the São Paulo Research Foundation (FAPESP) and the German Academic Exchange Service (DAAD).

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  1. Chair of Didactics of Physics, Technische Universität, Dresden, Germany

    Olaf Uhden & Gesche Pospiech

  2. School of Education, University of São Paulo, São Paulo, Brazil

    Ricardo Karam & Maurício Pietrocola

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  1. Olaf Uhden
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  3. Maurício Pietrocola
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Correspondence to Olaf Uhden.

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The authors Olaf Uhden, Ricardo Karam contributed equally to this work.

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Uhden, O., Karam, R., Pietrocola, M. et al. Modelling Mathematical Reasoning in Physics Education. Sci & Educ 21, 485–506 (2012). https://doi.org/10.1007/s11191-011-9396-6

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