Science & Education

, Volume 24, Issue 5–6, pp 661–698 | Cite as

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

  • Ricardo KaramEmail author
  • Olaf Krey


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.


Teacher Student Pedagogical Content Knowledge Incline Plane Circular Motion Epistemological Belief 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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).


  1. Ainsworth, S. E. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198.CrossRefGoogle Scholar
  2. Angell, C., Guttersrud, O., Henriksen, E. K., & Isnes, A. (2004). Physics: Frightful, but fun. Pupils’ and teachers’ views of physics and physics teaching. Science Education, 88(5), 683–706.Google Scholar
  3. Bagno, E., Berger, H., & Eylon, B. S. (2008). Meeting the challenge of students’ understanding of formulae in high-school physics: A learning tool. Physics Education, 43(1), 75–82.CrossRefGoogle Scholar
  4. Barth, M. (1992). Brechungsgesetz/Lichtmodell: Ein historischer Zugang. Praxis der Naturwissenschaften Physik, 8(41), 41.Google Scholar
  5. Bing, T., & Redish, E. F. (2009). Analyzing problem solving using math in physics: Epistemological framing via warrants. Physical Review Special Topics Physics Education Research, 5, 020108.CrossRefGoogle Scholar
  6. Bochner, S. (1963). The significance of some basic mathematical conceptions for physics. Isis, 54(2), 179–205.CrossRefGoogle Scholar
  7. Chevallard, Y. (1991). La transposition didactique—Du savoir savant au savoir enseigné. Grenoble: La Pensée sauvage.Google Scholar
  8. Clement, J. (1982). Student preconceptions in introductory mechanics. American Journal of Physics, 50, 66–71.CrossRefGoogle Scholar
  9. Corrao, C. (2012). Gamow on Newton: Another look at centripetal acceleration. The Physics Teacher, 50(3), 176.CrossRefGoogle Scholar
  10. Dittmann, H., Näpfel, H., & Schneider, W. B. (1989). Die zerrechnete Physik. In W.B. Schneider (Ed.) Wege in der Physikdidaktik (pp. 41–46). Band 1. Palm und Enke.Google Scholar
  11. Domert, D., Airey, J., Linder, C., & Kung, R. L. (2007). An exploration of university physics students’ epistemological mindsets towards the understanding of physics equations. NorDiNa - Nordic Studies in Science Education, 3(1), 15–28.Google Scholar
  12. Doorman, L. M. (2005). Modelling motion: From trace graphs to instantaneous change. Utrecht.Google Scholar
  13. Duit, R., Gropengießer, H., Kattmann, U., Komorek, M., & Parchmann, I. (2012). The model of educational reconstruction—A framework for improving teaching and learning science 1. In Science education research and practice in Europe (pp. 13-37). SensePublishers.Google Scholar
  14. Einstein, A. (1956). Mein Weltbild. Frankfurt: Ullstein Tachenbücher.Google Scholar
  15. Falk, G. (1990). Physik: Zahl und Realität. Die begrifflichen und mathematischen Grundlagen einer universellen quantitativen Naturbeschreibung. Basel: Birkhäuser.Google Scholar
  16. Feynman, R. P. (1985). The character of physical law. Cambridge, MA: The MIT Press.Google Scholar
  17. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman lectures on physics. Reading, MA: Addison-Wesley.Google Scholar
  18. Galilei, G. [1638] (1914). Dialogues concerning two new sciences. Translated from the Italian and Latin into English by Henry Crew and Alfonso de Salvio. With an Introduction by Antonio Favaro. New York: Macmillan.
  19. Galili, I., & Bar, V. (1992). Motion implies force: Where to expect vestiges of the misconception? International Journal of Science Education, 14(1), 63–81.CrossRefGoogle Scholar
  20. Gamow, G. (1962). Gravity. London: Heinemann Educational Books.Google Scholar
  21. Gindikin, S. G. (2007). Tales of mathematicians and physicists. New York: Springer.Google Scholar
  22. Gingras, Y. (2001). What did mathematics do to physics? History of Science, 39, 383–416.CrossRefGoogle Scholar
  23. Hammer, D. (1994). Epistemological beliefs in introductory physics. Cognition and Instruction, 12, 151–183.CrossRefGoogle Scholar
  24. Hechter, R. P. (2010). What does “I understand the equation” really mean? Physics Education, 45, 132.CrossRefGoogle Scholar
  25. Hestenes, D., Wells, M., & Swackhamer, G. (1992). Force concept inventory. The Physics Teacher, 30(3), 141–158.CrossRefGoogle Scholar
  26. Hofe, R. V. (1995). Grundvorstellungen mathematischer Inhalte. Heidelberg: Spektrum Akademischer Verlag.Google Scholar
  27. Karam, R. (2014). Framing the structural role of mathematics in physics lectures: A case study on electromagnetism. Physical Review Special Topics Physics Education Research, 10(1), 010119.CrossRefGoogle Scholar
  28. Krey, O. (2012). Zur Rolle der Mathematik in der Physik. Wissenschaftstheoretische Aspekte und Vorstellungen Physiklernender. Berlin: Logos Verlag.Google Scholar
  29. Lakoff, G. & Nunez, R. E. (2000). Where mathematics comes from. New York.Google Scholar
  30. Landy, D., & Goldstone, R. L. (2007a). Formal notations are diagrams: Evidence from a production task. Memory and Cognition, 35(8), 2033–2040.CrossRefGoogle Scholar
  31. Landy, D., & Goldstone, R. L. (2007b). How abstract is symbolic thought? Learning, Memory, 33(4), 720–733.CrossRefGoogle Scholar
  32. Lemke, J. (1998). Multiplying meaning: Visual and verbal semiotics in scientific text. In J. Martin & R. Veel (Eds.), Reading science (pp. 87–113). London: Routledge.Google Scholar
  33. Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. Lester (Ed.), Handbook for research on mathematics education (2nd ed., pp. 763–804). Charlotte, NC: Information Age Publishing.Google Scholar
  34. Levi, M. (2009). The mathematical mechanic: Using physical reasoning to solve problems. Princeton: Princeton University Press.CrossRefGoogle Scholar
  35. Mahajan, S. (2010). Street-fighting mathematics: The art of educated guessing and opportunistic problem solving. Cambridge: MIT Press.Google Scholar
  36. Matthews, M. (2000). Time for science education: How teaching the history and philosophy of pendulum motion can contribute to science literacy. New York: Kluwer Academic/Plenum Publishers.CrossRefGoogle Scholar
  37. Matthews, M. R., Gauld, C. F., & Stinner, A. (Eds.). (2005). The pendulum: Scientific, historical, philosophical and educational perspectives. New York: Springer.Google Scholar
  38. Mihas, P. (2008). Developing ideas of refraction, lenses and rainbow through the use of historical resources. Science & Education, 17(7), 751–777.CrossRefGoogle Scholar
  39. Newton, I., & Henry, R. C. (2000). Circular motion. American Journal of Physics, 68(7), 637–639.CrossRefGoogle Scholar
  40. Perkins, D., & Blythe, T. (1994). Putting understanding up front. Educational Leadership, 51(5), 4–7.Google Scholar
  41. Pfeffer, J. I. (1999). An alternative derivation of the kinetic theory of gases. Physics Education, 34(4), 237–239.CrossRefGoogle Scholar
  42. Pitucco, A. (1980). An approximation of a simple pendulum. The Physics Teacher, 18(9), 666.CrossRefGoogle Scholar
  43. Polya, G. (1954). Mathematics and plausible reasoning. Volume I. Induction and analogy in mathematics. New Jersey: Princeton University Press.Google Scholar
  44. Prain, V., & Tytler, R. (2013). Representing and learning in science. In R. Tytler, V. Prain, P. Hubber, & B. Waldrip (Eds.), Constructing representations to learn in science (pp. 1–14). Rotterdam: SensePublishers. doi: 10.1007/978-94-6209-203-7.CrossRefGoogle Scholar
  45. Rivadulla, A. (2005). Theoretical explanations in mathematical physics. In G. Boniolo, P. Budinich, & M. Trobok (Eds.), The role of mathematics in physical sciences (pp. 161–178). Dordrecht: Springer.CrossRefGoogle Scholar
  46. Romer, R. H. (1993). Reading the equations and confronting the phenomena—The delights and dilemmas of physics teaching. American Journal of Physics, 61(2), 128–142.Google Scholar
  47. Schecker, H. (1985) Das Schülervorverständnis zur Mechanik (Students’ frames of thinking in mechanics). Doctoral dissertation, University of Bremen.Google Scholar
  48. Schiffer, M. M., & Bowden, L. (1984). The role of mathematics in science. Washington, DC: MAA.Google Scholar
  49. Sherin, B. L. (2001). How students understand physics equations. Cognition and Instruction, 19(4), 479–541.CrossRefGoogle Scholar
  50. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.CrossRefGoogle Scholar
  51. Tuminaro, J., & Redish, E. F. (2007). Elements of a cognitive model of physics problem solving: Epistemic games. Physical Review Special Topics Physics Education Research, 3, 020101.CrossRefGoogle Scholar
  52. Viennot, L. (1979). Spontaneous reasoning in elementary dynamics. European Journal of Science Education, 1(2), 205–221.CrossRefGoogle Scholar
  53. Wagenschein, M. (1968). Verstehen lehren: Genetisch-Sokratisch-Exemplarisch. Weinheim: Beltz.Google Scholar
  54. Walsh, L., Howard, R., & Bowe, B. (2007). Phenomenographic study of students problem solving approaches in physics. Physical Review Special Topics-Physics Education Research, 3, 020108.CrossRefGoogle Scholar
  55. Wedemeyer, B. (1993). Centripetal acceleration: A simple derivation. The Physics Teacher, 31(4), 238–239.CrossRefGoogle Scholar
  56. Whewell, W. (1858). History of the inductive sciences, from the earliest to the present times (Vol. 2). New York: D. Appleton and company.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Science EducationUniversity of CopenhagenCopenhagenDenmark
  2. 2.Institut für Physik - Didaktik der PhysikMartin-Luther-Universität Halle-WittenbergHalleGermany

Personalised recommendations