Mathematics in Physics Education pp 335-353 | Cite as

# Taking the Phys-Math Interplay from Research into Practice

## Abstract

Physics and mathematics are heavily interwoven in the context of physics education at many levels. Research in physics education indicates that insufficient knowledge of the “Phys-Math” interplay may reflect on the quality of the learners’ explanations of physical phenomena, their ability to construct mathematical models of physical processes, or on their ability to describe the physical meaning of mathematical constructs (Clement et al. 1981; Cohen et al. 1983; Rozier S, Viennot L. Int J SciEdu 13:159–170, 1991; Rebmann and Viennot 1994; Bagno E, Eylon B,Berger H. Phys Edu 43(1):75–82, 2007; Redish EF, Smith KA. J Eng Edu97(3):295–307; Baumert et al. 2010; Zuccarini and Michelini 2014).

Studies on physics teachers’ pedagogical content knowledge (PCK) with regard to the “Phys-Math” interplay indicated that high school physics teachers employ complex two-way tracks between the two disciplines in order to support learners in constructing their knowledge and understanding of physics. These tracks construct different patterns, each of which addresses different teaching goals (Lehavi et al. 2013; Pospiech et al. 2014; Pospiech G, Eylon BS, Bagno E, Lehavi Y, Geyer MA. *The role of mathematics for physics teaching and understanding*. In The GIREP MPTL 2015 Conference Proceedings. Italian Physical Society. https://doi.org/10.1393/ncc/i2015-15110-6, 2015; Lehavi Y, Amit Yosovich A, Barak S. Sch Sci Rev 97(361):9–14, 2016a, Lehavi Y, Bagno E, Eylon B, Mualem R, Pospiech G, Böhm U, Krey O, Karam R. Classroom evidence of teachers’ PCK of the interplay of physics and mathematics. In: Greczyło T, Dębowska E (eds) Selected contributions from the International Conference GIREP EPEC 2015, Wrocław Poland, 6–10 July 2015, p 95–104, https://doi.org/10.1007/978-3-319-44887-9_8, 2016b).

Here we present two applications of the Phys-Math patterns used by physics teachers, which were identified as construction and application patterns. First, a strategy, which involves visual representations for explaining and predicting phenomena, was applied in the context of Newton’s laws and was shown to significantly advance JHS students’ performance in the Force Concept Inventory (FCI) test| (Mualem and Eylon 2007; Mualem R, Eylon BS. J Res Sci Teach 47(9):1094–1115. https://doi.org/10.1002/tea.20369). This strategy, however, was not discussed in the context of the Phys-Math interplay. Hereafter we will refer to this strategy as “Visual Mathematics” (VM), and we suggest that it will be used possibly as support for students in constructing a mathematical model for physical situations that can also assist them in solving problems. We will also reinterpret the strategy and its implications in the context of the Phys-Math interplay.

To further demonstrate the possible potential of the VM strategy, we will describe here how this strategy, which was developed and tested in the context of Newton’s laws, can be applied in a new context – teaching energy for the JHS level.

We claim that such a strategy can play an important role in teachers’ training and in fostering their Phys-Math PCK.

As a second application of the construction and the application patterns, we will demonstrate in the same teaching context (energy) how formulae can be constructed from experiments. More specifically, we will demonstrate how students can arrive from the results of experiments, similar to those conducted by Joule, to the formula for the energy change corresponding to a change in the height of an object.

## References

- Arons, A. B. (1999). Development of energy concepts in introductory physics courses.
*American Journal of Physics, 67*(12), 1063–1067.CrossRefGoogle Scholar - Bagno, E., Eylon, B., & Berger, H. (2007). Meeting the challenge of students’ understanding formulas in high-school physics – A learning tool.
*Physics Education, 43*(1), 75–82.CrossRefGoogle Scholar - Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., Klusmann, U., Krauss, S., Neubrand, M., & Tsai, Y.-M. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress.
*American Educational Research Journal, 47*(1), 133–180.CrossRefGoogle Scholar - Bevilacqua, F. (2014, June). Energy: Learning from the past.
*Science & Education, 23*(6), 1231–1243.CrossRefGoogle Scholar - Bécu-Robinault, K., & Tiberghien, A. (1998). Integrating experiments into the teaching of energy.
*International Journal of Science Education, 20*(1), 99–114.CrossRefGoogle Scholar - Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics.
*American Mathematical Monthly, 88*(4), 286–290.CrossRefGoogle Scholar - Cohen, R., Eylon, B., & Ganiel, U. (1983). Potential difference and current in simple electric circuits: A study of students’ concepts.
*American Journal of Physics, 51*(5), 407–412.CrossRefGoogle Scholar - Etkina, E. (2010). Pedagogical content knowledge and preparation of high school physics teachers.
*Physical Review Special Topics – Physics Education Research, 6*(2). Retrieved from http://www.univ-reims.fr/site/evenement/girep-icpe-mptl-2010-reims-international-conference/gallery_files/site/1/90/4401/22908/29702/30688.pdf - Eylon, B. S., & Lehavi, Y. (2010). Position paper: Energy as the language of changes. In P. R. L. Heron, M. Michelini, B. S. Eylon, Y. Lehavi, & A. Stefanel (Co-organizers),
*Teaching about energy. Which concepts should be taught at which educational level?*A workshop held within: GIREP – ICPE – MPTL 2010, University of Reims, France.Google Scholar - Hake, R. R. (1998). Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses.
*American Journal of Physics, 66*(1), 64–74.CrossRefGoogle Scholar - Halloun, I. A., & Hestenes, D. (1985). The initial knowledge state of college physics students.
*American Journal of Physics, 53*(11), 1043–1055.CrossRefGoogle Scholar - Hull, M., Kuo, E., Gupta, A., & Elby, A. (2013). Problem-solving rubrics revisited: Attending to the blending of informal conceptual and formal mathematical reasoning.
*Physical Review Special Topics – Physics Education Research, 9*(1), 010105.CrossRefGoogle Scholar - Joule, J. P. (1850). On the mechanical equivalent of heat.
*Philosophical Transactions of the Royal Society of London Series A, 140*, 61–82. Retrieved from http://links.jstor.org/sici?sici=0261-0523%281850%29140%3C61%3AOTMEOH%3E2.0.CO%3B2-M - Karplus, R. (2003).
*Introductory physics – A model approach*. Captains Engineering Services, Inc.Google Scholar - Kuo, E., Hull, M. M., Gupta, A., & Elby, A. (2013). How students blend conceptual and formal mathematical reasoning in solving physics problems.
*Science Education, 97*, 32.CrossRefGoogle Scholar - Lehavi, Y., Bagno, E., Eylon, B. S., & Cohen, E. (2013).
*Can math for physics teachers impact their conceptual knowledge of physics*.Google Scholar - Lehavi, Y., Eylon, B. S., Hazan, A., Bamberger, Y., & Weizman, A. (2014). Focusing on changes in teaching about energy. In M. F. Taşar & G. Üniversitesi (Eds.),
*Proceedings of the World Conference on Physics Education*2012 (pp. 491–498), Istanbul, Turkey.Google Scholar - Lehavi, Y., Bagno, E., Eylon, B. S., Mualem, R., Pospiech, G., Böhm, U., et al. (2015). Towards a pck of physics and mathematics interplay. In C. Fazio & R. Sperandeo-Mineo (Eds.),
*The GIREP MPTL 2014 conference proceedings*. Dipartimento di Fisica e Chimica, Università degli Studi di Palermo.Google Scholar - Lehavi, Y., Amit Yosovich, A., & Barak, S. (2016a). Bringing Joule back to school.
*School Science Review, 97*(361), 9–14.Google Scholar - Lehavi, Y., Bagno, E., Eylon, B., Mualem, R., Pospiech, G., Böhm, U., Krey, O., & Karam, R. (2016b). Classroom evidence of teachers’ PCK of the interplay of physics and mathematics. In T. Greczyło & E. Dębowska (Eds.),
*Selected contributions from the International Conference GIREP EPEC*2015*, Wrocław Poland, 6–10 July 2015*(pp. 95–104). https://doi.org/10.1007/978-3-319-44887-9_8 Google Scholar - Lehavi, Y., et al. (2017). Classroom evidence of teachers’ PCK of the interplay of physics and mathematics. In T. Greczyło & E. Dębowska (Eds.),
*Key competences in physics teaching and learning. Springer proceedings in physics*(Vol. 190). Cham: Springer.Google Scholar - Lehavi, Y., & Eylon, B. S. (2018). Integrating science education research and history and philosophy of science in developing an energy curriculum. In
*History, philosophy and science teaching*(pp. 235–260). Cham: Springer.CrossRefGoogle Scholar - Magnusson, S., Krajcik, J., & Borko, H. (1999). Nature, sources and development of pedagogical content knowledge for science teaching. In J. Gess-Newsome & N. G. Lederman (Eds.),
*Examining pedagogical content knowledge: The construct and its implications for science education*(pp. 95–133). Dordrecht: Kluwer Academic Publishers.Google Scholar - McDermott, L. C. (1984, July). Research on conceptual understanding in mechanics.
*Physics Today, 37*(7), 24–32.CrossRefGoogle Scholar - Minstrell, J. (1983). Getting the facts straight.
*Science Teacher, 50*(1), 52–54.Google Scholar - Millar, R. (2014). Towards a research-informed teaching sequence for energy. In R. F. Chen, A. Eisenkraft, D. Fortus, J. Krajcik, K. Neumann, & J. C. Nordine (Eds.),
*Teaching and learning of energy in K-12 education*. New York: Springer.Google Scholar - Mualem, R., & Eylon, B. S. (2010). Junior high school physics: Using a qualitative strategy for successful problem solving.
*Journal of Research in Science Teaching, 47*(9), 1094–1115. https://doi.org/10.1002/tea.20369.CrossRefGoogle Scholar - Pospiech, G., Eylon, B. S., Bagno, E., Lehavi, Y., & Geyer, M. A. (2015). The role of mathematics for physics teaching and understanding. In
*The GIREP MPTL 2015 Conference Proceedings*. Italian Physical Society. https://doi.org/10.1393/ncc/i2015-15110-6. - Pugh, K. (2004). Newton’s laws beyond the classroom walls.
*Science Education, 88*(2), 182–195.CrossRefGoogle Scholar - Quinn, H. (2014). A physicist’s musings on teaching about energy. In R. Chen, A. Eisenkraft, D. Fortus, J. Krajcik, J. Nordine, & A. Scheff (Eds.),
*Teaching and learning of energy in K-12 education*(pp. 15–36). Cham, Heidelberg, New York, Dordrecht, London: Springer.CrossRefGoogle Scholar - Rebmann, G., & Viennot, L. (1994). Teaching algebraic coding: Stakes, difficulties, and suggestions.
*American Journal of Physics, 62*(8), 723–727.CrossRefGoogle Scholar - Redish, E. F. (1999). Millikan lecture 1998: Building a science of teaching physics.
*American Journal of Physics, 67*(7), 562–573.CrossRefGoogle Scholar - Redish, E. F., & Smith, K. A. (2008). Looking beyond content: Skill development for engineers.
*Journal of Engineering Education, 97*(3), 295–307.CrossRefGoogle Scholar - Reif, F. (1965).
*Fundamentals of statistical and thermal physics McGraw-Hill series in fundamentals of physics*. New York: Mcgraw-Hill Book Company.Google Scholar - Reif, F. (1967).
*Statistical physics: Berkeley physics course*(Vol. 5). New York: Mcgraw-Hill Book Company.Google Scholar - Reif, F. (1995, December). Understanding and teaching important scientific thought processes.
*Journal of Science Education and Technology, 4*(4), 261–282.CrossRefGoogle Scholar - Rozier, S., & Viennot, L. (1991). Students’ reasoning in thermodynamics.
*International Journal of Science Education, 13*, 159–170.CrossRefGoogle Scholar - Sichau, C. (2000). Practicing helps: Thermodynamics, history, and experiment.
*Science & Education, 9*(4), 389–398.CrossRefGoogle Scholar