Mathematics in Physics Education pp 127-151 | Cite as

# Blending Physical Knowledge with Mathematical Form in Physics Problem Solving

## Abstract

For physicists, equations are about more than computing physical quantities or constructing formal models; they are also about understanding. The conceptual systems physicists use to think about nature are made from many different resources, formal and not, working together and inextricably linked. By blending mathematical forms and physical intuition, physicists breathe meaning into the equations they use, and this process is fundamental to what it means for an expert to understand something. In contrast, in physics class, novice students often treat mathematics as only a calculational tool, isolating it from their rich knowledge of the physical world. We are interested in cases where students break that pattern by reading, manipulating, and building equations meaningfully rather than purely formally. To find examples of this and explore the diversity of ways students combine formal and intuitive resources, we conducted problem-solving interviews with students in an introductory physics for life sciences class. During the interviews, we scaffolded student use of strategies which call for both formal and intuitive reasoning, such as “examine the extreme cases” and “think about the dimensions”. We use the analytic framework of epistemic games to model how students used the strategies and how they accessed problem-solving resources, and we present evidence that novice students using these strategies accessed more expert-like conceptual systems than those typically described in problem-solving literature. They blended physical intuition with mathematical symbolic templates, reconceptualized the nature of variables and equations, and distinguished superficially-similar functional forms. Once introduced to a strategy, students sometimes applied it to new scenarios spontaneously or applied it in new ways to the present scenario, acknowledging it as a useful, general purpose problem-solving technique. Our data suggests that these strategies can potentially help novice students learn to develop and apply their physical intuition more effectively.

## References

- Adams, W. K., Perkins, K. K., Podolefsky, N. S., Dubson, M., Finkelstein, N. D., & Wieman, C. E. (2006). New instrument for measuring student beliefs about physics and learning physics: The Colorado learning attitudes about science survey.
*Physical Review Special Topicsphysics Education Research, 2*(1), 010101.CrossRefGoogle Scholar - Airey, J., & Linder, C. (2009). A disciplinary discourse perspective on university science learning: Achieving fluency in a critical constellation of modes.
*Journal of Research in Science Teaching, 46*(1), 27–49.CrossRefGoogle Scholar - Ataide, R., & Greca, I. (this volume). Pre-service physics teachers’ theorems-in-action about problem solving and its relation with epistemic views on the relationship between physics and mathematics in understanding physics. In G. Pospiech (Ed.),
*Mathematics in physics education research*. Cham: Springer.Google Scholar - Bender, C. M., & Orszag, S. A. (1999).
*Advanced mathematical methods for scientists and engineers I*. New York: Springer.CrossRefGoogle Scholar - Bing, T. J., & Redish, E. F. (2009). Analyzing problem solving using math in physics: Epistemological framing via warrants.
*Physical Review Special Topics-Physics Education Research, 5*(2), 020108.CrossRefGoogle Scholar - Bing, T. J., & Redish, E. F. (2012). Epistemic complexity and the journeyman-expert transition.
*Physical Review Special Topics-Physics Education Research, 8*(1), 010105.CrossRefGoogle Scholar - Bridgman, P. W. (1922).
*Dimensional analysis*. New Haven: Yale University Press.Google Scholar - Carlone, H. B., Scott, C. M., & Lowder, C. (2014). Becoming (less) scientific: A longitudinal study of students’ identity work from elementary to middle school science.
*Journal of Research in Science Teaching, 51*(7), 836–869.CrossRefGoogle Scholar - Chiang, E.
*Astronomy 250: Order-of-magnitude physics*. http://w.astro.berkeley.edu/ ~echiang/oom/oom.html . Accessed: 2017-05-04. - Clement, J. J., & Stephens, L. (2009).
*Extreme case reasoning and model based learning experts and students*. In Proceedings of the 2009 Annual Meeting of the National Association for Research in Science Learning.Google Scholar - Collins, A., & Ferguson, W. (1993). Epistemic forms and epistemic games: Structures and strategies to guide inquiry.
*Educational Psychologist, 28*(1), 25–42.CrossRefGoogle Scholar - Giancoli, D. C. (2000).
*Physics for scientists and engineers*(Vol. 3). Upper Saddle River: Prentice Hall.Google Scholar - Goffman, E. (1974).
*Frame analysis: An essay on the organization of experience*. Cambridge, MA: Harvard University Press.Google Scholar - Gupta, A., & Elby, A. (2011). Beyond epistemological deficits: Dynamic explanations of engineering students’ difficulties with mathematical sense-making.
*International Journal of Science Education, 33*(18), 2463–2488.CrossRefGoogle Scholar - Hammer, D. (2000, July). Student resources for learning introductory physics.
*American Journal of Physics*,*68*(S1):S52. ISSN: 00029505. doi: https://doi.org/10.1119/1.19520. http://link.aip.org/link/ ?AJP/68/S52/1{&}Agg=doi . CrossRefGoogle Scholar - Hammer, D., & Elby, A. (2003). Tapping epistemological resources for learning physics.
*The Journal of the Learning Sciences, 12*(1), 53–90.CrossRefGoogle Scholar - Hassani, S. (2013).
*Mathematical physics: A modern introduction to its foundations*. Cham: Springer.CrossRefGoogle Scholar - Heile, F. (2015).
*Why is the force between two charged particles eerily similar to the force between two large masses?*https://www.quora.com/Why-is-the-force-between-two-charged-particles-eerily-similar-to-the-force-between-two-large-ma answer/Frank-Heile. Online. Accessed 14 December 2015. - Hsu, L., Brewe, E., Foster, T. M., & Harper, K. A. (2004). Resource letter rps-1: Research in problem solving.
*American Journal of Physics, 72*(9), 1147–1156.CrossRefGoogle 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*(1), 32–57.CrossRefGoogle Scholar - Kustusch, M. B., Roundy, D., Dray, T., & Manogue, C. A. (2014). Partial derivative games in thermodynamics: A cognitive task analysis.
*Physical Review Special Topics – Physics Education Research, 10*(1), 1–16. ISSN: 15549178. 10.1103/PhysRevSTPER.10.010101.CrossRefGoogle Scholar - Mehan, H. (1979).
*Learning lessons*. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar - Meltzer, D. E., & Otero, V. K. (2015). A brief history of physics education in the United States.
*American Journal of Physics, 83*(5), 447–458.CrossRefGoogle Scholar - Modir, B., Irving, P. W., Wolf, S. F., & Sayre, E. C. (2014).
*Learning about the energy of a hurricane system through an estimation epistemic game*. In Proceedings of the 2014 physics education research conference, pp. 3–6. https://doi.org/10.1119/perc.2014.pr.044. - Morin, D. (2008).
*Introduction to classical mechanics: With problems and solutions*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Nave, R. (2017).
*Coulomb’s law*. http://hyperphysics.phy-astr.gsu.edu/hbase/ electric/elefor.html . Online. Accessed 10 May 2017. - Nearing, J. C. (2003).
*Mathematical tools for physics*. New York: Dover Publications.Google Scholar - Phinney, S.
*Ph 101 order of magnitude physics*. https://www.its.caltech.edu/oom/ . Accessed: 2017-05-04. - Redish, E. F. (2004).
*A theoretical framework for physics education research: Modeling student thinking*. arXiv preprint physics/0411149.Google Scholar - Redish, E. F., & Cooke, T. J. (2013). Learning each other’s ropes: Negotiating interdisciplinary authenticity.
*CBE-Life Sciences Education, 12*(2), 175–186.CrossRefGoogle Scholar - Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology.
*Science & Education, 24*(5–6), 561–590.CrossRefGoogle Scholar - Redish, E. F., Saul, J. M., & Steinberg, R. N. (1998). Student expectations in introductory physics.
*American Journal of Physics, 66*(3), 212–224.CrossRefGoogle Scholar - Redish, E. F., Singh, C., Sabella, M. , & Rebello, S. (2010).
*Introducing students to the culture of physics: Explicating elements of the hidden curriculum*. In AIP conference proceedings (Vol. 1289, pp. 49–52). AIP.Google Scholar - Redish, E. F., Bauer, C., Carleton, K. L., Cooke, T. J., Cooper, M., Crouch, C. H., Dreyfus, B. W., Geller, B. D., Giannini, J., Svoboda Gouvea, J., et al. (2014). Nexus/physics: An interdisciplinary repurposing of physics for biologists.
*American Journal of Physics, 82*(5), 368–377.CrossRefGoogle Scholar - Robinett, R. W. (2015). Dimensional analysis as the other language of physics.
*American Journal of Physics, 83*(4), 353–361.CrossRefGoogle Scholar - Russ, R. S., Coffey, J. E., Hammer, D., & Hutchison, P. (2009). Making classroom assessment more accountable to scientific reasoning: A case for attending to mechanistic thinking.
*Science Education, 93*(5), 875–891.CrossRefGoogle Scholar - Schoenfeld, A. H., & Sloane, A. H. (2016).
*Mathematical thinking and problem solving*. London: Routledge.CrossRefGoogle Scholar - Serway, R. A., & Jewett, J. W. (2004).
*Physics for scientists and engineers*. ThomsonBrooks/Cole, 6th edn. ISBN: 9780534408428, 0495142425, 0534408427, 9780495142423.Google Scholar - Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*(1), 1–36.CrossRefGoogle Scholar - Sherin, B. L. (2001). How students understand physics equations.
*Cognition and Instruction, 19*(4), 479–541.CrossRefGoogle Scholar - Spike, B. T., & Finkelstein, N. D. (2016). Design and application of a framework for examining the beliefs and practices of physics teaching assistants.
*Physical Review Physics Education Research, 12*(1), 010114.CrossRefGoogle Scholar - Tipler, P. A., & Mosca, G. (2007).
*Physics for scientists and engineers*. London: Macmillan.Google Scholar - Townsend, J. S. (2000).
*A modern approach to quantum mechanics*. Sausalito: University Science Books.Google Scholar - 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*(2), 020101.CrossRefGoogle Scholar - Weinberg, S. (1967). A model of leptons.
*Physical Review Letters, 19*(21), 1264.CrossRefGoogle Scholar - Weinstein, L. (2018).
*Fermi questions*. URL https://aapt.scitation.org/topic/ collections/fermi-questions . Online. Accessed 20 May 2018. - Wikipedia. (2016).
*Yukawa potential — wikipedia, the free encyclopedia.*https://en. wikipedia.org/w/index.php?title=Yukawa_potential&oldid=715463242 . Online. Accessed 3 October 2016. - Wilcox, B. R., Caballero, M. D., Rehn, D. A., & Pollock, S. J. (2013). Analytic framework for students’ use of mathematics in upper-division physics.
*Physical Review Special Topics Physics Education Research*,*9*(2). ISSN: 15549178. doi: https://doi.org/10.1103/PhysRevSTPER.9.020119. - Wittmann, M. C., & Black, K. E. (2015). Mathematical actions as procedural resources: An example from the separation of variables.
*Physical Review Special Topics – Physics Education Research*,*11*(2):020114. ISSN: 1554-9178. https://doi.org/10.1103/PhysRevSTPER.11.020114. http://link.aps.org/doi/10.1103/PhysRevSTPER.11.020114. - Zee, A. (2010).
*Quantum field theory in a nutshell*. Princeton: Princeton University Press.Google Scholar