- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Physics Education Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Students’ understanding of the 2D heat equation: An action, process, object, schema approach

Maria Al Dehaybes1, Johan Deprez2, Paul van Kampen3, and Mieke De Cock1

  • 1LESEC and Department of Physics and Astronomy, Faculty of Science, KU Leuven, 3001 Leuven, Belgium
  • 2LESEC and Department of Mathematics, Faculty of Science, KU Leuven, 3001 Leuven, Belgium
  • 3CASTeL and School of Physical Sciences, Dublin City University, Dublin 9, Ireland

PDF

Phys. Rev. Phys. Educ. Res. 22, 010139 – Published 1 May, 2026

DOI: https://doi.org/10.1103/w1xc-mxjh

Abstract

In this paper, we use the action, process, object, schema theoretical framework to validate a hypothetical learning trajectory of the 2D heat equation based on a preliminary genetic decomposition that stresses the conceptual understanding of its mathematical formulation. We design questions to probe specific mental constructions of the preliminary genetic decomposition. We interviewed eight second-year B.Sc. students enrolled in engineering, physics, or twin (mathematics and physics) majors. Our findings indicate that students engage with many predicted mental constructions. In particular, the coordination and encapsulation of two process conceptions of the Laplacian of temperature improve understanding, although doing so is challenging. Other parts of the genetic decomposition require refinements. These include mental constructions related to the temperature distribution function, heat flow, and temperature gradient.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (32)

  1. B. R. Wilcox, M. D. Caballero, D. A. Rehn, and S. J. Pollock, Analytic framework for students’ use of mathematics in upper-division physics, Phys. Rev. ST Phys. Educ. Res. 9, 020119 (2013).
  2. M. D. Caballero, B. R. Wilcox, L. Doughty, and S. J. Pollock, Unpacking students’ use of mathematics in upper-division physics: Where do we go from here?, Eur. J. Phys. 36, 065004 (2015).
  3. R. R. Bajracharya, V. L. Sealey, and J. R. Thompson, Student understanding of the sign of negative definite integrals in mathematics and physics, Int. J. Res. Undergrad. Math. Educ. 9, 62 (2023).
  4. R. Karam, O. Uhden, and D. Höttecke, The “math as prerequisite” illusion: Historical considerations, and implications for physics teaching BT—Mathematics in physics education, in Mathematics in Physics Education, edited by G. Pospiech, M. Michelini, and B.-S. Eylon (Springer International Publishing, Cham, 2019), pp. 37–52, 10.1007/978-3-030-04627-9_2.
  5. O. Uhden, R. Karam, M. Pietrocola, and G. Pospiech, Modelling mathematical reasoning in physics education, Sci. Educ. 21, 485 (2012).
  6. L. Bollen, P. Van Kampen, and M. De Cock, Students’ difficulties with vector calculus in electrodynamics, Phys. Rev. ST Phys. Educ. Res. 11, 020129 (2015).
  7. M. De Cock, Representation use and strategy choice in physics problem solving, Phys. Rev. ST Phys. Educ. Res. 8, 020117 (2012).
  8. Concepts, Strategies and Models to Enhance Physics Teaching and Learning, edited by E. Mcloughlin and P. Van Kampen (Springer Nature, Switzerland, 2019).
  9. M. De Cock, T. Deprez, S. Gijsen, and J. Deprez, Investigating student understanding of cross products in a mathematical and two electromagnetism contexts, Phys. Rev. Phys. Educ. Res. 15, 020132 (2019).
  10. M. Al Dehaybes, J. Deprez, P. van Kampen, and M. De Cock, Students’ understanding of Laplacian, and gradient in mathematics, and physics context, in The Learning, and Teaching of Calculus Across Disciplines, edited by T. Dreyfus, A. S. Gonzalez-Marin, J. Monaghan, E. Nardi, and P. W. Thompson (2023), pp. 85–88 [Online], Available at https://matriccalcconf2.sciencesconf.org/.
  11. Van Den Eynde Sofie, Blending of mathematics and physics Undergraduate students’ reasoning in the context of the heat equation (KU Leuven, 2021) [Online], Available: https://kuleuven.limo.libis.be/permalink/32KUL_KUL/cro8e8/alma9993054108401488%0A
  12. M. Asiala, A. Brown, D. Devries, D. Mathews, and K. Thomas, A framework for research and curriculum development in undergraduate mathematics education, in Research in Collegiate Mathematics Education. II (1996), pp. 1–32, 10.1090/cbmath/006/01.
  13. M. Al Dehaybes, J. Deprez, P. van Kampen, and M. De Cock, Extending APOS to physics education: A genetic decomposition of the 2D heat equation, Phys. Rev. Phys. Educ. Res. 21, 020132 (2025).
  14. I. Arnon, J. Cotrill, E. Dubinsky, A. Oktaç, S. Roa Fuentes, M. Trigueros, and K. Weller, APOS Theory: A framework for research and curriculum development, in Mathematics Education (Springer, New York, 2014), 10.1007/978-1-4614-7966-6.
  15. R. Martínez-Planell and M. Trigueros, Using cycles of research in APOS: The case of functions of two variables, J. Math. Behav. 55, 100687 (2019).
  16. S. Van Den Eynde, B. P. Schermerhorn, J. Deprez, M. Goedhart, J. R. Thompson, and M. De Cock, Dynamic conceptual blending analysis to model student reasoning processes while integrating mathematics and physics: A case study in the context of the heat equation, Phys. Rev. Phys. Educ. Res. 16, 010114 (2020).
  17. W. M. Christensen and J. R. Thompson, Investigating student understanding of physics concepts and the underlying calculus concepts in thermodynamics, in Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education, Raleigh, NC (2010), http://sigmaa.maa.org/rume/crume2010/Archive/Thompson%20&%20Christensen.pdf.
  18. J. Tuminaro, A cognitive framework for analyzing and describing introductory students’ use and understanding of mathematics in physics, doctoral thesis, University of Maryland, College Park, MD, 2004.
  19. E. F. Redish, Problem solving and the use of math in physics courses, in Proceedings of the World View on Physics Education in 2005, Delhi (Cornell University, 2006), arXiv:physics/0608268.
  20. L. Bollen, P. van Kampen, C. Baily, and M. De Cock, Qualitative investigation into students’ use of divergence and curl in electromagnetism, Phys. Rev. Phys. Educ. Res. 12, 020134 (2016).
  21. S. Van den Eynde, J. Deprez, M. Goedhart, and M. De Cock, Undergraduate students’ difficulties with boundary conditions for the diffusion equation, Int. J. Math. Educ. Sci. Technol. 53, 2176 (2022).
  22. K. Carlton, Teaching about heat and temperature, Phys. Educ. 35, 101 (2000).
  23. F. N. Nabilah, J. Maknun, M. Muslim, A. Samsudin, L. Hasanah, and A. Suhandi, Eleventh-grade student’s conceptions about temperature and heat, J. Phys. Conf. Ser. 1280, 052067 (2019).
  24. J. Yeo, E. Lim, K. C. D. Tan, and Y. S. Ong, The efficacy of an image-to-writing approach to learning abstract scientific concepts: Temperature and heat, Int. J. Sci. Math. Educ. 19, 21 (2021).
  25. D. T. Brookes and E. Etkina, The importance of language in students’ reasoning about heat in thermodynamic processes, Int. J. Sci. Educ. 37, 759 (2015).
  26. S. Van den Eynde, M. Goedhart, J. Deprez, and M. De Cock, Role of graphs in blending physical and mathematical meaning of partial derivatives in the context of the heat equation, Int. J. Sci. Math. Educ. 21, 25 (2023).
  27. S. Ainsworth, DeFT: A conceptual framework for considering learning with multiple representations, Learn. Instr. 16, 183 (2006).
  28. M. Trigueros and R. Martínez-Planell, Geometrical representations in the learning of two-variable functions, Educ. Stud. Math. 73, 3 (2010).
  29. R. Martínez-Planell and M. Trigueros Gaisman, Students’ understanding of the general notion of a function of two variables, Educ. Stud. Math. 81, 365 (2012).
  30. R. Martínez-Planell and M. T. Gaisman, Graphs of functions of two variables: Results from the design of instruction, Int. J. Math. Educ. Sci. Technol. 44, 663 (2013).
  31. M. Al Dehaybes, J. Deprez, P. van Kampen, and M. De Cock, Students’ understanding of two-variable calculus concepts in mathematics, and physics contexts. II. The gradient, and the Laplacian, Phys. Rev. Phys. Educ. Res. 21, 010132 (2025).
  32. P. Klein, J. Viiri, and J. Kuhn, Visual cues improve students’ understanding of divergence and curl: Evidence from eye movements during reading and problem solving, Phys. Rev. Phys. Educ. Res. 15, 010126 (2019).

Outline

Information

Phys. Rev. Phys. Educ. Res. 22, 010139– Published 1 May, 2026
  • Received 28 August 2025
  • Accepted 20 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation