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Should quantum physics be approached with mathematical formalism in school? Effects of instructional approach on functional understanding under cognitive and affective controls

Fabian Hennig*, Kristóf Tóth, Joaquin Veith, Malte Ubben, and Philipp Bitzenbauer

  • *Contact author: fabian.hennig@uni-leipzig.de
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Phys. Rev. Phys. Educ. Res. 22, 020158 – Published 5 November, 2026

DOI: https://doi.org/10.1103/6vyb-dbvj

Abstract

Evidence from physics education research suggests that the introduction of mathematical formalism in quantum physics instruction can support the development of students’ conceptual understanding, particularly by promoting functional ways of reasoning about quantum phenomena. However, empirical findings on the effectiveness of such approaches at the secondary school level remain inconclusive, and little is known about the extent to which learning outcomes are shaped by instructional design as opposed to learners’ affective and cognitive characteristics. This study addresses this gap by investigating the relative influence of instructional approach on students’ understanding of single-photon states, while controlling for prior knowledge, cognitive load, situational interest, and physics self-concept. A quasiexperimental pre-post design was implemented with N=172 upper secondary students, comparing a qualitative instructional condition with an approach supplemented by a reduced Dirac formalism. Learning outcomes were assessed using the framework of Functional Fidelity and Fidelity of Gestalt. A multivariate analysis of covariance, followed by univariate analyses of covariance, was conducted to examine the contributions of instructional and learner-related variables. The findings indicate that affective variables emerge as strong predictors of functional understanding and suggest that learning in quantum physics is driven less by instructional format than by learners’ affective and cognitive prerequisites. In particular, fostering interest appears to be a key mechanism for supporting the development of functional understanding. The results highlight the importance of integrating affective dimensions into instructional design and provide a basis for future research on the interaction between instructional approaches and learner characteristics in quantum physics education.

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References (82)

  1. G. Pospiech, Quantum cryptography as an approach for teaching quantum physics, in Teaching-Learning Contemporary Physics: From Research to Practice (Springer, New York, 2021), pp. 19–31.
  2. H. K. E. Stadermann, E. van den Berg, and M. J. Goedhart, Analysis of secondary school quantum physics curricula of 15 different countries: Different perspectives on a challenging topic, Phys. Rev. Phys. Educ. Res. 15, 010130 (2019).
  3. M. Ubben, Typisierung Des Verständnisses Mentaler Modelle Mittels Empirischer Datenerhebung Am Beispiel der Quantenphysik (Volume 304) (Logos Verlag Berlin, Berlin, Germany, 2020).
  4. K. Krijtenburg-Lewerissa, H. J. Pol, A. Brinkman, and W. R. van Joolingen, Insights into teaching quantum mechanics in secondary and lower undergraduate education, Phys. Rev. Phys. Educ. Res. 13, 010109 (2017).
  5. F. Hennig, K. Tóth, J. Veith, and P. Bitzenbauer, Introducing quantum physics concepts and dirac notation at the secondary school level: Insights into student reasoning from an acceptance survey, Phys. Rev. Phys. Educ. Res. 20, 020147 (2024).
  6. T. Bouchée, L. de Putter - Smits, M. Thurlings, and B. Pepin, Towards a better understanding of conceptual difficulties in introductory quantum physics courses, Stud. Sci. Educ. 58, 183 (2022).
  7. A. Messiah, Quantum Mechanics: Quantum Mechanics (North-Holland publishing company, Amsterdam, 1964), Vol. 1.
  8. C. Stefani and G. Tsaparlis, Students’ levels of explanations, models, and misconceptions in basic quantum chemistry: A phenomenographic study, J. Res. Sci. Teach. 46, 520 (2009).
  9. M. S. Ubben and S. Heusler, Gestalt and functionality as independent dimensions of mental models in science, Res. Sci. Educ. 51, 1349 (2019).
  10. M. S. Ubben and P. Bitzenbauer, Two cognitive dimensions of students’ mental models in science: Fidelity of gestalt and functional fidelity, Educ. Sci. 12, 163 (2022).
  11. P. Bitzenbauer and M. S. Ubben, The structure of learners’ perceptions of models (not only) in quantum physics: spotlight on fidelity of gestalt and functional fidelity, EPJ Quantum Technol. 12, 16 (2025)
  12. M. Ubben and P. Bitzenbauer, Exploring the relationship between students’ conceptual understanding and model thinking in quantum optics, Front. Quantum Sci. Technol. 2, 1207619 (2023).
  13. C. Baily and N. D. Finkelstein, Teaching and understanding of quantum interpretations in modern physics courses, Phys. Rev. Phys. Educ. Res. 11, 020124 (2015).
  14. S. B. McKagan, K. K. Perkins, and C. E. Wieman, Developing and researching phet simulations for teaching quantum mechanics, Phys. Rev. ST Phys. Educ. Res. 4, 010103 (2008).
  15. C. Wieman and K. Perkins, Transforming physics education, Phys. Today 58, 36 (2005).
  16. M. S. Ubben, J. M. Veith, A. Merzel, and P. Bitzenbauer, Quantum science in a nutshell: Fostering students’ functional understanding of models, Front. Educ. 8, 1192708 (2023).
  17. F. Hennig, K. Tóth, J. Veith, and P. Bitzenbauer, Mathematical sense making of quantum phenomena using dirac notation: Its effect on secondary school students’ functional thinking about photons, EPJ Quantum Technol. 11, 61 (2024).
  18. C. Baily and N. D. Finkelstein, Refined characterization of student perspectives on quantum physics, Phys. Rev. ST Phys. Educ. Res. 6, 010101 (2010).
  19. J. Sweller, Cognitive load during problem solving: Effects on learning, Cogn. Sci. 12, 257 (1988).
  20. F. Paas, A. Renkl, and J. Sweller, Cognitive load theory and instructional design: Recent developments, Educ. Psychol. 38, 1 (2010).
  21. S. Hidi and K. A. Renninger, The four-phase model of interest development, Educ. Psychol. 41, 111 (2006).
  22. H. W. Marsh and R. G. Craven, Reciprocal effects of self-concept and performance from a multidimensional perspective: Beyond seductive pleasure and unidimensional perspectives, Perspect. Psychol. Sci. 1, 133 (2006).
  23. J. D. Gifford and N. D. Finkelstein, Categorical framework for mathematical sense making in physics, Phys. Rev. Phys. Educ. Res. 16, 020121 (2020).
  24. M. Michelini, S. Faletič, and G. Pospiech, Approaches and teaching resources for teacher education in quantum physics, in Physics Teacher Education: More About What Matters, edited by J. Borg Marks and P. Galea (Springer Nature Switzerland, Cham, 2023), pp. 77–91.
  25. R. Scholz, S. Wessnigk, and K.-A. Weber, A classical to quantum transition via key experiments, Eur. J. Phys. 41, 055304 (2020).
  26. P. P. Angara, U. Stege, A. MacLean, H. A. Muller, and T. Markham, Teaching quantum computing to high-school-aged youth: A hands-on approach, IEEE Trans. Quantum Eng. 3, 1 (2021).
  27. S. Faletič, A double well on-line simulation and activities for active learning of introductory quantum mechanics, Eur. J. Phys. 41, 045706 (2020).
  28. F. Hennig, K. Tóth, M. Förster, and P. Bitzenbauer, A new teaching-learning sequence to promote secondary school students’ learning of quantum physics using dirac notation, Phys. Educ. 59, 045007 (2024).
  29. K. Tóth and P. Bitzenbauer, A teaching approach to quantum computing at the secondary school level, Eurasia J. Math. Sci. Technol. Educ. 21, em2736 (2025).
  30. P. A. M. Dirac, A new notation for quantum mechanics, in Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge University Press, Cambridge, England, 1939), Vol. 35, pp. 416–418.
  31. E. Gire and E. Price, Structural features of algebraic quantum notations, Phys. Rev. ST Phys. Educ. Res. 11, 020109 (2015).
  32. W. Dür and S. Heusler, What we can learn about quantum physics from a single qubit, arXiv:1312.1463.
  33. P. Bitzenbauer, S. Faletič, M. Michelini, K. Tóth, and G. Pospiech, Design and evaluation of a questionnaire to assess learners’ understanding of quantum measurement in different two-state contexts: The context matters, Phys. Rev. Phys. Educ. Res. 20, 020136 (2024).
  34. K. Tóth, S. Faletič, M. Michelini, G. Pospiech, A. Betti, M. Nicolini, M. Parmiggiani, J. Veith, and P. Bitzenbauer, Investigating the effect of two-state approaches on students’ understanding of quantum measurement: A quasiexperimental field study, Phys. Rev. Phys. Educ. Res. 21, 020142 (2025).
  35. M. Michelini and A. Stefanel, A path to build basic quantum mechanics ideas in the context of light polarization and learning outcomes of secondary students, in Journal of Physics: Conference Series (IOP Publishing, Bristol, UK, 2021), Vol. 1929, p. 012052.
  36. K. Tóth, M. Michelini, and P. Bitzenbauer, From light polarization to quantum physics: Supporting lower secondary school students’ transition from gestalt to functional thinking, Eurasia J. Math. Sci. Technol. Educ. 20, em2449 (2024).
  37. G. Zuccarini, C. Sutrini, M. Bondani, C. Macchiavello, and M. Malgieri, Teaching quantum information science to secondary school students with photon polarization and which-path encoding, EPJ Quantum Technol. 11, 74 (2024).
  38. K. Tóth and T. Tél, Quantum uncertainty: What to teach?, Phys. Educ. 58, 025019 (2023).
  39. M. Bondani, Single-photon mach-zehnder interferometry for high schools, in Journal of Physics: Conference Series (IOP Publishing, Bristol, UK, 2021), Vol. 1929, p. 012055.
  40. R. Müller and O. Mishina, Milq—quantum physics in secondary school, in Teaching-Learning Contemporary Physics: From Research to Practice (Springer, New York, 2021), pp. 33–45.
  41. R. Sayer, A. Maries, and C. Singh, Quantum interactive learning tutorial on the double-slit experiment to improve student understanding of quantum mechanics, Phys. Rev. Phys. Educ. Res. 13, 010123 (2017).
  42. E. van den Berg, A. van Rossum, J. Grijsen, H. Pol, and J. van der Veen, Teaching particle-wave duality with double-slit single-photon interference in Dutch secondary schools, in Research and Innovation in Physics Education: Two Sides of the Same Coin, edited by J. Guisasola and K. Zuza (Springer International Publishing, Cham, 2020), pp. 135–143.
  43. M. Förster and G. Pospiech, Zur rolle mathematischer repräsentationen für das verständnis quantenphysikalischer prinzipien, PhyDid B-Didaktik der Physik-Beiträge Zur DPG-Frühjahrstagung (2024).
  44. R. P. Feynman, QED: The Strange Theory of Light and Matter (Princeton University Press, Princeton, New Jersey, USA, 2014).
  45. K. A. Renninger and S. Hidi, Revisiting the conceptualization, measurement, and generation of interest, Educ. Psychol. 46, 168 (2011).
  46. J. I. Rotgans and H. G. Schmidt, Situational interest and academic achievement in the active-learning classroom, Learn. Instr. 21, 58 (2011).
  47. M. Ainley, S. Hidi, and D. Berndorff, Interest, learning, and the psychological processes that mediate their relationship, J. Educ. Psychol. 94, 545 (2002).
  48. M. Niemivirta, A. Tapola, H. Tuominen, and J. Viljaranta, Developmental trajectories of school-beginners’ ability self-concept, intrinsic value and performance in mathematics, Br. J. Educ. Psychol. 94, 441 (2024).
  49. A. Grund, I. M. Pit-ten Cate, and A. Fischbach, (How) do self-concept, interest, and conscientiousness function together in academic motivation? A typological approach developed and replicated in two large-scale samples, Learn. Instr. 90, 101868 (2024).
  50. H. W. Marsh and R. Shavelson, Self-concept: Its multifaceted, hierarchical structure, Educ. Psychol. 20, 107 (1985).
  51. B. C. Hansford and J. A. Hattie, The relationship between self and achievement/performance measures, Rev. Educ. Res. 52, 123 (1982).
  52. H. W. Marsh, R. Pekrun, K. Murayama, A. K. Arens, P. D. Parker, J. Guo, and T. Dicke, An integrated model of academic self-concept development: Academic self-concept, grades, test scores, and tracking over 6 years., Dev. Psychol. 54, 263 (2018).
  53. J. Woithe, A. Müller, S. Schmeling, and J. Kuhn, Motivational outcomes of the science outreach lab S’Cool LAB at CERN: A multilevel analysis, J. Res. Sci. Teach. 59, 930 (2022).
  54. D. Anttila, A. Lehtinen, and P. Koskinen, Can a one-day event trigger interest in quantum physics at the university level?, Eur. J. Phys. 45, 045708 (2024).
  55. M. Darienzo, A. M. Kelly, D. Schneble, and T.-C. Wei, Student attitudes toward quantum information science and technology in a high school outreach program, Phys. Rev. Phys. Educ. Res. 20, 020126 (2024).
  56. M. T. H. Chi, P. J. Feltovich, and R. Glaser, Categorization and representation of physics problems by experts and novices, Cogn. Sci. 5, 121 (2010).
  57. P. A. Alexander, Domain knowledge: Evolving themes and emerging concerns, Educ. Psychol. 27, 33 (1992).
  58. J. Sweller, J. J. G. van Merrienboer, and F. G. W. C. Paas, Cognitive architecture and instructional design, Educ. Psychol. Rev. 10, 251 (1998).
  59. J. Sweller, Cognitive load theory, in Psychology of Learning and Motivation (Elsevier, New York, 2011), Vol. 55, pp. 37–76.
  60. J. J. G. van Merriënboer and J. Sweller, Cognitive load theory and complex learning: Recent developments and future directions, Educ. Psychol. Rev. 17, 147 (2005).
  61. B. A. Simonsmeier, M. Flaig, A. Deiglmayr, L. Schalk, and M. Schneider, Domain-specific prior knowledge and learning: A meta-analysis, Educ. Psychol. 57, 31 (2022).
  62. T. Binder, A. Sandmann, B. Sures, G. Friege, H. Theyssen, and P. Schmiemann, Assessing prior knowledge types as predictors of academic achievement in the introductory phase of biology and physics study programmes using logistic regression, Int. J. STEM Educ. 6, 33 (2019).
  63. D. P. Ausubel, J. D. Novak, H. Hanesian et al., Educational Psychology: A Cognitive View (Holt, Rinehart and Winston, New York, 1978).
  64. T. G. K. Bryce and E. J. Blown, Ausubel’s meaningful learning re-visited, Curr. Psychol. 43, 4579 (2024).
  65. R. Duit and D. F. Treagust, Conceptual change: A powerful framework for improving science teaching and learning, Int. J. Sci. Educ. 25, 671 (2003).
  66. S. Vosniadou et al., International Handbook of Research on Conceptual Change (Routledge, New York, 2008), Vol. 259.
  67. T. Anderson and J. Shattuck, Design-based research: A decade of progress in education research?, Educ. Res. 41, 16 (2012).
  68. P. Bronner, A. Strunz, C. Silberhorn, and J.-P. Meyn, Interactive screen experiments with single photons, Eur. J. Phys. 30, 345 (2009).
  69. J. Leppink, F. Paas, C. P. M. Van der Vleuten, T. Van Gog, and J. J. G. Van Merriënboer, Development of an instrument for measuring different types of cognitive load, Behav. Res. Methods 45, 1058 (2013).
  70. P. Häussler and L. Hoffmann, An intervention study to enhance girls’ interest, self-concept, and achievement in physics classes, J. Res. Sci. Teach. 39, 870 (2002).
  71. J. Woithe, Designing, Measuring and Modelling the Impact of the Hands-on Particle Physics Learning Laboratory S’Cool LAB at CERN (VerlagDr. Hut, Munich, Germany, 2020).
  72. J. Stevens et al., Applied Multivariate Statistics for the Social Sciences (Lawrence Erlbaum Associates, Mahwah, NJ, 2002), Vol. 4.
  73. A. Field, Discovering Statistics Using IBM SPSS Statistics (Sage Publications Limited, London, UK, 2024).
  74. C. L. Olson, Comparative robustness of six tests in multivariate analysis of variance, J. Am. Stat. Assoc. 69, 894 (1974).
  75. P. Justice, E. Marshman, and C. Singh, Impact of mathematical reasoning on students’ understanding of quantum optics, in 2022 Physics Education Research Conference Proceedings. College Park: American Association of Physics Teachers (American Association of Physics Teachers (AAPT), College Park, MD, USA, 2022), pp. 261–266.
  76. H. W. Marsh and A. J. Martin, Academic self-concept and academic achievement: Relations and causal ordering, Br. J. Educ. Psychol. 81, 59 (2011).
  77. J. S. Eccles and A. Wigfield, Motivational beliefs, values, and goals, Annu. Rev. Psychol. 53, 109 (2002).
  78. U. Schiefele, Interest, learning, and motivation, Educ. Psychol. 26, 299 (1991).
  79. L. K. Fryer and M. Ainley, Supporting interest in a study domain: A longitudinal test of the interplay between interest, utility-value, and competence beliefs, Learn. Instr. 60, 252 (2019).
  80. K. Schnitzler, D. Holzberger, and T. Seidel, All better than being disengaged, Eur. J. Psychol. Educ. 36, 627 (2021).
  81. S. Majidy, Addressing misconceptions in university physics: A review and experiences from quantum physics educators, arXiv:2405.20923.
  82. M. Knogler, J. M. Harackiewicz, A. Gegenfurtner, and D. Lewalter, How situational is situational interest? investigating the longitudinal structure of situational interest, Contemp. Educ. Psychol. 43, 39 (2015).

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Phys. Rev. Phys. Educ. Res. 22, 020158– Published 5 November, 2026
  • Received 16 April 2026
  • Accepted 29 September 2026
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