- 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

Pedagogical moves related to analogy that support a unified understanding of eigentheory concepts in a quantum mechanics class

Kaitlyn Stephens Serbin

Megan Wawro

PDF

Phys. Rev. Phys. Educ. Res. 20, 020137 – Published 30 October, 2024

DOI: https://doi.org/10.1103/PhysRevPhysEducRes.20.020137

Abstract

[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] It is beneficial for quantum mechanics students to have a unified understanding of eigentheory concepts, so they can recognize the shared structure of mathematized phenomena from the different quantum mechanical systems of spin, energy, or position and recognize those as instantiations of the same overarching concept. Quantum mechanics instructors should, therefore, provide opportunities for their class community to develop a shared unified understanding of eigentheory concepts. One such opportunity can arise by engaging students in analogizing eigentheory concepts in one context with those from another context. We investigate the pedagogical moves related to analogies that can be used by a quantum mechanics course instructor to support a class community in developing a shared unified understanding of eigenequations. We analyze classroom data to characterize an instructor’s pedagogical moves as he engaged students in analogical reasoning. Some moves include posing tasks conducive to analogizing; preparing, soliciting, and scaffolding students’ participation in analogical reasoning; using deictic gestures and inscriptions; juxtaposing symbols representing the analogized concepts; and explicitly highlighting the sameness of the analogized concepts. We exemplify these pedagogical moves with analytical descriptions of illustrative class episodes. We discuss how these pedagogical moves can support the class community’s expansion of their common ground by fostering the development of the class’s shared unified understanding of eigentheory concepts.

View figure in article

Physics Subject Headings (PhySH)

Collections

This article appears in the following collection:

Focused Collection in Investigating and Improving Quantum Education through Research

Focused Collection in Investigating and Improving Quantum Education through Research

Article Text

References (53)

  1. Y. Lee and M. K. Heid, Developing a structural perspective and its role in connecting school algebra and abstract algebra: A factorization example, in Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers, edited by N. Wasserman, Research in Mathematics Education (Springer, Cham, 2018), pp. 291–318, 10.1007/978-3-319-99214-3_14.
  2. K. S. Serbin, Prospective teachers’ unified understandings of the structure of identities, J. Math. Behav. 70, 101066 (2023).
  3. M. Zandieh, J. Ellis, and C. Rasmussen, A characterization of a unified notion of mathematical function: The case of high school function and linear transformation, Educ. Stud. Math. 95, 21 (2017).
  4. S. Bagley, C. Rasmussen, and M. Zandieh, Inverse, composition, and identity: The case of function and linear transformation, J. Math. Behav. 37, 36 (2015).
  5. J. P. Cook, K. Melhuish, and R. Uscanga, Reasoning productively across algebraic contexts: Students develop coordinated notions of inverse, J. Math. Behav. 72, 101099 (2023).
  6. K. Melhuish, K. Lew, M. D. Hicks, and S. S. Kandasamy, Abstract algebra students’ evoked concept images for functions and homomorphisms, J. Math. Behav. 60, 100806 (2020).
  7. D. Gentner, Structure-mapping: A theoretical framework, Cogn. Sci. 7, 155 (1983).
  8. S. M. Glynn, The teaching with analogies model, in Children’s Comprehension of Text: Research into Practice, edited by K. D. Muth (International Reading Association, Newark, Delaware, 1989), pp. 185–205.
  9. L. E. Richland, O. Zur, and K. J. Holyoak, Cognitive supports for analogies in the mathematics classroom, Science 316, 1128 (2007).
  10. K. N. Begolli and L. E. Richland, Teaching mathematics by comparison: Analog visibility as a double-edged sword, J. Educ. Psychol. 108, 194 (2016).
  11. P. C. Dawkins and K. H. Roh, Promoting metalinguistic and metamathematical reasoning in proof-oriented mathematics courses: A method and a framework, Int. J. Res. Undergrad. Math. Educ. 2, 197 (2016).
  12. M. D. Hicks, Fostering productive ways of thinking associated with analogical reasoning in advanced mathematics, Learn. Math. 42, 10 (2022).
  13. N. S. Podolefsky and N. D. Finkelstein, Analogical scaffolding and the learning of abstract ideas in physics: An example from electromagnetic waves, Phys. Rev. ST Phys. Educ. Res. 3, 010109 (2007).
  14. L. E. Richland and K. N. Begolli, Analogy and higher order thinking: Learning mathematics as an example, Policy Insights Behav. Brain Sci. 3, 160 (2016).
  15. L. E. Richland, K. J. Holyoak, and J. W. Stigler, Analogy use in eighth-grade mathematics classrooms, Cognit. Instr. 22, 37 (2004).
  16. P. G. Sidney and M. W. Alibali, Making connections in math: Activating a prior knowledge analogue matters for learning, J. Cognit. Dev. 16, 160 (2015).
  17. G. B. Saxe and A. M. Farid, The interplay between individual and collective activity: An analysis of classroom discussions about the Sierpinski triangle, Int. J. Res. Undergrad. Math. Educ. 9, 632 (2021).
  18. E. Marshman and C. Singh, Investigating and improving student understanding of quantum mechanical observables and their corresponding operators in Dirac notation, Eur. J. Phys. 39, 015707 (2017).
  19. A. Pina, Z. Topdemir, and J. R. Thompson, Student understanding of eigenvalue equations in quantum mechanics: Symbolic forms and sensemaking analysis, Phys. Rev. Phys. Educ. Res. 20, 010153 (2024).
  20. B. W. Dreyfus, A. Elby, A. Gupta, and E. R. Sohr, Mathematical sense-making in quantum mechanics: An initial peek, Phys. Rev. Phys. Educ. Res. 13, 020141 (2017).
  21. M. Wawro, A. Pina, J. R. Thompson, Z. Topdemir, and K. Watson, Student interpretations of eigenequations in linear algebra and quantum mechanics, Int. J. Res. Undergrad. Math. Educ. (2024), 10.1007/s40753-024-00241-7.
  22. E. Knuth, M. Alibali, N. McNeil, A. Weinberg, and A. Stephens, Middle school students’ understanding of core algebraic concepts: Equivalence and variable, ZDM Int. J. Math. Educ. 37, 68 (2005).
  23. B. L. Sherin, How students understand physics equations, Cognit. Instr. 19, 479 (2001).
  24. E. Gire and C. Manogue, Making sense of quantum operators, eigenstates and quantum measurements, AIP Conf. Proc. 1413, 195 (2012).
  25. G. Karakok, Making connections among representations of eigenvector: What sort of a beast is it?, ZDM Math. Educ. 51, 1141 (2019).
  26. P. Her and M. Loverude, Examining student understanding of matrix algebra and eigentheory, presented at PER Conf. 2020, virtual conference, 10.1119/perc.2020.pr.Her.
  27. O. Uhden, R. Karam, M. Pietrocola, and G. Pospiech, Modelling mathematical reasoning in physics education, Sci. Educ. 21, 485 (2012).
  28. K. S. Serbin, Prospective teachers’ knowledge of secondary and abstract algebra and their use of this knowledge while noticing students’ mathematical thinking, Doctoral dissertation, Virginia Tech, 2021.
  29. K. S. Serbin, Y. Bae, and S. Espinosa, Guided reinvention of the definitions of reducibles and irreducibles, in Proceedings of the 26th Annual Conference on Research in Undergraduate Mathematics Education, edited by S. Cook, B. Katz, and D. Moore-Russo (Special Interest Group of the Mathematical Association of America for Research in Undergraduate Mathematics Education, Washington, DC, 2024).
  30. B. P. Schermerhorn, G. Corsiglia, H. Sadaghiani, G. Passante, and S. Pollock, From Cartesian coordinates to Hilbert space: Supporting student understanding of basis in quantum mechanics, Phys. Rev. Phys. Educ. Res. 18, 010145 (2022).
  31. 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).
  32. A. B. Markman and D. Gentner, Structure mapping in the comparison process, Am. J. Psychol. 113, 501 (2000).
  33. K. J. Holyoak and P. Thagard, Analogical mapping by constraint satisfaction, Cogn. Sci. 13, 295 (1989).
  34. L. R. Novick and K. J. Holyoak, Mathematical problem solving by analogy, J. Exp. Psychol. Learn. Memory Cogn. 17, 398 (1991).
  35. M. D. Hicks, Developing a framework for characterizing student analogical activity in mathematics, in Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mexico, edited by A. I. Sacristán, J. C. Cortés-Zavala, and P. M. Ruiz-Arias (Mathematics Education Across Cultures, 2020), pp. 914–921.
  36. N. Didiş, The analysis of analogy use in the teaching of introductory quantum theory, Chem. Educ. Res. Pract. 16, 355 (2015).
  37. E. Marshman and C. Singh, Improving student understanding of Dirac notation by using analogical reasoning in the context of a three-dimensional vector space, presented at PER Conf. 2020, virtual conference, 10.1119/perc.2020.pr. Marshman.
  38. J. Clement, The role of explanatory models in teaching for conceptual change, in International Handbook of Research on Conceptual Change (Taylor & Francis, United Kingdom, 2008), pp. 417–452.
  39. X. Vamvakoussi, Using analogies to facilitate conceptual change in mathematics learning, ZDM Math. Educ. 49, 497 (2017).
  40. A. B. Ellis and P. Grinstead, Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations, J. Math. Behav. 27, 277 (2008).
  41. H. H. Clark, Using Language (Cambridge University Press, United Kingdom, 1996).
  42. G. B. Saxe, K. D. Kirby, M. Le, Y. Sitabkhan, and B. Kang, Understanding learning across lessons in classroom communities: A multi-leveled analytic approach, in Approaches to Qualitative Research in Mathematics Education, edited by A. Bikner-Ahsbahs, C. Knipping, and N. Presmeg (Springer, Dordrecht, 2015), pp. 253–318, 10.1007/978-94-017-9181-6_11.
  43. G. B. Saxe, Cognition, development, and cultural practices, New Direct. Child Adolescent Dev. 83, 19 (1999).
  44. D. McIntyre, Quantum Mechanics: A Paradigms Approach (Addison-Wesley, Boston, MA, 2012).
  45. K. S. Serbin and M. Wawro, The inextricability of students’ mathematical and physical reasoning in quantum mechanics problems, Int. J. Res. Undergrad. Math. Educ. 10, 57 (2022).
  46. K. S. Serbin, M. Wawro, and R. Storms, Characterizations of student, instructor, and textbook discourse related to basis and change of basis in quantum mechanics, Phys. Rev. Phys. Educ. Res. 17, 010140 (2021).
  47. K. S. Serbin, B. J. Sanchez-Robayo, J. Truman, K. Watson, and M. Wawro, Characterizing quantum physics students’ conceptual and procedural knowledge of the characteristic equation, J. Math. Behav. 58, 100777 (2020).
  48. M. Wawro and K. Serbin, “What makes it eigen-esque-ish?”: Eigentheory development in a quantum mechanics course, in Proceedings of the 25th Annual Conference on Research in Undergraduate Mathematics Education, edited by S. Cook, B. Katz, and D. Moore-Russo (The Special Interest Group of the Mathematical Association of America for Research in Undergraduate Mathematics Education, Washington, DC, 2023), pp. 991–998.
  49. M. Wawro and K. S. Serbin, “What makes it eigen-esque-ish?”: A form-function analysis of the development of eigentheory concepts in a quantum mechanics course (to be published).
  50. M. B. Miles, A. M. Huberman, and J. Saldaña, Qualitative Data Analysis: A Methods Sourcebook, 3rd ed. (SAGE, Thousand Oaks, CA, 2013).
  51. K. Serbin, S. Espinosa, and E. Johnson, “When are we ever going to need abstract algebra?”: Pedagogical moves that evoke prospective teachers’ intellectual needs (to be published).
  52. J. Maxwell, Qualitative Research Design: An Interactive Approach (SAGE, Thousand Oaks, CA, 2013).
  53. D. McNeill, Hand and Mind: What Gestures Reveal About Thought (University of Chicago Press, Chigago, IL, 1992).

Outline

Information

Phys. Rev. Phys. Educ. Res. 20, 020137– Published 30 October, 2024
  • Received 7 June 2024
  • Accepted 30 September 2024
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