Eva-Maria Ternblad and Betty Tärning
pp. 327 – 362, download
(https://doi.org/10.55612/s-5002-068-011)
Abstract
The use of digital learning environments has exploded during recent years, transforming learning at its core. At the same time, we know little about how this transformation affects younger students, still about to grasp basic concepts in STEM-subjects. This study explores the difference between interaction with physical or virtual representations when learning about areas of parallelograms. 94 middle-school students participated in the experiment, designed as an ordinary classroom activity. The students interacted with a physical deck of cards and a plastic frame – or with virtual representations of these objects – during two lessons. After that they took a test. The results reveal that even if there were no significant difference between the two conditions when it comes to applying the correct formula for areas of parallelograms, the students in the physical condition better grasped the concepts of height and base than the students in the virtual condition. This knowledge, in turn, correlated positively to the understanding of the area formula for parallelograms. These findings indicate that specific actions and/or interactions, facilitated by a physical material and the use of pen and paper, may be truly beneficial for acquiring knowledge in geometry and spatial reasoning. .
Keywords: Elementary education, Mathematics, Virtual interaction, Physical interaction, Embedded cognition and learning.
References
1. Thisgaard M., Makransky G.: Virtual learning simulations in high school: Effects on cognitive and non-cognitive outcomes and implications on the development of STEM academic and career choice. Frontiers in psychology, 8, p. 805 (2017) https://doi.org/10.3389/fpsyg.2017.00805
2. Zhai X., Zhang M., Li M. Zhang X.: Understanding the relationship between levels of mobile technology use in high school physics classrooms and the learning outcome. British journal of educational technology, 50(2), pp. 750–766 (2019) https://doi.org/10.1111/bjet.12700
3. Peramunugamage A., Ratnayake U. W., Karunanayaka S. P.: Systematic review on mobile collaborative learning for engineering education. Journal of Computers in Education, 10(1), 83106 (2023) https://doi.org/10.1007/s40692-022-00223-1
4. Shumway J. F., Welch L. E., Kozlowski J. S., Clarke-Midura J., Lee V. R.: Kindergarten students’ mathematics knowledge at work: the mathematics for programming robot toys. Mathematical Thinking and Learning, 25(4), pp. 380–408 (2023) https://doi.org/10.1080/10986065.2021.1982666
5. Wilensky U., Papert S.: Restructurations: Reformulations of knowledge disciplines through new representational forms. Constructionism, 17(2010), pp. 1–15 (2010)
6. Hwang W. Y., Hu S. S.: Analysis of peer learning behaviors using multiple representations in virtual reality and their impacts on geometry problem solving. Computers & Education, 62, pp. 308–319 (2013)
https://doi.org/10.1016/j.compedu.2012.10.005
7. Ng O. L., Shi L., Ting F.: Exploring differences in primary students’ geometry learning outcomes in two technology-enhanced environments: dynamic geometry and 3D printing. International Journal of STEM Education, 7(1), p. 50 (2020) https://doi.org/10.1186/s40594-020-00244-1
8. Sayeki Y, Ueno N., Nagasaka T. Mediation as a generative model for obtaining an area. Learning and Instruction, 1(3), pp. 229–242 (1991) https://doi.org/10.1016/0959-4752(91)90005-S
9. Palmer S. E.: Fundamental aspects of cognitive representation. In Rosch, E., & Lloyd, B. B. (Eds.). Cognition and categorization (pp. 259-303), Routledge, New York, (1978) https://doi.org/10.4324/9781032633275-13
10. Marton F.: Necessary conditions of learning. Routledge, New York, (2014) https://doi.org/10.4324/9781315816876
11. Stull A. T., Hegarty M.: Model manipulation and learning: Fostering representational competence with virtual and concrete models. Journal of Educational Psychology, 108(4), p. 509 (2016) https://doi.org/10.1037/edu0000077
12. Carbonneau K. J., Marley S. C., Selig J. P.: A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), p. 380 (2013) https://doi.org/10.1037/a0031084
13. Kollöffel B., De Jong T.: Conceptual understanding of electrical circuits in secondary vocational engineering education: Combining traditional instruction with inquiry learning in a virtual lab. Journal of engineering education, 102(3), pp. 375–393 (2013) https://doi.org/10.1002/jee.20022
14. Evangelou F., Kotsis K.: Real vs virtual physics experiments: comparison of learning outcomes among fifth grade primary school students. A case on the concept of frictional force. International Journal of Science Education, 41(3), pp. 330–348 (2019) https://doi.org/10.1080/09500693.2018.1549760
15. Wörner S., Kuhn J., Scheiter K.: The best of two worlds: A systematic review on combining real and virtual experiments in science education. Review of Educational Research, 92(6), 911952 (2022) https://doi.org/10.3102/00346543221079417
16. Zacharia Z. C., Olympiou G.: Physical versus virtual manipulative experimentation in physics learning. Learning and instruction, 21(3), 317–331 (2011) https://doi.org/10.1016/j.learninstruc.2010.03.001
17. Brinson J. R.: Learning outcome achievement in non-traditional (virtual and remote) versus traditional (hands-on) laboratories: A review of the empirical research. Computers & Education, 87, pp. 218–237 (2015)
https://doi.org/10.1016/j.compedu.2015.07.003
18. Rau M. A.: Comparing Multiple Theories about Learning with Physical and Virtual Representations: Conflicting or Complementary Effects? Educational Psychology Review, 32, pp. 297–325 (2020) https://doi.org/10.1007/s10648-020-09517-1
19. Pande P., Chandrasekharan S.: Representational competence: towards a distributed and embodied cognition account. Studies in science education, 53(1), pp.1–43 (2017) https://doi.org/10.1080/03057267.2017.1248627
20. Pavlou Y., Zacharia Z. C., Papaevripidou M.: Comparing the impact of physical and virtual manipulatives in different science domains among preschoolers. Science Education, 1–29 (2024) https://doi.org/10.1002/sce.21869
21. Sarama J., Clements D. H.: ‘Concrete’ computer manipulatives in mathematics education. Child Development Perspectives, 3(3), pp.145–150 (2009) https://doi.org/10.1111/j.1750-8606.2009.00095.x
22. Clark R. E.: Media will never influence learning. Educational Technology Research & Development, 42(2), pp. 11–29 (1994) https://doi.org/10.1007/BF02299088
23. Newcombe N. S.: A plea for spatial literacy. The Chronicle of Higher Education, 52(26), B20 (2006)
24. Hohol M. Foundations of geometric cognition. Routledge, New York, (2019) https://doi.org/10.4324/9780429056291
25. Izard V., Pica, P., Spelke E. S., Dehaene S.: Flexible intuitions of Euclidean geometry in an Amazonian indigene group. Proceedings of the National Academy of Sciences of the United States of America, 108, pp. 9782–9787 (2011)
https://doi.org/10.1073/pnas.1016686108
26. Jones K., Tzekaki M.: Research on the teaching and learning of geometry. In A. Gutiérrez, G. Leder & P. Boero (Eds.), The Second Handbook of Research on the Psychology of Mathematics Education: The Journey Continues (pp. 109–149). Rotterdam: Sense, (2016) https://doi.org/10.1007/978-94-6300-561-6_4
27. Battista M. T.: Representations and cognitive objects in modern school geometry. Research on technology and the teaching and learning of mathematics: Cases and perspectives, 2, pp. 341–362 (2008) https://doi.org/10.1108/978-1-60752-953-820251016
28. Clements D. H., Battista M. T.: Geometry and spatial reasoning. Handbook of research on mathematics teaching and learning, pp. 420–464 (1992)
https://doi.org/10.1108/978-1-60752-874-620251022
29. Van Hiele P. M.: Structure and insight: A theory of mathematics education. Academic Press, Orlando, (1986)
30. Güler M., Bütüner S. Ö., Danişman Ş., Gürsoy K.: A meta-analysis of the impact of mobile learning on mathematics achievement. Education and Information Technologies, pp. 1–21 (2022).
31. Juandi D., Kusumah Y. S., Tamur M., Perbowo K. S., Siagian M. D., Sulastri R., Negara H. R. P.: The Effectiveness of Dynamic Geometry Software Applications in Learning Mathematics: A Meta-Analysis Study. International Journal of Interactive Mobile Technologies (iJIM), 15(02), pp. 18–37 (2021)
https://doi.org/10.3991/ijim.v15i02.18853
32. Löwing M.: Grundläggande geometri. Matematikdidaktik för lärare. Studentlitteratur, Lund, (2020)
33. Wares A.: Paper folding and trigonometric ratios. International Journal of Mathematical Education in Science and Technology, 50(4), pp. 636–641 (2019) https://doi.org/10.1080/0020739X.2018.1500655
34. Leung A.: Empowering learning with rich mathematical experience: reflections on a primary lesson on area and perimeter. International Journal for Mathematics Teaching and Learning, 45(23), pp. 10–29 (2010)
35. Rellensmann J., Schukajlow S., Leopold C.: Measuring and investigating strategic knowledge about drawing to solve geometry modelling problems. ZDM, 52, pp. 97–110 (2020) https://doi.org/10.1007/s11858-019-01085-1
36. Schwartz D. L., Goldstone R.: Learning as coordination: Cognitive psychology and education. In Corno, L. & Andermann, E. M. (Eds.) Handbook of educational psychology (pp. 75–89). Routledge, New York, (2015)
37. Gal H., Linchevski L.: To see or not to see: analyzing difficulties in geometry from the perspective of visual perception. Educational studies in mathematics, 74, pp. 163–183 (2010) https://doi.org/10.1007/s10649-010-9232-y
38. Slors M.: Symbiotic cognition as an alternative for socially extended cognition. Philosophical psychology, 32(8), pp. 1179–1203 (2019) https://doi.org/10.1080/09515089.2019.1679591
39. Clark A.: Intrinsic content, active memory and the extended mind. Analysis, 65(1), pp. 1–11 (2005) https://doi.org/10.1093/analys/65.1.1
40. Clark A.: Embodied, situated, and distributed cognition. A companion to cognitive science, pp. 506–517 (2017) https://doi.org/10.1002/9781405164535.ch39
41. Pouw W. T., Van Gog T., Paas F.: An embedded and embodied cognition review of instructional manipulatives. Educational Psychology Review, 26(1), pp. 51–72 (2014) https://doi.org/10.1007/s10648-014-9255-5
42. Chandrasekharan S., Nersessian N. J.: Building cognition: The construction of computational representations for scientific discovery. Cognitive science, 39(8), pp. 1727–1763 (2015) https://doi.org/10.1111/cogs.12203
43. Rahaman J., Agrawal H., Srivastava N., Chandrasekharan S.: Recombinant enaction: Manipulatives generate new procedures in the imagination, by extending and recombining action spaces. Cognitive science, 42(2), pp. 370–415 (2018) https://doi.org/10.1111/cogs.12518
44. Nuthall G.: The hidden lives of learners. NZCER Press. (2007)
45. Hattie J.: Visible learning: A synthesis of over 800 meta-analyses relating to achievement. Routledge (2008)
46. Kluger A.N., DeNisi A.: Feedback interventions: toward the understanding of a double-edged sword. Curr. Dir. Psychol. Sci. 7(3), pp. 67–72 (1998) https://doi.org/10.1111/1467-8721.ep10772989
47. Shute V. J.: Focus on formative feedback. Review of educational research, 78(1), pp. 153–189 (2008) https://doi.org/10.3102/0034654307313795
48. Rau M. A.: Conditions for the effectiveness of multiple visual representations in enhancing STEM learning. Educational Psychology Review, 29(4), pp. 717–761 (2017) https://doi.org/10.1007/s10648-016-9365-3
49. Andersen K.: Cavalieri’s method of indivisibles. Archive for history of exact sciences, 31(4), pp. 291–367 (1985) https://doi.org/10.1007/BF00348519
50. Proulx J., Pimm D.: Algebraic Formulas, Geometric Awareness and Cavalieri’s Principle. For the Learning of Mathematics 28(2), pp. 17–24 (2008)
51. Ternblad E. M.: Understanding Areas of Parallelograms Through Virtual Geometrical Representations: A Pilot Study. In Proceedings of the 30th International Conference on Computers in Education (ICCE) 2022, pp. 206–209 (2022)
52. Ternblad E. M.: What If Interaction Fails? A Comparison of a Virtual and a Physical Learning Environment for Learning About Areas of Parallelograms. In Proceedings of the 17th International Conference of the Learning Sciences-ICLS 2023, pp. 1062—1065, International Society of the Learning Sciences. (2023)
https://doi.org/10.22318/icls2023.788401
53. Schenke K., Redman E. J., Chung G. K., Chang S. M., Feng T., Parks C. B., Roberts J. D.: Does “Measure Up!” measure up? Evaluation of an iPad app to teach preschoolers measurement concepts. Computers & Education, 146, 103749 (2020) https://doi.org/10.1016/j.compedu.2019.103749
54. Sinclair N., Moss J., Hawes Z., Stephenson C.: Learning through and from drawing in early years geometry. In Mix, K. S. & Battista, M. T. (Eds.), Visualizing mathematics: The role of spatial reasoning in mathematical thought, pp. 229–252 Cham: Springer, (2018) https://doi.org/10.1007/978-3-319-98767-5_11
55. Magana A. J., Balachandran S.: Students’ development of representational competence through the sense of touch. Journal of Science Education and Technology, 26(3), pp.332–346 (2017) https://doi.org/10.1007/s10956-016-9682-9
56. Skulmowski A., Pradel S., Kühnert T., Brunnett G., Rey G. D.: Embodied learning using a tangible user interface: the effects of haptic perception and selective pointing on a spatial learning task. Computers & Education, 92, pp. 64–75 (2016) https://doi.org/10.1016/j.compedu.2015.10.011
57. Cisek P.: Cortical mechanisms of action selection: the affordance competition hypothesis. Philosophical Transactions of the Royal Society B: Biological Sciences, 362(1485), pp. 1585–1599 (2007) https://doi.org/10.1098/rstb.2007.2054
58. Kirsh D.: The intelligent use of space. Artificial intelligence, 73(1-2), pp. 31–68 (1995) https://doi.org/10.1016/0004-3702(94)00017-U
59. Willingham D. T.: Why don’t students like school?: A cognitive scientist answers questions about how the mind works and what it means for the classroom. John Wiley & Sons, Hoboken, (2021).
60. Kirschner P. A., Hendrick C.: How learning happens: Seminal works in educational psychology and what they mean in practice. Routledge, New York, (2024) https://doi.org/10.4324/9781003395713
61. Kirchner P. A., Sweller J., Clark R. E.: Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 4(2), pp. 75–86 (2006) https://doi.org/10.1207/s15326985ep4102_1
62. Kirschner P. A., Hendrick C., Heal J.: Instructional Illusions. Hachette UK, (2025)
63. Komatsu K., Jones K.: Interplay between paper-and-pencil activity and dynamic-geometryenvironment use during generalisation and proving. Digital Experiences in Mathematics Education, 6(2), pp. 123–143 (2020) https://doi.org/10.1007/s40751-020-00067-3
64. Gonzalez C., Anderson J., Culmer P., Burke M. R., Mon-Williams M., Wilkie R. M.: Is tracing or copying better when learning to reproduce a pattern?. Experimental Brain Research, 208(3), pp. 459–465 (2011) https://doi.org/10.1007/s00221-010-2482-1