According to general relativity (GR), we live in a four-dimensional curved universe. Since the human mind cannot visualize those four dimensions, a popular analogy compares the universe to a two-dimensional rubber sheet distorted by massive objects. This analogy is often used when teaching GR to upper secondary and undergraduate physics students. However, physicists and physics educators criticize the analogy for being inaccurate and for introducing conceptual conflicts. Addressing these criticisms, we analyze the rubber sheet analogy through systematic metaphor analysis of textbooks and research literature, and present an empirical analysis of upper secondary school students’ use and understanding of the analogy. Taking a theoretical perspective of embodied cognition allows us to account for the relationship between the experiential and sensory aspects of the metaphor in relation to the abstract nature of spacetime. We employ methods of metaphor and thematic analysis to study written accounts of small groups of 97 students (18–19 years old) who worked with a collaborative online learning environment as part of their regular physics lessons in five classes in Norway. Students generated conceptual metaphors found in the literature as well as novel ones that led to different conceptions of gravity than those held by experts in the field. Even though most students showed awareness of some limitations of the analogy, we observed a conflict between students’ embodied understanding of gravity and the abstract description of GR. This conflict might add to the common perception of GR being counterintuitive. In making explicit strengths and weaknesses of the rubber sheet analogy and learners’ conceptual difficulties, our results offer guidance for teaching GR. More generally, these findings contribute to the epistemological implications of employing specific scientific metaphors in classrooms.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
The original Norwegian “å se for seg” can be translated as “to visualize,” “to envision,” “to see in your mind’s eye” or more literally “to see in front of you.” In our translations, we chose the expression “to visualize.”
Abbott, B. P., Abbott, R., Abbott, T. D., Abernathy, M. R., Acernese, F., Ackley, K., et al. (2016). Observation of gravitational waves from a binary black hole merger. Physical Review Letters, 116(6), 061102.
Amin, T. G., Jeppsson, F., & Haglund, J. (2015). Conceptual metaphor and embodied cognition in science learning: introduction to special issue. International Journal of Science Education, 37(5–6), 745–758.
Aubusson, P. J., Harrison, A. G., & Ritchie, S. M. (2006). Metaphor and analogy in science education (Vol. 30). Berlin: Springer.
Baldy, E. (2007). A new educational perspective for teaching gravity. International Journal of Science Education, 29(14), 1767–1788.
Bandyopadhyay, A., & Kumar, A. (2010a). Probing students’ ideas of the principle of equivalence. European Journal of Physics, 32(1), 139–159.
Bandyopadhyay, A., & Kumar, A. (2010b). Probing students’ understanding of some conceptual themes in general relativity. Physical Review Special Topics - Physics Education Research, 6(2), 020104.
Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3, 77–101.
Callin, N. P., Pålsgård, J., Stadsnes, R., & Tellefsen, C. W. (2012). ERGO Fysikk 2. Oslo: Aschehoug.
Casey, E. S. (1979). Imagining. Bloomington: Indiana University Press.
Chandler, M. (1994). Philosophy of gravity: intuitions of four-dimensional curved spacetime. Science & Education, 3(2), 155–176.
Chen, Y.-C., Park S., & Hand, B. (2016). Examining the use of talk and writing for students’ development of scientific conceptual knowledge through constructing and critiquing arguments. Cognition and Instruction, 34(2), 100–147.
Dimitriadi, K., & Halkia, K. (2012). Secondary students’ understanding of basic ideas of special relativity. International Journal of Science Education, 34(16), 2565–2582.
diSessa, A. A. (1981). An elementary formalism for general relativity. American Journal of Physics, 49(1981), 401.
Einstein, A. (1915). Grundgedanken der allgemeinen Relativitätstheorie und Anwendung dieser Theorie in der Astronomie [Fundamental Ideas of the General Theory of Relativity and the Application of this Theory in Astronomy]. Preussische Akademie Der Wissenschaften, Satzungsberichte, 1(1), 315.
Einstein, A. (1917). Über die spezielle und die allgemeine Relativitätstheorie [Relativity: the special and general theory]. Braunschweig: Vieweg.
Farr, B., Schelbert, G., & Trouille, L. (2012). Gravitational wave science in the high school classroom. American Journal of Physics, 80(10), 898.
Gentner, D., Bowdle, B., Wolff, P., & Boronat, C. (2001). Metaphor is like analogy. In D. Gentner, K. J. Holyoak, & B. N. Kokinov (Eds.), The analogical mind: perspectives from cognitive science (pp. 199–253). Cambridge, MA: MIT.
Gilbert, J. (2004). Models and modelling: routes to more authentic science education. International Journal of Science and Mathematics Education, 2(2), 115–130.
Gilbert, J. (Ed.). (2005). Visualization in science education. Visualization in science education Vol. 1. Dordrecht: Springer.
Gould, R. R. (2016). Why does a ball fall?: A new visualization for Einstein’s model of gravity. American Journal of Physics, 84(5), 396–402.
Greene, B. (2010). The elegant universe. New York: W W Norton & Co Inc..
Haglund, J. (2017). Good use of a ‘bad’ metaphor—entropy as disorder. Science & Education, 26(3–4), 205–214.
Harrison, A. G., & Treagust, D. F. (2006). Teaching and learning with analogies: friend or foe. In A. G. Harrison & S. M. Ritchie (Eds.), Metaphor & analogy in science education (pp. 11–24). Dordrecht, Netherlands: Springer.
Hartle, J. B. (2005). General relativity in the undergraduate physics curriculum. American Journal of Physics, 14(2006), 9.
Henriksen, E. K., Bungum, B., Angell, C., Tellefsen, C. W., Frågåt, T., & Vetleseter Bøe, M. (2014). Relativity, quantum physics and philosophy in the upper secondary curriculum: challenges, opportunities and proposed approaches. Physics Education, 49(6), 678–684.
Hentschel, K. (Ed.). (1998). The collected papers of Albert Einstein, Volume 8 (English) The Berlin Years: Correspondence, 1914–1918. Princeton, New Jersey: Princeton University Press.
Hesse, M. (1952). Operational definition and analogy in physical theories. British Journal for the Philosophy of Science, 2(8), 281–294.
Hesse, M. (1953). Models in physics. British Journal for the Philosophy of Science, 4, 98–214.
Jerstad, P., Sletbak, B., Grimenes, A. A., Renstrøm, R., Holm, O. B., & Nymo, M. (2014). RomStoffTid Fysikk 2. Oslo: Cappelen Damm.
Kampourakis, K. (2016). The bad use of metaphors and the use of bad metaphors. Science & Education, 25(9–10), 947–949.
Kapon, S., & DiSessa, A. A. (2012). Reasoning through instructional analogies. Cognition and Instruction, 30(3), 261–310.
Kaur, T., Blair, D., Moschilla, J., Stannard, W., & Zadnik, M. (2017a). Teaching Einsteinian physics at schools: models and analogies—Part 1 (Manuscript draft). Retrieved from http://arxiv.org/abs/1704.02058
Kaur, T., Blair, D., Moschilla, J., Stannard, W., & Zadnik, M. (2017b). Teaching Einsteinian physics at schools: Part 3, review of research outcomes. Physics Education, 52(6). https://doi.org/10.1088/1361-6552/aa83dd
Kersting, M., Henriksen, E. K., Bøe, M. V., & Angell, C. (2018). General relativity in upper secondary school: design and evaluation of an online learning environment using the model of educational reconstruction. Physsical Review Physics Education Research, 14(1), 010130–1–010130-18. https://doi.org/10.1103/PhysRevPhysEducRes.14.010130.
Kind, P. M., & Kind, V. (2007). Creativity in science education: perspectives and challenges for developing school science. Studies in Science Education, 43(1), 1–37.
Krijtenburg-Lewerissa, K., Pol, H. J., Brinkman, A., & Van Joolingen, W. R. (2017). Insights into teaching quantum mechanics in secondary and lower undergraduate education: a literature review. Physical Review Physics Education Research, 13(1), 010109-21. https://doi.org/10.1103/PhysRevPhysEducRes.13.010109.
Lakoff, G., & Johnson, M. (2003). Metaphors we live by (2nd ed.). Chicago: University of Chicago Press.
Lancor, R. (2014a). Using metaphor theory to examine conceptions of energy in biology, chemistry, and physics. Science & Education, 23(6), 1245–1267.
Lancor, R. (2014b). Using student-generated analogies to investigate conceptions of energy: a multidisciplinary study. International Journal of Science Education, 36(1), 1–23.
Lemke, J. L. (1990). Talking science: language, learning, and values. Norwood, New Jersey: Ablex Publishing Corporation.
Levrini, O. (2014). The role of history and philosophy in research on teaching and learning of relativity. In M. R. Matthews (Ed.), International handbook of research in history, philosophy and science teaching (pp. 157–181). Dordrecht: Springer.
Levrini, O., & DiSessa, A. A. (2008). How students learn from multiple contexts and definitions: proper time as a coordination class. Physical Review Special Topics - Physics Education Research, 4(1), 1–18..
Mach, E. (1893). In T. J. McCormack (Ed.), The science of mechanics. Chicago: Open Court Publishing Company.
Middleton, C. A., & Weller, D. (2016). Elliptical-like orbits on a warped spandex fabric: a theoretical/experimental undergraduate research project. American Journal of Physics, 84(4), 284–292.
Nemirovsky, R., Kelton, M. L., & Rhodehamel, B. (2012). Gesture and imagination on the constitution and uses of phantasms. Gesture, 12(2), 130–165.
Niebert, K., & Gropengießer, H. (2014). Understanding the greenhouse effect by embodiment—analysing and using students’ and scientists’ conceptual resources. International Journal of Science Education, 36(2), 277–303.
Niebert, K., Marsch, S., & Treagust, D. F. (2012). Understanding needs embodiment: a theory-guided reanalysis of the role of metaphors and analogies in understanding science. Science Education, 96(1), 849–877.
Pitts, M., Venville, G., Blair, D., & Zadnik, M. (2014). An exploratory study to investigate the impact of an enrichment program on aspects of Einsteinian physics on year 6 students. Research in Science Education, 44(3), 363–388.
Poincaré, H. (1898). La mesure du temps [The measure of time]. Revue de Métaphysique et de Morale, 6(1), 1–13.
Price, R. H. (2016). Spatial curvature, spacetime curvature, and gravity. American Journal of Physics, 84(8), 588–592.
Rasmussen, I., & Ludvigsen, S. (2010). Learning with computer tools and environments: a sociocultural perspective. In C. W & J. K. S. K. Littleton (Eds.), International handbook of psychology in education (pp. 399–435). Bingley, UK: Emerald.
Reichenbach, H. (1928). Philosophie der Raum-Zeit-Lehre [The philosophy of space and time]. New York: Dover Publications.
Roth, W. M., & Lawless, D. (2002). Science, culture, and the emergence of language. Science Education, 86(3), 368–385.
Russell, B. (1925). In K.-P. Trench & Trubner (Eds.), ABC of relativity. London: The ABC of Relativity.
Sartre, J.-P. (2004). The imaginary. A phenomenological psychology of the imagination. New York: Routledge.
Schmitt, R. (2005). Systematic metaphor analysis as a method of qualitative research. The Qualitative Report, 10(2), 358–394.
Schön, D. A. (1979). Generative metaphor: a perspective problem-setting in social policy. In A. Ortony (Ed.), Metaphor and thought (pp. 254–283). Cambridge, England: Cambridge University Press.
Scott, P., & Mortimer, E. (2005). Meaning making in high school science classrooms: a framework for analysing meaning making interactions. Research and the Quality of Science Education, 7, 395–406.
Silva, C. C. (2007). The role of models and analogies in the electromagnetic theory: a historical case study. Science & Education, 16, 835–848.
Stadermann, H. K. E., & Goedhart, M. J. (2017). Comparison and analysis of quantum physics in secondary school curricula of 13 different countries. Presentation at GIREP 2017, Dublin, Ireland.
Stannard, W., Blair, D., Zadnik, M., & Kaur, T. (2017). Why did the apple fall? A new model to explain Einstein’s gravity. European Journal of Physics, 38(1), 015603.
Steier, R., & Kersting, M. (n.d.). Metaimagining and embodied conceptions of spacetime (Manuscript under Review).
Stinner, A. (2003). Scientific method, imagination and the teaching of physics. Physics World, 59(6), 335–346.
The Norwegian Directorate for Education and Training. (2006). Physics—programme subject in programmes for specialization in general studies. Retrieved from http://www.udir.no/kl06/fys1-01/Hele/Kompetansemaal/fysikk-2/?lplang=eng
Thorne, K. (2009). Warping spacetime. In G. W. Gibbons, E. P. S. Shellard, & S. J. Rankin (Eds.), The future of theoretical physics and cosmology: celebrating Stephen Hawking’s contributions to physics. Cambridge: Cambridge University Press.
Treagust, D. F., & Duit, R. (2015). On the significance of conceptual metaphors in teaching and learning science: Commentary on Lancor; Niebert and Gropengiesser; and Fuchs. International Journal of Science Education, 37(5–6), 958–965.
Vailati, E. (1997). Leibniz and Clarke: a study of their correspondence. In Oxford and New York: Oxford University Press.
Velentzas, A., & Halkia, K. (2013). The use of thought experiments in teaching physics to upper secondary-level students: two examples from the theory of relativity. International Journal of Science Education, 35(18), 3026–3049.
Viereck, G. S. (1929). What life means to Einstein. The Saturday Evening Post. http://doi.org/http://www.saturdayeveningpost.com/wp-content/uploads/satevepost/what_life_means_to_einstein.pdf
Vygotsky, L. (1962). Thought and language. Cambridge: M.I.T. Press.
Watkins, T. R. (2014). Gravity & Einstein: assessing the rubber sheet analogy in undergraduate conceptual physics (Master Thesis). Boise State University.
Zahn, C., & Kraus, U. (2014). Sector models—a toolkit for teaching general relativity. Part 1: curved spaces and spacetimes. European Journal of Physics, 35, 055020.
This work was supported by the Research Council of Norway (ProjectNo. 246723) and the Olav Thon Foundation.
Conflict of Interest
The authors declare that they have no conflict of interest.
About this article
Cite this article
Kersting, M., Steier, R. Understanding Curved Spacetime. Sci & Educ 27, 593–623 (2018). https://doi.org/10.1007/s11191-018-9997-4