Skip to main content
SpringerLink
Log in

Using analogies to facilitate conceptual change in mathematics learning

  • Original Article
  • Published:
ZDM Aims and scope Submit manuscript

Abstract

The problem of adverse effects of prior knowledge in mathematics learning has been amply documented and theorized by mathematics educators as well as cognitive/developmental psychologists. This problem emerges when students’ prior knowledge about a mathematical notion comes in contrast with new information coming from instruction, giving rise to systematic errors. Conceptual change perspectives on mathematics learning suggest that in such cases reorganization of students’ prior knowledge is necessary. Analogical reasoning, in particular cross-domain mapping, is considered an important mechanism for conceptual restructuring. However, the use of analogies in instruction is often found ineffective, mainly because the structural similarity between two domains is obscure for students. To deal with this problem, John Clement and his colleagues developed the bridging strategy that uses multiple analogies to bring students to pay attention to the structural similarity that often goes unnoticed. This paper focuses on the cross-domain mapping between number and the (geometrical) line that has been instrumental in the development of the number concept. I summarize findings of a series of studies that investigated students’ understandings of density in arithmetical and geometrical contexts from a conceptual change perspective; and I discuss how this research-based evidence was used to design an intervention study that used the analogy “numbers are points on the number line”, and a bridging analogy (“the number line is like an imaginary rubber band that never breaks, no matter how much it is stressed”) with the aim of bringing the notion of density within the grasp of secondary students.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. There is a debate regarding the relation between analogy and metaphor. It is beyond the scope of this paper to enter this discussion. Following Bowdle and Gentner (2005), we take (conceptual) metaphor to be a species of analogy.

References

  • Alcock, L., Ansari, B., Batchelor, S., Bissona, M. J., De Smedt, B., Gilmore C. ,..., & Webe, K (2016). Challenges in mathematical cognition: a collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20–41. doi:10.5964/jnc.v2i1.10.

    Article  Google Scholar 

  • Anderson, J.R., Reder, L.M., & Simon, H.A. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review. Retrieved on 2016 August 15 from http://act-r.psy.cmu.edu/?post_type=publications&p=13741.

  • Berch, D. B. (2016). Disciplinary differences between cognitive psychology and mathematics education: A developmental disconnection syndrome. Journal of Numerical Cognition, 2(1), 42–47. doi:10.5964/jnc.v2i1.23.

    Article  Google Scholar 

  • Bowdle, B. F., & Gentner, D. (2005). The career of metaphor. Psychological Review, 112(1), 193–216. doi:10.1037/0033-295X.112.1.193.

    Article  Google Scholar 

  • Bransford, J. D., Brown, A. L., & Cocking, R. R. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.

    Google Scholar 

  • Brousseau, G. (2002). Theory of didactical situations in mathematics (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. and Trans.). New York: Kluwer Academic Publishers.

    Google Scholar 

  • Brown, D. E., & Clement, J. (1989). Overcoming misconceptions via analogical reasoning: Abstract transfer versus explanatory model construction. Instructional Science, 18, 237–261. doi:10.1007/BF00118013.

    Article  Google Scholar 

  • Bryce, T., & Macmillan, K. (2005). Encouraging conceptual change: the use of bridging analogies in the teaching of action–reaction forces and the ‘at rest’condition in physics. International Journal of Science Education, 27, 737–763. doi:10.1080/09500690500038132.

    Article  Google Scholar 

  • Carey, S. (2004). Bootstrapping and the origin of concepts. Daedalus, 133, 59–68. doi:10.1162/001152604772746701.

    Article  Google Scholar 

  • Clement, J. (1993). Using bridging analogies and anchoring intuitions to deal with students’preconceptions in physics. Journal of Research in Science Teaching, 30, 1241–1257.

    Article  Google Scholar 

  • Clement, J. (2008). The role of explanatory models in teaching for conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (1st ed., pp. 417–452). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Dantzig, T. (2005). Number: The language of science (4th edn.). New York: Pi Press.

    Google Scholar 

  • De Smedt, B., & Verschaffel, L. (2010). Travelling down the road: From cognitive neuroscience to mathematics education …and back. ZDM—The International Journal on Mathematics Education, 42, 649–654. doi:10.1007/s11858-010-0282-5.

    Article  Google Scholar 

  • Donovan, M. S., & Bransford, J. D. (2005). How students learn: History, mathematics, and science in the classroom. Washington, DC: The National Academies Press.

    Google Scholar 

  • Doritou, M., & Gray, E. (2009). Teachers’ subject knowledge: the number line representation. Paper presented at 6th Conference of the European society for Research in Mathematics Education (CERME 6), Lyon, France.

  • Duit, R. (1991). On the role of analogies and metaphors in learning science. Science Education, 75, 649–672. doi:10.1002/sce.3730750606.

    Article  Google Scholar 

  • Dunbar, K. (2001). The analogical paradox: Why analogy is so easy in naturalistic settings yet so difficult in the psychological laboratory. In D. Gentner, K. J. Holyoak & B. N. Kokinov (Eds.), The analogical mind: Perspectives from cognitive science (pp. 313–3340). Cambridge: The MIT Press.

    Google Scholar 

  • English, L. D. (1997). Analogies, metaphors, and images: Vehicles for mathematical reasoning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 3–18). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Gelman, R. (1990). First principles organize attention to and learning about relevant data: Number and animate-inanimate distinction as examples. Cognitive Science, 14, 79–106. doi:10.1207/s15516709cog1401_5.

    Article  Google Scholar 

  • Gelman, R. (2000). The epigenesis of mathematical thinking. Journal of Applied Developmental Psychology, 21, 27–37. doi:10.1016/S0193-3973(99)00048-9.

    Article  Google Scholar 

  • Gentner, D., Brem, S., Ferguson, R. W., Markman, A. B., Levidow, B. B., Wolff, P., & Forbus, K. D. (1997). Analogical reasoning and conceptual change: A case study of Johannes Kepler. Journal of the Learning Sciences, 6(1), 3–40. doi:10.1207/s15327809jls0601_2.

    Article  Google Scholar 

  • Gentner, D., & Wolff, P. (2000). Metaphor and knowledge change. In E. Dietrich & A. Markman (Eds.), Conceptual change in humans and machines (pp. 295–342). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Giannakoulias, E., Souyoul, A., & Zachariades, T. (2007). Students’ thinking about fundamental real numbers properties. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 416–425). Cyprus: ERME, Department of Education, University of Cyprus.

    Google Scholar 

  • Hartnett, P., & Gelman, R. (1998). Early understandings of numbers: Paths or barriers to the construction of new understandings? Learning and Instruction, 8 ,341–374. doi:10.1016/S0959-4752(97)00026-1.

    Article  Google Scholar 

  • Jacob, N. J., Vallentin, D., & Nieder, A. (2012). Relating magnitudes: The brain’s code for proportions. Trends in Cognitive Science, 16, 157–166. doi:10.1016/j.tics.2012.02.002.

    Article  Google Scholar 

  • Kilpatrick, J. (2014). History of research in mathematics education. In S. Lehrman (Ed.), Encyclopedia of mathematics education (pp. 267–271). London: Springer.

    Google Scholar 

  • Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding + it up. Helping children learn mathematics. Washington, DC: National Academy Press.

    Google Scholar 

  • Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

    Google Scholar 

  • Limón, M. (2001). On the cognitive conflict as an instructional strategy for conceptual change: A critical appraisal. Learning and Instruction, 11, 357–380. doi:10.1016/S0959-4752(00)00037-2.

    Article  Google Scholar 

  • McMullen, J., Laakkonen, E., Hannula-Sormunen, M., & Lehtinen, E. (2015). Modeling the developmental trajectories of rational number concept (s). Learning and Instruction, 37, 14–20. doi:10.1016/j.learninstruc.2013.12.004.

    Article  Google Scholar 

  • Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In M. Limon & L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 233–258). Dordrecht: Kluwer.

    Google Scholar 

  • Merenluoto, K., & Lehtinen, E. (2004). Number concept and conceptual change: Towards a systemic model of the processes of change. Learning and Instruction, 14, 519–534. doi:10.1016/j.learninstruc.2004.06.016.

    Article  Google Scholar 

  • Moss, J. (2005). Pipes, tubes, and beakers: New approaches to teaching the rational number system. In M. S. Donovan & J. D. Bransford (Eds.), How students learn: Mathematics in the classroom (pp. 121–162). Washington, DC: National Academic Press.

    Google Scholar 

  • Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52. doi:10.1207/s15326985ep4001_3.

    Article  Google Scholar 

  • Núñez, R., & Lakoff, G. (2005). The cognitive foundations of mathematics: The role of conceptual metaphor. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 109–124). New York, NY: Psychology Press.

    Google Scholar 

  • Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accommodation of a scientific conception: Towards a theory of conceptual change. Science Education, 66, 211–227. doi:10.1002/sce.3730660207.

    Article  Google Scholar 

  • Resnick, L., & Singer, J. (1993). Protoquantitative origins of ratio reasoning. In T. Carpenter, E. Fennema & T. Romberg (Eds.), Rational numbers: An integration of research (pp. 107–130). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Resnick, L. B. (2006). The dilemma of mathematical intuition in learning. In J. Novotná, H. Moraová, M. Krátká & N. Stehliková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 173–175). Prague: PME.

    Google Scholar 

  • Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128–1129. doi:10.1126/science.1142103.

    Article  Google Scholar 

  • Savinainen, A., Scott, P., & Viiri, J. (2005). Using a bridging representation and social interactions to foster conceptual change: Designing and evaluating an instructional sequence for Newton’s third law. Science Education, 89 (2), 175–195. doi:10.1002/sce.20037.

    Article  Google Scholar 

  • Sbaragli, S. (2006). Primary school teachers’ beliefs and change of beliefs on mathematical infinity. Mediterranean Journal for Research in Mathematics Education, 5(2), 49–76.

    Google Scholar 

  • Schneider, M., Vamvakoussi, X., & Van Dooren, W. (2012). Conceptual change. In N. M. Seel (Ed.), Encyclopedia of the sciences of learning (pp. 735–738). New York: Springer.

    Google Scholar 

  • Schoenfeld, A. H. (Ed.). (1987). Cognitive science and mathematics education. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Siegler, R. S. (2016). Magnitude knowledge: The common core of numerical development. Developmental Science, 19, 341–361. doi:10.1111/desc.12395.

    Article  Google Scholar 

  • Smith, C. L., Solomon, G. E. A., & Carey, S. (2005). Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology, 51, 101–140. doi:10.1016/j.cogpsych.2005.03.001.

    Article  Google Scholar 

  • Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115–163. doi:10.1207/s15327809jls0302_1.

    Article  Google Scholar 

  • Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York, NY: Springer.

    Book  Google Scholar 

  • Vamvakoussi, X. (2010). The ‘numbers are points οn the line’ analogy: Does it have an instructional value? In L. Verschaffel, E. De Corte, T. de Jong & J. Elen (Eds.), Use of external representations in reasoning and problem solving: Analysis and improvement. New Perspectives on Learning and Instruction Series (pp. 209–224). New York, NY: Routlege.

    Google Scholar 

  • Vamvakoussi, X. (2015). The development of rational number knowledge: Old topic, new insights. Learning and Instruction, 37, 50–55. doi:10.1016/j.learninstruc.2015.01.002.

    Article  Google Scholar 

  • Vamvakoussi, X., Christou, K. P., & Van Dooren, W. (2011). What fills the gap between the discrete and the dense? Greek and Flemish students’ understanding of density. Learning & Instruction, 21, 676–685. doi:10.1016/j.learninstruc.2011.03.005.

    Article  Google Scholar 

  • Vamvakoussi, X., Kargiotakis, G., Kollias, Mamalougos, N. G., & Vosniadou, S. (2003). Collaborative modelling of rational numbers. In B. Wasson, S. Ludvigsen & U. Hoppe (Eds.), Designing for change in networked learning environments—Proceedings of the International Conference on Computer Support for Collaborative Learning (pp. 103–107). Dordrecht: Kluwer.

    Google Scholar 

  • Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for a natural number bias in adults. The Journal of Mathematical Behavior, 31, 344–355. doi:10.1016/j.jmathb.2012.02.001.

    Article  Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14, 453–467. doi:10.1016/j.learninstruc.2004.06.013.

    Article  Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2007). How many numbers in an interval? Presuppositions, synthetic models and the effect of the number line. Ιn S. Vosniadou. In A. Baltas & X. Vamvakoussi (Eds.), Reframing the conceptual change approach in learning and instruction (pp. 267–283). Oxford: Elsevier.

    Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181–209. doi:10.1080/07370001003676603.

    Article  Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2012). Bridging the gap between the dense and the discrete: the number line and the “rubber line” bridging analogy. Mathematical Thinking and Learning, 14, 265–284. doi:10.1080/10986065.2012.717378.

    Article  Google Scholar 

  • Vamvakoussi, X., Vosniadou, S., & Van Dooren, W. (2013). The framework theory approach applied to mathematics learning. In S. Vosniadou (Ed.), International handbook of research on conceptual change (2nd Ed.) (pp. 305–321). New York, NY: Routledge.

    Google Scholar 

  • Van Dooren, W., Vamvakoussi, X., & Verschaffel, L. (2013). Mind the gap–Task design principles to achieve conceptual change in rational number understanding. In C. Margolinas (Ed.), Task design in mathematics education: Proceedings of ICMI Study 22 (pp. 519–527). Oxford: International Commission on Mathematical Instruction.

    Google Scholar 

  • Verschaffel, L., & Vosniadou, S. (Guest Eds.). (2004). The conceptual change approach to mathematics learning and teaching [Special Issue]. Learning and Instruction, 14(5).

  • Vosniadou, S. (1989). Analogical reasoning as a mechanism in knowledge acquisition: A developmental perspective. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 413–436). Cambridge, MA: Cambridge University Press.

    Chapter  Google Scholar 

  • Vosniadou, S., Ioannides, C., Dimitrakopoulou, A., & Papademetriou, E. (2001). Designing learning environments to promote conceptual change in science. Learning and Instruction, 11, 381–419. doi:10.1016/S0959-4752(00)00038-4.

    Article  Google Scholar 

  • Vosniadou, S., Vamvakoussi, X., & Skopeliti, I. (2008). Τhe framework theory approach to conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (1st ed., pp. 3–34). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Yilmaz, S., Eryilmaz, A., & Geban, O. (2006). Assessing the impact of bridging analogies in mechanics. School Science and Mathematics, 106(6), 220–230. doi:10.1111/j.1949-8594.2006.tb17911.x.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Early Childhood Education, University of Ioannina, Building: Metavatiko, University Campus, 45110, Ioannina, Greece

    Xenia Vamvakoussi

Authors
  1. Xenia Vamvakoussi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xenia Vamvakoussi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vamvakoussi, X. Using analogies to facilitate conceptual change in mathematics learning. ZDM Mathematics Education 49, 497–507 (2017). https://doi.org/10.1007/s11858-017-0857-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-017-0857-5

Keywords

Search

Navigation