Skip to main content
Log in

Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: a critical review

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

Abstract

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

Notes

  1. As it can be noted, our definition of flexible representational choice is inspired on a definition of adaptive strategy choice. We are aware of the fact that some authors do not consider the terms flexibility and adaptivity as being synonyms, but for the sake of simplicity we have decided to use them as synonyms in this article. For an explanation regarding the different conceptualisations of these terms, we refer to Verschaffel, Luwel, Torbeyns, and Van Dooren (2009).

References

  • Acevedo Nistal, A., Clarebout, G., Elen, J., Van Dooren, W., & Verschaffel, L. (2008, September). Representational flexibility in the domain of linear functions: A choice/no-choice study. Paper presented at an international workshop of the FWO scientific network on developing critical and flexible thinking: Use of (external) representations in mathematical and scientific reasoning and problem solving: Analysis and improvement. Leuven: FWO Scientific Network.

  • Ainsworth, S. (1999). Designing effective multi-representational learning environments (SRC Tech. Rep. No. 58). Nottingham: University of Nottingham.

  • Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11, 25–61. doi:10.1207/S15327809JLS1101_2.

    Article  Google Scholar 

  • Ainsworth, S. E., Bibby, P. A., & Wood, D. J. (1998). Analysing the costs and benefits of multi-representational learning environments. In M. W. van Someren, P. Reimann, H. P. A. Boshuizen, & T. de Jong (Eds.), Learning with multiple representations (pp. 120–134). Amsterdam: Pergamon.

    Google Scholar 

  • Borba, M. C., & Confrey, J. (1996). A student’s construction of transformations of functions in a multiple representational environment. Educational Studies in Mathematics, 31, 319–337. doi:10.1007/BF00376325.

    Article  Google Scholar 

  • Cheng, P. C. H. (2000). Unlocking conceptual learning in mathematics and science with effective representational systems. Computers & Education, 33, 109–130. doi:10.1016/S0360-1315(99)00028-7.

    Article  Google Scholar 

  • Confrey, J., & Maloney, A. (1996). Function probe. Communications of the ACM, 39(8), 86–87. doi:10.1145/232014.232036.

    Article  Google Scholar 

  • Cox, R. (1996). Analytical reasoning with multiple external representations. Unpublished doctoral dissertation, Department of Artificial Intelligence, University of Edinburgh, Scotland.

  • DeBellis, V. A., & Goldin, G. A. (2006). Affect and meta-affect in mathematical problem-solving: A representational perspective. Educational Studies in Mathematics, 63(2), 131–147. doi:10.1007/s10649-006-9026-4.

    Article  Google Scholar 

  • de Jong, T., Ainsworth, S., Dobson, M., van der Hulst, A., Levonen, J., & Reimann, P. (1998). Acquiring knowledge in science and math: The use of multiple representations in technology based learning environments. In M. W. van Someren, P. Reimann, H. P. A. Boshuizen, T. de Jong, et al. (Eds.), Learning with multiple representations (pp. 9–40). Amsterdam: Pergamon.

    Google Scholar 

  • Demetriou, A. (1998). Cognitive development. In A. Demetriou, W. Doise, & C. Van Lieshout (Eds.), Life-span developmental psychology (pp. 179–270). Chichester: Wiley.

    Google Scholar 

  • Dienes, Z. P. (1960). Building up mathematics. London: Anchor Press, Hutchinson Educational.

    Google Scholar 

  • diSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10, 117–160.

    Google Scholar 

  • diSessa, A. A., & Sherin, B. L. (2000). Meta-representation: An introduction. Journal of Mathematical Behavior, 19, 385–398.

    Article  Google Scholar 

  • Duval, R. (2002). The cognitive analysis of problems of comprehension in the learning of mathematics. Mediterranean Journal for Research in Mathematics Education, 1(2), 1–16.

    Google Scholar 

  • Elia, I., Panaoura, A., Eracleous, A., & Gagatsis, A. (2007). Relations between secondary pupils’ conceptions about functions and problem solving in different representations. International Journal of Science and Mathematics Education, 5, 533–556.

    Article  Google Scholar 

  • Elia, I., Panaoura, A., Gagatsis, A., Gravvani, K., & Spyrou, P. (2006). An empirical four-dimensional model for the understanding of function. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education: Vol. 3 (pp. 137–142). Prague: Charles University.

  • Ellis, S. (1997). Strategy choice in sociocultural context. Developmental Review, 17, 490–524.

    Article  Google Scholar 

  • Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17, 105–121.

    Article  Google Scholar 

  • Gagatsis, A., Christou, C., & Elia, I. (2004). The nature of multiple representations in developing mathematical relationships. Quaderni di Ricerca in Didattica, 14, 150–159.

    Google Scholar 

  • Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24, 645–657.

    Article  Google Scholar 

  • Gilmore, D. J., & Green, T. R. G. (1984). Comprehension and recall of miniature programs. International Journal of Man-Machine Studies, 21, 31–48.

    Article  Google Scholar 

  • Goldin, G. A. (2000). Affective pathways and representation in mathematical problem solving. Mathematical Thinking and Learning, 2(3), 209–219.

    Article  Google Scholar 

  • Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W. Gary Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 275–285). Reston, NJ: NCTM.

    Google Scholar 

  • Grawemeyer, B. (2006). Evaluation of ERST—An external representation selection tutor. In D. Barker-Plummer, R. Cox, & N. Swoboda (Eds.), Diagrammatic representation and inference: 4th international conference, diagrams 2006, Stanford, CA, USA, June 28–30, 2006, proceedings (pp. 154–167). New York: Springer.

  • Greeno, J. G., & Hall, R. P. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78(5), 361–367.

    Google Scholar 

  • Gruber, H., Graf, M., Mandl, H., Renkl, A., & Stark, R. (1995, August). Fostering applicable knowledge by multiple perspectives and guided problem solving. Paper presented at the 6th conference of the European Association for Research on Learning and Instruction, Nijmegen, The Netherlands.

  • Hollands, J. G., & Spence, I. (1998). Judging proportion with graphs: The summation model. Applied Cognitive Psychology, 12, 173–190.

    Article  Google Scholar 

  • Kaput, J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515–556). New York: Macmillan.

    Google Scholar 

  • Kollöffel, B., de Jong, T., & Eysink, T. (2005, August). The effects of representational format in simulation-based inquiry learning. Paper presented at the 11th conference of the European Association for Research on Learning and Instruction, Nicosia, Cyprus.

  • Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60, 1–64.

    Google Scholar 

  • Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NY: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Luwel, K., Torbeyns, J., Schillemans, V., Onghena, P., & Verschaffel, L. (in press). Strengths and weaknesses of the choice/no-choice method in research on strategy use. European Psychologist.

  • Meyer, J. (2000). Performance with tables and graphs: Effects of training and a visual search model. Ergonomics, 43, 1840–1865.

    Article  Google Scholar 

  • Meyer, J., Shinar, D., & Leiser, D. (1997). Multiple factors that determine performance with tables and graphs. Human Factors: The Journal of the Human Factors and Ergonomics Society, 39, 268–286.

    Article  Google Scholar 

  • Nunes, T. N., Schliemann, A. D., & Carraher, D. W. (1993). Street mathematics and school mathematics. New York: Cambridge University Press.

    Google Scholar 

  • Ozgun-Koca, S. A. (2004). The effects of multiple linked representations on students’ learning of linear relationships. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 26, 82–90.

    Google Scholar 

  • Roth, W. M., & Bowen, G. M. (2001). Professionals read graphs: A semiotic analysis. Journal for Research in Mathematics Education, 32, 159–194.

    Article  Google Scholar 

  • Siegler, R. S., & Lemaire, P. (1997). Older and younger adults’ strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology: General, 126, 71–92.

    Article  Google Scholar 

  • Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59–84). Washington, DC: The Mathematical Association of America.

    Google Scholar 

  • Sparrow, J. A. (1989). Graphical displays in information systems: Some data properties influencing the effectiveness of alternative formats. Behaviour & Information Technology, 8, 43–56.

    Article  Google Scholar 

  • Spiro, R. J., & Jehng, J. C. (1990). Cognitive flexibility and hypertext: Theory and technology for the nonlinear and multidimensional traversal of complex subject matter. In D. Nix & R. Spiro (Eds.), Cognition, education, and multimedia: Exploring ideas in high technology (pp. 163–205). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Tabachneck, H. J. M., Koedinger, K. R., & Nathan, M. J. (1994). Towards a theoretical account of strategy use and sense making in mathematical problem solving. In A. Ram & K. Eiselt (Eds.), 16th annual conference of the Cognitive Science Society (pp. 836–841). Hillsdale, NJ: Lawrence Erlbaum Associates.

  • Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 7(2), 71–91.

    Google Scholar 

  • Uesaka, Y., & Manalo, E. (2006). Active comparison as a means of promoting the development of abstract conditional knowledge and appropriate choice of diagrams in math word problem solving. In D. Barker-Plummer, R. Cox, & N. Swoboda (Eds.), Diagrammatic representation and inference: 4th international conference, diagrams 2006, Stanford, CA, USA, June 28–30, 2006, proceedings (pp. 181–195). New York: Springer.

  • Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359

    Article  Google Scholar 

  • Vessey, I. (1991). Cognitive fit: A theory-based analysis of the graph versus tables literature. Decision Sciences, 22, 219–240.

    Article  Google Scholar 

  • Warner, L. B. (2005). Behaviors that indicate mathematical flexible thought. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick.

  • Watson, C. J., & Driver, R. W. (1983). The influence of computer graphics on the recall of information. MIS Quarterly, 7(1), 45–53.

    Article  Google Scholar 

  • Wickens, C. D., & Andre, A. D. (1990). Proximity compatibility and information display: Effects of color, space and objectness on information integration. Human Factors, 32, 61–78.

    Google Scholar 

  • Yerushalmy, M. (1991). Student perceptions of aspects of algebraic function using multiple representation software. Journal of Computer Assisted Learning, 7, 42–57.

    Article  Google Scholar 

  • Yerushalmy, M. (2006). Slower algebra students meet faster tools: Solving algebra word problems with graphing software. Journal for Research in Mathematics Education, 37, 356–387.

    Article  Google Scholar 

Download references

Acknowledgments

This research was partially supported by Grant GOA 2006/01 “Developing adaptive expertise in mathematics education” from the Research Fund K. U. Leuven, Belgium, and by Grant G.0637.09 “Representational adaptivity in mathematical thinking and learning: Analysis and improvement” of the Research Foundation, Flanders.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ana Acevedo Nistal.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Acevedo Nistal, A., Van Dooren, W., Clarebout, G. et al. Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: a critical review. ZDM Mathematics Education 41, 627–636 (2009). https://doi.org/10.1007/s11858-009-0189-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-009-0189-1

Keywords

Navigation