Abstract
Mathematics education research is far from consensus on the roles visualization can play in the teaching and learning of mathematics. This chapter offers similarly diverse perspectives: Kupferman illustrates a university teacher’s endeavour to integrate visualization in teaching with an example of introducing the formal definition of limit to Year 1 students. He concludes that the benefits of a visually rich approach cannot be taken for granted, especially when students are not yet accustomed to it. To bring visualization into students’ mathematical ‘custom’ Presmeg calls for teaching visuality, recognising that the relationship between logical and visual thinking in mathematics is not polarized but orthogonal, and reminding us that effective teaching of visuality originates in teachers whose own preferences are mixed and flexible. Analogously, Nardi calls for a new didactical contract that makes the rules about visualization explicit to learners, while recognising that a deliberate ‘fuzziness’ of this contract can also allow the manoeuvring that is often so potent in mathematics. Much like Kupferman, and in support of Presmeg’s call for teaching visuality, Hershkowitz, through examples, acknowledges visualization as one of the languages of mathematics and as one of several ways of thinking mathematically. To be expressed, visual thinking needs a language, visual or other; and visual language, to be meaningful, needs to be attached to some conceptual entity. Finally, Yerushalmy picks up Hershkowitz’s cue for meaningful integration of visualization into teaching with examples, such as interactive diagrams in algebra, that illustrate the challenges, affordances and profound epistemological shifts inherent in visually sensitive curriculum design.
With contributions by
Rina Hershkowitz, Weizmann Institute of Science, Revohot, Israel
Raz Kupferman, Hebrew University of Jerusalem, Jerusalem, Israel
Norma Presmeg, Illinois State University, Normal, USA
Michal Yerushalmy, University of Haifa, Haifa, Israel
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References
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241.
Aspinwall, L., Shaw, K. L., & Presmeg, N. C. (1997). Uncontrollable mental imagery: graphical connections between a function and its derivative. Educational Studies in Mathematics, 33, 301–317.
Bishop, A. J. (1980). Spatial abilities and mathematics education—a review. Educational Studies in Mathematics, 11, 257–269.
Biza, I., Nardi, E., & Zachariades, T. (2009). Teacher beliefs and the didactic contract on visualization. For the Learning of Mathematics, 29(3), 31–36.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer Academic.
Byers, W. (2007). How mathematicians think: using ambiguity, contradiction, and paradox to create mathematics. Princeton: Princeton University Press.
Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th conference of the international group for the psychology of mathematics education (Vol. 1, pp. 33–48).
Dreyfus, T., Nardi, E., & Leikin, R. (2012). Forms of proof and proving. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education—the 19th international commission for mathematics instruction study (pp. 111–120). New York: Springer.
Duval, R. (1998). Geometry from a cognitive point a view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 37–52). Dordrecht: Kluwer Academic.
Duval, R. (1999). Representation, vision and visualization: cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proc. of the 21st conf. of the North American chapter of the int. group for the psychology of mathematics education (Vol. 1, pp. 3–26). Cuernavaca: PME-NA.
Eisenberg, T. (1994). On understanding the reluctance to visualize. ZDM. Zentralblatt für Didaktik der Mathematik, 26, 109–113.
Fischbein, E. (1987). Intuition in science and mathematics: an educational approach. Dordrecht: Reidel.
Giaquinto, M. (2007). Visual thinking in mathematics. New York: Oxford University Press.
Hanna, G., & Sidoli, N. (2007). Visualisation and proof: a brief survey of philosophical perspectives. ZDM. The International Journal on Mathematics Education, 39, 73–78.
Hershkowitz, R., & Markovits, Z. (1992). Conquer math concepts by developing visual thinking. The Arithmetic Teacher, 39, 38–41.
Hershkowitz, R., Arcavi, A., & Bruckheimer, M. (2001). Reflections on the status and nature of visual reasoning—the case of matches. International Journal of Mathematical Education in Science and Technology, 32, 255–265.
Iannone, P. (2009). Concept usage in proof production: mathematicians’ perspectives. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), ICMI study 19: proof and proving in mathematics education (Vol. 1, pp. 220–225). Taipei: National Taiwan Normal University.
Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.
Mancosu, P., Jorgensen, K. F., & Pedersen, S. A. (Eds.) (2005). Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer.
Naftaliev, E., & Yerushalmy, M. (2009). Interactive diagrams: alternative approach to the design of algebra inquiry. In M. Tzekaki, M. Kaldrimidou, & H. Sakonides (Eds.), Proceedings of the 33rd conference of the international group of psychology of mathematics education (Vol. 4, pp. 185–192). Thessaloniki: PME.
Nardi, E. (2008). Amongst mathematicians: teaching and learning mathematics at university level. New York: Springer.
Nardi, E. (2009). ‘Because this is how mathematicians work!’ ‘Pictures’ and the creative fuzziness of the didactical contract at university level. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), ICMI study 19: proof and proving in mathematics education (Vol. 2, pp. 112–117). Taipei: National Taiwan Normal University.
Nardi, E., Jaworski, B., & Hegedus, S. (2005). A spectrum of pedagogical awareness for undergraduate mathematics: from ‘tricks’ to ‘techniques’. Journal for Research in Mathematics Education, 36, 284–316.
Nardi, E., Biza, I., & Zachariades, T. (2012). ‘Warrant’ revisited: integrating teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation. Educational Studies in Mathematics, 79, 157–173.
Paivio, A. (1971). Imagery and verbal processes. New York: Holt, Rinehart & Winston.
Parzysz, B. (1988). ‘Knowing’ vs. ‘Seeing’: problems of the plane representation of space geometry figures. Educational Studies in Mathematics, 19, 79–92.
Piaget, J., & Inhelder, B. (1971). Mental imagery and the child. London: Routledge/Kegan Paul.
Presmeg, N. C. (1985). The role of visually mediated process in high school mathematics: a classroom investigation. Unpublished Ph.D. dissertation, University of Cambridge, England.
Presmeg, N. C. (2006a). Research on visualization in learning and teaching mathematics: emergence from psychology. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205–235). Rotterdam: Sense Publishers.
Presmeg, N. C. (2006b). A semiotic view of the role of imagery and inscriptions in mathematics teaching and learning. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th annual meeting of the international group for the psychology of mathematics education (Vol. 1, pp. 19–34). Prague: PME.
Presmeg, N. C. (2008a). An overarching theory for research on visualization in mathematics education. Plenary paper. In Proceedings of topic study group 20, visualization in the teaching and learning of mathematics 11th international Congress on mathematics education (ICME-11), Monterrey, Mexico, July 6–13, 2008. Published on the ICME-11 web site: http://tsg.icme11.org (TSG20). Accessed 6 June 2013.
Presmeg, N. C. (2008b). The power and perils of metaphor in making internal connections in trigonometry and geometry. In Proceedings of the 5th conference of the European society for research in mathematics, Larnaca, Cyprus, February 22–26, 2007 (pp. 161–170) (CD published March 2008).
Radford, L. (2012). Early algebraic thinking: epistemological, semiotic, and developmental issues. In 12th international Congress on mathematical education, 8–15 July 2012, COEX, Seoul, Korea.
Razel, M., & Eylon, B. (1990). Development of visual cognition: transfer effects of the Agam program. Journal of Applied Developmental Psychology, 11, 459–485.
Roh, K. (2010). An empirical study of students’ understanding of a logical structure in the definition of limit via the ε-strip activity. Educational Studies in Mathematics, 73, 263–279.
Schwartz, J. L. (2011). Unsolving linear and quadratic equations. Poincare Institute for Mathematics Education, Tufts University, Medford, MA.
Steen, L. A. (1988). The science of patterns. Science, 240, 611–616.
Stenning, K., & Lemon, O. (2001). Aligning logical and psychological perspectives on diagrammatic reasoning. Artificial Intelligence Review, 15, 29–62.
Suwarsono, S. (1982). Visual imagery in the thinking of seventh grade students. Unpublished Ph.D. dissertation, Monash University, Australia.
Tall, D. (Ed.) (1991). Advanced mathematical thinking. Dordrecht: Kluwer Academic.
Vinner, S., Hershkowitz, R., & Bruckheimer, M. (1981). Some cognitive factors as causes of mistakes in addition of fractions. Journal for Research in Mathematics Education, 12, 70–76.
Visual Math: Algebra (1995/2005). Center for educational technology, Ramat-Aviv. http://www.cet.ac.il/math-international/first.htm. Accessed 6 June 2013. http://www.cet.ac.il/math/function/english. Accessed 6 June 2013.
Whiteley, W. (2004). To see like a mathematician. In G. Malcolm (Ed.), Multidisciplinary approaches to visual representations and interpretations (Vol. 2, pp. 279–291). London: Elsevier.
Whiteley, W. (2009). Refutations: the role of counter-examples in developing proof. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), ICMI study 19: proof and proving in mathematics education (Vol. 2, pp. 257–262). Taipei: National Taiwan Normal University.
Yerushalmy, M. (2005). Challenging known transitions: learning and teaching algebra with technology. For the Learning of Mathematics, 25(3), 37–42.
Yerushalmy, M., & Botzer, G. (2011). Teaching secondary mathematics in the mobile age. In O. Zaslavsky & P. Sullivan (Eds.), Mathematics teacher education series: Vol. 6. Constructing knowledge for teaching secondary mathematics tasks to enhance prospective and practicing teacher learning (pp. 191–208). New York: Springer.
Yerushalmy, M., & Gilead, S. (1999). Structures of constant rate word problems: a functional approach analysis. Educational Studies in Mathematics, 39, 185–203.
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Nardi, E. (2014). Reflections on Visualization in Mathematics and in Mathematics Education. In: Fried, M., Dreyfus, T. (eds) Mathematics & Mathematics Education: Searching for Common Ground. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7473-5_12
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