Abstract
The aim of this paper is to propose a theoretical model to analyze prospective teachers’ reasoning and knowledge of real numbers, and to provide an empirical verification of it. The model is based on Sierpinska’s theory of theoretical thinking. Data were collected from 59 prospective teachers through a written test and interviews. The data indicated that mathematical tasks on real numbers, based on Sierpinska’s theory, could be categorized according to whether they require reflective, systemic or analytic thinking. Analysis of the data identified three different groups of prospective teachers reflecting different types of theoretical thinking about real numbers. The interviews confirmed the empirical data from the written test, and provided a better insight into the thinking and characteristic features of the prospective teachers in each group. The analysis also indicated that the participants were more successful in tasks requiring systemic and analytic thinking, and only when this was achieved were they able to solve problems which required reflective thinking. Implications for teaching related to the findings of the study are discussed.
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References
Arcavi, A., Bruckheimer, M., & Ben-Zvi, R. (1987). History of mathematics for teachers: The case of irrational numbers. For the Learning of Mathematics, 7(2), 18–23.
Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht: Reidel.
Fischbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational number in high-school student and prospective teachers. Educational Studies in Mathematics, 29(1), 29–44.
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). Larnaka, Cyprus: Department of Education, University of Cyprus.
Malara, N. (2001). From fractions to rational numbers in their structure: Outlines for an innovative didactical strategy and the question of density. In J. Novotná (Ed.), Proceedings of the 2nd Conference of the European Society for Research Mathematics Education (pp. 35–46). Praga: Univerzita Karlova v Praze, Pedagogická Faculta.
Marcoulides, G. A., & Kyriakides, L. (2010). Structural equation modelling techniques. In B. Creemers, L. Kyriakides, & P. Sammons (Eds.), Methodological advances in educational effectiveness research (Quantitative methodology series) (pp. 277–302). London, UK and New York, USA: Routledge.
Marcoulides, G. A., & Schumacker, R. E. (1996). Advanced structural equation modeling. New Jersey: Lawrence Erlbaum Associates.
Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In M. Limón & L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 233–257). Dordrecht: Kluwer.
Merenluoto, K., & Lehtinen, E. (2004). Number concept and conceptual change: Towards a systemic model of the processes of change. Learning and Instruction, 14(5), 519–534.
Muthén, B. (2004). Latent variable analysis: Growth mixture modeling and related techniques for longitudinal data. In D. Kaplan (Ed.), Handbook of quantitative methodology for the social sciences (pp. 345–368). Newbury Park, CA: Sage.
Muthén, L. K., & Muthén, B. O. (1998). Mplus user’s guide. Los Angeles, CA: Muthén & Muthén.
National Council of Teachers of Mathematics (NCTM). (2006). Navigating through number and operations in grades 9–12. Reston, VA: NCTM.
Neumann, R. (1998). Students’ ideas on the density of fractions. In H. G. Weigand, A. Peter-Koop, N. Neil, K. Reiss, G. Törner, & B. Wollring (Eds.), Proceedings of the Annual Meeting of the Gesellschaft fur Didaktik der Mathematik on Didactics of Mathematics (pp. 97–104). Munich: Gesellschaft fur Didaktik der Mathematik.
Richardson, V. (Ed.). (2001). Handbook of research teaching (4th ed.). Washington, DC: American Educational Research Association.
Shirpley, B. (2000). A new inferential test for path models based on directed acyclic graphs. Structural Equation Modeling, 7(2), 206–218.
Sierpinska, A. (1994). Understanding in mathematics. London: The Falmer Press.
Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J. L. Dorier (Ed.), On the teaching of linear algebra (pp. 209–246). Dordrecht: Kluwer.
Sierpinska, A. (2005). On practical and theoretical thinking and other false dichotomies in mathematics education. In M. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and sign - grounding mathematics education (pp. 117–135). New York: Springer.
Sierpinska, A., Bobos, A., & Pruncut, A. (2011). Teaching absolute value inequalities to mature students. Educational Studies in Mathematics, 78(3), 275–305.
Sierpinska, A., Nnadozie, A., & Oktaç, A. (2002). A study of relationships between theoretical thinking and high achievement in linear algebra. Retrieved May 14, 2012, from Anna Sierpinska http://annasierpinska.wkrib.com/pdf/Sierpinska-TT-Report.pdf
Sirotic, N., & Zazkis, R. (2004). Irrational numbers: Dimensions of knowledge. In D. E. McDougall & J. A. Ross (Eds.), Proceedings of the 26th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 1 (pp. 171–178). Ontario: Ontario Institute for Studies in Education, University of Toronto.
Sirotic, N., & Zazkis, R. (2007). Irrational number: The gap between formal and intuitive knowledge. Educational Studies in Mathematics, 65(1), 49–76.
Tirosh, D., Fischbein, E., Graeber, A., & Wilson, J. W. (1998). Prospective elementary teachers’ conceptions of rational numbers. Retrieved January 31, 2012, from http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html
Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453–467.
Vamvakoussi, X., & Vosniadou, S. (2007). How many numbers are there in a rational number interval? Constraints, synthetic models and the effect of the number line. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds.), Reframing the conceptual change approach in learning and instruction (pp. 265–282). The Netherlands: Elsevier.
Zazkis, R. (2005). Representing numbers: Prime and irrational. International Journal of Mathematical Education in Science and Technology, 36(2–3), 207–217.
Zazkis, R., & Sirotic, N. (2004). Making sense of irrational numbers: Focusing on representation. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 497–504). Norway: Bergen University College.
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Zachariades, T., Christou, C. & Pitta-Pantazi, D. Reflective, systemic and analytic thinking in real numbers. Educ Stud Math 82, 5–22 (2013). https://doi.org/10.1007/s10649-012-9413-y
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DOI: https://doi.org/10.1007/s10649-012-9413-y