Abstract
In this chapter some special features of mathematical knowledge are considered in order to better understand the nature of conceptual change in this domain. In learning mathematics, every extension to the number concept demands, not only accepting new concepts, but new logic as well. This new logic more or less contradicts the prior fundamental logic of natural numbers. Therefore, misconceptions and learning difficulties are possible at every enlargement. To understand the problems students have in the conceptual change pertaining to the enlargement of the number concept a test was administered to 564 students (mean age 17.3) from randomly selected Finnish upper secondary schools. The test included identification, classification and construction problems in the domain of rational and real numbers. We found that changes of number conceptions, which was measured through questions in the domain of rational and real numbers, was not adequately carried out by the majority of the students who had just finished their first calculus class. While working on the tasks on the more advanced numbers they spontaneously used the logic and general presumptions of natural numbers or based their answers on their everyday intuition. The number concept of the majority of these students seemed to be based on the spontaneous logic of natural numbers but had also fragmented pieces of more advanced numbers. The students tended to overestimate the certainty of their answers when they used the logic of natural numbers even if it was erroneous.
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References
Boyer, C. B. (1959). The history of calculus and its conceptual development (2nd ed.). New York: Dover Publications.
Carey, S., & Spelke, E. (1994). Domain-specific knowledge and conceptual change. In L. A. Hirschfeld & S. A. Gelman (Eds.), Mapping the mind. Domain specificity in cognition and culture (pp. 169–200). Cambridge, MA: Cambridge University Press.
Chi, M. (1992). Conceptual Change within and across Ontological Categories: Examples from Learning and Discovery in Science. In R. Giere (Ed.) Cognitive models of science. Minnesota Studies in the Philosophy of Science (pp. 129–186). Minneapolis, MN: University of Minnesota Press.
Chi, M., Slotta, J., & de Leeuw, N. (1994). From things to process: A theory of conceptual change for learning science concepts. Learning and Instruction, 26, 27–43.
Cornu, B. (1991). Limits, In Tall, D. (Ed.) Advanced mathematical thinking (pp. 153–166). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Dantzig, T. (1954). Number, the language of science (4th ed.) New York: The Free Press.
Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall, D. (Ed.) Advanced mathematical thinking (pp. 25–41). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.) Advanced mathematical thinking (pp. 95–123). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Fischbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational numbers in high-school students and prospective teachers. Educational Studies in Mathematics, 29, 29–44.
Fischbein, E. (1987). Intuition in science and mathematics. An educational approach. Dordrecht, The Netherlands: D. Reidel Publishing Company.
Gallistel, G. R,. & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43–74.
Gelman, R., & Brenneman, K. (1994). First principles can support both universal and culture-specific learning about number and music. In L. A. Hirschfeld, & S. A. Gelman (Eds.) Mapping the mind. Domain specificy in cognition and culture (pp 369–390). Cambridge: Cambridge University Press.
Goodson-Espy, T. (1998). The roles of reification and reflective abstraction in the development of abstract thought: transition from arithmetic to algebra. Educational Studies in mathematics, 36, 219–245.
Cifarelli, V. (1988). The role of abstraction as a learning process in mathematical problem solving. Unpublished doctoral dissertation, Purdue University, IN.
Hartnett, P., & Gelman, R. (1998). Early understanding of numbers: paths or barriers to the construction of new understandings? Learning and Instruction, 8, 341–374.
Harel, G., & Kaput, J. (1991). The role of conceptual entities and their symbols in building advanced mathematical concepts. In D. Tall (Ed.) Advanced mathematical thinking (pp. 82–94). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Hatano, G. (1996). A conception of knowledge acquisition and its implications for mathematics education. In P. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.) Theories of mathematical learning (pp. 197–217). Mahwah, NJ: Lawrence Erlbaum Associates.
Kaput, J. (1994). Democratizing access to calculus: New routes to old roots. In A. J. Schoenfeld (Ed.) Mathematical thinking and problem solving (pp. 77–155). Hillsdale, NJ: Lawrence Erlbaum Associates.
Karmiloff-Smith, A. 1995. Beyond modularity. A developmental perspective on cognitive science. Cambridge, MA: MIT Press.
Kieren, T. (1992). Rational and fractional numbers as mathematical and personal knowledge: implications for curriculum and instruction. In G. Leinhardt, R. Putnam, & R. Hattrup (Eds.) Analysis of arithmetic for mathematics teaching (pp. 323–371). Hillsdale, NJ: Lawrence Erlbaum Associates.
Kline, M. (1980). Mathematics. The loss ofcertainty. New York: Oxford University Press.
Landau, E. (1960). Foundations of analysis. The arithmetic of whole, rational, irrational and complex numbers (2nd ed.). New York: Chelsea Publishing.
Lehtinen, E. (1998, November). Conceptual change in mathematics. Paper presented at the Second European Symposium on Conceptual Change. Madrid, Spain.
Lehtinen, E., Merenluoto, K., & Kasanen, E (1997). Conceptual change in mathematics: From rational to (un)real numbers. European Journal of Psychology of Education, 12, 131–145.
Lehtinen, E., & Ohlsson, S. (1999, August). The function of abstraction in deep learning. Paper presented at the 8th European Conference for Research on Learning and Instruction, Göteborg, Sweden.
Merenluoto K., & Lehtinen E. (2000) The “conflicting” concepts of continuity and limit, a conceptual change perspective: Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education, T. Nakahara & M. Koyama (Eds), Hiroshima University, 3, 303–310.
Neumann, R. (1998). Schülervorstellungen bezüglich der Dichtheit von Bruchzahlen. Mathematica didadtica. Zeitschrift für Didaktik der Mathematik. 21, 109–119.
Ohlsson, S., & Lehtinen, E. (1997). Abstraction and the acquisition of complex ideas. International Journal of Educational Research, 27, 37–48.
Russell, B. (1993). Introduction to mathematical philosophy. New York: Dover Publications.
Russell, B. (1996). The principles of mathematics (2nd ed.) London: Norton & Company.
Rudin, W. (1953). Principles of mathematical analysis. London: McGraw-Hill
diSessa, A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10, 105–225.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.
Sfard, A, & Linchevski, L. (1994). The gains and pitfalls of reification — the case of algebra. Educational Studies in Mathematics, 26, 191–228
Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397.
Spelke, S. E. 1991. Physical knowledge in infancy: Reflections on Piaget’s theory. In S. Carey & R. Gelman (Eds.) The epigenesis of mind. Essays on biology and cognition (pp. 133–170). Hillsdale: NJ: Lawrence Erlbaum Associates.
Stafilidou, M., & Vosniadou, S. (1999, August). Children’s believes about the mathematical concept of fraction. Abstracts. Proceedings of the European Conference for Research on Learning and Instruction, Göteborg, Sweden.
Starkey, P., Spelke, E,. & Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36, 97–127.
Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31, 258–276.
Tall, D. & Schwarzenberger, R. (1978). Conflict in the learning of real numbers and limits. Mathematics Teaching 82, 44–49.
Tall, D. (1991). The Psychology of Advanced Mathematical Thinking. In Tall, D, (Ed.) Advanced mathematical thinking (pp. 3–24). Dordrecht, The Netherlands: Kluwer Academic publishers.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.
Vosniadou, S. (1994). Capturing and modelling the process of conceptual change. Learning and Instruction, 4, 45–69.
Vosniadou, S. (1996). Towards a revised cognitive psychology for new advances in learning and instruction. Learning and Instruction, 6, 95–109.
Vosniadou, S., & Ioannides, C. (1998). From conceptual development to science Education: a psychological point of view. International Journal of Science Education, 20, 1213–1230.
Williams, S. R. (1991). Models of limit held by college students. Journal for Research in Mathematics Education, 22, 219–236.
Wistedt, I., & Martinsson, M. (1996). Orchestrating a mathematical theme: eleven year olds discuss the problem of infinity. Learning and Instruction, 6, 173–185.
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Merenluoto, K., Lehtinen, E. (2002). Conceptual Change in Mathematics: Understanding the Real Numbers. In: Limón, M., Mason, L. (eds) Reconsidering Conceptual Change: Issues in Theory and Practice. Springer, Dordrecht. https://doi.org/10.1007/0-306-47637-1_13
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DOI: https://doi.org/10.1007/0-306-47637-1_13
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