The quality of conceptual change in mathematics – the case of number concept
DOI:
https://doi.org/10.7146/nomad.v9i2.147124Abstract
In this article, the main results are presented from a number concept test in which 538 students from 24 randomly selected Finnish upper secondary schools took part. The test included identification, classification and construction problems in the domain of rational and real numbers. In addition, the students were asked to explain their answers and estimate their certainty about them. The theories of conceptual change and of mathematics concept formation were used to categorize students’ explanations into different levels. The results indicate the clearly restricted nature of students’ prior thinking of whole numbers and of their everyday experiences of counting and continuity. On the basis of the results, we claim that the problems that students have with these difficult concepts are not only due to the complexity or abstract nature of the concepts to be learned, but also to the quality of their prior knowledge, which is not sufficiently taken into account in traditional teaching.
References
Boyer, C. B. (1949). The history of calculus and its conceptual development. New York: Dover Publications.
Carey, S. (1985). Conceptual change in childhood. Cambridge, MA: MIT Press.
Carey, S., & Spelke, E. (1994). Domain-specific knowledge and conceptual change. In L. Hirsfeld & S. Gelman (Eds.), Mapping the mind. Domain specificy in cognition and culture (pp. 169-200). Cambridge, MA: Cambridge University Press. https://doi.org/10.1017/CBO9780511752902.008
Chi, M. T. H., Slotta, J. D., & de Leeuw, N. (1994). From things to process: A theory of conceptual change for learning science concepts. Learning and Instruction, 4, 27-43. https://doi.org/10.1016/0959-4752(94)90017-5
Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47203-1_10
Dantzig, T. (1930/1954). Number, the language of science. New York: The Free Press. https://doi.org/10.2307/2224269
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-123). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47203-1_7
Fischbein, E. (1987). Intuition in science and mathematics. An educational approach. Dordrecht: D. Reidel Publishing Company.
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. https://doi.org/10.1007/BF01273899
Gallistel, G. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43-74. https://doi.org/10.1016/0010-0277(92)90050-R
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 (3), 219-245. https://doi.org/10.1023/A:1003473509628
Hatano, G. (1996). A conception of knowledge acquisition and its implications for mathematics education. In L. P Steffe, P. Neshner, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 197-218). Hillsdale: Lawrence Erlbaum Associates.
Hatano, G., & Inagaki, K. (1998). Qualitative changes in intuitive biology. European Journal of Psychology of Education, 12, 111-130. https://doi.org/10.1007/BF03173080
Ioannides, C. & Vosniadou, S. (2002). The changing meanings of force. Cognitive Science Quarterly, 2 (1), 5-62.
Karmiloff-Smith, A. (1995). Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: MIT Press.
Landau, E. (1951/ 1960). Foundations of analysis. The arithmetic of whole, rational, irrational and complex numbers. New York: Chelsea Publishing.
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. 233-258). Dordrecht: Kluwer Academic publishers.
Neumann, R. (1998). Schülervorstellungen bezüglich der Dichtheit von Bruchzahlen. Mathematica Didactica. Zeitschrift für Didaktik der Mathematik, 21(1), 109-119.
Piaget, J., (1980/1974). Adaptation and intelligence. Chicago: University of Chicago Press.
Posner, G., Strike, K., Hewson, P., & Gertzog, W. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66(2), 211-227. https://doi.org/10.1002/sce.3730660207
Schwarzenberger, R., & Tall, D. (1978). Conflict in the learning of real numbers and limits. Mathematics Teaching, 82, 44-49.
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. https://doi.org/10.1007/BF00302715
Sfard, A. & Linchevski, L. (1994). The gains and pitfalls of reification - the case of algebra. Educational Studies in Mathematics, 26, 191-228. https://doi.org/10.1007/BF01273663
Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371-397. https://doi.org/10.1007/BF00240986
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). Hillsadale, NJ: Lawrence Erlbaum.
Starkey, P., Spelke, E., & Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36, 97-127. https://doi.org/10.1016/0010-0277(90)90001-Z
Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31(3), 258-276. https://doi.org/10.2307/749807
Vamvakoussi, X. & Vosniadou, S. (2002). Conceptual change in mathematics: From the set of natural to the set of rational numbers. In S. Lehti & K. Merenluoto (Eds.), Proceedings of the third European Symposium on Conceptual Change (pp. 56-61). University of Turku.
Verschaffel, L. DeCorte, E., & Van Coillie, V. (1988). Specifying the multi- plier effect on children's solutions of simple multiplication word problems. In A. Borbas (Ed.), Proceedings of the annual conference of the international group for the Psychology of Mathematics Education (pp. 617-624). Veszprem: OOK printing house.
Vosniadou, S. (1994). Capturing and modelling the process of conceptual change. Learning and Instruction, 4, 45-69. https://doi.org/10.1016/0959-4752(94)90018-3
Vosniadou, S. (1999). Conceptual change research: state of art and future directions. In W. Schnotz, S. Vosniadou & M. Carretero (Eds.), New perspectives on conceptual change (pp. 3-14). Oxford: Elsevier Science.
Williams, S. R. (1991). Models of limit held by college students. Journal for Research in Mathematics Education, 22(3), 219-236. https://doi.org/10.2307/749075
Wittgenstein, L. (1969). Über gewissheit / On certainty. Oxford: Basil Blackwell.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
