Abstract
In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant similarity to valid examples of the concept, and consequently are more often mistakenly identified as examples. We describe and discuss these notions and present a study regarding kindergarten children’s grasp of nonexamples of triangles.
Similar content being viewed by others
Notes
Kindergarten children were labeled K1-K65.
References
Attneave, F. (1957). Transfer of experience with a class schema to identification of patterns and shapes. Journal of Experimental Psychology, 54, 81–88.
Battista, M. T. (2007). The development of geometric and spatial thinking. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843–908). Reston, VA: National Council of Teachers of Mathematics.
Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th PME International Conference (vol. 1, pp. 126–154). Czech Republic: PME.
Burger, W., & Shaughnessy, J. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17(1), 31–48.
Clements, D. H. (2003). Teaching and learning geometry. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 151–178). Reston, VA: National Council of Teachers of Mathematics.
Clements, D., Swaminathan, S., Hannibal, M., & Sarama, J. (1999). Young children’s concepts of shape. Journal for Research in Mathematics Education, 30(2), 192–212. doi:10.2307/749610.
Cohen, M., & Carpenter, J. (1980). The effects of non-examples in geometrical concept acquisition. International Journal of Mathematical Education in Science and Technology, 11(2), 259–263.
Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht, the Netherlands: Reidel.
Fischbein, E. (1993). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. Scholz, R. Strässer, & B. Winkelmann (Eds.), Didactics of Mathematics as a Scientific Discipline (pp. 231–245). Dordrecht, The Netherlands: Kluwer.
Hannibal, M. (1999). Young children’s developing understanding of geometric shapes. Teaching Children Mathematics, 5(6), 353–357.
Hasegawa, J. (1997). Concept formation of triangles and quadrilaterals in the second grade. Educational Studies in Mathematics, 32, 157–179.
Hershkowitz, R. (1989). Visualization in geometry – two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.
Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition (pp. 70–95). Cambridge, UK: Cambridge University Press.
Hershkowitz, R., & Vinner, S. (1983). The role of critical and non-critical attributes in the concept image of geometrical concepts. In R. Hershkowitz (Ed.), Proceedings of the 7th PME International Conference (pp. 223–228). Rehovot, Israel: Weizmann Institute of Science.
Kellogg, R. (1980). Feature frequency and hypothesis testing in the acquisition of rule-governed concepts. Memory & Cognition, 8, 297–303.
Klausmeier, H., & Feldman, K. (1975). Effects of a definition and a varying number of examples and nonexamples on concept attainment. Journal of Educational Psychology, 67, 174–178.
Klausmeier, H., & Sipple, T. (1980). Learning and teaching concepts. New York: Academic.
Linchevsky, L., Vinner, S., & Karsenty, R. (1992). To be or not to be minimal? Student teachers’ views about definitions in geometry. In W. Geeslin, & K. Graham (Eds.), Proceedings of the 16th PME International Conference (vol. 2, pp. 48–55). New Hampshire, MA: University of New Hampshire.
Markman, E. (1989). Categorization and naming in children. Massachusetts: MIT.
Markman, E., & Watchtel, G. (1988). Children’s use of mutual exclusivity to constrain the meaning of words. Cognitive Psychology, 20(2), 121–157.
McKinney, C., Larkins, A., Ford, M., & Davis III, J. (1983). The effectiveness of three methods of teaching social studies concepts to fourth-grade students: an aptitude-treatment interaction study. American Educational Research Journal, 20, 663–670.
Petty, O., & Jansson, L. (1987). Sequencing examples and non-examples to facilitate concept attainment. Journal for Research in Mathematics Education, 18(2), 112–125.
Posner, M. I., & Keele, S. W. (1968). On the genesis of abstract ideas. Journal of Experimental Psychology, 77, 353–363.
Reed, S. K. (1972). Pattern recognition and categorization. Cognitive Psychology, 3, 382–407.
Rosch, E. (1973). Natural Categories. Cognitive Psychology, 4, 328–350.
Rosch, E., & Mervis, C. (1975). Family resemblances: studies in the internal structure of categories. Cognitive Psychology, 7, 773–605.
Schwarz, B., & Hershkowitz, R. (1999). Prototypes: brakes or levers in learning the function concept? Journal for Research in Mathematics Education, 30, 362–389.
Shaughnessy, J., & Burger, W. (1985). Spadework prior to deduction in geometry. Mathematics Teacher, 78(6), 419–427.
Smith, E., & Medin, D. (1981). Categories and concepts. Cambridge, MA: Harvard University.
Smith, E., Shoben, E., & Rips, L. (1974). Structure and process in semantic memory: a featural model for semantic decisions. Psychological Review, 81, 214–241.
van Hiele, P. M., & van Hiele, D. (1958). A method of initiation into geometry. In H. Freudenthal (Ed.), Report on methods of initiation into geometry. Groningen: Walters.
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht, The Netherlands: Kluwer.
Vinner, S., & Hershkowitz, R. (1980). Concept images and common cognitive paths in the development of some simple geometric concepts. In R. Karplus (Ed.), Proceedings of the 4th PME International Conference, 177–184.
Watson, A., & Mason, J. (2005). Mathematics as a Constructive Activity: Learners generating examples. Mahwah: Erlbaum.
Waxman, S. (1999). The dubbin ceremony revisited: Object naming and categorization in infancy and early childhood. In D. Medin, & S. Atran (Eds.), Folkbiology. Cambridge, MA: MIT.
Waxman, S., & Braun, I. (2005). Consistent (but not variable) names as invitations to form object categories: new evidence from 12-month-old infants. Cognition, 95(3), B59–B68.
Wilson, S. (1986). Feature frequency and the use of negative instances in a geometric task. Journal for Research in Mathematics Education, 17, 130–139.
Wilson, S. (1990). Inconsistent ideas related to definitions and examples. Focus on Learning Problems in Mathematics, 12, 31–47.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tsamir, P., Tirosh, D. & Levenson, E. Intuitive nonexamples: the case of triangles. Educ Stud Math 69, 81–95 (2008). https://doi.org/10.1007/s10649-008-9133-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-008-9133-5