Engaging Young Children with Mathematical Activities Involving Different Representations: Triangles, Patterns, and Counting Objects
Keywords:
counting, multiple representations, repeating patterns, triangles, young children
Abstract
This paper synthesises research from three separate studies, analysing how different representations of a mathematical concept may affect young children’s engagement with mathematical activities. Children between five and seven years old engaged in counting objects, identifying triangles and completing repeating patterns. The implementation of three counting principles were investigated: the one-to-one principle, the stable-order principle and the cardinal principal. Children’s reasoning when identifying triangles was analysed in terms of visual, critical and non-critical attribute reasoning. With regard to repeating patterns, we analyse children’s references to the minimal unit of repeat of the pattern. Results are discussed in terms of three functions of multiple external representations: to complement, to constrain and to construct.
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References
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Tirosh, D., Tsamir, P., Levenson, E., Tabach, M., & Barkai, R. (2013). Two children, three tasks, one set of figures: Highlighting different elements of children’s geometric knowledge. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the eighth congress of the European society for research in mathematics education (CERME 8) (pp. 2228–2237). Ankara: Middle East Technical University and ERME.
Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive nonexamples: The case of triangles. Educational Studies in Mathematics, 69(2), 81–95.
Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (2017). Repeating patterns in kindergarten: Findings from children’s enactments of two activities. Educational Studies in Mathematics, 96(1), 83–99.
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 (pp. 67–80). Groningen: Walters.
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 (pp. 177–184). Berkley, CA.
Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking
and algebraic notation. Educational Studies in Mathematics, 49(3), 379–402.
Baroody, A. J. (1987). Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York, NY: Teacher’s College Press.
Burger, W., & Shaughnessy, J. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17(1), 31–48.
Dienes, Z. P. (1969). Building Up Mathematics. London, UK: Hutchison Education.
Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.
Fuson, K. C. (1991). Children’s early counting: Saying the number-word sequence, counting objects, and understanding cardinality. In K. Durkin & B. Shire (Eds.), Language and mathematical education (pp. 27–39). Milton Keynes, UK: Open University Press.
Geary, D. C., Bow-Thomas, C. C., & Yao, Y. (1992). Counting knowledge and skill in cognitive addition: A comparison of normal and mathematically disabled children. Journal of Experimental Child Psychology, 54(3), 372–391.
Gelman, R., & Gallistel, C. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.
Griffin, S. (2004). Building number sense with Number Worlds: A mathematics program for young children. Early Childhood Research Quarterly, 19(1), 173–180.
Han, Y., & Ginsburg, H. P. (2001). Chinese and English mathematics language: The relation between linguistic clarity and mathematics performance. Mathematical Thinking and Learning, 3(2-3), 201–220.
Hannibal, M. (1999). Young children’s developing understanding of geometric shapes. Teaching Children Mathematics, 5(6), 353–357.
Hershkowitz, R. (1989). Visualization in geometry – two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in mathematics, 47(2), 175–197.
Rittle-Johnson, B., Fyfe, E. R., McLean, L. E., & McEldoon, K. L. (2013). Emerging understanding of patterning in 4-year-olds. Journal of Cognition and Development, 14(3), 376–396.
Sarama, J., & Clements, D. (2009). Early childhood mathematics education research: Learning trajectories for young children. London, UK: Routledge.
Tirosh, D., Tsamir, P., Levenson, E., Tabach, M., & Barkai, R. (2013). Two children, three tasks, one set of figures: Highlighting different elements of children’s geometric knowledge. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the eighth congress of the European society for research in mathematics education (CERME 8) (pp. 2228–2237). Ankara: Middle East Technical University and ERME.
Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive nonexamples: The case of triangles. Educational Studies in Mathematics, 69(2), 81–95.
Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (2017). Repeating patterns in kindergarten: Findings from children’s enactments of two activities. Educational Studies in Mathematics, 96(1), 83–99.
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 (pp. 67–80). Groningen: Walters.
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 (pp. 177–184). Berkley, CA.
Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking
and algebraic notation. Educational Studies in Mathematics, 49(3), 379–402.
Published
2018-06-27
How to Cite
Tirosh, D., Tsamir, P., Barkai, R., & Levenson, E. (2018). Engaging Young Children with Mathematical Activities Involving Different Representations: Triangles, Patterns, and Counting Objects. Center for Educational Policy Studies Journal, 8(2), 9–30. https://doi.org/10.26529/cepsj.271
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