, Volume 47, Issue 3, pp 497–509 | Cite as

Early-years teachers’ concept images and concept definitions: triangles, circles, and cylinders

  • Pessia Tsamir
  • Dina Tirosh
  • Esther Levenson
  • Ruthi Barkai
  • Michal Tabach
Original Article


This study investigates practicing early-years teachers’ concept images and concept definitions for triangles, circles, and cylinders. Teachers were requested to define each figure and then to identify various examples and non-examples of the figure. Teachers’ use of correct and precise mathematical language and reference to critical and non-critical attributes was also investigated. Results indicated that, in general, teachers were able to identify examples and non-examples of triangles and define triangles, were able to identify examples and non-examples of circles but had difficulties defining circles, and had some difficulties in both identifying examples and non-examples of cylinders and defining cylinders. Possible reasons for these results are discussed.


Critical Attribute Concept Definition Concept Image Mathematical Language Everyday Language 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research was supported by THE ISRAEL SCIENCE FOUNDATION (Grant No. 654/10).


  1. Attneave, F. (1957). Transfer of experience with a class schema to identification of patterns and shapes. Journal of Experimental Psychology, 54, 81–88.CrossRefGoogle Scholar
  2. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29, 14–22.Google Scholar
  3. Ball, D., Thames, M., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 59(5), 389–407.CrossRefGoogle Scholar
  4. Blömeke, S., & Delaney, S. (2012). Assessment of teacher knowledge across countries: a review of the state of research. ZDM - The International Journal on Mathematics Education, 44(3), 223–247.CrossRefGoogle Scholar
  5. Burger, W., & Shaughnessy, J. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17(1), 31–48.CrossRefGoogle Scholar
  6. Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38(2), 136–163.Google Scholar
  7. Clements, D. H., & Sarama, J. (2011). Early childhood teacher education: the case of geometry. Journal of Mathematics Teacher Education, 14(2), 133–148.CrossRefGoogle Scholar
  8. 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.CrossRefGoogle Scholar
  9. Delaney, S. (2012). A validation study of the use of mathematical knowledge for teaching measures in Ireland. ZDM - The International Journal on Mathematics Education, 44(3), 427–441.CrossRefGoogle Scholar
  10. Fischbein, E. (1993). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. Scholz, R. Straber, & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 231–245). Dordrecht: Kluwer.Google Scholar
  11. Fujita, T., & Jones, K. (2006). Primary trainee teachers’ understanding of basic geometrical figures in Scotland. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings 30th conference of the International Group for the Psychology of Mathematics Education (PME30) (Vol. 3, pp. 129–136). Prague, Czech Republic.Google Scholar
  12. Ginsburg, H. P., Kaplan, R. G., Cannon, J., Cordero, M. I., Eisenband, J. G., Galanter, M., et al. (2006). Helping early childhood educators to teach mathematics. In M. Zaslow & I. Martinez-Beck (Eds.), Critical issues in early childhood professional development (pp. 171–202). Baltimore: Paul H. Brookes.Google Scholar
  13. Ginsburg, H. P., Lee, J. S., & Boyd, J. S. (2008). Mathematics education for young children: What it is and how to promote it. Social Policy Report, XXII(I), 1–22.Google Scholar
  14. Hershkowitz, R. (1989). Visualization in geometry—two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.Google Scholar
  15. Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition (pp. 70–95). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  16. Inan, H. Z., & Dogan-Temur, O. (2010). Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research. ZDM - The International Journal on Mathematics Education, 42(5), 457–468.CrossRefGoogle Scholar
  17. Israel National Mathematics Preschool Curriculum (INMPC) (2008). Accessed 6 Oct 2014.
  18. Klausmeier, H., & Sipple, T. (1980). Learning and teaching concepts. New York: Academic Press.Google Scholar
  19. Klibanoff, R. S., Levine, S. C., Huttenlocher, J., Vasilyeva, M., & Hedges, L. V. (2006). Preschool children’s mathematical knowledge: The effect of teacher “math talk.”. Developmental Psychology, 42(1), 59.CrossRefGoogle Scholar
  20. Levenson, E., Tirosh, D., & Tsamir, P. (2011). Preschool geometry: Theory, research, and practical perspectives. Rotterdam: Sense.CrossRefGoogle Scholar
  21. Moss, J., Hawes, Z., Naqvi, S., & Caswell, B. (2015). Adapting Japanese Lesson Study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study. ZDM - The International Journal on Mathematics Education, 47(3).Google Scholar
  22. National Council of Teachers of Mathematics. (2006). Curriculum focal points for Prekindergarten through Grade 8 Mathematics: A quest for coherence. Reston: National Council of Teachers of Mathematics.Google Scholar
  23. Ouvrier-Buffet, C. (2006). Exploring mathematical definition construction processes. Educational Studies in Mathematics, 63(3), 259–282.CrossRefGoogle Scholar
  24. Rosch, E. (1973). Natural categories. Cognitive Psychology, 4, 328–350.CrossRefGoogle Scholar
  25. Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4–14.CrossRefGoogle Scholar
  26. Smith, E., Shoben, E., & Rips, L. (1974). Structure and process in semantic memory: a featural model for semantic decisions. Psychological Review, 81, 214–241.CrossRefGoogle Scholar
  27. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.CrossRefGoogle Scholar
  28. Tirosh, D., & Tsamir, P. (2008). Starting right: Mathematics in preschool. Unpublished research report. In Hebrew.Google Scholar
  29. Tirosh, D., Tsamir, P., & Levenson, E. (2011). Using theories to build kindergarten teachers’ mathematical knowledge for teaching. In K. Ruthven & T. Rowland (Eds.), Mathematical knowledge in teaching (pp. 231–250). Dordrecht: Springer.CrossRefGoogle Scholar
  30. Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive non-examples: the case of triangles. Educational Studies in Mathematics, 69(2), 81–95.CrossRefGoogle Scholar
  31. van Dormolen, J., & Arcavi, A. (2000). What is a circle? Mathematics in School, 29(5), 15–19.Google Scholar
  32. 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.Google Scholar
  33. Vandell, D. L., Belsky, J., Burchinal, M., Steinberg, L., & Vandergrift, N. (2010). Do effects of early child care extend to age 15 years? Results from the NICHD study of early child care and youth development. Child Development, 81(3), 737–756.CrossRefGoogle Scholar
  34. 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: Kluwer.Google Scholar
  35. Vinner, S. (2011). The role of examples in the learning of mathematics and in everyday thought processes. ZDM - The International Journal on Mathematics Education, 43(2), 247–256.CrossRefGoogle Scholar
  36. 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.Google Scholar
  37. Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: a case of a square. Educational Studies in Mathematics, 69(2), 131–148.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2014

Authors and Affiliations

  • Pessia Tsamir
    • 1
  • Dina Tirosh
    • 1
  • Esther Levenson
    • 1
  • Ruthi Barkai
    • 1
  • Michal Tabach
    • 1
  1. 1.Tel Aviv UniversityTel AvivIsrael

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