Skip to main content
Log in

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

  • Original Article
  • Published:
ZDM Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Attneave, F. (1957). Transfer of experience with a class schema to identification of patterns and shapes. Journal of Experimental Psychology, 54, 81–88.

    Article  Google Scholar 

  • 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 

  • Ball, D., Thames, M., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 59(5), 389–407.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Burger, W., & Shaughnessy, J. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17(1), 31–48.

    Article  Google Scholar 

  • 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 

  • Clements, D. H., & Sarama, J. (2011). Early childhood teacher education: the case of geometry. Journal of Mathematics Teacher Education, 14(2), 133–148.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

  • 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 

  • 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.

  • Hershkowitz, R. (1989). Visualization in geometry—two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.

    Google Scholar 

  • Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition (pp. 70–95). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Israel National Mathematics Preschool Curriculum (INMPC) (2008). http://meyda.education.gov.il/files/Tochniyot_Limudim/KdamYesodi/Math1.pdf. Accessed 6 Oct 2014.

  • Klausmeier, H., & Sipple, T. (1980). Learning and teaching concepts. New York: Academic Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Levenson, E., Tirosh, D., & Tsamir, P. (2011). Preschool geometry: Theory, research, and practical perspectives. Rotterdam: Sense.

    Book  Google Scholar 

  • 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).

  • 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 

  • Ouvrier-Buffet, C. (2006). Exploring mathematical definition construction processes. Educational Studies in Mathematics, 63(3), 259–282.

    Article  Google Scholar 

  • Rosch, E. (1973). Natural categories. Cognitive Psychology, 4, 328–350.

    Article  Google Scholar 

  • Shulman, L. S. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

    Article  Google Scholar 

  • Smith, E., Shoben, E., & Rips, L. (1974). Structure and process in semantic memory: a featural model for semantic decisions. Psychological Review, 81, 214–241.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Tirosh, D., & Tsamir, P. (2008). Starting right: Mathematics in preschool. Unpublished research report. In Hebrew.

  • 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.

    Chapter  Google Scholar 

  • Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive non-examples: the case of triangles. Educational Studies in Mathematics, 69(2), 81–95.

    Article  Google Scholar 

  • van Dormolen, J., & Arcavi, A. (2000). What is a circle? Mathematics in School, 29(5), 15–19.

    Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

  • Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: a case of a square. Educational Studies in Mathematics, 69(2), 131–148.

    Article  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Esther Levenson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsamir, P., Tirosh, D., Levenson, E. et al. Early-years teachers’ concept images and concept definitions: triangles, circles, and cylinders. ZDM Mathematics Education 47, 497–509 (2015). https://doi.org/10.1007/s11858-014-0641-8

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-014-0641-8

Keywords

Navigation