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

## 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.

### References

- 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 - 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - Levenson, E., Tirosh, D., & Tsamir, P. (2011).
*Preschool geometry: Theory, research, and practical perspectives*. Rotterdam: Sense.CrossRefGoogle 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).Google Scholar - 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.CrossRefGoogle Scholar - Shulman, L. S. (1986). Those who understand: knowledge growth in teaching.
*Educational Researcher,**15*(2), 4–14.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar - Tirosh, D., & Tsamir, P. (2008). Starting right: Mathematics in preschool. Unpublished research report. In Hebrew.Google Scholar
- 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 - Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive non-examples: the case of triangles.
*Educational Studies in Mathematics,**69*(2), 81–95.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.Google Scholar - Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: a case of a square.
*Educational Studies in Mathematics,**69*(2), 131–148.CrossRefGoogle Scholar