ZDM

, Volume 47, Issue 3, pp 519–529

Geometry in the early years: a commentary

Commentary Paper

Abstract

The primary goal of this paper is to provide a commentary on the teaching and learning of geometry in the early years of schooling with the set of papers in this issue as a guiding factor. It is structured around issues about geometry education of young learners, such as: what should we teach in geometry and why; representation of geometrical ideas; the teaching and learning of geometry; and assessment of children’s learning in geometry. The author outlines his views based on the literature and the papers in this issue and concludes with an outlook on the future teaching and learning of geometry in schools.

Keywords

Geometry Spatial reasoning Early childhood 

References

  1. Bartolini-Bussi, M. & Baccaglini-Frank, A. (2015). Geometry in early years: sowing seeds for a mathematical definition of squares and rectangles. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-014-0636-5.
  2. 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: National Council of Teachers of Mathematics.Google Scholar
  3. Bruce, C. & Hawes, Z. (2015). The role of 2D and 3D mental rotations in mathematics for young children: What is it? Why does it matter? And what can we do about it? ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-014-0637-4.
  4. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook for research in teaching and learning of mathematics (pp. 420–464). New York: Macmillan.Google Scholar
  5. Clements, D.H., Natasi, B.K., & Swaminathan, S. (1993). Young children and computers: crossroads and directions from research. Young Children, 56–64.Google Scholar
  6. Clements, D. H., Swaminathan, S., & Hannibal, M. A. Z. (1999). Young children’s concepts of shapes. Journal for Research in Mathematics Education, 30(2), 192–212.CrossRefGoogle Scholar
  7. Crowley, M. L. (1987). The van Hiele model of the development of geometric thought. In M. M. Lindquist, & A. P. Shulte (Eds.), Learning and teaching geometry, K-12. 1987 Yearbook of the National Council of Teachers of Mathematics (pp. 1–16). Reston: NCTM.Google Scholar
  8. De Moor, E. (2005). Domain description geometry. In M. van den Heuvel-Panhuizen & K. Buys (Eds.), Young children learn measurement and geometry (pp. 115–144). The Netherlands: Freudenthal Institute, Utrecht University.Google Scholar
  9. Diezmann, C. M., & English, L. D. (2001). Promoting the use of diagrams as tools for thinking. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics: 2001 NCTM Yearbook (pp. 77–89). Reston: NCTM.Google Scholar
  10. Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century: an ICMI study (pp. 37–51). Dordrecht: Kluwer Academic Publishers.Google Scholar
  11. Duval, R. (1999). Representation, vision and visualization: cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt, & M. Santos (Eds.), Proceedings of the 21st annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 3–26). Columbus: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar
  12. Duval, R. (2005). Les conditions cognitives de l’apprentissage de la géométrie : développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements. Annales de Didactique et de Sciences Cognitives, 10, 5–53.Google Scholar
  13. Fleer, M. (2010). Early learning and development: cultural-historical concepts in play. Cambridge: Cambridge University Press.Google Scholar
  14. Fleer, M., & Quiñones, G. (2013). An assessment perezhivanie: building an assessment pedagogy for, with and of early childhood science learning. In D. Corrigan, et al. (Eds.), Valuing assessment in science education: pedagogy, curriculum, policy (pp. 231–247). Dordrecht: Springer.CrossRefGoogle Scholar
  15. Fuys, D. J., & Liebov, A. K. (1993). Geometry and spatial sense. In R. J. Jensen (Ed.), Research ideas for the classroom: early childhood mathematics (pp. 195–222). London: Simon & Schuster/Prentice Hall International.Google Scholar
  16. Ginsburg, H. P., Cannon, J., Eisenband, J., & Pappas, S. (2006). Mathematical thinking and learning. In K. McCartney & D. Phillips (Eds.), Blackwell handbook of early childhood development (pp. 208–229). Massachusetts: Blackwell.CrossRefGoogle Scholar
  17. Guitiérrez, A., Jaime, A., & Fortuny, J. M. (1991). An alternative paradigm to assess the acquisition of van Hiele levels. Journal for Research in Mathematics Education, 22(3), 237–251.CrossRefGoogle Scholar
  18. Hallowell, D., Okamoto, Y, Romo, L., & LaJoy, J. (2015). First-grader’s spatial-mathematical reasoning about plane and solid shapes and their representations. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-015-0664-9.
  19. Hershkowitz, R. (1989). Visualization in geometry: two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.Google Scholar
  20. Hershkowitz, R., Parzysz, B., & Van Dormorlen, J. (1996). Space and shape. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education: Part 1 (pp. 161–204). Dordrecht: Kluwer Academic Publishers.Google Scholar
  21. Hoffer, A. (1981). Geometry is more than proof. Mathematics Teacher, 74, 11–18.Google Scholar
  22. Kaur, H. (2015). Two aspects of young children’s thinking about different types of dynamic triangles: prototypicality and inclusion. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-014-0658-z.
  23. Koehler, M., & Mishra, P. (2009). What is technological pedagogical content knowledge (TPACK)? Contemporary Issues in Technology and Teacher Education, 9(1), 60–70.Google Scholar
  24. Kotsopoulos, D., Cordy, M., & Langemeyer, M. (2015). Children’s understanding of large-scale mapping tasks: an analysis of talk, drawings, and gesture. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-014-0661-4.
  25. Lowrie, T. (2002a). Making connections between simulated and “real” worlds: young children interpreting computer representations. In B. Barton, K. C. Irwin, M. Pfannkuch, & M. O. J. Thomas (Eds.), Mathematics education in the South Pacific (Proceedings of the 25th annual conference of the Mathematics Education Research Group of Australasia, Auckland, pp. 441–448). Sydney: MERGA.Google Scholar
  26. Lowrie, T. (2002b). Visual and spatial reasoning: young children playing computers. In D. Edge, & B. H. Yeap (Eds.), Mathematics education for a knowledge-based era (Proceedings of the 2nd East Asia regional conference on mathematics education, Singapore, vol. 2, pp. 440–446). Singapore: National Institute of Education.Google Scholar
  27. Mamolo, A., Ruttenberg-Rozen, R., & Whitelely, W. (2015). Developing a network of and for geometric reasoning. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-014-0654-3.
  28. Mesquita, A. L. (1998). On conceptual obstacles linked with external representations in geometry. Journal of Mathematical Behavior, 17(2), 183–195.CrossRefGoogle Scholar
  29. 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 Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-015-0679-2.
  30. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: NCTM.Google Scholar
  31. National Research Council (NRC) (2006). Learning to think spatially: GIS as a support system in the K-12 curriculum. Washington, DC: National Academy Press. http://www.nap.edu/download.php?record_id=11019.
  32. Ng, O., & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-014-0660-5.
  33. Olive, J. (1991). Logo programming and geometric understanding: an in-depth study. Journal for Research in Mathematics Education, 22(2), 90–111.CrossRefGoogle Scholar
  34. Organisation for Economic Cooperation and Development. (2001). Starting strong: early childhood education and care. Paris: OECD.Google Scholar
  35. Pandiscio, E., & Orton, R. E. (1998). Geometry and metacognition: an analysis of Piaget’s and van Hiele’s perspectives. Focus on Learning Problems in Mathematics, 20(2&3), 78–87.Google Scholar
  36. Perry, B., & Dockett, S. (2002). Young children’s access to powerful mathematical ideas. In L. D. Enhlish (Ed.), Handbook of international research in mathematics education: direction for the 21st century (pp. 81–111). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  37. Piaget, J., & Inhelder, B. (1967). The child’s conception of space. New York: W.W. Norton.Google Scholar
  38. Sinclair, N., & Bruce, C. (2015). New opportunities in geometry at the primary school. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-015-0693-4.
  39. Soury-Lavergne, S., & Maschietto, M. (2015). Articulation of spatial and geometrical knowledge in problem solving with technology at primary school. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-015-0694-3.
  40. Thom, J., & McGarvey, L. (2015). The act and artifact of drawing(s): observing geometric thinking with, in, and through children’s drawings. ZDM Mathematics Education, 47(3) (this issue).Google Scholar
  41. Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (2015). Early-years teachers’ concept images and concept definitions: triangles, circles, and cylinders. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-014-0641-8.
  42. Usiskin, Z. (1987). Resolving the continuing dilemmas in school geometry. In M. M. Lindquist & A. P. Shulte (Eds.), Learning and teaching geometry, K-12:1987 Yearbook (pp. 17–31). Reston: National Council of Teachers of Mathematics.Google Scholar
  43. Van den Heuvel-Panhuizen, M. (1996). Assessment and realistic mathematics education. Utrecht: Centre for Science and Mathematics Education, Freudenthal Institute.Google Scholar
  44. Van den Heuvel-Panhuizen, M., Iliade, E., & Robitzsch (2015). Kindergartner’s performance in two types of imaginary perspective taking. ZDM Mathematics Education, 47(3) (this issue). doi:10.1007/s11858-015-0677-4.
  45. Van Hiele, P. M. (1986). Structure and insight. Orlando: Academic Press.Google Scholar
  46. Vinner, S., & Hershkowitz, R. (1980). Concept images and common cognitive paths in the development of some simple geometrical concepts. In R. Karplus (Ed.), Proceedings of the 4th international conference for the psychology of mathematics education (pp. 177–184). Berkeley: University of California.Google Scholar

Copyright information

© FIZ Karlsruhe 2015

Authors and Affiliations

  1. 1.Nanyang Technological UniversitySingaporeSingapore

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