Journal of Mathematics Teacher Education

, Volume 14, Issue 2, pp 133–148 | Cite as

Early childhood teacher education: the case of geometry



For early childhood, the domain of geometry and spatial reasoning is an important area of mathematics learning. Unfortunately, geometry and spatial thinking are often ignored or minimized in early education. We build a case for the importance of geometry and spatial thinking, review research on professional development for these teachers, and describe a series of research and development projects based on this body of knowledge. We conclude that research-based models hold the potential to make a significant difference in the learning of young children by catalyzing substantive change in the knowledge and beliefs of their teachers.


Scaling up professional development Geometry Spatial reasoning Learning trajectories Early childhood 



The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant No. R305K05157 to the University at Buffalo, State University of New York, D. H. Clements, J. Sarama, and J. Lee, “Scaling Up TRIAD: Teaching Early Mathematics for Understanding with Trajectories and Technologies.” Work on the research was also supported in part by the National Science Foundation under Grant No. ESI-9730804 to D. H. Clements and J. Sarama “Building Blocks-Foundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development,” and the IERI under NSF Grant No. REC-0228440, “Scaling Up the Implementation of a Pre-Kindergarten Mathematics Curricula: Teaching for Understanding with Trajectories and Technologies.” Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.


  1. Ansari, D., Donlan, C., Thomas, M. S. C., Ewing, S. A., Peen, T., & Karmiloff-Smith, A. (2003). What makes counting count? Verbal and visuo-spatial contributions to typical and atypical number development. Journal of Experimental Child Psychology, 85, 50–62.CrossRefGoogle Scholar
  2. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241.CrossRefGoogle Scholar
  3. Ball, D. L., & Cohen, D. K. (1999). Instruction, capacity, and improvement. Philadelphia, PA: Consortium for Policy Research in Education, University of Pennsylvania.Google Scholar
  4. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 5, 389–407.CrossRefGoogle Scholar
  5. Beaton, A. E., Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., Kelly, D. L., & Smith, T. A. (1996, January 19, 1997). Mathematics achievement in the middle school years: IEA’s Third International Mathematics and Science Study (TIMSS). Retrieved February 1, 1997, from
  6. Bronowski, J. (1947). Mathematics. In D. Thompson & J. Reeves (Eds.), The quality of education: Methods and purposes in the secondary curriculum (pp. 179–195). London: Frederick Muller.Google Scholar
  7. Brown, M., Blondel, E., Simon, S., & Black, P. (1995). Progression in measuring. Research Papers in Education, 10(2), 143–170.CrossRefGoogle Scholar
  8. Campbell, P. F., & Rowan, T. E. (1995). Project IMPACT: Increasing the mathematical power of all children and teachers (Phase One Implementation Final Report). Baltimore, MD: Center for Mathematics Education, University of Maryland.Google Scholar
  9. Carpenter, T. P., Fennema, E. H., Peterson, P. L., & Carey, D. A. (1988). Teacher’s pedagogical content knowledge of students’ problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19, 385–401.CrossRefGoogle Scholar
  10. Casey, M. B., & Erkut, S. (2005, April). Early spatial interventions benefit girls and boys. Paper presented at the Biennial Meeting of the Society for Research in Child Development, Atlanta, GA.Google Scholar
  11. Clarke, B. A. (2004). A shape is not defined by its shape: Developing young children’s geometric understanding. Journal of Australian Research in Early Childhood Education, 11(2), 110–127.Google Scholar
  12. Clements, D. H. (2003). Teaching and learning geometry. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 151–178). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  13. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York: Macmillan.Google Scholar
  14. Clements, D. H., & Sarama, J. (2004a). Hypothetical learning trajectories (special issue). Mathematical Thinking and Learning, 6(2), 81–260.CrossRefGoogle Scholar
  15. Clements, D. H., & Sarama, J. (2004b). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6, 81–89.CrossRefGoogle Scholar
  16. Clements, D. H., & Sarama, J. (2007a). Building blocks—SRA real math, Grade PreK. Columbus, OH: SRA/McGraw-Hill.Google Scholar
  17. Clements, D. H., & Sarama, J. (2007b). Early childhood mathematics learning. In F. K. Lester Jr (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1 (pp. 461–555). New York: Information Age Publishing.Google Scholar
  18. Clements, D. H., & Sarama, J. (2007c). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38, 136–163.Google Scholar
  19. Clements, D. H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal, 45, 443–494.CrossRefGoogle Scholar
  20. Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. Journal for Research in Mathematics Education, 42(2), 127–166.Google Scholar
  21. Clements, D. H., Swaminathan, S., Hannibal, M. A. Z., & Sarama, J. (1999). Young children’s concepts of shape. Journal for Research in Mathematics Education, 30, 192–212.CrossRefGoogle Scholar
  22. Clements, D. H., Wilson, D. C., & Sarama, J. (2004). Young children’s composition of geometric figures: A learning trajectory. Mathematical Thinking and Learning, 6, 163–184.CrossRefGoogle Scholar
  23. Cooper, R. (1998). Socio-cultural and within-school factors that affect the quality of implementation of school-wide programs. (Tech. Rep. No. 28): John Hopkins University, Center for Research on the Education of Students Placed at Risk.Google Scholar
  24. Dehaene, S., Izard, V., Pica, P., & Spelke, E. S. (2006). Core knowledge of geometry in an Amazonian indigene group. Science, 311, 381–384.CrossRefGoogle Scholar
  25. Denton, C. A., & Fletcher, J. M. (2003). Scaling reading intervention. In B. R. Foorman (Ed.), Preventing and remediating reading difficulties: Bringing science to scale (pp. 445–463). Baltimore, MD: York Press.Google Scholar
  26. Early, D., Maxwell, K. L., Burchinal, M. R., Alva, S., Bender, R. H., Bryant, D., et al. (2007). Teachers’ education, classroom quality, and young children’s academic skills: Results from seven studies of preschool programs. Child Development, 78, 558–580.CrossRefGoogle Scholar
  27. Elmore, R. F. (1996). Getting to scale with good educational practices. Harvard Educational Review, 66, 1–25.Google Scholar
  28. Epstein, J. L. (1987). Toward a theory of family-school connections: Teacher practices and parent involvement. In K. Hurrelman, F. Kaufmann, & F. Losel (Eds.), Social intervention: Potential and constraints (pp. 121–136). New York: DeGruyter.Google Scholar
  29. Feltovich, P. J., Spiro, R. J., & Coulson, R. L. (1997). Issues of expert flexibility in contexts characterized by complexity and change. In P. J. Feltovich, K. M. Ford, & R. R. Hoffman (Eds.), Expertise in context: Human and machine (pp. 125–146). Cambridge, Mass.: The MIT Press.Google Scholar
  30. Fennema, E. H., Carpenter, T. P., Frank, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403–434.CrossRefGoogle Scholar
  31. Fujita, T., & Jones, K. (2006a). Primary trainee teachers’ knowledge of parallelograms. Proceedings of the British Society for Research into Learning Mathematics, 26(2), 25–30.Google Scholar
  32. Fujita, T., & Jones, K. (2006b). 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
  33. Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education Monograph Series, 3, 1–198.Google Scholar
  34. Gagatsis, A. (2003). Young children’s understanding of geometric shapes: The role of geometric models. European Early Childhood Education Research Journal, 11, 43–62.CrossRefGoogle Scholar
  35. Gagatsis, A., & Patronis, T. (1990). Using geometrical models in a process of reflective thinking in learning and teaching mathematics. Educational Studies in Mathematics, 21, 29–54.CrossRefGoogle Scholar
  36. Geary, D. C. (2007). An evolutionary perspective on learning disability in mathematics. Developmental Neuropsychology, 32, 471–519.Google Scholar
  37. Ginsburg, A., Cooke, G., Leinwand, S., Noell, J., & Pollock, E. (2005). Reassessing U.S. International Mathematics Performance: New Findings from the 2003 TIMSS and PISA. Washington, DC: American Institutes for Research.Google Scholar
  38. 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, MD: Paul H. Brookes.Google Scholar
  39. Hall, G. E., & Hord, S. M. (2001). Implementing change: Patterns, principles, and potholes. Boston, MA: Allyn and Bacon.Google Scholar
  40. Han, Y., & Ginsburg, H. P. (2001). Chinese and English mathematics language: The relation between linguistic clarity and mathematics performance. Mathematical Thinking and Learning, 3, 201–220.CrossRefGoogle Scholar
  41. Hannibal, M. A. Z., & Clements, D. H. (2010). Young children’s understanding of basic geometric shapes. Manuscript submitted for publication.Google Scholar
  42. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371–406.CrossRefGoogle Scholar
  43. Jacobson, C., & Lehrer, R. (2000). Teacher appropriation and students learning of geometry through design. Journal for Research in Mathematics Education, 31, 71–88.CrossRefGoogle Scholar
  44. Jones, K. (2000). Teacher knowledge and professional development in geometry. Proceedings of the British Society for Research into Learning Mathematics, 20(3), 109–114.Google Scholar
  45. Jones, K., Mooney, C., & Harries, T. (2002). Trainee primary teachers’ knowledge of geometry for teaching. Proceedings of the British Society for Research into Learning Mathematics, 22(2), 95–100.Google Scholar
  46. Klingner, J. K., Ahwee, S., Pilonieta, P., & Menendez, R. (2003). Barriers and facilitators in scaling up research-based practices. Exceptional Children, 69, 411–429.Google Scholar
  47. Lappan, G. (1999). Geometry: The forgotten strand. NCTM News Bulletin, 36(5), 3.Google Scholar
  48. Larsen, K. (2005). Stephen Hawking: A biography. Westport, CT: Greenwood Press.Google Scholar
  49. Mullis, I. V. S., Martin, M. O., Beaton, A. E., Gonzalez, E. J., Kelly, D. L., & Smith, T. A. (1997). Mathematics achievement in the primary school years: IEA’s third international mathematics and science study (TIMSS). Chestnut Hill, MA: Center for the Study of Testing, Evaluation, and Educational Policy, Boston College.Google Scholar
  50. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, Office of Planning, Evaluation and Policy Development.Google Scholar
  51. NCTM. (1991). Professional standards for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  52. NCTM. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  53. Newcombe, N. S., & Huttenlocher, J. (2000). Making space: The development of spatial representation and reasoning. Cambridge, MA: MIT Press.Google Scholar
  54. Owens, K. (1999). The role of visualization in young students’ learning. In O. Zaslavsky (Ed.), Proceedings of the 23rd conference of the international group for the psychology of mathematics education (Vol. 1, pp. 220–234). Haifa, Isreal: Technion.Google Scholar
  55. Peterson, P. L., Carpenter, T. P., & Fennema, E. H. (1989). Teachers’ knowledge of students’ knowledge in mathematics problem solving: Correlational and case analyses. Journal of Educational Psychology, 81, 558–569.CrossRefGoogle Scholar
  56. Pinel, P., Piazza, D., Le Bihan, D., & Dehaene, S. (2004). Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron, 41, 983–993.CrossRefGoogle Scholar
  57. Raudenbush, S. W. (2008). Advancing educational policy by advancing research on instruction. American Educational Research Journal, 45, 206–230.CrossRefGoogle Scholar
  58. Rogers, E. M. (2003). Diffusion of innovations (4th ed.). New York: The Free Press.Google Scholar
  59. Sanders, W. L., & Rivers, J. C. (1996). Cumulative and residual effects of teachers on future student academic achievement (Research Progress Report). Knoxville, TN: University of Tennessee Value-Added Research and Assessment Center.Google Scholar
  60. Sarama, J. (2002). Listening to teachers: Planning for professional development. Teaching Children Mathematics, 9, 36–39.Google Scholar
  61. Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.Google Scholar
  62. Sarama, J., Clements, D. H., & Henry, J. J. (1998). Network of influences in an implementation of a mathematics curriculum innovation. International Journal of Computers for Mathematical Learning, 3, 113–148.CrossRefGoogle Scholar
  63. Sarama, J., Clements, D. H., Starkey, P., Klein, A., & Wakeley, A. (2008). Scaling up the implementation of a pre-kindergarten mathematics curriculum: Teaching for understanding with trajectories and technologies. Journal of Research on Educational Effectiveness, 1, 89–119.CrossRefGoogle Scholar
  64. Sarama, J., Clements, D. H., & Vukelic, E. B. (1996). The role of a computer manipulative in fostering specific psychological/mathematical processes. In E. Jakubowski, D. Watkins & H. Biske (Eds.), Proceedings of the 18th annual meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 567–572). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar
  65. Sarama, J., & DiBiase, A.-M. (2004). The professional development challenge in preschool mathematics. In D. H. Clements, J. Sarama, & A.-M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 415–446). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  66. Shepard, R. N. (1978). Externalization of mental images and the act of creation. In B. S. Randhawa & W. E. Coffman (Eds.), Visual learning, thinking, and communication (pp. 133–189). New York: Academic Press.Google Scholar
  67. Shepard, R. N., & Cooper, L. A. (1982). Mental images and their transformations. Cambridge, MA: The MIT Press.Google Scholar
  68. Showers, B., Joyce, B., & Bennett, B. (1987). Synthesis of research on staff development: A framework for future study and a state-of-the-art analysis. Educational Leadership, 45(3), 77–87.Google Scholar
  69. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.CrossRefGoogle Scholar
  70. Smith, I. (1964). Spatial ability. San Diego: Knapp.Google Scholar
  71. Smith, C. L., Wiser, M., Anderson, C. W., & Krajcik, J. S. (2006). Implications of research on children’s learning for standards and assessment: A proposed learning progression for matter and the atomic-molecular theory. Measurement: Interdisciplinary Research & Perspective, 14(1&2), 1–98.Google Scholar
  72. Sowder, J. T. (2007). The mathematical education and development of teachers. In F. K. Lester Jr (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1) (pp. 157–223). New York: Information Age Publishing.Google Scholar
  73. Swafford, J., Jones, G. A., & Thorton, C. A. (1997). Increased knowledge in geometry and instructional practice. Journal for Research in Mathematics Education, 28, 467–483.CrossRefGoogle Scholar
  74. Tatsuoka, K. K., Corter, J. E., & Tatsuoka, C. (2004). Patterns of diagnosed mathematical content and process skills in TIMSS-R across a sample of 20 countries. American Educational Research Journal, 41, 901–926.CrossRefGoogle Scholar
  75. Thomas, B. (1982). An abstract of kindergarten teachers’ elicitation and utilization of children’s prior knowledge in the teaching of shape concepts: Unpublished manuscript, School of Education, Health, Nursing, and Arts Professions, New York University.Google Scholar
  76. van der Sandt, S. (2007). Pre-service geometry education in South Africa: A typical case? IUMPST: The Journal (, 1, 1–9.
  77. van der Sandt, S., & Nieuwoudt, H. D. (2004). Prospective mathematics teachers’ geometry content knowledge: A typical case? In S. Nieuwoudt, S. Froneman & P. Nkhoma (Eds.), Proceedings of tenth national congress of the association for mathematics education of South Africa (pp. 250–262). Potchefstroom: AMESA.Google Scholar
  78. van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Orlando, FL: Academic Press.Google Scholar
  79. 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 fourth international conference for the psychology of mathematics education (pp. 177–184). Berkeley, CA: Lawrence Hall of Science, University of California.Google Scholar
  80. Wheatley, G. H., Brown, D. L., & Solano, A. (1994). Long term relationship between spatial ability and mathematical knowledge. In D. Kirshner (Ed.), Proceedings of the sixteenth annual meeting North American chapter of the international group for the psychology of mathematics education (Vol. 1, pp. 225–231). Baton Rouge, LA: Louisiana State University.Google Scholar
  81. Wright, S. P., Horn, S. P., & Sanders, W. L. (1997). Teacher and classroom context effects on student achievement: Implications for teacher evaluation. Journal of Personnel Evaluation in Education, 11, 57–67.CrossRefGoogle Scholar
  82. Zorzi, M., Priftis, K., & Umiltà, C. (2002). Brain damage: Neglect disrupts the mental number line. Nature, 417, 138.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Graduate School of EducationUniversity at Buffalo, State University of New YorkBuffaloUSA

Personalised recommendations