Educational Studies in Mathematics

, Volume 96, Issue 3, pp 289–304 | Cite as

Observing and analyzing children’s mathematical development, based on action theory

  • M. J. A. Bunck
  • E. Terlien
  • M. van Groenestijn
  • S. W. M. Toll
  • J. E. H. Van Luit


Children who experience difficulties with learning mathematics should be taught by teachers who focus on the child’s best way of learning. Analyses of the mathematical difficulties are necessary for fine-tuning mathematics education to the needs of these children. For this reason, an instrument for Observing and Analyzing children’s Mathematical Development (OAMD), based on action theory, has been developed. The use of levels of action is a new insight in the diagnostic process. Using the OAMD makes it possible to explore and analyze a child’s knowledge, proficiency and possible difficulties on four levels of acting in the domain of Number (counting, addition, subtraction, multiplication and division). The research concerns children from kindergarten up to grade three. In this article, we will discuss the purpose and the construction of the instrument with the focus on the usability of the OAMD. The study examines the quality and the diagnostic value of the instrument by means of the internal consistency, the test–retest reliability and the construct validity. The analyses show positive results. The conclusion of the research is that the OAMD is a suitable instrument for the analysis of numerical development of young children and their possible difficulties in this domain.


Elementary mathematics Mathematical development Mathematical difficulties Levels of numerical actions Diagnostic instrument Special educational needs 


  1. Arievitch, I. M., & Haenen, J. P. P. (2005). Connecting sociocultural theory and educational practice: Galperin’s approach. Educational Psychologist, 40, 155–165. doi: 10.1207/s15326985ep4003_2 CrossRefGoogle Scholar
  2. Aunola, K., Leskinen, E., Lerkkanen, M., & Nurmi, J. (2004). Developmental dynamics of math performances from preschool to grade 2. Journal of Educational Psychology, 96, 699–713. doi: 10.1037/0022-0663.96.4.699 CrossRefGoogle Scholar
  3. Baroody, A. J., Eiland, M., & Thompson, B. (2009). Fostering at-risk preschoolers’ number sense. Early Education and Development, 20, 80–128. doi: 10.1080/10409280802206619 CrossRefGoogle Scholar
  4. Berch, D. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38, 333–339. doi: 10.1177/00222194050380040901
  5. Butterworth, B. (2003). Dyscalculia screener. London, UK: GL Assessment.Google Scholar
  6. Clements, D. H., & Sarama, J. (2007). Early childhood mathematics. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 461–555). Charlotte, NC: Information Age Publishing.Google Scholar
  7. Cobb, P., Wood, T., & Yackel, E. (1992). A constructivist approach to second grade mathematics. In E. Von Glasersfeld (Ed.), Radical constructivism in mathematical education (pp. 157–176). Dordrecht, the Netherlands: Kluwer Academic Publishers.Google Scholar
  8. Desoete, A., Ceulemans, A., De Weerdt, F., & Pieters, S. (2012). Can we predict mathematical learning disabilities from symbolic and non-symbolic comparison tasks in kindergarten? Findings from a longitudinal study. British Journal of Educational Psychology, 82, 64–81. doi: 10.1348/2044-8279.002002 CrossRefGoogle Scholar
  9. De Vos, T. (2010). TTA: Tempo test Automatiseren [Arithmetic automatization test]. Amsterdam, The Netherlands: Boom test uitgevers.Google Scholar
  10. Early Numeracy Research Project. (2002). Final report. Retrieved from
  11. Gal’perin, P. J. (1969). Stages in the development of mental acts. In M. Cole & I. Maltzman (Eds.), A handbook of contemporary soviet psychology (pp. 249–273). New York, NY: Basic Books.Google Scholar
  12. Geary, D. C. (2013). Early foundations for mathematics learning and their relations to learning disabilities. Current Directions in Psychological Science, 22, 23–27. doi: 10.1177/0963721412469398 CrossRefGoogle Scholar
  13. Geary, D. C., Hamson, C. O., & Hoard, M. K. (2000). Numerical and arithmetical cognition: A longitudinal study of process and concept deficits in children with learning disability. Journal of Experimental Child Psychology, 77, 236–263. doi: 10.1006/jecp.2000.2561 CrossRefGoogle Scholar
  14. Geary, D. C., & Hoard, M. K. (2005). Learning disabilities in arithmetic and mathematics: Theoretical and empirical perspectives. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 253–268). New York, NY: Psychology Press. doi: 10.1006/jecp.2000.2561 Google Scholar
  15. Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293–304. doi: 10.1177/00222194050380040301 CrossRefGoogle Scholar
  16. Ginsburg, H. P., & Baroody, A. J. (2003). Test of early mathematics ability - third edition (TEMA-3). Austin, TX: PRO-ED.Google Scholar
  17. Goswami, U. (2008). Cognitive development: The learning brain. Hove, UK: Psychology Press.Google Scholar
  18. Gravemeijer, K. P. E. (1994). Developing realistic mathematics education. Utrecht, The Netherlands: Freudenthal Instituut.Google Scholar
  19. Gravemeijer, K. P. E. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1, 155–177.CrossRefGoogle Scholar
  20. Grégoire, J., Noël, M., & Van Nieuwenhoven, C. (2004). TEDI-MATH. Amsterdam, The Netherlands: Pearson.Google Scholar
  21. Haenen, J. (2001). Outlining the teaching–learning process: Piotr Gal’perin’s contribution. Learning and Instruction, 11, 157–170. doi: 10.1016/S0959-4752(00)00020-7 CrossRefGoogle Scholar
  22. Janssen, J., Scheltens, F., & Kraemer, J. M. (2005a). Leerling- en onderwijsvolgsysteem rekenen-wiskunde groep 3 [Student and education monitoring system mathematics grade 1]. Arnhem, The Netherlands: Cito.Google Scholar
  23. Janssen, J., Scheltens, F., & Kraemer, J. M. (2005b). Leerling- en onderwijsvolgsysteem rekenen-wiskunde groep 4 [Student and education monitoring system mathematics grade 2]. Arnhem, The Netherlands: Cito.Google Scholar
  24. Janssen, J., Scheltens, F., & Kraemer, J. M. (2006). Leerling- en onderwijsvolgsysteem rekenen-wiskunde groep 5 [Student and education monitoring system mathematics grade 3]. Arnhem, The Netherlands: Cito.Google Scholar
  25. Janssen, J., Verhelst, N., Engelen, R., & Scheltens, F. (2010). Wetenschappelijke verantwoording van de toetsen LOVS Rekenen-Wiskunde voor groep 3 tot en met 8 [Scientific justification of the mathematics test for grade 1 until grade 6]. Arnhem, The Netherlands: Cito.Google Scholar
  26. Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2009). Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology, 45, 850–867. doi: 10.1037/a0014939 CrossRefGoogle Scholar
  27. Koerhuis, I. (2010). Rekenen voor kleuters [Mathematics for kindergarten]. Arnhem, The Netherlands: Cito.Google Scholar
  28. Koerhuis, I., & Keuning, J. (2011). Wetenschappelijke verantwoording van de toetsen Rekenen voor kleuters [Scientific justification of the mathematics test for kindergarten]. Arnhem, The Netherlands: Cito.Google Scholar
  29. Kolkman, M. E., Kroesbergen, E. H., & Leseman, P. M. (2013). Early numerical development and the role of non-symbolic and symbolic skills. Learning and Instruction, 25, 95–03. doi: 10.1016/j.learninstruc.2012.12.001 CrossRefGoogle Scholar
  30. Kroesbergen, E. H., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs: A meta-analysis. Remedial and Special Education, 24, 97–114. doi: 10.1177/07419325030240020501 CrossRefGoogle Scholar
  31. Kroesbergen, E. H., Van Luit, J. E. H., Van Lieshout, E. C. D. M., Van Loosbroek, E., & Van de Rijt, B. A. M. (2009). Individual differences in early numeracy: The role of executive functions and subitizing. Journal of Psychoeducational Assessment, 27, 226–236. doi: 10.1177/0734282908330586 CrossRefGoogle Scholar
  32. Mazzocco, M. M. M. (2007). Defining and differentiating mathematical learning disabilities and difficulties. In D. B. Berch & M. M. M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 29–47). Baltimore, MD: Paul H. Brooks.Google Scholar
  33. Mazzocco, M. M. M., & Thompson, R. E. (2005). Kindergarten predictors of math learning disability. Learning Disabilities Research and Practice, 20, 142–155. doi: 10.1111/j.1540-5826.2005.00129.x CrossRefGoogle Scholar
  34. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.Google Scholar
  35. Navarro, J. I., Aguilar, M., Marchena, E., Ruiz, G., Menacho, I., & Van Luit, J. E. H. (2012). Longitudinal study of low and high achievers in early mathematics. British Journal of Educational Psychology, 82, 28–41. doi: 10.1111/j.2044-8279.2011.02043.x CrossRefGoogle Scholar
  36. Nunes, T., & Bryant, P. (1996). Children doing mathematics. Oxford, UK: Blackwell.Google Scholar
  37. Opfer, J. E., & Siegler, R. S. (2012). Development of quantitative thinking. In K. J. Holyoak & R. G. Morrison (Eds.), The Oxford handbook of thinking and reasoning (pp. 585–605). UK: Oxford University Press. doi: 10.1093/oxfordhb/9780199734689.013.0030 Google Scholar
  38. Van de Rijt, B. A. M., Van Luit, J. E. H., & Pennings, A. H. (1999). The construction of the Utrecht early mathematical competence scales. Educational and Psychological Measurement, 59, 289–309. doi: 10.1177/0013164499592006 CrossRefGoogle Scholar
  39. Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54, 9–35.CrossRefGoogle Scholar
  40. Van Groenestijn, M. J. A. (2002). A gateway to numeracy. A study of numeracy in adult basic education. Utrecht, The Netherlands: CD-β Press.Google Scholar
  41. Van Groenestijn, M., Borghouts, C., & Janssen, C. (2011). Protocol ernstige rekenwiskunde-problemen en dyscalculie [Protocol severe mathematical problems and dyscalculia]. Assen, The Netherlands: Van Gorcum.Google Scholar
  42. Van Luit, J. E. H. (2015). Good math education in kindergarten cannot prevent dyscalculia. Revista de Psicología y Educación / Journal of Psychology and Education, 10, 43–60.Google Scholar
  43. Veraksa, N. E., & Van Oers, B. (2011). Early childhood education from a Russian perspective. International Journal of Early Years Education, 19, 5–17.CrossRefGoogle Scholar
  44. Von Aster, M., Weinhold, M., & Horn, R. (2005). Zareki-R. Neuropsychologische Testbatterie für Zahlenverarbeitung und Rechnen bei Kindern [Zareki-R. Neuropsychological mathematics test for children]. Frankfurt am Main, Germany: Pearson.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • M. J. A. Bunck
    • 1
  • E. Terlien
    • 1
  • M. van Groenestijn
    • 1
  • S. W. M. Toll
    • 2
  • J. E. H. Van Luit
    • 2
  1. 1.Hogeschool UtrechtUniversity of Applied SciencesUtrechtNetherlands
  2. 2.Utrecht UniversityUtrechtNetherlands

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