Understanding of number concepts in low attaining 7–9 year olds: Part I. Development of descriptive framework and diagnostic instrument

Abstract

Three studies were carried out into the development of number concepts in pupils aged 7 to 9 years who were considered to be ‘low attainers’ in mathematics. This paper reports the search for a descriptive framework, the development of a diagnostic assessment instrument and a longitudinal study. A subsequent paper (Denvir and Brown, 1986) reports two teaching studies. Support was found for the two main hypotheses namely that:

  1. (i)

    A framework can be identified which describes the orders in which children acquire number concepts

  2. (ii)

    This framework can be used to develop a diagnostic assessment instrument which will provide a description of pupils' understanding of number.

The aspects of number which were considered were counting, addition, subtraction and place value. The number skills which children had grasped were inferred from solutions and strategies offered by them to questions posed in a series of interviews.

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References

  1. BednarzN. and JanvierB.: 1982, ‘The understanding of numeration’, Educational Studies in Mathematics 13, 33–57.

    Google Scholar 

  2. BennettN., DesforgesC., CockburnA., and WilkinsonB.: 1984, The Quality of Pupil Learning Experiences, Erlbaum, New Jersey.

    Google Scholar 

  3. Brown, M.: 1981, ‘Levels of understanding of number operation, place value and decimals in secondary school children’, unpublished Doctoral Dissertation, University of London, Chelsea College.

  4. CarpenterT. P. and MoserJ. M.: 1979, ‘An investigation of the learning of addition and subtraction’, Theoretical Paper No. 79, Madison, Wisconsin Research and Development Center for Individual Schooling.

    Google Scholar 

  5. CarpenterT. P. and MoserJ. M.: 1982, ‘The development of addition and subtraction problem solving skills’, in T. P.Carpenter, J. M.Moser and T. A.Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, New Jersey, pp. 9–24.

    Google Scholar 

  6. CarpenterT. P. and MoserJ. M.: 1983, ‘Acquisition of addition and subtraction concepts’, in R.Lesh and M.Landau (eds.), Acquisition of Mathematics Concepts and Processes, Academic Press, New York, pp. 7–44.

    Google Scholar 

  7. ComitiC.: 1981, ‘Les premieres acquisitions de la notion de nombre par l'enfant’, Educational Studies in Mathematics 11, 301–318.

    Google Scholar 

  8. Denvir, B. and Brown, M.: 1986, ‘Understanding of number concepts in low attaining 7–9 year olds, part II: The teaching studies’, Educational Studies in Mathematics 17 (2) (in press).

  9. DenvirB., StolzC., and BrownM.: 1982, ‘Low attainers in mathematics 5–16: Policies and practices in schools’, Schools Council Working Paper 72, Methuen Educational, London.

    Google Scholar 

  10. Descoudres, A.: 1921, ‘Le development de l'enfant de deux a sept ans’, Delachaux et Niestle, C.A., Paris.

  11. FusonK. C.: 1982, ‘Analysis of the counting on procedure’, in T. P.Carpenter, J. M.Moser and T. A.Romberg (ed.), Addition and Subtraction: A Cognitive Perspective, Elbaum, New Jersey, pp. 67–81.

    Google Scholar 

  12. FusonK. C., RichardsJ. and BriarsD. J.: 1982, ‘The acquisition and elaboration of the number word sequence’, in C. J.Brainerd (ed.), Children's Logical and Mathematical Cognition, Springer Verlag, New York, pp. 33–91.

    Google Scholar 

  13. GelmanR. and GallistelC. R.: 1978, The Child's Understanding of Number, Harvard University Press, Cambridge, Mass.

    Google Scholar 

  14. HughesM.: 1981, ‘Can preschool children add and subtract?’ Educational Psychology 1, 207–219.

    Google Scholar 

  15. Loevinger, J.: 1947, ‘A systematic approach to the construction and evaluation of tests of ability’, Psychological Monographs 61, No. 4. American Psychological Association.

  16. MatthewsJ.: 1983, ‘A subtraction experiment with six and seven year old children’, Educational Studies in Mathematics 14, 139–154.

    Google Scholar 

  17. NesherP.: 1982, ‘Levels of description in the analysis of addition and subtraction word problems’, in T. P.Carpenter, J. M.Moser and T. A.Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, New Jersey, pp. 25–38.

    Google Scholar 

  18. N.F.E.R.: 1969–80, ‘Basic Mathematics Tests’, National Foundation for Educational Research. NFER-Nelson, London.

    Google Scholar 

  19. SchaefferB., EgglestonV. H., and ScottJ. L.: 1974, ‘Number development in young children’, Cognitive Psychology 6, 357–379.

    Google Scholar 

  20. SteffeL. P.: 1983, ‘Children's algorithms as schemes’, Educational Studies in Mathematics 14, 109–125.

    Google Scholar 

  21. SteffeL. P., vonGlaserfeldE., RichardsJ., and CobbP.: 1983, Children's Counting Types: Philosophy, Theory and Application, Praeger, New York.

    Google Scholar 

  22. SteffeL. P. and JohnsonD. C.: 1971, ‘Problem solving performance of first-grade children’, Journal for Research in Mathematics Education 2, 50–64.

    Google Scholar 

  23. ResnickL. B.: 1983, ‘A development theory of number understanding’, in H. P.Ginsberg (ed.), The Development of Mathematical Thinking, Academic Press, New York, pp. 109–151.

    Google Scholar 

  24. RileyM. S., GreenoJ. G., and HellerJ. I.: 1983, ‘Development of children's ability in arithmetic’, in H. P.Ginsburg (ed.), The Development of Mathematical Thinking, Academic Press, New York, pp. 153–196.

    Google Scholar 

  25. VergnaudG.: 1982, ‘A classification of cognitive tasks and operations of thought involved in addition and subtraction problems’, in T. P.Carpenter, J. M.Moser, and T. A.Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, New Jersey, pp. 35–59.

    Google Scholar 

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Denvir, B., Brown, M. Understanding of number concepts in low attaining 7–9 year olds: Part I. Development of descriptive framework and diagnostic instrument. Educ Stud Math 17, 15–36 (1986). https://doi.org/10.1007/BF00302376

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Keywords

  • Longitudinal Study
  • Assessment Instrument
  • Main Hypothesis
  • Subsequent Paper
  • Diagnostic Assessment