Abstract
Earlier research by the author indicated that many below average attainers do not remember number facts and use alternative strategies to obtain solutions to basic arithmetical problems. These alternatives were frequently seen as the ‘best way’ of finding a solution.
This paper considers the relationship between the various strategies used by mixed ability children aged 7 to 12. An analysis of alternatives suggests that the selection is not underpinned by regression through the learning sequence, but by regression dominated by the child's preference for certain strategies over others. Through the evaluation of a hierarchy of preferences, divergence between the strategies available to the less able and the more able child is revealed. The alternative strategies used are based either on counting — procedural strategies, or on the use of selected known knowledge — deductive strategies. Above average children have both available as alternatives; evidence of deduction is rare amongst below average children. The more able child appears to build up a growing body of known facts from which new known facts are deduced. Less able children — relying mainly on procedural strategies — do not appear to have this feedback loop available to them.
The paper contends that, for some children, procedural methods do not encourage the need to remember; the procedure provides security. On the other hand, deductive methods initially enhance the ability to remember other basic facts and eventually help children make extensive use of facts that are known to remove the need to remember new ones. More able children appear to be doing a qualitatively different sort of mathematics than the less able.
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References
Byers, V. and Erlwanger, S.: 1985, ‘Memory in mathematical understanding’, Educational Studies in Mathematics 16, 259–281.
Carpenter, T. P., Hiebert, J., and Moser, J. M.: 1981, ‘Problem structure and first-grade children's initial solution processes for simple addition and subtraction problems’, Journal for Research in Mathematics Education 12, 27–39.
Carpenter, T. P., Hiebert, J., and Moser, J. M.: 1982, ‘Cognitive development and children's solutions to verbal arithmetic problems’, Journal for Research in Mathematics Education 13, 83–98.
Cohen, L. and Manion, L.: 1985, Research Methods in Education, 2nd ed., Croom Helm Ltd., London.
Fuson, K. C., Richards, J., and Briars, D. J.: 1982, ‘The acquisition and elaboration of the number word sequence’, In Brainerd, C. (ed.), Progress in Cognitive Development: Vol. 1, Children's Logical and Mathematical Cognition, Springer-Verlag, New York.
Fuson, K. C.: 1982, An analysis of the counting-on solution procedure in addition’, In T.Carpenter et al. (eds.), Addition and Subtraction: A Cognitive Perspective, Laurence Erlbaum, New Jersey.
Gray, E. M.: ‘Barriers to numerical competence created by efforts to understand’, in preparation.
Gray, E. M.: 1987, ‘Slow learning children and their range of risk in basic addition calculations’, Proceedings of BSRLM Day Conference, University of London Institute of Education, November 7th 1987, 8.
Gray, E. M.: 1988, ‘“O God — It's Take Away”: An examination of the strategies that below average children use to solve basic subtraction problems’, Proceedings of BSRLM Conference, University of Warwick, May 1988, pp. 27–29.
Groen, G. J. and Resnick, L. B.: 1977, ‘Can preschool children invent addition algorithms’, Journal of Educational Psychology 69, 645–652.
Herscovics, N. and Bergeron, J. C.: 1983, ‘Models of Understanding’, Zentralblatt für Didaktik der Mathematik 83, 75–83.
McEvoy, J.: 1989, “From counting to arithmetic: The development of early number skills’, British Journal of Special Education 16, 107–110.
Rosenthal, D. J. and Resnick, L. B.: 1974, ‘Children's solution processes in arithmetic word problems’, Journal of Educational Psychology 66, 817–825.
Secada, G. W., Fuson, C. K., and Hall, J. W.: 1983, ‘The transition from counting-all to counting-on in addition’, Journal for Research in Mathematics Education 14, 47–57.
Skemp, R. R.: 1976, ‘Relational understanding and instrumental understanding’, Mathematics Teaching 77, 20–26.
Steffe, L., vonGlaserfeld, E., Richards, J., and Cobb, P.: 1983, Children's Counting Types: Philosophy, Theory and Applications, Praeger, New York.
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Gray, E.M. An analysis of diverging approaches to simple arithmetic: Preference and its consequences. Educ Stud Math 22, 551–574 (1991). https://doi.org/10.1007/BF00312715
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DOI: https://doi.org/10.1007/BF00312715