Skip to main content
SpringerLink
Log in

The use of mathematical symbolism in problem solving: An empirical study carried out in grade one in the French community of Belgium

  • Published:
European Journal of Psychology of Education Aims and scope Submit manuscript

Abstract

This article relates to an empirical study based on the use of mathematical symbolism in problem solving. Twenty-five pupils were interviewed individually at the end of grade one; each of them was asked to solve and symbolize 14 different problems. In their classical curriculum, these pupils have received a traditional education based on a “top-down” approach (an approach that is still applied within the French Community of Belgium): conventional symbols are presented to the pupils immediately with an explanation of what they represent and how they should be used. Teaching then focuses on calculation techniques (considered as a pre-requisite for solving problems). The results presented here show the abilities (and difficulties) demonstrated by the children in making connections between the conventional symbolism taught in class and the informal approaches they develop when faced with the problems that are put to them. The limits of the “top-down” approach are then discussed as opposed to the more innovative “bottom-up” type approaches, such as those developed by supporters of Realistic Mathematics Educations in particular.

Résumé

Cet article relate une étude empirique centrée sur l’utilisation du symbolisme mathématique en résolution de problèmes. Vingt-cinq élèves ont été interviewés individuellement en fin de première année primaire; ils ont chacun été amenés à résoudre et à symboliser 14 problèmes différents. Dans leur curriculum classique, ces élèves ont reçu un enseignement traditionnel basé sur une approche de type “top-down” (approache encore couramment développée en Communauté française de Belgique): les symboles conventionnels sont proposés d’emblée aux élèves à qui on explique ce qu’ils représentent et comment ils doivent les utiliser. L’enseignement se focalise alors sur les techniques de calculs (considérées comme un pré requis à la résolution de problèmes). Les résultats présentés ici montrent les capacités (et difficultés) démontrés par les élèves pour créer des connexions entre le symbolisme conventionnel enseigné en classe et les démarches informelles qu’ils développent face aux problèmes qui leur sont proposés. Les limites de l’approche “top-down” sont alors discutées en opposition avec des approches plus novatrices de type “bottom-up”, telles que celles développées par les tenants de la Realistic Mathematics Education notamment.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Barrouillet, P., & Camos, V. (2002).Savoires, savoir-faire arithmétiques et leurs déficiences. Paris: Rapport pour le Ministère de la Recherche. Programme cognitique, école et sciences cognitives, document non publié.

    Google Scholar 

  • Bebout, H.C. (1990). Children’s symbolic representation of addition and subtraction word problems.Journal for Research in Mathematics Education, 2(2), 123–131.

    Article  Google Scholar 

  • Bednarz, N., Dufour-Janvier, B., Poirier, L., & Bacon, L. (1993). Socioconstructivist viewpoint on the use of symbolism in mathematics education.Alberta Journal of Educational Research, 39, 41–58.

    Google Scholar 

  • Booth, L.R. (1984).Algebra: Children’s strategies and errors. Oxford, UK: Nfer-Nelson Publishing Company.

    Google Scholar 

  • Carey, D.A. (1991). Number sentences: Linking addition and subtraction word problems and symbols.Journal for Research in Mathematics Education, 22(4), 266–280.

    Article  Google Scholar 

  • Carpenter, T.P., & Moser, J.M. (1982). The development of addition and subtraction problem solving skills. In T.P. Carpenter, J.M. Moser, & T.A. Romberg (Eds.),Addition and Subtraction. A cognitive perspective (pp. 9–24). Hillsdale, NJ: Lawrence Erlbaum Associates Publishers.

    Google Scholar 

  • Carpenter, T.P., & Moser, J.M. (1984). The acquisition of addition and subtraction concepts in grade one through three.Journal for Research in Mathematics Education, 15(3), 179–202.

    Article  Google Scholar 

  • Carpenter, T.P., Moser, J.M., & Bebout, H.C.88. Representation of addition and substraction word problems.Journal for Research in Mathematics Education,4, 345–357.

  • Cobb, P. (2000). From representation to symbolizing: Introductory comments on semiotics and mathematical learning. In P. Cobbs, E. Yackel, & K. Mc Clain (Eds.),Symbolising and communicating in mathematics classrooms. Perspectives on discourse, tools and instructional design (pp. 17–36). Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.

    Google Scholar 

  • Cobb, P., Yackel, E., & Mc Clain, K. (2000, Eds.).Symbolizing and communicating in mathematics classrooms. Perspectives on discourse, tools and instructional design. Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.

    Google Scholar 

  • De Corte, E., & Verschaffel, L. (1985a). Beginning first grader’s initial representation of arithmetic word problems.Journal of Mathematical Behavior, 4, 3–21.

    Google Scholar 

  • De Corte, E., & Verschaffel, L. (1985b). Writing number sentences to represent addition and substraction problems. In J.K. Damarin & M. Shelton (Eds.)Proceeding of the Seventh Annual Meeting of North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 50–56). Columbus, USA: University of Colombus.

    Google Scholar 

  • De Corte, E., Greer, B., & Verschaffel, L. (1996). Mathematics teaching and learning. In D.C. Berliner & R.C. Calfee (Eds.),Handbook of Educational Psychology (pp. 491–549). New York: MacMillan.

    Google Scholar 

  • Dellarosa-Cummins, D. (1991). Children interpretation of arithmetic word-problems.Cognition and Instruction, 8(3), 261–289.

    Article  Google Scholar 

  • Fagnant, A. (2002a).Quelle compréhension du symbolisme mathématique au travers de la résolution de problèmes arithmétiques? Thèse de doctorat non publiée. Université. Belgique.

  • Fagnant, A. (2002b). Mathematical symbolism: A feature responsible for superficial approaches? In A.D. Cockburn & E. Nardi (Eds.),Proceedings of the 26th Annual Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 345–352). Norwich, UK: University of East Anglia.

    Google Scholar 

  • Fagnant, A. (2005). Résoudre et symboliser des problèmes additifs et soustractifs en début d’enseignement primaire. In M. Crahay, E. De Corte, J. Grégoire, & L. Verschaffel (Eds.),Enseignement et apprentissage des mathématiques: Que disent les recherches? (pp.131–150). Bruxelles: De Boeck.

    Google Scholar 

  • Fayol, M. (1990).L’enfant et le nombre. Paris: Delachaux et Niestlé.

    Google Scholar 

  • Fuson, K.C. (1992). Research on whole number addition and subtraction. In D.A. Grows (Ed.),Handbook of research on mathematics teaching and learning (pp. 243–275). New York: MacMillan.

    Google Scholar 

  • Gravemeijer, K. (1997). Mediating between concrete and abstract. In T. Nunes & P. Bryant (Eds.),Learning and teaching mathematics. An international perspective (pp. 315–345). Hove, East Sussex: Psychology Press Ltd.

    Google Scholar 

  • Gravemeijer, K. (2002). Preamble: From models to modelling. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffell (Eds.),Symbolizing, modeling and tool use in mathematics education (pp. 7–22), Dordrecht, The Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000).Symbolizing, modelling, and instructional design. In P. Cobbs, E. Yackel, & K. Mc Clain (Eds.),Symbolising and communicating in mathematics classrooms. Perspectives on discourse, tools and instructional design (pp. 225–273). Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.

    Google Scholar 

  • Gravemeijer, K., Lehrer, R., van Oers, B., & Verschaffel, L. (2002). Introduction and overview. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.),Symbolizing, modeling and tool use in mathematics education (pp. 1–5), Dordrecht, The Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Julo, J. (1995).Représentation des problèmes et réussite en mathémtiques. Un apport de la psychologie cognitive à l’enseignement. PU de Rennes: Psyhcologies.

    Google Scholar 

  • Lewis, A.B., & Mayer, R.F. (1987). Student’s miscomprehension of relational statements in arithmetic word problems.Journal of Educational psychology, 75(4), 363–371.

    Article  Google Scholar 

  • Nunes, T. (1997). Systems of signs and mathematical reasoning. In T. Nunes & P. Bryant (Eds.),Learning and teaching mathematics: An international perspective (pp. 29–44), Hove, East Sussex: Psychology Press Ltd.

    Google Scholar 

  • Riley, M.S., Greeno, J.G., & Heller, J.I. (1983). Development of children’s problem-solving ability in arithmetic. In H.P. Ginsburg (Ed.),The development of mathematical thinking (pp. 153–196). New York: Academic Press.

    Google Scholar 

  • Sfard, A. (2000). Symbolizing mathematical reality into being. Or how mathematical discourse and mathematical objects create each other. In P. Cobbs, E. Yackel, & M. Mc Clain (Eds.),Symbolising and communicating in mathematics classrooms. Perspectives on discourse, tools and instructional design (pp. 37–98). Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.

    Google Scholar 

  • Stern, E. (1993). What makes certain arithmetic word problems involving the comparison of sets so difficult for children?Journal of Educational Psychology, 85(1), 7–23.

    Article  Google Scholar 

  • Vergnaud, G. (1982). A classification of cognitive tasks and operation of thought involved in addition and subtraction problems. In T.P. Carpenter, J.M. Moser, & T.A. Romberg (Eds.),Addition and Subtraction. A cognitive perspective (pp. 39–59). Hillsdale, NJ: Lawrence Erlbaum Associates Publishers.

    Google Scholar 

  • Verschaffel, L., & De Corte, E. (1997). Word problems: A vehicle for promoting authentic mathematical understanding and problem solving in the primary school? In T. Nunes & P. Bryant (Eds.),Learning and teaching mathematics: An international perspective (pp. 69–97). Hove, East Sussex: Psychology Press Ltd.

    Google Scholar 

  • Verschaffel, L., Greer, B., & De Corte, E. (2000).Making sense of word problems. Lisse, The Netherlands: Swets & Zeitlinger.

    Google Scholar 

  • Yackel, E. (2000). Introduction: Perspectives on semiotics and instructional design. In P. Cobbs, E. Yackel, & K. Mc Clain (Eds.),Symbolising and communicating in mathematics classrooms. Perspectives on discourse, tools and instructional design (pp. 1–15). Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Service de Pédagogie Expérimentale, Université de Liège, Bd du Rectorat, 5, B32, B4000, Liège, Belgique

    Annick Fagnant

Authors
  1. Annick Fagnant
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Annick Fagnant.

Additional information

This paper is based an a research project which was financed by the National Found for Scientific Research.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fagnant, A. The use of mathematical symbolism in problem solving: An empirical study carried out in grade one in the French community of Belgium. Eur J Psychol Educ 20, 355–367 (2005). https://doi.org/10.1007/BF03173562

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03173562

Key words

Search

Navigation