Skip to main content
SpringerLink
Log in

A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, some of them are specific to mathematical activity? Starting from the paramount importance of semiotic representation for any mathematical activity, we put forward a classification of the various registers of semiotic representations that are mobilized in mathematical processes. Thus, we can reveal two types of transformation of semiotic representations: treatment and conversion. These two types correspond to quite different cognitive processes. They are two separate sources of incomprehension in the learning of mathematics. If treatment is the more important from a mathematical point of view, conversion is basically the deciding factor for learning. Supporting empirical data, at any level of curriculum and for any area of mathematics, can be widely and methodologically gathered: some empirical evidence is presented in this paper.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

  • Descartes, R.: 1954, The Geometry of Rene Descartes (translated from French and Latin), Dover, New York.

  • Duval, R.: 1983, ‘L'obstacle du dédoublement des objets mathématiques’, Educational Studies in Mathematics 14(4), 385–414.

    Article  Google Scholar 

  • Duval, R.: 1988, ‘Graphiques et équations: l'articulation de deux registres’, Annales de Didactique et de Sciences Cognitives 1, 235–253.

    Google Scholar 

  • Duval, R.: 1991, ‘Structure du raisonnement déductif et apprentissage de la Démonstration’, Educational Studies in Mathematics 22(3), 233–261.

    Article  Google Scholar 

  • Duval, R.: 1993, ‘Registres de représentation sémiotique et fonctionnement cognitif de la pensée’, Annales de Didactique et de Sciences Cognitives 5, 37–65.

    Google Scholar 

  • Duval, R.: 1995a, ‘Geometrical Pictures: Kinds of representation and specific processing’, in R. Suttherland and J. Mason (eds.), Exploiting Mental Imagery with Computers in Mathematics Education, Springer, Berlin, pp. 142–157.

    Google Scholar 

  • Duval, R.: 1995b, Sémiosis et pensée humaine, Berne, Peter Lang.

    Google Scholar 

  • Duval, R.: 1996a, ‘Quel cognitif retenir en didactique des mathématiques?’ Recherches en Didactique des Mathématiques 16(3), 349–382.

    Google Scholar 

  • Duval, R.: 1996b, ‘Les représentations graphiques: fonctionnement et conditions de leur apprentissage’, in Actes de la 46ème Rencontre Internationale de la CIEAEM, tome 1, 3–15, Université Paul Sabatier, Toulouse, pp. 3–15.

  • Duval, R.: 1998a, ‘Signe et objet (I): trois grandes étapes dans la problématique des rapports entre représentation et objet’, Annales de Didactique et de Sciences Cognitives 6, 139–163.

    Google Scholar 

  • Duval, R.: 1998b, ‘Geometry from a cognitive point a view’, in C. Mammana and V. Villani (eds.), Perspectives on the Teaching of Geometry for the 21st Century, Kluwer Academic Publishers, Dordrecht, pp. 37–52.

    Google Scholar 

  • Duval, R.: 2000a, ‘Basic issues for research in mathematics education’, in T. Nakahara and M. Koyama (eds.), Proceedings of the 24th Conference of PME, 1, Nishiki Print Co. Ltd., Hiroshima, pp. 55–69.

  • Duval, R.: 2000b, ‘Ecriture, raisonnement et découverte de la démonstration en mathématiques’, Recherches en Didactique des Mathématiques 20(2), 135– 170.

    Google Scholar 

  • Duval, R.: 2002, ‘L'apprentissage de l'algèbre et le problème cognitif de la désignation des objets’, in J.Ph. Drouhard and M. Maurel (eds.), Actes des Séminaires SFIDA, 13–16 (IV), IREM, Nice, pp. 67–94.

  • Duval, R.: 2003, ‘Langage(s) et représentation(s) dans l'enseignement des mathématiques: deux pratiques et une troisième’, in M. Kourkoulos, G. Toulis and C. Tzanakis (eds.), Proceedings 3rd Colloquium on the Didactics of Mathematics, University of Crete, Rethymnon, pp. 13–33.

  • Duval, R. (ed.): 1999, Conversion et articulation des représentations analogiques, Séminaire I.U.F.M., D.R.E.D., Villeneuve d'Ascq.

  • Hitt, F. (ed.), 2002, Representations and Mathematics Visualization, North American Chapter of IGPME, Cinvestav-IPN, Mexico.

  • Hitt, F.: 2003, ‘Le caractère fonctionnel des représentations’, Annales de Didactique et de Sciences Cognitives 8, 255–271.

    Google Scholar 

  • Frege, G.: 1971, ‘Sens et dénotation’, in Ecrits logiques et philosophiques (translated by C. Imbert), Seuil, Paris.

  • Kant, E.: 1956, Kritik der reinen Vernunft, Felix Meiner, Hamburg.

    Google Scholar 

  • MEN: 1997, ‘Évaluation CE2-6ème Résultats nationaux-Septembre 1996’, Les Dossiers 79, Ministère de l'Education Nationale, Paris.

  • MEN: 1998, ‘Évaluation CE2-6ème Repères nationaux-Septembre 1997’, Les Dossiers 100, Ministère de l'Education Nationale, Paris.

  • MEN: 1999, ‘Évaluation CE2-6ème Repères nationaux-Septembre 1998’, Les Dossiers 111, Ministère de l'Education Nationale, Paris.

  • Pavlopoulou, K.: 1993, ‘Un problème décisif pour l'apprentissage de l'algèbre linéaire: la coordination des registres de représentation’, Annales de Didactique et de Sciences Cognitives 5, 67–93.

    Google Scholar 

  • CP = Peirce, C.S.: 1931, Collected Papers, II, Elements of Logic, Harvard University Press, Cambridge, Massachusetts.

  • Piaget, J.: 1923, Le langage et la pensée chez l'enfant, Delachaux et Niestlé,Neuchâtel.

  • Piaget, J.: 1926, La représentation du monde chez l'enfant, Alcan, Paris.

    Google Scholar 

  • Piaget, J.: 1967, Biologie et connaissance, Gallimard, Paris.

    Google Scholar 

  • Piaget, J.: 1973, Introduction à l'épistémologie génétique, 1, La pensée mathématique, P.U.F., Paris.

    Google Scholar 

  • Pluvinage, F.: 1990, ‘Didactique de la résolution de problèmes’, Annales de Didactique et de Sciences Cognitives 3, 7–34.

    Google Scholar 

  • Rommevaux, M.P.: 1998, ‘Le discernement des plans dans une situation tridimensionnelle’, Annales de Didactique et de Sciences Cognitives 6, 27–65.

    Google Scholar 

  • Saussure (de), F.: 1973 (1915), Cours de linguistique générale, Payot, Paris.

    Google Scholar 

  • Schoenfeld, A.H.: 1986, ‘On having and using Geometric Knowledge’, in J. Hiebert (ed.), Conceptual and Procedural Knowledge: The Case of Mathematics, Lawrence Erlbaum Associates, Hillsdale, NJ, pp. 225–264.

    Google Scholar 

  • Vygotski, L.S.: 1985, Pensée et langage, Editions Sociales, Paris.

Download references

Author information

Authors and Affiliations

  1. Laboratoire Mutations des Systèmes Educatifs, Maison de la Recherche, Université du Littoral, 17 rue du Puits d'Amour B.P. 751, 62231, Boulogne-sur-Mer Cedex, France

    Raymond Duval

Authors
  1. Raymond Duval
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raymond Duval.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Duval, R. A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educ Stud Math 61, 103–131 (2006). https://doi.org/10.1007/s10649-006-0400-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-006-0400-z

Key Words

Search

Navigation