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Problem solving in France: didactic and curricular perspectives

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Abstract

In this paper, we address the issue of problem solving in classrooms in France through two different and complementary approaches: didactic research and curricular choices. These two approaches correspond to two different, but not independent perspectives on problem solving and we investigate the existing links between them. We show that in France, the solving of problems is given a central role both in didactic research and curricular choices and that problem solving, as generally understood, is an object taking controversial positions, and we try to elucidate the rationale behind such positions. This paper is structured into two main parts: the first part devoted to didactic research, the second part to curricula, and one last part, discussion, wherein we question didactic research and curricula as regards their potential for influencing the reality of problem solving in classroom practices.

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Notes

  1. This text entitled “Invitation to Didactique”, accessible on the web, is a very clear introduction to the TDS. Its first chapter includes all the notions mentioned in this article.

  2. According to the TDS, “each item of mathematical knowledge can be characterized by a (or some) adidactical situation(s) which preserve(s) meaning”. These are called fundamental situations (Brousseau 1997, p. 30).

  3. The term adidactic labels a situation wherein the students behave as epistemic subjects: they forget at least for a while that the problem proposed to them has been designed by the teacher with a particular didactical goal, and accept the mathematical responsibility given to them by the teacher (devolution process).

  4. The adidactic milieu denotes the elements with which the students interact in an adidactic situation. It includes material and symbolic artefacts, and generally other students.

  5. The didactic contract denotes the implicit set of expectations that the teacher and the students have of each other regarding mathematical knowledge. The progression of knowledge necessarily involves ruptures and renegotiations of the didactic contract.

  6. FUG concepts are defined as Formalizing, Unifying and Generalizing concepts.

  7. Information about this project can be found in its website: (http://www.mathsamodeler.net/).

  8. The use of the world “tool” in this excerpt can be linked to the distinction introduced by Douady (1986) between the tool and object dimensions of mathematics concepts and the fact that most concepts appear first as implicit tools of mathematical activity before becoming official objects of the mathematics edifice. On the basis of this epistemological analysis, Douady has built a didactical construction known as the tool-object dialectics that many French researchers use jointly with the TDS.

  9. It must be stressed nevertheless that the expression problem-situation will not disappear from the educational discourse and is still used at primary and secondary level.

  10. The term noosphere denotes the interface between didactic systems stricto sensu, and the outside world. The noosphere involves all those having curricular responsibilities, the authors of textbooks and didactic material, the members of educational commissions and associations…

  11. IREM: Research Institute in Mathematics Education. A national commission created in 1975: the COPIRELEM coordinates the IREMs activities dealing with primary education. It plays a crucial role in the diffusion of research and action-research results, through its annual conferences and seminars, its various publications, its regular contacts with the Ministry of Education, and the involvement of some of its members in governmental groups of experts. Information about the IREMs and the COPIRELEM can be found on the national website of the IREMs: http://www.irem.univ-mrs.fr.

  12. INRP: National Institute for Pedagogical Research.

  13. These categories are the following: Exercices d’exposition, problèmes ou exercices de recherche, exercices didactiques, exécution de tâches techniques, manipulations, applications des mathématiques, tests.

  14. The quotations come from the second edition of the book published in 1976, much more accessible than the original one.

  15. In the official syllabus, the expression problem-situation, present in the preparatory texts, has nearly disappeared at the benefit of the expression “situation creating a problem”.

  16. This single example contrasts with the rich set of open problems proposed in the book. The part devoted to problem-situations in this book is in fact inspired by a text on learning situations and problem situations published in the first book produced by the Commission Inter-IREM Premier Cycle (1986–1989): Suivi scientifique Sixième - 1985–1986 resulting from the pre-experimentation of the new Collège syllabus imparted to the IREMs by the Ministry of Education from 1985 to 1989.

  17. through the IREM publications but also the strong engagement of IREMs in in-service teacher training programs.

  18. ERMEL: Equipe de recherche sur l’Enseignement des mathématiques à l’École Élémentaire.

  19. COREM: Centre pour l’Observation et la Recherche sur l’Enseignement des Mathématiques, created by Brousseau and attached to the IREM of Bordeaux.

  20. APMEP: Association des Professeurs de Mathématiques de l’Enseignement Public.

  21. This situation is also described in (Warfield, 2006), pp. 55–57.

  22. Les maths à la découverte du monde CE1 (176 pp). Editions Hachette 2004.

  23. EuroMaths CE1 student textbook (160 pp), teacher textbook (255 pp). Editions Hatier 2001.

References

  • Arsac, G., Germain, G., & Mante, M. (1988). Problème ouvert et situation problème. Lyon: IREM de Lyon.

    Google Scholar 

  • Balmes, R. M., & Coppé, S. (1999). Les activités d’aide à la résolution de problèmes dans les manuels de cycle 3. Grand N, 63, 39–58.

    Google Scholar 

  • Berthelot, R., & Salin, M. H. (1998). The role of pupils’ spatial knowledge in the elementary teaching of geometry. In D. Villani (Ed.), Perspectives on the teaching of geometry for the 21st century (pp. 71–78). New ICMI Study Series 5, Dordrecht: Kluwer.

    Google Scholar 

  • Bonafé, F., Chevalier, A., Combes, M. C., Deville, A., Dray, L., Robert, J.P., & Sauter, M. (2002). Les narrations de recherche: de l’école primaire au lycée. IREM de Montpellier and APMEP.

  • Bosch, M., Fonseca, C., & Gascón, J. (2004). Incompletitud de las organizaciones matematicas locales en las instituciones escolares. Recherches en Didactique des mathématiques, 24(2.3), 205–250.

    Google Scholar 

  • Bosch, M., Chevallard, Y., & Gascón, J. (2005). Science or Magic? The use of models and theories in didactics of mathematics. In M. Bosch (Ed.), Proceedings of the IVth Congress of the European Society for Research in Mathematics Education (CERME 4) (pp. 1254–1263). Barcelona: Universitat Ramon Llull Editions.

  • Bosch, M., & Gascón, J. (2006). Twenty-five years of the didactic transposition. ICMI Bulletin, 58, 51–65. (http://www.mathunion.org/ICMI/bulletin/).

    Google Scholar 

  • Brousseau, N., & Brousseau, G. (1987). Rationnels et décimaux dans la scolarité obligatoire. IREM de Bordeaux.

  • Brousseau, G. (1997). Theory of Didactical Situations in Mathematics (1970–1990). Dordrecht: Kluwer.

    Google Scholar 

  • Brousseau, G. (2006). Mathematics, didactical engineering and observation. Lecture presented at the PME 30 conference, Prague, July 2006. Retrieved from (http://www.math.washington.edu/∼warfield/Didactique.html).

  • Chevalier, A., & Sauter, M. (1992). Narration de recherche. IREM de Montpellier.

  • Chevallard, Y. (1985). La transposition didactique (1991). Grenoble: La Pensée Sauvage.

    Google Scholar 

  • Chevallard, Y. (1992). Concepts fondamentaux de la didactique. Perspectives apportées par une approche anthropologique. Recherches en Didactique des mathématiques, 12(1), 73–112.

    Google Scholar 

  • Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des mathématiques, 19(2), 221–266.

    Google Scholar 

  • Chevallard, Y. (2002). Organiser l’étude. Cours 1—Structures & Fonctions. Cours 3 - Écologie & Régulation. In J. L. Dorier et al. (Eds), Actes de la XI ème École d’été de Didactique des Mathématiques (pp. 3–22 and 41–56). Grenoble: La Pensée Sauvage.

  • Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In M. Bosch (Ed.), Proceedings of the IVth Congress of the European Society for Research in Mathematics Education (CERME 4) (pp. 22–30). Barcelona: Universitat Ramon Llull Editions.

  • Commission Inter-IREM Premier Cycle (1986–1989). Suivis Scientifiques. Lyon: IREM de Lyon.

  • COPIRELEM (1987). Aides pédagogiques pour le cycle moyen. Situations problèmes. Elem-math IX. Brochure 64. Paris: APMEP.

    Google Scholar 

  • COPIRELEM, (2003). Concertum. Dix ans de formation de professeurs des écoles en mathématiques. Paris: Editions ARPEME. (http://www.arpeme.com).

  • Dorier, J. L. (Ed.) (2000). On the teaching of linear algebra. Dordrecht: Kluwer.

  • Douady, R. (1986). Jeux de cadre et dialectique outil-objet. Recherches en Didactique des mathématiques, 7(2), 5–32.

    Google Scholar 

  • ERMEL (1977–1982). Apprentissages mathématiques à l’école élémentaire (CP to CM2). Paris: Hatier.

  • ERMEL (1991–1999). Apprentissages numériques et résolution de problèmes (CP to CM2). Paris: Hatier.

  • Gascón, J. (1993). Desarrollo del conocimiento matemático y analysis didáctico: del patrón de analisis-síntesis a la genesis del lenguaje algebraico. Recherches en Didactique des mathématiques, 13(3), 295–332.

    Google Scholar 

  • Gascón, J. (1998). Evolución de la didáctica de la matemática como disciplina científica. Recherches en Didactique des mathématiques, 18(1), 7–34.

    Google Scholar 

  • Glaeser, G. (1976) Le livre du problème. Pédagogie de l’exercice et du problème. Paris: Editions CEDIC (second edition).

  • Grenier, D., & Payan, C. (1998). Spécificités de la preuve et de la modélisation en mathématiques Discrètes. Recherches en Didactique des mathématiques, 18(2), 59–100.

    Google Scholar 

  • Grenier, D., & Payan, C. (2002). Situations de recherche en «classe»: essais de caractérisation et propositions de modélisation. In V. Durand-Guerrier, & C. Tisseron (Eds.), Actes du séminaire national de Didactique des mathématiques 2002 (pp. 189–204). Paris: IREM Paris 7.

    Google Scholar 

  • Houdement, C. (1999). Le choix des problèmes pour «la résolution de problèmes». Grand N, 63, 59–76.

    Google Scholar 

  • IREM de Lyon (1983). 250 problèmes pour nos élèves. Lyon: IREM de Lyon.

    Google Scholar 

  • Laborde, C., & Perrin-Glorian, M.J. (Eds.) (2005). Teaching situations as object of research: empirical studies within theoretical perspectives. Educational Studies in Mathematics, 59(1–2–3) (special issue).

  • Legrand, M. (1993). Débat scientifique en classe de mathématiques. Repères-IREM, 10, 123–158.

    Google Scholar 

  • Masselot, P. (2000). De la formation initiale en didactique des mathématiques (en centre IUFM) aux pratiques quotidiennes en classe. Thèse de Didactique des Mathématiques. Paris: Université Paris 7.

    Google Scholar 

  • MENJS (1991). Les cycles à l’école primaire. Paris: Hachette et CNDP.

    Google Scholar 

  • MEN (1995). Les programmes à l’école primaire. Paris: Hachette et CNDP.

    Google Scholar 

  • MEN (2002a). Programmes du primaire (2002). Retrieved from (http://www.cndp.fr/doc_administrative/programmes/accueil.htm).

  • MEN (2002b). Documents d’application des programmes. Mathématiques. Cycle 2 (37 pages), Cycle 3 (47 pages). Paris: Scéren.CNDP.

    Google Scholar 

  • MEN (2005). Documents d’accompagnement des programmes. Mathématiques (96 pages). Paris: Scéren.CNDP.

    Google Scholar 

  • MEN, (2006). L’enseignement des mathématiques au cycle 3 de l’école primaire. Rapport à monsieur le ministre de l’éducation nationale, de l’enseignement supérieur et de la recherche. Retreived from http://www.education.gouv.fr/cid4172/l-enseignement-des-mathematiques-au-cycle-3-de-l-ecole-primaire.html.

  • MEN (2007). Programmes du secondaire (2002). Retrieved from (http://eduscol.education.fr/D0015/LLPHPR01.htm).

  • Ouvrier-Buffet, C. (2003). Construction de définitions/construction de concepts: vers une situation fondamentale pour la construction de définitions en mathématiques. Thèse de Didactique des Mathématiques. Grenoble: Université Joseph Fourier.

    Google Scholar 

  • Perrin-Glorian, M. J. (1994). Théorie des Situations Didactiques: Naissance, Développement, Perspectives. In: M. Artigue, et al. (Eds.), Vingt Ans de Didactique des Mathématiques en France (pp. 97–147). Grenoble: La Pensée Sauvage.

    Google Scholar 

  • Polya, G. (1945). How to solve it? Princeton: Princeton University Press.

    Google Scholar 

  • Robert, A., & Robinet J. (1996). Prise en compte du méta en didactique des mathématiques. Recherches en Didactique des mathématiques, 16(2),145–176.

    Google Scholar 

  • Robert, A., & Rogalski, J. (2002). Le système cohérent et complexe des pratiques des enseignants de mathématiques: une double approche. Revue Canadienne de l’enseignement des sciences, des mathématiques, de la technologie, 12(4), 505–528.

    Google Scholar 

  • Robert, A., & Rogalski, J. (2005). A cross-analysis of the mathematics’ teacher activity. An example in a French 10th-grade class. Educational Studies in Mathematics, 59(1–2–3), 259–298.

    Google Scholar 

  • Roditi, E. (2001). L’enseignement des nombres décimaux en sixième. Étude de pratiques ordinaires? Thèse de Didactique des Mathématiques. Paris: Université de Paris 7.

    Google Scholar 

  • Rodriguez, E., Bosch, M., & Gascón, J. (2007). An anthropological approach to metacognition: the “study and research courses”. Communication to CERME 5. Retrieved from (http://www.cyprusisland.com/cerme/group11.htm).

  • Sarrazy, B. (1997). Sens et situations: une mise en question de l’enseignement des stratégies méta-cognitives en mathématiques. Recherches en Didactique des mathématiques, 17(2), 135–166.

    Google Scholar 

  • Sauter, M. (2000). Formation de l’esprit scientifique avec les narrations de recherche au cycle central du collège. Repères-IREM, 39, 7–20.

    Google Scholar 

  • Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando: Academic Press.

    Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense-making in mathematics. In: D. Grouws (Eds.), Handbook for Research on Mathematics Teaching and Learning (pp. 334–370). New York: MacMillan.

    Google Scholar 

  • Sierpinska, A. (2006). Entre l’idéal et la réalité de l’enseignement mathématique. Annales de Didactique et de Sciences Cognitives, 11, 5–40.

    Google Scholar 

  • Vergnaud, G. (1990). La théorie des champs conceptuels. Recherches en Didactique des Mathématiques, 10(2.3), 133–170.

    Google Scholar 

  • Vergnes, D. (2001). Les effets d’un stage de formation en géométrie. Recherches en Didactique des Mathématiques, 21(1.2), 99–122.

    Google Scholar 

  • Warfield, V. M. (2006). Invitation to didactique. Retrieved from (http://www.math.washington.edu/∼warfield/Didactique.html).

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Authors and Affiliations

  1. Université Paris 7 & Equipe DIDIREM, Université Paris 7, case 7018, 2 Place Jussieu, 75251, Paris cedex 05, France

    Michèle Artigue

  2. IUFM de Haute Normandie & Equipe DIDIREM, IUFM de Haute Normandie, BP 18, 76131, Mont Saint Aignan Cedex, France

    Catherine Houdement

Authors
  1. Michèle Artigue
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  2. Catherine Houdement
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Correspondence to Michèle Artigue.

Appendix

Appendix

1.1 Appendix 1: Les maths à la découverte du monde CE1—Student worksheet—Éditions Hachette 2004. P. 96

From sums to products: multiplication

Observe how the number of mail bags is computed and do the same with the trucks.

figure a

1.2 Appendix 2: EuroMaths CE1 – Student textbook. Éditions Hatier 2001. P 86

Rules for the game: three players. Each player randomly chooses a grid, throws the die and colors the number of rows given by the die. The player who has colored the biggest number of squares wins 1 point. Players play five successive games.

figure b

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Artigue, M., Houdement, C. Problem solving in France: didactic and curricular perspectives. ZDM Mathematics Education 39, 365–382 (2007). https://doi.org/10.1007/s11858-007-0048-x

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