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Didactics of Mathematics: Concepts, Roots, Interactions and Dynamics from France

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Tools and Mathematics

Part of the book series: Mathematics Education Library ((MELI,volume 110))

Abstract

This chapter analyses specificities of the French field of ‘didactics of mathematics’, questioning its reasons, tracing the geneses of concepts related to artefacts and following influences on, and interactions with the international communities of research. This complex genesis is traced in four sections: a first section on the roots of the didactics of mathematics in France, a second section on two founding theoretical frameworks (the theory of didactical situations of Brousseau, and the theory of conceptual fields of Vergnaud), a third section on the anthropological approach of Chevallard, a fourth focusing on specific approaches dedicated to artefacts and resources in mathematics education. Beyond historical and cultural specificities, the chapter aims to evidence the potential of interactions between teachers and researchers, as well as interactions between researchers in mathematics and mathematics education for improving our understanding of learning and teaching issues in mathematics.

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Notes

  1. 1.

    I have chosen to give sometimes long quotations, keeping the words and the language—sometimes in French—of these authors, in order to allow the reader to have a direct contact with their works.

  2. 2.

    I would like to thank Janine Rogalski and Rudolf Straesser for their comments and advices.

  3. 3.

    CREM: Commission de réflexion sur l’enseignement des mathématiques.

  4. 4.

    The digital copies of the journal can be found at http://www.unige.ch/math/EnsMath/EM_fr/welcome.html.

  5. 5.

    Informations from the ICMI website http://www.mathunion.org/icmi/icmi/executive-committee/past-executive-committees/.

  6. 6.

    The Musée pédagogique is the forerunner of the National Institute for Pedagogical Research, which became in 2010 the French Institute of Education.

  7. 7.

    Note that this did not come, for the 1967 reform, from the whole Bourbaki group, but only from its members interested in changing the teaching at a secondary level. Houzel (2004, p. 57) wrote, on this particular question: “In the late 60s, a reform movement in secondary mathematics education has been launched in most countries and this movement has unfortunately claimed Bourbaki. From it, came what was called the ‘new math’, whose harmfulness is no longer in doubt. But it is unfair to shift the burden to Bourbaki, whose only fault was to ignore the problem of Dieudonné propaganda rather dangerous to teachers (our translation)”

  8. 8.

    It is indeed significant that the first ICMI study, 1985) was dedicated to the Influence of Computers and Informatics on the Mathematics and its Teaching (Cornu & Ralston, 1992).

  9. 9.

    In 1910 was also created the association « L’école émancipée » [The emancipated school], gathering pedagogical activist teachers and revolutionary syndicalists.

  10. 10.

    The democratic goal of the French republic was to move from a schooling system founded on ‘orders’ (schools for people vs. schools for upper classes) to a schooling system founded on ‘levels’ (primary level vs. secondary level), i.e. a same school for each student: ‘l’école unique’. Several successive laws (1959, 1963, 1966) constituted a progress towards this objective, never fully achieved.

  11. 11.

    http://faculty.washington.edu/warfield/guy-brousseau.com

  12. 12.

    Extract of the third part of the preliminary report of the commission Lichnerowicz, published in the « Bulletin of APMEP », no. 258.

  13. 13.

    Artigue and Douady (p. 85) underline that “the expression didactics of mathematics has been introduced by Klein in 1910”.

  14. 14.

    French national centre for scientific research.

  15. 15.

    This growth mirrors the growth of mathematics education as a field noted in Sect. 7.2.

  16. 16.

    I have chosen to give, in this section, recent references to the work of Brousseau and Vergnaud, offering a more synthetic view on their work, but it has to be clear that the foundations of their theories come from the 1970s.

  17. 17.

    http://guy-brousseau.com/le-corem/

  18. 18.

    In a recent paper, Brousseau (2012) came back to the “psychological and didactical roots” of his theory, acknowledging the importance, among other researchers, of Greco and Piaget. He has developed a website http://faculty.washington.edu/warfield/guy-brousseau.com where could be found various elements grounding his approach.

  19. 19.

    For distinguishing Brousseau’s notion from the non-specific, standard uses of the word ‘situation’, she chose to capitalize this as soon as it is used in the frame of the Theory of didactical situations. I have not retained here such a convention: it is enough to consider that, in this section related to Brousseau’s theory, the word ‘situation’ is used with respect to this theory.

  20. 20.

    In French in the text/

  21. 21.

    In French in the text.

  22. 22.

    This proposition results also of interactions with a French didactician, Alain Mercier I want to thank here.

  23. 23.

    More on Vergnaud theory and publications can be found on the French mathematics didactics website: http://www.ardm.eu/contenu/gérard-vergnaud-english.

  24. 24.

    Chevallard offers, on his website, most of his publications: http://yves.chevallard.free.fr.

  25. 25.

    Translation resulting from interactions between Marianna Bosch and John Monaghan.

  26. 26.

    For a discussion on this point, see Bosch and Chevallard (1999, p. 29).

  27. 27.

    Contrary to what is said in the previous quote of Guin and Trouche (1999), corresponding to a previous step of the genesis instrumental approach of didactics, an artefact is not necessarily material. It can be also symbolic, as an algorithm, or a language. Its structural characteristic is to be a result of human activity, and to be potentially engaged in new activity.

  28. 28.

    For example, Hembree and Dessart (1986), see Sect. 13.2.

  29. 29.

    The notion of ‘problematic’ comes from the French ‘problématique’, well defined by Edward Said: ‘The idea of beginning, indeed the act of beginning [a research], necessarily involves an act of delimitation by which something is cut out of a great mass of material, separated from the mass, and made to stand for, as well as be, a starting point, a beginning; […] such notion of inaugural delimitation is Louis Althusser’s idea of the problematic, […] is something given rise to by analysis’ (Said 1978, p. 24).

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Trouche, L. (2016). Didactics of Mathematics: Concepts, Roots, Interactions and Dynamics from France. In: Tools and Mathematics. Mathematics Education Library, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-319-02396-0_10

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