Skip to main content

Advertisement

SpringerLink
Log in

Structuralist Praxeologies as a Research Program on the Teaching and Learning of Abstract Algebra

  • Published:
International Journal of Research in Undergraduate Mathematics Education Aims and scope Submit manuscript

Abstract

Research reported in this article has been conducted from the theoretical perspective of the Anthropological Theory of the Didactic developed by Y. Chevallard and his collaborators, and from the institutional perspective of university mathematics education in France. It focuses on the teaching and learning of algebraic structures. The article introduces the notion of structuralist praxeology in the theory of teaching and learning of Abstract Algebra. The notion is illustrated by detailed descriptions of structuralist praxeologies related to the arithmetic of abstract rings, identified, on the one hand, in the history of mathematics and in Abstract Algebra textbooks and, on the other, in interactions among a group of learners of Abstract Algebra on an online discussion forum. The theoretical (and practical) issues regarding the teaching and learning of the structuralist praxeologies are discussed in the context of the difficulties that learners experience in the transition from undergraduate and graduate studies in mathematics at university. The article concludes with an outline of a research program based on further praxeological analyses of the tasks, methods, methodologies and theories characteristic of structuralist thinking in a teaching and learning environment.

Résumé Cet article rend compte d’une recherche menée dans la perspective théorique de la Théorie Anthroplogique du Didactique développée par Y. Chevallard et ses collaborateurs, et dans le contexte institutionnel de l’enseignement supérieur en France. Le sujet d’étude est l’enseignement et l’apprentissage des structures algébriques. L’article introduit la notion de praxéologie structuraliste en didactique de l’algébre abstraite. Le propos est illustré en détaillant les praxéologies structuralistes en arithmétique des anneaux abstraits, sur la base d’une étude historique et de manuels, d’une part, et à travers l’étude des interactions d’un groupe d’apprenants sur un forum de mathématiques d’autre part. Les enjeux didactiques de la notion de praxéologie structuraliste sont discutés en relation avec les difficultés identifiées dans l’enseignement et l’apprentissage de l’algébre abstraite, à la transition entre Licence et Master de mathématiques à l’université. Enfin, l’article trace les grandes lignes d’un programme de recherches fondé sur la poursuite des analyses des tâches, des méthodes, méthodologies et théories caractéristiques de la pensée structuraliste dans un contexte d’enseignement-apprentissage.

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

Fig. 1

Similar content being viewed by others

Notes

  1. A metaphor from biology in which bodies of knowledge are seen as living entities that develop inside institutions.

References

  • Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s practice: the case of limits of functions in spanish hish schools. Educational Studies in Mathematics, 59, 235–268.

    Article  Google Scholar 

  • Barquero, B., Bosch, M., & Gascón, J. (2013). The ecological dimension in the teaching of mathematical modeling at university. Recherches en Didactique des Mathématiques, 33(3), 307–338.

    Google Scholar 

  • Beaulieu, L. (1993). A parisian café and ten proto–bourbaki meetings 1934–35. The Mathematical Intelligencer, 51(1), 27–35.

    Google Scholar 

  • Benis-Sinaceur, H. (2010). Emmy Noether et l’école algébrique allemande dans le premier tiers du XXe siècle: mathématiques, style de pensée et philosophie. Lyon: Communication aux Journées Noether.

    Google Scholar 

  • Benis-Sinaceur, H. (2014). Facets and levels of mathematical abstraction. Philosophia Scientiæ, 18(1), 81–112.

    Article  Google Scholar 

  • Bosch, M., & Gascón, J. (2014) In Bikner-Ahsbahs, A., Prediger, S., & Bosch, M. (Eds.), Introduction to the anthropological theory of the didactic, (pp. 67–83). Berlin: Springer.

  • Bosch, M., & Winsløw, C. (2016). Linking problem solving and learning content: the challenges of self–sustained study and research processes. Recherches en Didactique des Mathématiques, 35(3), 357–401.

    Google Scholar 

  • Bourbaki, N. (1950). The architecture of mathematics. American Mathematical Monthly, 4(57), 221–232.

    Article  Google Scholar 

  • Brousseau, G. (1997). Theory of didactic situations in mathematics. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Chevallard, Y. (2006). Steps toward a new epistemology in mathematics education. In Bosch, M. (Ed.) Proceedings of the 4th Conference of the European Society for Research in Mathematics Education (CERME 4) (pp. 21–30). FUNDEMIIQS: Barcelona.

  • Chevallard, Y. (2015). Teaching Mathematics in Tomorrow’s Society: A Case for an Oncoming Counter Paradigm. In Cho, S (Ed.) The Proceedings of the 12th International Congress on Mathematical Education: Springer, Cham.

  • Colmez, P. (2011). Eléments d’analyze et d’algèbre (et de théorie des nombres). Palaiseau: Les éditions de l’Ecole Polytechnique.

    Google Scholar 

  • Corry, L. (1996). Modern algebra and the rise of mathematical structures. 2nd edition (2004). Birkhäuser: Basel.

    Google Scholar 

  • Cuoco, A., & Rotman, J. (2013). Learning modern algebra from early attempts to prove Fermat’s last theorem. Washington: MAA textbooks.

    Google Scholar 

  • Dorier, J. -L. (1995). Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics, 29, 175–197.

    Article  Google Scholar 

  • Douady, R. (1991). Tool, object, setting, window: elements for analysing and constructing didactical situations in mathematics. In Bishop, A., Mellin–Olsen, S., & van Dormolen, J. (Eds.) Mathematical Knowledge: Its Growth Through Teaching (pp. 109–132). Dordrecht: Springer.

  • Dubinsky, E. (1991). Reflective abstraction in mathematical thinking. In Tall, D. O. (Ed.) Advanced Mathematical Thinking (pp. 95–123). Dordrecht: Kluwer Academic Publisher.

  • Durand-Guerrier, V., Hausberger, T., & Spitalas, C. (2015). Définitions et exemples : prérequis pour l’apprentissage de l’algèbre moderne. Annales de Didactique et de Sciences Cognitives, 20, 101–148.

    Google Scholar 

  • Florensa, I., Bosch, M., & Gascón, J. (2015). The epistemological dimension in didactics: Two problematic issues. In Krainer, K., & Vondrová, N. (Eds.) Proceedings of the ninth congress of the european society for research in mathematics educationp (pp. 2635–2641). Prague: Faculty of Education, Charles University.

  • Francinou, S., & Gianella, H. (1994). Exercices de mathématiques pour l’agrégation Algèbre I. Paris: Masson.

    Google Scholar 

  • Gueudet, G. (2008). Investigating the secondary–tertiary transition. Educational Studies in Mathematics, 67(3), 237–254.

    Article  Google Scholar 

  • Guin, D. (2013). Algèbre II : Anneaux modules et algèbre multilinéaire. Paris: EDP Sciences.

    Google Scholar 

  • Hasse, H. (1930). Die moderne algebraische Methode. Jahresbericht der Deutschen Mathematiker-Vereinigung, 39, 22–34.

    Google Scholar 

  • Hausberger, T. (2012). Le challenge de la pensée structuraliste dans l’apprentissage de l’algèbre abstraite : une approche épistémologique. In Dorie, J. -L., & Coutat, S. (Eds.) Enseignement des mathématiques et contrat social, Enjeux et défis pour le 21e siècle, Actes du colloque EMF2012 (pp. 425–434). Université de Genève: Genève.

  • Hausberger, T. (2013). On the concept of (homo)morphism: a key notion in the learning of Abstract Algebra. In Ubuz, B., Haser, C., & Mariotti, M. A. (Eds.) Proceedings of the Eight Congress of the European Society for Research in Mathematics Education (pp. 2346–2355). Ankara: Middle East Technical University.

  • Hausberger, T. (2016a). Comment développer des praxéologies structuralistes en Algèbre Abstraite ? Recherches en Didactique des Mathématiques, 36(1), 97–142.

  • Hausberger, T. (2016b). Enseignement et apprentissage de l’algèbre abstraite à l’université et premiers éléments d’une didactique du structuralisme algébrique: études croisées en didactique et épistémologique des mathématiques. Note de synthèse pour l’habilitation à diriger des recherches. Montpellier: Université de Montpellier. Retrieved from https://hal.archives--ouvertes.fr/tel--01408565.

  • Hausberger, T. (2016c). Enseignement et apprentissage de l’algèbre abstraite à l’université : vers un paradigme du questionnement du monde. Castro-Urdiales: Communication at the 5th International Congress on ATD.

  • Hausberger, T. (2016d). A propos des praxéologies structuralistes en algèbre abstraite. In Nardi, E., Winsløw, C., & Hausberger, T. (Eds.) Proceedings of the 1st Congress of the International Network for Didactic Research in University Mathematics (pp. 296–305). Montpellier: University of Montpellier and INDRUM.

  • Hausberger, T. (2017a). The (homo)morphism concept: didactic transposition, metadiscourse and thematization. International Journal of Research in Undergraduate Mathematics Education, 3(3), 417–443.

  • Hausberger, T. (2017b). La dialectique objets-structures comme cadre de référence pour une étude didactique du structuralisme algébrique. Education et Didactique. To appear, 11(2).

  • Jovignot, J., Hausberger, T., & Durand-Guerrier, V. (2017). Praxeological analysis: the case of ideals in Ring Theory. Dublin: Paper presented at the CERME10 conference.

    Google Scholar 

  • Kondratieva, M., & Winsløw, C. (2017). A praxeological approach to Klein’s plan B: cross–cutting from Calculus to Fourier Analysis. Dublin: Paper presented at the CERME10 conference.

    Google Scholar 

  • Lakatos, I. (1976). Proofs and Refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Leron, U., & Dubinsky, E. (1995). An Abstract Algebra story. American Mathematical Monthly, 102(3), 227–242.

    Article  Google Scholar 

  • Mandelbrojt, S. (1952). Pourquoi je fais des mathématiques. Revue de métaphysique et de morale, 57(4), 422–429.

    Google Scholar 

  • Nardi, E. (2000). Mathematics undergrates’ responses to semantic abbreviations, geometric images and multi-level abstractions in group theory. Educational Studies in Mathematics, 34, 169–189.

    Article  Google Scholar 

  • Perrin, D. (1996). Cours d’algèbre. Paris: Editions Ellipses.

    Google Scholar 

  • Piaget, R., & Garcia, J. (1989). Psychogenesis and the history of science. New York: Columbia.

    Google Scholar 

  • Robert, A. (1987). De quelques spécificités de l’enseignement des mathématiques dans l’enseignement post-obligatoire. In Cahier de didactique des mathématiques 47, (p. 7). Paris: IREM de Paris.

    Google Scholar 

  • Sfard, A. (1991). On the dual nature of mathematical conceptions: reflection on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.

    Article  Google Scholar 

  • van der Waerden, B.L. (1930). Moderne Algebra 2 vols. Berlin: Springer.

    Book  Google Scholar 

  • Winsløw, C. (2008). Transformer la théorie en tâches : la transition du concret à l’abstrait en analyze réelle. In Rouchier, R., & et al. (Eds.) Actes de la XIIIème Ecole d’Eté de Didactique des Mathématiques (pp. 1–12 Cédérom). La Pensée Sauvage: Grenoble.

  • Winsløw, C., Barquero, B., De Vleeschouwer, M., & Hardy, N. (2014). Study and research courses as an epistemological model for didactics. Research in Mathematics Education, 16(2), 95–111.

    Article  Google Scholar 

  • Winsløw, C., Matheron, Y., & Mercier, A. (2013). Study and research courses as an epistemological model for didactics. Educational Studies in Mathematics, 83 (2), 267–284.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Montpelliérain Alexander Grothendieck, CNRS, Université de Montpellier, Montpellier, France

    Thomas Hausberger

Authors
  1. Thomas Hausberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Hausberger.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hausberger, T. Structuralist Praxeologies as a Research Program on the Teaching and Learning of Abstract Algebra. Int. J. Res. Undergrad. Math. Ed. 4, 74–93 (2018). https://doi.org/10.1007/s40753-017-0063-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40753-017-0063-4

Keywords

Search

Navigation