Abstract
What is questioned and what is taken for granted when carrying out research into the teaching of a given mathematical topic such as Group Theory? This paper presents two different questioning procedures using the methodological tools provided by the Anthropological Theory of the Didactic (ATD). The first one, leading to an undergraduate instructional proposal called “workshop of practices”, is based on the praxeological analysis of the official content to be taught, assuming the prevailing pedagogical paradigm in tertiary education, which the ATD characterizes as “visiting works”. This also seems to be the procedure many investigations adopted with regard to the teaching of Group Theory, centered on the construction or reinvention of concepts. The transition towards the inquiry-based paradigm of “questioning the world” requires another questioning strategy, to put the raisons d’être and uses of the mathematical works to be taught at the core of the analysis.
Resume
Qu’est-ce que l’on questionne et qu’est-ce que l’on assume dans la recherche sur l’enseignement d’un thème mathématique donné comme la théorie des groupes? Cet article présente deux procédures de questionnement différentes à partir des outils méthodologiques de la théorie anthropologique du didactique (TAD). La première, qui a conduit à une proposition d’enseignement pour premier cycle universitaire appelé « atelier de pratiques », est. fondée sur une analyse praxéologique des contenus à enseigner, en assumant le paradigme pédagogique dominant dans l’enseignement universitaire que la TAD caractérise comme la « visite des œuvres». Celle-ci semble être aussi l’approche adoptée par nombre de recherches sur l’enseignement de la théorie des groupes, centrées sur la construction ou réinvention de concepts. La transition vers le paradigme du « questionnement du monde » requiert une autre stratégie de questionnement pour situer au centre de l’analyse les raisons d’être des œuvres mathématiques à enseigner. Mots-clé: théorie anthropologique du didactique, théorie des groupes, praxéologies, enquête, questionnement épistémologique
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alcock, L., Brown, G., & Dunning, C. (2015). Independent study workbooks for proofs in group theory. International Journal of Research in Undergraduate Mathematics Education, 1(1), 3–26.
Asher, M. (2002). Mathematics elsewhere: An exploration of ideas across cultures. Princeton, NJ: Princeton University Press.
Asiala, M., Dubinksy, E., Mathews, D. M., Morics, S., & Oktaç, A. (1997). Development of students’ understanding of cosets, normality, and quotient groups. The Journal of Mathematical Behavior, 16(3), 241–309.
Bosch, M., & Gascón, J. (1993). Prácticas en Matemáticas: el trabajo de la técnica. In T. Rojano & L. Puig (Eds.), Memorias del Tercer Simposio Internacional sobre Investigación en Educación Matemática (pp. 141–152). México, DF: Sección Matemática Educativa del CINVESTAV.
Bosch, M., & Gascón, J. (1994). La integración del momento de la técnica en el proceso de estudio de campos de problemas de matemáticas. Enseñanza de las Ciencias, 12(3), 314–332.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer.
Burn, B. (1996). What are the fundamental concepts of group theory? Educational Studies in Mathematics, 31, 371–378.
Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In M. Bosch (Ed.), Proceedings of the 4th Conference of the European Society for Research in Mathematics Education (CERME 4) (pp. 21–30). Barcelona: FUNDEMI-IQS.
Chevallard, Y. (2015). Teaching mathematics in Tomorrow’s society: A case for an oncoming counter paradigm. In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education (pp. 173–187). Dordrecht: Springer.
Dorier, J.-L. (1995). Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics, 29(2), 175–197.
Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267–305.
Freudenthal, H. (1973). What groups mean in mathematics and what they should mean in mathematical education. In A. G. Howson (Ed.), Developments in mathematical education, (pp. 101–114). Proceeding of the 2nd International Congress on Mathematics Education. Cambridge: Cambridge University Press.
Gascón, J. (2004). Efectos del “autismo temático” sobre el estudio de la Geometría en Secundaria. Parte II. La clasificación de los cuadriláteros convexos. Suma. Revista sobre la enseñanza y el aprendizaje de las matemáticas, 45, 41–52.
Hausberger, T. (2013). On the concept of (homo)morphism: A key notion in the learning of abstract algebra. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of CERME8 (pp. 2346–2355). Ankara: Middle East Technical University.
Klein, F. (2004). Elementary mathematics from an advanced standpoint: Geometry. New York: Dover.
Lakatos, I. (1976). A renaissance of empiricism in the recent philosophy of mathematics. The British Journal for the Philosophy of Science, 27(3), 201–223.
Larsen, S. P. (2013). A local instructional theory for the guided reinvention of the group and isomorphism concepts. The Journal of Mathematical Behavior, 32(4), 712–725.
Nardi, E. (2000). Mathematics undergraduates’ responses to semantic abbreviations, ‘geometric’ images and multi-level abstractions in group theory. Educational Studies in Mathematics, 43, 169–189.
Pólya, G., & Reade, R. C. (1987). Combinatorial enumeration of groups, graphs, and chemical compounds. New York: Springer-Verlag.
Robert, A. (1998). Outils d’analyse des contenus mathématiques à enseigner au lycée et à l’université. Recherches en Didactique des Mathématique, 18(2), 139–190.
Robert, A., & Robinet, J. (1996). Prise en compte du méta en didactique des mathématiques. Recherches en Didactique des Mathématique, 16(2), 145–176.
Vergnaud, G. (1990). La théorie des champs conceptuels. Recherches en Didactique des Mathématique, 10(2.3), 133–170.
Weber, K., & Larsen, S. (2008). Teaching and learning group theory. In M. Carlson & C. Rasmussen (Eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics Education (pp. 139–152). Washington, DC: Mathematical Association of America.
Weyl, H. (1952). Symmetry. Princeton, NJ: Princeton University Press.
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.
Acknowledgements
Funded by the Spanish R&D project EDU2015-69865-C3-1-R (MINECO/FEDER, UE) and by the Fundación Séneca (Región de Murcia) 19880/GERM/15.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Rights and permissions
About this article
Cite this article
Bosch, M., Gascón, J. & Nicolás, P. Questioning Mathematical Knowledge in Different Didactic Paradigms: the Case of Group Theory. Int. J. Res. Undergrad. Math. Ed. 4, 23–37 (2018). https://doi.org/10.1007/s40753-018-0072-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40753-018-0072-y