Abstract
Since its emergence in the early 80s with the study of didactic transposition processes, the Anthropological Theory of the Didactic maintains a privileged relationship with school algebra and its diffusion, both in school and outside school. I have chosen this case study to introduce the main “gestures of research” promoted by this framework and the methodological tools used to help researchers detach from the dominant viewpoints of the institutions where teaching and learning processes take place or which affect these processes in the distance. The construction of alternative reference models concerning school algebra and teaching and learning processes leads to some recent teaching experiences that break down the established didactic contracts, raising new research questions that need more in-depth analysis in the way opened by the “procognitive paradigm”.
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References
Anghileri, J. (2006). Teaching number sense (2nd ed.). London: Continuum International Publishing Group.
Artigue, M., Assude, T., Grugeon, B., & Lenfant, A. (2001). Teaching and learning algebra: approaching complexity through complementary perspectives. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra Proceedings of the 12th ICMI Study Conference (Vol. 1, pp. 21–32). Melbourne: The University of Melbourne.
Assude, T. (1993). Écologie de l’objet «racine carrée» et analyse du curriculum. Petit x, 35, 43–58.
Atiyah, M. (2001). Mathematics in the 20th century. American Mathematical Monthly, 108(7), 654–666.
Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s practice. The case of limits of functions in Spanish High Schools. Educational Studies in Mathematics, 59, 235–268.
Berger, P. L., & Luckmann, T. (1966). The social construction of reality: A treatise in the sociology of knowledge. Garden City, NY: Doubleday.
Blanc, J.-F., & Soler, S. (1933). L’algèbre à l’école primaire. Marseilles: Ferran.
Bolea, P. (2003). El proceso de algebrización de organizaciones matemáticas escolares. Monografía del Seminario Matemático García de Galdeano, 29. Zaragoza, Spain: Departamento de Matemáticas, Universidad de Zaragoza.
Bolea, P., Bosch, M., & Gascón, J. (2001a). La transposición didáctica de organizaciones matemáticas en proceso de algebrización. Recherches en Didactique des Mathématiques, 21(3), 247–304.
Bolea, P., Bosch, M., & Gascón, J. (2001b). ¿Cómo se construyen los problemas en Didáctica de las Matemáticas? Educación Matemática, 13(3), 22–63.
Bolea, P., Bosch, M., & Gascón, J. (2004). Why is modelling not included in the teaching of algebra at secondary school? Quaderni di Ricerca in Didattica, 14, 125–133.
Bosch, M. (1994). La dimensión ostensiva en la actividad matemática. El caso de la proporcionalidad (Doctoral dissertation). Universitat Autònoma de Barcelona, Spain.
Bosch, M. (2010). L’écologie des parcours d’étude et de recherche au secondaire. In G. Gueudet, G. Aldon, J. Douaire, & J. Traglova (Eds.), Apprendre, enseigner, se former en mathématiques: quels effets des ressources? Actes des Journées mathématiques de l’INRP. Lyon, France: Éditions de l’INRP.
Bosch, M., & Chevallard, Y. (1999). La sensibilité de l’activité mathématique aux ostensifs. Recherches en didactique des mathématiques, 19(1), 77–124.
Bosch, M., & Gascón, J. (2005). La praxéologie comme unité d’analyse des processus didactiques. In Mercier, A. & et Margolinas, C. (Coord.) Balises en Didactique des Mathématiques (pp. 107–122), Grenoble, France: La Pensée sauvage.
Bosch, M., Gascón, J., Ruiz Olarría, A., Artaud, M., Bronner, A., Chevallard, Y., Cirade, G., Ladage, C., & Larguier M. (2011). Un panorama de la TAD. An overview on ATD (CRM documents, Vol. 10). Bellaterra (Barcelona, Spain): Centre de Recerca Matemàtica.
Bourdieu, P., Chamboredon, J. C., & Passeron, J. C. (1968). Le métier de sociologue: Préalables épistémologiques. Paris: Mouton de Gruyter.
Bronner, A., Larguier, M., Artaud, M., Bosch, M., Chevallard, Y., Cirade, G., & Ladage, C. (2010). Diffuser les mathématiques (et les autres savoirs) comme outils de connaissance et d’action. Montpellier: IUFM.
Carraher, D. W., Schliemann, A. D., & Brizuela, B. M. (2000). Early algebra, early arithmetic: Treating operations as functions. The Twenty-second Annual Meeting of the North American Chapter of the International Group for th Psychology of Mathematics Education, Tucson, AZ.
Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Aritmetic an algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87–115.
Chevallard, Y. (1984). Le passage de l’arithmétique a l’algébrique dans l’enseignement des mathématiques au collège—Première partie. L’évolution de la transposition didactique, Petit x, 5, 51–94.
Chevallard, Y. (1985). La Transposition Didactique. Du savoir savant au savoir enseigné. Grenoble: La Pensée Sauvage (2nd edn., 1991).
Chevallard, Y. (1989a). Le passage de l’arithmétique à l’algébrique dans l’enseignement des mathématiques au collège—Deuxième partie: Perspectives curriculaires : la notion de modelisation. Petit x, 19, 45–75.
Chevallard, Y. (1989b). Le passage de l’arithmétique à l’algébrique dans l’enseignement des mathématiques au collège—Troisième partie: Perspectives curriculaires: voies d’attaque et problèmes didactiques. Petit x, 25, 5–38.
Chevallard, Y. (1989c). On didactic transposition theory: some introductory notes. In Proceedings of the International Symposium on Selected Domains of Research and Development in Mathematics Education (Bratislava, 3–7 August 1988), pp. 51–62.
Chevallard, Y. (1990). Le passage de l’arithmétique a l’algébrique dans l’enseignement des mathématiques au collège—Troisième partie. Perspectives curriculaires : vois d’attaque et problèmes didactiques, Petit x, 23, 5–38.
Chevallard, Y. (1994). Enseignement de l’algèbre et transposition didactique. Rendiconti del seminario matematico Università e Politecnico Torino, 52(2), 175–234.
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221–266.
Chevallard, Y. (2002). Organiser l’étude. 3. Écologie & regulation. In J.-L. Dorier, Artaud, M., Artigue, M., Berthelot, R. & Floris, R. (Eds), Actes de la 11 e École d’Été de didactique des mathématiques (pp. 41–56). Grenoble: La Pensée sauvage.
Chevallard, Y. (2005). Didactique et formation des enseignants. In B. David (Ed.), Impulsions 4 (pp. 215–231). Lyon: INRP.
Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In Bosch, M. (Ed.), Proceedings of the IV Congress of the European Society for Research in Mathematics Education (pp. 21–30). Barcelona: FUNDEMI-IQS.
Chevallard, Y. (2007). Passé et present de la Théorie Anthropologique de Didactique. In A. Estepa, L. Ruiz, & F. J. García (Eds.), Sociedad, escuela y matemáticas. Aportaciones de la Teoría Antropológica de lo Didáctico (TAD) (pp. 705–746). Jaén, Spain: Publicaciones de la Universidad de Jaén.
Chevallard, Y. (2010). La notion d’ingénierie didactique, un concept à refonder. Questionnement et éléments de réponse à partir de la TAD. In C. Margolinas, M. Abboud-Blanchard, L. Bueno-Ravel, N. Douek, A. Fluckiger, P. Gibel, F. Vanderbrouck, & F. Wozniak (Coord.), En amont et en aval des ingénieries didactiques (pp. 705–746). Grenoble: La Pensée sauvage.
Chevallard, Y., & Bosch, M. (2012). L’algèbre entre effacement et réaffirmation. Aspects critiques de l’offre scolaire d’algèbre. In L. Coulange, J.-P. Drouhard, J.-L. Dorier, & A. Robert (coord.) Enseignement de l’algèbre élémentaire. Bilan et perspectives. Recherches en Didactique des Mathématiques, Special Issue (pp. 13–33).
Cid, E., & Bolea, P. (2010). Diseño de un modelo epistemológico de referencia para introducir los números negativos en un entorno algebraico. In A. Bronner, M. Larguier, M. Artaud, M. Bosch, Y. Chevallard, G. Cirade, & C. Ladage (Eds.), Diffuser les mathématiques (et les autres savoirs) comme outils de connaissance et d’action (pp. 575–594). Montpellier: Université de Montpellier.
Cid, E., & Ruiz Munzón, N. (2011). Actividades de studio e investigación para introducer los numeros negativos en un entorno algebraico. In M. Bosch, J. Gascón, A. Ruiz Olarría, M. Artaud, A. Bronner, Y. Chevallard, G. Cirade, C. Ladage, & M. Larguier (Eds.), Un panorama de la TAD. An overview on ATD (CRM Documents, Vol. 10, pp. 579–604). Bellaterra (Barcelona, Spain): Centre de Recerca Matemàtica.
Coulange, L. (2001a). Enseigner les systèmes d’équations en Troisième. Une étude économique et écologique. Recherches en Didactique des Mathématiques, 21(3), 305–353.
Coulange, L. (2001b). Evolutions du passage arithmétique-algèbre dans les manuels et les programmes du 20ème siècle. Contraintes et espaces de liberté pour le professeur. Petit x., 57, 61–78.
Derrida, J. (1967). De la grammatologie. Paris: Les Editions de Minuit.
Douglas, M. (1986). How institutions think. Syracuse, NY: Syracuse University Press.
Elias, N. (1978). What is sociology? London: Hutchinson.
Elias, N. (1987). Involvement and detachment. Contributions to the sociology of knowledge. Oxford: Blackwell.
Estepa, A., García, F. J., & Ruiz-Higueras, L. (2007). Sociedad, escuela y matemáticas. Aportaciones de la Teoría Antropológica de lo Didáctico (TAD). Jaén, Spain: Publicaciones de la Universidad de Jaén.
García, F. J., Gascón, J., Ruiz Higueras, L., & Bosch, M. (2006). Mathematical modelling as a tool for the connection of school mathematics. ZDM International Journal on Mathematics Education, 38(3), 226–246.
Gascón, J. (1993). Desarrollo del conocimiento matemático y análisis didáctico: Del patrón análisis-síntesis a la génesis del lenguaje algebraico. Recherches en didactique des mathématiques, 13(3), 295–332.
Gascón, J. (1999). La naturaleza prealgebraica de la matemática escolar. Educación Matemática, 11(1), 77–88.
Gascón, J. (2011). Las tres dimensiones fundamentales de un problema didáctico. El caso del álgebra elemental. Revista Latinoamericana de Investigación en Matemática Educativa, RELIME, 14(2), 203–231.
Grugeon, B. (1995). Étude des rapports institutionnels et des rapports personnels des élèves à l’algèbre élémentaire dans la transition entre deux cycles d’enseignement: BEP et Première G (Doctoral dissertation). Université de Paris 7, Paris.
Havelock, E. A. (1963). Preface to plato. Cambridge, MA: Harvard University Press.
Kaput, J. J. (1983). Errors in translations to algebraic equations: roots and implications. Focus on Learning Problems in Mathematics, 5, 63–78.
Kaput, J. J. (1998). Transforming Algebra from and Engine of Inequity to an Engine of Mathematical Power by “Algebrafying” the K-12 Curriculum. In The Nature and Role of Algebra in the K-14 Curriculum (pp. 25–26). Washington, DC: National Council of Teachers of Mathematics and the Mathematical Sciences Education Board, National Research Council.
Kaput, J. J. (2000). Transforming algebra from an engine of inequality to an engine of mathematical power by “algebrafying” the K-12 curriculum. National Center for Improving Student Learning and Achievement in Mathematics and Science, Dartmouth, MA.
Malara, N. A. (2003). Dialectics between theory and practice: theoretical issues and aspects of practice from an Early Algebra Project. In N. Pateman, G. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education North America (Vol. 1, pp. 33–48).
OECD. (2009). Learning mathematics for life: A perspective from PISA. programme for international student assessment. http://www.oecd.org/dataoecd/53/32/44203966.pdf
Ong, W. J. (1982). Orality and literacy. The technologizing of the world. New York: Routledge.
Pujnachov, I. V., & Popov, I. P. (2008). Matemáticas sin formulas: Libro 1 y 2. Moscow: URSS.
Ruiz-Munzón, N. (2010). La introducción del álgebra elemental y su desarrollo hacia la modelización functional (Doctoral dissertation). Spain: Universitat Autònoma de Barcelona.
Ruiz-Munzón, N., Bosch, M., & Gascón, J. (2007). The functional algebraic modelling at secondary level. In D. Pitta-Panzati & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 2170–2179). Nicosia: University of Cyprus.
Ruiz-Munzón, N., Bosch, M., & Gascón, J. (2011). Un modelo epistemológico de referencia del álgebra como instrumento de modelización. In M. Bosch, J. Gascón, A. Ruiz Olarría, M. Artaud, A. Bronner, Y. Chevallard, G. Cirade, C. Ladage, & M. Larguier (Eds.), Un panorama de la TAD. An overview on ATD (CRM Documents, Vol. 10, pp. 743–765). Bellaterra (Barcelona, Spain): Centre de Recerca Matemàtica.
Ruiz-Munzón, N., Matheron, Y., Bosch, M., & Gascón, J. (2012). Autour de l’algèbre : les entiers relatifs et la modélisation algébrico-fonctionnelle. In L. Coulange, J.-P. Drouhard, J.-L. Dorier, & A. Robert (coord.) Enseignement de l’algèbre élémentaire. Bilan et perspectives. Recherches en Didactique des Mathématiques, special issue (pp. 81–101).
Schoenfeld, A. H. (2011). Reflections of an accidental theorist. In M. Pitici (Ed.), The best writing on mathematics (pp. 219–235). Princeton, NJ: Princeton University Press.
Sierra, T., Bosch, M., & Gascón, J. (2011). La formación matemático-didáctica del maestro de Educación Infantil: el caso de «cómo enseñar a contar». Revista de Educación, 357, 231–256.
Subramaniam, K., & Banerjee, R. (2004). Teaching arithmetic and algebraic expressions. In M. Johnsen & A. Berit (Eds.), Proceedings of the 28th International Group for the Psychology of Mathematics Education (Vol. 3, pp. 121–128). Bergen: Bergen University College.
Warren, E. (2004). Generalizing arithmetic: supporting the process in the early years. In M. Johnsen, & A. Berit (Eds.), Proceedings of the 28th International Group for the Psychology of Mathematics Education (Vol. 4, pp. 417–424). Bergen: Bergen University College.
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Bosch, M. (2015). Doing Research Within the Anthropological Theory of the Didactic: The Case of School Algebra. In: Cho, S. (eds) Selected Regular Lectures from the 12th International Congress on Mathematical Education. Springer, Cham. https://doi.org/10.1007/978-3-319-17187-6_4
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