From global to local aspects of Klein’s second discontinuity

• Carl Winslow ^[1] ; Rongrong Huo ^[1]
  1. [1] University of Copenhagen

     University of Copenhagen

     Dinamarca

     
• Localización: Recherches en Didactique des Mathématiques, ISSN 0246-9367, Vol. 45, Nº 1, 2025, págs. 69-94
• Idioma: inglés
• Títulos paralelos:
  • De aspectos globales a locales de la segunda discontinuidad de Klein
  • La double discontinuité de Klein: de perspectives globales à perspectives locales
• Enlaces
  • Texto completo
• Resumen
  • español

    La cuestión global de cómo identificar, desarrollar y evaluar el conocimiento matemático que es relevante para futuros profesores de secundaria ha sido central en la emergencia de la investigación en educación matemática desde sus inicios. Revisamos partes de esta historia desde el punto de vista de la teoría antropológica de lo didáctico, y en particular la noción de relaciones con las praxeologías matemáticas que son sostenidas por ciertas posiciones dentro de las instituciones escolares y universitarias. También consideramos un caso moderno, donde las preguntas surgen de manera muy práctica: ¿cómo cerrar la brecha entre los cursos de matemáticas de pregrado estándar y un modelo de números reales y funciones relevante para la escuela? Mostramos cómo tanto los aspectos teóricos como prácticos de esta cuestión más local surgen en un curso de culminación para estudiantes con aproximadamente dos años de experiencia en matemáticas de pregrado.

  • English

    The global question of how to identify, develop and assess mathematical knowledge that is relevant to future secondary school teachers, has been central in the emergence of mathematics education research from early on. We review parts of this history from the viewpoint of the anthropological theory of the didactic, and in particular the notion of relationships to mathematical praxeologies that are held by certain positions within school and university institutions. We also consider a modern case, where the questions arise in a very practical sense: how to bridge the gap between standard undergraduate mathematics courses and a school relevant model of real numbers and functions? We show how both theoretical and practical aspects of this more local question arise in a so-called capstone course for students with about two years of undergraduate mathematics experience.

  • français

    La question globale d’identifier, développer et évaluer les connaissances mathématiques qui sont pertinentes pour les futurs enseignants du secondaire, a été depuis les débuts un levier central dans l’émergence de recherches en didactique des mathématiques. Nous exposons des éléments historiques de cette question du point de vue de la théorie anthropologique du didactique, et en particulier la notion de rapport aux praxéologies mathématiques entretenu par certaines positions au sein des institutions scolaires et universitaires. Nous examinons aussi un cas moderne où ces questions apparaissent d’une manière plus pratique : comment combler le fossé entre une licence générale en mathématiques et des conceptions des nombres réels et des fonctions d’une variable réelle qui sera pertinente pour l’enseignement secondaire ? Nous montrons comment les aspects théoriques et pratiques de cette question plus locale apparaissent dans un cours de synthèse pour des futurs enseignants, qui ont passé deux ans de cours mathématiques universitaires.

• Referencias bibliográficas
  • BEGLE, E. G. (1972) Teacher knowledge and student achievement in algebra. SMSG Reports, No. 9. Stanford: School Mathematics Study Group. Online...
  • CHEVALLARD, Y. (1992). Fundamental concepts in didactics : perspectives provided by an anthropological approach. In R. Douady & A. Mercier...
  • CHEVALLARD, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologie du didactique. Recherches en Didactique des Mathématiques...
  • BOSCH, M., HAUSBERGER, T., HOCHMUTH, R., KONDRATIEVA, M. & WINSLØW, C. (2021). External Didactic Transposition in Undergraduate Mathematics....
  • EISENBERG, T. (1977). Begle Revisited: Teacher Knowledge and Student Achievement in Algebra. Journal for Research in Mathematics Education...
  • GONZÁLEZ-MARTÍN, A., GIRALDO, V. & SOUTO A. (2013). The introduction of real numbers in secondary education: an institutional analysis...
  • GOLDBERG, M. (2006). Computing logarithms digit-by-digit. International Journal of Mathematics Education in Science and Technology 37(1),...
  • GRØNBÆK, N. & WINSLØW, C. (2007). Thematic projects: a format to further and assess advanced student work in undergraduate mathematics. Recherches...
  • HILL, H. C., ROWAN, B., & BALL, D. L. (2005). Effects of Teachers’ Mathematical Knowledge for Teaching on Student Achievement. American...
  • KILPATRICK, J. (2019). A double discontinuity and a triple approach: Felix Klein’s perspective on mathematics teacher education. In H. G. Weigand,...
  • KLEIN, F. (2016). Elementary mathematics from a higher standpoint (G. SCHUBRING, Trans.). Berlin: Springer. (Original work published 1908)
  • KLINE, M. (1973). Why Johnny Can't Add: The Failure of the New Math. New York: St. Martin's Press.
  • KRAINER, K., HSIEH, F.-J., PECK, R. & TATTO, M. (2015). The TEDS-M: important issues, results and questions. In S.J. Cho (ed.), The Proceedings...
  • MONK, D. H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review...
  • OECD (2014). Education at a Glance 2014: OECD Indicators. OECD Publishing, Paris.
  • SCHMIDT, W., BURROUGHS, N. & COGAN, L. (2013). World Class Standards for Preparing Teachers of Mathematics. Working paper, Michigan State...
  • SULTAN, A. & ARTZT, A. (2018). The mathematics that every secondary school math teacher needs to know (2nd edition). New York: Routledge.
  • TATTO M. (ed., 2014). The Teacher Education and Development Study in Mathematics (TEDS-M). Policy, Practice, and Readiness to Teach Primary...
  • WEBER, C. (2016). Making logarithms accessible – operational and structural basic models for logaritms. Journal für Mathematikdidaktik.
  • WEIGAND, H.-G., MCCALLUM, W., MENGHINI, M., NEUBRAND, M. AND SCHUBRING, G. (2019). The legacy of Felix Klein. Springer Nature.
  • WINSLØW, C. (2013). The transition from university to high school and the case of exponential functions. In: B. Ubuz, Ç. Haser, M. Mariotti (Eds.),...
  • WINSLØW, C. & GRØNBÆK, N. (2014). Klein’s double discontinuity revisited: contemporary challenges for universities preparing teachers...
  • WINSLØW, C. & HUO, R. (2023). Task design for Klein’s second discontinuity. In M. Trigueros, B. Barquero, R. Hochmuth & J. Peters (Eds.),...