Adding diversity to mathematical connections to counter Klein’s second discontinuity

• Nicholas H. Wasserman ^[1]
  1. [1] Columbia University

     Columbia University

     Estados Unidos

     
• Localización: Recherches en Didactique des Mathématiques, ISSN 0246-9367, Vol. 45, Nº 1, 2025, págs. 5-34
• Idioma: inglés
• Títulos paralelos:
  • Agregar diversidad a las conexiones matemáticas para contrarrestar la segunda discontinuidad de Klein
  • Ajouter de la diversité aux connexions mathématiques pour contrer la seconde discontinuité de Klein
• Enlaces
  • Texto completo
• Resumen
  • español

    Para los profesores que intentan que los cursos de matemáticas universitarias sean relevantes para los futuros profesores de secundaria, hacerlo generalmente implica establecer conexiones entre el contenido de matemáticas universitarias y el contenido de matemáticas escolares–en un intento de contrarrestar lo que Felix Klein denominó una “doble discontinuidad.” En este artículo, considero la naturaleza de las conexiones matemáticas que unen estos dos dominios y las distinciones comunes que se hacen en la literatura existente entre ellos, como la direccionalidad. A través de este análisis, señalo otro aspecto de estas conexiones que se ha dejado implícito: las matemáticas universitarias se enmarcan principalmente (y razonablemente) como un superconjunto del contenido de matemáticas escolares. En este artículo considero alternativas, en particular la espiritualización de conexiones que invierten esta conexión relacional típica–es decir, una conexión relacional de subconjunto–y ejemplifico estas conexiones con conceptos de cursos universitarios como análisis real y álgebra abstracta. Luego, considero la lógica para hacerlo en términos de la formación de profesores de secundaria y las formas en que diversificar nuestro marco de conexiones de esta manera puede usarse para ayudar a contrarrestar la segunda discontinuidad de Klein.

  • English

    For instructors that try to make university mathematics courses relevant to future secondary school teachers, doing so generally involves making connections between university mathematics content and school mathematics content–in attempts to counter what Felix Klein referred to as a “double discontinuity.” In this paper, I consider the nature of the mathematical connections that bridge these two domains, and common distinctions made in extant literature between them, such as directionality. Through this analysis, I point out another aspect of these connections that has been left implicit: university mathematics is primarily–and reasonably–framed as a superset of school mathematics content. In this paper I consider alternatives, in particular conceptualizing connections that invert this typical relational connection–i.e., a subset relational connection–and I exemplify these connections with concepts from university courses such as real analysis and abstract algebra. Then, I consider the rationale for doing so in terms of secondary teacher education, and the ways that diversifying our framework of connections in this way can be used to help counter Klein’s second discontinuity.

  • français

    Pour les enseignants qui tentent de rendre les cours de mathématiques universitaires pertinents pour les futurs enseignants du secondaire, cela implique généralement d’établir des liens entre le contenu des mathématiques universitaires et celui des mathématiques scolaires, dans une ten-tative de contrer ce que Felix Klein appelle une « double discontinuité ». Dans cet article, j’exa-mine la nature des connexions mathématiques qui relient ces deux domaines et les distinctions courantes faites entre eux dans la littérature existante, telle que la « directionality ». À travers cette analyse, je pointe un autre aspect de ces connexions mathématiques qui est resté impli-cite : les mathématiques universitaires sont principalement – et raisonnablement – présentées comme un sur-ensemble du contenu des mathématiques scolaires. Dans cet article, j’envisage des alternatives, en particulier la conceptualisation de connexions mathématiques qui inversent cette connexion relationnelle typique – c’est-à-dire une connexion relationnelle de sous-ensemble – et j’illustre ces connexions avec des concepts issus de cours universitaires tels que l’analyse réelle et l’algèbre abstraite. J’examine ensuite la justification de cette démarche en termes de formation des enseignants du secondaire et les façons dont la diversification de notre cadre de connexions mathématiques peut ainsi être utilisée pour aider à contrer la seconde discontinuité de Klein.

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