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Can students justify the correctness of an arithmetic algorithm?: A case-study at the primary-secondary transition

  • Autores: Renaud Chorlay
  • Localización: Recherches en didactique des mathématiques, ISSN 0246-9367, Vol. 41, Nº 2, 2021, págs. 177-216
  • Idioma: inglés
  • Títulos paralelos:
    • ¿Pueden los alumnos de tercer ciclo justificar la correción de una técnica operatoria?
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  • Resumen
    • español

      El objetivo de este estudio es determinar en qué medida las técnicas operativas proporcionan situaciones propicias para la argumentación en las clases ordinarias. Pedimos a los alumnos del 3º ciclo (de 4º a 6º curso de primaria) que justificaran la corrección de una técnica que no conocían para dividir enteros entre dos, pero cuya justificación se basa en conocimientos potencialmente disponibles en este nivel. Partiendo de los resultados consolidados en la investigación sobre los números enteros, pretendemos mostrar que la entrada de los alumnos en la argumentación combinando la herramienta diseño de N. Balacheff y las aportaciones de la historia de las matemáticas sobre la expresión y justificación de los algoritmos. Los resultados muestran que (1) los alumnos del 3º ciclo pueden considerar las técnicas operativas como objetos de estudio y no sólo como herramientas de uso, y que (2) las diferentes concepciones válidas de los números enteros ofrecen diferentes recursos argumentativos y conducen a argumentos de desigual fuerza epistémica.

    • English

      This study aims to explore to what extent arithmetic algorithms provide opportunities for argumentation in ordinary classrooms. We designed an experiment in which French students in grades 5 and 6 were asked to describe and justify an algorithm for halving whole numbers which they had not studied earlier, but whose correctness rests on mathematical facts with which they ought to be familiar. Drawing on established results on the teaching and learning of whole number arithmetic, we account for students’ engagement in justification by combining a didactical theory aimed for the analysis of explanation and proof – namely Balacheff’s notion of conception – and inputs from the history of mathematics on algorithms, their expression and their justification. The results show that (1) young students can regard arithmetic algorithms as objects to study and not only as tools to use, and that (2) different valid conceptions of whole numbers provide unequal opportunities for argumentation and lead to arguments of unequal epistemic strengths.

    • français

      Cette étude vise à déterminer dans quelle mesure les techniques opératoires fournissent des situations propices à l’argumentation dans les classes ordinaires. Nous avons demandé à des élèves de cycle 3 de justifier la correction d’une technique de division des entiers par deux qu’ils ne connaissaient pas, mais dont la justification repose sur des connaissances potentiellement disponibles à ce niveau. Faisant fond sur des résultats de recherche stabilisés concernant les nombres entiers, nous visons à rendre compte de l’entrée éventuelle des élèves dans l’argumentation en combinant l’outil conception de Balacheff et des apports de l’histoire des mathématiques sur l’expression et la justification des algorithmes. Les résultats montrent que (1) les élèves de cycle 3 peuvent considérer les techniques opératoires comme des objets à étudier et pas seulement comme des outils à utiliser, et que (2) différentes conceptions valides des entiers offrent des ressources argumentatives différentes et conduisent à des arguments de force épistémique inégales.

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