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Un obstacle épistémologique soulevé par des « découpages infinis » des surfaces et des solides

Maggy Schneider
Maggy Schneider
  • Retour au sommaire : RDM Vol. 11/2.3
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Schneider, M. (1991). Un obstacle épistémologique soulevé par des « découpages infinis » des surfaces et des solides. Recherches En Didactique Des Mathématiques, 11(2.3), 241–294. https://revue-rdm.com/1991/un-obstacle-epistemologique/ Cite

Résumé

Plusieurs erreurs d’élèves de 15 à 18 ans dans des calculs d’aires et de volumes révèlent des conceptions erronées telles que : à une réunion d’indivisibles disjoints correspond toujours une somme de mesures.

Une même interprétation: l’obstacle de l’hétérogénéité des dimensions rend compte de ces conceptions mêlant des grandeurs de dimensions distinctes (des solides avec des surfaces ou des surfaces avec des lignes). En gros, cet obstacle consiste en glissements inconscients et indus entre le domaine des grandeurs et celui de leurs mesures.

Plusieurs caractéristiques de cet obstacle attestent de son origine épistémologique. Une étude critique de celles-ci contribue à préciser le concept d’obstacle épistémologique.

Abstract

In keeping with the logic of euclidian geometry traditional teaching presented the homothety or similarity in a « intarfigural » way based on the properties of similar figures. Afterwards geometry with figures disappeared from the programs at the beginning of secondary school, and was replaced by geometry by transformations. Finally subsequent simplifications of the programs propose only the study of transformations, without going as far as to restore geometry with figures. The validity of such selections is not so clear. An epistemological analysis shows in fact that its development, from ancient Greece to the beginning of the 20th century, is marked by stages which should be taken into account in the teaching of homothety.

Our first hypothesis is the necessity for the acquisition of the idea of homothety of an initial experience of the different types of figures which can occur. This requires figurative analysis which without being of a strictly mathematical nature has mathematical purposes.

Our second hypothesis is that for the exploration of relations that exist between the figurative and numerical aspects, we must first separate and then articulate these two aspects in the study work of students. In this way we can organize teaching which leads to much better results, and which takes hardly more time than that commonly accepted for the teaching of this subject.

Resumen

Varios errores cometidos por alumnos de 15 a 18 años en cálculos de áreas y volúmenes muestran concepciones erróneas así como: una reunión de « indivisibles » se traduce siempre por una suma de medidas.

Una sola interpretación: el obstáculo de heterogeneidad de las dimensiones explica estas concepciones mezelando cantidades de dimensiones distinctas (volúmenes con áreas y áreas con líneas). En claro, consiste en derivaciones inconcientes e indebidas, en la manera de pensar de los alumnos, entre el campo de las cantidades y el campo de las medidas.

Varias características de este obstáculo indican su origen epistemológico. Un estudio crítico de estas características contribuye a precisar el concepto de obstáculo epistemológico.


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