We state and justify a conjecture concerning the comparative complexity of the representation of real numbers on the line versus other representations. The Introduction provides the setting of the subject and states the conjecture. Part 1 deals with the controversial line's «nature» and depicts several interpretations of the straight line. Part 2 analyses the particular phenomenology related to the assignment of real numbers to the line's points, in order to extract some limitations of such an assignment. Part 3 compares several representations of real numbers with the representation on the line, and presents a set of distinctive features for the latter. This paper, theoretically oriented, comes from a broader research work addressed to uncover epistemological obstacles related to the representation of real numbers on the straight line. We are doing a work field, involving students of several levels, on this subject.
RESUMEN• Esta investigación describe conexiones o nexos entre algunas «celdas» de la estructura cognitiva (Vinner, 1991) y los fenómenos, en el sentido de Freudenthal (1983), organizados por una definición de límite finito de una sucesión, así como los fenómenos organizados por una sucesión de Cauchy (Claros, 2010). Estas conexiones surgieron cuando afrontamos el objetivo de proponer secuencias didácticas destinadas a abordar en el aula el límite finito de una sucesión y las sucesiones de Cauchy. Estas secuencias didácticas apelan al uso de los fenómenos de aproximación intuitiva y retroalimentación descritos por Claros (2010) y los sitúan como elementos indispensables en la construcción de un concepto imagen de una sucesión convergente o de una imagen de la demostración (Kidron y Dreyfuss, 2014). Estas dos «celdas» de la estructura cognitiva ayudan en la elaboración del concepto definición de una sucesión con límite.PALABRAS CLAVE: límite; sucesión de Cauchy; imagen de la demostración; concepto definición; fenómenos.ABSTRACT • This research describes how two «cells» of the cognitive structure (Vinner, 1991) connect to the phenomena, in Freudenthal's (1983) sense, organized by a definition of finite limit of a sequence and the phenomena organized by a Cauchy sequence (Claros, 2010). These connections arose when we faced, as a target, to the introduction of didactic sequences or lesson plans to teach the finite limit of a sequence and Cauchy sequences. These didactic sequences appeal to the use of intuitive approach and feedback phenomena described by Claros (2010) and focuses them as essential elements in building a concept image of a convergent sequence or an image proof of it (Kidron and Dreyfuss, 2014). These two «cells» of the cognitive structure help in developing the concept definition of a sequence with limit.
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