Exploración del uso de los lenguajes natural y simbólico en la enseñanza de Matemática superior

ABSTRACT Introducción Los estudiantes, mayoritariamente, consideran que los conceptos y propiedades trabajadas en un curso son parte de una verdad matemática que deben aprender por medio del docente, a quien conciben como el poseedor de esta verdad y el encargado de transmitírsela utilizando lenguaje natural y simbólico. El primero generalmente se usa en el medio oral, siendo pocas las veces que el docente lo utiliza por escrito en el pizarrón, opuesto a lo que ocurre con el lenguaje simbólico. En un proyecto de investigación en curso 2 nos propusimos explorar el uso -del docente-de los lenguajes natural y simbólico en sus medios oral y escrito al enseñar un recorte de la verdad matemática. El recorte incluye conceptos y propiedades de límite, continuidad y derivabilidad pues demandan esfuerzo para su comprensión y conllevan notación simbólica compleja. Para recolectar datos a partir de los cuales se pudiera obtener información de los lenguajes natural y simbólico en ambos medios (oral y escrito en el pizarrón), el instrumento elegido fue la observación no participante de clases de Análisis Matemático del primer año del nivel superior. Con el fin de captar la simultaneidad entre lo oral y escrito, se filmaron con audio dichas clases. Presentamos en este trabajo los primeros avances realizados en esta línea, limitando nuestros ejemplos –por cuestión de espacio-a conceptos vinculados con la noción de límite, previos a su enseñanza, que extrajimos de una de las clases observadas. Marco teórico El marco teórico utilizado toma centralmente a) elementos de las Teorías de la Verdad en Matemática y en la clase de Matemática ([7], [1], [12]); b) lenguajes y registros, y c) 1 Parte de este trabajo cuenta con financiamiento de la Universidad Abierta Interamericana. 2 Proyecto de tesis doctoral en curso "Un estudio sobre el uso del lenguaje natural y simbólico en la enseñanza y el aprendizaje de Matemática superior". Cristina Camós es alumna del Doctorado en Ciencias, Mención: Didáctica de las Ciencias Formales -Matemática-de la Universidad Nacional de Catamarca, Argentina.

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    ABSTRACT: This paper describes an analysis of student constructions of formal theory in university mathematics. After a preliminary study to establish initial categories for consideration, a main study followed students through a twenty-week Real Analysis course, interviewing individuals at regular intervals to plot the growth of their knowledge construction. By focusing on the students constructions of definitions, arguments and images, two distinct modes of operation emerged—giving meaning to the definitions and resulting theory by building from earlier concept images, and extracting meaning from the formal definition through formal deduction. Both routes may be successful or unsuccessful in constructing the formal theory. Advanced mathematical thinking is so vast an enterprise that different individuals focus on different kinds of activities. One mathematician might focus on "thinking hard about a somewhat vague and uncertain situation, trying to guess what might be found out, and only then finally reaching definitions and the definitive theorems and proofs." Another may extend formal theory already developed by "getting and understanding the needed definitions, working with them to see what could be calculated and what might be true to finally come up with new 'structure theorems'," (MacLane, 1994, p. 190–191). The division of labour between those "guided by intuition" and those "preoccupied with logic" was noted by Poincaré (1913), citing Riemann as an intuitive thinker who "calls geometry to his aid" and Hermite as a logical thinker who "never evoked a sensuous image" in mathematical conversation (p, 212). So how can we expect students to fully understand all the processes of advanced mathematical thinking when mathematicians themselves must specialise in only part of the total enterprise? This research project began with a preliminary study analysing written work and interviews with students to establish basic categories for analysis. It was founded on theory in the literature of advanced mathematical thinking (e.g. Tall, 1991, and subsequent developments). Few of the students concerned proved to have a grasp of the formal theory, exhibiting imagery already studied in the literature. The main study was designed to cover a wider spectrum of students, including highly successful ones. Students were interviewed at intervals on seven occasions through a twenty week first year course on Real Analysis. The methodology uses a form of theory construction following the style of Strauss (1987), Strauss & Corbin (1990). It begins by reviewing data and attempting to categorise it, re-evaluating the categorisations to fit the data collected until it falls into a natural structure that is grounded in the available data.
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    ABSTRACT: Mathematical thinking often extrapolates beyond the practical experience of the individual. Limiting processes are a case in point. To understand the nature of thinking processes it is insufficient just to analyse the mathematics, one must try to understand the thought processes themselves. This is of vital importance when we consider mathematical intuition, where thinking does not proceed along logical lines. In this paper we build on suggestions of Hebb and others as to how the brain functions and develop these ideas to give a description of mathematical intuition in cognitive terms. In the particular case of limiting processes we summarize various results which demonstrate the manner in which such processes can be naturally extrapolated to give intuitions of infinity quite different from cardinal infinity. In the choice of an appropriate terminology to describe the cognitive aspects I have been fortunate to be able to work with Dr S. Vinner and to develop a model of conceptual thinking including some of his ideas. This puts forward possible reasons why an individual can on different occasions have apparently conflicting intuitions and yet sense no cognitive conflict, yet on other occasions cognitive conflict can occur without any explicit reasons being apparent. The various intuitions of infinity are rich in such conflicts. In the first section of the paper I concentrate on cognitive ideas, introducing the formulation developed with Vinner in 搂2.Then there follows a section which briefly describes a formal notion of infinity quite different from cardinal infinity. In 搂4 we will consider various examples of infinite processes and indicate how conflicting intuitions of infinity can arise. All of these intuitions are natural extrapolations of certain parts of finite experience and some of them include a reciprocal idea of the infinitesimally small. The threads are drawn together in the final section when we review the general notion of intuition in the light of the particular examples described in the paper.
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    ABSTRACT: The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition. The development of limits and continuity, as taught in secondary school and university, are considered. Various investigations are reported which demonstrate individual concept images differing from the formal theory and containing factors which cause cognitive conflict.
    Educational Studies in Mathematics 05/1981; 12(2). DOI:10.1007/BF00305619


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