Proving an existence theorem is less intuitive than proving other theorems. This article presents a semiotic analysis of significant fragments of classroom meaning-making which took place during the class-session in which the existence of the midpoint of a linesegment was proven. The purpose of the analysis is twofold. First follow the evolution of students' conceptualization when constructing a geometric object that has to satisfy two conditions to guarantee its existence within the Euclidean geometric system. An object must be created satisfying one condition that should lead to the fulfillment of the other. Since the construction is not intuitive it generates a dilemma as to which condition can be validly assigned initially. Usually, the students' spontaneous procedure is to force the conditions on a randomly chosen object. Thus, the second goal is to highlight the need for the teacher's mediation so the students understand the strategy to prove existence theorems. In the analysis, we use a model of conceptualization and interpretation based on the Peircean triadic SIGN.
Resumen En este artículo se presenta una relatión entre tipos de problemas abiertos de conjeturación en geometría, preferiblemente abordados en Entornos de Geometría Dinámica (EGD), y clases de argumentos producidos. Se expone cómo cada tipo de problema provoca la producción de argumentos (inductivos, abductivos o deductivos) durante el proceso de resolución. Para apoyar la idea, se exponen estrategias de solución producidas por estudiantes de un curso de geometría plana de un programa de formación inicial de profesores de matemáticas (Universidad Pedagógica Nacional, Colombia), cuando abordan problemas que involucran el objeto mediatriz de un segmento. Usando el modelo de argumento propuesto por Toulmin para analizar dichas estrategias, se identifican los tipos de argumentos asociados a cada tipo de problema. Se señala cómo la tipología de problemas puede contribuir al conocimiento didáctico-matemático del profesor de matemáticas.
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