Colección Digital Eudoxus 

Abstract: The last decade has seen the development in France of a significant body of research into the teaching and learning of mathematics in CAS environments. As part of this, French researchers have reflected on issues of instrumentation, and the dialectics between conceptual and technical work in mathematics. The reflection presented here is more than a personal one it is based on the collaboration and dialogues that I have been involved in during the nineties. After a short introduction, I briefly present the main theoretical frameworks which we have used and developed in the French research: the anthropological approach in didactics initiated by Chevallard, and the theory of instrumentation developed in cognitive ergonomics. Turning to the CAS research, I show how these frameworks have allowed us to approach important issues as regards the educational use of CAS technology, focusing on the following points: the unexpected complexity of instrumental genesis, the mathematical needs of instrumentation, the status of instrumented techniques, the problems arising from their connection with paper & pencil techniques, and their institutional management.

Abstract: The concept of meaning, which is frequently used in an informal way in didactic research, is a central and controversial subject in philosophy, logic, semiotics nd other sciences and technologies which are concerned in human cognition. The analysis of this concept from a didactical point of view could be useful to understand the relationship between the different theoretical frameworks in mathematics education and to throw a new light upon some research questions, particularly those referred to the assessment of knowledge. In this work, the aforementioned analysis is approached and a pragmatic theory of the meaning of mathematical objects is presented, in which a triple conditioning -institutional, personal and temporal- of the concept of meaning is established. The relationship between the proposed notion of meaning and those of conception and "rapport a l'objet" is also studied. La nocion de significado, utilizada con frecuencia de modo informal en los estudios didacticos, es un tema central controvertido en filosofia, logica, semiotica y demas ciencias y tecnologias interesadas en la cognicion humana. El analisis de esta nocion desde un punto de vista didactico puede ayudar a comprender las relaciones entre las distintas formulaciones teoricas en esta disciplina y permitir estudiar bajo una nueva perspectiva las cuestiones de investigacion, particularmente las referidas a la evaluacion de los conocimientos. En este trabajo se aborda el mencionado analisis y se presenta una teoria pragmatica del significado de los objetos matematicos en la que se propone para el mismo un triple condicionamiento: institucional, personal y temporal. Se estudian, asimismo, las conexiones entre la nocion de significado propuesta y las de concepcion y relacion al objeto. La notion de signifie, utilisee frequement de maniere informell dans les etudes didactiques, es un theme central controverse en philosophie, logique, semiotique et dans les autres sciences et technologies concernees par la cognition humaine. L'analyse de cette notion dans une perspective didactique peut aider a comprendre les relations entre differents cadres theoriques de cette discipline et permettre de formuler de nouvelles questions de recherche. Ceci nous parait concerner tout particulierment l'evaluation des connaisances. Dans le travail qui fait l'objet du present article, on aborde cette question en developpant une theorie pragmatique du signifie des objets mathematiques, theorie dans laquelle on propose pour ce dernier un triple conditionnement: personnel, institutionnel et temporel. On etudie egalement les connexions entre la notion proposee de signifie et celles de conception et de rapport a l'objet.

Abstract: En este trabajo se describe una tecnica de analisis de los procesos de ensenanza y aprendizaje de las matematicas, que permite determinar los significados institucionales y personales puestos en juego e identificar posibles conflictos semioticos en la interaccion didactica. La tecnica se basa en un modelo ontologico y semiotico para la cognicion matematica que se presenta previamente y se ejemplifica mediante el analisis del proceso de estudio propuesto para la mediana en un libro de texto, y de las respuestas de una estudiante a una prueba de evaluacion, aplicada tras la realizacion de dicho proceso de estudio. In this article we describe a technique to analyse mathematics teaching and learning processes, which permits the characterisation of the institutional and personal meanings involved and the identification of possible semiotic conflict in the didactic interaction. The technique is based on an ontological and semiotic model of mathematics cognition, which is previously presented. An application to analyse the study process of the median in a textbook, and the student’s answers to an assessment test given after the study process is also included. Ce travail decrit une technique d’analyse des processus d’enseignement et d’apprentissage des mathematiques qui permet de determiner les significations institutionnelles et personnelles mises en jeu dans l’interaction didactique et d’y identifier les possibles conflits semiotiques. La technique est fondee sur un modele ontologique et semiotique de la cognition mathematique que l’on presente initialement et que l’on illustre a travers, d’une part, l’analyse d’un processus d’etude de la mediane propose par un manuel et, d’autre part, les reponses d’une etudiante a une epreuve d’evaluation passee apres la realisation du processus d’etude considere.