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An ICMI Regional Conference: EM 2000. Grenoble (France), 15 - 17 July 2000
Bernard Cornue, Grenoble (France); Pierre Jullien, Meyreuil (France); Colette Laborde, Grenoble (France); Guy Nöel, Mons (Belgium)

This text is a report of an international conference about mathematics education in French-speaking countries in the XXth century and prospects for the beginning of the XXIst century which was organised by the CFEM in Grenoble from 15^th to 17^th July, 2000.
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Pourquoi, pour qui enseigner les mathématiques? Une mise en perspective historique de l'évolution des programmes, au XX e siècle, en Belgique
Guy Nöel, Mons  (Belgium)

Why teach mathematics at all? A historical outline of the development of education programmes in Belgium in the 20^th century. The evolution of education in Belgium is described from 1830 to our days. The mastery of education has always been a subject of disputes between the political forces in presence. With the flow of years, more attention is paid to social considerations and the necessity of a more democratic educational system arises. Simultaneously, the pedagogical ideas change and active methods of teaching are promoted, although not always used. Different school systems still coexist. However, except for philosophical and religious education, the differences between them tend to diminish. This is in particular the case for mathematics teaching whose evolution is also described with some details.
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Y a-t-il lieu d’envisager des Mathématiques Post-modernes  ?
Nicolas Bouleau, Paris (France)

Is there a place for modelling in post-modern mathematics? In order to give a right place to modeling as a major connexion of mathematics to economic activities, we emphasize the contrast between modern mathematics and polysemic mahematics. The philosophies of Popper and Quine are opposed in accordance to the lignt they bring to modeling. That point of view leads to see modeling as talking a language with the three syntaxical, semantical and pragmatical components. This reinforces the importance of teaching the critic of modeling and the critic by non conformist modeling.
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EM 2000 Mathématiques, quel avenir?
Jean-Pierre Kahane, Paris ( France)

EM 2000 – the future of mathematics. Delivered as the closing session, this talk was supposed to be an opening towards the future, both of children and humankind. There will be problems and challenges. How is math involved? Math has permanent values, recognized by other scientists, and also stable notions, some of them going back to the ancient Greeks. However math is in a perpetual motion. Old notions get a new look, new notions appear, as well as new relations with other sciences, international relations, including developing countries, new trends and a new conception of mathematical sciences. Math teaching should express both permanence and mobility of the subject, utility and beauty. Informatics, probability and statistics, geometry and all kinds of computing are subjects of reports under preparation, for a long term view of math education.
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La preuve en mathématique
Maria Alessandra Mariotti, Pisa (Italy)

Mathematical proof. Proof and deductive method in mathematics have their origin in the classic model of exposition developed by Euclid in his famous book on Elements. The attitude of mathematicians towards this method has certainly evolved in the past centuries, but the relationship between understanding and acceptability of mathematical statements has not dramatically changed and still constitutes a characterising element of this discipline.This paper is aimed at explaining and discussing some aspects which may be considered at the origin of difficulties related to proof; in particular, it focusses on the tension between two poles, that of production and that of systematisation of mathematical knowledge. Some examples drawn from different research projects are presented with the aim of illustrating the complementarity of various aspects and problems concerning proof. In the first part, the theoretical construct of Cognitive Unity is used to analyse and interpret the relationship between the production of a conjecture and its proof. In the second part, we present two long-term teaching experiments that are part of a research project whose principal goal is to introduce pupils to theoretical thinking and to study the role of particular microworlds in the teaching/learning processes. Assuming a Vygotskian perspective, attention is focussed on the semiotic mediation accomplished through cultural artefacts; in the case of Geometry the microworld is Cabri-géomètr, in the case of algebra the microworld is the prototype "L'Algebrista" (designed and realized by our team).
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Pourquoi et pour qui enseigner les mathématiques? Une mise en perspective historique de l’évolution des programmes au Québec au XXème siècle
Nadine Bednarz, Montréal (Canada)

Why teach mathematics at all? A historical outline of the development of teaching programmes in Quebec in the 20^th century. The history of reform in the teaching of mathematics in Quebec since the start of the 20th Century has been strongly influenced by a social and political context that gives meaning to its particular orientation. In this paper, we attempt to trace the important moments in the evolution of the teaching of mathematics through the roles and missions that successive governments have given to schooling and subsequent curricular reforms. Our analysis shows that the teaching of mathematics has evolved from an essentially practical role, prior to 1945, to a double role that is both practical and cultural, in the 1950s, with the balance tipped in favor of the practical role. In the 1960s, the political will for universal education and, more recently, the concern over forming persons who can adapt to a constantly evolving society, are gradually pushing the Quebec curricula towards major changes. Mathematics is presented as a powerful tool that gives one a handle on reality, a complement of culture and an important language that is essential to the communication of ideas. Throughout the various reforms, these two finalities--the practical and the cultural--are constantly present, varying in importance and meaning with the successive roles attributed to schools. Our analysis shows a strong pragmatic anchoring of the first mathematics programs in Quebec, one that has not disappeared over time. The gradual curricular changes show, however, a progressive integration of the cultural formation role of mathematics.
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Pourquoi, pour qui enseigner les mathématiques ? Une mise en perspective historique de l’évolution des programmes de mathématiques dans la société française au XXe siècle.
Hélène Gispert, Paris (France)

Why teach mathematics at all? A historical outline of the development of mathematics teaching programmes in France in the 20^th century. Evolutions of mathematical curricula in French society have been marked by the successive answers institutions have given to the following question since one century: Why and whom teaching mathematics ? Here I present two of these, one given in 1908 and one in 1967. Each symbolises a breaking period of reforms in secondary mathematics teaching in France. We will see in the two first part of this paper that they belong to two different worlds, with social, institutional, ideological and mathematical specific features. In the third part, I’ll focus on geometry, showing the effects of the different answers concerning the public and the aims of mathematical teaching.
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Perspectives sur les recherches en didactique des mathématiques
Anna Sierpinska, Montréal (Canada)

Approaches to research in mathematics education. The paper is a review of chosen approaches to research in mathematics education in several countries: Germany, France, United States, Russia, Poland, Canada. The review is done in the literary form of a satire, in which a character is taken on a voyage to a variety of "islands" representing different re-search interests and methodologies in mathematics education. The story is a parody of Homer's Odyssee, and the main cha-racter is called Odysseus. Odysseus' role is played by the famous arithmetic problem about a team of an unknown number of scythers who are given the task of scything two meadows one of which is double the size of the other. As the problem travels from one "island" to another, mathematics educators do different things to and with the problem and it is solved is a variety of ways. The main text of the paper reads as a story and there are no explicit references and names of authors, whose work is only alluded to. However, the solution to all allusions, i.e. explicit references, can be found in the footnotes.
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L'enseignement des mathématiques dans les pays francophones d'Afrique et de l'Ocean Indien
Saliou TOURE, Cocody-Abidjan (Ivory Coast)

Mathematics teaching in the French-speaking countries of Africa and the Indian Ocean.We examine Mathematics teaching in the French-speaking countries of Africa and the Indian Ocean, starting from the consequences of the Colonial Period. At that time, education was mainly aimed at preparing the civil servants., and there were no organized structure for teaching. When they became independent, these countries started with the French system and methods, but they progressively realized that it was not totally adapted to the aims and specificities of such countries. So progressively new systems and curricula were designed. In this paper, we describe some examples, and give some trends in the development of Mathematics education in Africa and Indian Ocean, and perspectives for the future.
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L’enseignement des mathématiques dans l’enseignement secondaire maghrébin
Mohamed Akkar, Bordeaux (France)

Mathematics teaching in secondary schools of the Maghreb region. The goal of this talk is to study the following questions. Does the mathematics teaching in the secondary schools in the Maghreb prepare to University studies and more specifically does it initiate students to modern science and technology ? Is anyone able to understand mathematics or is mathematics only accessible to the happy few. Is it a means of selection? Is mathematics omnipresent in our modern society? What relationship can one hope to find between mathematics and other disciplines? Has mathematics evolved to such an abstract and formal state that it seems difficult relate it to any other topic? All these questions are discussed in relationship with the particular problems in the Maghreb, namely the mathematics program as taught today in these countries.
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