ABSTRACT In this paper, we analyse the conditions and constraints which might favour, or on the c... more ABSTRACT In this paper, we analyse the conditions and constraints which might favour, or on the contrary hinder, a large-scale implementation of inquiry-based mathematics and science education, on the basis of our work within the PRIMAS project in 12 European countries. As a complement to the approach through the analysis of teachers’ beliefs and practices (see Engeln et al. in ZDM Int J Math Educ 45(6), this issue, 2013), we tackle this issue from a systemic institutional perspective. Indeed, in our approach, we consider teachers as actors of institutions, representing some disciplines, embedded in a school system, sharing some common pedagogical issues, in relation to society. Our sources of information are easily accessible public documents. With a theoretical background from Chevallard’s anthropological theory of didactics, we organized our analysis according to four levels of institutional organization that co-determine both content and didactical aspects in the teaching of mathematics and sciences: society, school, pedagogy and disciplinary. Our approach is systemic in the sense that we do not focus on teachers as individuals, nor on the curricula, the organization of teachers’ training or the textbooks themselves. Rather, we trace the way the conditions and constraints are operative, provide the main results of our analysis and draw out a few perspectives according to our four levels of didactical determination. Finally, in the conclusion, we reflect on the limits and potential of our analysis.
This paper issued from a doctorate in process deals with the relation between mathematics and phy... more This paper issued from a doctorate in process deals with the relation between mathematics and physics about vectors / forces-velocity and translation / movement of translation. In the framework of TAD we address the issue at the level of a problem of the profession. We first give a brief account of the evolution of the teaching of vector in mathematics and physics since the end of 19th century in France. Then we give some elements of the institutional and personal relation of teachers of both subjects to the objects at stake in our study
Venue prioritairement des pays nordiques, slaves et anglo-saxons, la methode connue sous le nom a... more Venue prioritairement des pays nordiques, slaves et anglo-saxons, la methode connue sous le nom anglais de Inquiry Based Learning (IBL) se traduit habituellement en francais par demarche d’investigation. En mathematiques, il ne semble pas evident de distinguer l’IBL de la resolution de probleme. Dans ce texte prospectif, nous nous proposons de regarder comment la demarche de resolution de probleme et/ou d’investigation est abordee dans les programmes les plus recents de mathematiques de l’ecole primaire et secondaire inferieur en France et en Suisse romande. Nous recensons ensuite divers travaux de recherche en didactique des mathematiques et de recherches actions essentiellement francophones sur ces methodes, en evoquant des travaux plus internationaux sur la modelisation et les rapports aux autres disciplines. Nous conclurons par quelques pistes sur la formation des enseignants.
Cet article fait une synthèse de différents travaux portant sur des recherches de didactique des ... more Cet article fait une synthèse de différents travaux portant sur des recherches de didactique des mathématiques à propos de l’enseignement de l’algèbre linéaire en première année d’université scientifique. Nous présenterons d’abord brièvement un travail au cœur d’une vaste réforme de cet enseignement aux USA. Ensuite à travers quatre travaux menés au Canada, au Brésil ou en France, nous examinerons la question des changements de registres, de cadres, de niveaux de description, de points de vue et de modes de raisonnement dans une perspective de « flexibilité cognitive ». Enfin, nous présenterons les principaux résultats d’une équipe française, dont les travaux didactiques s’appuient à la fois sur une analyse historique et épistémologique et sur l’expérimentation sur plusieurs années d’un projet long d’enseignement
L'algebre lineaire est un domaine central dans l'edifice des mathematiques actuelles. Or ... more L'algebre lineaire est un domaine central dans l'edifice des mathematiques actuelles. Or l'experience montre que des difficultes souvent persistantes emaillent l'enseignement a l'universite des premieres notions de la theorie des espaces vectoriels. Notre recherche propose un travail de defrichement preparant a une etude didactique de ce probleme. Dans un premier temps, nous presentons une analyse historique de la genese des concepts elementaires d'algebre lineaire en essayant de degager ainsi leur nature epistemologique. Nous montrons egalement en quoi cette etude est un point d'appui pour un travail didactique. Ensuite nous essayons, a travers une analyse de productions d'etudiants, de mieux cerner la nature de l'enseignement des debuts d'algebre lineaire tel qu'il existe, et d'expliciter les procedures employees par les etudiants en fonction des taches qu'on leur propose et de certaines de leurs connaissances anterieures. Cette etude s'accompagne egalement d'une reflexion sur le type de methodologie employee et sur l'apport des outils de la didactique. Enfin nous donnons quelques propositions globales pour l'enseignement de l'algebre lineaire aux debutants. A travers ce travail, nous soulevons egalement le probleme de l'adaptation de certaines theories didactiques aux specificites de l'enseignement superieur et de la nature epistemologique des concepts d'algebre lineaire; en reponse a cette interrogation, nous proposons de facon concrete l'utilisation du levier meta-mathematique, dans l'elaboration de sequences d'enseignement
The issue of mathematics education being an academic field has to do not only with a structural a... more The issue of mathematics education being an academic field has to do not only with a structural aspect of research but also with the question of the legitimacy of a theoretical positioning about issues within the scope of mathematics education.
Teaching Mathematics as to be Meaningful – Foregrounding Play and Children’s Perspectives
Considering a semiotic perspective this paper focuses on the mathematical actions and gestures of... more Considering a semiotic perspective this paper focuses on the mathematical actions and gestures of a 4-year-old boy on variously shaped and colored wooden figures in different amounts while he interacts with a group of another three kids and an accompanying person in a German kindergarten. The paper investigates the boy’s elaboration of a first trial of counting all green and blue dogs in front of him and his development of a colored pattern in gesture and action while he participates in the ongoing interaction in his speech. Even if the originally assumed mathematical area was numbers and operations, the situation was an open offer for the learners with the freedom to address different mathematical ideas. By dint of a semiotic analysis, it is shown that the boy uses his multimodal interaction skills in different ways, focusing same or different mathematical ideas of number determination in different modes. Theoretically the first empirical findings give rise to potentially rethink t...
Mathematics is often seen as a very specific subject, either by students, parents, media or even ... more Mathematics is often seen as a very specific subject, either by students, parents, media or even mathematicians themselves. In many contexts, mathematics is feared and seen as a subject for selection, disconnected from interesting applications to real life. Moreover, the structure of teaching institutions, in many cases, makes the collaboration between teachers from different disciplines very difficult. At the same time, mathematics is more and more invisible in everyday life, since high technology tends to hide the mathematics necessary for its creation in sophisticated black boxes. As a result, it is quite a challenge to give an adequate answer to those who, legitimately, wonder what mathematics is useful for. Our purpose in this paper is to see how mathematics in different curricula is actually connected to other disciplines and to give propositions to make this connection more effi- cient for the benefit of both mathematics and other subjects. The results presented here emerged ...
ABSTRACT In this paper, we analyse the conditions and constraints which might favour, or on the c... more ABSTRACT In this paper, we analyse the conditions and constraints which might favour, or on the contrary hinder, a large-scale implementation of inquiry-based mathematics and science education, on the basis of our work within the PRIMAS project in 12 European countries. As a complement to the approach through the analysis of teachers’ beliefs and practices (see Engeln et al. in ZDM Int J Math Educ 45(6), this issue, 2013), we tackle this issue from a systemic institutional perspective. Indeed, in our approach, we consider teachers as actors of institutions, representing some disciplines, embedded in a school system, sharing some common pedagogical issues, in relation to society. Our sources of information are easily accessible public documents. With a theoretical background from Chevallard’s anthropological theory of didactics, we organized our analysis according to four levels of institutional organization that co-determine both content and didactical aspects in the teaching of mathematics and sciences: society, school, pedagogy and disciplinary. Our approach is systemic in the sense that we do not focus on teachers as individuals, nor on the curricula, the organization of teachers’ training or the textbooks themselves. Rather, we trace the way the conditions and constraints are operative, provide the main results of our analysis and draw out a few perspectives according to our four levels of didactical determination. Finally, in the conclusion, we reflect on the limits and potential of our analysis.
This paper issued from a doctorate in process deals with the relation between mathematics and phy... more This paper issued from a doctorate in process deals with the relation between mathematics and physics about vectors / forces-velocity and translation / movement of translation. In the framework of TAD we address the issue at the level of a problem of the profession. We first give a brief account of the evolution of the teaching of vector in mathematics and physics since the end of 19th century in France. Then we give some elements of the institutional and personal relation of teachers of both subjects to the objects at stake in our study
Venue prioritairement des pays nordiques, slaves et anglo-saxons, la methode connue sous le nom a... more Venue prioritairement des pays nordiques, slaves et anglo-saxons, la methode connue sous le nom anglais de Inquiry Based Learning (IBL) se traduit habituellement en francais par demarche d’investigation. En mathematiques, il ne semble pas evident de distinguer l’IBL de la resolution de probleme. Dans ce texte prospectif, nous nous proposons de regarder comment la demarche de resolution de probleme et/ou d’investigation est abordee dans les programmes les plus recents de mathematiques de l’ecole primaire et secondaire inferieur en France et en Suisse romande. Nous recensons ensuite divers travaux de recherche en didactique des mathematiques et de recherches actions essentiellement francophones sur ces methodes, en evoquant des travaux plus internationaux sur la modelisation et les rapports aux autres disciplines. Nous conclurons par quelques pistes sur la formation des enseignants.
Cet article fait une synthèse de différents travaux portant sur des recherches de didactique des ... more Cet article fait une synthèse de différents travaux portant sur des recherches de didactique des mathématiques à propos de l’enseignement de l’algèbre linéaire en première année d’université scientifique. Nous présenterons d’abord brièvement un travail au cœur d’une vaste réforme de cet enseignement aux USA. Ensuite à travers quatre travaux menés au Canada, au Brésil ou en France, nous examinerons la question des changements de registres, de cadres, de niveaux de description, de points de vue et de modes de raisonnement dans une perspective de « flexibilité cognitive ». Enfin, nous présenterons les principaux résultats d’une équipe française, dont les travaux didactiques s’appuient à la fois sur une analyse historique et épistémologique et sur l’expérimentation sur plusieurs années d’un projet long d’enseignement
L'algebre lineaire est un domaine central dans l'edifice des mathematiques actuelles. Or ... more L'algebre lineaire est un domaine central dans l'edifice des mathematiques actuelles. Or l'experience montre que des difficultes souvent persistantes emaillent l'enseignement a l'universite des premieres notions de la theorie des espaces vectoriels. Notre recherche propose un travail de defrichement preparant a une etude didactique de ce probleme. Dans un premier temps, nous presentons une analyse historique de la genese des concepts elementaires d'algebre lineaire en essayant de degager ainsi leur nature epistemologique. Nous montrons egalement en quoi cette etude est un point d'appui pour un travail didactique. Ensuite nous essayons, a travers une analyse de productions d'etudiants, de mieux cerner la nature de l'enseignement des debuts d'algebre lineaire tel qu'il existe, et d'expliciter les procedures employees par les etudiants en fonction des taches qu'on leur propose et de certaines de leurs connaissances anterieures. Cette etude s'accompagne egalement d'une reflexion sur le type de methodologie employee et sur l'apport des outils de la didactique. Enfin nous donnons quelques propositions globales pour l'enseignement de l'algebre lineaire aux debutants. A travers ce travail, nous soulevons egalement le probleme de l'adaptation de certaines theories didactiques aux specificites de l'enseignement superieur et de la nature epistemologique des concepts d'algebre lineaire; en reponse a cette interrogation, nous proposons de facon concrete l'utilisation du levier meta-mathematique, dans l'elaboration de sequences d'enseignement
The issue of mathematics education being an academic field has to do not only with a structural a... more The issue of mathematics education being an academic field has to do not only with a structural aspect of research but also with the question of the legitimacy of a theoretical positioning about issues within the scope of mathematics education.
Teaching Mathematics as to be Meaningful – Foregrounding Play and Children’s Perspectives
Considering a semiotic perspective this paper focuses on the mathematical actions and gestures of... more Considering a semiotic perspective this paper focuses on the mathematical actions and gestures of a 4-year-old boy on variously shaped and colored wooden figures in different amounts while he interacts with a group of another three kids and an accompanying person in a German kindergarten. The paper investigates the boy’s elaboration of a first trial of counting all green and blue dogs in front of him and his development of a colored pattern in gesture and action while he participates in the ongoing interaction in his speech. Even if the originally assumed mathematical area was numbers and operations, the situation was an open offer for the learners with the freedom to address different mathematical ideas. By dint of a semiotic analysis, it is shown that the boy uses his multimodal interaction skills in different ways, focusing same or different mathematical ideas of number determination in different modes. Theoretically the first empirical findings give rise to potentially rethink t...
Mathematics is often seen as a very specific subject, either by students, parents, media or even ... more Mathematics is often seen as a very specific subject, either by students, parents, media or even mathematicians themselves. In many contexts, mathematics is feared and seen as a subject for selection, disconnected from interesting applications to real life. Moreover, the structure of teaching institutions, in many cases, makes the collaboration between teachers from different disciplines very difficult. At the same time, mathematics is more and more invisible in everyday life, since high technology tends to hide the mathematics necessary for its creation in sophisticated black boxes. As a result, it is quite a challenge to give an adequate answer to those who, legitimately, wonder what mathematics is useful for. Our purpose in this paper is to see how mathematics in different curricula is actually connected to other disciplines and to give propositions to make this connection more effi- cient for the benefit of both mathematics and other subjects. The results presented here emerged ...
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