Dans la pratique usuelle, on considere que les representations geometriques des nombres rationnel... more Dans la pratique usuelle, on considere que les representations geometriques des nombres rationnels ne sont que des illustrations aidant a comprendre ce que l'on envisage comme les veritables objets d'apprentissage : le sens des ecritures fractionnaires et decimales, construit a partir de leur action sur les grandeurs familieres aux eleves, et leur traitement par les operations elementaires. Mais on sait que l'apprentissage ainsi conduit se heurte a des obstacles resistants. Pour les depasser, il nous est apparu necessaire de proposer une introduction des rationnels qui s'appuie sur un autre registre d'expression. Dans ce but, nous avons choisi un mode de representation geometrique et l'avons constitue en veritable registre, profitant, pour ce faire, de specificites de l'outil informatique. Le recours a un registre geometrique s'imposait, mais les dimensions 1 et 2 se trouvaient en concurrence. Si la dimension 1 presente un cout didactique non negligea...
Les nombres rationnels sont aujourd'hui l'objet d'ingenieries qui distinguent en gene... more Les nombres rationnels sont aujourd'hui l'objet d'ingenieries qui distinguent en general deux phases d'apprentissage : un temps long pour traiter rhetoriquement une classe de problemes a des fins de conceptualisation, un temps bref pour assimiler les notations symboliques a des fins de communication et traitements. Des avancees importantes ont deja ete obtenues grâce a une telle demarche. Mais la discrimination des caracteristiques propres a chaque systeme exprimant les nombres, notamment celles des ecritures fractionnaires et decimales, reste difficile. Une autre difficulte importante est de reconnaitre dans les objets mathematiques ainsi – symboliquement – exprimes ceux dont on parlait lors de la phase de conceptualisation. Pour surmonter ces difficultes, nous avons propose une introduction aux rationnels qui privilegie un systeme d'expression au moyen de droites graduees. L'elaboration de ce support, plonge dans un environnement informatique permettant l&#...
Introduction This study relies on a previous survey that I conducted with 120 seventh-graders two... more Introduction This study relies on a previous survey that I conducted with 120 seventh-graders two years ago. A questionnaire presented for solution five ratio problems based on the same mathematical framework: working out a fourth proportional in combining the same type of data. But each problem referred to a different physico-empirical context (e.g. mixture and enlargement). For a given student, significant variations were found, from one problem to another, both in the success rates and in the procedures used. It seems therefore necessary to better take into account the physical context, and to classify the different types of ratio problems one student can face at this scholastic level, according to physical references. It seems also desirable to better organize, in the teaching, the articulations between and within the physical and mathematical domains. Now, in order to better articulate, it is first advisable to better separate, that is, better point out: differences between phy...
Rational numbers today are the object of a variety of teaching models which generally distinguish... more Rational numbers today are the object of a variety of teaching models which generally distinguish two learning phases : a longer phase to treat rhetorically a specific class of problems in order to permit the pupil to conceptualise them, a shorter phase to assimilate the symbols needed to communicate and process them. Significant progress has already been made thanks to this approach. Pupils still have trouble, however, discriminating between the various systems expressing numbers, particularly the notation of fractions and decimals. Another major problem is recognising in the symbolically expressed mathematical objects, those discussed during the conceptualisation phase. In order to overcome these difficulties, we propose an introduction to rational numbers using graduated lines as the main system of expression. A certain number of parameters were retained in the development of this teaching aid, used in connection with an interactive computer program. The first of these is that pu...
This paper relies on observations, lead in a didactical engineering context (Adjiage & Rauscher, ... more This paper relies on observations, lead in a didactical engineering context (Adjiage & Rauscher, 2013), of a sequence devoted to solving a modelling problem at grade 5. Here, we reconsider this study from the MWS standpoint (Kuzniak, 2011, p. 20). This allows us to reinterpret phenomena and to sketch features of an improved teaching schema. We thus: analyse student failures in terms of a lack in their empirical and theoretical system of reference; infer new guidelines for the appropriate MWS; interpret student difficulties in terms of dissociation of the visual and experimental geneses; analyse student progress in terms of the ability to integrate these two geneses, stemming from specific resources of writing (Duval 2001, p. 191)
This paper addresses the experimental phase of a didactical engineering bearing on a modelling pr... more This paper addresses the experimental phase of a didactical engineering bearing on a modelling problem solving sequence at grade 5. We examine the improvement of the observed students through the evolution of their explanatory texts, paying special attention to their explicit arguments and justifications. Our main hypothesis is that sharing and revising peers’ writings, particularly through a practice mobilizing the specific resources of writing (Duval 2001), helps students develop solutions involving a modelling process. This hypothesis is validated for a little more than two thirds of the students
This paper will only consider the numerical domain. We will first present and analyse two current... more This paper will only consider the numerical domain. We will first present and analyse two current trends related to mathematical assessment, and their consequences on curricula. We will then propose alternatives for defining competences on which curricula and thus assessments may be based. In this purpose we will be first led to specify: what are mathematical objects and what sort of syntax is requested to link and process them at different scholastic levels; secondly, what notion of competence is relevant for determining what means mastering these objects, their relationships and processing. We lastly define four levels of competence, from primary school to the end of secondary school, for designing curricula and testing student achievements
This study concerns the field of mathematics education. Today, for almost any technology, attaini... more This study concerns the field of mathematics education. Today, for almost any technology, attaining the most advanced level relies on using digital systems. Therefore, we focus on number acquisition and use, emphasize major discussions about related topics, and introduce our personal contribution. We consider three areas: numbers in society, at school, and in the field of education. Numbers in society Modern society needs two kinds of number users: the first has to deal with numbers, the second has to work with numbers. According to the framework for PISA assessment, dealing with numbers concerns every citizen facing situations in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem. Working with numbers is what a professional (marketing specialist, engineer, physician, artist…) does when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. This brings two types of questioning: what does competenc...
Étude comparée de deux pratiques enseignantes lors d'une séquence de résolution d'un problème de ... more Étude comparée de deux pratiques enseignantes lors d'une séquence de résolution d'un problème de modélisation A comparative study of two teaching practices during a sequence of modeling problem solving
This paper summarizes our analysis of the complexity of ratio problems at Grades 6 and 7, and rep... more This paper summarizes our analysis of the complexity of ratio problems at Grades 6 and 7, and reports a two-year experiment related to the teaching and learning of rational numbers and proportionality in these grades. Two classes were followed and observed. Part of the teaching material was common to both classes, mainly the objectives and the corpus of ratio problems in a physical context. But in one class, here called “Partial-experiment”, the learning environment was exclusively a paper-pencil one and the teacher followed his usual method in designing and conducting teaching sequences. In the other class, here called “Full-experiment”, the teaching was based on a framework, emerging from our analysis of complexity of ratio problems, involving precise guidelines and a specific computer environment. Using a pre-test and a post-test, we observed clear progress in both classes compared to a sample of “standard” pupils. Our comparative pupil-oriented study indicates more complete improvement in the “full-experiment” class, i.e., a better acquisition of fractions and their use for solving usual proportionality problems. The average pupil’s progress is greater in the “full experiment”, with the pupils who were initially high- or low-level attainers benefiting the most from the “full-experiment”.
Pour éviter les pièges de l'emphase et du convenu, je souhaitais être bref dans l'expression de m... more Pour éviter les pièges de l'emphase et du convenu, je souhaitais être bref dans l'expression de ma reconnaissance. Je n'y suis pas parvenu. Cela tient sans doute à la diversité intellectuelle, morale ou matérielle (une qualification n'excluant pas l'autre) des aides qui m'ont été dispensées pour la réalisation de cette thèse.
This study relies on a previous survey that I conducted with 120 seventh-graders two years ago. A... more This study relies on a previous survey that I conducted with 120 seventh-graders two years ago. A questionnaire presented for solution five ratio problems based on the same mathematical framework: working out a fourth proportional in combining the same type of data. But each problem referred to a different physico-empirical context (e.g. mixture and enlargement). For a given student, significant variations were found, from one problem to another, both in the success rates and in the procedures used. It seems therefore necessary to better take into account the physical context, and to classify the different types of ratio problems one student can face at this scholastic level, according to physical references. It seems also desirable to better organize, in the teaching, the articulations between and within the physical and mathematical domains. Now, in order to better articulate, it is first advisable to better separate, that is, better point out: differences between physical investigations (multi-sensorial with or without the real or virtual use of instruments) and mathematical ones (formal expressions and processing); differences between the diverse ratio problems on one hand and between the means of expression of rational numbers on the other hand. So that the complexity of ratio problems at Grades 6 and 7 is declined in three levels of separation/articulation. Noelting (1980), or Vergnaud (1983) emphasized the importance of physical references in solving ratio problems. I just try here to specify these references and how the mathematical ways of expressing ratios match to them.
Dans la pratique usuelle, on considere que les representations geometriques des nombres rationnel... more Dans la pratique usuelle, on considere que les representations geometriques des nombres rationnels ne sont que des illustrations aidant a comprendre ce que l'on envisage comme les veritables objets d'apprentissage : le sens des ecritures fractionnaires et decimales, construit a partir de leur action sur les grandeurs familieres aux eleves, et leur traitement par les operations elementaires. Mais on sait que l'apprentissage ainsi conduit se heurte a des obstacles resistants. Pour les depasser, il nous est apparu necessaire de proposer une introduction des rationnels qui s'appuie sur un autre registre d'expression. Dans ce but, nous avons choisi un mode de representation geometrique et l'avons constitue en veritable registre, profitant, pour ce faire, de specificites de l'outil informatique. Le recours a un registre geometrique s'imposait, mais les dimensions 1 et 2 se trouvaient en concurrence. Si la dimension 1 presente un cout didactique non negligea...
Les nombres rationnels sont aujourd'hui l'objet d'ingenieries qui distinguent en gene... more Les nombres rationnels sont aujourd'hui l'objet d'ingenieries qui distinguent en general deux phases d'apprentissage : un temps long pour traiter rhetoriquement une classe de problemes a des fins de conceptualisation, un temps bref pour assimiler les notations symboliques a des fins de communication et traitements. Des avancees importantes ont deja ete obtenues grâce a une telle demarche. Mais la discrimination des caracteristiques propres a chaque systeme exprimant les nombres, notamment celles des ecritures fractionnaires et decimales, reste difficile. Une autre difficulte importante est de reconnaitre dans les objets mathematiques ainsi – symboliquement – exprimes ceux dont on parlait lors de la phase de conceptualisation. Pour surmonter ces difficultes, nous avons propose une introduction aux rationnels qui privilegie un systeme d'expression au moyen de droites graduees. L'elaboration de ce support, plonge dans un environnement informatique permettant l&#...
Introduction This study relies on a previous survey that I conducted with 120 seventh-graders two... more Introduction This study relies on a previous survey that I conducted with 120 seventh-graders two years ago. A questionnaire presented for solution five ratio problems based on the same mathematical framework: working out a fourth proportional in combining the same type of data. But each problem referred to a different physico-empirical context (e.g. mixture and enlargement). For a given student, significant variations were found, from one problem to another, both in the success rates and in the procedures used. It seems therefore necessary to better take into account the physical context, and to classify the different types of ratio problems one student can face at this scholastic level, according to physical references. It seems also desirable to better organize, in the teaching, the articulations between and within the physical and mathematical domains. Now, in order to better articulate, it is first advisable to better separate, that is, better point out: differences between phy...
Rational numbers today are the object of a variety of teaching models which generally distinguish... more Rational numbers today are the object of a variety of teaching models which generally distinguish two learning phases : a longer phase to treat rhetorically a specific class of problems in order to permit the pupil to conceptualise them, a shorter phase to assimilate the symbols needed to communicate and process them. Significant progress has already been made thanks to this approach. Pupils still have trouble, however, discriminating between the various systems expressing numbers, particularly the notation of fractions and decimals. Another major problem is recognising in the symbolically expressed mathematical objects, those discussed during the conceptualisation phase. In order to overcome these difficulties, we propose an introduction to rational numbers using graduated lines as the main system of expression. A certain number of parameters were retained in the development of this teaching aid, used in connection with an interactive computer program. The first of these is that pu...
This paper relies on observations, lead in a didactical engineering context (Adjiage & Rauscher, ... more This paper relies on observations, lead in a didactical engineering context (Adjiage & Rauscher, 2013), of a sequence devoted to solving a modelling problem at grade 5. Here, we reconsider this study from the MWS standpoint (Kuzniak, 2011, p. 20). This allows us to reinterpret phenomena and to sketch features of an improved teaching schema. We thus: analyse student failures in terms of a lack in their empirical and theoretical system of reference; infer new guidelines for the appropriate MWS; interpret student difficulties in terms of dissociation of the visual and experimental geneses; analyse student progress in terms of the ability to integrate these two geneses, stemming from specific resources of writing (Duval 2001, p. 191)
This paper addresses the experimental phase of a didactical engineering bearing on a modelling pr... more This paper addresses the experimental phase of a didactical engineering bearing on a modelling problem solving sequence at grade 5. We examine the improvement of the observed students through the evolution of their explanatory texts, paying special attention to their explicit arguments and justifications. Our main hypothesis is that sharing and revising peers’ writings, particularly through a practice mobilizing the specific resources of writing (Duval 2001), helps students develop solutions involving a modelling process. This hypothesis is validated for a little more than two thirds of the students
This paper will only consider the numerical domain. We will first present and analyse two current... more This paper will only consider the numerical domain. We will first present and analyse two current trends related to mathematical assessment, and their consequences on curricula. We will then propose alternatives for defining competences on which curricula and thus assessments may be based. In this purpose we will be first led to specify: what are mathematical objects and what sort of syntax is requested to link and process them at different scholastic levels; secondly, what notion of competence is relevant for determining what means mastering these objects, their relationships and processing. We lastly define four levels of competence, from primary school to the end of secondary school, for designing curricula and testing student achievements
This study concerns the field of mathematics education. Today, for almost any technology, attaini... more This study concerns the field of mathematics education. Today, for almost any technology, attaining the most advanced level relies on using digital systems. Therefore, we focus on number acquisition and use, emphasize major discussions about related topics, and introduce our personal contribution. We consider three areas: numbers in society, at school, and in the field of education. Numbers in society Modern society needs two kinds of number users: the first has to deal with numbers, the second has to work with numbers. According to the framework for PISA assessment, dealing with numbers concerns every citizen facing situations in which the use of a quantitative… reasoning… would help clarify, formulate or solve a problem. Working with numbers is what a professional (marketing specialist, engineer, physician, artist…) does when he has to cope with numerical theoretical frameworks, or when he collaborates with a mathematician. This brings two types of questioning: what does competenc...
Étude comparée de deux pratiques enseignantes lors d'une séquence de résolution d'un problème de ... more Étude comparée de deux pratiques enseignantes lors d'une séquence de résolution d'un problème de modélisation A comparative study of two teaching practices during a sequence of modeling problem solving
This paper summarizes our analysis of the complexity of ratio problems at Grades 6 and 7, and rep... more This paper summarizes our analysis of the complexity of ratio problems at Grades 6 and 7, and reports a two-year experiment related to the teaching and learning of rational numbers and proportionality in these grades. Two classes were followed and observed. Part of the teaching material was common to both classes, mainly the objectives and the corpus of ratio problems in a physical context. But in one class, here called “Partial-experiment”, the learning environment was exclusively a paper-pencil one and the teacher followed his usual method in designing and conducting teaching sequences. In the other class, here called “Full-experiment”, the teaching was based on a framework, emerging from our analysis of complexity of ratio problems, involving precise guidelines and a specific computer environment. Using a pre-test and a post-test, we observed clear progress in both classes compared to a sample of “standard” pupils. Our comparative pupil-oriented study indicates more complete improvement in the “full-experiment” class, i.e., a better acquisition of fractions and their use for solving usual proportionality problems. The average pupil’s progress is greater in the “full experiment”, with the pupils who were initially high- or low-level attainers benefiting the most from the “full-experiment”.
Pour éviter les pièges de l'emphase et du convenu, je souhaitais être bref dans l'expression de m... more Pour éviter les pièges de l'emphase et du convenu, je souhaitais être bref dans l'expression de ma reconnaissance. Je n'y suis pas parvenu. Cela tient sans doute à la diversité intellectuelle, morale ou matérielle (une qualification n'excluant pas l'autre) des aides qui m'ont été dispensées pour la réalisation de cette thèse.
This study relies on a previous survey that I conducted with 120 seventh-graders two years ago. A... more This study relies on a previous survey that I conducted with 120 seventh-graders two years ago. A questionnaire presented for solution five ratio problems based on the same mathematical framework: working out a fourth proportional in combining the same type of data. But each problem referred to a different physico-empirical context (e.g. mixture and enlargement). For a given student, significant variations were found, from one problem to another, both in the success rates and in the procedures used. It seems therefore necessary to better take into account the physical context, and to classify the different types of ratio problems one student can face at this scholastic level, according to physical references. It seems also desirable to better organize, in the teaching, the articulations between and within the physical and mathematical domains. Now, in order to better articulate, it is first advisable to better separate, that is, better point out: differences between physical investigations (multi-sensorial with or without the real or virtual use of instruments) and mathematical ones (formal expressions and processing); differences between the diverse ratio problems on one hand and between the means of expression of rational numbers on the other hand. So that the complexity of ratio problems at Grades 6 and 7 is declined in three levels of separation/articulation. Noelting (1980), or Vergnaud (1983) emphasized the importance of physical references in solving ratio problems. I just try here to specify these references and how the mathematical ways of expressing ratios match to them.
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Papers by Robert ADJIAGE