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Connectedness of Problems and Impasse Resolution in the Solving Process in Geometry: A Major Educational Challenge
Abstract
Our contribution shows the anticipated effect of what we call connected problems in developing the competencies of students and their acquisition of mathematical knowledge. Whilst our theoretical approach focuses on didactic and cognitive interactions, we give special attention to a model to reason about learners’ conceptions, and the ideas of mathematical working space and zone of proximal development, in order to explore how connected problems can help to resolve moments of impasse of a student when solving a proof problem in geometry. In particular, we discuss how the notion of interaction moves our theoretical framework closer to the methodological challenges raised in the QED-Tutrix research project jointly being realized in didactics of mathematics and computer engineering.
... Indeed, some problem bases are presented in the form of sequences, the order of resolution having been determined in advance, while others adapt the sequence of problems to being solved in accordance with the assessment made by the machine. Some systems take advantage of the link between an obstacle and knowledge, in a way reminiscent of the epistemological obstacle of Bachelard (1967), in order to show how connected problems can help to resolve a student's moments of impasse when solving problems of proof (Richard, Gagnon & Fortuny, 2018). The validity of the process of assessment depends, of course, on the actions of the user, but it is above all a matter of didactic epistemology which deals with decision-making and the complexity of connectedness. ...
This chapter focuses on the particularities of mathematical work in the digital age. It opens with historical considerations that relate to the development of mathematical work to do arithmetic when symbolic or mechanical tools and algorithmic methods were used. He goes on to define the new mathematical work by introducing concepts of reference in the interaction between human and machine. The notion of valence of mathematical work makes it possible to account for the operating domain of interactions as well as for the possible adaptations of an evolving subject-milieu system, whether for the accomplishment of a task or in the course of learning mathematics. A difference between the work of the designer and that of a user is established, especially in terms of the effects on reasoning, proof, modelling activity and the creation of digital artefacts. The adaptation in a process of idoneity between the teaching and the learning project, as well as between the intention of the designer and the realisation by a user, is discussed.
... It is then by developing ideas already expounded in our previous work, including the recent paper The Concept of Proof in the Light of Mathematical Work (Richard, Oller, & Meavilla, 2016), and adopting the conclusions of our current research projects on the design of the tutorial system QED-Tutrix in high school geometry (Richard, Gagnon, & Fortuny, 2018;Font, Richard, & Gagnon, 2018), and the use of automated reasoning tools in teacher training (Kovács, Richard, Recio, & Vélez, 2017), that we address the question of instrumental proofs, keeping in mind that the interaction between subject and milieu 1 is a unit of epistemic necessity. With this consideration, the subject can be either a reader, considering traditional proofs, or the user of software or of a mathematical machine. ...
Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof, reasoning and epistemic necessity, taking advantage of the possibilities offered by the development geneses and fibrations in an instrumented perspective. The coordination of the semiotic, discursive and instrumental geneses of the working space founds discursive-graphic proofs, mechanical proofs and algorithmic proofs that are activated at school in the subject-milieu interactions. We end with a discussion on some consequences of the computer-assisted modelling of the learning conditions of mathematics, and we conclude on a necessary reconciliation of heuristics and validation. Notre article vise à définir la notion de preuve instrumentale en partant de considérations didactiques, épistémologiques et cognitives. Nous soulevons des enjeux et des défis liés à ce type de preuve au regard du travail mathématique et de la pensée mathématique. La théorie des espaces de travail mathématique sert de charpente sur laquelle nous abordons des questions sur la preuve, le raisonnement et la nécessité épistémique, profitant des possibilités qu'offrent le développement des genèses et des fibrations dans une perspective instrumentée. La coordination des genèses sémiotique, discursive et instrumentale de l'espace de travail fondent des preuves discursivo-graphiques, des preuves mécaniques et des preuves algorithmiques qui s'activent à l'école dans l'interaction sujet-milieu. Nous terminons par une discussion de quelques conséquences de la modélisation des conditions d'apprentissage des mathématiques assisté par des dispositifs informatiques, et nous concluons sur un rapprochement nécessaire entre heuristique et validation.
Il est essentiel de se pencher sur les interactions entre l’intelligence artificielle (IA) et la didactique, encore plus à notre époque où l’impact de l’IA sur la société et l’économie est aussi profond. Tout d’abord, nous remettons en question la notion dite d’intelligence et
les préjugés qu’elle peut susciter lorsqu’on réfléchit à l’IA et à ses définitions. Ensuite, nous analysons les liens potentiels entre l’IA et la didactique des mathématiques. Pour ce faire, nous examinons des exemples de projets en cours dans le monde francophone permettant de dresser un état des lieux des aspects actuellement développés. Par la suite, nous explorons les cadres théoriques de la didactique des mathématiques et leur articulation avec l’IA. Enfin,
nous abordons les questions et les défis soulevés par l’utilisation de l’IA, tout en offrant des perspectives prometteuses pour l’avenir.
When considering the mathematics competences referential (Niss & Højgaard, 2019), whether from Québec Education Program, the Standards of the National Council of Teachers of Mathematics or some large international studies such as the Programme for International StudentStudent Assessment, reasoning, and mathematical proofs are always at the forefront. Although the notion of proof is limited, in elementary school, to mathematical reasoning and conviction, and in post-secondary education, to demonstration and written communication, the evolution of the treatment reserved for deductive reasoning in secondary school has the appearance, depending on the region, of an eternally shilly-shallying between valorization and marginalization.
Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof, reasoning and epistemic necessity, taking advantage of the possibilities offered by the development geneses and fibrations in an instrumented perspective. The coordination of the semiotic, discursive and instrumental geneses of the working space founds discursive-graphic proofs, mechanical proofs and algorithmic proofs that are activated at school in the subject-milieu interactions. We end with a discussion on some consequences of the computer-assisted modelling of the learning conditions of mathematics, and we conclude on a necessary reconciliation of heuristics and validation.
The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the resolution of demonstration problems. It focuses on specific goals : 1) to allow the student to freely explore the problem and its figure, 2) to accept proofs elements in any order, 3) to handle a variety of proofs, which can be customized by the teacher, and 4) to be able to help the student at any step of the resolution of the problem, if the need arises. The software is also independent from the intervention of the teacher. QED-Tutrix offers an interesting approach to geometry education, but is currently crippled by the lengthiness of the process of implementing new problems, a task that must still be done manually. Therefore, one of the main focuses of the QED-Tutrix' research team is to ease the implementation of new problems, by automating the tedious step of finding all possible proofs for a given problem. This automation must follow fundamental constraints in order to create problems compatible with QED-Tutrix : 1) readability of the proofs, 2) accessibility at a high school level, and 3) possibility for the teacher to modify the parameters defining the "acceptability" of a proof. We present in this paper the result of our preliminary exploration of possible avenues for this task. Automated theorem proving in geometry is a widely studied subject, and various provers exist. However, our constraints are quite specific and some adaptation would be required to use an existing prover. We have therefore implemented a prototype of automated prover to suit our needs. The future goal is to compare performances and usability in our specific use-case between the existing provers and our implementation.
Este estudio forma parte de una investigación1 en curso sobre la interpretación del comportamiento de los estudiantes de Bachillerato Tecnológico en la resolución de problemas de geometría plana, mediante el análisis de la relación entre el uso de GeoGebra2, la resolución en lápiz y papel y el pensamiento geométrico. El marco teórico se basa principalmente en la teoría de la instrumentación de Rabardel (2001). Proponemos un análisis de los grados de adquisición de los procesos de instrumentación e instrumentalización de los alumnos, las estrategias de resolución en ambos medios y las interacciones entre los distintos agentes involucrados. Pretendemos buscar una relación entre las concepciones de los alumnos y las técnicas que utilizan en las estrategias de resolución de problemas.
Teachers try to help their students learn. But why do they make the particular teaching choices they do? What resources do they draw upon? What accounts for the success or failure of their efforts? In How We Think, esteemed scholar and mathematician, Alan H. Schoenfeld, proposes a groundbreaking theory and model for how we think and act in the classroom and beyond. Based on thirty years of research on problem solving and teaching, Schoenfeld provides compelling evidence for a concrete approach that describes how teachers, and individuals more generally, navigate their way through in-the-moment decision-making in well-practiced domains. Applying his theoretical model to detailed representations and analyses of teachers at work as well as of professionals outside education, Schoenfeld argues that understanding and recognizing the goal-oriented patterns of our day to day decisions can help identify what makes effective or ineffective behavior in the classroom and beyond.
In the Geometrical Working Space (GWS) theoretical framework, the differentiation of geometrical approaches is based on the geometrical paradigms notion. Due to this notion, it is possible not only to point out the epistemological differences in the test approaches, but also to understand and explain variations in the instrumental and figural geneses. The extension of the GWS’s theoretical framework to the Mathematical Working Spaces draws attention on the simultaneous use of several domains of mathematics in the mathematical work. We analyze the problems initially lying on a geometrical support, but which solution can be expressed in another mathematical domain that, form a educational point of view, is not, thus, necessarily situated neither to the same paradigmatic level, nor to the same didactical or pedagogical level of elaboration. These differences in level can be source of misunderstandings and dysfunction in the academic practice. © 2014, Comite Latinoamericano de Matematica Educativa. All rights reserved.
Tutoring System for Improvement of Argumentative Competencies in High School Mathematics This article tries to show how secondary school pupils can improve their argument skills with the help of tutorial systems intended for learning geometry. After having established the conceptual framework at the intersection of mathematics teaching and computer environments for human learning, the article compares the heuristic and discursive features of some tutorial systems, including the systems developed by our research team. It then deals with the issue of the complementary nature of knowledge and skills in order to aim at a strategy for assessing argument skills based on relationships in the subject-environment system. The text particularly includes cognitive control, semiotic and situational structures associated with the development of argument skills in such environments. It also tackles the specific nature of reference knowledge, the decontextualization of learning, the instrumentation of resources, the idea of mathematical proof and the role of teaching agents. Résumé. Cet article vise à montrer comment l'élève de l'école secondaire peut améliorer ses compétences argumentatives à l'aide de systèmes tutoriels destinés à l'apprentissage de la géométrie. Après avoir situé le cadre conceptuel à l'intersection de la didactique des mathématiques et des environnements informatiques d'apprentissage humain, l'article compare les caractéristiques heuristiques et discursives de quelques systèmes tutoriels, dont les systèmes développés par notre équipe de recherche. Il traite ensuite la question de complémentarité entre connaissances et compétences pour se diriger vers une stratégie d'évaluation des compétences argumentatives sur la base des rapports du système sujet-milieu. Le texte intègre, en particulier, les structures de contrôle cognitif, sémiotique et situationnel associées au développement d'une compétence argumentative dans de tels environnements. Il aborde aussi la spécificité des connaissances de référence, la décontextualisation des apprentissages, l'instrumentation des ressources, l'idée de démonstration mathématique ainsi que le rôle d'agents didactiques.
Dynamic figures in a mathematical workspace for the learning of geometrical properties. Our paper aims at showing how dynamic figures are useful in the learning of the use of geometrical properties at a high school level, in continuity with the practices inherited during the primary school education. After considering the general contextual rooting of the problem situations while comparing geometrical reality with educational institution, we focus on the student-milieu system and on the connections between the reasoning and the operational dynamic figure. We then present a research framework in order to analyze a geometrical workspace dedicated towards the learning of the use of properties. The workspace is presented in what it has of generic to enhance such learning and it enables us to conclude by some theoretical remarks on the components from the suitable geometrical working space. Résumé. Notre article vise à montrer comment les figures dynamiques sont utiles pour l'apprentissage des propriétés géométriques de l'école secondaire, en continuité avec les habitudes héritées du primaire. Après avoir considéré l'enracinement contextuel des situations-problèmes en général et mis la réalité géométrique au regard de l'institution scolaire, nous centrons notre propos sur le système sujet-milieu et sur les rapports du raisonnement à la figure dynamique opératoire. Nous situons ensuite un dispositif de recherche afin d'analyser un espace de travail géométrique idoine orienté vers l'apprentissage des propriétés. L'espace de travail est présenté en ce qu'il a de générique pour enclencher un tel apprentissage et il nous permet de conclure par quelques remarques théoriques sur les constituants de l'espace de travail géométrique. Mots-clés. Didactique des mathématiques, géométrie dynamique, raisonnement, espaces de travail géométrique, apprentissage des propriétés géométriques.
Résumé
Cet article vise à montrer l’utilité d’une approche fonctionnelle-structurelle pour l’étude des manuels scolaires en mathématiques. L’approche s’inspire de trois sources : la théorie des fonctions du langage de Duval, le modèle des fonctions du langage dans la communication de Jakobson et le modèle des structures sémiotiques développé par Richard. Après avoir présenté l’approche dans la première partie, nous l’appliquons ensuite à l’analyse de deux courts textes, tirés des manuels de géométrie qui ont été en usage au Québec, dans les écoles secondaires de langue française. L’intention adidactique qui se dégage des deux textes montre comment les moyens sémiotiques mobilisés sont mis au sevice de la qualité de la communication avec l’éventuel lecteur.
Our paper aims at showing how the instrumented learning of the properties in Euclidean geometry can be introduced in continuity with the mathematical competences developed at the elementary school. The principle founder of our research is based on the relation of subordination between the constraints of a property, posed using dynamic geometry software, and a necessary conclusion. Starting from an experimentation carried out in classes of 12-14 years old in France and in Quebec, our study presents a critical analysis of results where the ultimate objective of the activities suggested to the students relates as well to the significance of elementary properties, as on the understanding of the necessity of the link between the antecedents to the consequents of a deduction. Before ending by the didactic consequences of our approach, the text introduces the concept of instrumented figural inference as a means, employed by certain students, to justify a step of structured reasoning. A reconciling of the semiotic, instrumental and discursive aspects is offered throughout the paper.
Cours à l'école d'été de didactique des mathématiques, 2003, présentant le modèle cK¢ et le cadre théorique dans lequel il a été construit, ainsi que divers exemples.
Our paper presents a project that involves two research questions: does the choice of a related problem by the tutorial system allow the problem solving process which is blocked for the student to be restarted? What information about learning do related problems returned by the system provide us? We answer the first question according to the didactic engineering, whose mode of validation is internal and based on the confrontation between an a priori analysis and an a posteriori analysis that relies on data from experiments in schools. We consider the student as a subject whose adaptation processes are conditioned by the problem and the possible interactions with the computer environment, and also by his knowledge, usually implicit, of the institutional norms that condition his relationship with geometry. Choosing a set of good problems within the system is therefore an essential element of the learning model. Since the source of a problem depends on the student's actions with the computer tool, it is necessary to wait and see what are the related to problems that are returned to him before being able to identify patterns and assess the learning. With the simultaneity of collecting and analysing interactions in each class, we answer the second question according to a grounded theory analysis. By approaching the problems posed by the system and the designs in play at learning blockages, our analysis links the characteristics of problems to the design components in order to theorize on the decisional, epistemological, representational, didactic and instrumental aspects of the subject-milieu system in interaction.
Cet article développe les éléments d'un cadre théorique utilisé pour l'étude de l'enseignement de la géométrie élémentaire notamment en formation des enseignants. Les notions de paradigmes géométriques et d'espace de travail de la géométrie font l'objet d'une présentation détaillée. La géométrie élémentaire est ainsi vue comme éclatée en trois paradigmes géométriques différents. Ces derniers approfondissent la notion de cadre géométrique due à Douady. Ils sont ensuite mis en relation avec les niveaux de Van Hiele. La notion d'espace de travail permet d'étudier la spécificité de l'activité du géomètre qu'il soit expert ou apprenti. Une articulation avec l'approche cognitive de Duval est envisagée.
This article aims at legitimating the use of the graphic registers in the written expression of mathematical reasoning. More
than the defence of this legitimacy, the article shows also the risks of its use: the problems of communication which it causes
as well as the difficulties, for the teachers, to evaluate the reasoning of the pupils when they integrate these registers
into their solutions, and for the pupils, to understand the mathematical texts which use these registers in their discursive
expansions differently than optional illustration. With the concept of graphic expansion and figural inference, the article
claims to supplement the inferences in Raymond Duval's discursive plans whilst respecting the reasoning functional definition.
Having considered the cognitive and semiotic aspects in the definition of figure, the article shows the use of drawings in
the reasoning structure of a pupil at secondary level. Figural inference born from this study is examined then defined in
terms of function, structure and quality.
This paper aims at showing the didactic and theoretical-based perspectives in the experimental development of the geogebraTUTOR
system (GGBT) in interaction with the students. As a research and technological realization developed in a convergent way
between mathematical education and computer science, GGBT is an intelligent tutorial system, which supports the student in
the solving of complex problems at a high school level by assuring the management of discursive messages as well as the management
of problem situations. By situating the learning model upstream and the diagnostic model downstream, GGBT proposes to act
on the development of mathematical competencies by controlling the acquisition of knowledge in the interaction between the
student and the milieu, which allows for the adaptation of the instructional design (learning opportunities) according to
the instrumented actions of the student. The inferential and construction graphs, a structured bridge (interface) between
the contextualized world of didactical contracts and the formal computer science models, structure GGBT. This way allows for
the tutorial action to adjust itself to the competential habits conveyed by a certain classroom of students and to be enriched
by the research results in mathematical education.
KeywordsMathematical education (didactics of mathematics)–Intelligent tutorial system–Mathematical competencies–Computer science models (informatics)–Geometry learning–Dynamic geometry software–Student–milieu cognitive interactions
Ce travail se situe, en Sciences de l'Éducation, dans le champ des didactiques, c'est-à-dire dans celui de l'étude de la transmission sociale des savoirs culturels, notamment dans le cadre scolaire, plus précisément, il cherche à développer la théorie des situations didactiques, qui prend son origine dans le travail de Guy Brousseau. La première partie montre comment l'existence de ce que j'ai appelé les phases de conclusion permet une première intelligibilité du rôle du professeur dans les classes ordinaires. Le travail du professeur n'apparaît plus comme principalement un travail en classe : beaucoup des ressources et des déterminations de la situation du professeur proviennent de ressources et de contraintes en amont. La deuxième partie est centrée sur le modèle de la structuration du milieu,. Du point de vue de l'étude du professeur, les cinq niveaux de détermination dégagés par l'étude de la structuration du milieu permettent de décrire d'une façon fine les connaissances en jeu dans une situation du professeur au sens large. La troisième et dernière partie cherche à problématiser la difficile rencontre entre le professeur et les élèves autour des savoirs à enseigner et à apprendre. L'analyse des situations montre qu'il existe des bifurcations didactiques, qui sont un candidat pour comprendre, à l'échelle « microscopique », les phénomènes de différenciations scolaires connus au niveau macroscopique et sociologique.
This article proposes a state of the art of tutorial systems for high school planar geometry proof learning. The chosen approach is part of exploring the research problem that motivates the development, by our research team, of a tutorial system named QED-Tutrix, that we will present in another paper. In the following article, a synthesis and a comparison of existing tutorial systems is carried out on the basis of a set of original indicators highlighting the differences between the systems covered by our analysis. Each indicator aims to describe the functioning of the studied software according to the geometric work made possible at their interface. Eleven tutorial systems are compared according to their integration of a geometric figure, the structure they impose on the student's reasoning and the tutorial intervention they offer.
Our article aims to show how illuminating mathematical work as a concept from didactics of mathematics is useful in understanding issues relating to proving and learning of proof, with or without technology. After posing our hypotheses on the relationship of proof with mathematical work, the pedagogical intent of historical elements of geometry and the use of modern technical tools, we present reference contexts and situations around the property of the tangent. These contexts and situations allow for a comparison of the validation modes, the type of epistemic necessity at stake and certain underlying discourse peculiarities. We introduce the idea of a “valence of mathematical work” and we interpret in an a priori approach the main interactions that could maintain a model user-reader in a mathematical working space. We pay special attention to an extract of Elémens de Géométrie by Alexis Claude Clairaut, with one of the reference contexts revisiting his problem with the assistance of dynamic geometry software.
Central to caring professions such as teaching is the need to notice and be sensitive to the experiences of pupils and teachers. Starting from this position, Researching Your Own Practice demonstrates that in order to develop your professional practice you must first develop your own sensitivities and awareness. One must be attuned to fresh possibilities when they are needed and be alert to such a need through awareness of what is happening at any given time. By giving a full explanation of this theory and a guide to its implementation, this book provides a practical approach to becoming more methodical and systematic in professional development. It also gives the reader a basis for turning professional development into practitioner research, as well as giving advice on how noticing can be used to improve any research, or be used as a research paradigm in its own right. The discipline of noticing is a groundbreaking approach to professional development and research, based upon noticing a possibility for the future, noticing a possibility in the present moment and reflecting back on what has been noticed before in order to prepare for the future. John Mason, one of the discipline's most authoritative exponents, provides us here with a clear, persuasive and practical guide to its understanding and implementation.
Introduction à l’étude de l’enseignement du raisonnement et de la preuve: Les paradoxes
- G Brousseau
QED-Tutrix: Système tutoriel intelligent pour l’accompagnement d’élèves en situation de résolution de problèmes de démonstration en géométrie plane
- N Leduc
Comment poser et résoudre un problème, 2e éd
- G Pölya
Raisonnement et stratégies de preuve dans l’enseignement des mathématiques
- P R Richard
- PR Richard
GeoGebraTUTOR: Modélisation d’un système tutoriel autonome pour l’accompagnement d’élèves en situation de résolution de problèmes de démonstration en géométrie plane et genèse d’un espace de travail géométrique idoine
- M Tessier-Baillargeon
Gestion interactive de problèmes en géométrie pour le développement des compétences des élèves et l’acquisition du savoir mathématique
- P R Richard
- M Gagnon
- J M Fortuny
Pensée et langage. Paris: La dispute
- L S Vygotsky
- LS Vygotsky
Programme de formation de l’école québécoise
- Méls