Viviane Durand-Guerrier

• Université de Montpellier

Dans ce texte introductif, nous présentons tout d’abord le contexte historique et philosophique de l’émergence de l’infini potentiel et l’infini actuel, puis les débats autour de l’infini actuel cantorien. Nous abordons ensuite la question du jeu entre infini potentiel et infini actuel en mathématiques, et les spécificités liées au numérique. Nous...

This text introduces the first of two volumes following the DEMIMES (Didactics and Epistémology of Mathematics and their Interactions for Mediation and Post-secondary Teaching) thematic school held in Autrans (France) from April 4 to 7, 2022. Ce texte introduit le premier des deux volumes qui font suite à la tenue de l’école thématique DEMIMES (Did...

This paper presents a comparative study of two surprisingly different systems of preparing teachers for lower secondary level teaching of mathematics, namely those of Denmark and France. We first describe these systems differences succinctly. The main part of the paper reports on a qualitative study of how final year teacher students in the two cou...

Recent research in university mathematics education has moved beyond the traditional focus on the transition from secondary to tertiary education and students' understanding of introductory courses such as pre-calculus and calculus. There is growing interest in the challenges students face as they move into more advanced mathematics courses that re...

Understanding the concept of completeness for an ordered field is known to be difficult for many university mathematics students. We hypothesise that the variety of possible axioms of completeness for the set of real numbers is one of the sources of difficulties as is the lack of understanding of the raison d’être of these axioms. In this paper, we...

In this paper, we present an activity to introduce the idea of public-key cryptography and to make pre-service STEM teachers explore fundamental informatics and mathematical concepts and methods. We follow the Theory of Didactical Situations within the Didactical Engineering methodology (both widely used in mathematics education research) to design...

Dans cet article nous présentons une étude exploratoire conduite auprès d'étudiants de master se destinant au métier de professeur de mathématiques du second degré sur les conflits cognitifs susceptibles d'apparaître au sujet d'ensembles ordonnés de nombres qui sont à la fois denses et dénombrables. Nous présentons tout d'abord une revue de littéra...

The present project is a result of a collaboration of research groups of five European academic institutions specialised in science, mathematics and computer science education. The project aims to the design and development of interdisciplinary teaching modules for pre-service teacher training in advanced STEM topics, as well as in curricular inter...

In this paper, we support the claim that although mathematics aims at using non-ambiguous definitions and statements, the polysemy of natural language invites itself into the mathematics classroom, generating misunderstandings with students that are often not recognised by teachers. We provide two examples showing that the logical analysis is a too...

Parmi les recherches en didactique des mathématiques consacrées à l’enseignement supérieur, ondistingue deux types d’initiatives : celles qui favorisent le développement des résultats de rechercheet celles qui promeuvent les réflexions sur l’enseignement des mathématiques. Dans les deux cas, desdidacticien.ne.s des mathématiques et des chercheur.e....

Fresh university students face from the very beginning the need to study and develop by themselves more and more complex reasoning and proofs, which they had little opportunity to make in their secondary studies, including in the scientific tracks. In this article, we focus on the notion of implication. We first come back to the main known difficul...

We compare the pencil/paper proofs and formal proof of two traditional proofs in high school geometry. We highlight the fact that slightly different formulations or proofs can lead to difficulties in the formalization. We discuss the challenges and impact on both mathematical teaching and on the design of AI tools for mathematical education.

This contribution takes place in the context of a research project on the epistemological and didactical issues of interactions between mathematics and computer science. We make the hypothesis that, with the introduction of digital tools and computer science content in most curricula, significantly taking into account the epistemology of mathematic...

Various studies about the convergence of functions and sequences in first year of university

This chapter presents the French didactic tradition. It first describes the emergence and development of this tradition according to four key features (role of mathematics and mathematicians, role of theories, role of design of teaching and learning environments, and role of empirical research), and illustrates it through two case studies respectiv...

Résumé – Nous nous intéressons à l’enseignement et l’apprentissage de l’infini en classe de mathématiques en considérant les différences et les relations entre infini potentiel et infini actuel. Nous présentons les principaux éléments de notre étude philosophique, épistémologique et didactique, ainsi que trois situations visant à conduire un travai...

In this paper we study cognitive conflicts on the issue of number sets being dense, ordered and denumerable. We first provide historical-epistemological background related to these notions. Then we consider the cognitive conflicts under the lenses of concept image and concept definition, which we use to analyse empirical data collected in order to...

In this chapter, we propose a mathematical and epistemological study about two classical constructions of the real number system, by Dedekind (cuts ) and Cantor (Cauchy sequences ), and the associated proofs of its completeness . In addition, we present two contrasting constructions leaning on decimal expansions. Our analysis points out that Dedeki...

Cet ouvrage explore le rôle que peut jouer l’épistémologie en didactique des sciences expérimentales et des mathématiques. Il offre une vue d’ensemble sur cette question ainsi qu’une série d’études de cas portant sur des contenus d’enseignement variés et couvrant différents niveaux d’enseignement, depuis l’école primaire jusqu’à l’université. Les c...

Modelling is a complex and difficult process when studying phenomena from real-life situations. However, it is an important part of the work of mathematicians that needs to be addressed in the teaching of mathematics. We have studied two different types of problems, the first within mathematics and the second in the form of “realistic fictions”, th...

Dans cette communication, nous décrivons et argumentons l'adaptation de deux ingénieries didactiques développées au début des années quatre-vingt en France pour un public d'étudiants actuels de première année de licence ou de niveau équivalent en Sciences. La première (Robert 1983) vise à favoriser l'entrée des étudiants dans un point de vue concep...

Modeling is a complex and difficult process as long as the studied phenomena come from real situations. However it's an important part of the work of mathematicians that needs to be addressed in the teaching of mathematics. We have experienced two different types of problems, the first within mathematics and the second in form of " realistic fictio...

This paper discusses the use of the Theory of Didactic Situations (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of Didactical Engineering (DE), particularl...

Enhancing the teaching of mathematics in ways that support the development of students’ competence in argumentation and proof calls for increasing teachers’ awareness of the crucial role played by logical reasoning in proof. The article examines the relevance of, and interest in, teaching logic in order to foster competence with proof in the mathem...

Studying the relationships between argumentation and proof could help teachers and students deal with the tension between the process by which a student develops a proof and the requirements the teacher places on the final product. This tension results from the need for students to experience freedom and flexibility during an initial exploratory ph...

Math-Bridge is based on an ITS and is an e-learning tool for mathematics that can be used in various pedagogical scenarios: acquiring knew mathematical knowledge, revising known concepts, deepening knowledge or training competencies. We carried out an experiment testing Math-Bridge in a short revision session with 96 first-year students before thei...

In this paper, we discuss the idea of a gap between argumentation and proof, an idea we think to be prevailing in the educational institution. Our claim is that the only use of propositional calculus is insufficient to the analysis of the validation process in mathematics and could artificially reinforce that idea of a gap. This claim can be unders...

In mathematics education, it is often said that mathematical statements are necessarily either true or false. It is also well known that this idea presents a great deal of difficulty for many students. Many authors as well as researchers in psychology and mathematics education emphasize the difference between common sense and mathematical logic. In...

In this paper, we intend to provide theoretical arguments for the importance of taking account in quantification matters while analysing proofs in a didactic perspective, not only at tertiary level, where various research are still available, but also at secondary level and we argue that natural deduction in predicate calculus is a relevant logical...

It is widely attested that university students face considerable difficulties with reasoning in analysis, especially when dealing with statements involving two different quantifiers. We focus in this paper on a specific mistake which appears in proofs where one applies twice or more a statement of the kind “for all X, there exists Y such that R(X,...

Implication is at the very heart of mathematical reasoning. As many authors have shown, pupils and students experience serious difficulties in using it in a suitable manner. In this paper, we support the thesis that these difficulties are closely related with the complexity of this notion. In order to study this complexity,we refer to Tarski's sema...

In this paper, we intend to show, on various examples, the relevance of predicate calculus, and specially the model-theoretic approach introduced by Tarski, for a didactical analyse of mathematical reasoning and proofs. The main interest of this framework is to help us in elucidating the relationship between syntax, semantic and pragmatic as define...

In order to understand the policy issues at stake in teaching at University level in France, it is necessary to have an overview of the overall organization of the tertiary education sector. This is because universities in France do not represent the only education options in post-secondary education. We will start by a brief account of the organiz...

Friday the 30th I'm going to present today a numerical resource for maths teachers ; one more ? perhaps, but this one is the result of a work which rely on theoretical hypothesis and want to be a part of the teachers milieu in a situation of construction. As you can hear, I'm referring to the theory of didactical situations of [Brousseau, 2004] and...

Résumé : Dans cette communication, nous montrerons que les pratiques ordinaires des enseignants de mathématiques concernant les questions de logique peuvent être mises en relation avec les difficultés des étudiants, et nous donnerons quelques pistes permettant de travailler sur l'articulation entre logique et raisonnement mathématique dans une pers...

I will start from the main question addressed in the draft of the working group: "How do (should) the goals of school mathematics reflect the nature of disciplinary thinking and practice?" Then I will propose to move from this question to another one, which is: "How school mathematics reveal or enlighten some mathematical practises that have to be...

VII, sous la direction de Serge GRIGORIEFF, dans le cadre du séminaire de Théorie Descriptive des Ensembles. -AGREGATION de Mathématiques, option Probabilités et Statistiques, en Juillet 1986. -DIPLOME D'ETUDES APPROFONDIES de Didactique des Disciplines Scientifiques, obtenu en Octobre 1991, avec la mention BIEN à l'Université LYON I, sous la direc...

In this communication we intend to present a digital resource the aim of which is to give aid to teachers to use research problems in their classes; in a first part we are going to present the theoretical framework which was used by the team in the conception of the resource and the consequences on its model; we will present the results of a study...

Cette note de synthèse rédigée en vue de soutenir une habilitation à diriger des recherches (HDR) a comme but de proposer une relecture de l'ensemble de travaux de recherche que je conduis depuis une quinzaine d'année sur l'articulation entre la logique et le raisonnement mathématique à la lumière de la théorie élémentaire des modèles de Tarski, qu...