Le contrat et la coutume : deux registres des interactions didactiques

International audienceNous examinerons, en nous appuyant sur les résultats d'une recherche expérimentale conduite dans le cadre de la théorie des situations didactiques, les contraintes liées aux caractéristiques sociales des situations pour l'apprentissage de la preuve au collège, notamment en abordant le problème de la nature et des moyens de leur régulation

Sur la forme de la boule unit\'{e} de la norme stable unidimensionnelle

For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then the realizable forms as stable ball of Riemannan manifolds of dimension larger than three. For a Riemannian manifold $(M, g)$ fixed, we show that a broad class of polytopes can appear as stable ball of metrics in the conformal class of $g$. We use for that a polyhedral technique.Comment: 13 pages, in Frenc

L'apprentissage de l'itération dans deux environnements informatiques

Cet article présente une étude comparée de l'apprentissage de l'itération dans deux environnements informatiques différents : Pascal et Multiplan. Cette étude a été réalisée avec des élèves de troisième (fin de la scolarité obligatoire) d'un collège français. Les AA. ont pu dégager des analogies importantes entre les deux environnements à propos de l'itération. Mais l'hypothèse consistant à prévoir un transfert des connaissances d'un logiciel à l'autre ne s'est pas trouvée confirmée, ce qui confirme la complexité cognitive du concept d'itération, déjà mise en évidence dans d'autres recherche

Design approaches in technology enhanced learning

Design is a critical to the successful development of any interactive learning environment (ILE). Moreover, in technology enhanced learning (TEL), the design process requires input from many diverse areas of expertise. As such, anyone undertaking tool development is required to directly address the design challenge from multiple perspectives. We provide a motivation and rationale for design approaches for learning technologies that draws upon Simon's seminal proposition of Design Science (Simon, 1969). We then review the application of Design Experiments (Brown, 1992) and Design Patterns (Alexander et al., 1977) and argue that a patterns approach has the potential to address many of the critical challenges faced by learning technologists

Aubry sets vs Mather sets in two degrees of freedom

We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in that case the Aubry set and the Mather set almost always coincide.Comment: Revised and expanded version. New proof of Lemma 2.3 (formerly Lemma 14

CROO: A universal infrastructure and protocol to detect identity fraud

Identity fraud (IDF) may be defined as unauthorized exploitation of credential information through the use of false identity. We propose CROO, a universal (i.e. generic) infrastructure and protocol to either prevent IDF (by detecting attempts thereof), or limit its consequences (by identifying cases of previously undetected IDF). CROO is a capture resilient one-time password scheme, whereby each user must carry a personal trusted device used to generate one-time passwords (OTPs) verified by online trusted parties. Multiple trusted parties may be used for increased scalability. OTPs can be used regardless of a transaction’s purpose (e.g. user authentication or financial payment), associated credentials, and online or on-site nature; this makes CROO a universal scheme. OTPs are not sent in cleartext; they are used as keys to compute MACs of hashed transaction information, in a manner allowing OTP-verifying parties to confirm that given user credentials (i.e. OTP-keyed MACs) correspond to claimed hashed transaction details. Hashing transaction details increases user privacy. Each OTP is generated from a PIN-encrypted non-verifiable key; this makes users’ devices resilient to off-line PIN-guessing attacks. CROO’s credentials can be formatted as existing user credentials (e.g. credit cards or driver’s licenses)

The French Didactic Tradition in Mathematics

This chapter presents the French didactic tradition. It first describes theemergence and development of this tradition according to four key features (role ofmathematics and mathematicians, role of theories, role of design of teaching andlearning environments, and role of empirical research), and illustrates it through two case studies respectively devoted to research carried out within this traditionon algebra and on line symmetry-reflection. It then questions the influence of thistradition through the contributions of four researchers from Germany, Italy, Mexicoand Tunisia, before ending with a short epilogue

Changing classroom culture, curricula, and instruction for proof and proving: how amenable to scaling up, practicable for curricular integration, and capable of producing long-lasting effects are current interventions?

This paper is a commentary on the classroom interventions on the teaching and learning of proof reported in the seven empirical papers in this special issue. The seven papers show potential to enhance student learning in an area of mathematics that is not only notoriously difficult for students to learn and for teachers to teach, but also critically important to knowing and doing mathematics. Although the seven papers, and the intervention studies they report, vary in many ways—student population, content domain, goals and duration of the intervention, and theoretical perspectives, to name a few—they all provide valuable insight into ways in which classroom experiences might be designed to positively influence students’ learning to prove. In our commentary, we highlight the contributions and promise of the interventions in terms of whether and how they present capacity to change the classroom culture, the curriculum, or instruction. In doing so, we distinguish between works that aim to enhance students’ preparedness for, and competence in, proof and proving and works that explicitly foster appreciation for the need and importance of proof and proving. Finally, we also discuss briefly the interventions along three dimensions: how amenable to scaling up, how practicable for curricular integration, and how capable of producing long-lasting effects these interventions are

Bridging knowing and proving in mathematics An essay from a didactical perspective

Text of a talk at the conference "Explanation and Proof in Mathematics: Philosophical and Educational Perspective" held in Essen in November 2006International audienceThe learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory