Objet du séminaire

La transition entre l’enseignement secondaire et l’enseignement universitaire (au sens de l’enseignement post-bac) fait l’objet depuis longtemps de travaux de recherche dans les différents domaines mathématiques (Gueudet, 2008). Plus récemment, des travaux ont été conduits sur ce que l’on appelle dans la littérature la « double discontinuité de Klein » (Winsløw & Grønbæk, 2013). Ces  recherches s’inscrivent dans le point de vue développé par Felix Klein qui soulignait que les mathématiques universitaires et les mathématiques scolaires semblent n’avoir que peu de connexions (Kilpatrick, 2019).

Cette question est toujours d’actualité (e.g. Winsløw & Grønbæk, 2013, Gueudet & al., 2016), d’autant que de nombreux futurs professeurs de mathématiques partagent l’idée selon laquelle les mathématiques universitaires ne répondent pas au besoin de leur future profession (Gueudet & al., 2016).  Il y a donc un double enjeu ; d’une part, mettre en évidence pour les étudiants les liens entre les mathématiques universitaires et les mathématiques scolaires ; d’autre part, permettre aux futurs professeurs d’accéder à des outils efficaces pour leur travail didactique.

L’objectif du séminaire est de présenter des travaux récents sur ces questions, notamment ceux conduits par les jeunes chercheurs, incluant les doctorants, et centrés sur des domaines mathématiques bien identifiés (Analyse, Géométrie, Arithmétique, Probabilités, Mathématiques discrètes, logique, Topologie etc..). Des ouvertures sur la formation des enseignants du supérieur pourront être considérées.

Références

Gueudet, G. (2008). Investigating the secondary-tertiary transition. Educational Studies in Mathematics , 67 (3), 237-254.

Kilpatrick, J. (2019). A double Discontinuity and a Triple Approach : Felix Klein’s perspective on Mathematics Teacher Education. In H.G. Weigand., W. McCallum, M. Menghini, M. Neubrad, Schubring, G. (eds) The Legacy of Felix Klein. ICME 13 Monographs. Springer, 215-225.

Gueudet, G., Bosch, M., diSessa, A., Kwon, O.N. & Verschaffel, L.  (2016). Transitions in Mathematics Education. Springer, 2016, ICME 13 Topical survey, Gabriele Kaiser.

Winsløw, C. & Grønbæk, N. (2014). Klein’s double discontinuity revisited: contemporary challenges for universities preparing teachers to teach calculus. Recherches en Didactique des Mathématiques, 34/1, 59-86.

Responsables du séminaire

Le séminaire est conjointement organisé par le Laboratoire de Didactique André Revuz (LDAR, EA 44E4) et l'Institut Montpelliérain Alexander Grothendieck (IMAG, UMR 5149).

·         Responsable du séminaire : Nicolas Genier-Boley, Université de Rouen Normandie, France.

·         Co-responsable : Viviane Durand-Guerrier, Université de Montpellier, France.

Organisation pratique du séminaire

Le séminaire se déroulera en ligne entre septembre 2021 et décembre 2022 à raison d'une séance par mois. Sauf exception, cette séance se tiendra le second lundi de chaque mois de 14h à 16h, heure de Paris (UTC+2 entre mars et octobre, UTC+1 entre octobre et mars).

Chaque séance sera constituée d'une heure de présentation par l'intervenant·e, suivie d'une heure de discussion. La langue du séminaire sera l’anglais.

Contact

Les personnes intéressées par ce séminaire sont invitées à écrire à Nicolas Grenier-Boley (nicolas.grenier-boley@univ-rouen.fr), afin d'être inscrites sur la liste de diffusion du séminaire et de recevoir le lien de visio-conférence à chaque séance.

Séances du séminaire

lundi 13 septembre 2021, 14h-16h (UTC+2)

Carl Winsløw (Université de Copenhague) : "The real numbers as a key challenge in upper secondary teacher education"

Abstract: Upper secondary mathematics teacher education was the context in which Felix Klein formulated his famous second discontinuity problem (beginning with his 1872 inaugural lecture at Erlangen). This problem exists today in much the same form, and consists in the gap between the mathematical education delivered in universities, and the task of teaching in (high) schools. The problem has probably even widened today. Here, we shall focus on one central mathematical object - the real number system - and on what can be done at University to develop students’ relationship to this object so that it is both grounded in present-day mathematics, and adapted to the needs of teaching in present-day high schools. In particular we will attend to the role of computer tools and programming in both contexts.

lundi 11 octobre 2021, 14h-16h (UTC+2)

Nicolas Grenier-Boley (Université de Rouen Normandie): "Choosing relevant content for Klein's second transition? The case of bilinear or multilinear algebra"

Abstract: The design and implementation of dedicated research to Klein's second transition necessarily brings the question of the choice of relevant mathematical content on which to focus. In this presentation, we will propose to explore this problem within the framework of the French educational system with regard to the case of bilinear or multilinear algebra. The presentation will be based on examples that will highlight some challenges for such a program, whether for researchers, future teachers or students.

lundi 8 novembre 2021, 14h-16h (UTC+1)

Max Hoffmann (Université de Paderborn) : "Geometry for Student Teachers: A holistic course concept as a contribution to overcoming the second discontinuity"

Abstract: At Paderborn University, student teachers (for the German "Gymnasium") attend some of their mathematics courses not together with Bachelor of Science students. This provides the opportunity to design and research course concepts which implement profession orientation in a holistic way. One of these courses is the "Geometry for Student Teachers". First, I will present theoretical considerations on the functions of academic mathematics competencies for professional teacher action and derive design principles for the implementation of profession orientation in mathematics courses for student teachers. Based on this, I will introduce the design of the course. Finally, I will describe in detail the development of selected interface learning opportunities, discuss real student work and present results of the accompanying research.

lundi 13 décembre 2021, 14h-16h (UTC+1)

Katalin Gosztonyi (Université Eötvös Loránd) : "Discrete mathematics in the secondary curriculum and in teacher education in Hungary : issues and perspectives for the second transition"

Abstract: In Hungary, discrete mathematics has important traditions in research as well as in mathematics education. In mathematics education, several domains of discrete mathematics appear in, the curricula, and especially combinatorics is systematically present from primary school to the baccalaureate. By consequence, discrete mathematics constitutes also an autonomous domain in teacher education.

In the first part of the presentation, I will briefly present the place and role of discrete mathematics in school curricula. In the second part, I will talk about discrete mathematics courses in teacher education, explaining aspects of continuity and rupture with secondary school education. In the third part, I will discuss questions related to our teacher education course on the teaching of discrete mathematics, our current developments and endeavours to prepare teachers for the second transition.

lundi 10 janvier 2022, 14h-16h (UTC+1)

Laura Branchetti (Université de Milan) : "The interplay between Mathematics and Physics in secondary teacher-students education: activities and research developed within an interdisciplinary research team"

Abstract: Since 2014, in collaboration with a small group of researchers in Physics and Mathematics education and teachers-researchers (led by Prof. Olivia Levrini), I have been investigating the issues arisen by interdisciplinarity in secondary teaching, with particular attention to high school. The main motivations were the well-known difficulties emerged in research in physics education due to mathematics and the feeling that the research in didactics of these two disciplines had much more in common than it was actually represented in research and institutions in Italy. In the Italian context these connections are even more important to explore since teachers with a background in mathematics or physics can teach both disciplines at secondary school and the mathematical curriculum includes explicitly interactions with physics. Discussing and exploring the existing literature we realized that the main limitation of the existing studies was the “deformation” of the main aims and values of the other discipline once the issue of interdisciplinarity was formulated in a disciplinary way (Physics in Mathematics or Mathematics in Physics). This phenomenon affects also the teacher-students in the Klein's second transition, since they are taught in institutional contexts where epistemologies and identities of discipline are conceived as clearly separated in teaching (in particular at the university). Indeed, often the interdisciplinarity that characterizes the discipline at the research level is not presented to Bachelor or Master students. Teacher-students education is thus a key action to reach innovation. We designed activities for teacher-students impemlented in courses attended by students with a background in both disciplines and carried out research starting from our implementations. In 2019 we developed an international research project with the Universities of Montpellier, Crete and Barcelona, that was funded by the Erasmus project, enlarging the perspective to an international context.
In this contribution I will resume the main issues, the theoretical framework and some examples of design, implementation and analysis of modules for teacher-students, facing in particular the issue of Klein's second transition.

lundi 7 mars 2022, 14h-16h (UTC+1)

Thomas Bauer (Université de Marburg) & Eva Müller-Hill (Université de Rostock) : "Approaches in pre-service teacher education to counter Klein's double discontinuity"

Abstract: Felix Klein's observation of double discontinuity poses several challenges at several stages of pre-service teacher education: 1) At the transition from school to university, how can students be provided with the experience that school mathematics and university mathematics are interrelated and useful for each other? 2) In mathematics content courses, how can students experience mathematics in such a way that they are both willing and able to make good use of academic knowledge later in their job? 3) And finally, how can students learn to make full use of their knowledge (math content and math education) when planning classroom implementations?

In the talk we present approaches to these challenges from our teaching practice: With respect to the first discontinuity, we show how the concept of "interface tasks" in real analysis can help to strengthen the relationship between school and university mathematics. We present design principles and empirical results from the use of these tasks. Concerning the second discontinuity, we present math education course designs that relate proof and argumentation as core mathematical activities to implementations in the classroom. We use an activity theoretic framework to uncover and explain phenomena observed in students' classroom implementations.

lundi 28 mars 2022, 14h-16h (UTC+2)

Avenilde Romo (Center for Research and Advanced Studies of the National Polytechnic Institute, Mexique) : "Klein's second discontinuity as seen through the lens of a Mexican program of professional development for mathematics teachers in service"

Abstract : There is no specific training for future high school and university mathematics teachers in Mexico. Most mathematics teachers at this level are mathematicians, physicists, engineers or professionals with a five-year undergraduate degree in an area with a specific mathematical, scientific orientation who have a vocation and interest in teaching. Given the need to have specialised teachers, different professional training programmes have emerged to address the didactic needs of these in-service teachers. In particular, in 2000, was created the programme of professional development for mathematics teachers in service at the Instituto Politécnico Nacional, where I have worked for ten years. This seminar session will reflect, partially, how this programme addresses Klein's second discontinuity. To do so, I will consider some examples of the courses offered in a master's program and also the master's thesis developed by the Mexican and Latin American teachers of mathematics in service.

lundi 13 juin 2022, 14h-16h (UTC+2)

Virginie Deloustal-Jorrand (Université Claude Bernard Lyon 1) & Zoé Mesnil (Université Paris Cité) : "Spécificités des connaissances en logique et conséquences sur la double discontinuité de Klein"

Résumé. La logique mathématique a une place particulière et transversale au sein des mathématiques dans la mesure où elle permet de décrire, de contrôler et de valider ce que l'on fait dans l’activité mathématique. Son enseignement, au sein des mathématiques, semble donc pertinent pour aider à comprendre et utiliser le langage mathématique, nécessaire à l’activité mathématique, et pour produire des preuves.

En France comme dans d'autres pays, dans l'enseignement supérieur, différents cours sont proposés pour enseigner la preuve, en faisant plus ou moins de référence à la logique mathématique. Nous présenterons dans une première partie de l'exposé quelques résultats des travaux du thème « Logique » du groupe de recherche DEMIPS (Didactique et Épistémologie des Mathématiques, lien avec l'Informatique et la Physique dans le Supérieur, https://demips.math.cnrs.fr/) : à partir d'entretiens conduits avec des enseignant·e·s de tels cours, nous montrerons les différents choix effectués et les différentes épistémologies relatives à la preuve qui les sous-tendent.

Au lycée français, la logique a fait un timide retour dans les programmes depuis 2009. Comment les étudiant·e·s futur·e·s enseignant·e·s peuvent-ils et peuvent-elles alors s'appuyer sur ces connaissances et les transposer afin d'enseigner à leurs élèves celles qui leur sont nécessaires à leur niveau ? Nous essayerons de répondre à cette question de la seconde discontinuité de Klein en présentant, d’une part, certains résultats de didactique sur les difficultés des étudiant·e·s et, d’autre part, des propositions de situations de formation.

lundi 11 juillet 2022, 11h-13h (UTC+2)

Oh Nam Kwon (Université nationale de Séoul) : "Didactic Transposition in College and School Mathematics: Novice Teachers’ Understanding of Factorization"

Résumé. The factorization of polynomials is a basic concept for solving problems in various domains of school mathematics. It is relevant to the greatest common divisors and least common multiples of polynomials, and is needed to solve equations and inequalities. However, although the concept of factorization comprises important content, it is recognized only as procedural knowledge required to solve equations. Thus, students memorize the formula as the inverse process of expansion without necessarily understanding its importance. From an advanced standpoint, polynomial factorization in school mathematics could be related to the unique factorization domain (UFD). Thus, I examine the understanding of polynomial factorization of novice teachers with relatively vivid advanced mathematical experience, in order to compare the relationship between school and college mathematics, and to investigate what influence these teachers’ advanced mathematical experience has on their teaching of polynomial factorization.

In the first part of the presentation, I will present the curricula for mathematics teacher education and teacher employment tests focusing on teacher knowledge in Korean context. In the second part of the presentation, I will examine how the content of school mathematics is described from an advanced standpoint and how teachers utilize their college mathematics knowledge in the context of school mathematics. Focusing on the UFD in college mathematics and polynomial factorization in school mathematics, I will analyze school and algebra textbooks, and explored teacher knowledge of the UFD and how teachers’ factorization concepts occur in the teaching context in particular the issue of Klein's second transition.

lundi 12 septembre 2022, 14h-16h (UTC+2)

Nick Wasserman (Université Columbia): "Upgrading Learning for Teachers in Real Analysis: A Look at Diversifying Mathematical Connections to Counter Klein's Second Discontinuity"

Abstract: Secondary mathematics teachers in the United States often have to take a course in Real Analysis as part of their degree program. Many see little value in these advanced courses. This talk explores ideas and results from one project, Upgrading Learning for Teachers in Real Analysis (ULTRA), in which modules for a real analysis course were designed to make more explicit the connection to school mathematics teaching. The talk will explore the overarching instructional model for module design and, in particular, discuss how diversifying the kinds of mathematical connections between school and university mathematics was used to help counter Klein's second discontinuity. 

lundi 10 octobre 2022, 14h-16h (UTC+2)

Lisa Hefendehl-Hebeker (Université de Duisburg-Essen) : "Epistemological demands of the second discontinuity"

Résumé : Mathematics is an organ of knowledge and an infinit refinement of language. It rises out of ordinary language and imagination like a plant out of the soil, and its roots are numbers and simple spatial ideas" (Kähler, 1955). In comparison with their experiences at school, university students of mathematics must cope with a faster pace, a broader scope of material and a higher degree of abstraction and formalization. In addition, they must attain professional habits with accompanying attitudes and norms. When they return to the school as teachers, they must go through a reverse process and go back to the roots. But they should do this in a professional way that is suitable for further developing the naïve view of the learners. The presentation will address the epistemological requirements in this regard.

lundi 7 novembre 2022, 14h-16h (UTC+1)

Gaëtan Planchon (Université de Montpellier) : "Une ingénierie didactique sur l'intégrale pour aborder la seconde discontinuité de Klein"

Résumé :

Avant d'aborder le cycle de formation des enseignants de mathématiques, les étudiants, futurs enseignants de mathématiques, sont confrontés à différentes théories de l’intégration (Riemann, Lebesgue, théorie de la mesure). Ce faisant, ils éprouvent souvent des difficultés à percevoir les liens entre ces théories et l’intégrale enseignée au lycée, ce qui illustre le problème de la seconde discontinuité de Klein.

Dans cet exposé, nous présenterons quelques éléments d’une ingénierie expérimentée à Montpellier qui vise à atténuer cette discontinuité en rendant visibles des liens entre la théorie de la mesure et la notion intuitive d’aire qui fonde l’intégrale au lycée. La conception de cette ingénierie s'appuie sur la modélisation de la seconde discontinuité de Klein proposée par Winslow dans le cadre de la Théorie Anthropologique du Didactique et la description des connaissances en termes de modèles praxéologiques dominants, dans les différentes institutions (lycée et université). Nous utilisons également les outils de TAD relatifs au paradigme du questionnement du monde pour effectuer des analyses (a priori et a posteriori) des tâches proposées aux étudiants. Ceci nous conduit à formaliser le processus de mise en relation des connaissances du lycée et de l'université en termes de développement de nouvelles praxéologies, appelées praxéologies de Klein, en appui sur les exemples présents dans notre dispositif.

lundi 21 novembre 2022, 14h-16h (UTC+1)

Rina Zazkis (Université Simon Fraser) : "Accepting and enhancing mathematical knowledge via scripting tasks"

Résumé : How can a teacher get a “scan” of a group’s knowledge and understanding of a mathematical topic in order to plan for, or adjust, subsequent instruction? To address this question, I introduce the notion of “lesson play” and its evolution to “script writing” or “scripting”. I short, scripting involves teachers in writing a dialogue between a teacher and students based on a particular mathematical issue. The teacher-character in such a dialogue demonstrates how the script-writer envisions a particular mathematical situation. I will exemplify how the instruction that follows students scripts can be adjusted to connect school teaching to ideas of undergraduate mathematics.  

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Nicolas Grenier-Boley
Laboratoire de Didactique André Revuz (LDAR, EA 4434)
Université de Rouen Normandie
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