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Synthesis and perspectives in didactics of mathematics
Special issue coordinated by Cécile Ouvrier-Buffet, Fabrice Vandebrouck and Laurent Vivier
1. André Rouchier : d’un siècle à l’autre, le projet demeure
Hommage à André Rouchier, professeur en formation des maitres à l’IUFM, spécialiste de didactique des mathématiques et fondateur de la revue Recherches en didactique des mathématiques (RDM).
2. Situations for the learning of proof in mathematics: A review of research and open questions
Research on the epistemic, logical, and discursive complexity of proof learning has generated a wealth of literature over the past two decades. The results contribute to a more precise understanding of the difficulties encountered by students and those encountered by teachers. They support the design of situations, especially validation situations in the sense of the Theory of Didactical Situations (TDS—Brousseau, 1998), in which proof is a problem-solving tool. However, there remains the difficulty of grasping proof as an object, recognizing its mathematical specificities and institutionalizing it as such. This is the topic of this text. This text complements those given at the Séminaire national de didactique des mathématiques in 2017 and at CORFEM in 2019. The common theme of these three presentations was the learning and the teaching of proof before the introduction of mathematical proof as the canonical form of proof in mathematics. After an introduction recalling the institutional and scientific context, the first part (sections 2 to 4) is devoted to a review of the state of research, taking into account the reports of outstanding work from different approaches. In the second part (section 5), proposals are made to provide a basis for future research. The conclusion considers the issues raised by the need for situation-specific engineering to encourage and support the genesis and recognition of proof norms in the mathematics classroom, prior to the explicit teaching […]
3. La contribution des environnements technologiques à l’enseignement de la preuve
This contribution discusses the use of a dynamic geometry environment to promote students’ introduction to mathematical proof. Within the framework of semiotic mediation theory, I explore, on the one hand, the link between available computer tools and the personal meanings emerging from their use in classroom activities and, on the other hand, the mathematical notions that are the subject of teaching. The discussion uses three interrelated perspectives – epistemological, cognitive and didactic – to elaborate the outcomes of a number of long-term teaching experiences in secondary classrooms. Illustrative examples are presented, drawn from research studies conducted in previous years and still ongoing.
4. The teaching of mathematical modelling and the mathematical work: A dynamic that is problematic
The first part of the course considers the reasons for the interest in mathematical modelling and the resulting approach to teaching mathematics. The study focuses on approaches based on the modelling cycle associated with Northern European theories on modelling in education in relation to competency-based teaching. Additionally, the text reflects on our experience of teaching modelling as part of the master’s degree in didactics at the University of Paris-Diderot since 2000. By emphasising the role of mathematisation and models, we question and discuss the reality of the mathematical work developed by teachers and students in this manner. In the second part of the course, we deal with the theme of horizontal mathematisation and the chaining of models introduced in the context of Realistic Mathematics Education. Drawing on the theory of Mathematical Work Spaces (MWS), we then show how we consider the link between modelling activity and the training of mathematical work. Recent research in this area focused on mathematisation, the interplay between alternative models, and the connection between MWSs. The dual cognitive and epistemological approach to modelling activities is adopted. This aspect is illustrated, explored, and discussed in the workshop associated with the course.
5. Mathematical modelling as a field of research: advances in the ecological analysis
This paper focuses on the recent evolution of the field of mathematical modelling research and how different research questions have been formulated and addressed by international theoretical frameworks. This paper focuses on the “epistemological”, “economic” and “ecological” dimensions of these different theoretical approaches to mathematical modelling. We will see that the ecological issue is still largely absent from many approaches. This will allow us, in a second step, to show some tools of the anthropological theory of the didactic (ATD) such as the study and research paths (SRP) that we have been proposed to facilitate the design, implementation and analysis of modelling practices in different school levels and in teacher education.
6. Digital resources for mathematics teaching and learning
For several decades, research in mathematics education has investigated the place and the role of digital technologies in the teaching and learning of mathematics. The development of the Internet which has significantly increased the availability and accessibility of resources has given a new direction to research with the emergence of a "resource" approach to mathematics education. In this lecture, we first draw up a panorama of research on digital technologies, both “old” (dynamic geometry software, computer algebra software, graphic and symbolic calculators, etc.) and more recent (mobile, tactile technology, augmented and virtual reality). We show how the development of this research is accompanied by the emergence of new theoretical frameworks and concepts to address the specific issues raised by the use of digital technology. The second part of the lecture is devoted to more recent research on digital resources, their different conceptualizations and issues concerning their design, dissemination, evaluation, and appropriation.
7. Mathematics teachers in the time of digital technologies: a theoretical framework adapting the double approach to study their activity
This text deals with the mathematics teacher’s activity in digital technologies-based sessions. It is composed of two parts. The first part outlines recent studies and theoretical models that have been developed to study the teaching dimension in learning and teaching within environments integrating these technologies. The second presents a theoretical framework for the study of the teacher's activity in ordinary classroom situations: the DAaT (Double Approach applied to Technology). This framework is derived from the Double Didactic and Ergonomic Approach (Robert and Rogalski, 2002) while introducing adaptations and adding complementary constructs in order to better consider the specificity of the studied environments. The theoretical concepts thus developed are presented and discussed. The conclusion returns to the usefulness of the theoretical and methodological tools provided by the research in this field for the enrichment of teacher practices or for teacher education