On dialectic and dynamic links between the Mathematical Working Space model and practice in the teaching and learning of mathematics

Alain Kuzniak (1) , Assia Nechache (2)

1 LDAR (EA_4434) - Laboratoire de Didactique André Revuz
2 UPD7 - Université Paris Diderot - Paris 7

In this communication, we address the specific relationships between the Mathematical Working Space model (MWS model) and practice in the teaching and learning of mathematics. The strong and positive interactions existing between these two aspects are illustrated with two examples from geometry and probability teaching. They show how some theoretical constructs as MWS diagram can enlighten practice and, conversely, how studies on practice nourish the model with new tools such as "comics", "complete mathematical work" or "emblematic tasks". A decade ago, the Mathematical Working Space (MWS) model has been introduced as a theoretical and methodological framework dedicated to identify and shape the mathematical work in schooling. Developed by researchers working collaboratively in various countries with, sometimes, very different educational approaches in Europe (France, Spain, Cyprus), Latin America (Chile, Mexico…) and North America (Canada), the model is deeply rooted in the teaching of mathematics in real classrooms. This communication aims at showing dynamics and dialectics between the MWS model and relevant questions to education practice. After a short presentation of the MWS model, we show first how it can be used to deal with the question of planning series of tasks in the teaching and learning of geometry. That leads us to introduce two new constructs: the methodological tool of "comics" used to describe the evolution and circulation of the mathematical work and the more theoretical idea of "complete mathematical work" which allows qualifying the final nature of this circulation within the diagram. Then, these two new tools are used for the study of the teaching of probability and statistics. In France, this teaching is relatively new and it is now initiated on modeling tasks with use of technological tools. This leads us to check the relevance of the above constructs and identify two kinds of incompleteness of mathematical work and, in addition, to draw out certain specific tasks, named "emblematic tasks". Designed on these former results, our present research aims at tracking transformations made by teachers when they adapt these "emblematic tasks" to their classrooms. A short insight in the MWS model Extending the research work developed by Houdement and Kuzniak (2006) in didactics of geometry, the MWS model emerged during the last decade. The model, especially in geometry, had already been presented during former CERME meetings (Kuzniak and Nechache, 2015). A recent issue of ZDM-Mathematics Education (48-6, 2016) is devoted to this model and we refer the reader to this issue for further details and discussions about the MWS model. Some elements of the introduction (Kuzniak, Tanguay and Elia, 2016) to this issue are used to present the framework.

Mots clés: Mathematical work; MWS model; emblematic task; mathematics teaching

Domaines: Mathématiques [math]; Sciences de l'Homme et Société

Alain Kuzniak, Assia Nechache. On dialectic and dynamic links between the Mathematical Working Space model and practice in the teaching and learning of mathematics. CERME 10, Feb 2017, Dublin, Ireland. ⟨hal-01948867⟩