THE MATHEMATISATION PROCESS IN THE DEVELOPMENT OF AN EXTRA-MATHEMATICAL PROBLEM FOR PRIMARY SCHOOLS
Résumé
The aim of our research question is to study how, in developing an extra-mathematical situation, the dialectical relationships between horizontal mathematisation and vertical mathematisation are considered by the teachers and trainers in the Léa MaPcv working group (Mathematisation of Concrete Problems in Video Format, Mathématisation de problèmes concrets en vidéos (MaPcv) — LéA (ens-lyon.fr)) . We hypothesise H1 that putting them in the position of designers of a problem rooted in reality, presented in video format will lead them to go back-and-forth between these two aspects of mathematisation. In particular, we assume that the group will be torn between the need to provide an authentic context and the need to make mathematical work accessible to primary school pupils. This problem-creation activity also leads us to consider elements inherent to problem posing, which has received increasing attention over the last decade as an important mathematical process in its own right, and also as an approach to promoting other skills (Cai and Leikin, 2020). Broadly speaking, the term ‘problem posing’ in mathematics education refers to two processes: the development of new problems and the reformulation of existing problems. When developing the problem, the group decided not to explicitly formulate a question, but to ensure that the elements provided by the video strongly suggested the question they wanted the students to address mathematically. The stage of developing a problem already involves activities that are necessary for solving the problem at a later stage (Baumanns & Rott, 2024). Consequently, with regard to the present research question, it seems relevant to us, in both the development and solution of the modelling problem designed by the Léa MaPcV working group, to analyse the activities inherent in the mathematisation process. In other words, our aim is to highlight the choices made by the design team during the development of the problem in video format by comparing them with their expectations regarding the activity to be conducted by the pupils. We hypothesise H2 that the pupils will not necessarily engage in the mathematisation process intended by the designers. Indeed, previous studies (Yvain-Prébiski, 2023) have shown that during the horizontal mathematisation process, the choice of relevant measures and contextual elements is made in relation to the question to be studied. In the proposed video, however, the pupils will have to choose the question to be addressed themselves, placing them in a ‘problem posing’ type of activity. The students' modelling activity will therefore be intrinsically linked to the question they will ask themselves. Our hypothesis H2 is reinforced by the recent work of Hartmann & al. (2022) who demonstrate that posing a problem linked to modelling (i.e. asking a question in our study) involves an understanding activity. The designers of a problem have their own understanding of the situation created. We link this understanding activity to the work involved in horizontal mathematisation. Indeed, the choice of fragments of reality which the designers decide to include in their problem is driven by the question they plan to study. The ensuing problem exploration phase undertaken by the solver will be based on this understanding.