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The Mathematical Working space (MWS) was developed to better understand the didactic
issues around mathematical work in a school environment by Kuzniak et al (2016). The MWS has two levels: one is a cognitive nature in relation to the student and another is an
epistemological nature in relation to the mathematical content studied. The MWS diagram was transformed by adding an epistemological plane corresponding to the rationality framework of physics (Moutet 2018a, 2019) or of chemistry (Moutet, 2018b).
A first teaching sequence developed by Moutet (2018a, 2019) is destined for students in the final year of secondary school (grade 12) in France, on the topic of special relativity following the work of de Hosson (2010). The Minkowski diagram is used with the GeoGebra dynamic geometry software. The work on the chronological inversion of events in two reference frameworks in a relativistic context is done with students with problem solving. Another problem, including an experimental session in chemistry, is also studied (Moutet, 2018b). The construction of graphs allows both to work on the notion of stoichiometry with GeoGebra and to deduce the mass concentration of a pharmaceutical product. The methodological framework used is didactic engineering. Data collections can be videos, audio recordings or GeoGebra files. We used the modelling cycle proposed by Blum and Leiss (2005) to position the teaching sequences studied. We carried out a preliminary study of a physics sequence by studying the transition from the real model to the real results and a chemistry sequence covering the complete modelling cycle from the real situation to the real results.
Two research questions guided this work: 1) How does the extended MWS framework allow the analysis of the sets of rationality frameworks between mathematics and physics or chemistry, during a sequence with students in the final year of secondary school via a geometric approach? 2) To what extent does the analysis of the use of dynamic geometry software by the extended MWS framework, show that it promotes a conceptualisation in students?
It's possible to propose new strategies when using of GeoGebra with another register of
semiotic representation when working with problems solving. The extended MWS model
makes it possible to build detailed analyses of student's work in physics or chemistry.----------------------------------------------------------------------------------------------------------------------
Blum, W., Leiss, D. (2005). « Filling up » - the problem of independence-preserving teacher interventions in lessons with demanding modelling tasks. In M. Bosch (Ed.) Proceedings for the CERME 4, Spain. 1623–1633.
de Hosson, C., Kermen, I., & Parizot, E. (2010). Exploring students’ understanding of reference frames and time in Galilean and special relativity. European Journal of Physics, 31, 1527–1538.
Kuzniak, A., Tanguay, D., & Elia, I. (2016). Mathematical Working Spaces in schooling: an introduction. ZDM mathematics Education, 48, 721–737.
Moutet, L. (2018a). Analysis of a teaching sequence of special relativity: the contribution of the extended MWS model. Annales de didactique et de sciences cognitive, 23, 107–136.
Moutet, L. (2018b). The extended theoretical framework of Mathematical Working Space: potentialities in physics and chemistry. Sixth Symposium of Mathematical Work – ETM6, 13-18 December 2018, Valparaiso, Chili.
Moutet, L. (2019). The extended theoretical framework of Mathematical Working Space (extended MWS): potentialities in physics. CERME11, 6-10 February 2019, Utrecht, the Netherlands.

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