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The aim is to show how the extended Mathematical Working Space (extended MWS) theoretical framework makes it possible to analyse the tasks implemented during a few stages of a modelling cycle in physics. The study begins with a special relativity teaching sequence using a diagrammatic approach in “Terminale S” in France (grade 12). The analysis using the extended MWS theoretical framework allows to highlight the learning advantages of this diagrammatic approach during a complete didactic engineering. This work was proposed for TWG6 to CERME11.
Keywords: Mathematics, physics, extended MWS, modelling.

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... Each genesis can be viewed as the activation of a pole of the epistemological plane by an individual or a collective subject for the corresponding process in the cognitive plane and is analyzed with regard to the subject's cognition. Moutet (2019) considered an extended working space with a single cognitive plane, and two distinct epistemological planes, one in physics and the other in mathematics. Distinguishing two epistemological planes is consistent with the fact that in physics and mathematics, the three poles of the epistemological planes are specific to each field. ...

... Each genesis can be viewed as the activation of a pole of the epistemological plane by an individual or a collective subject for the corresponding process in the cognitive plane and is analyzed with regard to the subject's cognition. Moutet (2019) considered an extended working space with a single cognitive plane, and two distinct epistemological planes, one in physics and the other in mathematics. Distinguishing two epistemological planes is consistent with the fact that in physics and mathematics, the three poles of the epistemological planes are specific to each field. ...

In this chapter, the guiding line is the mathematical work that students develop in modeling tasks. We report first on MWS studies adopting the idea of a modeling cycle. In these studies, the MWS framework gives insight into the processes involved in students’ work, and it also sheds light upon the complex relationship between reality and mathematics and the multifaceted relationship between the process of problem solving and the underlying educational goals. We question the consistency of a sharp separation between mathematics and reality, and assuming that mathematical modeling is what mathematicians do when they work on models, we look at this work outside of education. Rather than steps of translation between reality and mathematics, we have to think of modeling as a coupling of reality and mathematics that should allow students to develop insight into, and understanding of, both mathematics and reality. We also look to epistemological studies that distinguish between modeling and mathematization, and characterize modeling by (1) plurality of models (2) operativity (3) subjective and social interpretation. The plurality of models for a given reality has been exploited in research studies to design tasks that put at stake transitions or coordination between specific domains corresponding to different a priori suitable working spaces. In these studies, mathematical work contributes to clarify and operationalize models as well as to give meaning to abstract mathematical notions. A MWS perspective could then break with a conception of modeling as an activity pursued individually and for individual competencies. In addition, considering the three dimensions of a working space should help to avoid a reduction of modeling to a translation and of mathematics to a language.KeywordsMathematical working spacesConnected working spacesModeling in educationPlurality of modelsOperativity of modelsSocial aspects of modeling

The aim is to show how the extended mathematical working space (extended MWS) theoretical framework can be used to analyse the tasksTasks implemented during a few stages of a modelling cycleModelling cycle in a chemical problem. This chapter studies a teachingTeaching sequence, including an experimental session in chemistry and graph construction for students in the last year of secondary school (grade 12) in France. The extended MWS theoretical framework makes it possible to study the multidisciplinary aspect of the different tasksTasks that students must perform when working on problem solvingProblem solving.

Our interest is in the development of modeling activities that link knowledge in various mathematical fields and other scientific disciplines. Looking at mathematics education, research we highlight the benefits and limitations of “problem-solving” and of approaches that involve translation between the real world and mathematics. Science education seldom takes mathematics into account. Its contribution is the conception of an “empirical referent”. Using epistemological studies, we highlight examples of modeling situations supported by a plurality of models and an organization of work that allows students to engage with them. To do this we adopt the theoretical framework of connected working spaces. A case study shows how students create linkages between the workspaces and the benefits of the connections on their understanding of concepts at stake.

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