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The theoretical framework of the Extended Mathematical Working Space (Extended MWS) allows to analyse the tasks implemented during a few steps of a physics or chemistry modelling cycle. The analysis of the work is also carried out with the anthropological theory of the didactic (ATD) to compare the two theoretical frameworks in the a priori analysis of the tasks to be performed by the students. Then, only the extended MWS
model will be used for post-analysis. A sequence of teaching special relativity using a diagrammatic approach in the final year of high school in France (grade 12) is first studied. The Minkowski diagram is used with the GeoGebra dynamic geometry software. The work on the chronological inversion of events in two reference frames is done with students during problem solving in a relativistic context. The analysis using the extended
MWS theoretical framework makes it possible to highlight the learning advantages of this diagrammatic approach during a complete didactic engineering. The theoretical framework of the extended MWS is also used in a teaching sequence dealing with the chemistry of solutions in secondary education, which includes an experimental part. 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 also didactic engineering. We will see that it is possible to propose new strategies when using GeoGebra with another semiotic representation register to work on problem solving.

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... En cuanto al esquema elaborado por Moutet (2019) en comparación con el propuesto en este documento podemos afirmar que ambos proporcionan visiones similares del trabajo matemático que se hace en la física, sin embargo, la consideramos poco práctica, dado que pareciera que ciertos procesos no pertenecen al plano epistemológico de las matemáticas, por lo que tienen que esquivarlo. Moutet (2019) También se considera que no basta con extender el enfoque del ETM, dado que en el estudio de la física se deben considerar otros elementos, como los que ya se han descrito en las secciones anteriores de este mismo capítulo. La interrelación que existe entre ambas ciencias y la necesidad de incluir dos formas de trabajo en un solo esquema propician que su interpretación se vuelva un tanto compleja. ...

In the teaching and learning of physics, the importance of experimental activities is recognized, because they allow the student to make observations of real and experiential events. Students face problems of interpretation that go beyond the application of formulae, where symbolic mathematical representations usually lack meaning. When requesting the mathematical description of a phenomenon, students face the problem of giving meaning to the signs that come from the accepted mathematical model. For this reason, we propose in the framework of the theory of Mathematical Working Spaces the study of physics within a Physical Mathematical Working Space (PMWS). In this thesis, we study dynamics and show how it can be applied to other knowledge domains. Moreover, for the appropriation of a mathematical model by the student, it is preferable that this model is not transmitted, but built with the participation of the individual. In this sense, in addition to experimentation, it becomes useful to explore a phenomenon with the help of simulations, which allow us to control variables, modify parameters, coordinate representations, and make simplifications. It is then discussed how the cognitive processes of the students are carried out, how the components of the PMWS are related, and how the articulation between the epistemological and cognitive plane occurs, using as a task the mathematical description of a phenomenon of inertia. Peer-to-peer discussion is recognized as a fundamental activity that allows generating a school science community in which argumentation processes are developed, which allow us to validate the proposed mathematical model.

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