Project

# Interdisciplinary modeling

Goal: I'm interested in mathematical-physical or mathematical-chemical relationships when solving modeling problems. I have developed a theoretical tool to analyse the tasks implemented: the extended MWS framework.

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## Project log

Implementing old physics syllabuses in a 12th grade science major in 2012 introduced elements of knowledge derived from the theory of special relativity. This chapter focuses on part of the second pilot sequence preparatory to engineering. A “graphic object” was built by the pupils step by step in order to more easily appropriate the notions of special relativity. The chapter analyzes transcripts of the pupils' work, revealing current teaching practice. Macro‐didactic and micro‐didactic hypotheses were established during the design of the sequence within the framework of didactic engineering. The teaching sequence was for the most part transcribed and the content analyzed by breaking down the corpus into elements of meaning by sentence unit. An analysis grid was designed to perform the a posteriori analysis of discussion between the teacher and the pupils. It took into account the achievements, the blockages and the various inputs of the teacher and also the registers implemented during the discussion.
The aim is to show the analysis of a problem solving using the theoretical framework of the extended mathematical workspace (Extended MWS) and the Blum and Leiss modeling cycle with a multidisciplinary approach (contribution of physics and mathematics). This problem-solving study the possibility of producing an intense magnetic field using a wire winding for use in a medical imaging device for example. The fields of electromagnetism and calorimetry are used in physics and that of algebra in mathematics. The extended MWS framework makes it possible to analyze academic tasks by considering the relationships between the cognitive plane of students, the epistemological plane of mathematics and that of physics. The whole activity proposed to grade 12 students in France can be described by three successive complete modelling cycles. The articulation of the different planes is studied according to the stage of the modelling cycle.
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.
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.
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.