Interdisciplinary modeling

Reflective analysis is an activity that is widely emphasised in the learning processes of trainee teachers in France. It remains important throughout a teacher's career in order to favour their creative potential. It corresponds to competence 14 of the reference frame of competences for teaching and education professions in France (MEN, 2013). JIGSAW (Aronson, 2002) is a technique that allows teachers to encourage relationships between students in order to solve complex tasks in some cases. The use of JIGSAW requires the teacher to decompose the final task performed by the students. It is therefore interesting to train teachers in the use of theoretical tools that facilitate reflective analysis in order to enable them to break down a complex task more easily to more accessible intermediate tasks. Diagrams used in the theoretical framework of extended Mathematical Working Space (Moutet, 2021) make it possible to understand the articulation, for a given task, between mathematics, physics and chemistry. A hybrid distance learning sequence for teachers is analysed in this article. The aim was to provide theoretical content on modelling (Blum & Leiss, 2005) and to enable the use of algorithms to be associated with modelling tasks (Lagrange, 2021). This training course introduced students to JIGSAW and to tools for reflective analysis of school tasks in order to help them design JIGSAW-type sequences. Only 30% of the trainees proposed a JIGSAW sequence skeleton during the asynchronous phase. The health context and the 100% distance-learning format may partially explain this finding. Similarly, 65% of the trainees followed the second part of the training in synchronous distance learning, partly because of technical problems with the video-conferencing platform used. This article aims to answer the following research question: How can theoretical tools originally designed for the research support teachers' reflective analysis? The theoretical framework used is that of the extended MWS (Moutet, 2021). The first results show that the implementation of reflective analysis tools initially developed for research is relatively difficult for teachers, but it appears that they are able to appropriate and carry out correct analyses with these tools, which is encouraging.

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.

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.

Il s’agit de montrer comment le cadre théorique de l’ETM étendu est utilisé pour analyser les tâches mises en œuvre lors du processus de modélisation. Le cadre de l’ETM étendu permet de montrer, au travers de l’exemple d’une séquence d’enseignement de relativité restreinte en terminale S en France (grade 12), quelles sont les interactions entre le plan cognitif et les plans épistémologiques de la physique ou des mathématiques. Les genèses, l’association de genèses et l’interaction entre les différents plans peuvent être explicitées pour certaines étapes du cycle de modélisation.

Analysis of a special relativity teaching sequence: The contribution of the extended MWS. The aim is to analyse the tasks performed during the modelling process of a special relativity teaching sequence in a “Terminale S” class in France (grade 12). Didactic engineering will be the methodological framework chosen for this study. Three theoretical frameworks will be used (DST, ADT and extended MWS) during the a priori analyses of the tasks to be performed in this sequence. The extended MWS framework will only be used for a posteriori analyses. Moutet, L. (2018). Analyse d’une sequence d’enseignement de la relativité restreinte : l’apport du modèle de l’ETM étendu. Annales de didactique et de sciences cognitives, 23, 107–136.

Le cadre théorique de l’ETM étendu permet d’analyser les tâches mises en œuvre lors de quelques étapes d’un cycle de modélisation en physique ou en chimie. Une séquence d’enseignement de relativité restreinte utilisant une approche diagrammatique en terminale S en France (Grade 12) est tout d’abord étudiée. L’analyse utilisant le cadre théorique de l’ETM étendu permet de mettre en valeur les avantages sur les apprentissages de cette approche diagrammatique lors d’une ingénierie didactique complète. Le cadre théorique de l’ETM étendu est également utilisé lors d’une analyse a priori de tâches mises en œuvre en chimie lors de la réalisation d’un dosage en chimie des solutions dans l’enseignement secondaire. L’analyse a priori utilisant le cadre théorique de l’ETM étendu permet de proposer de nouvelles stratégies d’apprentissage. La notion d’ETP personnel est également introduite dans cet article.

Abstract – The theoretical frame of the extended MWS allows to analyse the tasks operated during the process of modelling. It allows to show, through the example of a special relativity teaching sequence in a grade 12 class in France, which are the interactions between the cognitive plane and the epistemological planes of the physics or the mathematics. The geneses, the association of geneses and the interaction between differents plans can be clarified for certain stages of the cycle of modelling. Keywords: Mathematics, physics, extended MWS, special relativity, modelling. Moutet, L. (2019). Le cadre théorique de l’ETM étendu : Analyse d’une séquence utilisant la relativité restreinte. In M. Abboud (Éd.), Actes du colloque EMF 2018 (433-440). Paris : IREM de Paris.

We tried to develop and test several activities using a register based on diagrams for teaching the special theory of relativity to S class of twelfth graders. The graphic approach may result it complications in learning. However, its educational potential can turn out to be more beneficial. An epistemological study on diagrams used in special relativity allowed us to report important links between mathematics and the genesis of the special theory of relativity. This is the case of the Minkowski diagram. We were also interested in two other diagrams, Brehme and Loedel, which were developed much more later for teaching purposes. Following experimental sessions, we developed a new theoretical frame to comprehensively analyse the interactions developed by students to solve a problem using diagrams in special relativity. We modified the mathematical working spaces (MWS) by adding a new frame of rationality to the existing mathematic workspace to physics. The extended frame of the MWS allowed us to plan several versions of sequences proposed to the students and realize a priori analysis of their difficulty level and a posteriori study by analysing pupils' works. We have considered several works of student groups during a sequence using the Minkowski diagram with GeoGebra, a graphic simulation software. It allowed us to estimate the degree of control of the Minkowski diagram for every student, both from the frame of rationality of the mathematics and the physical sciences’ point of view. The results are promising and they tend to show a real appropriation of the concepts of the special theory of relativity with an approach using diagrams.