# Modeling in Education: New Perspectives Opened by the Theory of Mathematical Working Spaces

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In this paper, we identify various forms of geometric work carried out by student teachers who were asked to perform a geometric task for the estimation of a land area. The theory of Mathematical Working Spaces is used to analyze and characterize the work produced. This study provides evidence that students developed forms of geometric work that are compliant with at least two distinct geometric paradigms, one characterized by the utilization of measuring and drawing tools and the other by a property-based discourse on proof. Significantly, a sizable number of students also developed work forms that do not correspond to any geometrical paradigm. A broader purpose of this paper is to highlight three criteria born by the theory and shown to be useful for the description and evaluation of geometric work: compliance, completeness, and correctness.

In this work we have applied the thematic analysis methodology to extract the most relevant conceptions about mathematical modelling practices using the collected opinions from a interdisciplinary team of researchers. The main feature of the team is the use of mathematical modelling in its practice to solve applied problems of the real world. So, five conceptions are obtained, and then all of them were recognized in two of the works published by Poincare's. Consequently, these conceptions represent some key components of the interdisciplinary work supported by mathematical modelling practice.

The notion of mathematicalmodelling has multiple interpretations in research literature and has been included in curriculum documents in various ways. This study presents a thematic analysis of nine professional modellers’ conceptions of the notion of mathematical modelling, representing scholarly knowledge. The result is organised according to four main aspects of modelling, that is description, understanding, abstractionand negotiation, representing key features of mathematical models and how people are affected by, or engage in, modelling work. The paper concludes with a discussion of some potential benefits for mathematics education that might be drawn from the study fromthe perspective of the ‘gap’ between mathematical modelling in educational and non-educational contexts, in terms of curriculum design and teaching practice.

Educational research literature on mathematical modelling is extensive. However, not much attention has been paid to empirical investigations of its scholarly knowledge from the perspective of didactic transposition processes. This paper reports from an interview study of mathematical modelling activities involving nine professional model constructors. The research question was: How can mathematical modelling by professional mathematical model constructors be characterised? The analysis of our interview data inspired by the coding procedure of grounded theory led us to the description of three main types of modelling activities as a characterisation of mathematical modelling as a professional task. In data-generated modelling the models are developed principally from quantitative data drawing on no or only some assumed knowledge of the system being modelled, while in theory-generated modelling the models are developed based on established theory. In the third activity, model-generated modelling, the development of new models is based on already established models. For all types, the use of computer support and communication between clients, constructors and other experts are central aspects. Finally, the three types of modelling activities are related to existing theoretical descriptions of mathematical modelling and the relevance of the study for mathematical modelling in education is discussed.

The starting point of this article is the question, “how might we inform an epistemology of numeracy from the point of view of better preparing young people for workplace competence?” To inform thinking illustrative data from two projects that researched into mathematics in workplace activity and the teaching and learning of modelling in the classroom is used albeit, by necessity, briefly. Analysis draws attention to the crucial role that understanding the structure of the contextual situation plays in developing a mathematical model of this. It is at this coupling of reality and mathematics that insight into, and understanding of, both mathematics and reality can be developed – or not. This important issue is illustrated with reference to two specific workplace situations that draw on understanding of fraction as gradient and explore how we might use a model of this to scaffold understanding of both reality and mathematics and how each might support the other. With a focus on model formulation during classroom activity attention is then drawn to how students tend to work towards reaching a solution to a particular problem with the consequence that their mathematical representation of the reality does not easily allow for consideration of variation of key factors. In conclusion a research agenda is proposed that seeks to inform an epistemology of numeracy by focussing on numerate activity at the nexus of reality and mathematics by (i) structuring mathematical knowledge using mathematical models that might be used to provide insight into a range of models of workplace realities, and (ii) requiring student engagement with repeated use of models in ways that emphasise exploration of variability in key factors of the realities that the models represent.

Mathematical Modelling: Can It Be Taught And Learnt?
Werner Blum & Rita Borromeo Ferri
Journal of Mathematical Modelling and Application 2009, Vol. 1, No. 1, 45-58
Abstract
Mathematical modelling (the process of translating between the real world and mathematics in both directions) is one of the topics in mathematics education that has been discussed and propagated most intensely during the last few decades. In classroom practice all over the world, however, modelling still has a far less prominent role than is desirable. The main reason for this gap between the goals of
the educational debate and everyday school practice is that modelling is difficult both for students’ and for teachers. In our paper, we will show examples of how students and teachers deal with demanding modelling tasks. We will refer both to results from our own projects DISUM and COM² as well as to empirical findings from various other research studies. First, we will present some examples of students’ difficulties with modelling tasks and of students’ specific modelling routes when solving such tasks (also dependent on their mathematical thinking styles), and try to explain these difficulties by the cognitive demands of these tasks. We will emphasise that mathematical modelling has to be learnt specifically by students, and that modelling can indeed be learned if teaching obeys certain quality criteria, in particular maintaining a permanent balance between teacher's guidance and students’ independence. We will then show some examples of how teachers have successfully realised this subtle balance, and we will present interesting differences between individual teachers’ handling of modelling tasks. In the final part of our paper, we will draw some consequences from the reported empirical findings and formulate corresponding implications for teaching mathematical modelling. Eventually, we will present some encouraging results from a recent intervention study in the context of the DISUM project where it is demonstrated that appropriate learning environments may indeed lead to a higher and more enduring progress concerning students’ modelling competency.

Cet article introduit l'ouvrage La mathématisation comme problème. Dans un premier temps, j'explique en quoi la mathématisation peut être présentée comme un problème. Dans un deuxième temps, j'examine ce qu'on a appelé la "déraisonnable efficacité des mathématiques". Dans un troisième temps, j'explique en quoi il pourrait être bon d'identifier les formes de mathématisation. Finalement, je présente les articles de l'ouvrage autour des trois moments de la quantification, de la formalisation et de la modélisation.

Understanding how students construct abstract mathematical Knowledge is a central concern of research in mathematics education. Abstraction in Context (AiC) is a theoretical framework for studying students' processes of constructing abstract mathematical knowledge as it occurs in a context that includes specific mathematical, curricular and social components as well as a particular learning environment. The emergence of constructs that are new to a student is described and analyzed, according to AiC, by means of a model with three observable epistemic actions: Recognizing, Building-with and Constructing -the RBC-model. While being part of the theoretical framework, the RBC-model also serves as the main methodological tool of AiC.
In the first section of this chapter, we give an outline of the thoretical aspects of AiC as background to the description of the elements of our methodology in the second section, and their application to a specific example in the third section. In the concluding section, we close the circle by exhibiting the strong relationship of theory and methodology in AiC as it is mediated by the RBC-model.

Ce chapitre introductif propose une caractérisation large et souple pour la fonction générale des modèles en sciences. Il en déduit 20 fonctions non plus générales mais spécifiques pour les modèles. Les simulations sont rapprochées et distinguées des modèles. Trois types de simulations sur computer sont distingués. Pour finir, face à cette diversité compréhensible, il est suggéré qu’il peut paraître vain d'espérer que se construise un jour une épistémologie générale et théorique des modèles, comme il peut paraître vain d'espérer que se construise un discours critique uniforme à l'encontre des modèles et des simulations au vu de leurs incontestables - mais ponctuels - excès.

This article discusses the role of context problems, as they are used in the Dutch approach that is known as realistic mathematics
education (RME). In RME, context problems are intended for supporting a reinvention process that enables students to come
to grips with formal mathematics. This approach is primarily described from an instructional-design perspective. The instructional
designer tries to construe a route by which the conventional mathematics can be reinvented. Such a reinvention route will
be paved with context problems that offer the students opportunities for progressive mathematizing. Context problems are defined
as problems of which the problem situation is experientially real to the student. An RME design for a calculus course is taken
as an example, to illustrate that the theory based on the design heuristic using context problems and modeling, which was
developed for primary school mathematics, also fits an advanced topic such as calculus. Special attention is given to the
RME heuristic that refer to the role models can play in a shift from a model of situated activity to a model for mathematical
reasoning. In light of this model-of/model-for shift, it is argued that discrete functions and their graphs play a key role
as an intermediary between the context problems that have to be solved and the formal calculus that is developed.

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.

This paper aims at contributing to remedy the narrow treatment of functions at upper secondary level. Assuming that students make sense of functions by working on functional situations in distinctive settings, we propose to consider functional working spaces inspired by geometrical working spaces. We analyse a classroom situation based on a geometric optimization problem pointing out that no working space has been prepared by the teacher for students’ tasks outside algebra. We specify a dynamic geometry space, a measure space and an algebra space, with artefacts in each space and means for connecting these provided by Casyopée. The question at stake is then the functionality of this framework for implementing and analyzing classroom situations and for analyzing students’ and teachers’ evolution concerning functions, in terms of geneses relative to each space.

Présentation
Chapitre 1. La modélisation : une connaissance socialement située
Le trafic automobile
Le génie des matériaux
Un bassin versant
La production de connaissances
Questions de langage : les sciencettes
Références bibliographiques
Chapitre 2. Les mathématiques, ressource conceptuelle et syntaxique
Le calcul infinitésimal
Polysémie, localité de l’intuition
Le mathématicien décompilateur
Remarques pédagogiques
Références bibliographiques
Chapitre 3. Le métier de veille du chercheur
Popper : un critère devenu norme
Sociologie des sciences et excès du socio-centrisme
L’engagement critique du scientifique
La construction sociale de l’innocence du chercheur
Références bibliographiques
Chapitre 4. Modélisation de l’économie
Le programme de Georgescu-Roegen
La doctrine néoclassique
La décision décentralisée de Hayek et la modélisation
Le concept de modèle-commentaire
Références bibliographiques
Chapitre 5. Modèles et interprétation
Quine et la sous-détermination
Hans Jonas et Ulrich Beck : responsabilité et risque
Mill et la rationalité externe
Connaissance post-normale
Références bibliographiques
Chapitre 6. La contre-expertise : construction de co-vérités
Vers une linguistique des modèles
Analogies avec le projet d’architecture
Le modèle et ses ornières de pensée
La science et la culture
Construction de co-vérités
Références bibliographiques
Conclusion : il y a toujours plusieurs « presque vrais »
Références bibliographiques
Index thématique
Lien : http://www.quae.com/fr/r3436-la-modelisation-critique.html

In this paper, we shall report on some of the work that has been, and is being, done in the DISUM project. In §, we shall describe the starting point of DISUM, the SINUS project aimed at developing high-quality teaching. In §, we shall briefly describe the DISUM project itself, and in §3 we shall present and analyse a modelling task from DISUM, the “Sugarloaf” problem. How students dealt with this task will be the topic of §4, the core part of this paper. How experienced SINUS teachers dealt with this task in the classroom will be reported in §5. Finally, in §6, we shall briefly describe future plans for the DISUM project.

This paper is the Discussion Documentfor a forthcoming ICMI Study on Applications and Modelling in MathematicsEducation. As will be well-known, fromtime to time ICMI (the InternationalCommission on Mathematical Instruction)mounts specific studies
in order toinvestigate, both in depth and in detail,particular fields of interest inmathematics education. The purpose of
thisDiscussion Document is to raise someimportant issues related to the theory andpractice of teaching and learningmathematical
modelling and applications,and in particular to stimulate reactionsand contributions to these issues and tothe topic of applications
and modelling asa whole (see Section 4). Based onthese reactions and contributions, alimited number (approximately 75) ofparticipants will be invited to aconference
(the Study Conference)which is to take place in February 2004 inDortmund (Germany). Finally, using thecontributions to this conference, a bookwill
be produced (the Study Volume)whose content will reflect thestate-of-the-art in the topic ofapplications and modelling in mathematicseducation and suggest
directions for futuredevelopments in research and practice.The authors of this Discussion Document are themembers of the International ProgrammeCommittee for this ICMI Study. Thecommittee consists of 14 people from 12countries, listed at the end of Section4. The structure of the Document isas follows. In Section 1, we identifysome reasons why it seems appropriate tohold a study on applications and modelling.Section 2 sets a conceptual frameworkfor the theme of this Study, and Section3 contains a selection of importantissues, challenges and questions related tothis theme. In Section 4 we describepossible modes and ways of reacting to theDiscussion Document, and in the finalSection 5 we provide a shortbibliography relevant to the theme of thisStudy.

Understanding links between mathematics and engineering through mathematical modelling -The case of training civil engineers in a course of structural analysis

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Cosmes Aragón, S. E. & Montoya Delgadillo, E. (to appear). Understanding
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Les espaces de travail connectés : une perspective nouvelle pour la modélisation dans le secondaire?

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Lagrange, J.-B. (2021). Les espaces de travail connectés : une perspective
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Choosing a Mathematic Working Space in a modeling task: the influence of teaching

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