Séminaire InterLabos Aida 15 décembre 2017.

Conception et développement d’environnements numériques en didactique des Mathématiques.
n thème d’actualité ?

Résumé : Le travail de conception et de développement d’environnements numériques pour l'apprentissage,? l'enseignement et la formation a constitué une dimension très visible de la didactique des Mathématiques. Un jalon important a été il y a un peu plus de vingt ans le n° 14.1-2 de la revue Recherche en Didactique des Mathématiques consacré au thème « Didactique et intelligence artificielle ». Une résurgence a été le projet ReMath il y a maintenant presque 10 ans. Aujourd’hui le paysage des usages des TICE en mathématiques, semble dominé par un seul logiciel aux ambitions illimitées et cela contamine y compris la recherche en didactique.

L’équipe du LDAR a cependant fait l’hypothèse que ce travail de conception et de développement a continué et qu’il s’est élargi parallèlement au développement de nouvelles possibilités technologiques (Internet, résolveurs et moyens de simulation puissants maintenant accessibles aux élèves et enseignants…) et de nouvelles problématiques didactiques (nouvelles approches des contenus, attention portée à l’enseignant et à ses besoins, évaluation et personnalisation des apprentissages…) Une journée d’étude « Environnements numériques pour l'apprentissage,​ l'enseignement et la formation : perspectives didactiques sur la conception et le développement » a eu lieu en Mai 2016. Les contributions ont concerné une diversité de problématiques et de type d’environnements. L’exposé lors du séminaire tentera de dégager quelques traits saillants dans cette diversité et la discussion pourra viser à préciser comment ces travaux s’insèrent dans la recherche en EIAH et quels apports mutuels sont possibles.

• Les diapos

Department of Mathematics of the National and Kapodistrian University of Athens
Seminar 28/10/2017

Mathematical Working Spaces: designing and evaluating modelling based teaching/learning situations

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I am interested in mathematics taught at upper secondary level. The objective is to give meaning to this mathematics as a tool for understanding the sensible world. That is why I am interested in modeling processes involving models of a varied nature. On the one hand, it is important that students access mathematical models involving formalized and computationable objects, related for example to algebra or calculus, rather than purely numerical descriptions. On the other hand, it is also important that students encounter and use more informal models, often closer to the common experience and learn to compare and control the solutions obtained in both types of models in view of the context to be modeled.

While acknowledging the 'modelling cycle' (Blum, Galbraith, & Niss, 2007) as an overarching trajectory, with colleagues in the LDAR, we are interested in the diversity of models involved in this trajectory, the different domains, mathematical or not, to which they relate, the specific work in each model, and the connections that a student has to do to achieve  the modelling process.

The goal of this seminar is then to present the idea of Mathematical Working Spaces (Kuzniak, Tanguay, & Elia, 2016) as a construct useful for conceptualizing several aspects of the modelling process with regard both to epistemological and cognitive dimensions of teaching/learning. In the introduction, I will present a general framework and an example. Then I will present a classroom situation whose design and evaluation has been inspired by this framework

CIAEIM 69
July 16-19 2017 University of Berlin

Workshop

Subtheme 1: Mathematisation as a didactic principle

Mathematical Working Spaces: a construct to make sense of modelling based teaching/learning situations

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Authors : Alain Kuzniak & Jean-baptiste Lagrange. LDAR University Paris Diderot.

Our proposition deals with the following concerns as expressed in the discussion document. We emphasize particular keywords in bold font.

The mathematisation of social, economical and technological relations in the form of formal structures is a double-edged sword. On the one hand, it has proven effective and efficient in terms of developing more and more complex structures (…) On the other hand, once established as the standard (or only) way of describing, predicting and prescribing social, economic, ecologic, etc. processes, it severely reduces the possibilities of finding non-formal, non-quantifiable, nonmathematical solutions.

Mathematical Modelling (…) is another orientation for curriculum construction that attracts worldwide attention. Within Mathematical Modelling, the authenticity of everyday situations is of relevance. From these everyday situations a ‘real world model’ is generated and, further the ‘real world model’ is translated into a ‘mathematical model’, which can be used for calculation or other mathematical procedures.  

The goal of the proposed workshop is to present and discuss the idea of Mathematical Working Spaces (Kuzniak, Tanguay, Elia 2016) as a construct useful for conceptualizing several aspects of the modelling process with regard both to epistemological and cognitive dimensions of teaching/learning. In the introduction, we present a general framework and an example. Then we propose two phases of work: starting from a situation of modelling and reflecting on specific questions, and a concluding discussion in relationship with questions offered by the discussion document.  

Introduction

General Framework

We are interested in mathematics taught at upper secondary level. Our objective is to give meaning to these mathematics as a tool for understanding the sensible world. We are interested in modeling processes involving models of a varied nature. On the one hand, it is important that students access mathematical models involving formalized and computationable objects, like for example in algebra or calculus, rather than purely numerical descriptions. The goal is to give meaning to formalism and calculation as a tool for understanding the sensitive world and thus to remedy a weakness in teaching: the transmission of a meaningless formalism[1]. On the other hand, it is also important that students encounter and use more informal models, often closer to the common experience and learn to compare and control the solutions obtained in both types of models in view of the context to be modeled.

The context to be modeled is itself a space. It is the space where questions can be asked that motivate a representation in other spaces, the modeling spaces. As the announcement says, it is important that the questions at stake are "authentic", that is to say belong to a field of concern whose relevance students can grasp. However, there is no reason why these concerns should be limited to "everyday life". Actually, in most cases, the context motivating mathematical modelling at upper secondary level is not directly in students' everyday experience and special strategies are required to make them question this context. Think for instance of relativity, a model of the real world very far from everyday experience and without practical application, and which nevertheless can be a relevant topic for secondary students. Think also of vocational fields where mathematical models are often proposed, that are too simplistic to be relevant[2].

We see then modeling as the appropriation of spaces of different natures. Mathematical models are inscribed in modeling spaces based on specific theories, sign systems and artefacts. There are also modeling spaces where the theory remains more implicit, and where the rules relating to sign systems and artefacts are mainly understood in usage. These are the spaces where the more informal models that we mentioned earlier are inscribed. We propose, as in the example below, to consider these different spaces as "mathematical working spaces"

An example

A problem for a surveyor is the calculation of the surface area of a farmer's field. We consider a common case: flat quadrilateral fields. Assuming that the lengths of the sides and of one diagonal are known, it is possible to consider different spaces where models can answer the question. In the first one, we consider a sheet of paper, the scaling procedures, the instruments for drawing and measuring, and a formula for calculating the area of a triangle. In this first space, splitting the field into two triangles and measuring altitudes makes it possible to answer the question in a practical way. A second space is that of the elementary geometry where the so-called Heron formula makes it possible to calculate the area of ​​a triangle knowing the length of its sides without drawing or measurement. The two spaces share a common general strategy: splitting into two triangles, but do not share the other means of action, the justifications of these actions and the resulting conceptualizations.

Kuzniak (2013) shows how the framework of "mathematical workspaces" allows considering the specificities of each space and their complementarity. In the example, the first two modeling spaces do not necessarily organize themselves in a hierarchy where the mathematical model would have the pre-eminence. The "measuring" space allows the problem to be satisfactorily solved with limited theoretical apparatus. The "elementary geometry" space avoids drawing and measuring and therefore the accuracy is not limited by the measurement on a reduced scale and the imprecision of drawing. The procedure in this space allows automatization, for example by way of a program on a calculator[3]. The "measuring" space favors the use of instruments and therefore the associated genesis, while the "elementary geometry" space fosters the use of signs (semiotic genesis). In both spaces, discursive genesis may be called upon to justify the procedure used. We have noted that splitting into triangles is a procedure shared by the two spaces and can thus connects the two models.

Working theme 1: Mathematisation and demathematisation in understanding modelling processes

While we acknowledge the "modelling cycle" (Blum, Galbraith, & Niss 2007) as an overarching modelling trajectory, we also stress recursive processes inside this trajectory, designing, amending and fine-tuning diverse models inscribed in particular working spaces possibly related to different mathematical domains. These processes establish a continuous balance between mathematisation and demathematisation. This is illustrated by the following situation.

Situation

We consider here three dimensional models of sea floor areas. These models are very important for human activities and provide impressive pictures and videos. However, they are most often unquestioned relatively to the processes of data acquisition and treatment they result of, and then relatively to their accuracy. As a part of their project in an in-service master class, two students-teachers aimed at questioning both processes and at evaluating the model by confronting with reality. It means that they wanted to acquire and process data for a variety of configurations on the sea floor in order to compare to the original configuration. Configurations were chosen simple: flat floor, sloped floor, half sphere… The students had to recreate the data acquisition, of course not at sea!

Since "real life" data acquisition is made from a boat navigating an area and carrying a multi-beam echo sounder, they decided to build a mock-up traveling on two horizontal axes above a floor, using 5 sticks to represent beams. Moving the boat with 3 positions on one axis and 6 on the other, they obtained a 3x6x5 table of data. Then they put into operation various treatments inspired by relevant scientific documentation.

Activity

The participants in the workshop will first look at the treatments that have to be applied in order to get a 3D model. Then they will consider the interest of designing a mock-up and of acquiring data from this mock-up as compared to using data acquired in "real life". Finally they will discuss how data acquisition in the mockup and mathematical treatments imply two different working spaces whose interrelation can help understanding.

Working theme 2: Working spaces for organizing students' modelling work  

In this theme we propose to consider the classroom implementation of a modelling process involving models in several working spaces.

Situation

We consider here the shape of a main cable of a suspension bridge. Tasks for student at secondary level most often assume that the shape is a portion of a parabola, but do not question the reasons why it is quadratic whereas for instance cable-stayed bridges have straight cables. It seemed to us that these reasons are accessible to 12 grade students, with the condition that they understand the process of modelling that this result is issued of: the tension along the cable progresses linearly, and so does the slope.  Moreover, this process of modelling offers an opportunity for students to mobilize their knowledge in calculus at a synthetic level while investigating a real world situation.

We consider four models and the corresponding working spaces.                     

1. A discrete model derives from the finite number of suspensors: a main cable is represented by a broken line and the deck is represented by a collection of weights hanged on the cable by way of equally distant suspensors. The "static working space" organized around the first Newton Law, is suitable for a study of the tension along the cable.
2. Knowing the tension at each suspension point, it is possible to establish recurrence formulas for the coordinates of these points. Students work here in a "coordinate geometry" working space.
3. It is then interesting to program the recurrence formulas in order to graph the broken line. Using a dynamic programming environment, students can appreciate the role of parameters, like the constant value of the horizontal component of the tension. This is the algorithmic working space.
4. Given the big number of suspensors, one can look for a curve, limit of the broken line modeling the cable when this number tends to infinity. The work on this "continuous model" is done inside a "mathematical functions working space" governed by classical symbolism and rules in calculus.

Activity

The participants in the workshop will first characterize the four working spaces and the potential geneses inside each working space. Then, taking advantage of reports on classroom implementation (Lagrange et al. 2015), the participants will discuss how this characterization helps designing and evaluating classroom situations.

Conclusion

We will conclude by discussing with the participants how the work done helps to reflect on starting questions offered for subtheme 1 by the discussion document:

 What qualifies a real-world context as a point of departure and/or point of arrival of a didactic arrangement that builds on mathematisation?

• What are specific cognitive, social or discursive processes that occur in learning

environments that have mathematisation as a pivot?

• Which material arrangements support students' learning of mathematics by mathematisation (e.g. artefacts, physical experiences, learning spaces, etc. ).
• Which epistemologies of mathematics are built?

References

Blum W., Galbraith, P. L., & Niss, M (2007). Introduction. In W. Blum P. L. Galbraith, H Heml, &M Niss (Eds.), Modelling and applications in mathematics education (pp. 32). New York, NY: Springer.

Blum, W., & Ferri, R. B. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58.

Boero, P., & Morselli, F. (2009). Towards a comprehensive frame for the use of algebraic language in mathematicalmodelling and proving. Proceedings of PME 33, 2, 185-192.

Kuzniak, A. (2013) Teaching and learning geometry and beyond. Ubuz, Behiye (ed.) et al. Proceedings of CERME 8, Antalya, Turkey.

Kuzniak, A., Tanguay, D., & Elia, I. (2016). Mathematical working spaces in schooling: an introduction. ZDM Mathematics Education. doi:10.1007/s11858-016-0812-x.

Lagrange, J. B., Halbert, R., Le Bihan, C., Le Feuvre, B., Manens, M. C., Meyrier, X. & Minh, T. K. (2015). Investigation, communication et synthèse dans un travail mathématique: un dispositif en lycèe. Actes de la conférence EMF, Alger. http://emf2015.usthb.dz/actes/EMF2015GT10COMPLET.pdf.