Jean-Baptiste Lagrange

Jaime Huincahue

Giorgos Psycharis

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

Keywords: Mathematical Working Spaces, Connected Working Spaces, Modeling in education, Plurality of models, Operativity of models, Social aspects of modeling

In this chapter the guiding line is the mathematical work that students develop in modeling tasks. We report first on MWS studies adopting the idea of a modeling cycle. In these studies the MWS framework gives insight into the processes involved in students' work and it also sheds light upon the complex relationship between reality and mathematics, and the multifaceted relationship between the process of problem solving and the underlying educational goals. We question the consistency of a sharp separation between mathematics and reality and, assuming that mathematical modeling is what mathematicians do when they work on models, we look at this work outside of education. Rather than steps of translation between reality and mathematics, we have to think of modeling as a coupling of reality and mathematics that should allow students to develop insight into, and understanding of, both mathematics and reality. We also look to epistemological studies that distinguish between modeling and mathematization, and characterize modeling by (1) plurality of models (2) operativity (3) subjective and social interpretation. The plurality of models for a given reality has been exploited in research studies to design tasks that put at stake transitions or coordination between specific domains corresponding to different a priori suitable working spaces. In these studies mathematical work contributes to clarify and operationalize models as well as to give meaning to abstract mathematical notions. A MWS perspective could then break with a conception of modeling as an activity pursued individually and for individual competencies. In addition, considering the three dimensions of a working space should help to avoid a reduction of modeling to a translation, and of mathematics to a language.

Introduction: modeling and students' mathematical work

There are a variety of approaches to modeling in mathematics education that regard both educational aims (cognitive and social) and task conception. In this chapter the guiding line is the mathematical work that students develop in modeling tasks. How do researchers characterize this work and the way in which it contributes to students' development of their mathematical conceptualizations and personal and social capabilities? How does the theory of MWS with the idea of work and the three dimensions, semiotic, instrumental and discursive1 help this characterization? How does it help to analyze a priori the design of modeling situations and to make sense a posteriori of students' work in these situations?

Our investigation aims to characterize the specific contribution of MWS within the variety of approaches to educational modeling. We do not try to take into account all the frameworks in this variety. Throughout this chapter, we reference particular frameworks when necessary. As a start, we consider the idea of a "modeling cycle" [START_REF] Blum | Mathematical modeling: Can it be taught and learnt[END_REF] to describe students' potential and actual trajectories in problems involving "reality" and mathematics. In the rest of the chapter, we refer to other approaches, especially those to modeling in science and at work, and discuss learning objectives that various frameworks assign to modeling.

2 MWS and modeling cycle

Despite researchers' growing adoption of a MWS perspective, research studies about modeling with this perspective are still scarce. We review in this section a majority of these studies and notice that the idea of a modeling cycle is part of their theoretical framework together with MWS. We examine the contribution brought about by MWS in these research studies. Further, we discuss assumptions underlying this idea of a modeling cycle in the light of mathematical work in order to try to explore more deeply the potential of MWS.

3

The modeling cycle is organized in steps. Two initial steps consist of the elaboration of a real model, particularly by simplifying and structuring a real situation. It is followed by steps of translation, one of which associates the real model and a mathematical model. This is called mathematization. The next step, interpretation, associates real results and mathematical results. Among the existing MWS studies, Nechache (2018) analyzes a phase she named "describing reality" that corresponds to the first steps in the modeling cycle. Nechache's study deals with a random experiment, a repeated Bernoulli trial with a stopping rule. She demonstrates that building a real model requires introducing probabilistic hypothesis (equiprobability of events, independence of successive trials) and then more or less formal knowledge in probability. Figure 1 summarizes this analysis. The MWS framework allows Nechache to analyze this work as participating in the discursive dimension of a preprobabilistic working space. [START_REF] Blum | How do students' and teachers deal with modeling problems?[END_REF] also evidence the construction of geometric relationships in the step of simplifying and structuring in a task related to distances in a real life situation. In their work, mathematical work starts before the step that they label mathematization. This demonstrates that regardless of how it is labeled in the modeling cycle, the real model is actually a mathematical model, provided that mathematics are not restricted to typical content knowledge. Nechache also analyzes the process of mathematization in probability. This process goes from a real model to a probabilistic one, allowing pre-4 diction making. It actually involves the interaction of the preprobabilistic working space at stake in the previous phase, and a more standard probabilistic working space. She discusses three methods currently used in this process. The first method models the experiment by way of random variables and laws of probability. The work uses formalism and rules associated with random variables and is mostly in the discursive dimension. In the second method, the model is a weighted tree diagram and the work favors the semiotic and instrumental dimensions: Each node of the tree denotes a possible outcome at a given step. The computation of the probabilities follows "mechanical" rules rather than discursive reasoning. Finally, in the third method, the model is a simulation on a spreadsheet. Recognizing the spreadsheet's capabilities and obtaining adequate formulas and organization is not obvious, but requires reflection and combining instrumental and discursive dimensions. The outcome of Nechache's study was that the MWS framework allows an in depth a priori analysis of steps in the modeling cycle, and then of the stakes of modeling activity for students, especially with regard to mathematical conceptualizations.

Another study by [START_REF] Derouet | La fonction de densité au carrefour entre probabilités et analyse. Une ingénierie didactique en classe de terminale scientifique. Actes du séminaire national de didactique des mathématiques 2017[END_REF] analyzes a posteriori a process of mathematization in a classroom in light of MWS. The goal is to build a continuous probabilistic model for a phenomenon from statistical data. The analysis shows non trivial transitions between three domains (statistics, mathematical functions, and continuous probabilities), and the potential of the three dimensions of MWS for making sense of these transitions. Derouet explains particularly a step of mathematization using the diagram of figure 2. The students look for a function that best fits a histogram, namely, a candidate for a density function modeling the phenomenon. Derouet shows first a circulation between statistics and mathematical functions in the semiotic and instrumental dimensions. Students propose functions and the teacher operates a software displaying the graph of this functions and the histogram. Derouet notes that the work in the semiotic dimension (proposing functions that fit best) is at the students' initiative (in dark grey figure 2 left) while the teacher takes in charge most of the instrumental work (in light grey figure 2 left) . The discursive dimension of the continuous probabilities workspace is then solicited to check properties of the function with regard to probabilities (positivity, area under the curve) in connection with the semiotic work on mathematical functions (in dashed dark grey figure 2 right). Finally, there is work in the discursive dimension to adjust the function. This is performed primarily by the teacher (in light grey figure 2 right.) 5

In our interpretation, the outcome of the MWS framework is a fine grain analysis that brings deep insight into the stakes in this step of mathematization, i.e.: variety of knowledge activated by the students; choices about which actions should be left to the students as the teacher felt these were central to the situation; and the part the teacher takes in charge so that the situation progressed. Modeling is not seen as an individual endeavor to discover the right model, but rather as a social process where the parts respectively played by the teacher and the students are identified relatively to the dimensions. [START_REF] Derouet | Modeling tasks and mathematical work[END_REF] note that the use of the MWS framework can enrich and strengthen the analysis of the modeling process based on the study of a cycle in connection with the resolution of a problem. Beyond this, the authors note that in an educational context, a modeling task aims not just at solving a real problem, but more deeply at exploring and understanding the numerous uses of a mathematical notion. After solving a problem though modeling, students have to perform specific work in order to better understand the model and the mathematical objects in-volved in the solution. This thus enriches the MWS, in particular the theoretical referential in the discursive dimension.

Considering students' work in modeling situations, the MWS framework gives insight into the processes involved, and it also sheds light upon the complex relationship between reality and mathematics, and the multifaceted relationship between the process of problem solving and the underlying educational goals. This is good news but it also raises questions. If the steps of describing reality involve mathematical knowledge and the real model includes some mathematics, how is it consistent with a sharp separation between mathematics and reality? If working at the interface of reality and mathematics, as considered in probability theory by [START_REF] Derouet | La fonction de densité au carrefour entre probabilités et analyse. Une ingénierie didactique en classe de terminale scientifique. Actes du séminaire national de didactique des mathématiques 2017[END_REF] and by [START_REF] Nechache | L'articulation des genèses de l'Espace de Travail Mathématique dans la résolution de tâches probabilistes faisant appel à la modélisation[END_REF], involves non trivial transitions and choices of dimensions, how can we think of the processes involved beyond a simple translation? What are the cognitive processes in work that aims at understanding the model and the objects involved? How can they be described, with geneses or with other cognitive notions?

The work of mathematicians in the social activity of modeling

Traditionally, statistics and probabilities are domains favored by secondary curricula for a mathematical approach to real-world phenomena. From the years 2000 the interest for involving real-world contexts in a wider variety of domains of learning, beyond statistics and probabilities, grew in mathematics education, as witnessed by the introduction of the OECD "Programme for International Student Assessment" (PISA) including a mathematics literacy test asking students to apply their mathematical knowledge to solve problems set in real-world contexts. Regarding math education research, the ICMI Study 14: Applications and modeling in mathematics education [START_REF] Blum | ICMI Study 14: Applications and modeling in mathematics education -Discussion document[END_REF] was an important landmark. In it, Blum stressed that, "Today mathematical models and modeling have invaded a great variety of disciplines," and that "mathematising real situations as well as interpreting, reflecting and validating mathematical results in 'reality' are essential processes when solving literacy-oriented problems." The ICMI study started a very lively stream of educational research and put a growing focus on mathematics as an important activity in society. However nearly twenty years afterwards, 7 much remains to be done in order to conceptualize the complex relationship between mathematics and modeling.

Derouet's and Nechache's studies deal with statistics and probabilities, and rely on epistemological and didactical knowledge in order to study modeling processes: It is more or less common knowledge that random phenomena cannot be approached without some mathematical apparatus. In contrast, in other domains researchers most often deal with modeling by starting from a simple conception of reality and its relationship to mathematics. For instance, Blum and Ferri (2009, p. 45) explain that by reality they mean "the rest of the world outside mathematics including nature, society, everyday life and other scientific disciplines." While this conception avoids being locked into philosophical debates about mathematics and reality, it does not shed much light upon the variety of intervention of mathematics in domains of modeling. This chapter aims to help close this gap, as the idea of mathematical work is central here. Mathematical modeling is what mathematicians do when they work on models and thus, it is useful to look at this work outside of education as a source of inspiration. One way is to build upon observations of mathematical modelers' practices. Another possibility is to look at epistemological studies of modeling and consider these as source of second hand information about the work of modeling.

Insights from the workplace

Frejd and Bergsten (2016) take a didactical transposition approach. They typologize workplace practices around the relationship between a client in need of a model for practical application, the modeler, and an expert that brings the relevant knowledge. While arguing that the typology could be a starting point for a transposition into educational settings, they recognize that there is still a long way to go. Their study allows them to criticize "policy makers referring to PISA league tables as an argument for curricular reform without providing analyses of how mathematical modeling is conceptualized and operationalized" (p. 31). In further interpretation, [START_REF] Frejd | Professional Modellers' conceptions of the notion of mathematical modeling: ideas for education[END_REF] characterize mathematical modeling professional activity by four components: description, understanding, abstraction, and negotiation. From this characterization, they draw a "check-list" of didactic principles to be used in the design of modeling activities within mathematics teaching. An interesting principle is related to technology use "crucial for modeling work but also in 8 itself (involving) a modeling activity." This echoes the role of the computer as a key component in the instrumental dimension of the work in studies like Derouet's.

In another study, [START_REF] Huincahue | Interdisciplinarity, Mathematical modeling, and Poincare's work: Comparing conceptions about knowledge construction[END_REF] define mathematical modeling as an activity in which two or more disciplines converge. For them, a mathematical modeler necessarily works in interdisciplinary teams, each discipline bringing its own contribution, but without hierarchical disciplinary preferences. Their empirical study shows modeling as a cyclical process of dialogue between mathematicians and specialists of a domain (like epidemiology or agriculture). The dialogue aims at elaborating a common understanding while discussing ways to approach a problem. The work is always directed by the need of a solution in the domain, but from the mathematician's point of view, the actual outcome is the model(s) rather than the solution, and possibly a base for new developments in mathematics.

Finally, it is worth mentioning research focused on numeracy (i.e., reasoning and applying simple numerical concepts) from a modeling perspective. [START_REF] Wake | Preparing for workplace numeracy: A modeling perspective[END_REF] is concerned that modeling activities in schooling are very different as compared to mathematical activity at workplaces. Rather than steps of translation between reality and mathematics, he thinks of modeling as a coupling of reality and mathematics that allows students to develop insight into, and understanding of, both mathematics and reality. Building from this, Wake goes on to identify (ibid, p. 13) three objectives for teaching/learning. Namely, students should become "able to:

• develop critically and mathematically informed models of complex realities;

• both construct and deconstruct (...) models of complex situations;

• understand how the structure of models of (workplace) realities and the structure of their mathematical counterparts (inter-)relate;". Wake then adds that education must "• prioritize mathematical models that afford insight into typical structures important in workers' practices and that are found in and across workplaces;

• involve learners in repetitive use of models in ways that ensures they engage substantially with issues of variability." 9 11.3.2 Epistemological studies of modeling [START_REF] Lagrange | Les espaces de travail connectés : une perspective nouvelle pour la modélisation dans le secondaire[END_REF] draws from studies by Francophone authors, [START_REF] Roux | Pour une étude des formes de la mathématisation[END_REF], [START_REF] Bouleau | La modélisation critique[END_REF] and [START_REF] Varenne | Chapitre 1. Épistémologie des modèles et des simulations : tour d'horizon et tendances[END_REF]. The idea of mathematization is questioned. It is necessary to distinguish between mathematizing processes (quantification, formalization, inductive and deductive reasoning, etc.) and the mathematization of a domain, which systematizes these processes until building a theory where deductive proof is the unique source of truth. This is the case for many domains taught in physical sciences at secondary level (think for instance of Newton's theory of universal gravitation), but modeling does not necessarily require this systematization and often it is not possible (think for instance of climate).

While mathematizing processes can be important in modeling they do not by themselves differentiate modeling from mathematization. According to [START_REF] Roux | Pour une étude des formes de la mathématisation[END_REF], modeling can be characterized: (i) by locality: "whereas mathematization has a global aim, modeling is always local, and to say the least, fragmentary: the model assumes itself to be only one way among others to give an account not of reality, but of some of its aspects only";

(ii) by operativity: "whereas mathematization presents itself as a theory whose sole purpose is knowledge, modeling has an operative and interventionist component: a model is a device that allows one to act on what is being modeled, whether by supplementing, controlling or modifying."2 Drawing from [START_REF] Bouleau | La modélisation critique[END_REF], [START_REF] Lagrange | Les espaces de travail connectés : une perspective nouvelle pour la modélisation dans le secondaire[END_REF] extends this characterization:

A corollary of (i) is the multiplicity of models for a given reality. The example of automobile traffic helps to identify a variety of models each taking into account aspects of traffic complexity and measurement (…) Concerning (ii) the operative character of the model should not hide an "interpretative" component which concerns the subject engaged in modeling and thus its social context. This component distinguishes modeling from mathematization, where deductive reasoning sets aside societal issues.

Distinguishing between modeling and mathematization, epistemological studies thus allow characterization of modeling by (1) a plurality of models which makes possible to account for local aspects of a reality (2) 10 operativity and therefore proximity to simulation, (3) a subjective and social interpretative component. [START_REF] Varenne | Chapitre 1. Épistémologie des modèles et des simulations : tour d'horizon et tendances[END_REF] insisted on varied types of simulation. As Varenne put it, a simulation can be just a "phenomenological model" giving account of visible functioning, or it can be a "second order model" when it is based on a previous modeling. This is consistent with different roles that digital simulation can play in modeling [START_REF] Derouet | Modeling tasks and mathematical work[END_REF]. The first one occurs when building a real model with a simulation close to the initial situation (urn model or a calculator used to simulate rolls of dice or coin, in probability). The second role presupposes a stronger mathematical expertise in the MWS of the domain at stake as, for example, the implementation of an algorithm.

Finally for [START_REF] Bouleau | La modélisation critique[END_REF], the contribution of mathematics is in the work of the mathematician, that he describes as the search for simple ideas without which the models would be too complex to be interpreted. In other words, mathematicians quantify and formalize, but it is not for the sake of quantification and formalization, or to serve an idiosyncratic way to reach a solution, but as means to express simple ideas for other specialists. This is consistent with the Huincahue and Vilches (2019) observations reported above. Beyond simplification, the authors point out that mathematical modeling allows inexpensive experimental work as explained by a professional they observed: "Imagine the environmental engineers who model the movement of suspension particles from one continent to another … how can you measure that? Probe balloons in the stratosphere are very expensive ... then you model it!"

An example

Inspired by the above studies, [START_REF] Lagrange | Les espaces de travail connectés : une perspective nouvelle pour la modélisation dans le secondaire[END_REF] gives the example of a classroom situation for 12th grade students aiming at finding a function that models the main cable of a suspension bridge. In a first phase, the students looked at pictures of bridges and could easily conjecture that a parabola "fits" the cable and that they would be able to find a function for a given bridge by adjusting parameters. This would be a phenomenological model that possibly allows solving some concrete problems. In parallel, the students did not show a deep understanding of principles of bridge design. For instance, they overlooked the role of the suspenders (vertical cables equally spaced that connect the main cables and the deck). The goal of the situation was not principally to solve a problem, 11 but rather to understand principles of bridge design and to develop the associated capacities in mathematics and physics. Therefore, further phases of the situation offered students tasks to investigate four models of a suspension bridge and learn about the cable properties. A first model was based on a mock-up of the bridge. The deck was represented by a succession of equal weights, suspended to a cable at equal horizontal distance. Such a mock-up embodies the idea that there is no compression in the deck, a fundamental property of suspension bridges. It is abstracted into a physical model of the evolution of tensions along the cable by way of recurrence formulas. A second model was a broken line in analytic geometry. Recurrence between tensions at suspension points at equal horizontal distance allows computing recursively the coordinates of these points. As a third model, a computer simulation made this computation dynamically. It displayed the broken line with dynamical variations, when parameters like the number of suspension points and the value of the horizontal component of the tensions were animated. Finally, a continuous model was obtained by considering an infinite number of suspension points. It was a quadratic function, as conjectured by students, but obtained by using fundamental concepts in physics and mathematics in interaction rather than by phenomenological observation. In this model, too, the horizontal component of the tensions could be a parameter. This showed the importance of this quantity: suspension bridge designers have to consider that increasing the height of pillars reduces the tensions and then the necessary resistance of the cable and of the anchor points, but also increases cost and constraints. The professionals, thus, have to make a balance between those two parameters in a given geological and geographical configuration. Although the four models can be seen as organized linearly from the mock-up to the mathematical function, this organization cannot be thought of in a modeling cycle, as all four models are both real and 12 mathematical. In contrast, this characterization of four models echoes the above consideration of a plurality of models for a given phenomenon, as well as the role that mathematics plays in these models. Mathematizing occurs in each model. It simplifies, but does not hinder the physical nature of the bridge and the operativity of the model. It makes visible the parameters that are crucial for professional designers. The example is also consistent with the Huincahue and Vilches (2019) definition of mathematical modeling as an activity in which two or more disciplines converge, because a branch of physics (statics) is deeply at stake. Further, this example is also in line with Wakes' (2015) concern for modeling activities that couple reality and mathematics and reduce the gap between school and workplace.

Modeling: connecting a plurality of working spaces

In the preceding section the idea of work helped to overcome a too simple dichotomy between reality and mathematics and show the stakes brought about by insights from the workplace and from epistemology. In this section, we look more closely at the potential of the idea of working space. First, we consider the contribution of the three dimensions: semiotic, instrumental, and discursive. We begin with a famous citation from Galileo:

"Nature is written in that great book which ever is before our eyes-I mean the universe-but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language." Galileo's thinking is not far from the above-mentioned idea of mathematics as means to make the world intelligible, provided that "great book" and "language" are understood as metaphoric forms. Unfortunately, they are often taken literally, as mathematics is considered to be a language into which reality has to be translated. 3 A great strength of the MWS framework is that it supports avoiding this reduction. Mathematical work on a model is certainly not possible without means for representing. However, these means cannot be thought of as a standard preex-isting mathematical language, as they have specificities and their elaboration is part of the work. Moreover, the work on a model is not reduced to symbolic manipulation. All studies mentioned above, both at the workplace and in the educational context, consider instruments which are specialized for a domain and have a mathematical basis. Finally, working on a model also involves reasoning in both the domain of modeling and in mathematics. In short, MWS is a good candidate for conceptualizing the work on a model. This is because the three dimensions take into account respectively the semiotic systems, the instruments, and the modes of reasoning used in this work. Furthermore, the notion of space encompasses the specificities of these dimensions with regard to the model beyond their mathematical nature.

In the MWS framework, the plurality of models for a given reality has led authors to consider a plurality of working spaces. In the "Alphonso" task [START_REF] Kuzniak | Développer un travail géométrique complet et cohérent chez les étudiants de première année de master enseignement en France[END_REF][START_REF] Lagrange | Les espaces de travail connectés : une perspective nouvelle pour la modélisation dans le secondaire[END_REF], students have to compute the area of a piece of quadrilateral land in which the lengths of the sides are given. The observation shows each student building their own model after implicit assumptions. The student activity can be seen as the organization of personal geometrical working spaces adjusting their view of the task and their geometrical knowledge. This is also the case in the "Gutter" task, [START_REF] Montoya Delgadillo | Choosing a Mathematic Working Space in a modeling task: the influence of teaching[END_REF] presented in Chapter 5.2.2 of this book. The study reports on pre-service teachers solving an optimization problem: finding the best way to fold a sheet of metal in order to create a gutter that maximizes water flow. The pre-service teacher course was in four years numbered from one (younger students) to four (older students) and the study involved year one, two and four students. Only year four students had been taught calculus. The researchers observed differences between procedures favored on the one hand by year one and two students, and on the other hand by year fours students. They analyzed these differences with respect to students' personal working spaces. Year four students' personal spaces were close to a reference working space in calculus. In other words, these students recognized the task as a standard problem for which official calculus procedures exist and have to be applied. For them it was clear that the water flow depends on the choice of the length on one side, and this dependency was modeled as a quadratic function with the symbolism, instruments, and mode of reasoning attached to mathematical functions. In contrast, the other students' procedures varied. The researchers identified personal spaces the students arranged on purpose by using their knowledge in geometry and measure. Sometimes students based their answer on their (false) intuition 14 about a model of an optimal gutter without considering variation. For students who did consider variation, the model was not a function but rather a collection of configurations in which to pick up an optimum. In these studies, researchers interpret students' procedures by identifying different underlying models and associated personal working spaces corresponding to different understandings of a reality and to different conceptualizations in various mathematical domains.

The possibility of varied models and associated working spaces for a given reality was exploited in other studies to explicitly design tasks that put at stake transitions or coordination between specific domains corresponding to different a priori suitable working spaces. [START_REF] Derouet | La fonction de densité au carrefour entre probabilités et analyse. Une ingénierie didactique en classe de terminale scientifique. Actes du séminaire national de didactique des mathématiques 2017[END_REF], for example, considered suitable working spaces, respectively in statistics, mathematical functions, and continuous probabilities. We mentioned above the situation of modeling a suspension bridge proposed by [START_REF] Lagrange | Les espaces de travail connectés : une perspective nouvelle pour la modélisation dans le secondaire[END_REF] that draws from a plurality of models of the main cable. Lagrange explains that the work on each model is related to a specific working space with a specific system of symbols, specific instruments, and specific ways of reasoning. In a first phase of group-work, a study of each model was devoted to a specific group of students. A second phase mixed the groups so the students-each one an expert of a particular model, could find together links between the models and then make connections between the working spaces. The a posteriori analysis of an experiment was done by considering the various connections the students had made and the dimensions implied.

In these examples, the plurality of working spaces with no hierarchy between them echoes the work in interdisciplinary teams of professional modelers. Mathematics is involved neither as "the queen of sciences," nor as a pure "servant." Mathematical work clarifies and operationalizes models, but it also contributes to giving meaning to abstract mathematical notions.

Learning by modeling: questions for the MWS framework

In this part, our goals are to investigate questions dealing with the objectives assigned to modeling tasks or situations based upon the MWS framework. Further, we explore here the methods and concepts used in this framework to analyze a priori and a posteriori what students learn and how they learn. Objectives

We mentioned at the beginning of section 3 the stream of educational research following the 2002 ICMI study 14. With regard to objectives, this stream is characterized by a focus on modeling competences. It is reflected by the fact that six of the eight crucial issues proposed by [START_REF] Blum | ICMI Study 14: Applications and modeling in mathematics education -Discussion document[END_REF] mention these competences. Issue five stresses particularly that modeling activities "will have to 'compete' with other components of the mathematics curriculum." This is a difference with the studies adopting the MWS perspective. First, the situations investigated by Derouet (2019) and [START_REF] Nechache | L'articulation des genèses de l'Espace de Travail Mathématique dans la résolution de tâches probabilistes faisant appel à la modélisation[END_REF] are motivated by objectives clearly inscribed into the mathematics curriculum. Moutet ( 2019) is another example. His study puts the physics curriculum in view. It proposes and analyzes a teaching unit in relativity theory that puts at stake the physical notions of time and simultaneity. This is an example where mathematics is involved in order to make physical ideas accessible. As for Lagrange (2021) the objective is not that students reach a new curricular content, but rather that they understand more consistently notions already taught in calculus, algorithmics, analytical geometry, and physics. Also with regard to objectives, [START_REF] Psycharis | A framework for analysing students' learning of function at upper secondary level: Connected Working Spaces and Abstraction in Context[END_REF] analyze a situation that does not aim at specific curricular notions. This study focuses on developing students' functional thinking by working on covariation in four different working spaces. The proposed situation is based on the same gutter optimization task described above in our discussion of Montoya, Viola, and Vivier's 2017 research. However, there is a difference: Whereas Montoya, Viola, and Vivier characterize a posteriori personal working spaces by observing students' self-generated procedures, Psycharis, Kafetzopoulos, and Lagrange start from the a priori identification of four possible models. One first model is a sheet of paper that can be folded in order to make a mock-up of the gutter. The second is a dynamic geometry (DG) construction representing a section. The third model is also in DG but involves the covariation of measures. The fourth one is a mathematical function. Four working spaces are associated with the four models in order to analyze a priori and a posteriori students' work on the four models and then examine student understanding of covariation.

There is then a common concern in studies adopting a MWS perspective about making students consider notions taught in varied fields in synergy. This concern is reflected by the consideration of working spaces specific to each field. This focus on the knowledge to be taught clearly differentiates these studies from research putting at stake modeling competences. It could denote a similarity with theories such as realistic 16 mathematics education (RME). For RME, modeling is a means to bring students to mathematical concepts from a "reality" [START_REF] Gravemeijer | Context problems in realistic mathematics education: A calculus course as an example[END_REF]. Modeling intervenes in the RME approach at the moments of first meeting (of "reinvention") in mathematics. A model of some reality is built, and after that it becomes an abstract tool for doing mathematics. This tool, labeled model for, has a representational and an instrumental dimension derived from the model of, but representations and instruments belong to mathematics. [START_REF] Gravemeijer | Context problems in realistic mathematics education: A calculus course as an example[END_REF] explain that "students' final understanding of the formal mathematics should stay connected with (…) everyday-life phenomena." However, this connection is conceived as a background reference for students' practice of formal mathematics rather than as a relationship between a domain of reality and the mathematical knowledge at stake.

As a difference to RME, it seems that although modeling in the MWS framework is oriented towards mathematical knowledge, it has a potential for keeping a relationship between the mathematics learned and the various fields involved in the modelization and also, the social circumstances in which this relationship has been built. To go further, MWS studies on modeling could benefit from Wakes (2015, p. 684) reflection: (…) workplaces provide rich sources of (...) realities, (...) providing purpose and a meaningful context in which to understand and explore an emerging mathematical model. The mathematical models developed in this way have the potential to provide insight, crucially into both the reality and the mathematics in contrast to the RME approach that focused ultimately on understanding the mathematics.

In line with this, studies on modeling could break with a narrow conception in which modeling is an activity pursued individually and for individual competencies. As noted above, modeling at the workplace is a social activity and implies the identification of specific working spaces and the collaboration of specialists of each space. In this latter conception, the stake for students is not to reach individual abilities for solving real world problems, but rather to participate in a collective task. Here, students take advantage of existing knowledge and develop this knowledge by interaction with others.

Consequence on design: an example

In order to exemplify the opportunities brought by workplace contexts, let us consider an example from professional sea navigation. [START_REF] Vroutsis | Crossing the boundaries between school mathematics and workplace through authentic tasks[END_REF] proposed students a typical problem from this sector. Starting from a given position, a ship must pass through a dangerous area with underwater obstacles. The area was represented on a navigational chart and the captain had marked two landmarks on this chart as reference points. The students were asked to use the landmarks and specific professional measures (bearing, range, and horizontal angle) and tools (parallel ruler, divider) in order to find a way to steer the ship to ensure its safe passage. They then had to rediscover a navigational technique: describe an arc of a circle that avoids the obstacles by steering the ship and keeping a constant angle between the two landmarks. As a parallel aim, the students also learned about inscribed angles. The realistic context (original workplace problem), the practitioner's tools (measures) and the workplace constraints (rules) acting as boundary objects revealed the insufficiency of school taught knowledge. It thus highlights a discontinuity between formal mathematics and the genre of mathematics developed in the workplace. When we analyze this task, we see that the problem involves two working spaces: one is professional navigation and the other is geometry. The professional context marks the three dimensions -representations, instruments used, and modes of reasoning-of the navigation working space. Thus, it differs from a geometrical working space in the school context. A third working space, programming, could take into account the prevalence of programming in professional modeling and of electronic devices in contemporary navigation. In order to bring the working spaces to life and help students develop connections, group-work could be organized in phases like in the suspension bridge situation reported above. In a first phase of group-work, three different tasks could be distributed to groups of students. One task, proposed to groups of students labeled sailors, would be in a navigation working space. A professional would present the problem and the technique used to go through dangerous areas. Students would have to implement the technique to a practical case of navigation and discuss its effectiveness. Another task, proposed to groups of students labeled geometricians, would be in a geometrical working space. In this task, for two given points A and B, students would have to conjecture with the help of DG the positions of the points M, such as angle (MA, MB) would be a given constant. A third task could be proposed to groups of students labeled programmers. They would work on programming a trajectory for a point M with the constraint that angle (MA, MB) is constant.

In a second phase, groups would be comprised of one sailor, one programmer, and one geometrician. The tasks would be to explain and confront the work done by each participant in the first phase and then, write a mathematical justification of the navigational technique and of the program (fig. 11.4). A collective classroom discussion could follow that emphasizes the contribution of mathematics to professional techniques. A further discussion point would be the care that should be exercised when applying to the real world: Good sailors will constantly double check! With this type of design, modeling activities could articulate learning and socialization while contributing to the education of students as citizens and for their (future) professional/social life. This is a different way, as compared to modeling activities that aim primarily to foster individual modeling competencies.

19

Analyzing what students learn and how they learn

In the preceding sections of this chapter, we did not specifically separate epistemology and cognition. Furthermore, we characterized working spaces by the three dimensions (semiotic, instrumental, discursive). This is also the case of studies on modeling whose authors observe "connections" [START_REF] Lagrange | Les espaces de travail connectés : une perspective nouvelle pour la modélisation dans le secondaire[END_REF] or "circulations" [START_REF] Derouet | La fonction de densité au carrefour entre probabilités et analyse. Une ingénierie didactique en classe de terminale scientifique. Actes du séminaire national de didactique des mathématiques 2017[END_REF] between dimensions of the working spaces and analyze these connections or circulations as evidence of students' progress relative to the goals of the situation.

Beyond dimensions, the MWS theory (preceding chapters) considers three geneses that articulate poles of two planes labeled respectively epistemological and cognitive. The epistemological plane consists of human made entities (representamens, artefacts, rules of reasoning.) The cognitive plane is described in terms of processes (visualizing, constructing, proving). Each genesis can be viewed as the activation of a pole of the epistemological plane by an individual or a collective subject for the corresponding process in the cognitive plane and is analyzed with regard to the subject's cognition. [START_REF] Moutet | The extended theoretical framework of Mathematical Working Space (extended MWS): potentialities in physics[END_REF] considered an extended working space with a single cognitive plane, and two distinct epistemological planes, one in physics and the other in mathematics. Distinguishing two epistemological planes is consistent with the fact that in physics and mathematics, the three poles of the epistemological planes are specific to each field. A single cognitive plane presupposes that poles in the cognitive plane can be associated to poles of a mathematical epistemological plane, as well as to poles of a physical epistemological plane. This choice helped Moutet to identify how students develop knowledge through geneses involving alternatively poles of the mathematics epistemological plane and poles of the physics epistemological plane. With regard to modeling, Moutet's extended MWS is then centered on a single model (the Minkowski diagram) alternatively considered from a mathematical and a physical point of view. 4 Moutet also considered a model involving imaginary objects like a car traveling at nearly light speed.5 However, the elaboration of this model and the connection with the Minkowski model is not conceptualized in the MWS theory. This is different for Psycharis, Kafetzopoulos, and Lagrange ( 2021) who analyze how students progress in their understanding of covariation 20 by working on a succession of models. At the theoretical level, their study draws from [START_REF] Minh | Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level[END_REF] that introduced the Connected Working Spaces (CWS) framework. Minh and Lagrange were concerned that although at both the epistemological and cognitive levels, functions make sense because of their occurrence in many dissimilar settings, teaching actually favors algebraic representations. They developed the CWS framework in order to take the following fact into consideration: For students and teachers to fully consider each setting where functions can make sense, they have to think of it as a particular working space allowing a specific type of scientific work and specific conceptualizations. Their hypothesis is that adequate teaching situations should organize students' work in various non-algebraic and algebraic working spaces and allow connections between these.

As explained above, in the situation analyzed by [START_REF] Psycharis | A framework for analysing students' learning of function at upper secondary level: Connected Working Spaces and Abstraction in Context[END_REF], the students work in turn on four models of a gutter (sheet of paper, section in DG, measures in DG, mathematical function). Four working spaces are associated with the four models. The identification of connections between these working spaces in the three dimensions theorized in the CWS framework plays a role in the a priori and a posteriori analyses by focusing on key transitions occurring when students are introduced to a new working space and by specifying the dimensions involved. Psycharis et al. (ibid) present episodes in various phases of work. We take the example of a phase where students were introduced to the DG working space. We extract here first a dialogue between the teacher (T) and students (S) while working to construct a dynamic rectangle (ABCD) to model a section of the gutter, and then the authors' analysis in two parts.

The dialogue:

T: You will need one point for the lower part of the gutter and one point which describes the maximum folding. Then we need another point between these two points to describe every time the different folding, but first of all we need to find the restriction of the construction. S3: I propose to put point D in (0;0). S4: We have to create a point E as (0;10) in case we fold the metal plate in the middle so that we get a segment for positioning the free point C.

After creating C, the students observed the folding in order to find an expression for the x-coordinate of A. Most of the groups attempted to find it through solving the equation x+x+y=20 for y. Students from different groups commented: "I tried to create A, as (20-2*x ; 0) but it did 21 not work!", "We created A as (20-2*yC;0) and it worked!", "As for us, we created A as (20-2*DC;0) and it worked also!

The first part of the analysis uses the CWS framework. In order for students to take advantage of the DG possibilities they need to identify key elements of the model as geometrical objects using geometrical notations. The teacher's intervention is crucial, insisting on the choice of points defining a rectangle but the students also have their part: propositions for creating points E and D involving the constraints of the sheet, expression of the dependency of point A to point C. The work is in the instrumental dimension: adequate use of the instrument (DG) is at stake. The work also combines a semiotic dimension: students progressively integrate the notations of DG in order to express the x-coordinate of A.

For the authors, this first part of the analysis alone does not give precise account of what students learnt and how. That is why they choose to combine with the activity theory framework Abstraction in Context (AiC), see [START_REF] Dreyfus | The nested epistemic actions model for Abstraction in Context: Theory as methodological tool and methodological tool as theory[END_REF]. The abstraction in context theory was created to give account of conceptualization within contextualized tasks. This is how [START_REF] Psycharis | A framework for analysing students' learning of function at upper secondary level: Connected Working Spaces and Abstraction in Context[END_REF] introduce AiC:

AiC describes the process of abstraction by means of a model of three epistemic actions that researchers can be observe and analyze: recognizing (R), building-with (B) and constructing (C). Recognizing an already known mathematical concept, process or idea occurs when a student recognizes it as relevant in a given situation (...) Building-with comprises the combination of existing knowledge elements (i.e. recognised constructs) to achieve a goal (...) Constructing is carried out by integrating previous knowledge elements (constructions) by vertical mathematization to produce a new structure/construct (...)

Then the second part of their analysis uses AiC. Drawing from their experience with folding the paper sheet, the students see the need to define a rectangle through four points and also to distinguish the point that 'causes' the dynamic change of the construction (Recognizing). In order to organize the objects in the DG in a way consistent with the paper, the students make faulty and successful attempts to relate the coordinates of point A to measures dependent on point C (Building-with). In the end, different symbolizations for the coordinates of point A are suggested by different groups (Constructing) indicating students' progressive coordination of their preceding sensual manipulation of the paper sheet with the available notation structures of DG.

Analyses of the other steps confirm that at each step, students first connect a new working space to the former (Recognizing) and then de-velop conceptualizations (Building-with and Constructing) inside the new working space. Psycharis, Kafetzopoulos, and Lagrange conclude:

AiC is powerful here to make sense of students' progress but could not be put into operation without the structure provided by CWS. In addition, CWS distinguishes between three dimensions (instrumental, semiotic, and discursive) in students' work that sheds light upon the AiC process of abstraction.

Reporting on four studies in this section, we observed varied ways of dealing with students' cognition in modeling work. Two studies (Derouet, 2019 and Lagrange, 2021) assume that the connections or circulations between working spaces are evidence of students' progress. Another one [START_REF] Moutet | The extended theoretical framework of Mathematical Working Space (extended MWS): potentialities in physics[END_REF] takes the classical MWS approach in considering a cognitive plane and geneses. Whereas the fourth study [START_REF] Psycharis | A framework for analysing students' learning of function at upper secondary level: Connected Working Spaces and Abstraction in Context[END_REF], combines an activity theory framework with the CWS framework and concluded on a productive combination. There is no definite best method to address the question of what students learn and how they learn in modeling situations. However, it is promising to note that MWS offers internal means with the idea of the cognitive plane, as well as the ability to combine with a cognitive oriented framework.

Conclusion

We will conclude this study of the new perspectives opened by the working space theory by focusing on two fundamental constructs presented in this chapter that have proven productive: the idea of mathematical work and the three dimensions of this work (semiotic, instrumental and discursive). The idea of mathematical work helps to think of modeling without artificial separation between mathematics and reality. It then looks at mathematics as contributing to various models of a reality rather than as being confined to mathematizing reality into pure mathematical models. It also motivates consideration of modeling practices outside education. These practices offer a wide landscape with common trends. These include the plurality of models for a given reality where each model contains some mathematics, and the conception of modeling as a social activity involving specialists from different fields and people with various points of view. Models are plural relatively to the fields of knowledge involved (profession, physics, mathematics…) but also with regard to their operativeness. Analytic models coexist with algorithmic simulations, and different models benefit from mutual connections. The idea of a plurality of working spaces reflecting this plurality of models, of fields, and of points of view is productive in the MWS studies. Further, this idea allows addressing traditional mathematical fields. These include 1) probability, 2) topics involving physics and mathematics like relativity, and 3) wider issues like the understanding of covariation as a basis for functional thinking.

At present, social aspects of modeling are not very developed in math education. The plurality of working spaces can help in an educational context to organize in the classroom collaboration and a discussion between fields and between points of view. Inspiration can be found from situations of modeling at the workplace. In this conception, there is no opposition between objectives in terms of socialization and in terms of knowledge. Socialization is based on the recognition of various forms of knowledge and the subsequent necessity of discussion and collaboration. Considering the three dimensions of a working space helps to avoid a reduction of the modeling activity to a pure translation-and of mathematics to a language. As Psycharis, Kafetzopoulos, and Lagrange (ibid) say, the identification of these dimensions in each of the working spaces involved in a modeling situation provides the "structure" needed for well-grounded a priori and a posteriori analyses. Building upon this structure, it is possible to study students' cognition along the modeling activity. We noted above two possibilities. One is to try to characterize "geneses" associating poles of epistemological planes and a cognitive plane. The other is to combine with an activity theory framework.

Studies on modeling based on MWS are still small in number. Analyzing some of these helped us in this chapter to open new perspectives for the design of modeling situations and for making sense of students' learning in these situations. Beyond providing elements of a theoretical framework for researchers, we hope that this chapter will also be useful for mathematics educators at large at the methodological level. This could include thinking of contents or notions to be taught in relation with social settings in which these contents and notions exist and contribute in understanding the world. This could also include identifying varied models developed in these settings and their connections, conceiving spaces for working on each of these models and finally, organizing students' work and social interaction inside and across the working spaces. A wide field for experimentation and research opens up.

Fig. 11

 11 Fig.11.1 A modeling cycle in probability theory (adapted from Nechache 2018)

Fig. 11

 11 Fig. 11.2 A circulation between statistics, calculus and continuous probabilities working spaces.

Fig.11. 3

 3 Fig.11.3 The suspension bridge. A question and four models (adapted from Lagrange 2021).

Fig. 11

 11 Fig. 11.4 A suitable group-work organization for the sea navigation situation.

A MWS is organized on two different planes, epistemological and cognitive. Here the three "dimensions" involve both epistemology and cognition and helps us to characterize mathematical work in modeling without considering two-way processes between the planes (geneses). We discuss this later.

modeling in Education: New Perspectives Opened by the Theory of MWS

All translations from French sources are our own translations.

For instance, MEN/[START_REF]Les compétences mathématiques au lycée[END_REF], an official document for the French curriculum (lycée, grades 10 to 12) characterizes modeling as "Traduire en langage mathématique une situation réelle" (Translate a real situation into the mathematical language.)

For another example of an extended MWS about a model in engineering, see Cosmes Aragón, & Montoya Delgadillo (to appear).

 5 Paradoxically labeled "real model" in reference to Blum's modeling cycle.


Keywords: UAV, drone, deployment, potential field

Unmanned Aerial Vehicles (UAVs) can be a cost saving and easy to deploy solution to implement a temporary network infrastructure. They can act as access points in scenarios such as emergency situations, special events, or specific area monitoring.

Two main deployment families can be found in the literature. The first one, the location-based family, is based on the fundamental assumption that the network user positions are known. We do believe that this could not suit the most general scenarios. On the other hand, the location-independent family can not be as efficient as the first one. The main idea in this paper is to introduce a new crowd-based family, based on a probabilistic knowledge of user positions.

We then propose a self-deployment method built on a Coulomb's law analogy where users and UAVs act as electrical charges. Short range interactions are implemented through network sensing, while long range ones use a crowd-based approach. Some numerical results are depicted, showing the performance of this self-deploying mechanism as well as a comparison with a well-known clustering algorithm.

I. INTRODUCTION

In some scenarios where a network infrastructure needs to be deployed within short delays, Unmanned Aerial Vehicles (UAVs) can be an efficient solution.

Indeed, installing cables, antennas or satellite access require heavy investments. Besides, as drones are moving objects, they can reach areas inaccessible to humans, making them a practical solution for impractical zones. Of course, satellites can also fulfill this task, with a greater range of action, but cost, maintenance and deployment time are still problematic.

In this paper, we consider the deployment of a wireless mesh network of drones in a remote area where the quality of the network is too poor to establish communications. Thus, in this network of UAVs, the objective would be to convey voice, but also video or data from users in difficult to access areas. For instance in the context of a natural disaster in the mountains where the classical telecommunication infrastructure is no longer operable. Our aim is to be able to offer communication services to first responders, to facilitate the coordination and intervention. We would then have a network of drones that can be interconnected to the Internet via a satellite gateway.

The goal of this paper is to study a novel self-deploying mechanism for this drone network, capable of covering many users without knowing the exact initial user positions. The idea is to use a potential field like approach for the deployment. There are plenty of research done on mobile cells, access point or sensor deployment, but, as far as we know, limited research was done on drone deployment as a temporary access point without a priori information on user positions.

The paper is organized as follows. Section II presents a short state of the art of existing deployment methods. Section III introduces the scenario, offers a detailed picture of our method and the model implementation, Section IV presents the general results as well as a comparison between our model and a clustering algorithm, and in the end, some research tracks for future work and a conclusion are given.

II. RELATED WORKS

To better understand the deployment solutions already present in the literature, we classified them into two groups: location-based methods, i.e. requiring initial positions of nodes or users and location-independent algorithms.

A. Location-based deployment methods

In [START_REF] Bahc | On femto-cell deployment strategies for randomly distributed hotspots in cellular networks[END_REF] the authors propose a two-step approach, grid search, and enhancement prediction, to determine the best antenna positions for femtocell deployment. User locations and traffic demand have to be known in advance by the command center to maximize the gain, applying a centralized approach.

The authors of [START_REF] Chou | Mobile small cell deployment for next generation cellular networks[END_REF] are proposing another optimization approach called Mobile Small Cell Deployment (MSCD) that tries to maximize the total in-service time of mobile clients. The mechanism is a reduction for the facility location problem. After computing the best positions, the antennae will be placed randomly at some of these positions. Similar work is also presented in [START_REF] Qutqut | Dynamic small cell placement strategies for lte heterogeneous networks[END_REF] where the authors minimize macrocell resource usage by computing the best femtocell positions.

Wang et al. presented in [START_REF] Wang | Integrated coverage and connectivity configuration in wireless sensor networks[END_REF] an algorithm called Coverage Configuration Protocol (CCP) for sensor networks. It can determine the nodes' placement to maintain a configurable degree of coverage by intersecting the sensing circles. In [START_REF] Zhang | Maintaining sensing coverage and connectivity in large sensor networks[END_REF] another approach for sensor networks called Optimal Geographic Density Control (OGDC) tries to minimize the overlap between sensing disks to cover a given area.

All these approaches are centralized and static. To determine the optimal deployment, location calculations must be made in advance. If user hot-spots tend to change, the antennae positions have to be recomputed.

Besides, several placement strategies can adapt the deployment to mobile clients. These approaches were primarily based on drones as nomadic platforms. In [START_REF] Li | Drone-assisted public safety wireless broadband network[END_REF], [START_REF] Merwaday | Uav assisted heterogeneous networks for public safety communications[END_REF], [START_REF] Jiang | Dynamic uav relay positioning for the ground-to-air uplink[END_REF], [START_REF] Mozaffari | Optimal transport theory for power-efficient deployment of unmanned aerial vehicles[END_REF] Crowd-based positioning of UAVs as Access Points Dorin RAUTU, Riadh DHAOU and Emmanuel CHAPUT Universite ´ de Toulouse; INP; F-31071 Toulouse, France;

IRIT; F-31062 Toulouse, France {dorin.rautu, dhaou, chaput}@enseeiht.fr different optimal heuristics are presented for access point placement. The authors of [START_REF] Li | Drone-assisted public safety wireless broadband network[END_REF] search to place drones as repeaters, relaying access points to users. By solving the optimization problem, they maximize the end to end data rate. They examine the optimal position between access points and users. In [START_REF] Merwaday | Uav assisted heterogeneous networks for public safety communications[END_REF], the goal is to provide better coverage to public services; the access points are embedded on drones. By brute force search, the UAVs are placed in areas where the signal to noise ratio of base stations is weak. Knowing the user positions, they adjust the UAV placement to maximize the network throughput. The same technique is used in [START_REF] Jiang | Dynamic uav relay positioning for the ground-to-air uplink[END_REF] where the authors are maximizing the signal to noise ratio considering co-channel interference in a UAV mobile access network. In [START_REF] Mozaffari | Optimal transport theory for power-efficient deployment of unmanned aerial vehicles[END_REF], optimization heuristics are used to minimize the overall transmit power in a mobile network covering ground users. Even though the methods can adapt to user movement, the computation still has to be done in a centralized manner.

A different plan of attack for deploying nodes to cover users are the location-based clustering techniques. In [START_REF] Wang | Movement-assisted sensor deployment[END_REF], three deployment schemes are proposed based on Voronoi polygons. These solutions have the advantage to be distributed, each node calculates and moves to the most favorable position. In [START_REF] Heo | An intelligent deployment and clustering algorithm for a distributed mobile sensor network[END_REF], Intelligent Deployment and Clustering Algorithm (IDCA) is proposed using local clustering to place sensors uniformly by moving in an energy efficient manner.

Two similar optimization approaches are presented in [START_REF] Galkin | Deployment of uav-mounted access points according to spatial user locations in two-tier cellular networks[END_REF], [START_REF] Mozaffari | Mobile internet of things: Can uavs provide an energy-efficient mobile architecture?[END_REF] each of them based on k-means. The former shows an algorithm calculating UAV locations based on clustering. They minimize the overall users' distance to the cluster centroid. The latter computes k clusters by minimizing the squared distance between clients and drones, placing them in the center of clusters. Both have the benefit of adapting to user movement, but they use a centralized approach.

To summarize, there are many location-based algorithms developed either for mobile cells placement, wireless sensor networks or mobile access points using UAVs. They are either optimization based or clustering approaches, the main criteria for searching the best placement is either user oriented (inservice time, throughput), or network oriented (signal to noise ratio, interference, energy efficiency).

We do not consider using any of this solutions mainly because, in our system, the user positions are not known in advance and we consider that it is not feasible in real life. In the second part, some location-independent approaches are presented.

B. Location-independent methods

Regarding the solutions that do not use location information as input, [START_REF] Chen | On small cell network deployment: A comparative study of random and grid topologies[END_REF] presents a random mobile small cell placement approach. The idea is to place picocells in a given area, randomly. The authors compare this method to a grid placement showing that the performance loss is insignificant compared to planning costs for grid placement.

The authors of [START_REF] Mozaffari | Efficient deployment of multiple unmanned aerial vehicles for optimal wireless coverage[END_REF] propose to cover a given area, modeled as a circle by the use of the circle packing theory. The purpose is to place several drone access points as so each coverage radius helps to fill the area. By adapting the transmission power, they make sure that the circles don't overlap, so they do not create interference. A similar method was also presented in [START_REF] Liao | Two birds with one stone: Wireless access point deployment for both coverage and localization[END_REF]. Here, the authors optimize the number of wireless access points to fill the region, obtaining a total coverage. They are also using them to locate users. They control the localization accuracy by varying the AP's radius.

This kind of solutions cannot be put in place in our scenario as the number of UAV has to be significant to cover the entire area, wasting resources, in case users are concentrated in a single place, for example.

In wireless sensor network, another method is used besides those presented above. In [START_REF] Aitsaadi | Artificial potential field approach in wsn deployment: Cost, qom, connectivity, and lifetime constraints[END_REF], [START_REF] Mills | A brief survey of self-organization in wireless sensor networks[END_REF] another deployment scheme for sensors is presented, using the physical principle of potential fields. Like charges repel, and opposite charges attract. An optimization approach is used in [START_REF] Aitsaadi | Artificial potential field approach in wsn deployment: Cost, qom, connectivity, and lifetime constraints[END_REF]. The authors are using the potential field approach to maximize the distance between sensors and to minimize the repulsion interactions. A similar technique is used in [START_REF] Mills | A brief survey of self-organization in wireless sensor networks[END_REF], where the authors place sensors acting as electric charges influenced by other sensors or obstacles. They manage to cover the given area with minimal movement; the sensors move only when a force is exerted on them.

A summarized version of the related works is presented in figure 1.

Our objective is to build a versatile solution, and we believe that it should be crowd-based, distributed and dynamic. For these reasons, our method is mainly based on potential field law. In the articles cited above, this solution has been used to minimize sensor movement and overlap. In our case, we aim to use this technique to search and find users on the ground as well as covering them in the most efficient way. The energy saved by optimal moving is an interesting side effect that will be investigated later.

In order to evaluate the consequence of not knowing user positions, we will compare our solution with a location-based one. We have chosen to use k-means based one as our solution could be thought as a distributed flavor of such an algorithm. We then hope to minimize the sources of bias.

III. PROBLEM MODELING A. Scenario

Our primary objective is to use a fleet of UAVs as a backbone for network connectivity. Users are scattered in a given area, and they can move (depending on the scenario). For this reason, we believe that location-based methods do not suit such a scenario. Because of their movements, UAVs may not be able to maintain a permanent connectivity and, as a consequence, any centralized or even cooperative method seems impossible. Accordingly, we will propose a distributed method.

As an example, one could imagine an emergency situation (earthquakes, fire, nuclear disasters, etc.). The UAVs can be used to help the rescue teams providing communication means as well as for observation (cameras on board). Another scenario is the deployment of a cultural or sporting event.

. Classification of placement methods

There again, the UAVs will help the organizers by providing communications, observation means or data collection (from sensors on the ground or the participants).

B. Crowd-based methods

In the most general scenario, user positions are unknown. For this reason, we do not believe that location-based methods could wildly be used. On the other hand, most scenarios exhibit a non-uniform distribution.

Location-based methods are not aware of this behavior, and the performance may suffer from it. To circumvent this drawback, we introduce a new class: crowd-based methods. The basic idea is that, even though the actual position of each user cannot be known with precision, we have some information that can help us to determine areas where users can be found. Such gathering areas can be imagined around specific spots depending on the scenario. We can also imagine crowd movements such as supporters during a sporting event, or a herd in wildlife observation. In this paper, we will focus on scenarios where users are gathering around some spots called Point of Interest.

C. Our approach 1) User position:

To represent the crowd phenomenon, we introduce the notion of Point of Interest (POI). Users are supposed to be close to POIs. A Point of Interest could be any special spot, depending on the scenario (a command center in emergency situations, a refreshment point in a sporting event, etc.). The position of a particular user is not known, but the well-known position of the POI give some statistical information about user positions. An important property of a POI is its attractiveness, which measures how close users are from it. We will give a formal definition later on. 2) Introduction: To fulfill our objective, we need the UAVs to move toward the users. Of course, we want a cost-effective solution, so each UAV has to cover as many users as possible. We will use the Points of Interest to estimate users position; our solution is then a crowd-based one. Furthermore, we want the UAVs to be able to act autonomously. These constraints lead us to use a model based on electrically charged particles interactions.

3) Model description: We want to place UAVs as access points above a large number of users with random positions in a defined area. UAVs are thus supposed to get as close as possible to a maximum number of users. For this purpose, we have chosen to represent their interactions with the help of a model inspired by Coulomb's law. In this model, a user is described as a positive electric charge and a UAV as a negative electric charge. UAVs are then attracted by users.

There are, however, lots of fundamental differences with the well-known law of physics. These variations come from the objectives of our model:

• Users should not be repelled by one another, so there is no interaction between them in our model. • We want the UAVs to move towards the users, but the users are free to move without any constraints, therefore they are not attracted by the drones. • In a real world implementation, UAVs can not be aware of out of range users, so in our model interactions take place in the sensing range, named d. • To implement multiple channels "particles" come in different colors. Those with different colors can not interact, whatever their charge is. • Particles can spontaneously change their color. With this option, we plan to implement some form of load balancing.

Figure 2 illustrates this model. Three UAVs (named U i ) cover seven users (A to H). Each UAV is placed at the centroid of its users, as a consequence of the sum of the forces, they exert on it. U 2 has no interaction with other UAVs, while U 2 and U 3 repel. D could change its color from orange to blue (associating with U 1 ).

Please note that the defined sensing range d should be chosen in accordance with the physical network used. If a user is detected in this area, the UAV will be then attracted to it. However, by setting d = ∞ our solution becomes a location-based method, as the drones detect every user.

4) Initial position:

We imagine that the drones leave from the management center, placed arbitrarily at (0, 0). They will activate the sensing mechanism to search the users. Their goal is to fly towards the POIs, searching for users. As soon as a UAV finds someone, it will stop over to cover the users. The major risk is that they will get attracted by isolated users on their way to the Points of Interest.

Of course, this problem increases when the attractiveness of the POI decreases. The more the users are gathered around the POIs, the less isolated clients could attract a UAV.

For these reasons, we imagine varying the position where the drone activates the sensing mechanism. This will help us to analyze if the initial position of the drone impacts the results. We can also imagine activating the UAV either at a random position or when it arrives over a certain POI. We call these two versions C d,R , when the drones are activated at a random position and C d,P when they are activated at the POI. d is the sensing range, previously described.

5) Location based version:

We plan to study the difference between our probabilistic location-based model and a pure location-based model. We modified our algorithm, for comparison reasons, by increasing the sensing range to infinity. Hence, we introduce here C ∞ when our model has every user positions. We then compare our method with a clustering mechanism like k-means. Since the algorithm imposes a global knowledge of users locations, we cannot limit its sensing range. Similarly, we call it k ∞ . In either case, when each algorithm finishes placing the drones, we will measure the coverage rate by taking into account only the users that are in the sensing range of UAVs/clusters.

In the next part, we will present how this model is implemented in our simulator.

D. Model implementation

The purpose of this section is to expand on the implementation choices made for simulation purpose. Our simulator was created in Java for easy access to a graphical interface.

The basic idea is to simulate the attraction and repulsion forces between users and drones as well as the coverage of UAVs over the users. Our model can be adapted to whichever wireless physical technology as the "electrical field" emitted by charges is based either on the signal power or the media access control messages like Wi-Fi beacons, for instance. The fundamental principle of the developed model is that the UAVs will be attracted by users, having opposite charges, and repelled by other UAVs, like charges.

Each user has a weight, the quantity of charge. This feature can be used to prioritize users, by type of demanded service or importance, firefighters, organizers, etc. Earlier we introduced the sensing range d. If two drones sensing disks cross each other, a repulsive force will exert on these UAVs. Both forces are computed by using Coulomb's law taking into consideration both charges and the distance between the two of them. Each drone movement results from the sum of this two vectors, attractive and repulsive forces. We opted for this solution so if a UAV is connected to lots of users and another drone crosses its sensing range, it will remain static. Only the intruding UAV will repel.

In our implementation, POIs are placed by using a bidirectional uniform distribution. Of course, depending on the scenario, more suitable distributions could be used. As for the users, we use the following algorithm. Each user belongs to a specific POI (in this paper, users are equally distributed among POIs). A direction is randomly chosen with a uniform distribution [0, 2π]. The distance from the Point of Interest then follows a normal distribution with standard deviation σ. We want to focus on the impact of the "attractiveness" of the POI on the efficiency of the drone positioning algorithm. For this purpose, we define the covering range, equal to the sensing range d and a probability p that the distance between a user and its POI is lower than d.

For each simulation, σ is then chosen as a function of p and d. Figure 3 shows p out as a function of p, where p out is the probability that a user is at a distance < r from any POI. We can notice some side effects:

1) the area is finite; 2) a user can be far from its POI but close to another one. These effects lead to a p out value greater than p. This impact is more significant for small values. In this paper, we focus on scenarios where users cluster around the POI. As a consequence, we will not take into consideration the small values of p.

Other meaningful parameters regarding the implementation are presented in Table I. In the next section, we introduce the performance results of our method, as well as the results of the comparison between C d , C ∞ and k ∞ .

IV. SIMULATIONS AND RESULTS

In this section, different test-cases with their results and conclusions about the proposed model will be presented.

A. General results

To start with, we need to check if the drones are indeed trapped by isolated users. We start with 5 Points of Interest and 5 UAVs, departing from the management center at (0, 0). In the results presented in figure 4 we clearly see that our assumptions are confirmed. As the UAVs leave from the same spot flying towards the POIs, they get "stuck" covering isolated users around (0, 0). As a consequence, the coverage ratio is low, specially for low values of p.

The need for diversity regarding the UAVs' start points is thus confirmed. Of course, to limit the deployment costs, all the drones will still leave from (0, 0). We will activate the user detection on each drone when it is already flying. Either when it is in a random position or over the POI.

In the next part, we study the impact of location information on the deployment.

B. Location-based versus location independent

In this section, we analyze the performance of our crowdbased algorithm, C d , and a location-based version, C ∞ . We can add this information by increasing the sensing range to ∞. We will then look at both location-based schemes (C ∞,P , C ∞,R ) and crowd-based schemes (C d,P , C d,R ).

By analyzing the comparison between C d,P and C d,R , in figure 5, we notice that adding some information on potential user positions, by taking into account the POIs, increases the coverage probability by 20%, even though there are 50% chances that the users are located near the point of interest.

We could be tempted to think that having the exact location of each user could be an advantage for covering everyone.

Because each drone has a limited coverage range, on the other hand, having this information could be a disadvantage, as a UAV could thus move forward to an unreachable user. Looking at the comparison between C d,P and C ∞,R or C ∞,P we can make the following conclusion. Detecting users only when they are in the drone's sensing range has the same performances as knowing all the user positions from the beginning. The interest of gathering all the exact positions is minimal compared to the complexity of the gathering method.

The measurement of the coverage rate is done by taking into account only the users that are in the coverage range, as explained previously. By not having any information as input to our algorithm, C d,R , the coverage rate remains low in comparison to the other schemes, hence the importance of POIs.

C. Equivalence between C ∞ and k ∞ in a POI based location scenario

As mentioned earlier, we want to compare the proposed model to other solutions. Our work can also be seen as a method of user clustering. Our solution is then compared to a clustering algorithm like k-means, a classical wildly used location-based solution.

We extensively analyzed the coverage rate of both algorithms by varying between 5 and 30 the number of UAVs. With an infinite sensing range for the drones, C ∞,P we can be as close as possible to k-means, k ∞,P . In figure 6 we can see that our model is performing exactly as k-means in this situation. Both are having the same information, they can cover the users in the same proportion, even if they use different approaches.

We looked then to C d,P . In figure 7, we present the user coverage comparison between C d,P and k ∞,P . Globally, our algorithm, without exact knowledge of user positions, but drones activated at POI, performs well in comparison with the clustering mechanism. In this figure, we notice that the clustering method have slightly better results than our algorithm. The initial position information helps k ∞ to better place the clusters' centroids, by taking into account the local information.

As the difference regarding user coverage is fairly small between our model and the clustering algorithm, we consider that our contribution is substantial as it demands little resources, no exact location information and no centralized computations. In the next section, we present our conclusions on our work as well as some future works.

V. CONCLUSION AND FUTURE WORKS

In this paper, a novel method of node deployment for a network of UAVs used as access points is presented. By looking at the existing solutions of deployment, two major classes can be distinguished, location-based and locationindependent solutions. With regard to our scenario, we choose to introduce a deployment method that is neither locationbased, nor location-independent. For this reason, we propose to describe it as a "crowd-based" method.

We came up with a solution based on Coulomb's law. The users are positively charged, and the UAVs are negatively charged, being attracted to users. After presenting the model, some numerical results are described. Also, we compared our model to k-means, a location-based solution. By looking at the comparison, we notice that our solution performs very well, even compared to a location-based algorithm, the benefit being that our proposal demands no exact location information and no centralized computations are needed to deploy the nodes.

We are currently working on some experimentations by implementing our model on UAVs, to confirm these results. A real-world implementation will introduce some bias we want to evaluate: there is, of course, no Coulomb's law available, and we need to use network sensing to drive UAV mobility.

Of course, user mobility is an important point we will study. Being distributed and easy to implement, our solution should be efficient in a mobile scenario. We also plan to implement a covered users' threshold. Under this limit a UAV will not stop over some covered users, continuing to search better coverage positions.

Besides, we plan to test our model with multiple physical channels by using the color scheme previously described. Our solution is built to behave as a distributed clustering algorithm in such a situation.

Furthermore, as all simulations detailed above are using some users equally distributed between the points of interest, we can imagine introducing some weight on each POI to proportionally distribute users among them. Another idea is to do an extensive study on the behavior of our method in the case that there is no point of interest defined.

Fig. 1

 1 Fig. 1. Classification of placement methods

Fig. 2 .

 2 Fig. 2. Illustration of our model

Fig. 3 .

 3 Fig. 3. Difference between theoretical and actual p

Fig. 4 .Fig. 5 .

 45 Fig. 4. User coverage C d,0

Fig. 6 .Fig. 7 .

 67 Fig. 6. Coverage comparison C ∞,P and k ∞,P