Jean-Marie Lantau
email: lantau@mathematik.uni-kl.de

Martin Bracke
email: bracke@mathematik.uni-kl.de

Effective factors of a teacher training concerning the implementation of interdisciplinary STEM-modelling projects

Keywords: Teacher training, STEM-modelling projects, interview study

The modelling of the functional principle of a Segway requires an understanding of physical, technical and mathematical concepts, which are usually taught at university level. Nevertheless, the focus of two two-day-teacher trainings lied in the preparation of 11 teachers and 3 teacher students, such that they can implement interdisciplinary STEM-modelling projects with their students of Grade 10 to 12. Since 5 school projects were implemented by 7 teachers and 3 teacher students during summer 2018 we focus on the detection of effective factors and obstacles of the teacher training, concerning the implementation of the modelling projects. This paper describes the design of a one-year-study, especially the concept of the two-day teacher training and the resulting school projects. We also present first results of a qualitative study focusing on the effective factors of the teacher training based on guided interviews, which were conducted with all participants.

Introduction

The question of a pilot school project [START_REF] Lantau | Mathematische Modellierung eines Segways mit Umsetzung in der Schule als interdisziplinäre Projektarbeit[END_REF] with 12 students of Grade 12 was if a STEMmodelling project concerning the functional principle of a Segway can be implemented at school. The comprehension of the functional principle of a Segway requires knowledge about modelling a Segway as an inverted pendulum to detect physical forces and energies leading to a mathematical model of a linear state space system. This system is controllable and therefore stabilizable by feedback-or PID-control which can be implemented practically to control a Lego Mindstorms  Segway (Lantau, 2017). The students in the pilot project learned many mathematical and physical concepts, mostly at university level, which are used in practice to control a Segway. Furthermore, we received the feedback of the supervising teachers that this kind of STEM-modelling-project shows their students the important role of mathematics in real-life-problems, but that they are not able to supervise such a STEM-modelling project on their own. This observation lead to the research question of a PhD-project, of how teachers have to be trained such that they are able to supervise this kind of STEM-modelling project on their own or in a teacher team. By this mean, in 2017 two two-day teacher trainings were offered for 9 teachers. The teacher trainings content was designed to follow the modelling cycle of [START_REF] Blum | How do students and teachers deal with mathematical modelling problems[END_REF] and will be described in the section teacher training. Based on the teacher training, four school projects were implemented with students of Grade 12 (Lantau, 2017). Three of four school projects concentrated on the comprehension of the functional principle of a Segway, so the students learned how to use the PDcontrol theoretically and practically to counteract tangential forces to stabilize a Segway. The essence of the first year's PhD-project was that the design of the teacher training was successful in a way that teachers are able to supervise STEM-modelling projects with their students. Therefore, we focus on the analysis of effective factors and obstacles of the teacher training concerning the implementation of STEM-modelling projects at school with students of Grade 10 to 12 by teachers and teacher students.

Research focus

In recent years some projects and research studies concentrated on the professional development of teachers concerning the teaching of mathematical modelling in regular lessons. In their work, Blum and Borromeo-Ferri (2011) stresses that teachers need to be aware of an optimal balance between the maximal independence of the students and the minimal help of teachers. In addition, the results of the LEMA project show that a professional development course has a positive effect on selfefficacy and on pedagogical content knowledge of teachers [START_REF] Maaß | LEMA -Professional development of teachers in relation to mathematics modelling[END_REF]. Gastón and Lawrence (2015, p. 9) investigated in their work that "the research does not reveal one perfect method or set of techniques that can prepare each individual teacher to effectively teach mathematical modelling as a transferable process". Nevertheless, most of the professional development courses for mathematical modelling concentrate on teaching teachers to use mathematical modelling processes in regular lessons or small tasks. For despite, the long tradition of mathematical modelling in Kaiserslautern focuses on interdisciplinary and real-life STEMproblems which students usually investigate in project-based learning environments such as modelling days and weeks [START_REF] Bock | Applied school mathematics -Made in Kaiserslautern[END_REF] it was not clear how a teacher training has to be designed such that teachers are able to supervise interdisciplinary STEM-modelling projects. Since one first approach of a teacher training concerning the mathematical modelling of a Segway seemed to be successful (Lantau, 2017), the research focus was set on the detection of effective factors and obstacles of the teacher training. In our work we refer to the model of Lipowsky, in which he describes factors and constraints to explain the effectiveness of teacher training courses [START_REF] Lipowsky | Lernen im Beruf. Empirische Befunde zur Wirksamkeit von Lehrerfortbildungen[END_REF]. Quite similar to Lipowsky's model, [START_REF] Wilson | Professional development for science teachers[END_REF] and [START_REF] Desimone | Effects of professional development on teachers' instruction: Results from a three-year-longitudinal study[END_REF] investigated 5 core features for professional developments, namely the content focus, an active learning process, a collective participation, coherence and a significant duration of the teacher training.

Nevertheless, since it exists only few research results concerning teacher trainings for interdisciplinary STEM-modelling projects our research question is to explore the effective factors of the teacher training such that interdisciplinary STEM-modelling projects can be implemented at school. We want to answer our question by the help of a qualitative study in which we used guided interviews based on Lipowsky's model. At first, the participants were asked to reflect their meaning and experiences towards STEM-modelling projects and later the interview focuses on effective factors of the teacher training which encourages teachers to implement this STEM-modellingproject. Furthermore, teachers were asked to describe obstacles during the teacher training. The participants (11 teachers and 3 teacher students) of the teacher training were interviewed after the training and those who implemented their modelling project as a result of the teacher training in summer 2018 (7 teachers and 3 teacher students) were interviewed again after their projects to detect successful factors after the transfer process of the teacher training. The interviews will be analyzed by the method of content-structured analysis [START_REF] Kuckartz | Qualitiative Inhaltsanalyse. Methoden, Praxis[END_REF]. Figure 1 shows the design of the one-year-project.

Teacher training

The teacher training followed the intention that teachers learn the modelling process concerning the understanding of the functional principle of a Segway actively. Therefore, the training started by an experimental phase in which the teachers got the opportunity to drive with a Ninebot Mini Street to get aware of a Segway´s functional principle. After the teachers made their driving experiences, the control process of a Ninebot Mini Street and a Lego Mindstorms  Segway were compared. Based on their own experiences and observations of the control process, the teachers set up a model for the functional principle of the Segway. During this phase, the teacher recognized that they are in their students' role while modelling a Segway, so they reflected what kind of modelling approaches can students have when they model the functional principle of a Segway. In a following phase the teachers considered possible questions which they can implement with their students concerning the context Segway. For example, such a question could be the exploration of a shortest and fastest driving of a Segway in a given course. Further contexts are to investigate the effect of the center of mass' location for the control process or to calculate an optimal route for a Segway tourist tour. The collection of possible project ideas gave the teachers the opportunity to think about possible interdisciplinary modelling projects for their students. After further project ideas were collected, the focus of the teacher training was set on the control principle of a Segway, which is described in Figure 2. Figure 2 shows that the design of the teacher training followed the modelling cycle of [START_REF] Blum | How do students and teachers deal with mathematical modelling problems[END_REF], in which each step is connected to a module, which has either a physical, technical, or mathematical focus. By this design, a bilateral learning approach was offered: First, teachers experienced an interdisciplinary STEM-modelling process on their own (active learning) and secondly, they reflected on each module, whether it can be transferred into their school projects, based on the knowledge of their students (coherence). The different modules are described in the following.

Physics module: Situation and real model

A first model for the control process of a Segway is set up by an inverted pendulum, which contains two mass-centers, one for the wheel axis M and one for the center of mass of the Segway m. The mass centers are connected with a massless pole of length l. The aim of the control principle is to counteract acting tangential forces by a motor force u of the Segway. One can describe the inverted pendulum by two approaches. Once, by analysing acting tangential forces, leading to a nonlinear, inhomogeneous ordinary differential equation of second order:

The second approach is done by describing the movement of the inverted pendulum by the Euler-Lagrange-formalism [START_REF] Goldstein | Classical mechanics[END_REF] in which the Lagrange function (difference of kinetic and potential energy) for the generalized coordinates position x and angle  is set up:

In a second step the Euler-Lagrange equation is calculated for both generalized coordinates:

This process leads to two equations, one describing horizontal forces and the other describing tangential forces of an inverted pendulum:

Mathematics module: Mathematical model and results

A mathematical model for the inverted pendulum is generated of the physical model by applying

Taylor-series-approximations of first order around which leads to the linear state space system

In order to stabilize the linear state space system a first control technique, the PID-control [START_REF] Sontag | Mathematical control theory: Deterministic finite dimensional systems[END_REF]) can be applied, in which the motor force u is chosen as a weighted sum of the generalized coordinates (proportional control), its time-integral (integral control) and its time-derivative (derivative control). The parameters for the P-, I-, D-control part are usually determined experimentally regarding some conditions that have to be fulfilled to generate asymptotic stable solutions for the linearized system. Based on the PD-control approach, one can analytically solve a linear differential equation of second order to elaborate constraints for the control parameters k P and k D in order to stabilize the Segway. Furthermore, an explicit Euler-method visualizes the control process of a Segway, which was implemented in GeoGebra.

Technics module: Real results and situation model

Even though the implementation of a simulation in GeoGebra is some kind of a technical product, a more ambitious technical approach is to combine the theoretical results with a practical control of a Lego Mindstorms  Segway. This can either be done by using preconceived Lego Mindstorms  robots and balancing programs [START_REF] Valk | Lego-EV-3-Roboter: Bauen und Programmieren lernen mit LEGO MINDSTORMS EV3[END_REF], or by developing a practical control in Lego Mindstorms 

School projects

As a result of the two-day teacher training 5 school projects took place in summer 2018. The structure and content focus of each project was different based on the learning group, the supervision and the time framework of the project days. The following table summarizes the school projects briefly:

Preliminary results

In our point of view, considering the brief exposition of the resulting school projects, one can assess the design of the teacher training successful concerning the implementation of different interdisciplinary STEM-modelling projects by teachers and teacher students. The focus of research is set on the investigation of effective factors and obstacles of the teacher training (i), as well as of the school projects (ii), such that this quite ambitious modelling project can be implemented at school. Therefore, the participants were interviewed individually after the teacher training and after their school projects. The process of content-structuring qualitative content analysis has started by analyzing all interviews along the main categories of effective factors, obstacles and assessment of the teacher training and school projects, as well as along school dependent factors, or the meaning and experience towards modelling projects by teachers in their daily life work. In this section we will give an insight into preliminary results, mainly considering effective factors and obstacles.

Effective factors

The analysis of effective factors was done by analyzing once theoretical core features of professional development like the content focus of the training, the active learning during the teacher training, the coherence, and the collective participation [START_REF] Desimone | Effects of professional development on teachers' instruction: Results from a three-year-longitudinal study[END_REF][START_REF] Lipowsky | Lernen im Beruf. Empirische Befunde zur Wirksamkeit von Lehrerfortbildungen[END_REF][START_REF] Wilson | Professional development for science teachers[END_REF] and secondly indicate effective factors inductively by elaborating sub categories during the analysis of the interviews. The first results hint that most of the teachers who wanted to implement a STEM-modelling project with their students reported that it is very helpful for them that they know that the teacher trainer will support them during the project days. By this fact they know that they can ask the teacher trainer for support if they feel unsure in supervising the project. Furthermore, the teachers indicated that the mathematical and physical background of the STEM-modelling project is very profound, so that their students get an insight where mathematical concepts are used in real-life situations (content focus). On the other hand, based on the teacher training, they feel sure when elaborating the physical and mathematical concepts with their students (active learning). Some of the teachers reported that they had a lot of fun during the teacher training to learn mathematical and physical concepts of university level (content focus). Beside the theoretical background of the functional principle of a Segway, many teachers reported that the simulation of the functional principle in GeoGebra is very useful for their students, so that they get a better understanding of the mathematical background, when they can compare their theoretical results with a simulation (coherence). The most important factor concerning the implementation of a STEM-modelling project however is, that teachers have seen a concrete project concept for an authentic real-life context, which can be described by mathematical concepts, which their students are able to learn with their knowledge (content focus). Finally, teachers indicate that it was very helpful for them to explore a specific modelling project, which can be implemented at school (content focus).

Obstacles

Since the implementation of this STEM-modelling projects requires a profound mathematical, physical and technical knowledge, most of the teacher reported that this project is very ambitious and complex to implement with students of Grade 10 to 12. As the description of the physical and

Figure 1 :

 1 Figure 1: The study design

Figure 2 :

 2 Figure 2: The design of the teacher training

Table 1 : The resulting 5 school projects

 1

mathematical modules show, it does not surprise that teachers indicate that the advanced level of the project is an obstacle for them and their students. Furthermore, some teachers indicated that they don't have any experience concerning the work with Lego Mindstorms  . Since the teacher training was designed with phases in which the teacher could work on their own ideas, they got an insight in technical and theoretical obstacles during the working phase. In addition to that fact, almost every teacher reflected that it is quite hard to prepare the modelling project, since the possible questions, that students might ask themselves in the project can't be foreseen.

Acknowledgment

This work is partially supported by the European Social Fund (ESF) project SchuMaMoMINT and by U.EDU (Technische Universität Kaiserslautern) as part of the national and federal project "Qualitätsoffensive Lehrerbildung", funded by the Federal Ministry of Education and Research.

Discussion and Outlook

The short description of the school projects and the section of the preliminary results point out that interdisciplinary STEM-modelling-projects can be implemented by teachers after a respective training. Although the analysis process of the interviews by the method of qualitative content analysis has not been finished yet, there are some hints that some of the theoretical effective factors for professional development, namely content focus, active learning, collective participation and coherence [START_REF] Desimone | Effects of professional development on teachers' instruction: Results from a three-year-longitudinal study[END_REF][START_REF] Lipowsky | Lernen im Beruf. Empirische Befunde zur Wirksamkeit von Lehrerfortbildungen[END_REF][START_REF] Wilson | Professional development for science teachers[END_REF], were also effective factors for the implementation of interdisciplinary STEM-modelling-projects. The first results of the analysis show that main effective factors for the implementation of interdisciplinary STEM-modellingprojects are the collective participation at the teacher training and the following collaborative supervision (i), the possibility of implementing different projects based on the context Segway (ii), the exposition of a concrete modelling-project (iii), the motivating product of a Segway (iv) and the support of the teacher trainer during the school projects (v). Furthermore, the teachers indicated that the teacher training was a necessary condition for the implementation of their school projects. The participants of the teacher training reported the deep mathematical and physical knowledge, which is necessary to supervise the project (i) as the main obstacle. The main obstacles during their implementations were the supervision of open-structured project lessons (ii) and a non-fitting learning group for the interdisciplinary school project (iii). Besides the first analysis of effective factors and obstacles of the teacher training, one main open question still is how to support teachers such that they are confident enough to implement STEM-modelling projects on their own. In order to answer this question, we asked teachers during the interviews to report us necessary conditions to implement further STEM-modelling projects in future. Fortunately, almost every teacher who successfully implemented their project reported that he or she has the principle intention to implement further STEM-modelling projects in future. First hints for necessary conditions for the implementation of further STEM-modelling projects are the collective support of colleagues during the project days and the external support, for example by offering specific teacher trainings, or the external support during the project days. Finally, it has to be stressed that the focus of this work is not on the implementation of modelling activities in regular lessons, since from our experience this complex and interdisciplinary STEM-modelling project is recommended to be implemented in project days.


M Bertrand

A Guitierrez-Chavida

E Plasari

H Muhr

N Vigier

Thermodynamic study of uranium oxalate precipitation solubility and supersaturation ratio determination

Thermodynamic study of uranium oxalate precipitation: solubility and supersaturation ratio determination

M. Bertrand, A. Gutiérrez-Chavida, 1 E. Plasari, 1 H. Muhr, 2 N. Vigier CEA, DEN, RadioChemistry & Processes Department, B.P. 17171, 30207 Bagnols-sur-Cèze, France, tel +33 466796648, e-mail: murielle.bertrand@cea.fr; 1 Reactions and Process Engineering Laboratory, University of Lorraine, 1 rue Grandville, BP 20451, 54001 Nancy, France ; 2 AREVA 3747C-1, 1 Place Jean Millier, 92084 Paris La Défense, France Oxalic precipitation is usually applied to process radioactive wastes or to recover actinides from a multicomponent solution. The driving force of precipitation reactions is given by supersaturation, which characterizes the deviation from the equilibrium state. Precipitation mechanisms, such as nucleation, crystal growth or agglomeration, are governed by supersaturation, which is therefore a crucial parameter to determine for process control. This work focuses on the description of a thermodynamic study of uranium (IV) oxalate precipitation in order, on the one hand, to develop a solubility model and, on the other hand, to estimate supersaturation ratio.

Uranium (IV) oxalate can be precipitated by mixing an uranium (IV)-nitric acid solution and an oxalic acid solution according to the following equilibrium:

U(NO 3 ) 4 + 2 H 2 C 2 O 4 → U(C 2 O 4 ) 2 . 6H 2 O + 4 HNO 3 (1)
The supersaturation ratio (S) is defined using the reagent activities (a i ) as follows:

(2) where eq denotes the equilibrium state. As uranium (IV) ions can be oxidized by nitrous acid HNO 2 , present in small quantities in nitric acid media due to the dynamic equilibrium of this solution, hydrazine nitrate is added to stabilize the uranium (IV) ions for several months.

As very few information is available in the literature dealing with uranium oxalate [1,2], an experimental study was performed in order to determine solubilities over a wide range of conditions. The data acquisitions were carried out by dissolution of uranium oxalate powder (Fig1) in aqueous solutions with different nitric acid, oxalic acid and hydrazine nitrate concentrations at controlled temperature of 25°C and protected from light. Experimental results are of the same order of magnitude as the values of the literature (see Fig2) in spite of experimental differences between the studies (duration of the experiment, analytical methods…).

The mathematical treatment of experimental results allowed a thermodynamic model for both solubility and supersaturation ratio of uranium oxalate to be developed. The comparison between the model and the experimental data points out a good agreement as shown in the figure 3. The thermodynamic model was then applied to compare two different approaches to determine the supersaturation ratio described in the literature: either from the free ion concentration and the ionic activity coefficient (eq 1) or from the total concentration of species using the mean stoichiometric activity coefficients (eq 3):

(3)

The comparison leads to the very interesting conclusion that both approaches are equivalent as they result in the same supersaturation values as shown in figure 4.

  Fig1: Uranium (IV) oxalate crystals (SEM)

Fig3

  Fig3: Comparison experimental/modelling Fig 4: comparison equations (2) / (3)


Marta Pla-Castells
email: marta.pla@uv.esirene.ferrando@uv.es

Irene Ferrando

Irene Ferrando Downscaling

Downscaling and upscaling Fermi problems

Keywords: Problem solving, measurement techniques, early childhood education, real context problems, heuristic strategies

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

In the last years, tasks based on real contexts are becoming increasingly important in primary and secondary school curricula. Even so, due to the complexity of carrying out this type of tasks in a classroom, both by teachers and students [START_REF] Cabassut | Difficulties in Teaching Modelling: A French-Spanish Exploration[END_REF], there are few studies that analyze the implementation of this kind of task in the first years of primary school. All this makes of special interest the study of the concepts related to the measurement of magnitudes and the estimation of large quantities in a real context, using the so-called Fermi Problems.

Theoretical framework

Fermi problems

Jonas Ärlebäck (2009) defines Fermi problems as "open, non-standard problems requiring the students to make assumptions about the problem situation and estimate relevant quantities before engaging in, often, simple calculations". According to the research presented by Ärlebäck and Albarracín in the previous edition of CERME, this definition is the most used in mathematics education research literature. Here we will focus on a particular kind of such problems: Big numbers estimation problems (Albarracín & Gorgorió, 2013).

Twenty years ago, [START_REF] Clements | Teaching length measurement: Research challenges[END_REF] already claimed that children, in their first experiences with length measurement, should be given a variety of experiences in order to develop measurement strategies. According to [START_REF] Hogan | Quantitative estimation: One, two, or three abilities? Mathematical Thinking and Learning[END_REF], an important part of teaching measurement concerns the development of estimation. Nevertheless, there are still few studies that refer to the way estimation is worked in primary school classroom [START_REF] Pizarro | Primary teacher' approach to measurement estimation activities[END_REF].

In previous works, other authors have made experiences based on the use of Fermi problems with students of the higher levels of primary education [START_REF] Peter-Koop | Teaching and Understanding Mathematical Modelling through Fermi-Problems[END_REF]) and very recently with students of the first cycle of primary education [START_REF] Gómez Bedmarz | Estimación de grandes cantidades, en primaria[END_REF]. In the present work we have tried to take advantage of environments close to the students to introduce some elementary concepts related to the calculation of a magnitude that is often complicated and leads to confusion: the perimeter of a polygon (Mason, Stephen, & Watson, 2009). Ärlebäck, Doerr and O'Neil (2013, p. 316) gave importance to the development of sequences of modeling tasks so that through these sequences students "engage in multiple cycles of descriptions, interpretations, conjectures, and explanations that are iteratively refined while interacting with other students". In this way, by constructing a sequence of modeling tasks, we can deepen the study of complex concepts in the first years of primary school. Based on the previous works of Pólya (1973) and [START_REF] Stender | The use of heuristic strategies in modelling activities[END_REF] we have also focused on the use of heuristic strategies directed by teachers when solving complex modeling tasks. We propose a design of a sequence of tasks based on the heuristics recommendations provided by Pólya (1973, p. xvi): "If there is a problem you can't solve, then there is an easier problem you can solve: find it." [START_REF] Stender | The use of heuristic strategies in modelling activities[END_REF] affirm that, ideally, students should use heuristic strategies during the modeling process that teachers should convert into strategic interventions. Since our experience is addressed to very young students, our aim is to allow them to approach a complex problem using techniques based on downscaling and up-scaling the original problem as a heuristic strategy.

Modelling sequences

Research objectives

Our work is in an initial phase. We will give some details about the first phase of an empirical exploratory study where we intend to describe a technique, called downscaling-upscaling, to design sequences of Fermi problems in the first years of primary school. Since this is an early-stage investigation, our objective is limited to: (1) Describe in detail the aspects considered in the design of the sequence of tasks; (2) Describe a qualitative exploratory analysis of the results of implementation with a natural group of second grade students.

Methodology. Design of the empirical study

Sample

Our study focuses on a natural group of 21 second grade students of primary education who have not yet worked on the concepts of area and perimeter nor have ever faced a modeling task. They have already worked on arithmetic concepts like addition and subtraction but they have very little notions about multiplication. The activity was implemented in May 2017.

Treatment: downscalingupscaling design of tasks sequence

The starting point of the activity is a very general question that, a priori, the students are not able to answer. The objective of establishing an initial problem is to observe that a complex problem that seems difficult to tackleestimate the number of people necessary to surround the towncan be simplified if we try to solve similar smaller problems where the resolution strategies are equivalent. Following Pólya's heuristics, we propose them to "solve a simpler problem". In this case, what we try is to reduce (downscale) the scope of the initial question, first to the schoolyard and secondly to their classroom, giving them the opportunity to make measures in a very close context. Once they have attempted a solution of the simpler problem, we foster the generalization of the procedures developed in the first part of the sequence (there the upscaling process starts). In the following, we give the details of the activities developed during each session of the experience. 1st session: A "Big Problem". The introductory session aimed exclusively to present to the students the problem to be solved. Thus, this first activity started with an open question that students did not know how to answer: "If we asked all the inhabitants of Sueca1 to surround the town by standing next to each other, could they do it or would they have to ask for reinforcements from the people of the next town?" At this point we just wanted to catch the attention of the students and identify some ideas about perimeter and big numbers conceptions.

2nd session: Progressively reducing the "Big Problem". The first step was to propose to the students to make a smaller human chain. In this case, they were asked to think about how to estimate the number of students needed to surround the schoolyard. At this point, we just ask them to think about possible strategies to obtain, graphically, a procedure to get an estimation, not to measure or even compute. The second step was to answer to the smallest question: "How many small wooden cubes do we need to surround a large wooden board?" Here, we try to show them that the previous problems were the same as this one. In this case, we provide them with manipulative materials in order to foster measeure strategies based on the iteration of the unit as explained in [START_REF] Barrett | Children's unit concepts in measurement: a teaching experiment spanning grades 2 through 5[END_REF]. 3rd and 4th session: We solve the problem of the schoolyard. During the third and fourth sessions, the students worked on the rectangular schoolyard. So, based on the strategies worked on in the previous session, the students would try to solve a bigger problem. They had to estimate the number of students needed to surround the perimeter of the yard. In these sessions, we provide students with one-meter-long ropes.

5th session: We work on scaled ground plans. In order to answer the "Big problem", it is necessary to introduce the perimeter measurement on scaled plans. Therefore, the fifth session was devoted to work on a scale plan of a relatively small space close to students; a garden close to the school. We asked them to use the ruler and, using the scale and the estimations made in the schoolyard in the previous session, to estimate the number of people needed to surround the yard. 6th session: Solving the big problem. During this session each group of students was given a map in A3 format of the town of Sueca in a grid in order to finally give an answer to the "Big problem".

Data collection and data evaluation

The experience was organized sequentially through five activities that were carried out over six sessions of 50 minutes. Students worked sometimes individually and sometimes in 4 groups of 4 children and one group of 5 (we will specify this in the following section). The teacher who conducted the sequence had previous experience in modelling activities and was working under the supervision of two researchers. During the group work sessions, we video-taped two groups, and we audio-taped the other groups. In Session 1 and at the end of the Session 6 there were whole group discussions that were also video-recorded. In our analysis we have also considered the written productions (individual and grouped ones).

Since this is an exploratory study, we are limited to analyze the actions of students' groups to each of the tasks. In fact, the sample is not big enough to establish a grounded analysis. That is why we will not include any quantitative analysis; we'll just describe some aspects that we have considered important for further research. Our objective, at this point, is to identify whether the design of the sequence allows students to face, without blockages, the resolution of each task. We want also to identify, if possible, different strategies for solving equivalent problems and their evolution throughout the sequence. In the following section we summarize the description of the results for session.

Description of the results

First session. The objective of this session was to collect general impressions of the students about a problem of estimating a big number. It is not expected that they will base their answers in mathematical arguments. Here are some of the answers collected: "In the village there are too many people (S1)", "There won't be enough people (S2)", "There are many people who work (S3)", "The planet is very large and we would not be enough (S4)" and "If we put one next to the other and put also the dogs and cars, yes we will be enough (S5)".

Since students were not asked for more than an unjustified spontaneous response, none of the reflections contain references to neither calculations nor mathematical concepts. This first activity allows us to detect difficulties linked to the understanding of the question (it is not clear in the case of the student S5), problems associated with the concept of perimeter (this we observe, for example, in the answer of the student S4) and the process of obtaining an estimation (response of student S3).

Second session. 1st part: Let's reduce the problem: the schoolyard: "To surround the yard, are there enough children in this class? Or would we need to ask for help from children in the other classes?" The students, working individually in the classroom, represented graphically how they would make the estimation. In the productions shown in Figure , different schemes are observed. In the third one there is some confusion between the perimeter and the area since the student tries to cover all the space of the schoolyard with children and not just surround its perimeter. Another element detected in this part of the experience is the difficulty of the students to simultaneously handle elements of very different dimensions (the sides of the yard and the linear space occupied by a person). Indeed, when we asked the children to explain their answer from their drawings, the unanimous answer was that they had to count the drawn "dolls". Anyway, this part of the experience was rich to detect difficulties (linked to the difference between perimeter and area) and different levels in the resolutions.

As we have already noticed, the problem was still big for the students. Once all the students had graphically described their strategy to estimate the result of the problem of the schoolyard, workgroup began. At each group, students shared their ideas they previously represented. In order to help them better understand what they had done, they were encouraged to estimate the number of students needed to surround their tables. Here we observed that some students infer the most effective way to find the perimeter of a rectangular surface from the measurement of two unequal sides: add them and then make double. Once the notion of perimeter has been introduced intuitively, the second part of the session aimed to further reduce the initial problem.

2nd part: Deduction from manipulation. To answer the question "How many small wooden cubes do we need to surround a large wooden board?" only 5 small cubes were provided to the children. The boards provided to each group differed in their dimensions. In some of them the solutions were whole values, while in others there was too much or too little space, in order to observe how the students faced the problem.

Some groups proposed strategies based on the cyclic movements of the group of 5 cubes in order to count the number of cubes needed to cover one side. This strategy is interesting because the idea of an effective five-in-five counting is introduced and this improves the strategy of exhaustive counting. The concept of the measurement and estimation of the perimeter of a polygon was now formally introduced to the students.

3rd and 4th session: Each group was provided with a pair or ropes of the same length (one meter) in order to solve the problem of the number of children needed to surround the schoolyard. To clearly describe the two strategies observed, we will focus on the description of the groups that we will call Group A and Group B. One of the groups was totally lost and was not able to finish this activity.

The students of Group A counted effectively (making repeated sums of four-by-four) the number of children needed to cover the four sides of the rectangle that makes up the schoolyard using their bodies as a unit. They measured two unequal sides of the yard (in which they fit 29 and 38 children), then they doubled the values and then added both values. This group of students obtained as a result that 134 children were needed to surround the schoolyard.

Group B, although they obtained the measure in "children" of one of the sides of the yard, chose to measure the uneven side using the ropes. The students measured first with the ropes, and then they were urged to find out, from that measurement, the measure in "children". Thus, during the development of the activity, Group B students obtained a measure of "9 ropes and 3 children" for the minor side of the schoolyard. In this way, from their first measurements, the teacher asked the students to try to find out how many children fit in each string. This part is particularly interesting because two complex aspects are worked on in a simple manipulative way: the notion of linear density (how many children complete a rope?) and the change of units (how to transform a measure given in big unit -ropein a smaller one -child-?). The students estimated that they could cover each rope with 7 children. Thus, they faced the question: "If we know that 7 children fit in each string, how many children do we have in 9 strings + 3 children?" After discussing this problem, the students came to the conclusion that the operation to be performed was to multiply 9 × 7 and then add 3. In this way, measuring the two unequal sides of the schoolyard, they infer the total of children that they needed to surround it: they estimate 292.

5th session: Work on scaled ground plans. During this session the students worked individually, each of them was given a scaled ground plan, a 15 cm graduated ruler and a calculator. Since they had already worked with the concept of perimeter, the first part consisted in answering the question: "How many centimeters does the perimeter of the garden measure in the ground plan?"

We did not identify any problem in the use of the graduated ruler (each side had a whole number measure), but we observed different strategies to obtain the total perimeter in centimeters. Some students followed the strategy of measuring each of the sides and adding the successive measurements of the sides as they got them (using the calculator). Others preferred to take note of the partial measures and, finally, add them. Once they had obtained the perimeter of the garden on the ground plan, they had to infer the real perimeter from the use of the scale (each centimeter of the plane represents 5 meters of reality). With the help of the calculator and after a whole group discussion, the students deduced that they should multiply the measurement found in the plan by 5. Next, they were asked to estimate, based on the measure of the perimeter in meters (approximately 360 meters), the number of children needed to surround the garden. Since in the previous session they had found, experimentally, that in each one-meter string 7 children can fit in, they simply had to 360 by 7. The children were very surprised by the result: they could not imagine that 2520 children were needed to surround the entire perimeter of the garden! 6th session: Solving the big problem. Before starting to do the calculations, the difference between a map and reality was explained again. Some students had difficulties with this concept, but we observed that those that had understood this idea were able to explain it to their classmates. Anyway, the intention of using a grid was to facilitate the use of scales, so the students were told that each side of each square of the grid corresponded to 250 people in line. Figure shows a picture of the students working on the map. First, the students obtained an approximation to the perimeter of the town using the side of the square as unit. Since we told them that around 250 people fit in one square side of the grid, they just had to make one operation: multiply (using calculator) their previous result by 250.

The obtained results varied between 51500 and 64500 people.

At the end of the activity, it was important that the students could validate their solution. For this aim, we used the Google Maps tool that allows to measure distances on a plan. Erreur ! Source du renvoi introuvable.The perimeter of the town of Sueca measures approximately 10.5km, that is, 10500 meters. Since the students knew -from the resolution of the previous taskthat 7 children fit in each meter, it was enough to multiply by 7, being the final result 73500 children. Since the number of adults that fit online in a meter is surely lower than the number of children, the result obtained by the students working on the grid was quite approximate. Therefore, the inhabitants of Sueca (around 28000 inhabitants) would not be enough to surround the perimeter of the town.

Conclusions

The procedures associated with the measurement of magnitudes, particularly the measurement of lengths, are strongly linked to estimation processes. The choice of the appropriate unit, the use of a specific unit, the strategy of iterating a unit, more than one unit, changes of scale (moving from one unit to another), etc., are, among others, procedures associated with the act of measuring. But in addition, in real life problems, these procedures also appear when making estimations. That is why, in our current research -which begins with this work-we intend to design a sequence of tasks whose starting point is a problem of estimating large quantities that allows us to introduce, at an early age, procedures linked to the length measurement.

We have presented here the design of a sequence of tasks that starts from a Fermi problem that is unapproachable for students. Thus, from two elementary heuristics of Pólya's work, a sequence of tasks is constructed. The sequence begins with the simplification (named downscaling in our context) of a large problem. In this way, when students are able to develop a procedure in a reduced context, we encourage them to generalize (upscaling process, in our context) that procedure to apply it in a larger context. We have implemented the sequence with a natural group of 21 second grade students. A first qualitative analysis of this intervention allows us to identify some aspects of interest to work on in the future. The following are the most interesting ones. At the start of the sequence, in the second session, we have detected that the children have difficulties in differentiating the reality from the representations of the real world. Thus, they try to find out the number of children that fit on the perimeter of the track by simply counting the children (or their abstract representations) in their own drawings. This reminds us of the importance of introducing, even in a very simplified way, the idea of scale. We have also observed some difficulties in the process of using physical elements as units of measure. This is why we think that our proposal, and the future versions we'll make of it, has to foster students to make use of different units (cubes, children, ropes and, finally, graduated rules). In this sense, we have observed that, in the third and fourth session, a group of students was totally lost. The problem was not, apparently, the mathematical activity itself, but the fact that they were not used to work outside the classroom. This is an aspect that should be considered in our future work.

The proposed sequence has also allowed us to observe the evolution of the students. Given that this is not an isolated activity but a sequence, some of the students had the opportunity, in the next activity, to assimilate or discover the procedures they were working on. In addition, they were able to put them into practice in another context. This seems to us a strong point of our sequence, but we will have to test its efficiency by working with more students.

Furthermore, the sequence design aims to support the teacher's task that, in many cases, feels overwhelmed to bring this type of activity to the classroom. In this way, its own design serves the teacher as a guide to be able to offer what [START_REF] Stender | The use of heuristic strategies in modelling activities[END_REF] call strategic interventions. Finally, we want to highlight that the designed activity allows working the validation phase that, from our perspective, is fundamental when solving any problem of real context.

Figure 1 :

 1 Figure 1: Some of the schemes made by students in the first part of Session 2

Figure

  Figure 2: Working on the gridded map

A Spanish town with around

inhabitants.

Acknowledgements

Research supported by the project EDU2017-84377-R, Programa Retos de la Sociedad of the Agencia Española de Investigación and FEDER.


Amrit B Poudel
email: amrit.poudel@uia.no

Pauline Vos
email: pauline.vos@uia.no

F Leron Shults
email: leron.shults@uia.no

Students of Development Studies learning about modelling and simulations as a research approach in their discipline

Keywords: Development studies, mathematical modelling, meta-knowledge, Schelling's segregation model, simulation-based research

Researchers in the social sciences are increasingly using modelling and simulation (M&S) as a research approach. They create virtual worlds to discover relations across variables, and to test theories and potential policies. We introduced this research approach to students in the department of Development Studies at our university. The goal was to investigate the way in which such students can gain meta-knowledge about M&S-based research, that is, general knowledge about its nature and rationale. We organized a seminar to introduce the research approach and illustrated it with a simulation of the behaviour of agents with varying levels of tolerance towards their outgroup neighbours (based on Schelling's segregation model). We analysed students' interactions through a socio-cultural lens. Students were able to gain meta-knowledge about M&S-based research, which they judged as useful for their future as professionals when working on development projects.

Introduction

A growing number of universities are running programs in Development Studies, a relatively new, interdisciplinary field building on economic and social sciences. This discipline focuses on issues regarding regional, national, and global development, such as food security, health, energy, and migration. Graduates from Development Studies departments often find placement in organizations whose agendas relate to social responsibility, sustainability and economic development (e.g., UNESCO, FAO). Research published in the Journal of Development Studies utilizes both quantitative and qualitative methods to shed light on both macro and micro variables that impact economic and social development, typically focusing on less affluent regions. Increasingly, researchers in Development Studies use mathematical models to simulate complex social and economic systems. For example, [START_REF] Kumar | Caste and credit: A woeful tale?[END_REF] used survey data from bank loan applicants of various castes in rural India to create a model that enabled them to run simulations of different hypothetical scenarios; they found that lower loans were given to farmers from lower castes but, surprisingly, this discrimination did not affect owners of small farms. This research approach, in which modelling and simulation (M&S) is utilized, typically involves the following steps. Researchers begin with a real-world problem revealed by statistical, ethnographic or other analysis. Often, these problems are of particular interest to politicians and others attempting to improve the conditions for those in less affluent contexts. Researchers then identify relevant variables and construct a causal architecture that reflects both insights from theoretical literature and findings from empirical data. This work results in a mathematical or computational model. Using software such as NetLogo, they then simulate in the virtual world the phenomena observed in the real world. In this way, the real world data help to validate the model. However, the main goal of M&S-based research approaches is not to create models, but to answer 'what if?' questions by varying parameters in the model. Researchers can create a variety of scenarios and run a large number of simulations, often visualized in graphs, in order to discover complex interactions in the relevant socio-economic systems and, in some cases, to 'predict' the future behavior of those systems under certain conditions.

The steps in a M&S-based research approach can be illustrated by the modelling cycle in Figure 1, which is an adaptation from [START_REF] Greefrath | Mathematical modelling with digital tools: A study on mathematising with DGS[END_REF], and based on earlier work by [START_REF] Blum | Quality teaching of mathematical modelling: What do we know, what can we do[END_REF] and Kaiser (2014). Researchers start by investigating real world data and potential causalities to build a mathematical model, after which they run simulations to control whether the model aligns with the real data. They will iteratively improve the computational model until it fits the data, thereby repeatedly 'going through' the modelling cycle. In a subsequent phase, they ask 'what-if' questions and experiment, based on givens in the real-world (e.g., possible policy measures). By varying parameters in the model, and running new simulations, they obtain mathematical results that they translate into real results. After publishing, their results may be implemented and, possibly, solve real world development issues. [START_REF] Djohari | Breaking other people's toys': Reflections on teaching critical anthropology in development studies[END_REF] and [START_REF] Engel | What do they need to know? Core skills for postgraduate development studies students[END_REF], curricula emphasize the teaching of academic theories (e.g. social justice theories), critical and anti-colonial thinking, or skills useful for future development workers (e.g. project management). Most universities offering Development Studies include a course on research methods, typically focusing on qualitative methods. Only a few require statistics, since quantitative methods are known to be a hurdle for many social sciences students [START_REF] Onwuegbuzie | Academic procrastination and statistics anxiety[END_REF][START_REF] Zeidner | Statistics and mathematics anxiety in social science students: Some interesting parallels[END_REF]. At this point, curricula in Development Studies rarely incorporate the newer M&S-based research approaches, although these are increasingly used in this discipline. This lacuna in the curricula challenged us. We hypothesized that any Bachelor or Master's student could understand generally how social simulations can assist academic researchers in their research. In other words, they can gain a meta-knowledge of M&S-based research, that is, general knowledge about the nature of such research, its rationale, the way it is conducted, and the extent to which it can provide policy-relevant information. For our definition of meta-knowledge, we borrow from [START_REF] Brown | Developing the roots of modelling conceptions[END_REF], who also used the term meta-knowledge in relation to modelling. To explore our hypothesis, we organized a voluntary seminar aimed at giving students a 'feel' for the explanatory power of simulations, so they could gain meta-knowledge about the research approach without a technical introduction to the simulation software, the computer codes, etc.

Theoretical frame

We based our analysis of students' interactions in the seminar on Cultural-Historical Activity Theory (CHAT) [START_REF] Engeström | Learning by expanding: An activity-theoretical approach to developmental research[END_REF]. This theory focuses on the way in which, for example, the learning environment and students' social backgrounds interact with what students think and how they communicate. Within mathematics education, CHAT has proved useful in various studies; e.g., in research on how college students negotiate a workplace's and school mathematics' worlds [START_REF] Wake | Making sense of and with mathematics; the interface between academic mathematics and mathematics in practice[END_REF]. Following CHAT, the students in our seminar are understood as actors participating in different worlds. In the first place, they are participants in a learning context (in a Development Studies program, attending lectures, pursuing a degree). Second, they are oriented toward becoming professionals within a development organization (e.g. an urban planner in a less affluent country). Third, they might have the ambition to become member of a research community (within a university, publishing academic articles). Fourth, they are citizens in the real world (as consumers, migrants, etc.). Each of these worlds has its own conventions, norms, jargon, tools, etc.

When connecting the above CHAT-based worlds to insights from research on mathematical modelling education, we observe that in the modelling cycles of [START_REF] Blum | Quality teaching of mathematical modelling: What do we know, what can we do[END_REF] and [START_REF] Greefrath | Mathematical modelling with digital tools: A study on mathematising with DGS[END_REF], there are two worlds: the real world and the mathematical world. These are two worlds that both an M&S-researcher and a student in a mathematical modelling classroom negotiate. [START_REF] Doerr | Representations of modelling in mathematics education[END_REF] pointed out, describing modelling activities in terms of real and mathematical world is challenging. For example, the real world is far larger than the context of a modelling problem. Students and researchers participate in this larger world, and they may or may not have experiences with the problems addressed in M&S-research or in the classroom. Employees within development organizations also participate within this larger real world, but professionally they focus on a narrower world of specific problems in less affluent contexts. Researchers using M&S-based approaches operate primarily within the mathematical world of Figure 1; their work consists of identifying variables, creating relations between these, creating computer codes, running thousands of simulations, creating numerous graphs, and writing technical academic articles. So, although they work typically for the sake of the real world (global, national and regional development issues), the world of M&S-based research is mainly a mathematical world. The distinction between real world and mathematical world as depicted in Figure 1 has emerged from research into mathematical modelling in classroom contexts. However, this differs in several ways from the modelling activities of professional researchers. Students in classrooms often only 'go through' the modelling cycle once rather than several times, they use existing models rather than create new ones, they work with descriptive models rather than explanatory ones, they use educational digital tools if any (e.g., Geogebra) rather than professional computational programs, and their errors are less likely to have social and political implications [START_REF] Doerr | Representations of modelling in mathematics education[END_REF][START_REF] Vos | How real people really need mathematics in the real world"-Authenticity in mathematics education[END_REF]. In our study, we didn't ask students to engage in modelling activities, but rather to learn about the work of researchers utilizing M&S-based approaches. Therefore, we were not expecting to observe them operating in a mathematical world.

Our overarching research question was: to what extent can students in a Development Studies program gain meta-knowledge about the relevance of M&S for their discipline during a short intervention seminar? We had several sub-questions: to what extent can this interactive process enable these students 1) to understand the way in which these research approaches describe and explain social dynamics, 2) to grasp the basic benefits and limitations of M&S-based research, 3) to gain a sense of how researchers in Development Studies use such research approaches, and 4) to imagine themselves as future researchers using M&S-based approaches?

Methods

We used a design-based research approach for this project. This involved designing a seminar, implementing it, evaluating it, and then planning to repeat iteratively the intervention. Design-based research aims to improve educational practice in cases where new content is taught [START_REF] Plomp | Educational design research[END_REF]. The study reported here was the first of its kind; in forthcoming iterations, we intend to have improved seminars on the same topic with another group of students. In this study, the participants were three Nepali students from a master's program in Development Studies: we refer to them as Student B (female), Student K (female), and Student S (male). The first author of this paper was the leader of the seminar, which was conducted in Nepalese. The seminar was designed to last 3 hours and consisted of three sections. The first section introduced some relevant social problems (e.g., the 2015 earthquake and its social consequences, segregation and violence) and the impossibility of using experiments to study this sort of phenomena (i.e., exposing participants to exclusion or violence is unethical), and a first introduction to M&S-based research approaches. The second section involved a semi-guided activity, described in more detail below. The third section consisted of a discussion triggered by probing questions by the seminar leader. To illustrate the research approach, we included a hands-on simulation experience regarding an issue relevant for Development Studies: the migration and segregation of a city's inhabitants. In this part of the seminar we used an educational applet, which offers a simulation of the well-known Schelling Segregation model from Nobel Prize laureate Th. Schelling [START_REF] Schelling | Dynamic models of segregation[END_REF]. This applet, available from http://nifty.stanford.edu/2014/mccown-schelling-model-segregation, see Figure 2, begins with a random distribution of a population with two groups of agents (indicated by red and blue blocks). Depending on an agent's wish to live with same-colour neighbours (in other words: its tolerance for living with neighbours from the out-group), it will move to a new location. In Figure 2 the slider for similarity tolerance is set to 54%, which means that an agent is 'satisfied' when at least 54% of its neighbours share its colour. If the number of same-coloured neighbours falls below this threshold, an agent moves to an empty spot (a white block). The simulation famously shows that even with a relatively high level of tolerance at agent-level, clustering quickly begins and segregation takes over in the city.

Students' interactions were video recorded and transcribed. We analysed these in light of the theoretical frame by going through the transcripts and identifying utterances, in which the students positioned themselves in a world (for example, by their use of the term 'we' or by their description of experiences). We coded when the students engaged (1) as participants in a learning environment, (2) as future development professionals, (3) as potential M&S-researchers in Development Studies, or (4) as citizens in a dynamic society. The analysis resulted in clear, and sometimes multiple codes.

Results

The first part of the seminar was basically a lecture, which we didn't code due to the absence of students' utterances. The second part was a semi-guided activity with the Schelling Applet. The students sat together at one laptop, and interactively ran simulations to see the effects of varying the tolerance parameter. They tried many scenarios, and consistently found that raising agents' bias leads quickly leads to segregation, and that even relatively low levels achieved the same result, albeit more slowly. They also tried removing the empty spaces and discovered that no segregation could occur since "no options are available anymore" (Student S). They had experienced that in times of crises, people need to be tolerant: in the case of the Nepalese earthquake, people moved in with each other or lived peacefully in overcrowded tents. Student S was critical of the applet, commenting that in real life the space of a city is not restricted, and people would move beyond the city borders to build bigger houses. Student B mentioned that she knew of an influential person who moved to another place after the earthquake, after which his whole clan soon followed; in this case, the clustering tendency was already present before the segregation. In terms of the CHAT framework, the students were primarily participants in a learning environment (discovering the effects of changing sliders in the applet), but also expressed their real-world experiences as citizens. During this second part of the seminar, we observed the students speak neither as future professionals nor as researchers. The third part of the seminar was a discussion guided by probing questions, the first of which was: "what questions from Development Studies could be answered by studying virtual worlds?" Student S suggested that the different clusters of people could be studied with respect to their socio-economic status. The seminar leader realized that such a study would likely require a survey, rather than a simulation, but did not comment so the others could respond. Student B then said that simulations provide a dynamic visualization of phenomena and enable researchers to observe long-term changes visually. She suggested that simulations could be a medium for communication "for those who hate large data sets" and do not have a strong background in mathematics. At this stage, the students were participating as potential future Development Studies researchers, in a world in which they anticipated executing and publishing quantitative research.

To focus on their future professions, the seminar leader asked the students to think as urban planners; how might the latter make use of simulations? This triggered a lively discussion on how urban planners could promote a tolerant community. The students agreed that the Schelling Applet represented a certain underlying structure in society, although agents' movement in the real world is related not only to the colour of their neighbours but also to other factors, such as economic concerns, or a desire to live close to relatives. They then discussed ways in which a simulation of road networks could show how certain groups have better facilities (e.g., close to hospitals, accessible to firefighters), and how M&S-research could contribute to improve urban lives:

Student B:

We can find out the road conditions, specifically in Kathmandu. , we can study the trend of urbanizations. We can compare the situations. Find out the areas where more housing is needed.

Student B was imagining ways in which a simulation could help her understand and predict dynamics of urban life. This triggered the other students to identify additional scenarios, in which simulations could be used. Here, the students were thinking of themselves as potential researchers who might use M&S methods for urban planning. In addition to seeing themselves as citizens (travellers, migrants, etc.), they also perceived themselves as future professionals contributing to developing their communities through simulations to analyse and predict social dynamics.

Student B:

Here is a different thought… If I have a virtual Nepal, I think we can find vulnerable places for a natural disaster. We can find out how likely it is.

Student We see here that the students used "we," speaking as future policy makers who assist their societies prepare for natural disasters, set up health posts, or fight criminality. At the same time, they speak as researchers, using verbs such as "find." When asked to identify the limitations of M&S-based research, Student S pointed out that simulations do not produce realistic pictures, Student B wondered whether M&S methods were sufficiently scientific, and Student K asked about the expense and training required for creating models. However, Student B noted that a simulation's visualizations could be helpful for communicating with less-educated people. All three agreed that simulations can serve as a tool for prediction in guiding critical decisions, as well as facilitating understanding of the social dynamics of urban life, enabling governments to develop better policies.

Conclusion, discussion and recommendations

We observed that the students were largely able to understand the opportunities and challenges of studying social dynamics through M&S-based approaches, connecting it to prior knowledge of artificial intelligence systems that simulate future scenarios. They described possibilities for using simulations for planning roads or identifying places vulnerable to natural disaster or crime spots. Mostly, they expressed themselves as future professionals in development projects who would use results from M&S-based research. In the process, they shifted roles from learners, to citizens, and future development professionals, but not to researchers at a university using M&S-based approaches. Thus, the seminar assisted them in gaining meta-knowledge about the relevance of M&S for development professionals, but to a lesser extent for researchers in their discipline. The students understood that the Schelling Applet was an example of a simulation, which simplified real life processes, but that despite its limitations the simulation had explanatory power for certain social dynamics. Thus, the applet served as an educational tool helping students to transcend the learning environment into other worlds and to imagine what other simulations could look like when used by development professionals. However, students' erratic interchanging of terms like 'model' and 'simulation' showed that they had only a cursory sense of M&S-based approaches. Their capacity to gain meta-knowledge was restricted by their lack of experience in creating models and running simulations. Since we didn't ask students to engage in modelling activities, we kept them away from the mathematical world. So, they learned about M&S, but not the advanced aspects of the real work done by M&S-based researchers (identifying variables, creating relations, coding, etc.). How best to introduce novices to creating simulations remains an open question.

In a future iteration of the seminar we could put more emphasis on how and why researchers in Development Studies increasingly embrace M&S. Inspired by a comment from one student, we might also show how M&S provide a powerful tool for communication. Further, we could stress the way in which M&S-based approaches can capture link the micro-and macro-level (e.g., tolerance between individuals and the segregation of a city), as well as their relevance for their future as professionals studying issues such as urban planning, disaster preparation, or crime prediction.

CHAT provided a productive framework for understanding the way in which students engaged in the seminar as participants in different worlds. This study revealed that a seminar was sufficient for promoting meta-knowledge about the nature and relevance of M&S in Development Studies but highlighted the additional competencies that will be required if they pursue these approaches as professionals. These findings will help us improve future seminar iterations with other students.

Figure 1 :

 1 Figure 1: The modelling cycle, with visualizing and simulating digitally Curricula of Development Studies differ across universities. According to[START_REF] Djohari | Breaking other people's toys': Reflections on teaching critical anthropology in development studies[END_REF] and[START_REF] Engel | What do they need to know? Core skills for postgraduate development studies students[END_REF], curricula emphasize the teaching of academic theories (e.g. social justice theories), critical and anti-colonial thinking, or skills useful for future development workers (e.g. project management). Most universities offering Development Studies include a course on research methods, typically focusing on qualitative methods. Only a few require statistics, since quantitative methods are known to be a hurdle for many social sciences students[START_REF] Onwuegbuzie | Academic procrastination and statistics anxiety[END_REF][START_REF] Zeidner | Statistics and mathematics anxiety in social science students: Some interesting parallels[END_REF]. At this point, curricula in Development Studies rarely incorporate the newer M&S-based research approaches, although these are increasingly used in this discipline. This lacuna in the curricula challenged us. We hypothesized that any Bachelor or Master's student could understand generally how social simulations can assist academic researchers in their research. In other words, they can gain a meta-knowledge of M&S-based research, that is, general knowledge about the nature of such research, its rationale, the way it is conducted, and the extent to which it can provide policy-relevant information. For our definition of meta-knowledge, we borrow from[START_REF] Brown | Developing the roots of modelling conceptions[END_REF], who also used the term meta-knowledge in relation to modelling. To explore our hypothesis, we organized a voluntary seminar aimed at giving students a 'feel' for the

Figure 2 :

 2 Figure 2: The Schelling Applet, at the start (left), and after 26 simulation rounds (right)

  Because they are constructing roads in different places. I want to know whether it [the road network] is effective or not, basically, already at the planning stage. For example, roads in Kristiansand [Norway] are well planned. If you walk in this city, nobody gets lost. But in Kathmandu, we always get lost or run into a wall [dead-end]. With these [simulations]

	Leader:	How effective is that planning? Any examples?
	Student B:

  K:If we talk about health facilities, there is one health post in a VDC [Village Development Committee; Nepalese term for a rural organization unit]. Isn't that right? Any VDC has 9 wards and the health post will be in one ward. For [people in] other wards, it is far. So, if we can see distance virtually, then it will help us to decide whether there is a need for an additional health post. For example, in the Artificial Intelligence Systems course, we studied the PredPol model [an Artificial Intelligence system used by the police in Los Angeles]. If we borrowed the PredPol model, which will be helpful to identify key places where crime is increasing. It will be helpful to estimate sufficient armed forces for those identified places. Find out the crime spots observing past situations. This PredPol model is helpful to predict future crime using previous data.

	Student B:
	Student K:	A predictive tool
	Leader:	(…) Is that model a simulation?
	Student B:	I think it is a simulation model, because it helps us to predict.

  The students grasped that development professionals might implement the recommendations of M&S-based researchers even if they did not use the tools themselves. Finally, when asked if they could imagine themselves doing M&S-based research, Students K and S were silent, but Student B said "your presentation made clearer what a model is. Before coming here, I didn't know what a model is. I am interested."


Carlos Segura
email: carlos.segura@uv.es

Irene Ferrando
email: irene.ferrando@uv.es

Exploring pre-service teachers' flexibility in solving Fermi problems

Keywords: Flexibility, estimation, modelling, problem solving, pre-service teacher training

Theoretical framework and purposes

Realistic Fermi problems are open problems, connected to the real world, which require students to make assumptions about the situation of the problem and estimates about the relevant quantities, and whose solution is accessible to all students, admitting different levels of complexity (Ärlebäck, 2009). Students do not immediately have known procedures to solve this type of problems, so their solutions include a variety of strategies (Albarracín & Gorgorió, 2014). This exploratory work is a part of a research started two years ago focused on studying problem-solving process of Fermi problems based on obtaining an approximation to the number of elements in a bounded enclosure. In [START_REF] Ferrando | Análisis de los modelos matemáticos producidos durante la resolución de problemas de Fermi[END_REF] the solution processes of this class of problems have been analyzed (following [START_REF] Lesh | Problem solving, modeling, and local conceptual development[END_REF]) using a mathematical model characterization that includes a conceptual system for describing the relevant mathematical objects and their relations, and accompanying procedures for generating useful constructions, manipulations or predictions. This work has allowed us to identify three types of solution strategies for this class of Fermi problems: (i) based on the notion of population density; (ii) based on the use of a base unit; (iii) based on the use of a geometrical pattern (Ferrando et al., 2017, p. 229). [START_REF] Star | Flexibility in problem solving: The case of equation solving[END_REF] define flexibility in problem solving as knowledge of (a) multiple strategies and (b) the relative efficiency of these strategies. According to CCSSI (2010) flexibility in problem solving is related with mathematical competence. Nevertheless, teachers have still some difficulties to develop flexible reasoning when solving mathematical tasks. This leads us to consider whether pre-service primary school teachers know and use different strategies to solve Fermi problems. In particular, this study will address the following research questions: Q1) According to the different strategies identified in a previous study [START_REF] Ferrando | Análisis de los modelos matemáticos producidos durante la resolución de problemas de Fermi[END_REF], which characteristics of the context of the problem prompt particular solution strategies? Q2) Does the variation of these characteristics effect students to choose between several solution strategies, or do they always use the same one? Q3) In what way, when students confront their individual solutions in group and in the physical space posed by a problem, do they arrive at a consensual strategy?

Methodology

This study, which contained two phases, was carried out during the academic year 2017-2018 with n=112 students from the last course of primary education degree at the University of Valencia. In both phases they have been given the same sequence of problems.

Phase 1: Individual and purely estimative solution

In this phase of the study, students had to solve the sequence of problems individually. The physical spaces involved in the problems belong to the environment of the Education Faculty and are familiar to the students, but they do not have any data and they should only explain the procedures about how to get an estimation of the solution. The design of the sequence of problems has been carried out in collaboration with other researchers in the working group and has been based on the results obtained in [START_REF] Ferrando | Análisis de los modelos matemáticos producidos durante la resolución de problemas de Fermi[END_REF]. The analysis of the individual solutions in this phase allowed us to answer the Q1 and Q2.

Phase 2: Empirical solution by groups

In this phase, heterogeneous groups (depending on the solutions obtained in Phase 1) of 3 to 5 students are configured. They had to solve the same sequence of problems as in phase 1 in situ, but this time they must agree on a joint strategy. The analysis of their solutions allowed us to answer Q3 and also complete the answer to Q1. Indeed, we have identified what we call problem variables. These variables are related to the size and shape of the elements, the ratio of sizes between element and surface, the shape of the surface and the distribution of the elements on the surface.

Implications

This exploratory study is the starting point of a PhD thesis in which it is intended to analyse the knowledge, use and promotion of flexibility in solving realistic Fermi problem among future teachers. The results of this study can help to led up to formulate a specific definition of flexibility for the domain of the Realistic Fermi problems and to raise new questions such as: What type of instruction is most effective in promoting flexibility in this class of problems? What is the relationship between the validation of the solution and the choice of the most efficient strategy? When the future teachers solve this type of sequence, does it influence their conception of flexibility?

Acknowledgements

Research supported by the project EDU2017-84377-R, Programa Retos de la Sociedad of the Agencia Española de Investigación and FEDER.