Berta Barquero
email: bbarquero@ub.edu

Marianna Bosch
email: marianna.bosch@iqs.edu

Floriane Wozniak
email: floriane.wozniak@umontpellier.fr

Modelling praxeologies in teacher education: the cake box

Keywords: Modelling, teacher education, mathematical discourse, anthropological theory of the didactic, praxeologies

This article presents a teacher education activity with pre-service Primary school teachers who are asked to address modelling problems before describing their activity using new discursive and conceptual tools. The terminology of modelling and the questions-answers maps are proposed to enrich the logos of teachers' modelling praxeologies, that is, to provide future teachers with a discourse-a logos-to explain and analyse the modelling praxis.

Introduction: Teacher difficulties with the modelling discourse

When teaching mathematics is considered as the teaching of already-produced knowledge (concepts, procedures, properties, etc.), teachers dispose of discursive resources to talk about the content at stake: "Here we can use proportionality", "Be careful with the distributive law", "The problem leads to a quadratic equation", etc. All these resources have been elaborated during long processes of didactic transposition [START_REF] Chevallard | La transposition didactique -Du savoir savant au savoir enseigné[END_REF] and can be found in textbooks, syllabi, curriculum guidelines, etc. They belong to what is called the knowledge to be taught and appear as crucial tools for teachers and students to manage learning processes, to talk about what is already done, what is still to be done, what mathematical tools can be used, what difficulties might appear, what mistakes have been identified, and so on.

In contrast, in the case of teaching modelling, the content at stake is not only a set of mathematical contents that are to be known and used to solve problems. Modelling is also taught as a process to follow, where all the steps-and not only the final result-are equally important. In this case, teachers and students are not always provided with the appropriate words or discursive elements needed to describe the different steps of the process.

The importance of language in teaching and learning modelling has been pointed out by many researchers [START_REF] Barbosa | Mathematical modelling in classroom: A critical and discursive perspective[END_REF][START_REF] Borromeo-Ferri | Insights into teachers' unconscious behavior in modeling contexts[END_REF][START_REF] De Oliveira | Mathematical modeling, mathematical content and tensions in discourses[END_REF][START_REF] Doerr | What knowledge do teachers need for teaching mathematics through applications and modelling[END_REF]. For instance, focusing on the students' performances, [START_REF] Redmond | Exploring the relationship between mathematical modelling and classroom discourse[END_REF] propose to use "the principles of collective argumentation to enable students to analyse mathematical contexts, to synthesise strategies to mathematize these tasks, and to communicate solutions and conclusions to others." (p. 13). This paper addresses the problem of the teachers' lack of discursive tools to teach modelling processes and how it can be addressed through teacher education processes.

Research problem: mute modelling praxeologies

In this section, we analyse the lack of words in teaching modelling using the anthropological theory of the didactic (ATD). In spite of a diversity of representations [START_REF] Perrenet | The many faces of the mathematical modeling cycle[END_REF], there is a widespread consensus about the modelling process, and its decomposition in different steps synthesised in the diverse versions of modelling cycles, where models are built and used to produce knowledge about a given piece of reality considered as a system. As [START_REF] Serrano | Fitting models to data. The mathematising step in the modelling process[END_REF]Gascón (2010, p. 2193) explain: "the productivity of the model, that is, the fact that it produces new knowledge about the system, requires a certain 'fit' or 'adaptation' to the system. This process is rarely done once and for all. It requires a forth and back movement between the model and the system, in a sort of questions-answers or trial-error dynamics."

In the ATD, mathematical activities are described in terms of praxeologies. A praxeology is the connection between know-how (praxis), made of types of problems and techniques to solve them, and a discourse about the praxis (logos), composed of descriptions, justifications and theoretical organisations of the problems and techniques. Any human activity (solving an equation, driving a car, combing one's hair, etc.) can be assigned to a praxeology or a set of praxeologies, with a more or less developed praxis and logos. We use praxeologies to describe both the activity of modelling and the teaching of modelling. In a teaching situation, the mathematical knowledge at stake is commonly revealed through the teacher's discourse, which updates, describes, explains, justifies, questions and validates the knowledge aimed for. Wozniak (2012b, pp. 65-66) considers three kinds of mathematical praxeologies in teaching situations: "mute", "weak" and "strong" praxeologies. A "mute" praxeology is one with its praxis component visible only, that is, the types of tasks and techniques used to carry them out without any explicit discourse to describe them. In a "weak" praxeology, the logos component is visible through the parts of the technique used, but the discourse is limited to the description of the technique through an incomplete formulation. Finally, a "strong" praxeology links its two components, the praxis and the logos, dialectically.

In the case of modelling praxeologies, Wozniak (2012aWozniak ( , 2012b) ) speaks of mute praxeologies when the hypotheses about the initial system are not explicitly stated (delimitation of the system), the model used is prebuilt and the work within the model gives results that are not validated against the system. Weak modelling praxeologies appear when the hypotheses are formulated but their validity domain is not clearly discussed. In this case, the model is not necessarily prebuilt but will not be validated. To develop them into strong praxeologies, all steps of the modelling process should be present throughout a discourse that explains their roles and validates the whole process. Wozniak (2012a) uses the three kinds of praxeologies to analyse five primary school teachers working with a modelling problem. Students are presented the photograph of the boot of a giant's sculpture with some people around (Figure 1) and are asked to find the approximate giant's height. The main phenomena observed-and which can be found in other investigations about teaching modelling activities-can be summarised as follows:

-The modelling activity is not identified as knowledge at stake and the modelling process is never questioned or taken as an object of study.

-The delimitation of the system remains implicit and is not discussed; the hypotheses are not explicitly formulated and only the missing data are identified. -The discussion focuses on the choice of the values of the system variables to apply a preestablished model (proportionality). -All 5 teachers lead the study towards their own solution, and then reduce the study to the same kind of techniques. -In all but one case, a single model is used without discussing its legitimacy and validity.

Teachers do not use the properties of the system (realism) to evaluate the answers obtained and even directly propose the model to be used. -The fifth teacher includes discussions about the legitimacy of the measures estimated but does not talk about the domain of validity of the model used. Wozniak (2012a) concludes that the modelling praxeologies used in all these classrooms correspond to mute or weak praxeologies, which emphasizes the teachers' need of new knowledge and tools to deal with this kind of activities. In the following section, we propose an example of teacher education activity that aimed at transforming the mute praxeologies into voiced ones, that is, modelling praxeologies with a logos that give them voice to "talk about" the modelling praxis.

How to turn mute praxeologies into voiced ones: the cake box

The teacher education activity was designed and implemented by this paper's first author. It took place during eight 2-hour sessions with a group of 40 pre-service teachers in their 3 rd year of a double university degree in preschool and primary school education. Although this activity has been tested since the academic year 2012-2013, we here focus on its last implementation in 2017-2018. Its design follows the first three modules of the proposal of study and research paths for teacher education (SRP-TE) (Barquero, Bosch, & Romo, 2018, pp. 3-4).

In the first module, pre-service teachers were proposed a general teaching question: "How can modelling be introduced in Primary school education? Through what kind of modelling activities? With what kind of shared discourse about modelling?" In this particular implementation, we used the giant height activity as an introduction, to make the students familiar with a modelling activity.

The activity was also used to establish some common terms to refer to modelling: system, model, hypotheses, variables of the system, mathematical results, interpretation, model validation, etc. Some of the main results described in Wozniak (2012a) were also explained to the students. Then, in the second module, we started with the implementation of another modelling activity: the cake box (an adaptation of the activity described in [START_REF] Chappaz | Il était une fois … La boîte du pâtissier[END_REF]). The following three sessions were devoted to implement the cake box activity with the pre-service teachers assuming the role of students. The main goal was to make teachers carry out an unfamiliar mathematical activity that could, to a certain extent, exist in an ordinary classroom. The third module covered three sessions, in which students were asked to analyse the modelling process as it was experienced.

The cake box and the different modelling phases

The modelling activity starts by presenting the case of a baker who needs help for packing her cakes in boxes. She wants to use the same type of boxes she has been using (Figure 2). At this phase, all the students worked using manipulative and measuring techniques, by concretely building the box and measuring the sizes of the box using different instruments (paper grid, ruler, etc.). This experimental work facilitates to delimitate a rich system, which had many possibilities to be extended by considering other paper dimensions and raising new questions about the sizes of the resulting boxes. Students also agreed on which variables had to be considered, for the paper (width and length) and for the box (width, length and height), and how to agree on a common notation. Some of the more advanced questions came from the arithmetical comparison of the papers and resulting boxes' sizes: students started formulating the first hypotheses about the likely relationships between the different variables. Most of them referred to relationships of proportionality between the sizes of the paper and the box. Although, in most of the occasions, none of the groups working with these initial arithmetic-geometric models could validate their first hypotheses, the possibility to experimentally build new boxes helped them to refuse or refine the hypotheses, but not to validate them.

At the end of this first phase, some of the questions students raised were about a possible relationship of proportionality between the paper and the box sizes, and about how to get a particular sized box. When these types of questions appeared, the modelling activity moved to a second phase.

Second phase: It was assumed the baker gave the box sizes she needed and the question Q 2 was stated: Which are the initial paper sizes needed to build a box with some specific dimensions? Students propose to consider new auxiliary questions with particular sizes of boxes, special forms (squared-based boxes) and possible changes in specific dimensions of the box:

Q 2.1 : What initial paper sizes do we need get a box with base of 6cm x 13cm? Q 2.2 : How to get a box with a base of 8cm x 10cm and 3cm high? Q 2.3 : How to get boxes with a square base (such as: 5cm x 5cm, 8cm x 8cm, etc.)? Q 2.4 : How to modify the sheet of paper to get a box with the same base and different height?

Students started trying to use the same manipulative-measuring techniques developed in the first phase, by selecting box sizes close to the ones they looked for. Most of them proposed to apply models of proportionality to deduce the sizes of the paper, in many occasions expressed by a rule of three (Figure 3, left-hand side). When they estimated the sizes of the paper, they built the box to check if their deductions were true. When students checked that the "rule of three" did not work (as not all variables have relationships of proportionality), they looked for new models. The most common models that appeared were pre-algebraic ones obtained by opening the box, analysing the 2-dimensional geometrical pattern and describing the relationships (Figure 3, right-hand side). Some other groups proposed algebraic models, by assigning algebraic values to the different dimensions of the paper-box and looking for the algebraic equations synthesizing these relationships. Concerning the modelling process, two traits may be stressed. First, the modelling work developed in Phase 1 constitutes a productive system from which Phase 2 starts. That is, thanks to the experimental work developed, students have many paper-box sizes collected, which constitutes a rich experimental milieu from where some first hypotheses about the relationships were explicitly formulated and against which students could evaluate their answers. Second, in this phase, more advanced models were built by students given the failure of the proportionality model. Several questions about how to reformulate the hypotheses (Do any of the box or paper variables maintain a relationship of proportionality?), about the comparison of models (Are the pre-algebraic and algebraic models expressing the same?) and about the scope and limitations of the models (Can we follow working with the pre-algebraic models for all the box cases?) could be explicitly stated.

This second phase finished when students were able to predict the sizes of the paper/box, without manipulating the paper/box. The following questions were posed by the teacher:

Q final.1 : How can we predict the size(s) of the box (without building it) from the dimensions of the initial paper? Q final.2 : How to predict the size of an initial paper sheet given the dimensions of the box (without building it)?

Third phase: The aim of this phase was to prepare a letter-report as an answer to the baker's demand. A list of the cakes the baker wanted to pack, and their sizes, was provided. Students were asked to prepare the final report for the baker by letting her know about the paper sizes she had to order to pack all her cakes. Moreover, students were asked to add a tray for each box, following the same pattern of construction. At this stage, students had no necessity to build new models, but the main issue was to agree about the way to build the tray (How many cm do we have to leave between the box and the tray (margins)? Where to add these differences?). At this stage, they could better formulate the paper-box relationships, and took advantage of using the model (see Figure 4). Furthermore, some time could be devoted to work on how to report the results, given the extent of numerical work and results, with many variables, they had to communicate and organize. When experiencing the cake box activity, participants managed to assume the role of students and carried out the proposed modelling activity although, as they stated, they were not used to this kind of long and open activities. After each session, the working teams (3-4 persons) had to deliver a written report about the modelling activity developed. Participating in the classroom debate and writing the reports was not an easy task for them, as they were conscious they needed new terms to describe the process. At this stage, the previous work with the giant activity was very helpful, since modelling terminology was then introduced. Still, students had many difficulties with, for instance, formulating questions, describing and categorising the kind of models they were suggesting, accepting different possibilities of models and answers, reformulating hypotheses and models, etc.

Translation of the students' proposal:

BOX 2:

Base of the box: 17 cm x 17 cm Size of the paper: 51 cm x 34 cm TRAY:  To build the tray, we add 0.5cm to each side of the box base to guarantee that it fits correctly.  The tray base will be 17,5 cm x 17,5 cm  The paper dimension will be 52,5 cm x 35 cm

The instructor decided to follow three main strategies during Module 2. First, she fixed the structure of the reports in terms of: questions studied, data and variables selected, mathematical models considered, answers obtained, new questions opened. This helped participants explicitly describe all these elements. Second, during the debates, she asked participants to explain their proposals and to create names to designate the models they were using (arithmetic, geometric, pre-algebraic, algebraic, etc.). This illustrated that there was not only one-way to work on this activity and that several valid models could cohabit. Third, in the debates and reports, the instructor payed special attention to the questions the working teams were posing to help them in their formulations and in relating them to each other.

In Module 3, students were asked to provide an analysis of the modelling work previously carried out, using the session reports and class debates. The instructor proposed to do this work using questions-answers maps [START_REF] Barquero | Mathematical modelling in teacher education: dealing with institutional constraints[END_REF]. She proposed an initial map with the first questions and answers (Q-A) that appeared in the classroom and asked the students to complete it with the description of their own work. This way of describing the modelling process not only provided students with new terminology. It also appeared as an alternative way to talk about doing mathematics, breaking with the usual and "static" way of describing school activities, more focused on concepts, notions and techniques to the detriment of questions, models and provisional answers.

Students could also use the Q-A maps to distinguish between their own modelling trajectories-the path followed-and the more complete one they could describe after the work done in class (see Figure 5), thus evidencing the possibility of finding several possible solutions to the same problem.

Conclusion

We have seen how Wozniak (2012b) describes the problem of the teachers' lack of discursive tools to talk about modelling processes in terms of mute praxeologies, that is, praxeologies where the logos component remains implicit. This logos is part of the mathematical praxeologies to be taught related to modelling activities. However, it also seems to be a crucial didactic tool to teach modelling-therefore, part of the teacher's didactic praxeology. It is difficult for teachers to highlight some important modelling procedures when they are not familiar with any specific mathematical terminology to talk about models, systems to be modelled, possible fits (and misfits) between models and systems, etc. To start addressing this problem, we propose to use study and research paths for teacher education to elaborate new epistemological tools, such as the questions-answers maps, with the teachers as part of the modelling praxeologies to later on develop them as didactic tools to better describe and analyse the modelling activities carried out in classrooms. This strategy needs to be developed, but at the present, it seems helpful to fill in the gap between the traditional way of describing-and conceiving-school mathematics and the new epistemological and didactic necessities raised by the teaching of modelling processes.

Figure 1 :

 1 Figure 1: Picture of the Giant activity

  The following question opens the activity: "How can we build boxes to help the baker packing the variety of cakes she offers? Which relation does exist between the sizes of the initial material (paper or cardboard) and the dimensions of the resulting box?" From this initial question, the activity is structured in three phases depending on what is given as known (the sizes of the paper, box or cake) and what remains unknown.

Figure 2 :

 2 Figure 2: Instructions to build the box and examples of the resulting boxes First phase: We considered that the paper sizes (width and length) are given and focused on question Q 1 : Which are the dimensions of a box resulting from a paper whose sizes are fixed? Students started by considering some particular cases, such as: Q 1.1 : What dimensions of the box resulting from an A4 paper? Is there only one possible result? Q 1.2 : If we take an A5 (half A4), do we obtain a box measuring half of the previous one? Q 1.3 : Which box sizes do we get from a squared sheet? Do we get a squared-based box?

Figure 3 :

 3 Figure 3: Models of proportionality (left-hand), pre-algebraic-geometric models (right-hand)

Figure 4 :

 4 Figure 4: Pre-algebraic-geometric models for the box and the tray A progressive construction of the modelling discourse

Figure 5 :

 5 Figure 5: Example of questions-answer map used to analyse a modelling trajectory of a group

Acknowledgments

Funded by Ministerio de Ciencia, Innovación y Universidades with the research projects Retos Investigación of the Programa Estatal de I+D+I: EDU2015-69865-C3-1-R; RTI2018-101153-A-C22.


Andréane Bourges

Amélie Chardac

Aude Caussarieu

Nicolas Plihon

Nicolas Taberlet

Oscillations in a half-empty bottle

Oscillations in a half-empty bottle

Andréane Bourges, Amélie Chardac, Aude Caussarieu, Nicolas Plihon, Nicolas

I. INTRODUCTION

The phenomenon

When a half-empty bottle initially resting horizontally on a flat table is pushed, the fluid it contains is set into motion, which leads to an oscillatory rolling motion of the bottle. Figure 1 shows an example of the trajectory of a 1L glass bottle half-filled with water, manually pushed at the time indicated by the dashed line. The space-time diagram showing the position of the cap as a function of time clearly illustrates the intermittent overall motion of the bottle. In the particular case shown in Figure 1, the bottle undergoes a stop-and-go motion with a typical frequency of a few Hertz. The goal of the present paper is to understand and model this oscillatory motion.

The phenomenon of surface gravity waves on water and oscillations of fluid has been studied for a wide variety of container shapes [START_REF] Wehausen | Surface waves[END_REF][START_REF] Davis | [END_REF] and the dependence of frequency on curvature in a vertical cylinder is known 3 . However, when the container is subjected to an external force and free to move, the problem becomes more complex and the free surface of the fluid adopts different shapes, depending on the excitation and the geometry of the container. This issue is of great importance in a wide range of applications involving liquid transport, from the problems encountered by space agencies in aerospace vehicles 4,[START_REF] Dodge | The new" dynamic behavior of liquids in moving containers[END_REF] , to tank carriages on highway or rail roads [START_REF][END_REF]7 .

The motion of a soft-drink can on an incline was studied by Jackson et al. 8 . The authors used water (and varied the filling fraction of the can) as well as granular matter (lead shots and glass marbles) and proposed a model which describes the limiting cases of non-viscous and infinitely viscous fluids. Later, Lin 9 compared the rolling dynamics of cans fully filled with liquid water and solid ice, Ireson and Twidle 10 showed that shaking a can of soda noticeably affects its rolling speed and Micklavzina 11 investigated the influence of the fluid viscosity.

In this article the bottles are modeled by simpler cylindrical tubes. The main goal of this article is to study the oscillating speed of a half-empty bottle. The ratio of filling is therefore fixed to one half and the fluid used in all of our experiments is tap water.

What students can learn from this problem

The oscillatory motion of a bottle is a simple hands-on experiment which can illustrate important concepts in solid and fluid mechanics. The Euler-Lagrange equations 12 are used to investigate the motion of a coupled system, and the conservation of energy during an elastic shock allows one to predict the bounce of a bottle on a wall. Moreover, the phenomenon exposes hydrodynamics concepts to explain a fluid motion: characteristic times relevant to the phenomenon are identified, the validity of a potential flow approach 13 is discussed and the eigenmodes of fluid motion in a container are studied.

Outline of the paper

In order to model the overall oscillating motion of a half-empty cylinder, we will first focus in part II on the coupling between the water and the solid and show that their rotation are decoupled for short times. The motion of the fluid within a half-empty still cylinder, known as sloshing, is presented in part III, and allows one to define the moment of inertia of the water, which can be used in the equations of motion derived from the purely mechanical model developed in section IV. Finally, part V will show experimental results of oscillations obtained when a bottle bounces on a wall or rolls down a steep slope.

II. ROTATIONAL COUPLING

A. Race down a slope

The study of a bottle rolling down an incline [8][9][10][11] is extremely informative regarding the motion of the fluid inside the rolling container. Jackson et al. 8 in particular, showed that the velocity of an empty can is noticeably less than that of a can filled with water. This indicates that the relative moment of inertia of an empty can (i.e. normalized by mass × radius 2 ) must be larger than that of a full or half-empty can. Indeed, although the fluid clearly contributes to the overall mass of the system (and therefore its weight), it may not contribute significantly to the total moment of inertia since, in general, it is not in solid-body rotation within the container. We have reproduced the experiments by releasing four 'bottles' of identical radius (3 cm) and length (12 cm) on a 2 • slope: a hollow tube (or empty bottle), a solid Plexiglass cylinder, a half-empty bottle and a full bottle.

Note that on this gentle slope, all objects roll with no slip. A picture taken 2s after the start of the race is shown in figure 2 while figure 3 displays the positions in time of the various bottles, measured from particle tracking (using the 'Analyze Particle' tool of ImageJ, a free software developed by NIH, on a 1500 × 1000 pixels video at 30 FPS).

As expected, the hollow tube is visibly slower than the solid cylinder. Figure 3 shows that the velocity increases roughly linearly in time, which indicates that friction plays no role. The behavior of the half-empty and full bottles is remarkable: both roll down the slope at greater speeds than the hollow tube and the empty bottle, which indicates a smaller moment of inertia. Their motion is close to that of a simple friction-free sliding block (indicated by a dashed line in figure 3). Again, these results indicate that the water inside the bottle (whether halfempty or full) undergoes little rotation but is instead simply translated. In conclusion, this simple experiment shows that the rotation of the bottle is decoupled from that of the fluid (over the short duration of the race).

B. Diffusion time

Results from the previous paragraph indicate that a bottle can spin without inducing rotation within the water. This only holds for short times, less than the typical diffusion time of momentum. Indeed, over longer times the viscosity of the fluid should induce motion of the fluid. In this paragraph we discuss the momentum diffusion in a vertical rotating bottle. We have performed a simple experiment in which a tall cylinder (of height 400 mm and radius R = 50 mm) is placed on a rotating table (an old record-player). The motion of the water is followed using neutrally buoyant tracers (700 µm polystyrene beads) from a 1000 × 1000-pixel video recorded at 30 FPS. At time t = 0, the rotation is started (at 33 RPM). It takes about 30s for the fluid at R/2 (i.e. half-way from the wall to the center) to start rotating and 50s to reach a steady velocity. Even more interestingly, it takes typically 10s for the fluid near the bottle wall (at R/10) to be set into motion. Therefore, one can conclude that over short times (typically a few seconds) the vast majority of the fluid is not affected by the rotation of the bottle.

An estimate of the diffusion length of the velocity within the fluid (or its momentum), δ, over the duration, T , of a typical experiment can be given knowing the kinematic viscosity, ν, of the fluid [START_REF] Kambe | Elementary Fluid Mechanics[END_REF][START_REF] Mikhailov | Unified Analysis and Solutions of Heat and Mass Diffusion[END_REF] : δ = √ νT 3 mm for T = 10 s, which is compatible with the experimental data of figure 4 and negligible compared to the radius of the bottle. For an oscillatory motion of typical frequency 1/T = 3 Hz (see figure 1), the boundary layer is given by penetration length [START_REF] Kambe | Elementary Fluid Mechanics[END_REF][START_REF] Mikhailov | Unified Analysis and Solutions of Heat and Mass Diffusion[END_REF] (analogous to the electromag-netic skin-depth [START_REF] Macdougall | [END_REF][17][18] ): δ = νT /π 0.3 mm, which is an order of magnitude smaller.

In conclusion, we have shown that the rotation of the fluid and of the bottle are decoupled as long as the experiment lasts typically for less than 10 s.

III. SLOSH DYNAMICS IN A STILL BOTTLE

In this section the motion of the fluid in a still bottle is discussed. This slosh dynamics was studied as early as the XIX th century by Rayleigh [START_REF] Rayleigh | The theory of sound[END_REF] and a comprehensive review can be found in Lamb [START_REF] Lamb | Hydrodynamics[END_REF] and Ibrahim [START_REF] Ibrahim | Liquid sloshing dynamics: theory and applications[END_REF] .

A. Potential flow

In a horizontal cylinder the first eigenmode of sloshing displays a flat (although not constantly horizontal) free surface which oscillates up and down. Rayleigh [START_REF] Rayleigh | The theory of sound[END_REF] showed that under the assumption of a potential flow (where in particular the viscosity of the fluid can be neglected) the velocity field can be written with separated variables, i.e. as a product of a function of time and a second function of the position. This allows to define the moment of inertia for the rotation of the fluid around the center of the bottle:

I s = πρR 4 L 4 π 2 - 1 4 0.31mR 2 ,
where ρ is the density of the fluid and L the length of the bottle. The potential from which the velocity is derived is given in terms of a series [START_REF] Lamb | Hydrodynamics[END_REF][START_REF] Ibrahim | Liquid sloshing dynamics: theory and applications[END_REF] and the corresponding velocity field is plotted in figure 5b. A few comments can highlight the differences with a rigid-body rotation. First, the velocity clearly decays with increasing distance from the free surface whereas it increases linearly in the case of a rigid-body rotation. Second, one can notice that the streamlines are not half-circles centered on the center of the bottle. Instead, the streamlines are flatter near the free surface. Finally, the velocity at the center of the bottle is not zero. It is purely horizontal and oscillates back and forth. The theoretical velocity field is remarkably similar to the experimental observations. The picture in figure 5a shows neutrally buoyant markers indicating the motion of the fluid in a bottle pushed and held against a wall. This excellent agreement supports the assumption that the viscosity of the fluid plays no major role in the slosh dynamics, aside from the boundary layer discussed earlier.

B. Experimental validation

Experiments were performed by pushing a half-empty cylinder against a wall in order to induce fluid motion. The cylinder is then firmly held still against the wall and the slosh dynamics of the water is studied. The altitude of the free surface against the curved side of the cylinder was recorded (at a frame rate of 500 FPS) and measured through particle tracking (see inset in figure 6).

A simple way to check whether the potential flow is an accurate description of the actual flow is to measure the frequency of the oscillations. The Fourier transform of the signal is shown in figure 6. The frequencies of the first three modes of sloshing can be analytically predicted [START_REF] Ibrahim | Liquid sloshing dynamics: theory and applications[END_REF] (ω 1 1.17 g/R, ω 2 2.17 g/R and ω 3 2.82 g/R) and are indicated by red lines. A decay of the amplitude of oscillations is visible on the inset and is a long-term effect of the viscosity of the fluid. However, the agreement between the observed frequency and the prediction under the assumption of an ideal fluid is excellent, again supporting the hypothesis that the viscosity plays no major part in the flow (although it causes a slow damping). Moreover, even if the force initially applied to the bottle induces a sloshing motion which is more complex that an oscillating flat surface, the first sloshing mode is always dominant. The second or third modes could be observed if the cylinder were shaken at the corresponding resonance frequency, but in the case of an inherently asymmetrical initial push mode 1 always dominates. Therefore in the following only the first mode (in which the free surface remains flat although obviously not horizontal) will be considered.

IV. A SOLID BODY TOY MODEL

Having understood that the rotation of the fluid and of the bottle are decoupled over short times (less than 10s) one can propose a mechanical model for the motion of a half-empty bottle in which the fluid is seen as a solid half-cylinder. This simplification is obviously inappropriate (as discussed in section III) but allows for a derivation and analysis of the equations of motion of a half-cylinder, representing the water, mounted on wheels, representing the bottle (see figure 7a). The mass and moment of inertia of the water are denoted m and I G (computed around its center of mass, G) while those of the bottle alone are denoted M and J. The horizontal position of the bottle is noted x, and the inclination of the half-cylinder θ. The distance between the center of the bottle and the center of mass of the water is noted l = OG = 4R/3π. The model presented here is very similar to that of Jackson 8 in which further technical details can be found. Assuming a slip-free rotation of the bottle allows one to express its rotation speed: ẋ/R. If all sources of dissipation are neglected (air drag, solid and rolling friction, viscosity of the fluid etc) the equations of motion can be derived from the Euler-Lagrange equations using the kinetic energy T and potential energy V :

   T = 1 2 M ẋ2 + 1 2 J ẋ R 2 + 1 2 m ẋ + l θ 2 + 1 2 I G θ2 V = mgl(1 -cos θ).
(1) In the small-angle approximation, the equations of motion therefore read:

   M + J R 2 ẍ + m(ẍ + l θ) = 0, (I G + ml 2 ) θ = -mglθ -mlẍ.
(2) Interestingly these equations are equivalent to those of an analogue system consisting in a pendulum with an effective moment of inertia I eff = I G + ml 2 attached to a block sliding with no friction (see figure 7b) with an effective mass M eff = M + J/R 2 . Note that the effective moment of inertia, I eff , is simply equal to the moment of inertia of the half-cylinder around the center of the bottle O: I O . The equations of motion can be easily combined and solved (see 8 for details):

           θ = A sin(Ωt + ϕ) ẍ = -A m m + M ef f l sin(Ωt + ϕ)
where Ω -2 = l g

I O ml 2 - m M ef f + m . (3)
where A (in rad.s -1 ) and ϕ (in rad) are constants which depend on the initial conditions.

V. SLOSH DYNAMICS IN A ROLLING BOTTLE A. Adaptation of the theory to a rolling bottle

The results presented above can be combined to study the motion of a half-empty bottle. The moment of inertia O in equation 3 simply needs to be replaced by that of the first mode of sloshing given in section III:

I s = πρR 4 L 4 π 2 -1 4 .
Moreover, for simplicity, the moment of inertia of the bottle itself is assumed to be J = M R 2 and the effective mass defined in section IV is then simply M eff = 2M . The angular frequency of the oscillations is therefore given by:

Ω -2 = l g I s ml 2 - m 2M + m . ( 4
)
Note that the ratio I s /ml 2 is only a geometrical constant ( 1.72) since l = 4R/3π. Moreover, the right-hand term is ensured to be positive since the term in parenthesis can be rewritten as:

I G /ml 2 + 2M/(2M + m).
Knowing the initial conditions therefore allows one to determine the constants A and ϕ in equation 3 and to integrate them. The following paragraphs will discuss the motion in two different setups: a half-empty bottle bouncing on a wall and a half-empty bottle rolling down a steep slope.

B. Soft collision of a half-empty bottle

Initial conditions

As explained above, the motion strongly depends on initial conditions. When a bottle is manually pushed, it is difficult to properly determine what force (i.e. acceleration) or velocity is imposed and the corresponding initial conditions remain unclear. In order to study a well-defined set of initial conditions, a half-empty bottle rolling on a flat surface without any oscillations is sent to bounce on a wall (figure 8). The initial steady velocity, v 0 , is achieved by letting the bottle slowly accelerate on a gentle slope (less than 1 • ). The wall is made of soft wood in order to minimize the energy dissipation. Although the collision must be somewhat inelastic we found that an empty bottle loses less than 5% of its speed after a collision against the wall. Moreover, very little energy is dissipated in the fluid boundary layer. Therefore, all sources of dissipation during the collision are neglected.

When the bottle hits the wall (and while it remains in contact with the wall), the water rises as the initial kinetic energy is converted into potential gravitational energy. The surface then recoils back to horizontal at which time (t = 0) the bottle leaves the wall and rolls forward while the fluid keeps oscillating. However, an empty bottle tends to almost instantly bounce off the wall. There is therefore a competition between the relatively slow rise of the water (which tends to pull the system against the wall) and the rapid rebound of the bottle itself (which tends to pull the system away from the wall). If the mass of the water dominates, only the first mode of sloshing is triggered and a set of initial conditions can be determined. In this case, the initial kinetic energy of the bottle itself prior to the collision not only does not contribute to the rise of the water but instead must be subtracted. This procedure is a reproducible and controlled way to set initial conditions: x 0 = 0, ẋ0 = 0, θ 0 = 0 and where θ0 is given by

1 2 I s θ2 0 = 1 2 mv 2 0 - 1 2 M v 2 0 . (5)

Analytical solution

Under these conditions the motion of the fluid and of the bottle is given by the solutions of equation 3:

   θ = θ0 cos Ωt ẋ = β v 0 (m -M ) 1/2 2M + m (1 -cos Ωt), (6)
where β = 8 FIG. 8. A half-empty bottle is sent to bounce on a wall. During the impact the fluid is set into rotation which triggers oscillations in the velocity when the bottle leaves the wall.

Experimental validation

A cylindrical bottle of radius R = 50mm and length L = 70cm, made of Plexiglas, was used to validate the theoretical predictions. The mass of the empty bottle is M =0.910kg and the mass of water used (corresponding to a half-empty bottle) is m=2.780kg. The motion of the bottle was recorded at 30 FPS and the position was measured using particle tracking and the velocity was com- Velocity of the half-empty bottle experiencing a soft collision against a wall. The velocity is constant before the impact (first dashed line) and oscillations are triggered after the bottle leaves the wall (second dashed line). The red line shows the velocity predicted using equation 6. There are no fitting parameters and the agreement is excellent.

puted as its numerical derivative. The results are presented in figure 9. The velocity prior to the impact, v 0 , is rather constant (with small fluctuations, in part due to the image processing) while after the impact (indicated by the two dashed lines) the velocity largely oscillates. One can see that the bottle undergoes a stop-and-go motion since the velocity periodically reaches zero. This feature is visible on the space-time diagram in figure 1. This motion is also predicted by our theory (red line). It is worth noting that there are no fitting parameters. The initial velocity is measured as the average before the impact, and the amplitudes, frequency and phase are those predicted by equation 6. The duration of the impact itself is given by a half-period of the oscillations. The agreement between the experimental data and the theory is excellent, supporting the validity of the numerous hypotheses made (first mode of sloshing, small angles, energy conservation and inviscid fluid). Knowing the time of impact, one can therefore entirely determine the motion of the bottle after the bounce.

For a bottle initially (t = 0) at rest:

ẋ = M + m 2M + m gαt + m (M + m) (2M + m) 2 αlΩ sin Ωt ( 8
)
The velocity is composed of two separate terms: the first one is the average constant acceleration (smaller than αg as discussed in section II), while the second indicates the velocity fluctuations.

It may seem counter-intuitive that releasing the bottle initially at rest should trigger oscillations, but it can be understood with a hand-waving argument. In the moving non-inertial frame of reference of the bottle an inertial force applies to the water, shifting the equilibrium position to θ eq = αM/(M + m). The amplitude of the oscillations (of the inclination of the free surface) is therefore given by θ 0 -θ eq = αm/(M + m).

Experimental validation

A cylindrical bottle of radius R = 44mm and length L = 170mm, made of Plexiglas, was used to validate the theoretical predictions. The mass of the empty bottle is M = 0.140kg and the mass of water used (corresponding to a half-empty bottle) is m = 0.510kg. The motion of the bottle on a table inclined at an angle of α = 15 • was recorded at 500 FPS and the position was measured using particle tracking and the velocity was computed as its numerical derivative. The results are presented in figure 10. The experimental data confirms the theoretical predictions: the bottle rolls down the slope with an average constant acceleration and oscillations in speed are clearly visible. The experimental acceleration is in excellent agreement with the predicted value. However, for longer times a discrepancy appears and might be attributed to the air drag acting on the bottle or to the failure of the decoupling assumption. The velocity fluctuations, ∆ ẋ defined as the difference in velocity from the linear behavior, are shown in the bottom of Figure 10 where the red line shows the theoretical predictions. It should be emphasized once again that there are no fitting parameters: all values are deduced from the simple measurement of the masses and sizes. The amplitude, frequency and phase are those given by equation 8. The fluctuations are small (typically 5% of the speed) and very sensitive to the imperfections of either the table or the bottle. Still, the experimental data shows a good agreement with the theory. One can see that the amplitude decays but the initial magnitude of the first few oscillations is well captured while the initial phase and the frequency are very well predicted. Again, the agreement between these experimental results and the theory validates the seemingly strong assumptions made.

VI. CONCLUSION

In conclusion we were able to successfully model the motion of a half-empty bottle rolling on a flat surface. We showed that over the duration of a typical experiment the rotation of the water and of the bottle are decoupled, which can also be predicted from the Navier-Stokes equation. A simple mechanical toy-model allowed for the derivation of the equation of motion while the exact motion of the fluid is well described by the first mode of sloshing under the assumption of an inviscid fluid. Experiments performed on a bottle bouncing on a wall and on a steep slope provided experimental validation of the predicting power of the theoretical predictions.

The limitations of our work could deserve further attention. It is clear that the decoupling of the rotation of the fluid and the bottle should not hold for smaller radii: if a test-tube of radius R = 1cm is used, the diffusion time is of the order of a few seconds and the whole fluid can be set into a rigid-body rotation. The same conclusion can be drawn if a more viscous fluid is used: vegetable oils are typically 50 times more viscous than water, leading to a diffusion time of the order of one second. Finally we should recall that the equations were derived in the small-angle approximation and that an exact study of larger amplitudes might reveal surprising results.

The potential flow gives an accurate description of the motion of the water in the bottle but the effect of the viscosity is visible in figure 6. A comprehensive study of the viscous dissipation in the boundary layer might deserve further attention. Similarly, the effect of static and rolling friction of a bottle rolling on a surface, as well as that of drag caused by the surrounding air, could be studied to provide one with a more realistic description of the motion over long times.

It would be interesting to vary the filling fraction of the bottle. Obviously, there can be no oscillations in a bottle either completely full or empty. Therefore, there must exist an optimal filling fraction for the oscillatory motion studied in this paper. It is unclear whether the optimal value should correspond to a filling fraction of one half.

FIG. 1 .

 1 FIG. 1. Top: picture of a half-empty glass bottle. Bottom: space-time plot showing the motion of the cap after the bottle was given a push. The oscillations in speed, adding to the overall forward motion at constant velocity, are clearly visible. Diagram obtained from a 1500-pixel line video at 500 FPS.

FIG. 2 .

 2 FIG. 2. Picture showing four bottles racing down a 2 • slope. The initial positions are shown by the dashed line and the picture is taken after 2s of rolling. The race shows that the moments of inertia of half-empty and full bottles are lower than that of the solid cylinder, which indicates that the water undergoes little rotation but only a simple translation.

FIG. 3 .

 3 FIG. 3. Top: position along the 2 • slope for an empty, halfempty and full bottles and a solid cylinder. Bottom: velocity computed as the derivative of the position. The full and halfempty bottles are the fastest whereas the empty bottle is the slowest. The greater acceleration of the half-empty and full bottles is due to their reduced moment of inertia. The dashed line indicates a constant acceleration: g sin 2 • .

2 FIG. 4 .

 24 FIG. 4. Rotation speed of water (measured at R/10, R/5 and R/2 from the wall) in a tall vertical cylinder initially at rest at set into rotation at t = 0. The lines are guides for the eye.

Figure 4

 4 Figure4shows the time evolution of the local rotation speed of the water (normalized by that of the rotating support) measured at various distances from the outer wall (R/10 = 5mm, R/5 = 10mm and R/2 = 25mm). It takes about 30s for the fluid at R/2 (i.e. half-way from the wall to the center) to start rotating and 50s to reach a steady velocity. Even more interestingly, it takes typically 10s for the fluid near the bottle wall (at R/10) to be set into motion. Therefore, one can conclude that over short times (typically a few seconds) the vast majority of the fluid is not affected by the rotation of the bottle.An estimate of the diffusion length of the velocity within the fluid (or its momentum), δ, over the duration, T , of a typical experiment can be given knowing the kinematic viscosity, ν, of the fluid[START_REF] Kambe | Elementary Fluid Mechanics[END_REF][START_REF] Mikhailov | Unified Analysis and Solutions of Heat and Mass Diffusion[END_REF] : δ = √ νT 3 mm for T = 10 s, which is compatible with the experimental data of figure4and negligible compared to the radius of the bottle. For an oscillatory motion of typical frequency 1/T = 3 Hz (see figure1), the boundary layer is given by penetration length[START_REF] Kambe | Elementary Fluid Mechanics[END_REF][START_REF] Mikhailov | Unified Analysis and Solutions of Heat and Mass Diffusion[END_REF] (analogous to the electromag-

FIG. 5

 5 FIG.5. a. Picture of the sloshing motion of water with a still cylinder (radius R = 5cm). Exposure time = 1/10 s. The picture shows neutrally buoyant tracers (polystyrene spheres of density 1.06 g/cm 3 and diameter 700 µm). b. Velocity field computed from the potential flow given by Ibrahim[START_REF] Ibrahim | Liquid sloshing dynamics: theory and applications[END_REF] . Note that the velocity decreases with increasing distance from the free surface.

FIG. 6 .

 6 FIG.6. Fourier power spectrum of the motion of the free surface in a bottle first pushed and then held still. The experimental maximum coincides with the theoretical frequency of the first sloshing mode. The inset shows the damping of the amplitude over long times.

FIG. 7

 7 FIG. 7. a): Schematics of a half-cylinder mounted on wheels equivalent to a pendulum attached to a sliding block (b) with effective moment of inertia and masses.

  FIG.9. Velocity of the half-empty bottle experiencing a soft collision against a wall. The velocity is constant before the impact (first dashed line) and oscillations are triggered after the bottle leaves the wall (second dashed line). The red line shows the velocity predicted using equation 6. There are no fitting parameters and the agreement is excellent.

  C. Half-empty bottle on a steep slope1. Analytical solutionAnother protocol used to produce reproducible and controlled initial conditions consists in letting a halfempty bottle roll down an incline plane of slope α, which must remain small enough to ensure a slip-free rolling motion of the bottle. The inclination of the free surface, θ, is measured in the frame of reference on the plane. The gravitational potential energy of the bottle and the fluid needs to be included in the Euler-Lagrange equations which yield (in the small-angle approximation): 2M ẍ + m(ẍ + l θ) = (M + m)gα, I s θ = -mgl(θ -α) -mlẍ.

1 FIG. 10 .

 110 FIG.10. Top: velocity of a bottle released on a 15 • slope. On average the bottle undergoes a constant acceleration but oscillations in the velocity are visible. The red line shows the predicted average acceleration. Bottom: deviation from the constant acceleration highlighting the oscillations: experimental data (black) and theoretical prediction (red).

√ 2 1 √16-π 2 0.76. Note that neglecting the mass of the end-caps, the ratio of masses can be expressed in terms of thickness, e, and density (relative to that of water), d, of the bottle itself: M/m =

4e/R d.

ACKNOWLEDGMENTS

The authors acknowledge support from the PALSE program of the University of Lyon Saint-Etienne, the University Lyon Claude Bernard, the Société Française de Physique and from the École Normale Supérieure de Lyon and its Physics Department and Laboratoire de Physique. We are thankful to B. Capitaine for techni-cal help and A. Chemin, A. Guittonneau, C. Gouiller, C. Malciu and J. Sautel for fruitful discussions. The work presented here was done in preparation for the International Physicists Tournament (http://iptnet.info), a world-wide competition for undergraduate students. The authors are grateful to both local and international organizing committees of the International Physicists Tournament for having put together an exciting event.


Rita Borromeo Ferri
email: borromeo@mathematik.uni-kassel.de

Assessing Teaching Competencies for Mathematical Modelling

Keywords: Mathematical modelling, teaching competencies, assessment

The question of how mathematical modelling competencies can be assessed has been well investigated and discussed in the last decades within the modelling research field. The test instrument, originally from Haines and Crouch, was adapted for learners in secondary and highschool and as well for teachers in many empirical studies. The results of these investigations also helped to conceptualize the term of modelling competencies. Far more at the beginning of research efforts is the question of how teaching competencies for mathematical modelling, based on the model of Borromeo Ferri, can be assessed. The complexity of the four dimensions is challenging in order to develop a test instrument with the goal to see an increase of teaching competencies after a course on 'learning and teaching of mathematical modelling'. The paper describes recently developed test items used in an intervention study with prospective teachers, and presents first results of this pilot study.

Teaching competencies for mathematical modelling

Since the beginning of research in the field of mathematical modelling education it is of great interest to get insight into how teachers in school, but also lecturers at university, should teach mathematical modelling to their learners in the best way. Many best-practice university courses or seminars can be found in the literature concerning this aspect (e.g., Lesh & Doerr, 2003;Schwarz & Kaiser, 2004;[START_REF] Borromeo Ferri | Mathematical modelling in teacher education -Experiences from a modelling seminar[END_REF][START_REF] Maaß | LEMA -Professional development of teachers in relation to mathematics modelling[END_REF][START_REF] Burkhardt | Ways to teach modelling-A 50 year study[END_REF][START_REF] Vorhölter | Notwendige Kompetenzen zur Betreuung von Modellierungsaktivitäten aus Sicht von Studierenden -Evaluation und Weiterentwicklung eines Seminars zur Vermittlung modellierungsspezifischer Lehrerkompetenzen[END_REF]. The cited authors describe in their courses or training workshops for mathematical modelling in teacher education several activities and thus competencies, which teachers should learn for teaching modelling successfully. The variety of didactical approaches, contents and aims in these course descriptions are very broad. Thus the conceptualization and operationalization of teaching competencies in mathematical modelling was necessary from the author's perspective in order to get also a theoretical framework.

Based on a Designed-Based Research (DBL) approach, Borromeo Ferri (2018b) developed, within the last 10 years, a module for a modelling course for university and also for teacher training, which led to a model for competencies needed in teaching mathematical modelling (Borromeo Ferri 2014, 2018[START_REF] Borromeo Ferri | Mathematical modelling in teacher education -Experiences from a modelling seminar[END_REF]. The goal is that the lecturer of a university seminar for prospective math teachers or a teacher trainer covers all competencies of the model in a balanced way and sometimes with different emphasis due to the timeline of the course or the workshop. This works very well, because I practiced and investigated these courses since a long time. Figure 1 shows the model with its 4 dimensions and sub-competencies: The structure of my modelling courses is based on this model, which means going through each dimension starting with the theory and ending with discussing about assessment in modelling. Step after step the participants gain more competencies and understand how everything is linked together. Due to the aspect that participants should develop a modelling problem, teach it at school and finally reflect on it, a Theory-Practice Balance is given. This greatly helps to understand what modelling means in the school context and in particular that teaching modelling is possible. For more details concerning the full module, the used of modelling problems, materials or teaching methods, see Borromeo Ferri (2018a). The continuous modification of the modelling course is one aspect, a further aspect is of course to measure systematically how teaching competencies of the participants increase during and after the modelling course, which means at the end of a semester or after a teacher training of two full days. Based on the observations and evaluation sheets of the university students and teachers, I can definitely argue that an increase of teaching competencies takes place after these courses, but the goal is to assess this from a scientific and empiric point of view. This is a challenge when recognizing the complexity of the model. Klock et al. (in press) and [START_REF] Klock | Adaptive Interventionskompetenz in mathematischen Modellierungsprozessen -Erste Ergebnisse einer Interventionsstudie[END_REF] also developed a test instrument for facets of a modified version of the model of Borromeo Ferri. This test instrument is piloted and used already in intervention studies.

However, I developed items for the four dimensions of mathematical modelling teaching competencies based on my model in order to get a test instrument. These items were recently applied and piloted with a pre-and post-test design for a university modelling course for prospective secondary teachers, in order to get knowledge on how their teaching competencies increase. In the following, the item development is briefly described, which shall show the difficulties of building scales, but also some examples of final items are presented. Furthermore first results of the pilot study are shown and discussed.

Development of test items for the teaching competencies

When looking at the model for teaching competencies it becomes clear that items are needed, which should test both declarative and conceptual knowledge. The four dimensions build on each other and are connected. The diagnosis of mistakes in students' modelling process for example is much easier, if the teachers have knowledge about the modelling phases and about modelling tasks. Learning to diagnose does not only mean to give a definition of several types of diagnosis (product oriented/process oriented), but also to practice it with school students and also with concrete examples presented in video clips. The same applies for the topic teacher interventions while modelling. Although this is a strong part within the modelling course, it is not easy to test these competencies.

Within the first approach of test development and evaluation, most of the items were open. This test version was used for 24 prospective secondary and high-school teachers in their fourth semester at university. A prominent open question for the theoretical dimension is: "What is mathematical modelling?", because on the one hand one would expect some kind of definition and on the other hand it is interesting to see what prospective teachers really have understood, when they describe it before and at the end of the course. According to this, very broad answers were given at the end of the course. Often, modelling was described as a process referring to the modelling cycle, for example: "Mathematical modelling means to follow the steps of a modelling cycle." This is not wrong, but also not completely right, because the nature of mathematical modelling is not emphasized. In a further open item the students should draw a modelling cycle, write down and describe the several steps. This is a nice activity, but difficult for coding the results at the end. How to code only partly answers or not precise descriptions? In sum, this kind of test items by using open questions for the teaching dimensions offered a lot of insight of university students' increase of their teacher competencies only on the basis of their elaborate answers and statements from the pre-test to the post-test, but not in the sense of a classical test. Furthermore, the test time was about 45 to 60 minutes, because answers needed intensive writing and the test construction with its complex items does not allow to cover all sub-competencies within the dimensions. Thus the decision for a multiple choice format with closed items was made for assessing declarative and conceptual knowledge in a balanced and economic way for all teaching dimensions.

Like for the previous version, the construction of new test items was based on the content of the modelling course. All items of the new test instrument were developed in such a way that one has to choose a correct answer according to a statement or after the analysis of a written dialogue from learners while modelling. Options between two or five possibilities, depending on the item, are given to pick out the right answer or one has to write down the correct term for which is asked. Thus, this answering format offers the possibility for dichotomous coding. The table gives an overview about the number of items per scale (teaching dimension): Within the theoretical dimensions the testing of declarative knowledge with 26 items was in the foreground. In the following three examples items are shown.

Here, your theoretical knowledge is asked. Choose, if the statement is right (yes) or wrong (no):

yes no

The basis of mathematical modelling are problems from real life

The "Complexity" describes one criterion of a modelling problem For the scale "Task dimension" 11 items were developed, which tested declarative and conceptual knowledge. In order to assess the conceptual knowledge, a modelling problem was given with corresponding solutions of learners. It was asked to choose the right phase of the modelling process, where the solution belongs to. Because of the fact that only one answer from five options is right, one has to analyze the solution more in detail. Not only the knowledge about the terms and the phases is of importance, but to apply it based on own experiences during the course while solving and reflecting modelling problems. The "Instruction dimension" is, compared to the other dimensions, quite hard to assess. First of all it is not possible to test if an individual is really able to teach modelling in the class, nor if the lesson is good or bad. However items were developed, which asked for basics of quality teaching concerning modelling and furthermore about the aspects of teacher interventions as well. Hence 14 items are in the scale "Instruction dimension", which cover declarative and conceptual knowledge. Two items of this scale are shown in Figure 3.

Here, your knowledge about teaching modelling is asked. Choose, if the statement is right (yes) or wrong (no):

yes no

The introduction of modelling activities works with over-determined problems In particular the teacher interventions were tested along different short written conversations of learners while working on a modelling problem in the classroom. The levels or activators of the interventions, for example, had to be determined correctly.

Responsive interventions lead back to the teacher

The last dimension and scale is the "Diagnostic dimension" with 14 developed items. Like in the task and instruction dimension, both declarative and conceptual knowledge was tested with several items. To diagnose in which modelling phase learners worked and also problems, mistakes and misconceptions were in the focus for the item development. Again, items were used, which showed a conversation of learners working on a modelling problem. Then one right answer has to be chosen. A crucial part of the modelling course is the development of students' own modelling problems, which are finally taught in school. This includes besides planning, executing and reflecting on the lesson also to diagnose learners while they work on the problem. This kind of real life observation is connected with the usage of video-clips of learners while modelling within the course. Thus the diagnostic dimension is stressed as very important in the course, because it gives a résumé and the possibility to apply all the competencies learned within the previous dimensions.

What you test is what you get! -First Results of the Pilot Study

The test instrument was piloted in September 2018 in a modelling seminar with 17 prospecting secondary teachers at university in their third semester. This course was a blocked seminar with four sessions each three hours. Between the third and fourth session, the students taught their developed modelling lesson in school and observed learners during modelling activities. The pretest was done at the beginning of the first session and the post-test at the end of the fourth and last day of the course. In contrary to the first test version with open items, the new test time was now 25 minutes. All participants already had basic knowledge about mathematical modelling from one lecture in the first semester within the lecture-series with the topic "Introduction in Mathematics Education". In this lecture the university students learn about the modelling cycle and they also solve a modelling problem. However, only basics about modelling are stressed in this lecture and of course one goal of introducing modelling in this lecture-series is that the prospective teachers are interested to choose a modelling course in the upcoming semesters.

Due to the fact that n=17 is a very small sample in the current pilot study, however, the very first results are promising concerning the quality of the items and α-values. Thus for the upcoming testing in the winter term with a larger population, the test instrument will be checked and if needed, modified again. In the following descriptive analysis the results of the pre-and post-tests are presented. The results are based on a dichotomy coding procedure, showing the means of the dimensions before and after the treatment. Because of the fact that n is smaller than 21, a test of Gaussian distribution was necessary. Not all scales had Gaussian distribution and in order to test the level of significance, the Wilcoxon-Rank test [START_REF] Bortz | Verteilungsfreie Methoden in der Biostatistik[END_REF] was necessary. In Figure 5 the significance values become evident: The results show statistical significant differences after the treatment for three of the dimensions.

Although no significance value could be stated for the instruction dimension, an increase of teaching competencies becomes evident like for the rest of the dimensions as well. The modelling course as the dependent variable can be seen as an effective predictor, so that teaching competencies for mathematical modelling increase. The mean of several items also makes clear that the treatment and thus the structure and content of the modelling course focuses on the central aspects of teaching and learning mathematical modelling. The last item within the Theoretical Dimension for example shows two tasks and one has to decide, which of them is a modelling problem. This item could also be put in the Task Dimension, but in this case, this item focuses more on the question of what mathematical modelling means and what characterizes it. In the pre-test the mean was 0.65 and in the post-test 1.00. Although students had basic pre-knowledge about modelling, it was not a trivial question at the beginning. During the modelling course, the university students again recognized several criteria of (good) modelling problems and that a real context leads not automatically to the conclusion that one can speak of a modelling task.

The Diagnostic Dimension contains many items for testing the declarative knowledge. As described in the previous section, items were developed, which show a short conversation of learners working on a modelling problem. One right answer had to be chosen according to the modelling phase the learners in the presented scene worked primarily. An increase of the means becomes visible from pre to the post-test from 0.65 to 0.88 for such a diagnostic item or from 0.41 to 0.71 for a similar item. Beneath this testing, the university students also evaluated the course separately, which is a normal procedure in my seminars. The aspect of the Theory-Practice-Balance is always strongly emphasized from the prospective teachers (see Borromeo Ferri, 2018). In particular the development of an own modelling problem and teaching this at school is for most of the students crucial for understanding the nature of mathematical modelling.

Summary, Discussion and Outlook

The model of teaching competencies for mathematical modelling is based on a long-termed Designbased research approach. Until now a test instrument for the four teaching dimensions did not exist in order to assess the increase of these competencies in a modelling course by using a pre-and posttest design. In fact it became evident that a test instrument with open items offered a great insight into prospective math teachers thinking concerning their knowledge about mathematical modelling, but it was difficult to evaluate and the test time was too long. The goal was to develop items and scales in order to grasp each of the teaching dimension by reducing the test time.

The presented results are only based on a pilot-study with a small population of students and thus no generalization is possible at this time. However the results allowed an insight into the quality of the items on the one hand and on the other hand statistically significant differences can be stated for three of the four dimension concerning an increase of the teaching competencies on a descriptive level. The pilot study focuses on the test instrument and also on the course as a dependent variable with the hypothesis that prospective teachers, who participate in this seminar, get an increase of their teaching competencies. Hence, a control group is needed. These university students could attend for example a seminar on mathematical problem solving. Then both groups can be compared and a further hypothesis is that mathematical modelling needs more specific teaching competencies than mathematical problem solving, although both topics are very close. Definitely it is clear that the sample size must be enlarged anyway, in order to guarantee the reliability of the test instrument.

Normally the goal of a developed test instrument is that it is universally applicable. The test of modelling competency from Haines and Crouch is such an example. The idea of developing tasks for the sub-competencies of the modelling cycle was great and in the last decades, several researchers adapted these tasks for learners in secondary or high-school. That requires that learners at school or university students were taught in modelling in order to assess them. The same idea is behind the test instrument for the four dimensions of teaching competencies. You can only get what you test, when the contents of the modelling course are in line with the test items, if you want to measure for example an increase of these competencies. This means that the test instrument can, in principle, be used worldwide, but it will certainly work best if the contents of the modelling courses are very close to the one presented here. This is also a big limitation for this test instrument at this stage, but the goal is to use this test instrument together with colleagues from Germany and further countries in order to see if a transfer is possible.

Figure 1 :

 1 Figure 1: Model for competencies needed in teaching mathematical modelling (Borromeo Ferri, 2018)

Figure 2 :

 2 Figure 2: Example items of the scale "Theoretical dimension"

Figure 3 :

 3 Figure 3: Example items of the scale "Instruction dimension"

Figure 4 :

 4 Figure 4: Example item of the scale "Diagnose dimension"

Figure 5 :

 5 Figure 5: Results of pre-and post-test

Table 1 : Number of items per scales (4 teaching dimensions)

 1

		Theoretical D.	Task D.	Instruction D.	Diagnostic D.
	Number of items	26	11	14	14