Farzad Mirzaei

Amir Tavakoli

Zinalabedin Tashakori

Stability Analysis of TLP Floating Wind Turbine Under Severe Environmental Loading Condition

Keywords: TLP floating wind turbine, conservation of angular momentum, severe environmental condition

Offshore wind turbines are subjected to combined wind and wave loadings. Severe environmental condition is probable to happen in marine areas when the wind speed and the wave height is high. Offshore wind turbines require to be stable and not to fail dynamically in severe environmental conditions. For this purpose, using the conservation of momentum theory and Newton's law, the floating wind turbine is modeled and the structural equations of motion is derived. The aerodynamic loading is determined using the blade element momentum theory and the hydrodynamic loading is modeled using the Airy wave theory. To model the severe environmental condition, high values for wind speed and wave height are chosen. To evaluate the stability of the floating wind turbine, a Tension Leg Platform type of floating wind turbine is selected. The system is modeled and simulated for different misalignment angles of wind and wave loading. It can be resulted that, the TLP type floating wind turbine is stable in the presence of misalignment between wind and wave loadings and in severe environmental condition.

Introduction

Environmental pollution and global warming has increased demand for renewable energy. In this regard, different studies and majors have focused on renewable energies . Among different sources of renewable energy, wind energy is the most reliable and practical type of renewable energy which can be generated using wind turbines.

The offshore wind energy is moving towards the deeper waters where floating wind turbines can be installed and operated. TLP floating wind turbine are promising kind of floating wind turbines which can operate in intermediate water depths because the small motions and displacements of the FWT reduces the structural loading on the blades and the tower in comparison to other floating wind turbines [START_REF] Bachynski | Design considerations for tension leg platform wind turbines[END_REF].

Severe environmental conditions where the wind speed and the significant wave height has a large value is more common in deep water seas. It is crucial for the FWT to be stable even in high wind and wave loads. The wind and wave loads are not always aligned and in one direction which will cause the FWT to have motion in different directions.

Different studies have focused on modelling improvement, such as: increasing the number of DOFs [START_REF] Ramachandran | Fully Coupled Three-Dimensional Dynamic Response of a Tension-Leg Platform Floating Wind Turbine in Waves and Wind[END_REF], reducing the computational time [START_REF] Karimirad | A simplified method for coupled analysis of floating offshore wind turbines[END_REF], improving the modeling of mooring cables [START_REF] Karimirad | Modeling aspects of a floating wind turbine for coupled wave-wind-induced dynamic analyses[END_REF], considering the flexibility of components [START_REF] Jahangiri | Multibody Dynamics of a Floating Wind Turbine Considering the Flexibility Between Nacelle and Tower[END_REF] to make the system simulation more accurate. Different methods are used to model and simulate the floating wind turbines and its external loadings. Fluid-structure interaction method is implemented to study the dynamic response of the floating wind turbines [START_REF] Huang | Dynamic analysis of tension leg platform for offshore wind turbine support as fluid-structure interaction[END_REF].

Reducing the motion and vibrations of the offshore wind turbines improves the stability of these systems under various environmental conditions including the extreme environmental conditions. In this regard, different researches have been conducted to reduce the vibrations of offshore wind turbines including fixed and floating wind turbines. Passive tuned mass dampers have been installed in the nacelle, tower or in the blades to reduce the vibrations of nacelle, tower or blades [START_REF] Taherzadeh | A New Efficient Fuel Optimization in Blended Charge Depletion/Charge Sustenance Control Strategy for Plugin Hybrid Electric Vehicles[END_REF]- [START_REF] Colwell | Tuned liquid column dampers in offshore wind turbines for structural control[END_REF]. Active tuned mass dampers were shown to be more effective than the passive counterparts [START_REF] Stewart | The effect of actuator dynamics on active structural control of offshore wind turbines[END_REF], [START_REF] Brodersen | Active tuned mass damper for damping of offshore wind turbine vibrations[END_REF]. Due to the advantages of semi-active tuned mass dampers over the passive and active tuned mass dampers, researchers have focused on vibration reduction of offshore wind turbines by semi-active tuned mass dampers [START_REF] Dinh | Semi-Active Control of Vibrations of Spar Type Floating Offshore Wind Turbines[END_REF]- [START_REF] Sun | Mitigation of monopile offshore wind turbines under wind and wave loading Loading[END_REF]. These studies have been limited to environmental conditions where wind and wave loadings are considered to be aligned.

The probability of occurrence for wind and wave loading to be aligned is higher than any other misalignment. However, it is still crucial to study the dynamic responses of offshore wind turbines under misaligned wind and wave loading. Stewart and Lackner [START_REF] Stewart | The impact of passive tuned mass dampers and wind-wave misalignment on offshore wind turbine loads[END_REF] used two linear tuned mass dampers to reduce the vibrations of barge type floating wind turbine in the presence of wind-wave misalignment. A simplified structural model was used to evaluate the performance of the controlling method. Sun and Jahangiri [START_REF] Sun | Bi-directional vibration control of offshore wind turbines using a 3D pendulum tuned mass damper[END_REF] used a pendulum tuned mass damper to mitigate the vibrations of a fixed monopile offshore wind turbine. It should be noted that, the misalignment effect is much more important in floating wind turbines than the fixed counterparts.

Based on the literature review, the present paper models the floating wind turbine using the conservation of angular momentum theory and Newton's second law. Totally, the floating wind turbine has six unknown degrees of freedom including three translational (surge, sway and heave) and three rotational (pitch, roll and yaw) motions. Also, the system includes two known degrees of freedom, nacelle yaw and spin of the blades. The motion of the floating wind turbine is determined using the Euler angles with exact nonlinear coupling between the degrees of freedom. The aerodynamic loading is determined using blade element momentum method and the hydrodynamic loading is determined using the Airy wave theory [START_REF] Feili | Ranking of Suitable Renewable Energy Location Using AHP Method and Scoring Systems with Sustainable Development Perspective[END_REF], [START_REF] Rabiei | Top down strategy for renewable energy investment: sizing methodologies and Integrated Renewable Energy System models[END_REF]. Also, four representative misalignment angles and large value of wind speed and wave height have been used to study the effects of the misalignment angle on the dynamic response and the stability of the floating wind turbine. MATLAB codes have been developed and the system is simulated and solved in MATLAB. It can be concluded that,

Analytical model

In this section, the equations of motion for the floating wind turbine are derived by using conservation of angular momentum and Newton's second law. It is essential to introduce different coordinate systems and to determine the system's equations of motion with respect to the mentioned coordinates. Also, the external forces and moments, which are resulted from buoyancy, hydrodynamic, wind, and mooring cables are determined.

(x , y , z ) and 2 2 2 (x , y , z ) are body fixed and are located at first and second body's center of mass respectively. (x , y , z ) s s s is the coordinate system for the entire system, which is assumed to be parallel to 1 1 1 (x , y , z ) and is located on the system's center of mass. The coordinate systems is illustrated in Fig. 1. Newton's second law is applied to the system to determine the translational equations:

12 (m m ) a II s CC G F =+  (1)
where

I C F
 is the summation of all external forces. Also, 1 m and 2 m denote the mass of first and second body respectively. Additionally, the linear acceleration of the system which is shown as

a I s C G is equal to 1 2 3 ,, X X X   .
To determine the rotational equations of motion, the conservation of angular momentum theory is used as follows: The angular momentum of the system can be determined as follows:

' s s G s s H I H  =+ (3)
where s I is the inertia tensor and

' ( sin , cos , ) B B C yaw H I I I      =-
. The absolute time derivative of angular momentum of the system can be determined as:

' s s G s s s s H I I H  = + + (4)
By substituting the angular momentum of the system and its derivative into the newton's second law and conservation of angular momentum theory, the equations of the motion can be written in a matrix form as:

wave wind buoy mooring gravity MX CX KX Q Q Q Q Q + + = + + + + (5)
It should be noted that,

Q Q Q Q Q
are the force and moment vectors resulted from the external loadings.

Loading

The mooring forces, buoyancy force, hydrodynamic and wind loads are derived in this section

3.1.Wind Load

An approximate wind thrust force is calculated for the swept area of the blades according to the method, which was introduced in [38]: . To determine the relative velocity of the wind and the blades it is important to model the wind field profile and determine the velocity of the blades.

In this study, wind velocity is represented by summation of a constant velocity with a turbulent component. The logarithmic wind profile is used to calculate the mean wind velocity and the turbulent component is calculated using the IEC Kaimal spectrum. A three dimensional wind field profile is generated using the TurbSim program.

The velocity of the center of blade area can be calculated as:

/ () R s r s I G G s I t G G V V T r  → = +  (7)
where s G V is the linear velocity of the system which can be defined as: 1 2 3 ( , , )

X X X .
The wind force in the I C coordinate system and the moment in the s C coordinate system is determined as follows:

3.2.Wave load

The wave force acting on circular cylindrical structural members can be estimated using Morison's equation as follows:

I M d rt rt a t m n f C DV V C D V C D V     = - + (9)
where d C , a C and m C are the drag, added mass and inertia coefficient respectively. Also,  is the sea water density and D is the diameter of the hull. t V , n V and rt V are the normal component of wave acceleration, structural acceleration and velocity of the water particle relative to the tower. The normal component of water particle acceleration is determined by: ()

II n z z V u V u =  
. Parameter V represents the wave acceleration vector in the I C coordinate system and I z u is the unit vector along the central axis of the tower in the I C coordinate system, which is equal to:

(0,0,1)

I z s I uT → = .
The structural velocity and acceleration are calculated by the following equations:

1 1 1 2 3 / ( , , ) ( ) s C t s I B i G V X X X T r  → = +  (10) 1 1 1 1 1 1 1 2 3 / / ( , , ) ( ( )) ss C C C t s I B i G B B i G V X X X T r r    → = +  +   (11)
Where / s iG r is the vector radius from CM of the system to the segment with unit length. The normal velocity of the water particle relative to the tower is expressed as ()

II rt z rwt z V u V u =  
. The relative velocity of the wave to the segment of the submerged tower is shown as

rwt t V V V
=-in which V is the wave kinematic velocity.

To define the velocity and acceleration of the water particles as it was developed by [START_REF] Feili | Ranking of Suitable Renewable Energy Location Using AHP Method and Scoring Systems with Sustainable Development Perspective[END_REF], [START_REF] Feili | Prioritization of Renewable Energy Systems Using AHP Method with Economic Analysis Perspective in Iran[END_REF], it is considered that the sea waves have a regular sinusoidal shape. According to this theory the surface elevation, velocity and acceleration of particles are computed from Eqs. ( 12)-( 14):

( ) ( , ) cos 2 H y t ky t   =- (12) (0, , ) cosh ( ) cos(
) sinh cosh ( ) sin( ) sinh V u v H k x d u ky t T kd H k x d v ky t T kd     = +   =  -    +   =  -   (13) 2 2 2 2 (0, , ) 2 cosh ( ) sin(
) sinh 2 cosh ( ) cos( ) sinh V u v H k x d u ky t kd T H k x d v ky t kd T     =  +  =  -    +   = -  -   (14)
where H , T , k and  are wave height, time period of wave, wave number and angular wave frequency respectively. The wave force on the platform of the system and the wave moment in the s C system is computed by numerically integrating over the submerged length of the tower [START_REF] Feili | Life Cycle Assessment Of Municipal Solid Waste Systems To Prioritize And Compare Their Methods With Multi-Criteria Decision Making[END_REF], [START_REF] Hosseinabad | A System Dynamics Approach in Air Pollution Mitigation of Metropolitan Areas with Sustainable Development Perspective: A Case Study of Mexico City[END_REF] as follows: / ( ( )

s I wave M I wave i G I s M Ff M r T f dz → = =   (15)
The wave loading along surge and sway by considering the wind-wave misalignment angle,  , can be expressed as:

3.3.Mooring forces

A simplified mooring system is assumed to consist of four radial taut lines in which the tension vector can be expressed as:

/ 0/ / ( ( )) gA I l g A gA r ES F T r L Lr = + - (17)
The pretension of each mooring line is shown as 0

T

, also E , S and L are the Young's Modulus, line cross sectional area and the initial length of the line respectively. In addition, the resulting external moment is calculated from

A/ s I s sI l G C C M r T F → = .

3.4.Buoyancy Load

The instantaneous buoyancy of the floating system can be expressed as follows:

2 1 (0,0, )

I B F g r h  = ( 18
)
where  is the water density, r is the radius of the cylinder and 1 h is the instant submerged length of the cylinder, To determine the moments, the center of buoyancy should be calculated, if the distance between center of buoyancy and CM of the system is assumed as / ( , , )

s B G B B B r    =
, the distance can be expressed as:

    =-     =-    +  = -+ +   (19)

Simulation and Results

The NREL 5MW TLP type wind turbine model is used as a case study. The wind speed is considered to be 25 m/s with a turbulence intensity of 10%. The significant wave height is 10 m with a 10 s period. Fig. 2 shows the translational motion of the platform including sway, surge and heave motion under four different misalignment angles. It is obvious that when the misalignment angle is zero and the wind and wave loadings are aligned the displacement in surge direction is minimal. Also, it can be concluded that the TLP type FWT is stable in the severe environmental condition where the wind speed is considered to be 25 m/s and the wave height 10 m. Fig. 3 shows the rotational motion of the platform under four different misalignment angles. It is obvious that when the misalignment angle is zero and the wind and wave loadings are aligned the rotational motion in pitch direction is minimal. Also, it can be concluded that the TLP type FWT is rotationally stable in the severe environmental condition where the wind speed is considered to be 25 m/s and the wave height 10 m. It should be noted that the rotation in yaw direction is more than any other direction. Another approach for the analysis performed in this study is optimization approach, as discussed in [START_REF] Liu | Optimal Power Flow with Disjoint Prohibited Zones: New Formulation and Solutions[END_REF]- [START_REF] Tashakori | Effective Micro Grid Stability Under Excitation Limiters in Islanded and Connected Modes[END_REF].

Conclusion

In this paper, TLP type floating wind turbine is modeled using conservation of angular momentum theory and Newton's law with considering the misalignment effects between wind and wave loadings. Severe environmental conditions are most likely to happen in marine areas. For this purpose, the system is modeled and subjected to high wind speeds and wave heights. The proposed model includes six unknowns and two known degrees of freedom. The nonlinearity coupling between motions in all degrees of freedom is considered in the model. The aerodynamic loading is determined using the blade element momentum theory and the hydrodynamic loading is modeled using the Airy wave theory. To model the severe environmental condition, high values for wind speed and wave height are chosen. To evaluate the stability of the floating wind turbine, a Tension Leg Platform type of floating wind turbine is selected. The system is modeled and simulated for different misalignment angles of wind and wave loading. It can be resulted that, the TLP type floating wind turbine is stable in the presence of misalignment between wind and wave loadings and in severe environmental condition.

Fig. 1 .

 1 Fig. 1. Schematic image of the spar type FWT Six unknown degrees of freedom is related to the first body which includes three translational, and 3 rotational degrees of freedom. Additionally, two known degrees of freedom for rotor spinning,  ,and nacelle yaw,  which describes the relative motion of the second body to the first body.

G

  is the external moment resulted from external loadings. Also, is the change of momentum amplitude for the entire system and 1 1 C B  describes the absolute angular velocity. The total angular momentum is equal to the summation of angular momentum of each body.

  coefficient, air density and swept area of the blades respectively. The amplitude of the wind velocity relative to the nacelle is shown as rb V

Fig. 2 .

 2 Fig. 2. Translational motion of the floating wind turbine under different misalignment angles

Fig. 3 .

 3 Fig. 3. Rotational motion of the floating wind turbine under different misalignment angles


Berta Barquero
email: bbarquero@ub.edu

The ecological relativity of modelling practices: adaptations of an study and research path to different university settings

Keywords: Modelling, anthropological theory of the didactic, research and study path, ecology, institutional relativity

This paper focuses on the problem of the ecology of mathematical modelling practices at university level through the systematic variation of teaching institutions. Our aim is to deal with the variety of constraints appearing when modelling is implemented in university classrooms, and to study the way new teaching proposals can overcome them. Within the framework of the anthropological theory of the didactic, a teaching and learning proposal in terms of study and research paths in tertiary education shows new possibilities to surmount some of these constraints. The paper presents the design and successive adaptations of an SRP about an urban bikesharing system according to the specificities of different university institutions and the reactions obtained by the students and lecturers.

INTRODUCTION

The starting point of this research is delving into the problem of studying the variety of constraints appearing when mathematical modelling proposals are implemented in university classrooms, impeding their regular development, and to study the way new teaching proposals can overcome them. Several research projects have highlighted the existence of strong constraints impinging on the large-scale dissemination of mathematics as a modelling activity in current educational systems at all school levels [START_REF] Doerr | Models and Modelling Perspectives on Teaching and Learning Mathematics in the Twenty-First Century[END_REF][START_REF] Kaiser | Modeling in lower secondary mathematics classroom -problems and opportunities[END_REF]. We use the term ecology to refer to the institutional conditions allowing and the constraints hindering the way a given activity is produced, transposed, taught and learned in a given educational setting.

In previous research developed in the framework of the anthropological theory of the didactic (ATD), we propose the use of a general frame to detect and place the institutional constraints hindering the possible large-scale dissemination of modelling activities based on a hierarchy of levels of didactic co-determinacy [START_REF] Chevallard | Organiser l'étude 3. Ecologie & régulation [Organizing the study 3. Ecology & regulation[END_REF]. In [START_REF] Barquero | The ecological dimension in the teaching of modelling at university level[END_REF], we use this general frame to detect constraints appearing at different levels, from the specific ones related to how mathematical contents are proposed to be taught at school, to the more general ones regarding the general organisation of school activities and the role assigned to schools in our societies. This ecological analysis shows how institutional constraints are anchored in deep-rooted practices and are difficult -for teachers and also for researchers -to notice since they appeared as "the natural way of doing". For instance, [START_REF] Barquero | The ecological dimension in the teaching of modelling at university level[END_REF] characterise and empirically contrast the predominance of "applicationism" as the dominant way of interpreting, describing and conceptualizing mathematical modelling in natural sciences university degrees. Under its influence, modelling is understood as a mere application of previously constructed knowledge, as if the construction of knowledge were independent of its use. At a more general level, in many schools the prevailing pedagogy is still strongly influenced by the paradigm of "visiting works" [START_REF] Chevallard | Teaching Mathematics in Tomorrow's Society: A Case for an Oncoming Counter Paradigm[END_REF], according to which school knowledge organisations are presented as interesting monuments to visit, instead of as useful tools to provide answers to problematic questions.

In this paper, we focus on going one-step to study the ecological relativity of modelling practices in university institutions. As it is described in Castella (2004) and [START_REF] Sierra | Lo matemático en el diseño y análisis de organizaciones didácticas. Los sistemas de numeración y la medida de magnitudes[END_REF], each institution endures an institutional relation with knowledge, in particular, with mathematical knowledge. Consequently, each institution establishes a set of specific conditions and constraints that can favour or, on the contrary, prevent certain teaching and learning processes and knowledge constructions to be appropriately developed. It is in this aspect where we want to look more carefully. Therefore, we focus on analysing the emergence, persistence and scope of the conditions and constraints for development of modelling through a variation of university institution. In our research, we work on the use of the study and research paths (SRP) as epistemological and didactic model [START_REF] Chevallard | Teaching Mathematics in Tomorrow's Society: A Case for an Oncoming Counter Paradigm[END_REF]Winslow et al., 2013;[START_REF] Barquero | Linking transmission with inquiry at university level through study and research paths: The case of forecasting Facebook user growth[END_REF] where mathematics are conceived as a modelling tool for the study of problematic questions. We here present an SRP based on an urban bike-sharing system inaugurated in Barcelona in 2007 that has been experimented in three different university settings. The starting point of this SRP is the difficulty to get a homogeneous distribution of bicycles in a city with many sloping streets. We present the successive transformations of the SRP to three different university settings, according to the specificities of each institution, and to the reactions from students and lecturers. Some of the commonalities found show the stable constraints hindering the development of the SRP, whereas the differences detected bring new insights about the conditions to surmount them.

DESIGN OF AN SRP ABOUT A SHARING-BIKE SYSTEM

In the following we describe the initial design of the study and research path (SRP) about the sharing-bike system whose starting point is the generating questions (Q 0 ) about how to improve the distribution of bikes in the 'Bicing' system to provide a better service to users. When working with the a priori design of the SRP, there are foreseen several derived questions from Q 0 that needs from a progressive modelling process. In general terms, the modelling project was organised around the following questions that structured the two phases the Bicing project: Q (A) : How can we describe the daily flow of bikes between stations? What is the natural behaviour of the system when it is left alone (without redeployment)? Q (B) : How can we predict the bikes' redeployment needs? Which changes can be proposed to improve the current policy of bikes redeployment in the city? Linked to these questions, we consider real data from Bicing about the distribution of bikes among the different bikes' stations. We, the researchers and the experts who collaborated with us, agreed to organise these data in certain city areas according to the similarities different stations shared on the pattern of daily bikes trips and routes followed. Finally, we decided to present the data organised in six areas (as shown in Table 1), which corresponds to the origin-destination matrix (OD matrix) containing the potential number of daily bikes' uses. Each number {od ij } means the average of the amount of bike traveling in a day from area j and arriving to area i.

Table 1: Origin-Destination matrix with daily bikes' trips

To face the first question Q 0(A) , and going beyond the descriptive analysis of the data contained in the OD matrix, models based on recurrent sequences of order d > 1 can be considered, which are equivalent to matrix recurrent sequences X (n)= f (X(n-1))) where X (n) = (x 1 (n), x 2 (n), …, x 6 (n)) is the vector with the bike distribution in each of the six areas at time n. Next we summarize the a priori design in terms of hypothesis (H), questions (Q) and answers (A) delimited by the researchers about the models that might be used in an implementation of the SRP.

One of the easier assumptions we can work with is considering that:

H (A)1 :
There is no redeployment of bikes in the system and the bike flows between stations is the same every day. Q (A)1.1 : Then, if we deploy different amounts of bikes in each station, what will be the distribution of bikes after 1, 2, 3,…, n days?

The model that can be considered under these assumptions is:

X (n) = M.X (n-1) à X (n) = M n . X (0) for n > 0 (1)
where M is the transition matrix (or transition probability matrix) obtained from the OD matrix, where {m ij } is the percentage of transition between two areas. That is, the potential number of daily travels with origin in j and arriving to i {(od ij )} divided by total amount of departures from j (d(j)). When working with this first model, several questions can appear:

Q (A)1.2 :
Working with the transition matrix and with different X(0) at the beginning of the day, which traits from the trajectory of X(n) can be underlined?

Q (A)1.
3 : Does it exist any fixed point X f to which the sequence X(n) converges to? Do all X(n) converge towards a fixed point X f ? Is it possible to calculate X f in advance?

Q (A)1.4 : Which relation there exist between X f and the n-power of the transition matrix?

And, it can easily appear questions about the limitations of the hypothesis assumed and models built, such as:

Q (A)1.5 :
How can include other factors that are important for Bicing, such as: the total amount of trips made by a bike, the potential demand of bikes, the available bikes?

Introducing questions about how to improve our hypothesis and the models to be more realistic with the system we want to analyse can open many possibilities. One possible new reformulation of the hypothesis we can work with is:

H (A)2 :
We assume that (1) each bike trip takes about t minutes, (2) the entire fleet of bikes does not move every t min, (3) the total number of bikes that moves in period t depends on: (a) the potential demand for bike trips, and (b) the amount of bikes available.

At this point, there appear more complex models where it is important to frame the time t, for instance, t = 30 minutes (which it is the average of a bike trip in Bicing). Then, we can define B i (t) as the number of bikes in an area at time t and

B(t) = (B 1 (t), B 2 (t), B 3 (t), B 4 (t), B 5 (t), B 6 (t)
) as the vector with the bikes distribution in each area. Then, if we define the departures as

D(t) = (D 1 (t), D 2 (t), D 3 (t), D 4 (t), D 5 (t), D 6 (t)
) and the arrival as

A(t) = (A 1 (t), A 2 (t), A 3 (t), A 4 (t), A 5 (t), A 6 (t))
, B(t) can be modelled by:

B(t +1) = B(t) -D(t) + A(t +1) (2)
where D(t) = min [demand_trips(30 min), B(t)] and A(t+1) = M•D(t), with M the transition matrix in time periods t. When this second model is considered, several questions can guide the study process: 2), and considering different initial distribution of bikes at the beginning of the day B(0), which will be the bike distribution B(t) at the end of the day? And, if the system is left alone, after 2, 3, 4, …, 30 days?

Q (A)2.1 : Using this model (
Q (A)2.2 :
Which traits can we underline about the trajectory of B(t) through the simulation of model ( 2)? Are there also some fixed points to which the sequence B(t) converge?

Q (A)2.3 :
Is there any relation between the fixed points X f we reach with the ones detected with model (1)?

Q (A)2.4 :
Which relationship is there between the first and second models, defined in (1) and by (2)? Which of the two models do integrate more realistic conditions about Bicing?

In the next section we retake this a priori design of the SRP in terms of Q 0 and the likely hypothesis and derived questions Q (A).n to analyse the particular implementation of the SRP about Bicing project in the different university institutions. Besides underlying the adaptations that were necessary to the SRP in each university institution, we focus on the most important conditions (common or not) that favour the development of the SRP, and consequently of the modelling practice. In most of the occasions, these conditions and constraints were phrases by the students and lecturers involved in the implementations or by the survey and interview done at the end of each implementation.

ECOLOGICAL ANALYSIS OF THE SRP IN DIFFERENT UNIVERSITY INSTITUTIONS

First SRP adaptation: The 'Bicing project' at the University of Copenhagen

The first implementation of the SRP about the bikes' distribution in the Bicing system took place in the University of Copenhagen (UC). Twenty-three students participated in this implementation. They were taking the course called MathMod (Mathematical Modelling), which was an optional course in the third year of the Mathematics degree. The course run over seven weeks, plus two extra weeks to prepare their final team project. The course had three weekly sessions of two hours each. In general terms, the first session was a lecture, the second was a practical or exercise-based session to practice the content introduce in the previous lecture and, the third one, to work in teams in the computer room to simulate by Mapple some models introduced along the course or to work on the team final project. The teaching course proposal was based on the realization of four short projects (mini-projects), linked to some practical activities. These mini-projects mostly consisted of being introduced to some pre-existing models in the lectures sessions to then asked students to put them into practice in the practical sessions. Some example of the project composing the course are: "Mini-project 1: Using the Malthusian and logistic models to predict population evolution" or "Mini-project 4: The Lotka-Volterra models".

In the academic year 2009/10, the author of the paper participated in this course as researcher and the lecturer offered the opportunity of implementing the SRP about Bicing. It was integrated as the fifth (and last) project of the course. The SRP implementation ran over two weeks, with six sessions of two hours. At the end of each week, students working in teams had to deliver a report with their temporary results of the Bicing project. It was necessary to break with the above-mentioned organisation of the course sessions and to set up time for the presentations by the lecturer-researcher and for students' presentation. There, students could compare their proposals and to collectively agree how to follow. During the first week, once the generating question Q 0 was presented by the lecturer-researcher, students agreed to firstly focused on Q 0(A) from where students developed most of the path described in the previous section about model (1). In the second week, we (students and instructors) worked on how to reformulate the H (A)2 and Q (A)2 , as most of the groups noticed that in model 1 there were considered some unrealistic assumptions. Due to time restrictions, we could not go further the second model. Finally, each team had to deliver a report one week later the ending of the project with some suggestions for Bicing about how to improve their bike replacing system, Q 0(B) . Figure 1 summarizes the path followed in this first implementation. We counted on different conditions that favour that the SRP progress fruitfully. First, as it was the fifth project of the course, and the course was explicitly focus on modelling, students and lecturers shared a common discourse to refer to modelling. This was an important condition for modelling to be noticed [START_REF] Barquero | The ecological dimension in the teaching of modelling at university level[END_REF].

Secondly, the second mini-project was about Leslie matrices and transition matrices. It thus facilitated that students autonomously posed many new questions, such as Q and Q (A)1.4 and, thanks to the previous work developed with Maple, students easily worked on calculating and simulating sequences and studying their convergence. On the contrary, there were also some constraints that were manifested by students mostly at the beginning of the SRP implementation. When we started with the Bicing project, students were astonished by the new responsibilities that they were asked, such as: formulating hypothesis, looking for and building models, testing models' appropriateness, formulating new questions, writing a report without any predetermined structure, etc. Although their initial confusion, consequence of a big rupture with the didactic contract established in the course, they started assuming these new responsibilities. In the previous activities of the course, students were only asked to "apply" the models they had been introduced to. So that, breaking some rules of the didactic contract and make students responsible of several new tasks in the modelling process were the main constraints we had to surmount. In fact, the course organisation shown many traits (and constraints) derived from "applicationism" [START_REF] Barquero | The ecological dimension in the teaching of modelling at university level[END_REF]. For instance, it was assumed (throughout the course organisation) that the mathematical models had to be introduced in advanced and then applied to different situation, models that are rarely questioned and hardly reformulated. When the Bicing project started, many students' resistances appeared that reflected the implicit assumptions about what modelling was suppose to be and what we (as students and as lecturer) were asked to do. At the end of the course, when students were asked through a survey and with the interviews with some of them, they stated how interesting it was this last project for several reasons. Some of main reasons mentioned by the students were: the openness of the questions, the possibility to delimit the questions to face, the necessity of clearly understanding the modelling process (the hypothesis assumed, the models' construction and their validation), the possibility to compare teams' proposals and results with the rest of the groups who could have been working differently, possibilities to discuss the limitation of the models proposed and make them evolve.

Ecological relativity of the second SRP adaptation implemented at UAB

The second implementation of the SRP was the following academic year at the Universitat Autònoma de Barcelona (UAB). There was a course called "Mathematical modelling workshop" which started in 2009/10 with second-year students of Mathematics degree. It was the first edition of the course, which was compulsory, with a total of 45 students participating. The didactic organisation of the course was different from the previously described at the UC. The main aim of the course was to develop a project in working teams (composed of 4-5 students) that students selected from a list provided by the lecturers of the course. Running in parallel, there were planned some short activities about modelling. The first year this course was implemented, one of the modelling activities planned was the 'Bicing project'. It ran over 5 weeks, with two 2-hour sessions per week. We invested more than the double of time than in its first implementation. Similarly, students were asked that at the end of each week they had to deliver a report with a synthesis of their advances in term of: (a) questions they had focused on, (b) hypothesis assumed and mathematical models considered, (c) temporary answers and (d) new questions to follow with). At the end of the Bicing project, each working team had to deliver a final report as summary of the whole modelling work developed. In general terms, the modelling process students and instructors followed in this occasion was not so different concerning Q 0(A) , although now none of the students' working team tackled the second phase of the project with Q 0(B) , or posed any questions about the properties of the n-power of transition matrices, such as: One important novelty (and extension of the SRP) was that some students asked about the possibility of working with partial matrices, for instance, by considering different OD matrix to describe differently the bikes' flow in the morning and in the afternoon. Students had checked in the web how many bikes were available at different time frames and they had concluded that there were different patterns of bikes disposition depending in the daily time frame. The instructors asked to the experts we worked with about the possibility of having these new data. The external experts provided us two new matrices: one for the morning pattern, from 05:00h to 14:30h, and the other for the afternoon, from 14:30h to 00:00h. With this new data, the modelling process concerning Q 0(A) was extended towards the construction of a third model, built upon the two previous ones (1) and ( 2), and taking into account these two different OD matrices. Figure 2 summarizes the path followed in this occasion and the extension it supposed for the first phase of the Bicing project.

Q (A)1.4 or Q (A)2.3 .
If we focus on analysing the conditions and constraints we detected in the second implementation of the SRP, we have to mention that in this occasion it was the lecturer of the course who expressed more clearly some important constraints. He expressed, in an interview at the end of the implementation, that we had invested too much time with the project. He manifested that students needed to work more independently and there was no need of planning common discussions among all the working groups. His main request was to let students work independently and ask them to present their finding at the end of the course. Reactions that were on an opposite sense than the ones expressed by the Danish lecturer, who expressed that the activity was too open and too less guided for students. We can say that these reactions corresponded to their spontaneous teaching models that both lecturers implicitly defended. In this second implementation, it shared traits of a modernist teaching model (Gascón, 2001), by considering knowledge construction as an individual process, also private. That is why the lecturer preferred not planning any teaching device where to share and collectively talk about the modelling work developed, and where to question, debate and agree about the questions, tools and strategies to follow along the modelling process. As the course organisation at the UAB showed, each team was supposed to work most of the time independently in their project, and it was not until the end of the course when they explained their results. We could observe several inconveniences, linked to important constraints, which were more evident in the following courses when the lecturers planned short modelling activities as complement to the working group project of the course. First, students showed a lack of terminology and of a common discourse (shared with lecturers) to talk and write about the modelling activity developed. Second, the main outcome from the students modelling work was their final presentation of the project at the end of the course. It was delivered as a report that mostly contained the final models and models simulation, as if all the intermediate modelling work may remain in the private space of each group. Consequently, most of final reports showed a poor progression of the models considered and of the tools to contrast and validate them.

CONCLUSIONS AND DISCUSSION

It has to be highlighted that the two adaptations of the SRP presented in this paper were done under advantageous conditions. First, it was experienced with students of the Mathematics degree who were taking a course on mathematical modelling and with lecturers who are experts on modelling. Second, in both cases, the schedule and programme of the course were flexible and we had longer sessions (2-hour sessions two or three times per week) than the prevailing university settings use to offer.

Nevertheless, one could think that we may detect similar conditions and constraints in these two university setting, but it is important to see how different institutions established different relations with the knowledge at the stake, in this case, with the teaching of mathematics modelling. Then, for example, some conditions that appear in the first implementation can become strong constraint for the second one. For instance, it was the case of the necessity of sharing a common discourse to talk about and analyse modelling practices, which was an important condition underlined in the first implementation, becoming a constraints in the second one.

But, if we move away from these "optimal" university conditions, do we find similar constraints? Which of them are sensitive to be surmounted? How to overcome some of the most important constraints? To face these questions, and follow enquiring into the institutional relativity of the conditions favouring and the constraints hindering modelling practices, we proceeded with the third adaptation of the SRP. It was redesigned and later implemented with first-year university students of business and administration degree (4-year programme) in IQS School of Management of Universitat Ramon Llull in Barcelona (Spain) during the entire academic year 2013/14. In this occasion, the Bicing project was extended (called now "Cycling project") to become the central project developed along the three terms of the mathematics first-year course. The SRP was broken into three branches. The one described in this paper (in section 2) was implemented during the third term, only focusing on the first model (1). During the entire course, not only the initial structure of the SRP was extended, but also we pay special attention to which teaching devices and strategies could help to overcome some of the most common constraints for modelling and to create appropriate conditions for modelling and for the SRP. We are in the process of analysing them in depth with the aim of extending our knowledge about the ecology of the SRP and its institutional relativity.

Figure 1 :

 1 Figure 1: Summary of the path follow in the first implementation of the SRP at UC

Figure 2 :

 2 Figure 2: Summary of the path follow in the second adaptation of the SRP in UAB

ACKNOWLEDGMENTS

The research leading to these results has received funding from the Spanish R & D project: EDU2015-69865-C3-1-R and EDU2015-64646-P.


Helge Fredriksen
email: helge.fredriksen@uia.no

Analysing tasks for the Flipped Classroom from the perspective of Realistic Mathematical Education

Keywords: Flipped Classroom, RME, Mathematics for engineers, the role of digital and other resources in university mathematics education

The advent of Flipped Classroom as a framework for organizing the teaching and learning of mathematics has the potential to revitalize the attention to tasks as a vehicle for meaningful learning in tertiary education. Flipped Classroom is based on the idea of student active learning under close guidance of the university teacher. Due to the possibility to engage the students in meaningful discovery of how mathematics can relate to real-life situations during in-class sessions, the tasks are seen to have a central role in a successful implementation of Flipped Classrooms. This paper explores the realistic mathematics education (RME) as a theoretical framework for task design and analysis in the context of in-class flipped engineering mathematics classrooms, and I seek evidence of knowledge construction through analysing students' work on modelling the height of a rider in a double Ferris wheel.

INTRODUCTION

One of the fundamental principles of Flipped Classroom (FC) approaches in education is the opportunity it provides for a more interactive and meaningful use of class time. [START_REF] Bergmann | Flip Your Classroom, Reach Every Student in Every Class Every Day[END_REF]. The idea is to use valuable classroom time for the more advanced learning processes like analysing, evaluating, and modelling in a cooperative environment consisting of peers and tutors. A contrast to this form of in-class student participation can be the traditional university teaching style where the lecturer is more or less seeking to "transfer" the mathematics in a monologue fashion [START_REF] Sfard | University mathematics as a discoursewhy, how, and what for[END_REF].

Moving the lecturing part out of the classroom as a preparatory activity can help remedy the problem lecturers often feel is present during the delivery of traditional lectures, the one of disengaged and passive students. Rather, the social setting of the classroom can be used to let the students discuss, evaluate, try out, and receive assistance with their ideas and conjectures. However, such learning processes need a structuring element, which can be the tasks that students work with, preferably in groups. The purpose of this paper is to advance knowledge on task design and analysis in connection with FC implementation, specifically considering the theoretical framework of RME. I attempt to do this through a case study of students' work with a modelling task that follows the heuristics of RME.

Previous research

FC is an innovative teaching approach, initially described by [START_REF] Lage | Inverting the Classroom: A Gateway to Creating an Inclusive Learning Environment[END_REF] that has many implementations. As the popularity of FC teaching has caught up, interest has been considerable also for tertiary educational settings [START_REF] Wasserman | Exploring flipped classroom instruction in Calculus III[END_REF].

Various authors have done research on general issues of FC pedagogy [START_REF] Abeysekera | Motivation and cognitive load in the flipped classroom: definition, rationale and a call for research[END_REF][START_REF] Wan | Technology Integration and the Flipped Classroom[END_REF]. However, little research has emerged on analysing how task design should be addressed in connection to FC implementation. In particular, I was not able to find any research considering RME as a framework for analysing tasks in connection with FC in-class activities. Several articles seem to call for more research on providing insights on FC pedagogic heuristics [START_REF] Song | How" to Design, Implement and Evaluate the Flipped Classroom? -A Syntheis[END_REF][START_REF] Wan | Technology Integration and the Flipped Classroom[END_REF]. Task design is of particular importance for the in-class FC component, since the model of FC considers the classroom activities to be the arena for attaining the highest levels of skills and knowledge [START_REF] Bergmann | Flip Your Classroom, Reach Every Student in Every Class Every Day[END_REF]. The challenge is to bridge the out-of-class videos with these in-class activities in a meaningful way.

The particular task analysis performed in this paper is based upon students' group work with mathematizing the double Ferris wheel movement. A study using a similar type of problem is described in [START_REF] Sweeney | Reconceiving Modeling. An Embodied Cognition View of Modeling[END_REF]. They found that the students' bodily engagement using gestures and measurements by fingers facilitated a link between the movement of the rider and the mathematical model.

THEORETICAL FOUNDATIONS OF RME

According to [START_REF] Van Den Heuvel-Panhuizen | Realistic Mathematics Education[END_REF], RME is a domainspecific instruction theory for mathematics that focuses on rich, "realistic" situations serving to initiate development of mathematical concepts, tools and procedures. Core teaching heuristics give these directions:

 The activity principle emphasises students' direct participation throughout the learning process. Mathematics is considered to be best learned by doing mathematics, which is in accordance with student active learning that is considered a widely accepted principle of FC.  The reality principle expresses the importance of presenting students with reallife situations and problems that they can imagine and mathematize upon.  The level principle highlights the idea that students pass through various cognitive development phases, from informal context-related descriptions of the problem towards the use of more formal mathematical language. This process sets the stage for bridging student understanding that the model of the contextrelated situation at hand can become a model for similar kinds of problems.  The intertwinement principle emphasises that mathematical content domain should not be considered as isolated fragments, but rather seen as a connected whole. This principle supports task designs facilitating open problems that stimulate students' own thinking and reasoning about which mathematical solution techniques and mediating artefacts to employ.  The interactivity principle relates to the idea that learning mathematics is not purely an individual pursuit, but rather a social activity, where group work and whole-class discussions should be orchestrated by the teacher. This principle aligns well with socio-cultural learning theories, as also emphasised by [START_REF] Cobb | Learning from and Adapting the Theory of Realistic Mathematics Education[END_REF]. They consider semiotic mediation and cultural tools as important means of conveying mathematical meaning in an RME setting.  The guidance principle refers to the idea of "guided re-invention" of mathematics. Instructional sequences in task design can involve historical evolutionary steps in mathematics as inspiration for rich context problems [START_REF] Gravemeijer | How Emergent Models May Foster the Constitution of Formal Mathematics[END_REF].

Further, RME makes a distinction between horizontal and vertical mathematization.

According to Van den Heuvel-Panhuizen and Drijvers (2014), students use mathematics as a tool to understand and organize problems in a real-life context, typically during task solution. This is called horizontal mathematization. Vertical mathematization refers to mathematizing one's own mathematical activity to reach a higher level of abstraction.

The research question explored in this paper can be framed in the theory of RME:

"To what extent does RME task design facilitate students' modelling activities and knowledge construction in a FC context?"

METHODOLOGY

The data for this paper was collected from video filming the work of two groups of computer engineering students in their first semester of study. These student groups were part of a larger research project conducted at a university campus in Norway. FC teaching was performed throughout the whole year of study. I focus on one in-class session where students were modelling the double Ferris wheel.

The Flipped Classroom setting

Before this in-class session, the students had watched an out-of-class session of videos that introduced the unit circle and how the y-component would map to the sine function as the angle rotated (Figure 1). In addition to this, there were videos showing examples on how to make mathematical models using this function, and how one could sketch a particular sine function based on textual information. Lastly, they were showed examples of how to determine the period, amplitude, baseline and phase shift by considering various graphs of this function. These videos were provided using English as the spoken and written language, while students were using Norwegian as working language in class. The particular task that I refer to in this paper was given to the students in connection with the teaching of trigonometric functions during the second week of the semester.

In this task, they were asked to create a model of the evolution of the height of a rider on a double Ferris wheel. The double Ferris wheel was represented in an applet available for the students to watch on their computer (Figure 2). The Ferris wheel consisted of two separate wheels rotating at the same speed and centred at the end of a larger rotating bar as seen on Figure 2. The students had the possibility to study each rotation separately by starting/stopping each of them. The height measurement of an imagined rider called Marit (marked with a dot on one of the wheels), was available as a readout at all time. The interested reader can study the applet at http://sigmaa.maa.org/rume/crume2017/Applet.html. The task presented for the students was the following:

1. Create a representation of your choice that illustrates Marits' ride on the double-Ferris wheel 2. Create a sketch/graph of Marits' height versus time as she ride on the double Ferris-wheel 3. Create a function for Marits' height versus time as she travels on the Double-Ferris wheel. 4. Graph the functions you found in the previous problem using a graphing utility like geogebra. Does the graph of your function make sense? Is it similar to the graph you sketched?

The design is seen to follow the principles of RME. Firstly, the purpose of the task is to engage students in exploring the dynamics of a familiar situation; the Ferris wheel ride. This meets the demands of the activity and reality principles. Furthermore, the sequence of the task, asking students to initially make a representation of the movement, followed by sketching a graph, and finally creating a function, attempts at making the students move from more informal to formal mathematizing, respecting the level principle. In setting the stage for employing trigonometric functions for mathematical modelling, the intertwinement principle is considered. Further, the interactivity principle is kept through the group work context of the task. The guidance principle, however, will not apply to this task design. For most themes in engineering mathematics, a reinvention can be a considerably challenge to embed in task designs.

Students were placed in groups of 3 or 4 when working with the task, and a 90 minutes session was spent on the task. The author of the paper was the teacher throughout this session1 . The study was performed under a naturalistic research paradigm to ensure as realistic settings as possible for the observation of students' collaboration [START_REF] Moschkovich | Integrating a Naturalitic Paradigm Into Research on Mathematics and Science Cognition and Learning[END_REF]. The students were filmed using a high definition camera that followed the group work for the whole session. Prior to the filming, students had acknowledged their participation in the study through signing a letter of consent. The selected students had watched the videos beforehand. Statistics about which videos and how long the students had spent watching them were available in the tool I used for video-distribution to the students. I analysed each recording using descriptive accounts [START_REF] Miles | Qualitative data analysis: An expanded sourcebook[END_REF], where sessions of data are broken into separate entities of activity. Each such entity were subsequently analysed for content informing us on the research question, and I report on a variety of such episodes in what follows. In the aftermath of the last case, I performed an interview with two selected students to obtain deeper insights about certain episodes during the work with the task and general qualities about videos and the session.

Furthermore, the study was performed on the cohort of 2017/2018, a class consisting of 20 students. The two groups that I report from had highly varied backgrounds, most of them from a pre-calculus course comprising two years of mathematics from upper secondary into 1 year of study at the university. They were acquainted with trigonometric functions through this pre-calculus course before arriving at the engineering study. However, as one of the students mentioned in an interview afterwards, he only remembered sparsely what the various mathematical terms was named by, but claimed to be able to use them anyway. I filmed two of the groups to be capture some of the variation taking place inside the single classroom environment. I will refer to these groups as group 1 and group 2.

I specifically looked for traces of content from the videos, that is, the out-of-class component of FC when analysing students' discourse. Such content can be word use, theoretical elaborations like definitions and theorems, visually mediated ideas and specific examples.

RESULTS

The two groups seemed to develop a rather different focus in terms of how to approach the task. One group spent much more time on the formal mathematizing part of the task, compared to the other one that wanted to understand the movement from a more informal, empirical viewpoint.

Group 1

The students in this group followed the layout of the task in the sense that they attempted to sketch a graph of the movement before moving on towards formal mathematical modelling. This seemed to set the scene for a discussion among the group members on which physical properties were influencing the model. They constantly argued about where the "x-line/dotted line" should be, meaning the baseline, in addition to discussing how the various lengths of the rods and rotation times were influencing the amplitude and period. However, synthesizing the two movements into one seemed to form the biggest challenge for the group. For instance, the teacher was consulted on how one could decide the amplitude on a combined sine. After about 15 minutes of trying to comprehend this, the group moved on to a more formal mathematical phase of modelling. One of the students, let us call him Finnegan, appeared to get a breakthrough at about 17:40 (timestamp corresponds to video recording). Figure 3 show this situation where he is pointing to the printed unit circle that expresses various exact values of sine and cosine, while he states:

"I suggest that we make two functions that are zero when… They are not zero, but After this episode, the students seemed to prefer working in two pairs to discuss further details of the functions. Both pairs of students agreed that the model should consist of a sum of two different sine functions.

Analysis

We can see that Finnegan struggles with the correct phrasing of what he is trying to express, but the visual connection between the Ferris wheel circle and the unit circle is guiding Finnegan towards an understanding of phase shifting. The printout of the unit circle became an important mediating artefact for Finnegan, although shared with students earlier for solving trigonometric equations, not for modelling. They spent much of the time getting accustomed to the mathematical terminology with amplitude, period and, as one sees, phase shifts. The informal part of the task where students were asked to elaborate on a sketch of the function seemed to facilitate these discussions, much in line with the interactivity and level principles of RME.

I was unable to make any clear connection to the videos from the discourse of these students. During the interview with one of the students in this group, it was pointed out that using English as the formal language in the videos was difficult to handle. The other group that was filmed during this session utilized a variety of means to depict the movement of the Ferris wheel. In Figure 4 we can see two of the students working in parallel, where one is considering using a graphing calculator, while the other is attempting to graph the model on GeoGebra. On the right PC the applet with the simulation is displayed. Another member in this group even mapped the movement from the PC screen to a sheet of paper, letting the pen trace the movement of a point on the Ferris wheel, similar to what was reported in [START_REF] Sweeney | Reconceiving Modeling. An Embodied Cognition View of Modeling[END_REF]. The group also attempted other practical means of tracing the movement like taking screenshots in timed intervals and putting the variable height values displayed in the applet in a function table for later analysis. It was even suggested that they would let GeoGebra use these data-points to produce an expression for them using interpolation. After working with various such "empirical" approaches to the problem, mostly on an individual basis, one student arrived at an expression for the model at 49:16. He gave this explanation to the others in the group on how he was able to arrive at a summation of two sinusoidal functions:

"Analyse the small circle, height over time, that became f(x), then we did the same with the large one, called this g(x), then we made a new one h(x) consisting of the two of them together"

Analysis

The big surprise in this session was that the one student that had not prepared watching the videos seemed to be the leading the group discussions. It might seem that RME can act to remedy group dynamics where certain individuals come unprepared. The reality principle in RME stresses the importance of using real-life situations which is likely to provoke engagement, discussions and active participation during group work.

In this particular data set, it seemed hard to find a direct link between not being prepared by the out-of-class session and poor participation in the group work although this was a prevalent finding in other analyses of group work [START_REF] Fredriksen | Exploring Tensions in a Mathematical Course for Engineers utilizing a Flipped Classroom Approach[END_REF]).

An important observation that was made for both these groups was the physical arrangement of four students placed face to face, which made it harder to collaborate on the task. This was a setup originally meant to spur more discussion since the students would face each other. However, it turned out to be a poor design in this particular setting due to the importance of using the laptops for controlling the visual simulation in addition to the actual modelling in a digital graphical environment. It became apparent that two students would look at one PC, while the adjacent students would look at an opposing one, effectively hindering much of the discussion across the group (Figure 3 and Figure 4) The teacher recognized this, trying to convince students to look at the same PC, but they quickly went back to work in pairs again. One of the students also confirmed this observation during the post-interview. Thus, it seems that this configuration interfered with the interactivity principle of RME for the groups.

DISCUSSION

In this study, I investigated to what extent RME task design facilitates students' modelling activities and knowledge construction in a FC context.

The analysis found students were making active use of computers, calculators, printouts of the unit circle, internet, and even their mobile phones to support their exploration of the task. This can be seen to support the intertwinement principle: students were actively using various tools, in addition to the mathematics introduced in the videos, to explore the problem.

The second group seemed to have problems progressing towards the formal mathematizing part of the task. An observation made from studying the group dynamics was a quite individualistic working attitude. There seemed to be little collaboration on working towards a common model for the group, effectively violating the interactivity principle of RME. During the post-interview with one of the students in the group, I asked him if he shared this impression, which he confirmed, stating that some kind of internal competition among the members evolved.

The guidance principle of RME refers to students' re-invention of mathematics.

Although there are examples of RME facilitating such inductive discovery [START_REF] Gravemeijer | How Emergent Models May Foster the Constitution of Formal Mathematics[END_REF], we can conjecture that the FC out-of-class preparatory component turns this table. Through the videos, the students should have attained basic knowledge of mathematics that is to be articulated through the modelling activities inclass. Thus, instead of re-inventing the mathematics, they are rather re-employing it at a given situation and, through this, making conceptual ties towards it.

We can clearly see how the level principle of RME is at play in these observations. Initially, students are grappling with understanding the basics of the movement, using gestures, making sketches and talking to each other using informal language. The next stage of the task prompts students to express a formal mathematical description of the movement, which all students achieved to some extent. Although the students work was directed towards making a model of this particular movement, the whole-class discussion at the end of the session pointed towards similar situations that this could apply as a model for. This was also done in the videos prior to class.

The reality and activity principle seem to be well safeguarded in this task. This is also supported by the statements by one of the students during the informal interview conducted after the session:

"To see the task in a physical real framing was motivating"

Throughout the session, students used the familiar terms and concepts related to the sine function. Words like amplitude, phase shift, period and baseline were all referred to, and the students seemed to be able to connect these to the real-life situation of the Ferris wheel. If this really stems from watching the videos or originates from previous experience with the sine function, I have no direct evidence. This of course influences the capacity to answer the research question. Light on this may be shed by statements from the other student interviewed: "It's very nice that you can see the videos, think about it for some days, get it refreshed in the lesson and then start working with tasks related to it. This helped me a lot".

Although this is a statement about the general nature of the teaching layout, it nevertheless gives indication about FC as an instructional platform giving learners a potential for experiencing different motivations for mathematics.

CONCLUSION

When working with task design in mathematics, we employ various theoretical frameworks that has certain sets of design principles that the designer should adhere to [START_REF] Kieran | Frameworks and Principles for Task Design[END_REF]. We have seen that the Ferris wheel task design presented in this paper aligns well with the principles of RME for task design. However, designing such tasks for a FC pedagogic context will need an additional component, namely the adoption of the video preparation. Clearly, close attention to the pedagogical structure combining videos and tasks seem to be necessary for facilitating rich discussions, modelling activity and knowledge construction to take place in-class. This statement is supported by the cases that were analysed in this paper, where we saw students engaged in horizontal mathematization utilizing out-of-class preparation through videos and mathematical modelling of realistic situations.

Figure 1 :

 1 Figure 1: Mapping of unit circle y-component to the sine-function

Figure 2 :

 2 Figure 2: The double Ferris wheel simulation

  Then you would have made a function that is zero when the period…Well you have to make two functions…"

Figure 3 :

 3 Figure 3: Phase shift of the sine mediated through the unit circle

Figure 4 :

 4 Figure 4: Utilizing various digital tools to model

A pilot-study in a very similar setting was performed on the previous cohort of

This was part of a cooperation project with San Diego State University, where the research fellow Matt Voigt was responsible for the teaching material in addition to conducting the in-class teaching.

Acknowledgements

The author is indebted to Matt Voigt and his fellow research team at SDSU, who were helpful in supporting this research. I also wish to thank MatRIC for funding and assisting the cooperation between our institutions.


Shaista Kanwal
email: shaista.kanwal@uia.no

Engineering students' engagement with resources in an online learning environment

Keywords: Students' interactions with resources, the role of digital and other resources in university mathematics education, mathematics for engineers

In this paper, we investigate how undergraduate engineering students interact with an online learning environment provided to them in a Calculus course. The constituent resources of this environment include tutorial videos, textbook and MyMathLaban online interactive system for mathematics. A qualitative case study involving a small group of students has been conducted. We investigated which resources these students used and the manner in which they incorporated these resources in their online mathematical work.

INTRODUCTION

In recent years, digital resources are increasingly used for teaching and learning of mathematics [START_REF] Borba | Blended learning, e-learning and mobile learning in mathematics education[END_REF][START_REF] Pepin | Digital curriculum resources in mathematics education: foundations for change[END_REF]. The presence of wide range of digital resources in terms of their functionalities allows various possibilities of creating digital environments for students to learn mathematics. Each digital environment might afford unique interactive and learning opportunities; therefore, empirical research closely looking at students' engagement and the opportunities for their learning in such environments is well needed. This study deals with one digital learning environment provided to undergraduate engineering students for practicing mathematics. The aim is to explore students' interactions with the constituent resources of this environment to elucidate the learning opportunities in this environment. [START_REF] Adler | Conceptualising resources as a theme for teacher education[END_REF] introduced the term resource to embrace several agents such as physical, human and cultural tools and aids intervening in a teacher's activity. In this paper, however, we distinguish between digital and classical resources and focus on students' work with resources. The use of digital resources is relevant in the context of engineering mathematics in the sense that engineers during their professional activities rely on technology for solving mathematical tasks (van der Wal, Bakker, & Drijvers, 2017). The framework for mathematics curricula in engineering education (Alpers et al., 2013) recommends the use of technology aimed at fostering engineering students' mathematical competencies. In the next section, we present the theoretical framework, and the subsequent section contains introduction to the constituent resources of the online learning environment.

THEORETICAL PERSPECTIVE

In order to study students' interactions with the resources, we employ the documentational approach to didactics (Gueudet, Pepin, & Trouche, 2012;[START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF] which is grounded on Rabardel's work [START_REF] Rabardel | People and Technology: A cognitive approach to contemporary instruments (translation of Les Hommes et les Technologies[END_REF] and enlarges the instrumental approach [START_REF] Trouche | Managing the complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations[END_REF]) in mathematics education. One important distinction between the two approaches lies in the extension of the concept of artefact, in the former approach, to resource which allows considering wider set of materials intervening in the teachers' and students' activities. A resource can be conceptualised "as both noun and verb, as both object and action that we draw on in our various practices (Adler, 2000, p. 207)". Thus, the approach has the potential to take in consideration material, human and cultural resources such as language, time, mathematics teachers, etc. Moreover, a resource is never isolated but belongs to the wider set of resources [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF].

While one focus of this approach is on the teacher's work with the resources, the study of students' use of resources can provide the overview of their actual use [START_REF] Gueudet | Students' work in mathematics and resources mediation at university[END_REF]. Also, this approach has the potential to provide rich analyses if used to evaluate students' work in terms of interactions with different resource systems [START_REF] Trouche | From instrumental to documentational approach: towards a holistic perspective of teachers' resource systems in higher education[END_REF] or with a particular resource [START_REF] Aldon | Handheld calculators between instrument and document[END_REF]. We will employ this approach to analyse how students interact with available resources.

In particular, we analyse students' techniques when working digitally in mathematics [START_REF] Artigue | Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF]. A technique is perceived as "a manner of solving a task (Artigue, 2002, p. 248)". While students work on mathematical tasks in a digital environment, they might adopt paper and pencil based techniques or instrumented techniques. The obvious and easily observable objective of each technique is to reach the goal of the activity i.e. to produce the results whereas the contribution of a technique to the learning of involved mathematical concepts might not be easily recognisable. The former corresponds to pragmatic value while the latter corresponds to epistemological value liked to each technique.

We seek to explore the kind of techniques implemented by the students in the digital environment to make sense of how students interact with this environment while working on mathematical tasks. Furthermore, realisation of the values attached to the students' instrumented techniques will also help to understand the role of digital resources in their learning (Guin, Ruthven, & Trouche, 2005). There are several resources involved in present situation, therefore, we confine to the general features of corresponding techniques in the present paper. By this, we mean to consider students' general organisation of digital work with several resources related to all contents in a Calculus course. We ask the following question: How do engineering students incorporate resources during their work in an online learning environment?

THE SETTING

This study took place in a Norwegian public university during the spring of 2017. Undergraduate students enrolled in electronics engineering program participated in this study. In their Calculus course, students were offered an online learning environment such that they could work remotely by interacting with the provided resources. These resources were made available to them electronically to work and proceed through the course. There were no mandatory lectures, and they could access the lecturer in the case they needed additional support. The final examination was also in digital format where the students were allowed the access to tools and aids.

The resource system comprised MyMathLab environment, tutorial videos coupled with the notes, and the textbook. The students' homework and the formative assessments were administered online through MyMathLab system. MyMathLab is an interactive learning system for practicing mathematics online (figure 1). While this system provides an online platform for homework and assessments, it also facilitates students in solving the tasks by providing help and feedback. Students can seek help through utilising "help me solve this" or "view an example" functions in the system. The former lets the student solve a similar task by guiding on each step whereas the latter shows a similar worked-example. The interactive nature of MyMathLab system allows considering it as a resource which can potentially influence students' activity in this course. The tutorial videos are created by the lecturer, and recorded by using a document camera. Each video deals with a specific section in the book and is named accordingly. In these videos, the lecturer explained the topics in the book and worked through the relevant examples occasionally. The notes pertaining to the video tutorials were available online. The length of these videos varies depending on nature of the concerned topics. The tutorial videos replaced lectures and it was expected that students would watch the videos to learn mathematical topics. The textbook served as the central resource in the sense that MyMathLab and tutorial videos were based on contents in the book.

In this course, a compulsory task was the group project in which students were required to prepare a question bank related to integration. That question bank was needed to be programmed in the STACK environment, a computer aided assessment platform. Maxima is the programming language used in the STACK, thus they were required to learn Maxima to complete the project. The intention was to make students familiar with programming language and its use in mathematics.

RESEARCH DESIGN AND METHODS

The case study research design (Yin, 2013) has been followed in this study. A group of three students has been observed over the semester. The methods used to generate data include group observations, semi-structured interviews, individual weekly journals and field notes. Using multiple methods for data collection contributed to triangulation of data.

In order to be able to observe participants' activity, we requested them to work at campus each week for which they agreed. During these sessions, they worked on their routine work including homework and assessments. Video recordings of their group work accompanied with the screen recordings to follow the activity on their computer screens have been collected. Screen recordings of their individual work external to these group sessions have also been collected. Furthermore, weekly journals containing self-reports about their use of resources were included to get the detailed overview. The journal was provided to participants in tabular format which they filled and submitted electronically each week. In the journal, they were asked to specify the resources they used and state how the use of a particular resource helped them in their work each week. The semi-structured interviews were held occasionally to understand the emerging patterns in their use of resources. During the group work sessions, participants communicated in their native language whereas the interviews were held in English. Both the group sessions and the interviews were transcribed.

We analyse participants' weekly journals, a semi-structured interview in the middle of the semester, screen recordings, and the field notes for reporting on students' use of resources in their work. This interview is being counted on because the participants were inquired about the general manner in which they used the resources. The observations, screen recordings and the field notes are being counted on while identifying participants' techniques during their work.

ANALYSIS Participants' weekly self-reports about use of resources

Table 1 presents the overview of participants' use of several resources as they reported in their journals. The manner in which they used them in their work and their evaluations of resources have been extracted from their journal inscriptions.

Resource used

How they incorporated resources in their work

Comments about resources (if any)

Tutorial videos Watched to get information to complete homework

Easy to understand through videos

MatRIC videos

Skimmed through the video at amplified speed

Own note

Used the already solved similar problems in the notes, to recall the problems (methods for solution)

Textbook

Read through the book, found formulas to work on homework, got questions from book (during project)

Maxima

Programmed tasks in Maxima for the project, used while doing homework, solved tasks using Maxima Programming in Maxima is hard but when it is done, all the problems are easy to solve

WolframAlpha

Used as a shortcut to get answers, compared answers obtained from Maxima, got help with solving difficult tasks Easier to use than Maxima, Faster than using calculator, useful when the answer is in the form of expression instead of numbers

MyMathLab

Worked on homework, learnt specific topic, solved some questions with higher difficulty Powerful tool, easier to get help and information online

Internet Lecturer's notes

Tailored" for the tasks at hand, the most relevant piece of information

Youtube vidoes Watched Maxima tutorials

Mathway and other online calulators

Solved questions

Severely increase the probability to get the correct answer, and therefore the overall score.

STACKS

Made some questions in STACKS

The three participants, Tor, Per and Jan, used MyMathLab almost every week because homework and assessments were required to be done in this system. As regards the textbook, Tor did not report the textbook in the journals rather he used the lecturer's notes. While in Per and Jan's weekly reports, they pointed out few ways in which they used the textbook on different occasions. The textbook served as a source of getting questions, checking answers to those questions, getting help with formulas, and going through examples in the book. During their project work, they consulted the book to take questions and subsequently checked the answers for those questions.

The tutorial videos were reported to be used by Jan and Per during their work. Jan occasionally watched the videos and when specifying about the kind of help, he used the word understand linked with this resource such as "to try to understand how to calculate…" and "to understand the calculation behind the math". Per has also mentioned the use of videos and commented, "I easily understand it when someone explains me the way of solving a problem". Tor did not mention any tutorial video provided by the lecturer, however he watched few videos on other platforms, MatRIC TV (an online resource containing videos aiming to support students in their transition from high school to university) and YouTube, once for getting introduction to partial integration and at another occasion to learn Maximathe programming language.

It can be seen that participants used some other resources in their work such as online calculators, WolframAlpha, Maxima and internet (cf. Table 1). Tor named several online calculators including Mathway (https://www.mathway.com) and WolframAlpha (https://www.wolframalpha.com) to solve the tasks and to compare the answers they got in Maxima while working on the project. He mentioned that he used online calculators for saving time, however, he wrote, "I did not learn anything doing this, but it severely increases the probability to get the correct answer, and therefore the overall score". Wolfram Alpha has also been used by Per and Jan in order to verify whether the answers they got were correct. While working on the project, they picked some questions from the book and programmed in Maxima. To check the answers to those questions, they used WolframAlpha.

After completing the group project that involved learning Maxima, this programming language became an important resource for them to solve tasks in homework and assessments. Both Per and Jan began making programs for solving each task to liberate themselves from calculations. Per inscribed in a weekly journal, "(I) used Maxima to make a program to solve the problems in an easy way. This is hard to make, but when it is done, all the problems are easy to solve". Tor did not seem to use Maxima a lot, he spent some time on learning how to use Maxima for solving tasks in one week, and then spending some more time in the next week, he rather chose to focus on MyMathLab. He inscribed that, "it's (MyMathLab) a more powerful tool and it's easier to attain help and information online".

In response to a question about using videos in a semi-structured interview, Per explained his way of working on homework using the provided resources. Tor's response was somewhat similar as he replied:

Int: Did you use any video while working on last week's homework?

Tor:

No, I think MyMathLab seemed sufficient so far.

Int:

Ok. So which resource did you use for getting introduction to the new topic?

Tor: I tried first MyMathLab but it went fine so I just carried on. …I check the notes and watch the videos if I get stuck..

Int:

So, you turn to the videos when you get stuck.

Tor: When it is a new topic, then I just skim through his notes, but since we have integration from a couple of weeks now, I am pretty confident and go straight with it.

While Jan responded to the same question as follows.

Jan: I did not watch that many videos. I mostly use MyMathLab and just see the examples…and if I can't get it from there then I go to…to the book because it is faster… and eventually go to the videos if I do not get constructive help from there.

The participants preferred MyMathLab during their work for being the source of quick and most relevant help in comparison to the other available resources. This approach of working on the tasks saved them time and effort to search for the required piece of information from other resources such as the videos and the textbook. However, the use of MyMathLab can be considered more pragmatic as both Per and Jan mentioned that the kind of help they get from MyMathLab is in the form of examples which contributes more towards producing the results.

Another approach was to watch the videos when the help from MyMathLab was not sufficient as evident through participants' responses in the interview. The use of videos has not been preferred much but participants reported that they consulted the videos when they needed to understand something. As discussed earlier, the help and feedback in MyMathLab concern the task only as it offers the formula and solutionsteps for the task. They might have needed to consult the videos to learn the concepts involved in those tasks in case when just knowing the solution steps in a question did not work. In the journal data, Jan and Per wrote that they used the videos to understand thus it indicates the epistemic value linked to usage of videos.

Observing participants' activity helped in finding that the use of different resources affected their manner of working on tasks i.e. techniques. We seek to categorise the participants' techniques pertaining to different resources they used, and by considering their motives behind use of each resource helped in recognising the pragmatic and epistemic value of their techniques. It is found that they increasingly used the digital tools to solve the tasks in MyMathLab environment with the progression in the course. This led to the use of more instrumented techniques instead of paper and pencil techniques promoted in the lecturer's videos and through MyMathLab. For instance, Tor mentioned in his weekly journals and it is observed in the screen recordings of his individual work that he used several calculators to work on homework as well as assessments. The participants themselves perceived this technique of using online calculators to solve the task as pragmatic.

Two of the participants used Maxima in their work as evident from journals and could be seen through the screen recordings of their work. They wanted to be pragmatic in order to make their future work easier. Making programs for each task for the first time can not be considered as merely pragmatic as Per mentioned that he found it hard. The difficulty in making programs may be linked to their knowledge of programming in order to code mathematical tasks. However, the extent to which it contributes epistemically in learning mathematics is not covered in present paper.

DISCUSSION AND CONCLUSION

In this study, we observed how a small group of three students interacted with the resources when provided with an online learning environment in their Calculus course. The environment allowed self-regulated learning and students could work remotely on their homework and assessments. To make sense of the opportunities for students' learning with resources in this environment, we explored their manner of incorporating the resources in general organisation of their digital work. Furthermore, we discussed the epistemic and pragmatic potential of participants' techniques.

In terms of resource usage and the corresponding techniques, participants opted for the resources and the techniques which were pragmatic in terms of producing results for the assigned tasks. Pragmatic techniques involved the use of online calculators, using help in the MyMathLab to produce the results for tasks. Watching videos for learning mathematical concepts seemed to be time consuming and hence not preferred much. Participants appropriated the programming language to work on the tasks with the motive to be more pragmatic and produce results easily in their work.

An important factor which is likely to cause the preference for more pragmatic instrumented techniques was the online final examination where they could use the resources. As for students, it is quite important to prepare according to the examination to be able to score better.

This case study provides an example of a self-regulated learning environment created for students to work independently. Our findings suggest some general prospects which are worth paying attention when assigning online homework to students.

Combination of an online homework online examination is likely to cause students to use unexpected use of resources and techniques, for instance, online calculators and solution tools in the present case. This observation also relates to the nature of tasks posed in an online homework environment. Variety in the nature of tasks, such as open-ended tasks, may lead students to interact with resources epistemically.

Figure 1 .

 1 Figure 1. Interface of MyMathLab environment.

Table 1 : Overview of participants' use of resources.

 1

  These topics I think are quite hard to learn all by yourself. When I get a new topic, I first try to solve it myself, if I can't do that I try to look at the examples in MyMathLab… and if I don't completely understand the examples I take a look at Olav's (lecturer) video…mainly the examples' videos because then I get to see the practical kind of way to do..to solve questions. will try to do it myself because then I think I… remember and learn it the best because then I have to think and ….and if I can't do it that way…then I will try to look at example just to get a few hints. If that does not work then I watch the videos because I can't look at the notes (provided by the lecturer)…I have to get explanation of what he is doing step by step.

	Int:	How would you rank the provided resources? Which one do you first consult
		with?
	Per:	First, I

Per: