Adrien Barrot

Barrot Christine

Doucet Bertrand

Karam Wissam

Rodriguez Emmanuel
email: emmanuel.rodriguez@insa-toulouse.fr

IS THERE AN IDEAL ASSESSMENT METHOD TO EVALUATE ACADEMIC KNOWLEDGE, TEAM WORK AND INTERPERSONAL SKILLS IN A PBL PROCESS?

Keywords: Assessment, student frustration, team work, interpersonal skills, PBL

Issues concerning learning assessments are presented, extracted from a PBL experience. Aiming at improving the current PBL second session, proposals of solutions to the risk of too many assessments, the students' disappointment facing the trade-off between daily work and formal evaluation, and interpersonal skills and team work assessments, are explained and proposed for discussion.

INTRODUCTION

A Problem Based Learning (PBL) Mechanical Design course for students in second year of an engineering school (around 20-years old), has been run twice between September 2005 and June 2007. Considering all the areas that reflect the quality of a learning experience, the first year (2005)(2006) brought many satisfactions (student motivation, respect of objectives, etc.) and some disappointments (excessive time spent for homework, assessment frustration, etc.) [START_REF] Barrot | L'APP pour initier à la Conception Mécanique[END_REF]. Among these areas, there are learning assessments which are crucial for the learning success, and also a delicate part to manage for PBL designers and tutors. We encountered indeed some unsatisfying situations of assessment, which may be summarised by four questions:

i-How can a succession of assessments which blurs their individual meaning for the students be avoided? ii-How can the students' frustration linked to the gap between time spent during the project phase and the formal evaluation be limited? iii-How can the interpersonal skills be evaluated? iv-Should team work be assessed? Based on this first experiment and publications concerning this topic (Dutch et al, 2001), we have looked for improvements to answer these issues, in order to apply them during the second year (2006)(2007). Our solutions are presented and submitted for discussion.

II ISSUES ABOUT THE SUCCESSION OF ASSESSMENTS

Through publications and discussions with other teachers, it is not rare to see that evaluation is a difficult point to implement. Pesse wanted to put the stress on the evaluations by multiplying them [START_REF] Pesse | La mise en place de la pédagogie par projet à l'Ecole Nationale Supérieure des Arts et Industries Textiles[END_REF]. Thus knowledge is regularly evaluated four times during the course. However, the significant number of evaluations causes a decrease in the marks standard deviation and finally a good project is not clearly differentiated from a bad one. This problem is recurrent even if the evaluation is not marked. This problem is encountered for know-how as well as for interpersonal skills or teamwork. Let us show a concrete example. During our first experience in PBL, we wanted to control everything to avoid bad surprises. Thus, in order to check the teamwork progress, many exhaustive grids were elaborated. They include five main points:

General feelings, Communication between team-mates, Team organization, Knowledge sharing, Personal work and commitment in the group. In practice, students had to complete a grid every two courses. To avoid boredom, the grids were different each time, and their complexity was increased. But the effect was the opposite. The complexity of these grids, the great number of questions and the non-repeatability of the questions produced two consequences. On one hand, the students did not look at them and, on the other hand, the tutor did not use them to see whether the group worked in a suitable way. Consequently, the intention of guiding the students and diagnosing various problems managed to hide the true goal of these self-assessment grids. At the end of each restructuration lecture based on student questions, we distributed only one exercise on the same topic but related to another study case. The small number of exercises (only one per topic) generated student fear attributable to the false idea of insufficient knowledge acquisition. This year, to improve the system of self-assessment, we increased the number of exercises per topic and we used only one self-assessment grid which summarises the points presented previously. Always using the same grid facilitates filling it out for the students and the tutor's diagnosis. In addition, using the same grid allows a comparison during the course. As the course has not yet finished, we cannot say if the feelings of the students will be better. A question remains: "Is it possible to find appropriate evaluations? (a good compromise between type and numbers)"

III

STUDENTS' FRUSTRATION

In the first year, one of the main problems we were confronted with was the students' frustration caused by the final examination method. Knowing that sustained frustration impedes students' learning and efficiency in exams [START_REF] Chaffar | The Emotional Conditions of Learning[END_REF], we will present our formal assessment technique and the consequent student reaction. Then we will explain how we intend to minimize complaints and frustrations during the second year.

III.1 Past examination method and its criticism

During the first year of our course, students' grades were composed equally by the project mark and by a formal examination mark. One of the main consequences we were confronted with was the frustration caused by the above ratio. Students did not tolerate that project work -on which they spent more than 60 hours (around 40 hours in the classroom) -could have an equal share in the final grade with a threehour examination. Moreover, this feeling was strengthened by the type of final examination (classical exam) that they considered not related to the active learning method we tried to use with them during a full semester. The direct effect was that some students who got used to working with diverse references (especially books and internet) without time constraints were destabilized. This led to some emotional breakdowns during the exam: some walkouts one hour after the beginning and a case of crying that required a close attention of the tutors present.

III.2 Present intention of modification

In order to minimise the students' frustration in the second year, we have tried to work conjointly on the emotional and the examination sides. First, thanks to the improvement of our tutoring skills and by increasing the time spent with students, we give the students more rooms to voice their opinions. Frustration is then minimised because they feel understood by the tutors. The latter should also inform the students about the examination method fairly early during the semester so that they take it into consideration during the learning and studying process (selfdeveloping formula sheets, doing exercises). Then, during last year's final examination, many students complained about the amount of time needed to fully read and understand the exam's technical file. In fact, the exam subject was composed of two distinct parts: the technical file and the question sheet. Consequently this year, we intend to hand out the technical file up to two weeks before the examination date so that they find enough time to understand the drawings, the functioning of the mechanism and the French technical words (for foreign students). This should eliminate a part of their fear during the exam session.

By the above, we are trying to minimize frustration caused by an examination technique which does not fit in with the learning methods. But a huge question is still unsolved: "How can we, when a group work is done, evaluate the personal skills and motivation without passing by a formal examination?"

IV INTERPERSONAL SKILLS

Problem Based Learning often takes into consideration the students' interpersonal skills. In fact, in a labour market where these competences are the most demanded, the PBL method is an effective means to teach students the basics of interpersonal skills. However, there is not a unique and complete solution for their assessment, since this knowledge is particularly subjective. According to Dorval, [START_REF] Dorval | La perception des professeurs à l'évaluation du savoir-être dans un baccalauréat en APP[END_REF], a biology teacher, interpersonal skills are often ignored in different assessments, though students must acquire some notions as: Identifying, analysing information to solve a problem, Communicating an idea, explaining a concept to someone, Working in group. He considers that interpersonal skills should be individually evaluated and should represent twenty per cent of the final mark. This percentage is shared out in ten per cent given by the tutor, five per cent by the laboratory team, and five per cent by the peers. To help the assessment, five criteria are given to tutors and peers:

1. The student's contribution to the team efficacy. Does the student take part in an active way in the "life of the group"? 2. The student's autonomy. How does the student manage his working time?

Is he creative? 3. The reasoning and scientific responsibility. Does the student take into account the good hypothesis? Is he giving evidence of scientific rigor? 4. Personal development and social implication. Is the student curious? Is he able to work in team? 5. Attendance. Is the student present at the meetings? Is he late? This article is based on a study done by twelve teachers practising PBL. It highlights the difficulty to obtain homogeneous interpersonal skills marks between teachers. Moreover, on one hand, according to the students, the part of this knowledge in the final mark is underestimated and on the other hand, some teachers think it is too high, because they consider interpersonal skills useless. During our first experience, we put the stress on scientific knowledge. That is why interpersonal skills were not identified as a marked item. We considered that PBL could bring more in the interpersonal skills field but that it was not a good thing to mark them. In practice, each group was given a global mark covering different aspects of subject knowledge. Nevertheless, tutors could attribute an individual mark in order to take into account motivation or absenteeism. Of course, as Dorval showed, it is only one aspect of the interpersonal skills. During our second year of experience, we have decided to clarify the interpersonal skills mark. Considering that only what has been learned or acquired should be assessed, the evaluation of these skills is formative. Finally, we just insist on building awareness of interpersonal skills; the students have three more years to improve them before graduating. Thus, a specific informative document about interpersonal skills is distributed to the students. It is presented as a grid of twenty adjectives which enables students to self assess their behaviour (rigorous, curious, autonomous, creative, etc.). This grid is completed individually at the beginning and at the end of the course. Finally, in our future work, we wish to improve the interpersonal skills assessment taking it into account officially by the means of evaluation grids. For this, a precise understanding by the teacher is necessary which is not yet the case. A question is raised: "Is PBL appropriate to teach interpersonal skills?"

V TEAMWORK ASSESSMENT

Teamwork is one of the main features of our course. Firstly, as always in PBL, working in a group can help students in many ways (for example by lessening the sense of isolation [START_REF] Seymour | Talking about leaving : Factors contributing to high attrition rates among science, mathematics, and engineering undergraduate majors[END_REF] and improving academic achievements, persistence [START_REF] Johnson | Cooperative learning : Increasing college faculty instrucional productivity[END_REF][START_REF] Johnson | Maximising instruction throught cooperative learning[END_REF]Springer, Stane & Donovan, 1999), autonomy and social maturity [START_REF] Michaelson | Building learning teams: The key to harnessing the power of small groups in higher education[END_REF]). It can also be a cause of problems [START_REF] Felder | Navigating the bumpy road to student -centered instruction[END_REF] and failure [START_REF] Feichtner | Why some groups fails : A survey of students' experiences with learning groups[END_REF] if not carefully monitored. Secondly, mechanical design methods are almost all based on a group of designers. The group has different aims depending on the method: value analysis [START_REF] Chevallier | Produits et Analyse de la Valeur[END_REF] and functional analysis [START_REF] Tassinari | Pratique de l'Analyse Fonctionnelle[END_REF] use it as a way to include different points of view to find all the functions of a product, TRIZ [START_REF] Cavallucci | TRIZ : l'approche altshullerienne de la créativité[END_REF] and many other creativity methods [START_REF] Gogu | Méthodologie d'innovation : la résolution des problèmes créatifs[END_REF] use it as a way to improve lateral thinking (de Bono, 1967), concurrent engineering [START_REF] Prudhomme | Le processus de conception de systèmes mécaniques et son enseignement -La transposition didactique comme outil d'analyse épistémologique[END_REF] uses it to fasten the design phase of a product by doing all its sub-phases in parallel. Therefore teamwork is not only a way to improve the learning of students but also a way to acquire skills that are used daily by mechanical designers.

V.1 Aims of our PBL course concerning teamwork

Before choosing a specific kind of assessment, the aims of the course, have to be precisely defined (Duch & Groh, 2001). Indeed, the more precise our aims are, the more accurate our assessment will be. This is especially true concerning the acquisition of skills that cannot be easily classified as right or wrong. In our case, students are almost complete beginners concerning team work skills and methods. They have already worked on a problem based learning course during which they used group roles (Allen, Duch & Groh, 2001): leader, reporter and timekeeper. The fact is that, when asked, the students globally do not think that this team working method is efficient and really used by engineers. The aims of our PBL course are therefore modest. First, students should experience and analyse the benefit of working in groups when tasks can be done in parallel. This is the most easily demonstrated contribution of team work and therefore a way to stimulate them to get the group organised. Secondly, they should experience the benefit of a team facing a task where everybody can contribute something to the group. This is harder to achieve. Our students have different background knowledge, some of them having already taken mechanical design classes. Finally, the students should experience and discuss whether the use of group roles is an efficient team management tool.

V.2 First assessments

Team working skills are hard to assess due to their non-Boolean nature. Facing an issue, the use of team working methods cannot be easily demonstrated as being right or wrong. Such a situation has to be experienced and analysed so that the students can be convinced by the efficiency of these methods. Therefore, we think that team working is better assessed by a formative peer, team or self-assessment. Assessment is thus a way to analyse the situations students have experienced during the course. To reach our first aim, we planned to make the students work on a time consuming pre-project that cannot be done without distributing the different tasks composing the pre-project. Last year, we noticed that students understood the benefit of the group facing a problem too long to be done in time by a single person. Concerning our second and third aims, we used four different self-assessment grids in order to progressively increase their perception of issues and benefits of team working, team working skills and methods (Bourret & Moore, 2005;Marre, Hernandez, Hernandez, & Ferrer, 2005). As explained in §II these grids were underused. Therefore if the groups were efficient because of tutor interventions, students have not clearly analysed the benefits of bringing different ways of thinking together and the need for a group to use organisation tools.

V.3 Assessment modifications

Our first aim being achieved by our first self-assessment method, we decided to keep it. The assessment of the second and third aim are now simplified in a single grid as explained in §II to help the students to keep track of the evolution of their team working skills. Analysis of the situation is improved by tutor questioning on the answers given by the students on the grid. To help the students understand the different issues of team work and the benefit of using group roles, we chose to let each student become a tutor for 15 minutes. They first observe a working team with a tutor group analysis tool [START_REF] Milgrom | Le guide du tuteur en candidature FSA[END_REF]. Then, they analyse what the good behaviours and the problems of the observed group are, and intervene as a tutor by asking questions to the group to help them improve their team behaviour. This peer assessment shows very good results on the observing students: nearly all of them declare to understand that using group roles is important and that they will be more vigilant on their own and their group behaviours. Nevertheless, formative assessments have a lesser impact than normative ones. In order to lay emphasis on team work, should we grade it? And what means are the best to assess such skills?

VI CONCLUSION

Our first experience in a PBL led us to face four main issues concerning assessment: the possible inefficiency of a succession of assessments, the students' disappointment facing the trade-off between daily work and final evaluation, the assessments of interpersonal skills and team work. For each issue, ideas for solutions have been proposed, such as a unique self-assessment grid, a modification of the final exam, or the occasional students' activity as tutors. But other questions remain in these fields. Our search for improvement in these issues is continuing during this second year, and our ideas are currently being tested in a real and demanding way: by the students' direct reactions.


A A Lebedev

A M Strel'chuk

S Ortolland

C Raynaud

M.-L Locatelli

D Planson

J.-P Chante

The negative temperature coefficient of the breakdown voltage of SiC p-n structures and deep centers in SiC

In this paper negative temperature coefficient of the 6H-SiC diode breakdown voltage is considered. It is shown that the temperature dependence of the breakdown voltage value can be explained in terms of recharging of deep centers in the space charge region of the diode. Experiments made with boron doped 6H-SiC diodes are in good agreement with calculation.

Introduction

In Ref. 1 it was found that in SiC p-n structures in which the electric field is applied parallel to the crystallographic axis c the breakdown voltage shows a negative temperature coefficient (BVTC). In a number of studies [2,3] this fact was related to the crystal structure of hexagonal SiC polytypes and to the occurrence of a natural superlattice. In some studies [4], alongside with the negative BVTC, a positive BVTC was observed and it was suggested that the negative value of BVTC might be related to recharging of deep centers, as in the case of silicon-based p-n structures [5].

Samples

The present study was concerned with SiC p-n structures prepared by sublimation epitaxy, with boron diffusion carried out prior to forming mesas. The electrical field in diode the p-n junction of the diode was parallel to the c-axis of the crystal. The structures had a negative BVTC of about 2x10 -3 K -1 . The absolute value of BVTC depended on temperature and dropped by about an order of magnitude when the junctions were heated to 600K.

Theory and experiment

To find an explanation for the observed negative values of BVTC we considered, following the authors of refs. 4 and 5, the effect of recharging of deep centers. It is known that boron in SiC introduces two types of deep acceptor levels, the deeper level (D-center) having the following parameters:

E " + 0.58 eV, σ -= 1 -3 × 10 345 cm 8 , σ 9 = 1 -3 × 10 38: cm 8 The model we suggested is based on the recharging of deep acceptor levels in a lightly doped p-type intermediate layer near the metallurgical boundary of the p-n junction. We made two assumptions:

(i)
The region of avalanche multiplication is situated in the low doped p-type region near the metallurgical boundary of the p-n junction. (ii)

The concentration of deep acceptor centers is comparable to that of shallow acceptors.

Such a region can be formed through overcompensation of the n-type region during the diffusion of acceptor impurities (boron, for example). The presence of a region of this kind in our samples has been shown experimentally [START_REF] Lebedev | Strel'[END_REF]. The breakdown voltage of a p-n junction is given by the well-known equation:

U ?@ = ε B E C@ 8 2qN G (1)
where ε B is the statistic permittivity, Ecr is the critical electric field, q is the electron charge, Ni is the impurity concentration in the base region.

Taking into account deep acceptor levels in the low doped p-base, we can write :

U ?@ = ε B E C@ 8 2q(N J + KM) = U ?@M N1 + KM N J O P ( 2
)
where NS is the concentration of shallow acceptor impurities, K is the electron occupancy of the deep acceptor level, and K = (M-m)/M, where M is the total concentration of deep levels; m is the concentration of deep acceptors occupied by the holes, Ubro is the breakdown voltage of p-n junction without deep acceptors.

When U<<Ubr, there is practically no current through the p-n junction and all deep acceptors in the p-type layer are occupied by electrons (K=1). When U~U ?@ , avalanche multiplication occurs in the space charge region and holes are captured by deep acceptors. Because acceptors occupied by holes are neutral, K decreases and, consequently, the electrical field becomes weak. This leads to an increase in the observed value of the Ubr. It should be noted that the level occupancy depends on temperature and decreases with heating. For this reason, Ubr will decrease with temperature, so we can see negative BVTC.

Based on the Shockley-Read statistics we can write the following equation for the discharging rate of deep acceptors in our case :

dm dt = (M -m)Δpα --mα 9 Δn -α -β G m (3)
Here, Δp = Δn is the carriers concentration; α 9 (p) = σ 9 (p)V Y , where σ 9 (p) is the electron (hole) capture cross section of the deep acceptor, Vt is the thermal velocity; and 𝛽 [ = 𝑁 ] 𝑒 3 _à b , where NV is the density of state in the valence band, Ei is the ionisation energy of the deep acceptor, T is the absolute temperature; k is the Boltzmann constant.

For the case of constant current (dm/dt = 0), and taking into account that α -≫ α 9 , we obtain form (3) the following equation for K :

K = β G (Δp + β G ) 34 (4)
Equations ( 4) and (2) give for Ubr : Fig. 1 shows experimental and calculated values of F = Ubr/Ubro-1. The following parameters were used in calculation: Ubro = 800 V, M/NS = 0.65 and NV = 5.10 15 x T 3/2 cm -3 . As one can see in the figure the best agreement between experimental and theoretical results is obtained for Δp of about 10 11 -10 12 cm -3 . However the magnitude of Δp calculated from the value of the breakdown current as Δp =J/VSq (where J is the current density in the p-n junction and VS is the drift velocity) is 10 10 -10 11 cm -3 . This discrepancy can be explained, from our point of view, if we take into account the fact that the breakdown in SiC usually takes place in small areas (microplasma breakdown) whose size is several orders of magnitude less than the total area of the p-n junctions. But when we calculate the current density we take the total area of the p-n junction. In other words, actual current density (and the Δp value) may be higher by one order of magnitude in the breakdown areas which coincides with the value needed by the theory.

U ?@ = U ?@M 1 + Mβ G N J (Δp + β G ) (5)

Summary

The temperature dependence of BVTC calculated in terms of our model and known D-denter parameters shows a good agreement with experiments. Because the boron diffusion is widely used for protecting the periphery of SiC mesas from surface breakdown [1,[START_REF] Lebedev | Strel'[END_REF] and because boron (D-center) is a typical background impurity in SiC fabricated by various techniques [7], the problem of the BVTC sign can be, in our opinion, ultimately solved only by taking into account (or eliminating) the effect of D-centers in studies of breakdown voltage in SiC p-n structures.

Figure 1 :

 1 Figure 1: temperature dependence of calculated (1-3) and experimental (4-5) values of F. Calculations were made for the following 𝛥𝑝 values, cm -3 : 1-10 10 ; 2-10 11 ; 3-10 12 . The experimental values of Ubr correspond to breakdown currents: 4-100 mA; 5-500 mA.

Acknowledgements.

This work was supported in part by US Department of Defence and Researcher Center of Schneider Electric S. A. (France)


Nataly Essonnier
email: nataly.essonnier@gmail.com

Chronis Kynigos
email: kynigos@ppp.uoa.gr

Jana Trgalova
email: jana.trgalova@univ-lyon1.fr

Social creativity in the collaborative design of a digital resource embedding mathematics into a story

Keywords: Process of collaborative design, social creativity, digital resource

Our research was

Introduction

Our research is motivated by promoting creativity in teachers and students, collaborative work and the use of digital artefacts to solve real world or interdisciplinary problems, and considering mathematics teachers as designers [START_REF] Kynigos | Teachers as designers of digital educational resources for creative mathematical thinking[END_REF]. It was carried out in the framework of the MC2 European project (http://www.mc2-project.eu/, 2013-2016) aiming at developing a socio-technical environment facilitating and enhancing the design of digital resources, called cbooks (c for creative), and fostering students' creative mathematical thinking. The technical part of the environment consisted of the so called C-book technology embedding an authoring tool and a collaborative tool called CoICode1 . Four communities of designers established in four partner countries, France, Greece, Spain and United Kingdom, constituted the social part of the environment. The French community, on which the paper focuses, brought together educational designers from diverse mathematical fields with manifold expertise and knowledge (researchers, teachers, software designers, etc.) because the process of designing purposeful mathematical activities in multi-representational technical environment is complex. An aspect of this complexity comes from the interrelated processes of learning to purposefully use a new technology, of designing tasks for students to initiate purposeful mathematical activity, of collating the various artefacts for the activity, of supporting students to learn to use technology, and of articulating the teacher's role in supporting the students to navigate their respective routes through the various artefacts to include interaction with the technology (Clark-Wilson & Timotheus, 2013, p. 48). In this paper, we aim at highlighting factors that foster or hinder social creativity (SC) in the collaborative design of c-books. Studying these complex design processes required several theoretical perspectives leading us to network theories and concepts. We start by explaining our theoretical framework and the conceptualisation of social creativity. We pursue by refining our research questions and by exposing our methodology before presenting an analysis of one case study. We conclude with some salient results.

Theoretical framework

Collaborative design of creative digital educational resources within a socio-technical environment is a complex human activity intertwining manifold interactions and elements which are difficult to capture without any relevant lenses.

As a first lens, within the activity theory (AT), [START_REF] Engeström | Learning by expanding: An activity-theoretical approach to developmental research[END_REF]'s model of activity structure (Figure 1) helps us focus on particular elements, called entities, that intervene in the design activity: subject that is the group of designers, object that is the c-book being designed, community with which the designers' group interacts, mostly the designers of the C-book technology, rules that govern the design process, division of labour that occurs among the designers, tools and signs, such as artefacts used by the designers, mostly the C-book technology including the CoICode collaborative workspace, and outcome that is the designed resource, as well as learning that can occur during the design process. [START_REF] Engeström | Learning by expanding: An activity-theoretical approach to developmental research[END_REF]'s model highlights also interactions between these entities, which allows to describe and understand the context of the design process. [START_REF] Engeström | Expansive learning at work: Toward an activity theoretical reconceptualization[END_REF] expands his model and sheds light and awareness on the interactions occurring between different individual activity systems, i. e., when the subject is a single designer. It is illustrated by the minimal structure of two interacting individual activity systems (ibid.) which allows to bring to the fore the modifications of the object. The object moves from an initial state of un-reflected, situationally given "raw material" (object 1) to a collectively meaningful object constructed by the activity system (object 2), and to a potentially shared or jointly constructed object (object 3). However, the models do not allow us to understand the nature of the entities. In order to capture the nature of the subject and community entities, we draw on the concepts of community of practice (CoP) [START_REF] Wenger | Communities of practice: Learning, meaning, identity[END_REF] and community of interest (CoI) [START_REF] Fischer | Communities of interest: Learning through the interaction of multiple Knowledge Systems[END_REF]. Besides, the designers belong to various "worlds" according to their domains of expertise. These worlds have boundaries that "can be seen as a socio-cultural difference leading to discontinuity in action or interaction" (Akkerman & Bakker, 2011, p. 133). It is therefore important, while studying collaborative resource design, to pay particular attention to what happens on those boundaries. The boundary crossing approach (BC) (ibid.) enables to enlighten the weight of boundaries in the creative design process and the arising mechanisms of identification, coordination, reflection and transformation through the concepts of boundary object [START_REF] Star | Institutional ecology, 'translations' and boundary objects: amateurs and professionals in Berkeley's Museum of Vertebrate Zoology, 1907-1939[END_REF] and broker, bound to interactions between subjectcommunity -(artefact or object or outcome), or between two or more individual activity systems. Finally, in order to highlight the role of the teachers (subjects) involved in the design, their personal resources (artefacts) brought into play in the design, and the trajectory of the digital resource being designed, i.e. its different versions from mother resources to daughter resource [START_REF] Hammoud | Le travail collectif des enseignants en chimie comme levier pour la mise en oeuvre de démarches d'investigation et le développement de connaissances professionnelles[END_REF] (from object to outcome), we draw on the documentational approach to didactics (DAD) [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF]. To sum up, the AT was our main framework allowing to connect all others through the entities of the Engeström's activity system model. Based on this, we have been able to conceptualise our understanding of social creativity.

As a matter of fact, different types of creativity are reported in literature. We have chosen the littlec creativity [START_REF] Craft | Creativity across the primary curriculum: Framing and developing practice[END_REF], which is when a person realizes a new or improved way to approach an issue or accomplish a task. We draw also on the componential tradition of creativity assessment [START_REF] Hennessey | Consensual assessment[END_REF] to assess it and on the concept of communities of interest (CoI) for characterising our communities of designers. Thus, within the MC2 project, we define social creativity (SC) as the generation of ideas which: (a) stem from a combination of two or more individual activity systems, (b) result from various interactions among the CoI members, the C-book technology and tools, (c) are externalized in and through specific digital artefacts (including the cbooks), and (d) are considered by the CoI members to be novel, appropriate and usable [START_REF] Daskolia | D2.2: Operational definitions and criteria for measuring social creativity in the design of digital educational resources for CMT. Deliverable of the MC Squared project[END_REF]. Moreover, creativity, social or individual, can be modelled through phases of divergent (novel ideas) and convergent thinking (appropriate and usable ideas) [START_REF] Csikszentmihalyi | Creativity: Flow and the psychology of discovery and invention[END_REF] constituting a creativity cycle. Social creativity is characterised by the reification of creative ideas collectively elaborated, i.e. elaborated by at least two CoI members.

Research questions

In

Methodology

Our methodology relies on one case study, i.e., we analyse the collaborative design of one particular c-book within the French CoI. The data was mostly collected via CoICode that keeps traces of interactions in the form of ideas posted by each designer (Figure 3), but also through minutes of manifold meetings, exchanged emails and the designed c-books. Although the methodology comprises quantitative and qualitative methods, in this paper we only focus on the qualitative analyses aiming at getting a deeper insight into social creativity processes. The data analysis was processed in two phases. First, we use the AT to describe in detail the design activity context through the activity system entities in order to situate and understand the collaborative design context of the group of designers. Second, we use our network of theories to qualitatively analyse data and bring to the fore factors impacting the social creativity taking place among the designers.

A snippet of the collaborative design of the c-book 'Ski'

In this section, we analyse a few episodes of the collaborative design process of a c-book on ski touring. From the AT perspective, the object of the design process was to create meaningful mathematical activities linked to the risks of provoking an avalanche while practicing ski touring to foster students' mathematical learning and creativity (purposeful activities). The group of designers (subject, Figure 2) was composed of seven members of the French CoI (community), all closely related to mathematics education with knowledge on mathematics, technology and pedagogy, and sharing a constructivist background. Four members acted as the main designers: Nina (secondary mathematics and physics teacher and PhD student, with a good knowledge of the C-book technology (artefacts)) who moderated the group, Marie (researcher and mathematics teacher educator, expert in mathematical modelling), Fred (mathematics teacher educator, interested in real problems), and Jane (mathematics teacher educator, interested in digital technology in mathematics education) who played the role of a reviewer. The other three members, Marc (expert of digital technology such as dynamic geometry systems), Jean and Adam intervened on request or during CoI meetings (division of labour). The main designers' group was constituted in a way that at least one of them was at ease with digital technologies and especially with the C-book technology (rules). The designed c-book (outcome) should be used by a teacher of an associated CoP, called MPS, which Marie belongs to (rules). This CoP reflects on the use of mathematics for solving interdisciplinary problems with Grade 10 students.

In order to bring to the fore the trajectory of the c-book design process through its versions, we combined documentational approach to didactics, activity theory, social creativity and boundary crossing approach. We could identify four versions of the c-book. The initial version was constituted by Marie's personal resources on the theme of avalanche and skiing (mother resources = objet 1), as she has already worked on this topic within the MPS CoP and experimented some activities with Grade 10 students. From her numerous resources, Marie selected several that she instrumentalised through CoICode posts (object 2) (reflection mechanism = perspective making).

Then, a divergence phase was initiated by a brainstorming amongst the CoI members. It generated nine new directions on real situations and mathematical notions (reflection mechanism = perspective making) and some creative ideas, i.e. judged novel, appropriate and usable by the designers. The creative ideas were related to real situations such as the avalanche risk depending on a slope leading to considering angles, the measurement of the angle of a snowy slope with ski poles that requires knowledge of trigonometry, the shape of snow crystals leading to working on geometric figures, transformations and sequences. This phase was followed by a phase of convergence (identification, coordination and transformation mechanisms) characterised by the collective elaboration of some of these ideas with, for instance, some artefacts already known within the French CoI, such as GeoGebra or Cinderella, in order to simulate an avalanche to encourage students to make conjectures about the risks, or programming an avalanche victim search device in order to make students understand the notion of algorithm (objects 3). The two phases constituted a creativity cycle. They led to the second version of the c-book pictured through a defined structure and content based on the preparation of a ski tour as a guideline for the story that would guarantee a unity for the c-book. In the following creativity cycle, the creative ideas were mainly bound to the technology and designers' technical knowledge, such as suggesting guidelines how to use GeoGebra for creating snowflakes, devising relevant feedback for students in different ways to enrich the milieu of the c-book activities by using widgets that were not familiar within the French CoI, or considering the copy/paste possibilities inside the Chat tool to enable interactions between students or between the students and the teacher (social aspects). During the third creativity cycle, the main concern of the designers was the usability of the c-book by the teachers and students from Grade 10 to 12. The designers were also concerned by giving a unity between all the activities thanks to a story on ski touring and avalanche risk. Hence, the focus of the designers changed from one cycle to the other. In addition, we noticed that a divergent phase was often initiated by a review of the c-book version at stake by the reviewer or by a critical feedback from some CoI members, less involved in the design, during a meeting (reflection mechanism). Each version was the result of a creativity cycle, i.e., a divergent phase followed by a convergent one as modelled in Figure 3. Furthermore, we observed that the first creativity cycle generated ten creative ideas out of 14 ideas judged as creatives by the designers, which was more than in the other cycles. The analysis of this cycle highlights a positive effect of the compulsory use of CoICode (rules) during brainstorming amongst the designers. This was not the case in the other cycles where the designers used also other media to communicate and to record their interactions, such as emails or minutes from the diverse meetings. Indeed, when the CoI members used CoICode, they had to read the ideas of the others and to react on them. Each post in CoICode obliged a designer to reflect on his/her own ideas. The latter had to choose if the idea was new or rather a reaction to another one, and what kind of reaction (opposition, contribution or alternative). Such a structured reflexion turned out to boost SC.

Finally, in order to have a deeper understanding of SC we studied the path of a creative idea, i.e. its collective elaboration. The creative idea we analysed (circled with red in Figure 4) was chosen because of its strong collective elaboration. This idea was preceded by Marie's suggestion to create an activity on algorithms bound to the search of a victim of an avalanche (mother resource). Then, Fred's creative idea proposed to work on algorithm with the "GeoGebra turtle". Fred was referring to explicit algorithm (reflection mechanism). This suggestion sparkled the negotiations between Marc and Fred who expressed different perspectives of a widget around the idea of a "turtle" to model an avalanche victim research device. Marc understood and proposed a widget designed by the Greek CoI using logo language. But Fred did not know this widget. He knew another widget using Java Logo language, which could enable to create explicit algorithm but he was hesitating. Thus, they seemed to share a common frame of reference around turtle and logo, which enabled them to understand each other. The word "turtle" became a boundary object between them. Nevertheless, from the meetings' minutes, Nina the moderator found that Fred was in trouble because he could not manage to finalize his widget with GeoGebra (creative idea) that should allow students to work implicitly on algorithms. Hence, Nina asked Marc to help Fred. She identified difficulties (identification mechanism), and to overcome them, she solicited another member of the CoI (coordination mechanism). Marc, expert of GeoGebra, designed the widget, limited the time and added a score to foster affective aspects (transformation mechanism = hybridization, daughter resource). Moreover, Marc was eager to create an explicit algorithm with a widget designed within the Greek CoI, but he did not master the logo language required for working with the latter. He therefore asked the technological expert from the Greek CoI how to write the algorithm and he eventually created a new widget (transformation mechanism = hybridization, daughter resource).

He thus played a role of broker looking for knowledge in another CoI (community).

Conclusion

The analysis of the process of design of the c-book "ski" shows that some rules impacted positively social creativity among the c-book designers. One of such rules applies to the constitution of the designers' community that should be as diversified as possible. The diversity should be understood in terms of complementary expertise of its members (pedagogical, mathematical and especially technical) broadening the discussion meaning, but also in terms of roles to be assigned (moderator, designer and reviewer, division of labour). Indeed, the reviewer fostered a divergent phase and the moderator facilitated a convergent phase, supporting creativity cycle. In addition, the use of CoICode to communicate during brainstorming seemed also to enhance SC.

In addition, we noticed that some words (e.g., turtle) allowed the designers to propose various ideas in different perspectives from other contexts, from other cultures. Such words became boundary objects widening the CoI perspective. Therefore, within a CoI that designs c-books, we can say that potential boundary objects are malleable, transformable objects, compatible with the object of the design and belonging to the designers' common frame of reference. Boundary objects helped to extend the common frame of reference between the designers (coordination) by enlarging their perspectives.

Figure 1 .

 1 Figure1. The structure of an activity system(Engeström, 1987, p. 78)

Figure 2 :

 2 Figure 2: The designers of the c-book 'ski touring'

Figure 3 :

 3 Figure 3: Model of versioning of the c-book collaborative design process[START_REF] Essonnier | Etude de la conception collaborative de ressources numériques mathématiques au sein d'une communauté d'intérêt[END_REF]

Figure 4 :

 4 Figure 4: Extract of the CoICode workspace dedicated to the c-book 'Ski'

  this paper, we address the following research questions: in the collaborative digital resource design activity,  what interactions between rules, community and division of labour promote or hinder SC?  what types of objects have the potential to become boundary objects, under which conditions and how can they stimulate SC?  which members have the potential to become brokers, under what conditions and what is their role in stimulating SC?

CoICode is a communication environment part of the C-book technology, offering a workspace within which members of a community engaged in a c-book design can collaborate.

We found that the brokers, who were members of the CoI, were able to identify the missing knowledge for the c-book design, to look for it and then, to share this new knowledge within the French CoI. Thus, in our context, we can say that a broker is a member of the CoI who identifies missing knowledge within the CoI, then can find it in another related professional world in order to bring it to the CoI. Thus, the broker acts as a coordination mechanism allowing the CoI members to extend their common frame of reference with new knowledge.


Rune Herheim

Suela Kacerja

Building bridges between school mathematics and workplace mathematics

Keywords: Workplace mathematics, school mathematics, building bridges, transfer, relevance

In this paper, we investigate three examples from two articles written by lower-secondary students in which they identify school mathematics used in workplaces. Their articles are written as part of a project aiming to help students find mathematics relevant. The students provide documentation on advanced use of graphs, mean values, calculations, and formulas in two occupations, music management and physiotherapy. By using Evans ' (1999) approach to transfer, we identify similarities and differences between school mathematics and workplace mathematics. In line with Evans' approach, we aim at building bridges between school mathematics and workplace mathematics.

Introduction

When doing research and following up student teachers in school practice, we hear students say "maths has nothing to do with my life", "I don't need maths", and "maths is something we do in school". These statements are in line with Masingila's (2002, p. 37) study in which a majority of the students "think mathematics is synonymous with school mathematics". Wager (2012, p. 10) built on this by saying that students often do not see "the connection between the mathematics in which they engage in school and the mathematics they experience out-of-school". The national curriculum in Norway and other countries, e.g. the US common core state standards, specify that students should be able to apply the mathematics learnt in school to the workplace. However, as English (2010, p. 36) argued, "we need further knowledge on why students have difficulties in applying the mathematical concepts and abilities … outside of school".

When doing mathematics in school, students often ask "why do I have to learn this?" (Hernandez-Martinez & Vos, 2018, p. 245) or "when will I use this?" (Willoughby, 2017, p. 877). Answers like "you are going to have mathematics next year as well, you know" and "because we have a test coming up" (cf. [START_REF] Onion | What use is maths to me? A report on the outcomes from student focus groups[END_REF], do not address the core of the questions. On the contrary, they might explain the student statements in the first paragraph. Alrø, Skovsmose, and Valero (2009, p. 15) asked eighth grade students what they used mathematics for outside school, and the most frequent answer was "to do homework". If the only purpose of learning mathematics is to do homework or to be prepared for future teaching and tests, then it is easy to understand from such circular reasoning why students question the relevance of mathematics. Hernandez-Martinez and Vos (2018) argued that students need answers regarding how, why, where, and what mathematics will be relevant because it is so closely related to their motivation to learn and to what extent they see the usefulness of what they learn. Hernandez-Martinez and Vos see usefulness as a property of the mathematics being learnt, and relevance as the potential connection between the mathematics learnt, its usefulness, and the student.

It might seem odd that mathematics, a field of knowledge so vital for today's society, has difficulties being regarded as relevant. However, being able to give qualified answers regarding the relevance of mathematics involves detailed knowledge of the mathematics required to perform an occupation. One cannot expect teachers to investigate a range of different workplaces in order to always have an answer when students question the relevance of a mathematical topic. Gaining insight into what extent the mathematics learnt in school is relevant to life outside school is a challenge for teachers and educators, like FitzSimons and Boistrup (2017) and [START_REF] Nicol | Where's the Math? Prospective teachers visit the workplace[END_REF] found in their studies. This relevance challenge and the importance of connecting ideas and building on students' out-of-school experiences and understanding when working with mathematics in school, constitute the rationale behind the Workplace mathematics project referred to in this paper. About 60 lower-secondary students, three of their teachers, and a researcher (the first author of this paper) investigated what mathematics their parents/guardians needed in order to perform their work. The students then wrote articles about their findings with the help from their teachers and in collaboration with the researcher.

By investigating two of the student articles, this paper addresses the following research question: What school mathematics are the students able to identify in the workplaces? The analysis focuses on identifying and discussing similarities and differences between mathematics at school and at workplaces and provide a foundation for discussing potential transfer and relevance of school mathematics. We do this by presenting and extending the mathematics the students identified in two workplaces, music management and physiotherapy.

Theoretical perspectives and previous research

In the following, we clarify how the national curriculum links school mathematics with the working life and give a brief overview of perspectives and terms for out-of-school mathematics. We elaborate on the concept of workplace mathematics and on the gaps and potential transfer between school mathematics and workplace mathematics.

Links between school mathematics and working life in the national curriculum

In the Norwegian mathematics curriculum, mathematical competence is considered important for understanding and influencing what happens in society and for "pursuing further education and for participation in working life and recreational activities" (Ministry of Education and Research, 2010, p. 2). Problem solving and modelling are emphasised, and rich experiences with mathematics are regarded as an important foundation for lifelong learning. In addition, the ability to work with mathematical concepts, methods, and strategies to solve problems also "involves learning to pinpoint and describe situations where mathematics is involved" (p. 5). This is defined as an important part of mathematical competence. Thus, being able to identify the use of mathematics is considered as vital for strong mathematical competence in the national curriculum. The workplaces investigated in the project serve as rich contexts for identifying and making sense of the mathematics used.

Perspectives and terms for out-of-school mathematics

There are multiple perspectives and terms in mathematics education research regarding the contextualization of school mathematics and links between school mathematics and out-of-school experiences. Realistic mathematics is used to focus on creating meaningful and realistic situations in which students can learn mathematics [START_REF] Freudenthal | Revisiting mathematics education[END_REF]. Genuine or authentic mathematics concerns real-life situations brought into the mathematics classroom, strongly, but not necessarily, linked to mathematical modelling (e.g. [START_REF] Palm | Impact of authenticity on sense making in word problem solving[END_REF]. Everyday mathematics focuses on identifying everyday situations where mathematics is used (e.g. [START_REF] Nunes | Street mathematics and school mathematics[END_REF][START_REF] Masingila | Examining students' perceptions of their everyday mathematics practice[END_REF]. Workplace mathematics is used by researchers who study the mathematical knowledge and skills required to do certain work processes (e.g. [START_REF] Hoyles | Mathematical skills in the workplace[END_REF][START_REF] Wake | Using college mathematics in understanding workplace practice[END_REF][START_REF] Wedege | Menneskers matematikholdige kompetencer[END_REF]. The perspectives presented have a key aspect in commonthey study the potential links between school mathematics and out-of-school mathematics. The project presented in this paper takes the workplace mathematics perspective.

Workplace mathematics and transfer

In UK, two groups of researchers have investigated mathematics in workplaces from different points of view. [START_REF] Hoyles | Mathematical skills in the workplace[END_REF] studied seven different workplaces focusing on the mathematical knowledge and skills required of the employees and identified several mathematical skills as important in different occupations. [START_REF] Wake | Using college mathematics in understanding workplace practice[END_REF] identified general mathematical competences applied in many occupations that also could be incorporated in the mathematics curriculum. Summarized, the competences and skills identified by the two groups of researchers were using proportions and percentages, handling data, using graphs, diagrams, and other tools and interpret them, modelling and solving problems, calculating and using formulas, recognizing anomalous phenomena, and doing extrapolations.

On the other hand, there are also hindrances for identifying the use of mathematics outside school. Workplace mathematics is situated and has to be combined with knowledge about the work to perform required tasks [START_REF] Hoyles | Mathematical skills in the workplace[END_REF]. It can be hidden caused by automation of processes and division of labour, and the nature of school mathematics can cause challenges as well (Williams & Wake, 2007). There are several differences between school and workplace mathematics that can hinder students recognizing mathematics in workplaces, [START_REF] Wedege | Menneskers matematikholdige kompetencer[END_REF] argued. She identified seven categories that describe a gap several adults had experienced between workplace mathematics and school mathematics. The differences start at the number level. In the workplace, numbers have units of measurement, calculations have to be constructed, and there are often multiple solutions. In school, numbers often occur without units, the numbers as well as calculations are given, and there is only one correct solution. Solving tasks in the workplace requires collaboration and dealing with disturbing factors, it offers opportunities to use mathematics, and solutions have consequences. Solving tasks in school is often individual work without disturbing factors, and real-life contexts are used merely to create a need for using mathematics and solutions have therefore no practical consequences.

However, the hindrances are surmountable. Wake (2013, p. 315) proposed a new mathematics curriculum to overcome the challenge of transfer "by placing the study of how mathematics models reality at the core of the curriculum". This curriculum should give students the possibility to make sense of, and develop further, the models of others. Wake considered the research done in workplace mathematics as rich initial contexts to be used by teachers who support inquiry in the classroom. Evans (1999, p. 23), when writing about transfer between mathematics in school and everyday life, linked relevance with transfer by arguing that "if schooling is to be relevant … we need some account of how learning from the school can be … 'transferred', to other contexts". He emphasised the importance of building bridges for transfer between school and out-of-school activities by identifying fruitful overlapping and inter-relations, and by analysing similarities and differences.

Method

The background for the students articles: the Workplace mathematics project

The project was a collaboration among the teachers, the students, the students' parents/guardians, and the researcher. The teachers contacted the researcher because they wanted to do a project that could help their students see the relevance of mathematics. The teachers' idea of visiting parents/guardians' workplaces to document the use of mathematics was embraced by the students, and more than twenty parents/guardians attended an information/kick-off meeting. The parents/guardians were excited by the idea, they expressed a positive attitude towards mathematics, and agreed to be visited and interviewed. The meeting tuned them into the project and was important in order to go beyond Hoyles, Noss, Kent, and Bakker's (2013) finding that visible workplace mathematics tends to be fragmented, routine-based, and mostly simple calculations.

The students were trained to interview, document, and do pre-investigations by the researcher and the teachers. They were encouraged to gather information about the occupation, the particular workplace, and the interviewee's duties, and identify concrete examples of mathematics being used. The students were grouped in pairs and triads, they decided their approach and interview questions, and then they interviewed, observed, and collected documents and photos. One of the teachers or the researcher joined most of the groups. Many of the parents/guardians had prepared information and examples when the students came. In 2017, after working with data handling, collecting supplementary data, and editing articles according to feedback from the teachers, the researcher, and the parents/guardians, the first student articles were published in a national journal for mathematics teaching. The researcher finished the articles, extended some of the examples, and added reflections on links between the students' examples and the competency aims in the national curriculum.

Analytical approach

Evans' (1999) approach to transfer, in terms of similarities and differences, is used as analytical lens to investigate what school mathematics the students were able to identify in the work done by the physiotherapist and the music management duo. The identification of similarities are discussed according to the skills identified by [START_REF] Hoyles | Mathematical skills in the workplace[END_REF] and [START_REF] Wake | Using college mathematics in understanding workplace practice[END_REF], and the identification of differences are discussed according to [START_REF] Wedege | Menneskers matematikholdige kompetencer[END_REF]. These similarities and differences are used to shed light on a potential transfer of mathematics between school and workplaces. The examples in this paper are chosen because they are interesting and contain mathematical details that make it possible to identify links to school mathematics.

Description and analysis of workplace examples from students' articles

We now present three examples and discuss the mathematics involved. The first two examples come from a music management duo [START_REF] Tronstad | Matematikk i musikkbransjen[END_REF]. They work with several artists, and one of their main tasks is to help artists make a living of music. An essential question concerns deciding where artists should sell their music: streaming, CD, or both? Figure 1, made by the researcher based on information collected by the students, shows the sales in Norway for CDs and streaming. In 2011, the graphs intersected, and streamed music generated for the first time more money than CDs. The CD sales continued to drop, but in 2014, there seems to be a trend change.

To make a qualified decision on this requires critical reflections based on mathematical understanding by the music managers. In line with [START_REF] Hoyles | Mathematical skills in the workplace[END_REF] and Wake and Williams' ( 2001) findings, they have to handle data, understand and interpret a graph, and understand the point of intersection of the two curves in terms of income. They have to predict a trend and make a decision based on past datatypical tasks within predictive modelling. Based on these reflections, they can make knowledgebased decisions of where to sell their artist's music. However, in line with [START_REF] Wedege | Menneskers matematikholdige kompetencer[END_REF] findings on differences between mathematics in school and in workplaces, this workplace example includes disturbing factors and the use of measurement units. Furthermore, it actually requires use of mathematics, and it is a decision that has potentially huge economic consequences.

Another aspect is the income generated from having a song played on the radio. In one of the radio channels in Norway, artists earn 60 NOK/minute. If a song lasts three minutes and 23 seconds, you get the following calculation: 3 × 60 + (23/60) × 60 = 203 NOK. If the song is played 200 times/ month, it provides an income of 200 × 203 = 40600 NOK. This resemble common calculations in school. However, there are also several differences. The managers have to find many of the numbers themselves, set up an expression including fractions, calculate it correctly in order to know the potential profit, manage different measurement units such as seconds and minutes in the same formula, and deal with practical unit rates such as NOK/minute and times/month. Again, there is an extensive use of measurement units. To summarize, in these two examples, the music managers have to apply all the competences and skills on Hoyles et al. and Wake and Williams' list. In addition, they need to gain an overview of these elements as parts of a complex system where several factors can influence their decisions such as where to sell the music, marketing, and arranging tours.

The third example comes from a physiotherapist's work with elderly and their walking ability [START_REF] Berrefjord | Fysioterapi og matematikk[END_REF]. When a patient walks across a mat with sensors connected to a computer, a range of functional walking data is registered. The students identified a need for the physiotherapist to integrate mathematics and ICT skills as [START_REF] Hoyles | Mathematical skills in the workplace[END_REF] Collecting the ten step times in seconds, five for each foot, makes it possible to calculate the mean values and standard deviations (the physiotherapist and the researcher calculated the standard deviation because this is not part of the students' curriculum):

This example shows the need for dealing with decimal numbers, for setting up formulas and calculate correctly, for systematizing and presenting data, and for using statistical concepts such as mean and standard deviation. Standard deviation is frequently used by researchers to quantify the amount of variation of a set of data. Here, it gives the physiotherapist vital information about the extent to which a person's walk is unstable and influences how the physiotherapist should proceed with the physical treatment. Such use of standard deviation exemplifies that mathematics has practical and concrete consequences. The importance of correctly knowing and using mathematics becomes more visible for the students, and can help them find answers to their relevance questions.

Discussion and concluding comments

Building bridges between school mathematics and workplace mathematics, helping students see the relevance of mathematics as something more than just being able to do homework and prepare for tests, is a challenge documented by several researchers (e.g. [START_REF] Alrø | Matematik er noget man bruger til at lave lektier med. MONA -Matematikk-og naturfagsdidaktik[END_REF][START_REF] Onion | What use is maths to me? A report on the outcomes from student focus groups[END_REF].

Students struggle to see connections between school mathematics and out of school mathematic [START_REF] Hoyles | Mathematical skills in the workplace[END_REF][START_REF] Nicol | Where's the Math? Prospective teachers visit the workplace[END_REF][START_REF] Wager | Incorporating out-of-school mathematics: from cultural context to embedded practice[END_REF]Williams & Wake, 2007), and they find it difficult to apply mathematics outside school (e.g. [START_REF] English | Promoting student understanding through complex learning[END_REF]. Teachers have difficulties answering questions from students on why they "have to learn this" and when they will ever need it [START_REF] Hernandez-Martinez | Why do I have to learn this?" A case study on students' experiences of the relevance of mathematical modelling activities[END_REF]. A possible remedy is to include genuine real life tasks, e.g. tasks situated in workplace contexts, as resources more often in the mathematics curriculum. This is one way of overcoming the differences between workplace mathematics and school mathematics identified by [START_REF] Wedege | Menneskers matematikholdige kompetencer[END_REF].

In contrast to the above-mentioned challenges, the Workplace Mathematics project generated multiple examples of how school mathematics is necessary in several workplaces, e.g. the use of tables, graphs, units, mean values, calculations, and formulas. This is emphasised by the national curriculum as important for developing mathematical competence. Connections between school mathematics and its usefulness in real life were documented, including information about what kind of mathematics was used, where and how it was used, in addition to why it was needed in the workplaces.

By searching for similarities, the students also identified differences between school and workplace mathematics. The workplace tasks were more complex, had numerous units of measurement, there were several potential useful methods and not one definitive answer, and finally yet importantly, doing mathematics wrong could have severe consequences. Although Wedege argued that such differences can create a gap between school and workplace mathematics, they can also help students develop a more flexible, nuanced view on mathematics. Identifying similarities and differences makes it possible to build bridges between school mathematics and workplace mathematics, as [START_REF] Evans | Building bridges: Reflections on the problem of transfer of learning in mathematics[END_REF] suggested, and through this explicate the relevance of school mathematics.

The parents/guardians took part in a kick-off meeting, they were enthusiastic, and several of them were prepared when the students came. This was important for the documentation of numerous examples on how school mathematics is used in different workplaces. The students experienced that school mathematics was used and had practical consequences. Such outcomes can be used to positively influence students in viewing mathematics as relevant, and this constitutes a valuable contribution to their learning and motivation for school mathematics [START_REF] Hernandez-Martinez | Why do I have to learn this?" A case study on students' experiences of the relevance of mathematical modelling activities[END_REF].

Including links between school and workplaces in the mathematics curriculum can provide fruitful, joint reference points for students and teachers when doing tasks and make working with mathematics more meaningful. Furthermore, the examples can be used to build bridges between school and workplace mathematics by discussing e.g. the importance of choosing appropriate representations for mathematical concepts in a specific situation and moving flexibly between different representations. We therefore argue that engaging students to identify workplace mathematics is worthwhile. However, further research is needed to find ways to help students "explore how mathematics can be transformed to meet the needs of a range of diverse situations" (Wake, 2013, p. 315).

Figure 1 :

 1 Figure 1: The sales in Norway for CDs (graph I) and streaming (graph II) in million NOK

Table 1 : Step times

 1

	did. One


Rafalska Maryna
email: rafalska.maryna@gmail.com

Towards improving teaching and learning of algorithmics by means of resources design: a case of primary school education in France

Keywords: Algorithms, resources, primary education, teacher

This article is devoted to the problem of design of resources in algorithmics for teachers of primary school in France. We investigate what elements should contain such resource in order to support the learning intended by authors of the resource, during its implementation by teachers who works in different contexts. To contribute to this question, we study the design principles of a particular resource in algorithmics. We follow this resource from the moment of its creation by a researcher and an experienced teacher to its appropriation and usage by three ordinary teachers. Paper describes how the results of analysis of lessons observations, interviews and collective discussions with teachers pointed to missing elements of the resource and, hence, contributed in its enhancement. The results of the study aim to bring some elements for improving the teaching of algorithmics by means of resources and, as a result, to support the learning of the involved concepts.

Introduction

Algorithmics [START_REF] Lagrange | Algorithmics[END_REF] is more and more present in school education of different countries. In France, the elements of algorithmics and programming were introduced in primary and middle school curricula in 2016. In our previous work, we analysed the didactical transposition [START_REF] Chevallard | La transposition didactique -Du savoir savant au savoir enseigné[END_REF] of the main concepts of algorithmics in French curriculum of cycle 4 (grades 7-9) and showed its partial nature with a focus on the effectiveness aspect of algorithm and on the usage of algorithm as a tool [START_REF] Modeste | Algorithmics in secondary school: a comparative study between Ukraine and France[END_REF]. The following analysis of didactical transposition of concepts of algorithmics in French curriculum resources for cycle 3 (grades 4-6) showed the existence of the same orientations.

In French curriculum of cycle 3, content related to algorithmics and programming is included in the chapters "Mathematics" and "Sciences and technologies". For example, in the theme "Space and geometry", it is suggested to make an introduction to programming by using activities of spotting and displacement (coding the movement of a robot or a character on the screen) as well as geometrical activities (construction of simple figures and figures composed of simple figures). In the theme "Materials and technical objects", it is mentioned that the notion "algorithm" could be discovered by pupils using "visual applications". Curriculum guidelines for teachers (published by Ministry of education of France) highlight the importance of development of pupils' knowledge of "the basic principles of algorithmics and software design" and indicate that the usage of unplugged activities could be the first step for introducing pupils to the notion of algorithm. However, the concept of algorithm is not defined and it seems that the main learning objectives of such activities is the development of rigor. Most of given examples present algorithm through programming tasks, using robots or different environments (like Scratch, Scratch JR, Géotortue, etc.). Analysed textbooks for cycle 3 (in particular, collections Nathan and iParcours for grade 5, Delta for grade 6) follow the recommendations of the curriculum and teacher's guides, and propose mostly programming tasks (usually about drawing figures on the screen) and games in Scratch without distinguishing the notions of algorithm and program. Such focus on programming leaves out other important aspects of algorithm, for example, the role of algorithms for problem solving.

Interviews with teachers of primary school revealed that many of them have never been trained either in algorithmics or in programming, and don't understand completely the goals and relation of the new topic to other parts of the curriculum. To prepare their lessons, they tend to use curriculum texts, teachers' guides, textbooks and existing online resources (that also propose mostly tasks of programming in Scratch). In spite of changes in French curriculum made in 2016, there are still many teachers of primary school who have not yet made lessons in this topic. This leads to the situations when in the same class of grade 6 (the first year of middle school) there are pupils with different levels of knowledge in algorithmics and programming.

We make the hypothesis that one of the ways to overcome the institutional constrains, to improve the teaching and, in consequence, learning of algorithmics, is the development of resources for teachers that takes into account different aspects of the notion of algorithm. However, the existed studies, like [START_REF] Aldon | Des intentions de l'auteur aux usages en classe, première réflexion sur la cohérence des usages d'une ressource[END_REF], showed that a teacher in interactions with a resource can interpret it in different ways. Sometimes, the didactical situation designed by a teacher on the base of a resource has different intentions comparing to the intentions of the authors of the resource. Thus, the elaboration of resources in algorithmics requires the design of tasks as well as the development of resources design principles that could "assure" their distribution and following usage with the relevance to initial learning goals. The research question that we investigate in this article is the following:

What elements should contain a resource in algorithmics in order to support the learning intended by the authors of the resource, during its implementation by teachers who works in different contexts?

To contribute to this question, we investigate the design principles of a particular resource in algorithmics. In the next sections, we present the theoretical framework, the methodology choices and the project framework in which our study is anchored. Then, we propose some elements of data analysis and the corresponding results. In the last section, we answer the research question and draw our conclusions.

Theoretical framework

This study refers to the documentational approach to didactics [START_REF] Trouche | Documentational approach to didactics[END_REF], which proposes a holistic approach to teachers' work, taking into account the new universe of resources to teacher use, design and re-design. The central role in this approach is given to the notion of resource that is used in the sense of a tool that "re-source" the teacher's work [START_REF] Adler | Conceptualizing resources as a theme for teacher education[END_REF]. In addition to the material resources (like textbooks, curriculum materials, etc.), this approach also takes into consideration such resources as discussions with colleagues and researchers, students' answers, etc. Teacher integrate resources in their system of resources and design on their base his/her documents related to a certain class of situations. In [START_REF] Pepin | Refining teacher design capacity: Mathematics teachers' interactions with digital curriculum resources[END_REF], the notion of document is presented as a combination of the resources adapted and re-combined, the ways the teacher uses them (which include the stable organisations of associated activities and particular usages), and contain the 'knowledge' guiding the usages.

The documentational approach to didactics provides the tools for following the evolution of a resource during its adaptation and usage by a teacher. As in this study we are also interested in the results of resource implementation in class, we refer to the theoretical framework of the structuring of the milieu as well [START_REF] Brousseau | Théorie des situations didactiques[END_REF]; [START_REF] Margolinas | La structuration du milieu et ses apports dans l'analyse a posteriori des situations[END_REF][START_REF] Margolinas | Situations, milieux, connaissances : analyse de l'activité du professeur[END_REF]; [START_REF] Bloch | Un modèle d'analyse des raisonnements dans les situations didactiques. Etude des niveaux de preuve dans une situation d'enseignement de la notion de limite[END_REF]). It allows to analyze both a teacher's and pupils' positions in a didactical situation. Thus, taking the pupils' point of view leads to bottom-up or ascending analysis starting from the moment when a generic pupil is confronted with a material environment (milieu), i.e. without a priori didactical intentions, to the institutionalization of knowledge, including the phases of experience or action during which the knowledge at stake is encountered. Top-down or descending analysis corresponds to the teacher's point of view. It starts from the confrontation of a teacher with a construction milieu in an unfinalized situation, includes the phases of development of teacher's global project (concerning a theme of study that involves one or several lessons) and local project (concerning a particular lesson), and ends with the devolution and observation of pupils' activities in a-didactical situation.

In our paper, we also use the notion of didactical bifurcation proposed by [START_REF] Margolinas | Les bifurcations didactiques : Un phénomène révélé par l'analyse de la structuration du milieu[END_REF] for modelling the case when pupils confronted with a material milieu, invest themselves in a situation that is different from the one intended by a teacher. Margolinas distinguishes two types of marginal branches of a situation: a-didactical marginal branch (when pupils acquire new knowledge that is not in the teacher's project) and nildidactical marginal branch (when pupils don't acquire new knowledge). In analogy with the notion of didactical bifurcation, [START_REF] Aldon | Des intentions de l'auteur aux usages en classe, première réflexion sur la cohérence des usages d'une ressource[END_REF] proposed the notion of bifurcation of construction: when the construction situation (in which a teacher confronts with a resource and interprets its intentions for constructing his/her own project) carries the intentions distinct from those of the resource and, in consequence, it leads to a didactical situation different from the didactical situations potentially carried by the resource.

Methodology

Our study is anchored in the French PREMaTT project 1 . The objective of the project is to stimulate the collaborative design of resources for teaching, in a network of schools, supported by researchers and a monthly meeting in a "laboratory for innovative design" [START_REF] Trouche | Evidencing missing resources of the documentational approach to didactics. Towards ten programs of research and development for enriching this approach[END_REF]. In particular, in frame of the project the teachers and researchers worked on the design of tasks and resources for teaching of algorithmics aiming to make them useful for other teachers. Collaboration between researchers and practitioners is widely used in design-based research for bringing solutions to problems of practice (in particular, by producing new artefacts) and for contributing in the research by identification of design principles.

Our methodology is based on the confrontation of a priori analysis (built around the intentions of authors of the resource's, analysis of the resource and situations carried by it) with a posteriori 1 PREMaTT: thinking the resources of mathematics teachers in a time of transitions (http://ife.enslyon.fr/ife/recherche/groupes-de-travail/prematt) analysis (built on the observation and analysis of implemented situations by ordinary teachers) as well as on the analysis of revealed dysfunctions, if any.

For our study, we constituted two teams. The first one involved a researcher and an experienced teacher who had been working part-time at a research laboratory and part-time in a primary school in the city centre for many years. The team worked on the design of didactical situations aimed to familiarize pupils with different properties of the notion «algorithm» and to show the role of algorithms in problem solving. The effectiveness of the developed tasks for supporting the intended learning was validated in result of confrontation of a priori analysis with a posteriori analysis of situations implemented in a class of grade 5 (10 years old pupils) by the experienced teacher.

The scenario of lessons developed by the team was chosen as the first version of the resource to transmit to the ordinary teachers. This choice was based on the hypothesis that this scenario, which takes into account both learning constraints and constraints of teacher practice, would constitute a sufficient support for teachers in order to design their own lessons project that will foster the intended learning. It included the description of the developed tasks (in particular, necessary material support, problems for pupils to solve, guidelines for teachers concerning implementation of the situations in class) and information about managing the lessons (timing, forms of class organization of pupils, etc.).

The second team involved three teachers (Natalie, Victor, Ida) who had been working together at a primary school in a socially disadvantaged part of the city for more than 5 years. The teachers obtained their initial education in different subjects2 : Natalie in biology, Victor in literature and Ida in informatics. All of them at the time of the study taught in class of grade 5. Except Ida, who worked at the beginning of her career as a database operator for 10 years, the teachers have never been trained either in algorithmics or in programming. Natalie made a few lessons in programming in Scratch, while the others have never taught this topic. The teachers' mission was to prepare lessons using the scenario and to make them in their classes as well as to give the reflexive feedbacks during interviews and collective meetings with the first team. Ida, Natalie and Victor were also asked to elaborate a scenario that could be a possible continuation of the proposed one. The members of the first team didn't intervene in the work of the teachers (except a short presentation of the elaborated tasks by the researcher).

Collected data and their exploitation

For our study we collected the following data: answers of teachers to the questionnaire about their education, professional path, teaching experience, system of resources in mathematics and participation in collective work; video of collective work of Ida, Natalie and Victor before the implementation of the scenario in their classes; videos of the lessons of three teachers; pupils' written production; interviews with Ida, Natalie, Victor after the lessons; scenario proposed by three teachers as a possible continuation of the first one; videos of two collective meetings of the members of both teams (the first one, concerning the results of the implementation of the scenario in three classes; the second onea collective discussion about a possible continuation of the first scenario).

We used the lessons videos to make the a posteriori ascending analysis of didactical situations implemented by each teacher and confronted it with the a priori analysis. We were particularly interested in the episodes that show existence of pupils' activities that were not expected. In such cases we questioned the reasons of appearance of didactical bifurcations and identified the type of marginal branches (regarding the didactical project of the authors of the scenario). We used the video of class observations to analyse teachers' actions in the class (particularly, the devolution of the situations made by the teachers) and their possible influence on the appearance of alternative pupils' projections of a-didactical situation based on the material milieu. In the video of collective discussions of the teachers before and after experimentations as well as in the interviews with them, we were looking for episodes that could bring to light the logic of their actions in class and their choices made during the construction of the situations on the base of the scenario. To understand teachers' considerations about the role and possible place of constructed didactical situations in their global project of teaching algorithmics and programming, we analysed the scenario developed by Ida, Natalie and Victor as a possible continuation of the first one as well as the video of its collective discussion. The revealed dysfunctions in construction of didactical situations (bifurcations of constructions) as well as in their implementation in class allowed to revise the design choices made a priori and to identify what elements were missing in the resource.

Analysis of lessons observations

Due to space restrictions, we present in this section the elements of the analysis of Ida's, Natalie's and Victor's lessons and show how it contributed to the identification of missing elements of the resource. We start from a short description of the first two situations proposed in the scenario. Then, we show how Natalie, Ida and Victor made the devolution of the second situation in their classes and the corresponding pupils' activities.

The proposed scenario included the sequence of situations with a-didactical dimension [START_REF] Bloch | Dimension adidactique et connaissance nécessaire : un exemple de 'retournement' d'une situation[END_REF] during the work on which pupils "re-invent" sorting algorithms. Implementation of the situations doesn't require a computer and propose the following material support: playing cards and grid with defined places for putting cards on it in an aligned list.

In the first situation of the sequence, pupils choose randomly 7 cards from 13 cards of one colour and put them face down on the grid. The goal is to sort the cards by returning only two cards at a time. The situation is used to introduce the material milieu (cards, grid and three allowed operations: "to take two cards", "to put them in the right order", "to place them on an empty place of the grid") and prepare pupils for the following tasks. The target knowledge includes the understanding that at a time one can put two cards in "local" but not necessarily in "global" (between all cards) order, as well as the comprehension of the possibility to sort the cards in a finite number of permutations using the defined operations. The results of experimentations in the class of the teacher from the first team showed that most of pupils' procedures in this situation are based on the usage of the memory.

In the second situation, one pupil should sort the cards of another one by giving him instructions with respect to the allowed operations. The pupil, who has cards, doesn't show their values to the pupil who gives instructions. All cards could be turned over only after the "pupil-speaker" says "stop". The objective of the task (that imitate, in some sense, the relation between a human and a machine) is to foster the development by pupils of sorting strategies that will be generalised in the following situations.

Natalie made devolution of the second situation, emphasizing on the phrases that pupils can use for giving instructions to their classmates: "take card number … and number … ", "put them in the right order", "place them on the grid", "stop". She highlighted that the use of other sentences is not allowed. The observation of pupils' activities in this class showed that all pupils invested themselves in the main branch (regarding the didactical project of the authors of the scenario) and were working on the development of sorting strategies.

Ida and Victor, while explaining the challenge of the second situation, mentioned only that pupils "will do the same thing like in the first task, but this time, there is one pupil who gives instructions and another one who executes them". They didn't discuss with pupils the phrases that they are allowed to use for giving instructions, emphasising only on the necessity that "the 'executor' have to do exactly what the speaker says" and "that it is very important to be precise in the language". Victor gave in his class the following criterion of success in the task: "if the cards are in the right order, it means that 'the speaker' was very good". Such devolution of the situation led to its different interpretations by pupils and evoked pupils' activities which were not intended by the authors of the scenario. For example, part of pupils in the class of Ida, for whom it was not evident which phrases to use, invested themselves in the nildidactic branch. They used the commands "exchange the cards" or "put back without exchanging", factually trying to guess if the cards, they claimed to take, are in the right order or not. In the class of Victor, we observed one pupil who was searching for the optimal formulation of the sentence for sorting the cards of his classmate. He proposed the following solution: "you take two cards that you want, sort them and put them on the grid, and you continue like this until all cards will not be in the right order". This answer was not accepted by Victor which produced a protest from the pupil who was sure of the correctness of his answer.

Thus, in the classes of Ida and Victor we identified the marginal branches that were not expected by a priori analysis. It happened due to the way chosen by the teachers for the devolution of the situation. The following analysis of collected data showed the differences between Ida's and Victor's interpretations of the objectives of the first two situations with those intended by the authors of the scenario. For example, Victor, expecting that pupils would find the sorting strategies during the work on the first situation, saw the goal of the second one in the development of pupils' competencies to verbalise the solving procedures. This explains his choice of devolution for the second situation, when he left open the possibility of different formulations of pupils' strategies.

The obtained results of the analysis contributed to improve the resource. More specifically, the identified bifurcations of construction showed the necessity to describe the expected pupils' procedures in the first situation and the target knowledge in the second one as well as to provide the help for the devolution of the second situation (to insist on the "authorised" sentences for giving instructions with a link to the property of feasibility of algorithm).

Discussion and conclusion

Our study puts the light on certain difficulties that teachers of primary school could have in interactions with resources in algorithmics (and, more generally, with resources that involves the concepts in which teachers don't have enough knowledge) and their influence on pupils' learning. In the case of Victor, flaws in a priori analysis of the proposed situations (in particular, wrong anticipation of possible pupils' procedure and identification of target knowledge) led to the construction of a didactical situation distinct from the one supposed by the authors of the resource. In consequence, the evoked learning activities of pupils didn't comply with intended ones.

It seems, that more pertinent choice of devolution of the second situation made by Natalie could be explained by the fact that she had taught a few lessons in Scratch before the experimentations took place and, as she said, "the usage of the given phrases for sorting cards is similar to the usage of blocs in Scratch for constructing a program". However, we also identified a few dysfunctions in her class that could influent on the learning outcomes. For example, Natalie didn't recognise in the pupils' answers the "germs" of the sorting algorithm with which she was not familiarized before. Hence, she didn't take them into account in the validation phase. Another example, which is common for the lessons of three teachers, is the lack of decontextualization in the phases of conclusion.

From the interview with Ida, Natalie and Victor as well as the analysis of the scenario proposed by them as a possible continuation of the first one, we draw the conclusion that they don't see all didactical potentiality of the resource proposed by the first team, focusing mostly on the effectiveness aspect of algorithm [START_REF] Modeste | Enseigner l'algorithme pour quoi ? Quelles nouvelles questions pour les mathématiques ? Quels apports pour l'apprentissage de la preuve ?[END_REF]. Moreover, analysis of the collective discussion about the possible prolongation of the first scenario showed that the situations, even didactically pertinent, could be discarded by the teachers if they don't see the connection between the target knowledge and curriculum requirements.

Revealed dysfunctions pointed to key elements to transmit to teachers by means of a resource in algorithmics in order to support the intended learning: a priori didactical analysis of situations proposed by a resource; target knowledge and ways of its gradual decontextualization in class; principal results of experimentations carried out in class (e.g. the examples of pupils' procedures, frequent errors, etc.); information regarding the basic theoretical and epistemological aspects of concepts involved in a resource; possible place of a resource in learning progression with link to curriculum requirements. Due to the fact that the transmission of the mentioned elements by means of a particular resource could make it too long and, in a result, not usable by teachers, the question of the structure of a resource in algorithmics need to be addressed.

Results presented in this paper, concerned only the first cycle of experimentation with the resource. The next step is to refine the design principles of a resource in algorithmics via the following cycles of experimentations with the revised resource as well as with other resources. Future work is also likely to investigate the impact of the resources designed on the base of the results of the study, on teaching and learning algorithmics at primary school.

In France, in order to enter a teacher training program, candidates must have at least a Bachelor degree in any speciality.