text:
Gilbert Greefrath
email: greefrath@uni-muenster.de

Hans-Stefan Siller
email: siller@dmuw.de

Matthias Ludwig
email: ludwig@math.uni-frankfurt.de

Modelling problems in German grammar school leaving examinations (Abitur) -Theory and practice

Keywords: Modelling, examinations, Abitur, authenticity

Examples of mathematical problems in German school leaving examinations are examined in detail as to the extent to which they verify the competence of mathematical modelling. For this purpose, the relevance, the authenticity of the context, the openness of the task and the partial competencies as criteria are examined in more detail. It is shown that authentic contexts but no relevant contexts are contained in the problems. As a rule, the use of mathematics is not authentic and problems are not open. The partial competencies in modelling are quite unequally divided and are not covered completely. It is recommended to use all the criteria for the design of future examination questions.

Introduction

In Germany, an overall policy of the Conference of Education Ministers has been in place with respect to "Educational Monitoring" since 2006. The intent is to enhance competence orientation within the educational system. In mathematics, the focus also lies on the general competence in the field of modelling. From the year 2017 onwards, a task pool of Abitur examinations will be available out of which all Federal States can take problems for Abitur examinations. This is an important step towards improving the quality of examination problems and to align levels of requirements between the Federal States step by step. Such examination problems have been developed on the basis of educational standards and also contain, in particular, elements examining mathematical modelling. Studies regarding problems in written Abitur examinations are rare. Still today, it is obvious that there are very few theoretically and methodically profound and empirically substantiated approaches in respect of examination problems in the framework of the written Abitur examinations, which at times just focus on individual standard-related aspects (Kühn, 2016, p. 75). Therefore, this applies in particular to problems regarding mathematical modelling. At the core of mathematical modelling lie the translation of a problem from reality into mathematics, the work with mathematical methods and the application of mathematical solutions to the real problem. Usually, the entire modelling process is represented as a cycle. Idealisation here means that the representation itself is, again, a model. In Borromeo [START_REF] Borromeo-Ferri | Der Modellierungskreislauf unter kognitionspsychologischer Perspektive[END_REF] the creation of a mathematical model is contemplated in detail, representing the process of the individual, creating the model, from the starting situation to the mental representation as a first step. Blum et al. (2007) describe modelling competence as the capacity to execute the relevant process steps required when switching between reality and mathematical problem adequately as well as to analyse models or comparatively evaluate such models. This can be discussed in more detail on the basis of modelling cycles by Blum and Leiss (2005, p. 19). Thus, it is possible to analyse the various partial processes of modelling activities considering various details and other aspects. It is also possible to consider the capacity to implement such a partial process as a partial competence of modelling [START_REF] Kaiser | Modelling and modelling competencies in school[END_REF][START_REF] Maaß | Mathematisches Modellieren im Unterricht: Ergebnisse einer empirischen Studie[END_REF]. In consideration of the modelling cycle one could characterise such partial competences as shown by Greefrath and Vorhölter (2016, p. 19). We do not mention inner-mathematical working between the mathematical model and the mathematical results as a partial modelling competence since this is not specific for modelling processes.

Modelling competence

Characteristics of modelling problems

It is not always easy to select or develop the right modelling task. As an indication, characteristics may be specified that a modelling task should fulfil. In her comprehensive classification scheme for modelling problems, [START_REF] Maaß | Classification scheme for modelling tasks[END_REF] focuses in particular on the type of relation to reality, to the openness and the modelling activity as criteria for modelling problems. With respect to the focus on the modelling activity, partial competences of modelling will be observed more closely. As regards the relation to reality, the relevance and authenticity of the context will be examined more closely (cf. [START_REF] Greefrath | Mathematisches Modellieren -Eine Einführung in theoretische und didaktische Hintergründe[END_REF].

Relevance

Initially, we deal with the classification of reality-related problems handed down in German discussions. In connection with the factual context of problems, we traditionally talk about "embedded problems", problems formulated in text form and "factual issues". These types of problems provide facts about the relevance of the factual context used for students.

Embedded problems have no real relation to reality. The factual context is of no importance regarding the solution of problems and can be exchanged arbitrarily. The intention of embedded problems is the application and practice of numeracy skills. An embedded task as a modelling task is suitable only to a limited extent since the mathematical model is already implicitly contained in the problem. Word problems are problems formulated in text formsometimes complemented by figures. Similar to the embedded problems, the issue is basically exchangeable as reality is often represented in a very simplified manner. The intention is to promote mathematical skills. It is not really about an autonomous development of a mathematical model because of the lack of reality and the given simplifications. Nevertheless, one major problem for students is the translation of the text into the corresponding mathematical symbolism such as terms or equations. For this reason, the known term "mathematisation" in the field of modelling is normally used in this context (Schütte, 1994, p. 79). In the case of problems formulated in text form the mathematical problem dominates the embedding of it. Another focus then lies -depending on the specific problem -on the interpretation of the mathematical results in the factual situation and in the wording of a corresponding answer. The substantial and -particularly in connection with factual problemsfrequently exclusive treatment of problems formulated in text form in mathematics classes has been strongly criticised. One reason is the lack of genuine reference to reality. Another reason is the process of practising similar mathematical issues, formulated in text form, as a result of which a real reflection about the context used becomes redundant. In the case of factual issues a factual environmental problem is of primary importance. In this context, the function of "Factual Calculation" is described by [START_REF] Winter | Sachrechnen in der Grundschule[END_REF] as "Environmental Development". In this context, real data are frequently given in respect to which authentic questions are then asked. Since the processed issue plays a real role, information about the respective matters must be gathered and processed.

Therefore, the processing of factual issues is also to be considered as interdisciplinary or -ideallyeven as discipline-linking. In this context, the factual issues can be viewed as modelling problems [START_REF] Franke | Didaktik des Sachrechnens in der Grundschule[END_REF].

Authenticity

Authenticity is one central characteristic of modelling problems. Authenticity means both the authenticity of the extra-mathematical context and of the application of mathematics in this particular situation. The extra-mathematical context must be real and not be specially designed for a certain arithmetical problem. The application of mathematics in this situation must also be reasonable and realistic and should not just be used in mathematics lessons. Authentic modelling problems are problems that genuinely belong to an existing subject or problem area where they are accepted by people working in those areas [START_REF] Niss | Applications and modelling in school mathematics -Directions for future development[END_REF]. Thus, an authentic task becomes credible and at the same time a realistic task for students from an environmental point of view. Authenticity helps students take problems seriously and avoid superficial substitute processing strategies as in the case of embedded problems [START_REF] Palm | Features and impact of the authenticity of applied mathematical school tasks[END_REF]. In the case of authentic problems, students can assume that the things they are dealing with really exist and that the task they are presented with is a real task that finds its justification outside mathematics lessons as well (the task have an out-ofschool origin and a certification [START_REF] Vos | Authenticity in extra-curricular mathematics activities: researching authenticity as a social construct[END_REF]). Authenticity of problems helps students take such problems seriously. However, authenticity of problems does not mean that the problems are actually important for the students' present or future lives.

Openness

Open problems are those problems that -for instance -allow for more than one approach or solution. Openness enables students to choose their own approaches or solutions regarding the problems. There are various classifications of open problems. We limit ourselves here to the examination of openness by initial state, transformation and target state (cf. e.g. Wiegand & Blum 1999). Said classifications use the known problem-solving psychological description of a problem through its initial state, target state and a transformation that transfers the initial state into the target state and are not limited to modelling problems [START_REF] Klix | Bildungsstandards im Fach Mathematik für die Allgemeine Hochschulreife[END_REF]. Open problems are divided by the clarity of their initial and target statuses as well as by transformation. [START_REF] Maaß | Classification scheme for modelling tasks[END_REF] suggests another classification of open modelling problems based on [START_REF] Bruder | Konstruierenauswählenbegleiten. Über den Umgang mit Aufgaben[END_REF] which includes seven different types and which distinguishes between overdetermined and underdetermined problems.

Criticism concerning Abitur examination problems with modelling elements

The use of modelling in examination problems, however, is not unreservedly viewed positively. The fact that in many cases the relevance of the factual context used is not at the focus of examination problems, gave rise to (fundamental) criticism on the part of some expert representatives with regard to modelling in examinations. On the one hand, the criticism is directed against the fact that "modelling competence" is not at all examined by the problems (Kühnel 2015, p. 76). On the other hand other authors show the categorical refusal of modelling problems (cf. Bandelt & Weidl, 2015, p. 4). Strong criticism is also directed against the greater part of texts in examination problems (cf. (Jahnke et al. 2014, p. 120): "Instead of dealing with mathematical problems, A-level students have to tackle wording problems".

Criticism regarding examination problems can be absolutely justified, in particular if it is about embedded problems and not about factual problems in examinations (cf. Fig. 1). Henn and Müller (2013, p. 205) comment: "Unfortunately, most of the so-called "modelling problems" at schooland in particular in the Abitur exam are not at all modellings according to our way of thinking. Almost always one starts out from a more or less complex function equation, allegedly describing a ski jump, a tower, a playground or another construction. Now, with this function, a common functional examination is to be made. However, the whole thing is not a modelling task, but entirely a mathematical problem." This brief insight shows that the use of the term "modelling problem" -in particular in examinations -does not at all guarantee the clear characterisation of a certain type of problem. It should be considered, however, that the conditions given in examination or in normal lesson situations are different from each other (cf. Siller et al., 2016, p. 381/382). Ultimately, in examination situations the focus is on the term of "measuring" the students' performance as outlined by Siller et al. (2016, p. 384). Thus, less attention may be paid to a creative phase in a test situation. In addition, psychometric findings (cf. [START_REF] Rost | Psychometrische Modelle zur Überprüfung von Bildungsstandards anhand von Kompetenzmodellen[END_REF] reveal that -for the measurement of performance -such performance must be addressed explicitly and structured into small units in order to obtain a valid statement.

Research issue

Against the background of criticism directed towards current Abitur examination tasks and in consideration of the importance of competence of mathematical modelling for German levels of education and of international discussions as well as of the theoretically clarified criteria for modelling problems, we have to ask ourselves the question, how to evaluate the quality of existing Abitur examination problems with modelling elements in Germany. We limit ourselves here to the calculus subject area, which accounts for the largest part of the Abitur examination in mathematics in Germany. With regard to the criteria discussed above, the question is: How can German Abitur examination problems in calculus be assessed with regard to the partial competences of modelling, to relevance, to authenticity and to openness?

Research method

For preparing for the pool of Abitur examination problems which will be available from the year 2017 and from which all Federal States can easily draw Abitur examination problems, the Institute for Educational Quality Improvement in Berlin will provide a collection of examples (cf. www.iqb.hu-berlin.de/bista/abi). Such samples were chosen as a basis for the examination because they best fit with the Abitur examination problems of 2017. Among the sample problems there is a total of four examination problems for calculus for general grammar schools. Two of the problems are of an increased level and a further two are intended to be used in Computer-Algebra-Systems (CAS). All in all, the four examination problems include 50 items. Problems were evaluated by items according to different criteria. The selection of criteria follows the frequently quoted central characteristics of modelling problems (cf. [START_REF] Bruder | Grundfragen mathematikmethodischer Theoriebildung unter besonderer Berücksichtigung des Arbeitens mit Aufgaben[END_REF], Greefrath & Vorhölter 2016, p. 17, Maaß p. 22). Problems were discussed and evaluated per item and on the basis of Tab. 1 in the framework of a qualitative research process including three evaluators. Finally, all items could clearly be allocated the corresponding characteristics (cf. Bortz & Döring 2006)by using 0 (does not matche.g. Relevance 0 in Fig. 2) or 1 (matches the criteriae.g. Authenticity in Fig. 2). In this way, a detailed compilation of the examined items was created and an allocation to the aforementioned criteria became possible.

Results

In an initial assessment of the existing data material, we focus on the criteria reference to reality, authenticity, openness, relevance and modelling competence. Due to the small data basis, it was not possible to make valid quantitative statements. However, certain trends are visible. 20 items out of a total of 50 contain a reference to real life situations. The problems, though, included in the basic level more items with reference to reality (52 %) vs. items without reference to reality than in the higher level group (31 %). Correspondingly, on average, problems with the use of CAS contain more items with reference to reality (46 %) than in the group without the use of CAS (33 %).

Authenticity: If one looks at the problems considering the requirement that the respective items should contain realistic situations, our analysis reveals that such items appear both at the increased level and at the level where CAS are used. More than one half of the items with reference to reality also contain realistic situation at both levels. Regarding a realistic use of mathematics which from our point of view represents a further aspect of authenticity (cf. table 1), this has been identified in just three items. Students were required to explain in the factual context of a parabolic modelled trajectory that only certain parameter values are possible. It should be noted here that quadratic functions are used even by experts for modelling trajectories.

Openness and Relevance: Open problems have not been identified in the items examined. Also with respect to the relevance of the problem, no item can be identified in the present examination problems that would fulfil this requirement.

Modelling competences:

The analysis concerning the partial modelling competences show that the items normally address one of the partial competences. However, the partial problems are not designed in such a way that the sequence of an idealised modelling cycle reflects the appearance of the partial competences of simplifying, mathematising, interpreting and validating one after the other. Most items focus on mathematising, followed by required interpretations. Simplifying or structuring or the validation within one item is not required by the current examination problems. One item requires both mathematising and interpreting. This item requires the determination of an intersection point on the basis of a model given by a function equation with an unknown parameter with self-determined conditions interpreted in the framework of the factual context.

Discussion

Generally, it can be said that around 40% of the items examined include reference to reality. However, this does not mean that such items also contain modelling problems. The application of common criteria for modelling problems shows that we can talk about modelling problems only to a limited extent. Not a few items include an authentic factual situation, but a realistic use of mathematics is given just in exceptional cases. None of the problems is open or relevant. This is why criticism towards Abitur examination problems must be taken seriously if one must concludeas already concluded by [START_REF] Henn | Von der Welt zum Modell und zurück[END_REF].

It is comprehensible that in individual items only a partial modelling competence is dealt with due to the examination situation or to performance measurement, respectively. This can also be observed essentially in respect to the problems examined. It would be desirable though, that all partial competences of modelling were contained in a test problem and that this problem was tested in the typical sequence as in the modelling cycle. This is not the case with regard to the problems examined. There are even partial competences, which are not tested at all, such as simplification.

Numerous general conditions must be observed when preparing examination problems. In order to actually test the modelling competence, it is necessary not just to use problems with reference to reality which describe an authentic situation only partially and to test just a few competences of partial modelling competencies, but to start with an intensive and criteria-led development of modelling problems for an examination on the basis of the current educational standards.

In order to improve the competency modeling and the development of modelling problems for examinations, the criteria used here should also be used for the development of examination problems and for training problem developers. In order to increase the share of suitable modellings in examinations, it seems that a substantially greater proportion of reality-related problems or a substantially targeted development of suitable items is needed.

Figure 1 :

 1 Figure 1: Stimulus text of an examination task (www.iqb.hu-berlin.de)

Figure 2 :

 2 Figure 2: Part b) of an examination task (www.iqb.hu-berlin.de)

  Reference to reality Does the problem refer to an extra-mathematical factual context? Relevance Is the factual context relevant to the students (factual problem)? Authenticity Is the factual context authentically related to the actual situation? Authenticity Is the factual context authentical with regard to the use of mathematics? Openness Is there more than one possibility to solve the problem (solution variety)? Partial competences of modelling Which partial competences of modelling are required for dealing with the problem (simplifying / structuring, mathematising, interpreting, validating)?

Table 1 : Criteria used for the Assessment of Items

 1

Proceedings of CERME10


domain: math shs

text:
Rita Borromeo Ferri

Nicholas Mousoulides
email: n.mousoulides@ucy.ac.cy

Mathematical modelling as a prototype for interdisciplinary mathematics education? -Theoretical reflections

Keywords: Mathematical modelling, interdisciplinary mathematics education, theory development

In the last years the discussion for promoting Science, Technology, Engineering, and Mathematics (STEM) education became a central goal of educational policy in many countries worldwide, in an attempt to prepare students for a scientific and technological society. However, interdisciplinary mathematics teaching and learning is not limited to the "STE" and should include other disciplines across the curriculum. Mathematical modelling, as a mathematical practice and key competence within mathematics education standards could be interpreted as an excellent example for promoting not only modelling competencies, but also interdisciplinary mathematics education (IdME) in school. In this paper we focus theoretically on the question, 'Which core similarities and differences can be stated between the two fields along three perspectives?', by presenting a piece of theory describing the interplay between IdME and mathematical modelling.

Introduction

The purpose of the present study is to examine, from a theoretical point of view, the interplay between mathematical modelling and interdisciplinary mathematics teaching and learning, and to propose how mathematical modelling can promote interdisciplinary mathematics education. Following our theoretical approach, we present an example of an activity, based on our previous work, which exemplifies the key features and components of such a modelling activity that can promote interdisciplinary mathematics teaching and learning.

In the following sections, we present mathematical modelling as a means for teaching and learning mathematics through an interdisciplinary lens, by referring to the characteristics of modelling that make this approach feasible. We later present the teacher perspective on using modelling in promoting an interdisciplinary approach, and an example of an interdisciplinary modelling activity. We finally present and discuss a model of the interplay between mathematical modelling and Interdisciplinary Mathematics Education (IdME).

Mathematical modelling as means for interdisciplinary teaching and learning

In this section we firstly give a brief general view on mathematical modelling and interdisciplinary mathematics teaching and learning by defining the two fields and by posing some initial thoughts for consideration. Although these definitions show strong overlaps, one has to look deeper concerning their differences, to understand both fields as exclusive as well. To make this more transparent we discuss mathematical modelling as a means for interdisciplinary teaching and learning, along the following perspectives: (a) the modelling cycle perspective and the individual modelling routes, (b) the teachers' perspective and the cross-link approach, and (c) the Thematic Working Group 06 Proceedings of CERME10 interdisciplinary activities and the students' work. These perspectives are presented in the following sections.

General view and definitions

Whereas there is a strong consensus in the international discussion that mathematical modelling can be described as an activity that involves transitioning back and forth between reality and mathematics, the definition of interdisciplinary mathematics education is very vague. Recently several researchers from different disciplines, including both authors of this paper, published a monograph entitled "Interdisciplinary Mathematics Education -State of the Art" [START_REF] Williams | Interdisciplinary mathematics education: A state of the art[END_REF]. Without going into detail here, it became clear that describing a 'discipline' is much easier than to think about, if more disciplines could be multi-, inter-, trans-or meta -disciplinary. So what does interdisciplinary mean? An interesting paper by [START_REF] Nikitina | Three strategies for interdisciplinary teaching: contextualizing, conceptualizing, and problem-centering[END_REF] described three core approaches to the teaching of science and mathematics in integrative ways, that differ from one another in form and purpose. These three strategies, namely conceptualizing, contextualizing, and problem-centering, ask different questions of mathematics and science, and serve different learning goals. The authors discussed these strategies based on their empirical study and furthermore they claim that understanding the strengths and weaknesses of each strategy can help educators choose the optimal way to present their interdisciplinary material. In contrast, a brief pragmatic definition is formulated by [START_REF] Roth | Interdisciplinary approaches in mathematics education[END_REF], who stated that: "Interdisciplinarity denotes the fact, quality, or condition of two or more academic fields or branches of learning. Interdisciplinary projects tend to cross the traditional boundaries between academic disciplines" (p. 317). In the following we use this definition of IdME as a basis for our theoretical reflections in this paper. Following [START_REF] Roth | Interdisciplinary approaches in mathematics education[END_REF] definition, some thoughts arise by contrasting it with mathematical modelling: Without having a real life problem, mathematical modelling activities are not possible. At first, real life questions come out of another '(scientific) discipline' than mathematics. This makes sense and so we have the solution and found the overlap between these fields by arguing that mathematical modelling is the same as interdisciplinary mathematics and vice versa! -Stop, this would be too easy! Can we easily speak about 'disciplines' or is it better to say that real life questions of modelling problems come out of other 'realities'? Do modelling problems always include or promote further disciplines/ realities and what is the importance when having one or more of them? If there are other disciplines implicitly and explicitly distinguishable in a modelling problem, is it the teacher's goal to connect them and make them understandable for the students? Are modelling problems per se a prototype for interdisciplinary mathematics education? In the next section the cycle perspective gives some answers to the raised questions and again new thoughts are presented.

Cycle perspective and individual modelling routes

Recently the importance of modelling cycles, independent of type (see [START_REF] Borromeo Ferri | Theoretical and empirical differentiations of phases in the modelling process[END_REF], became clear for the teaching and learning of mathematical modelling in the classroom. In addition to promoting general modelling sub-competencies, fostering the meta-cognitive modelling competency can be seen as a strong goal in the learning process as well. Research on students' cognitive processes while modelling also showed that the individual's process of modelling is far from linear. So, individual modelling routes [START_REF] Borromeo Ferri | Modelling problems from a cognitive perspective[END_REF] better describe students' jumps backward and forward within the cycle. These jumps could be empirically reconstructed between phases, not only inside "reality" or "mathematics", but mostly between "reality" and "mathematics".

Looking from the cycle perspective on modelling and interdisciplinary mathematics we would like to formulate two main thoughts/ideas, where (b) is dependent on (a).

(a) The modelling cycles which were developed in the field of applied mathematics and mathematics education during the last decades have a strong focus on the mathematics itself of course [START_REF] Pollak | The interaction between mathematics and other school subjects[END_REF][START_REF] Blum | Anwendungsorientierter Mathematikunterricht in der didaktischen Diskussion[END_REF]. The interdisciplinary view is not explicitly visible. The often used terms in the modelling cycles, like "simplifying" or "working mathematically" do not imply that other disciplines are involved. This shows exclusiveness and no overlap when just focusing on the cycles. The same phenomenon can be observed when looking at modelling cycles in physics or chemistry (e.g. Goldhausen & DiFuccia, 2014), because they indeed focus on their discipline, but applying mathematics, if necessary, is only a side product.

(b) Considering that the interdisciplinary view is not explicitly visible in the known cycles, it is clear that it can only happen in connection with appropriate modelling problems. The individual has to interpret by reading the problem the other disciplines/realities that are included in the problem. This means that the extra-mathematical knowledge is not only limited to one's own experiences, but to the knowledge of other disciplines, like physics or ICT. One could argue that the stronger the "discipline knowledge" and the "mathematical knowledge" the better a student's modelling process will be. Speaking on an abstract level, the individual modelling routes are on a multi-dimensional level, when the other discipline(s) included in the modelling problems is understood by the students. From this point of view, we see strong overlaps between mathematical modelling and interdisciplinary education.

Teachers' perspective and the cross-link approach

With respect to the previous section we focus now on the teachers' perspective. If we want teachers to be qualified in interdisciplinary mathematics, is it expected from them to become experts, for example, in all of the four STEM (Science, Technology, Engineering, and Mathematics)-fields? This question was the starting point for our theoretical conceptualization of STEM from a teaching and educational perspective, which is also based on the existing debate in STEM-education. Borromeo Ferri and her colleagues (2016) distinguish between the single-field teaching approach and the cross-link field teaching approach. The first approach describes promoting a single discipline in school very deeply, like for example an engineering learning environment (e.g. [START_REF] English | STEM learning through engineering design: Fourth-grade students' investigations in aerospace[END_REF]. The other disciplines are not fundamentally included within this environment. The second approach means to promote multiple disciplines; at least two disciplines are promoted in one learning environment in order to cross-link these disciplines (see [START_REF] Star | Studying technology-based strategies for enhancing motivation in mathematics[END_REF] for an example). Within the empirical classroom study of the "Leonardo-da-Vinci Project" [START_REF] Borromeo Ferri | Inspired by Leonardo da Vinci-Stem learning for primary and secondary school with the cross-link approach[END_REF] mathematics, physics, engineering and art were explicitly included in one learning environment. Grade 9 students (14 year olds) built and modelled the Leonardo bridge in an Inquirybased Learning environment. The main goal of the lesson-unit was the permanent reflection of cross-linking the disciplines. On the basis of the empirical data and theoretical thoughts "cross-link" could be characterized as follows: One can speak from cross-linking, if at least two (scientific) disciplines are combined during one lesson or within the whole lesson-unit and are reflected with students on a metacognitive level [START_REF] Borromeo Ferri | Inspired by Leonardo da Vinci-Stem learning for primary and secondary school with the cross-link approach[END_REF]. Again, the main aspect of making several disciplines explicit is at the foreground. If the teacher decided to look at the Leonardo bridge from only a strong mathematical perspective by neglecting the other disciplines, it is also possible.

The students had the opportunity to understand and to model the bridge by also using and naming the other disciplines.

Interdisciplinary activities and students' work

A great number of research studies has focused on the development of activities and learning materials, following an interdisciplinary approach. In this paper, we focus on the development (and the characteristics) of learning activities that have adopted a modelling perspective (e.g. [START_REF] English | STEM learning through engineering design: Fourth-grade students' investigations in aerospace[END_REF][START_REF] English | Bridging STEM in a real-world problem[END_REF][START_REF] Mousoulides | A modelling perspective in designing interdisciplinary professional learning communities[END_REF]. Such activities are set within authentic contexts, and allow for students' multiple interpretations. With regards to mathematics, such activities provide students with opportunities to be engaged in important mathematical processes, such as describing, analysing, constructing, and reasoning [START_REF] Lesh | Beyond constructivism: A models and modelling perspective on mathematics problem solving, learning and teaching[END_REF].

Research in the field listed six design principles for developing such learning activities, following a modelling perspective. These design principles are based on the work of teachers and researchers and that have subsequently been refined by [START_REF] Lesh | Beyond constructivism: A models and modelling perspective on mathematics problem solving, learning and teaching[END_REF]. The 'Model Construction Principle' ensures that the solution requires the construction of an explicit description, explanation, procedure, or justified prediction for a given mathematically significant situation. The 'Reality Principle' requires that students can interpret the activity meaningfully from their different levels of mathematical ability and prior knowledge. The 'Self-Assessment Principle' ensures the inclusion of criteria that the students themselves can identify, and use to test and revise their ways of thinking. Specifically, the modelling activity should include information that students can use for assessing the usefulness of their solutions, for judging when and how their solutions need to be improved, and for knowing when they are finished. The 'Model Documentation Principle' ensures that while completing the modelling activity, the students are required to create some form of documentation that will reveal explicitly how they are thinking about the problem situation and their solutions. The fifth principle is the 'Construct Share-Ability and Re-Usability Principle', which requires students to produce share-able and re-usable solutions that can be used by others, beyond the immediate situation. The 'Effective Prototype Principle' ensures that the modelling activity is as simple as possible yet still mathematically significant. The goal is for students to develop solutions that will provide useful prototypes for interpreting other structurally similar situations. By adopting the principles mentioned above, Mousoulides and colleagues (e.g. [START_REF] English | Bridging STEM in a real-world problem[END_REF][START_REF] Mousoulides | A modelling perspective in designing interdisciplinary professional learning communities[END_REF][START_REF] Williams | Interdisciplinary mathematics education: A state of the art[END_REF] have developed a number of interdisciplinary modelling activities for students. These activities have been piloted and mainstream tested in various schools in a number of countries. Such an example, the 'How can I lose weight' activity is presented in the monograph by Williams and colleagues (2016). The activity, which targeted 11-12 year olds, focused on the balance between nutrition and physical activity for a healthy life. The activity required students to actively participate in the collection, presentation and interpretation of data regarding their nutrition and exercise habits. Based on an analysis of their own data, students had opportunities to explore the variables (and their dependencies) that may affect the amount of energy intake on a daily basis (e.g. height, mass, age) and suggested specific diet and exercise plans, always taking into consideration the need of balancing the two.

The activity consisted of three parts. In the first part, the case of Mary, a 14-year-old girl who cannot fit into her favourite clothes, was presented. The students then considered the general question, "Is not eating the best approach to losing weight?" Students, with teachers' support, quickly realised that the question needed to be refined in order to be answered meaningfully. On refining the question in their own ways, students acknowledged that real (actual) data on nutrition, and also on physical activity are needed. Students were then encouraged (by teachers) to work with their parents to collect the required data, through an anonymously completed questionnaire. Using their own data, students worked in groups to summarise their results, by categorising data into the different food categories (e.g. protein, carbohydrates, dairy products, fruits, vegetables, sweets, etc.), and by discussing the advantages and disadvantages of each category (Reality Principle). Students also explored trends and relationships in their data, by using a spreadsheet software (Self-Assessment Principle). An example of their work is presented in Figure 1.

During the second part of the activity teachers guided a student-centred exploration for identifying the factors that determine a person's daily calorie intake (age, gender, height and body mass). Students worked on analysing tables and graphs by using an applet software, designed to support the interdisciplinary activity. In the third part of the activity students worked on suggesting a balanced diet plan (for a single day), taking into consideration the daily amount of energy a person needs (Model Construction Principle). Students could use the provided 'food database' for creating the person's diet for a day (Model Documentation Principle) and then explore the appropriateness of the diet with regards to the calories taken and the food categories (Figures 2 and3). After completing the tasks and sharing their results in a whole class discussion, students then moved to the last part of the task, in which they designed their own balanced nutrition and physical activity case (Construct Share-Ability and Re-Usability Principle). The interdisciplinary nature of the activity focused on the role of the bridging concept (balance between calories intake and exercise) (Effective Prototype Principle). The quite complex activity setting provided opportunities for students to explore important concepts from mathematics and biology. The implementation of the activity revealed that both teachers and parents found the interdisciplinary nature of the activity challenging; for teachers, it provided a new way of thinking and working, while for students it provided a real world problem framework, in which they could explore and connect concepts from different, yet connected, school subjects.

Theorizing the interplay between mathematical modelling and interdisciplinary mathematics education

When summarizing the presented theoretical analysis, it becomes clear that there are strong overlaps between mathematical modelling and IdME. The real context of modelling problems, like the case presented in the previous section, in fact evokes interdisciplinary activities, but the teacher is at first the person who should make them more explicit to students, and finally actively connect the different fields, through her/his teaching. Although in some ways mathematical modelling could serve as a prototype for interdisciplinary mathematics education, mathematical modelling has its own conditions. Theorizing the interplay between mathematical modelling and IdME is a challenge, which we like to think about furthermore and also find an appropriate visualization. At this point we argue that mathematical modelling has its own "theory(-ies)", because the characterization of mathematical modelling, and further terms/ concepts/ processes like "problem understanding", and "validation", are part of the theory(-ies) of modelling. This can be seen as the theoretical part of mathematical modelling, based on the theoretical and empirical research in this field in the past decades. It is rather difficult to separate mathematical modelling from IdME, because on one hand mathematical modelling as itself is (can be) a part of IdME, but also on the other hand we can view mathematical modelling as a comprehensive research field. By adopting this modelling oriented approach in the "nutrition-exercise" case study, students could work in finding/proposing a model for balancing the intake-consumption of calories. For instance, students could be asked to propose models for different people (e.g. peers, professional athletes, teachers, parents), which balance their daily diets and their exercise habits. In doing so, the emphasis of the activity would not be within mathematics or biology, but rather on modelling.

IdME can be situated in its own field, if the (interdisciplinary) task does not fulfil the criteria of the modelling problems, e.g. when you have some kind of a "word problem". Not every interdisciplinary task, which has (some) mathematics in it, is a modelling problem per se. IdME can be done differently and cannot be connected with a modelling problem, so one can cross the disciplines of mathematics, and for example biology in a task, but only focusing on mathematics when dealing with the problem at last. This is what we mean with pure crossing disciplines. The interplay of modelling and IdME is clearly observed, when a real life question is embedded in a real modelling problem, in which students understand the context, recognise all the disciplines involved, and use or get to know about the extra-mathematical (other disciplines) knowledge. By adopting this perspective, in the Nutrition-Exercise case study students could work in solving a problem related to finding algebraic formulae for calculating the number of calories in various types of food and/or sport activities. In doing so, students have to work with both mathematical and scientific concepts, in solving the required problem, but the emphasis would be on the mathematical concepts or the biology ones (e.g. different types of food and relation to calories per gram, etc.).

There is a number of possible implications, especially for the teachers, in promoting both modelling competencies and IdME-competencies. By using modelling problems, teachers can work with their students in crossing the boundaries between disciplines. In doing so, interdisciplinary modelling activities can provide unique opportunities for teachers to collaborate, synthesize and integrate more interdisciplinary pedagogies and teaching methods in their teaching [START_REF] Berlin | Connecting school science and mathematics[END_REF], and for students to develop better and more coherent solutions for complex, yet interesting and real world based problems. Such approaches raise teachers and students' expectations and confidence in working in a more interdisciplinary way, and lessened their focus on the difficulties in using and working with interdisciplinary modelling activities.

Figure 1 :

 1 Figure 1: Student eating habits

Figure 2 :Figure 3 :

 23 Figure 2: Creating a person's diet

Proceedings of CERME10


text:
Berta Barquero
email: bbarquero@ub.edu

Nacho Monreal
email: jimonreal@tecnocampus.cat

Noemí Ruíz-Munzón

Levels of analysis of a mathematical modelling activity: Beyond the questions-answers dialectic

Keywords: Mathematical modelling, study and research paths, dialectics, university level

This paper presents the a posteriori analysis of an study and research path (SRP) on comparing reality versus forecasts of Facebook users, which appears as a teaching and learning proposal for mathematical modelling. We present the main elements of the SRP that have been designed through a virtual platform developed in the frame of the European project MCSquared and experienced in a first-year course at university in management sciences degrees. The three-layer analysis we present is based on three main study dialectics: the questions-answers, media-milieu and individualcollective dialectics, which are central for an SRP and for mathematical modelling. In particular, we focus our a posteriori analysis on how these three dialectics were fostered and which are the main weaknesses as well as strengths of the SRP experienced.

Introduction: The SRP as teaching proposal for mathematical modelling

During the last decades, researchers and practitioners agree that teaching should not be focused only on the formal transmission of knowledge, but also should provide students of the tools for enquiring into the study of real phenomena and integrate mathematics as an essential modelling tool. It is thus important to foster a change of the school paradigm, with new functionalities to mathematical knowledge, novel responsibilities to teachers and students, different ways of questioning mathematical knowledge, that is, moving from a school paradigm which most of the time focused on introducing students to already built mathematical knowledge devoid of its rationale to a paradigm of questioning it, 'questioning the world' in the words of [START_REF] Chevallard | Teaching mathematics in tomorrow's society: A case for an oncoming counter paradigm[END_REF]. In the particular case of the research on modelling and their applications and on inquiry-based approaches some big steps have been made showing how, under certain suitable conditions in different educational levels and curricular frames, modelling activities may be successfully put into practice [START_REF] Artigue | Conceptualizing inquiry-based education in mathematics[END_REF][START_REF] Burkhardt | Making mathematical literacy a reality in classrooms[END_REF]among others). Hence, the dissemination and long-term survival of activities based on modelling, enquiring and other innovating proposals are one of the main challenges for Mathematical Education. Therefore, to support and analyse any kind of alternative teaching proposal, researchers need reference models that allow them to analyse and evaluate the impact that these innovative teaching practices have on the way mathematics is conceived, on the nature of the didactic systems and milieus emerged through these practices, and on the conditions and constraints that help or hinder the viability of these practices.

In our research, developed in the framework of the anthropological theory of the didactic, we bet on the use of the study and research paths (SRP) as epistemological and didactic model [START_REF] Chevallard | Teaching mathematics in tomorrow's society: A case for an oncoming counter paradigm[END_REF]Winslow et al., 2013) to face the problem of moving towards a functional teaching of mathematics and, particularly, where mathematics are conceived as a modelling tool for the study of problematic questions.

Traits and levels of analysis of an SRP

According to [START_REF] Barquero | Didactic engineering as a research methodology: From fundamental situations to study and research paths[END_REF], the starting point of an SRP should be a 'lively' question of real interest for the community of study (students and teacher/s). The study of Q0, called the generating question, evolves and opens many other derived questions Q1, Q2,…, Qn. The continuous looking for answers to Q0 (and to its derivative questions) is the main purpose of the study and an end in itself. As a result, the study of Q0 and its derived questions Qi leads to successive temporary answers Ai that can be helpful in elaborating a final response R ♥ to Q0. These first characteristics can be associated to the first level of analysis of the SRP that we here consider, it consists in the dialectic establishing between the questions posed and the likely answers appearing (questions-answers dialectic) which also provide the basic structure of an SRP to be implemented and to be enriched after each implementation. This first layer can be linked to the cronogenesis dimension of the teaching and learning practices, referring to the evolution of questions to be faced and the necessary knowledge to be used.

Another central dimension for an SRP is the media-milieu dialectic, which constitutes the second level of analysis. As described in the aforementioned investigations, the implementation of an SRP can only be carried out if the students have some pre-established responses accessible through the different means of communication and diffusion (that is, the media), to elaborate the successive provisional answers Ai. These media are any source of information, such as: textbooks, treatises, research articles, class notes, or the teacher acting as main media. However, the answers provided are constructions that have been elaborated to provide answers to questions that are different to the ones that may be put forward throughout the mathematical modelling process. Thus they have to be deand re-constructed according to the new needs. Other types of milieus will therefore be necessary to test the validity and appropriateness of these answers. This second level of analysis put attention to the mesogenesis, that is, the evolution of the experimental milieu. Finally, we may consider the collective dimension of the research and study of questions in an SRP. This third dimension focuses on the roles and responsibilities that, far from the traditional didactic contract, students and teachers may assume in experiencing an SRP and how the individual work is shared, transferred and agreed with the wider community, and vice versa. We will denote this third level as the individual-collective dimension, which refers to the topogenesis.

Didactic analysis of a mathematical modelling process: The case of an SRP about comparing reality against forecast

We focus on analysing the case of an SRP on comparing forecasts against reality in the case of Facebook users' evolution. This SRP was designed in the frame of the European project MCSquared (http://www.mc2-project.eu), the goal of which is the design of innovating teaching proposals (the so-called c-book units) to foster creative mathematical thinking. This c-unit in particular has been produced by a group of five multiple background designers: 2 Maths Education researchers, 2 university lecturers, who were then in charge of its implementation, and an expert on modelling in the field of Operations Research, which enrich a lot the way to structure the teaching sequence to prompt mathematical modelling. The teaching proposal was made in the virtual socio-technical environment developed along the project, the c-book that integrates some narratives with several applets of different factories of different educational technologies. Next we present the details about the design of the SRP experienced, combining the virtual environment offered by the c-unit with face-to-face sessions during the winter term of the academic year 2015-16 with first-year students of Business Administration Degree and of Innovation Management (BAIM), all from the 'Escola Superior de Ciències Socials i de l'Empresa-Tecnocampus', Pompeu Fabra University.

The initial situation starts from a real news about a research developed by Princeton University in 2014, in which it was predicted that Facebook would lose the 80% of its users before 2017. Hence, the initial question Q0 presented to students is about: Can these forecasts be true? How can we model and fit real data about Facebook users' evolution to provide our forecast the short-and long-term evolution of the social network? How can we validate Princeton conclusions? The experimentation was structured in three interconnected phases linked to the generating question Q0, building up the a priori design of the SRP, then reflected in the design of the c-book unit. A first phase that focuses on the open research of real data about Facebook users, a second one focused on which mathematical models (mainly based on elementary functions) can provide a good fitting to real data, and a third one about the use of these models to provide the short-, medium-and long-term forecasts of Facebook users and about how to decide about best and most reliable model. The students, working in 'consultant teams' of 3-4 people, got the order from MS2 Consulting ('Mathematical Solutions Squared') previously described as Q0 and they were asked to deliver a final report by the end of their work as an oral presentation as response to the order. The implementation combined face-to-face sessions in the teaching device called 'Math modelling workshop' (in a total of six 90-minuts weekly sessions) for the miss-in-common of the junior consultant teams' partial reports, with work out of the classroom. The guiding c-unit that gives the workshop support was ready before starting in order to let the different teams attach their answers, pose new questions by providing an interacting device (as chats or shared spread-sheets) and use some applets designed especially to get through the different phases of the SRP. Next we sketch how different dialectics were prompted by both: (a) the c-unit design (by its initial design but also by the different changes that were introduced according to students' requirements: new questions and answers not envisioned, new media required, etc.) and (b) the didactic gestures and devices to manage its implementation.

The questions-answers dialectics

We can visualize the first level of description of the SRP designed and experienced as an arborescence of the questions that were proposed and faced and the answers foreseen and appeared. This questionsanswers structure constitutes a first layer of analysis of the process designed and of the trajectories followed in the implementation of the SRP. As introduced above the generating questions Q0 of the SRP on comparing forecasts against reality in the case of Facebook users' evolution was broken into three main derived questions (see Figure 1 for the SRP questions' organization), which guided the successive phases of its implementation. A first phase that focuses on the open research of real data about Facebook users, a second one focused on which mathematical models (mainly based on elementary functions) can provide a good fitting to real data, and a third and last one about the use of these models to provide the short-, medium-and long-term forecasts of Facebook users and about how to decide about best and most reliable model. Q2: Which mathematical models provide the best fitting of real data about FB users?  A2: Each consultant group is asked to propose and justify three mathematical models fitting real data.

Q2.1: Which models (based on elementary functions: linear, parabolic, exponential, etc.) may fit the data? Q2.2: How can the coefficients of the model be determined?

Q3: How can we decide about the 'best' models fitting data? Can we use this model to predict the future evolution of FB users?  A3: Need to create tools to justify why a mathematical model/s is/are the 'best' with respect to: (a) fitting data and (b) forecasting the evolution of FB users.

Q3.1: How can we compare the error committed between reality and forecasts provided by models? Q3.2: Can be the same model used for the short-and long-term forecasts?

Let us comment the main aspects of the a posteriori analysis of the experimentation referring, in particular, to the questions-answers dialectic level. About the first phase, we should remark the ease with which the students found real data about the evolution of the social net. The most format they found the information was by means of a graphical representation (for example, a bar chart). This fact strongly determined their analysis, since they focused mainly in the graphical analysis of the data growth tendency, but not in their numerical versant (variation tax, that appeared in a tangential way). Besides, the fact that many groups found the same data triggered an intense debate and interchange of ideas among them, which took us to consider a brainstorming session about the previous hypothesis in the classroom, and as a consequence, the duration of the first phase was extended from 3 to 4 sessions. Due to the wealth of answers collected by the teams during the brainstorming session we decided to ask the students to deliver a first report in a poster format, so that each team could synthesize their findings and share their conclusions at that moment. About the second phase of the SRP, since many groups worked finally with very similar data on the worldwide evolution of FB users, we made two new decisions: (a) give each team a second set of different data, corresponding to different geographical areas, in order to contrast their hypothesis and extend their study; and (b) ask for more than one fitting model for each data set. The analysis of the teams proposals made arise a non-expected aspect: many of them proposed using piecewise functions, so that the expected answers to Q2 about the consideration of models based elementary functions (linear, quadratic, exponential, etc.) was extended. Otherwise, during the brainstorming in the first phase the teams started enquiring in the history of FB and about the possible reasons of the changes in the tendency of the data or number of users (IPO, new rival social nets, purchases of the company, new developments, etc.), and also the moments of change of tendency. New questions and answers appeared at this stage about changing the fitting model in accordance to a particular action or decisions of FB. Concerning the third and last phase, we only dedicated two face-to-face sessions of the workshop (one with the whole group and the other for the consultant teams' doubts) and were not enough for a rich development of Q3. Although this time constraint, there were some applets designed and integrated in the c-unit to help on the simulation of models and its contrast to real data, as we explain in the following section about the media-milieu.

The media-milieu dialectic

Since we have the first layer of analysis of the SRP in terms of the arborescence of the questionsanswers, it is important to ask when, where and how questions can arise and answers can be developed. It is at this new level when there may appear the different elements taking part of the milieu, composed of varied elements: questions, temporary answers, pre-existing answers in or out school, means to validate answers, experimental data, etc., accessible through different kind of media (textbook, lectures, website resources, etc.). The relation among these elements can be analysed through the media-milieu dialectic. In our SRP it has been central the constant dialectic between the search for data (for instance, real data about Facebook users, or about the company changes) and of pre-existing answers (ways to organise data, common models to fit population evaluation, elementary functions, tools to control error, etc.) that exist in different media that were available to students, such as web resources, contents of Mathematics course, answers from lecturers from other courses; and the creation of the appropriate means (milieu) to integrate (or refuse) them it their SRP path study. Let us stress the importance of some of them.

In the first phase of the SRP, it was important to count on the help of the teacher of another course called 'Introduction to digital communities' (running in parallel to the workshop) who helped on providing a general sense and functionality to Q0 and to show how the students could look for real data about FB and some techniques to organise them. Also, having open access to the news and papers published by Princeton University about FB, as well as FB answer to Princeton or their monthly report about users growth was very useful for students. All these elements took part of the media accessible to students, at the time it enriches students' milieu mainly composed at this stage of the data sets that each team chose to work with, shared and debated with the whole class sessions. All these elements helped them to prepare a first report with the first temporary answer A1, in a poster format given in the c-book platform, and then shared and debated in the face-to-face session (see Figure 1, left side). With respect to the second phase, the a priori design of the c-unit contained some applets (designed with Geogebra) proposed to help students to explore different models based on elementary functions (Q2). These applets provided the main media for students to visualize data jointly with model simulation, and also took part of their milieu as main tools for contrasting, comparing and deciding on the 'best' models to choose. As aforementioned, in the SRP experienced students suggested using piecewise functions, which pushed designers to make changes in the questions and applets as these necessities were coming up. Moreover, some groups started to present new hypothesis about model to use with non-elementary functions (such as Gaussian function), most of which were part of their milieu because they had been introduced in previous courses. So that, designers had to quickly cover this demand by designing a new applet with Geogebra to let them manipulate also these types of mathematical models.

With respect to the last phase, despite of the lack of time, the study of Q3 that the consultant teams developed was in general very rich. Concerning Q3.1 about comparing the error committed by models' forecasts against reality, we decided to integrate a new applet from Cinderella (see Figure 2, on the right) to provide students with the main media, also milieu, to simulate models they had bet for and to be able to compare graphically and numerically the error between data and forecasts coming from models' simulation. Through this tool, one could obtain the numerical calculation and graphical representation of the punctual and the averaged error (absolute and quadratic error) at the time one can changed the parameters that define the model to obtain a better fitting. Although the several advantages that this applet provided to students, students assumed and used uncritically the tools proposed by the applet. The lack of time and these designers' decisions made that the media-milieu dialectic at this stage was not so rich as it could be.

The individual-collective dialectic

In this level of analysis we focus on the relation between the teaching devices habilitated for the implementation of the SRP and the changes on the traditional didactic contract that are necessary, that is, changes on the roles and responsibilities that both students and lecturers involved in this experience had to assume. This layer, closely related to the two previously introduced, provides a finer detail about how individuals and the group developed their work in the SRP. And, how all the actions and objects of study and research (looking for questions and answers, proposing new media, adopting external answers, enlarging the milieu, etc.) are shared, agreed and transformed from the individuals to the community. Although we will be not able to go very deep in this description, due to space restriction, let us stress some important features of the SRP about its collective dimension.

First, the devolution of Q0 was presented as an external order coming from MS2 Consulting ('Mathematical Solutions Squared') and students were asked to spend more than a month to prepare and deliver a final report by the end of their work as an oral presentation. The lecturer of the (official) Maths course now changed their role to become only a guide of the study and research process, transferring most of the responsibilities to students on elaborating their answer. Moreover at the end of the implementation, two external persons carried out the validation and evaluation of the final consultant team's answers, in coherence with what was asked at the beginning and trying to make students assume these responsibilities. Second, students were organised all the time in consultant teams with an autonomous functioning, who were asked since the beginning to jointly deliver a collective report, although in its evaluation each of them had to be responsible of explaining responses to one of the phases and all could be asked about any phase. To help on assuming this autonomy, lecturers took several decisions: (a) distribute different data sets to help them not to be very influenced by other teams work and rhythms; (b) use the workshop sessions for the common debate of groups work and to share the main advances in finding answers (although different) and new questions emerged, and (c) constantly introduce changes in the design of the c-unit design, which work as the shared support, with the new questions that had emerged and with the media that could help them at each step. For instance, at the last of phase 1, designers decided to integrate a poster format to fix the things students had to share with the rest of the groups and dedicate a workshop session when consultant's teams could explain their answers to Q1; or, as explained in the previous section, in phase 2 and 3 designers decided to create new applets to deal with new models proposal and to facilitate them the media and milieu to make the SRP progress.

Final remarks and conclusions

In this paper we focus on the case of an SRP on comparing forecasts against reality in the case of Facebook users' evolution to show the use of three dialectics: the one of the questions-answers, of the media-milieu and of the individual-collective, corresponding to the three complementary level of didactic analysis of teaching and learning processes [START_REF] Chevallard | Un concept en émergence : La dialectique des médias et des milieux[END_REF]. Besides their analytic use, they suppose a productive framework to enrich teaching and learning practices, in particular, on modelling.

In what concerns to the questions-answers dialectic, since the beginning of the workshop, the generating question Q0 about the controversy of the article by Princeton was adopted by the students with a great interest and, up to the end of the process, was kept alive. From its implementation we can underline very important conditions that were created. First, the flexibility of the lecturers and designers team that were opened to readjust the schedule according to students' team work, that is why we devoted more sessions to the first phase and consequently reducing the ones to the last phase. Furthermore, they were very attentive to integrate in the c-book unit all new questions and means that students asked for. Second, students were very active on workshop session to share their proposals from which many derived questions appeared, some of them planned in the a priori design, some others that extended the initial proposal. About the media-milieu dialectic, in the case of this SRP and with the support of the c-unit infrastructure, we took several decisions along the implementation of transforming the media offered to students to help them in the modelling process and also to observe the impact new media had on students' milieu. We may again insist on the importance of very important contributions, such as: collaboration with other subjects (as the one of 'Introduction to digital communities'), focusing some workshop sessions on the discussion external answers that students brought, or by the creation of widgets to foster students' experimental work, etc. Last but not least, about the individual-collective dimension, we may underline that students easily accept the request of presenting their final response in front of an external committee. The last session that we dedicated to these presentations brought to the light the richness and variety of answers given to the stated initial question, as well as the complementarities of consultant teams' answers. But, we are aware of the weakness and insufficiency of mechanism to collect individual and teams internal work, as most of the workshop sessions were dedicated to the common debate. This is one aspect to be improved in the following experimentations and to integrate means to get access to the dynamics established between the individual and collective work.

Fig 1 :

 1 Fig 1: Tree of questions and answers of the different phases of the SRPQ1: Which data sets about Facebook users are better to consider in our research?  A1: Each group look for the data set to be used and shared; the whole community agree on the terminology (year, period, units, etc.) and on the dependent and independent variables to consider.Q1.1: Which time intervals may be considered? Q1.2: How can data be well-organized? Q1.3: How to organise and visualise data? Q1.4: What can we say about the growth tendency of the data analysed?

Fig 2 :

 2 Fig 2: Example of page format and applets of the c-unit about Facebook users

Proceedings of CERME10

Thematic Working Group 06Proceedings of CERME10

Acknowledgments

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


text:
Berta Barquero
email: bbarquero@ub.edu

Susana Carreira

Gabriele Kaiser
email: gabriele.kaiser@uni-hamburg.de

Introduction to the papers of TWG06: Applications and modelling

Keywords: Mathematical modelling, empirical studies, theoretical approaches, modelling examples, usage of technology

Introduction

Mathematical modelling and its teaching at various educational levels are widely accepted all over the world. There is consensus of the need to integrate mathematical modelling and applications in curricula and this has already taken place in many European countries. However, there is no unanimity on how to integrate mathematical modelling and applications into the processes of teaching-and-learning mathematics. In addition, there is little secure empirical knowledge available on how to implement efficiently the necessary new learning environments. In its discussions at CERME 10, the thematic working group TWG 6 on modelling and applications aimed to contribute answers to these open questions and hence further develop the work from previous ERME conferences. The contributions discussed at the congress are characterized by a strong and fruitful diversity in the research questions considered, the school levels addressed and the theoretical approaches taken. On the whole, the papers address theoretical, methodological, empirical or developmental research on the teaching and learning of applications and modelling. The group involved 35 participants from 16 countriesparticipants were from Europe, and also from South and Central America. A total of 20 papers and four posters were presented and discussed during the working group sessions.

In the following, we describe all presented papers, although not all have been submitted for the proceedings, grouped around eight comprehensive themes, which refer to current issues within the teaching and learning of mathematical modelling.

As a first important theme, we identify the interplay between disciplines into modelling activities referring to the specificities of interdisciplinary modelling activities, especially in engineering teaching. The second theme relates to the connection of the problem solving perspective and mathematical modelling and to the development of problem solving strategies and competences, when students work in groups and develop individual competences. The third theme covers developing modelling strategies and competences, for example theoretical and empirical work focused on the analysis and use of heuristic strategies adopted by teachers and/or students which illuminates strategies used to foster students' performance when solving modelling problems. The fourth theme refers to the tools and methodologies used to analyse modelling processes, namely studies focusing on elaborating specific methodologies for analysing and evaluating modelling practices. The fifth theme focuses on teachers' beliefs in relation to the teaching of mathematical modelling, including, for example, research on the role of teachers to foster modelling practices. The sixth theme covers teachers' interventions in mathematical modelling. The seventh theme refers to experimental materials and technology in modelling, which covers two topics, a first primarily focused on the role of the auxiliary material and its impact on modelling and a second concerned with how to combine different resources with technology in concept development by means of real word contexts. The eighth theme refers to the assessment of modelling practices.

Overarching themes

Modelling and interdisciplinary teaching

This theme focuses on the interplay between disciplines in modelling activities. The first study by Borromeo and Mousoulides describes theoretical reflections about the differences and similarities between mathematical modelling and interdisciplinary mathematics education. Besides its focus on underlying similarities and differences, this paper provides some examples of projects that make connections explicit and how these may be useful for teachers while discussing modelling as a means for solving interdisciplinary problems. Within this thematic strand, other empirical works were presented based on particular case studies involving interdisciplinary projects with mathematical modelling as a central issue. In particular, Brake and Lantau described a pilot study of an interdisciplinary project used with experienced students of grade 12 based on modelling segways which supported them to build models based on linear systems of differential equations. Furthermore, Sala, Font, Barquero and Giménez reported on the design and analysis of an implemented interdisciplinary project, where mathematical modelling was embedded into an archaeological context. The study showed how the complementarity between two subjects (history and mathematics) can be an important tool in supporting modelling and inquiry. Besides the potential of designing interdisciplinary modelling situations, such as those presented in these papers, a couple of important question remain unresolved:

 What are the specificities of interdisciplinary mathematical modelling in relation to mathematical modelling more generally?  How is it possible to manage the interplay between mathematics, mathematical modelling and non-mathematical knowledge to enrich teaching practices in the learning of mathematics?  Can some analytical tools widespread in the research frame of modelling, such as the modelling cycle, be adapted to analyse mathematical modelling in interdisciplinary contexts? And, if yes, how?  What relationships exist between the mathematical modelling cycle and the inquiry process?  How do we best integrate different disciplines in developing modelling tasks?

Another important topic referred to modelling in the particular case of teaching of engineering. The papers from Romo, Tolentino and Romo-Vázquez, and Siero, Romo and Abundez reported intentions to design and analyse modelling activities for the mathematical education of teachers. Both studies were based on the Anthropological Theory of Didactics. They focused on analysing the roles and interplay of institutions in the educational programmes of engineers and on the institutional conditions that the design of the study and the research activities expose in the teaching of modelling in such contexts. Some of the questions discussed were:

 How can different institutions become involved in mathematical modelling in engineering education?  How can practical knowledge be grounded on mathematical knowledge?  How can the steps of an engineering project (design, mathematical model, prototype) be described? How are these steps interrelated? How are they connected to the modelling cycle?

Connection of the problem-solving perspective and mathematical modelling

The second theme refers to the connection between problem-solving activities and mathematical modelling. First, Clohessy and Johnson examined the relationships between the problem-solving performance of small groups with that of individual students in order to identify the influence of group work as an effective instructional strategy when teaching problem solving. Second, Karatas, Soyak and Alp presented an investigation about mathematical non-routine problem solving processes of fifth grade students in small groups. Their study aimed to determine problem solving behaviours within different episodes of problem solving. As both papers focused on the description and measurement of problem solving competences in small groups and individually, some common questions appeared:

 How might individual competences improve when students are participating in a group?  What instruments can we use to measure the improvement of individual competences before, during and after mathematical modelling processes? Which are the most valuable indicators we might use to measure these changes?  What are the cultural aspects that have most impact on planning the implementation of problem-solving activities?

Developing modelling strategies and competences

This third theme concerns the use of heuristic strategies to support modelling practices. The paper from Stender and Kaiser presented a study on the usage of heuristic strategies by students in school within modelling activities and the promotion of strategic help provided by academic tutors, who guide the modelling activity of the school students. The paper of Schmelzer and Schukajlow focused on the relationship between reading comprehension and mathematical modelling. The study identified strategies to help learners comprehend a modelling problem and described ways these strategies might be implemented in the classroom. The following important questions emerged from discussions of the working group:

 How far can heuristic strategies developed in the frame of problem solving be transferred to mathematical modelling?  Can their identification and characterisation be helpful in supporting teachers' strategic interventions?  How, in different teacher education programmes, may these heuristics be made explicit? If we make them explicit, do we risk narrowing the radius of action of teachers when guiding modelling activities?

 What are the effects of text length and superfluous elements on reading comprehension in modelling problems?  How can we prepare teachers to foster students' reading comprehension in modelling?

A closely related topic was that focused on the metacognition of modelling competencies as an essential part of developing competence in modelling. Vorhölter, Krüger and Wendt presented their results from a pilot study about the identification and measurement of metacognitive modelling competencies in small groups when working on modelling activities. The following relevant questions were discussed:

 How might we best define metacognitive competencies? How can their characterisation be used to evaluate students' development of competencies?  How is it possible to separate the individual progress of metacognitive competences from that of the collective group?  How might a detailed evaluation of students' teamwork enrich the understanding of metacognitive competences?

Analysis of modelling processes

This fourth theme addressed tools and methodologies used to analyse modelling processes. On the one hand, Barquero, Monreal and Ruíz-Munzón presented a study, proposed within the frame of the Anthropological Theory of Didactic, about how to forecast the increasing number of Facebook users. The analysis of the implemented research path was based on three dialectics essential for mathematical modelling: the questions-answers dialectic, that of the media-milieu and that of individual-collective dynamics. On the other hand, the paper from Delgadillo, Viola and Vivier presented an analysis of a modelling task in the context of pre-service teacher education based on the theory of the Mathematical Working Space. In this latter study, the modelling cycle was used as an essential tool to analyse the personal Mathematical Working Space of students solving a modelling task. Some questions appeared in the discussion of both papers:

 Which different dimensions, or levels, of analysis may be taken into account when analysing mathematical modelling practices?  Which are the most valuable observables (depending on the focus of study)?  Up to what point can tools for analysis be used as tools to help with designing mathematical activities? In which more general approaches to task-design do they appear?  How do we carry out analysis that takes into account both individual activity and collective interactions in modelling processes?

Teachers' beliefs on teaching modelling

This theme refers to a study on teachers' beliefs on modelling tasks. The paper from Ramirez explored mathematics teachers' beliefs about teaching and learning mathematical modelling and about modelling itself. It presented an exploratory study of responses from teachers collected in an online questionnaire related to the characteristics of modelling practices. Several questions came up about the relation of teachers' beliefs with their experience and knowledge about modelling:

 While analysing beliefs many aspects have to be taken into account, which are difficult to separate. Thus, what is the relation between ideas, beliefs, previous experiences, and modelling competences?  The role of teachers' beliefs in connection with teachers' knowledge is examined. Therefore, the question arises: How do we take into account teachers' knowledge about modelling?  The role of teachers as individuals within institutions is studied. Thus, which cultural and school conditions influence their opinions?

Teachers' interventions in teaching modelling

This section refers to teachers' interventions in teaching mathematical modelling. Ferrando, Donat, Diago and Puig presented an analysis of the different kinds of interventions teachers made during a project in which students worked on a modelling task about the intensity of sound distribution throughout a classroom. The study aimed to identify the influence of such interventions on students' learning opportunities. The following questions arose:

 How do available resources and means influence the openness of the task and students' possible responses?  Who validates the final answer in a project?

Experimental materials and technology in modelling

This theme focused on the role and use of experimental materials and of technology. Two topics were dealt with. The first focused on the role of auxiliary material and its impact on modelling and, the second concerned the combination of different resources with technology to assist concept development by means of real-word contexts. Guerrero-Ortiz, Mena and Morales discussed how the handling of auxiliary material can favour knowledge transfer between real world situations and mathematical models. The research, which was conducted within pre-service teacher education in Chile, presented insights into the design of modelling tasks and the affordances of auxiliary materials in supporting modelling. Carreira and Baioa base their research on an episode of a modelling activity with grade 9 students which aimed to reflect on the authenticity of the modelling task and to examine how students used experimental work to help them succeed in modelling activities. Some of the matters discussed were:

 How can auxiliary material support the learning process? Is it necessary to support students in its use or is it self-explanatory?  Does the auxiliary material simulate the real processes taking place in the real setting?  How far are problems authentic or meaningful to students?  To what extent can we include the way things are really done in the real world in modelling tasks for students?

Regarding the second topic, Karimianzade and Rafiepour presented a study about the introduction of decimal numbers. The study showed how different resources were introduced and how experimental work helped fifth grade students to develop their knowledge and understanding of decimal numbers in the context of measurement. Lieban and Lavicza reported about students' work in using dynamic geometry systems in a geometric modelling situation. They suggested the use of some new manipulative resources together with digital applets that progressively enrich the development of the modelling process by students. Several ideas were debated around the questions:

 Which alternative ways exist to introduce decimal or rational numbers taking into account the introduction of standard units?  How can we produce computer simulations of physical models?  What are the goals for the different people (students, teachers and researchers) involved in creating GeoGebra models?

Assessment of mathematical modelling

This last theme refers to the assessment of mathematical modelling. The paper of Greefrath, Siller and Ludwig analysed the official school leaving examination in Germany allowing university entrance, the so-called Abitur examination, which is supposed to contain elements that examine mathematical modelling. For this purpose, they based their analysis on certain criteria: reference to reality, relevance, authenticity, openness and partial competence of modelling, to analyse the potential of problems included in the official examination. In addition to the difficulties of deciding how to evaluate modelling practices in these kinds of official examinations, several questions were discussed:

 Which criteria can be used to describe good examination questions for modelling?  Can all the sub-competences of modelling be assessed within examination tasks?  How can we design school examination tasks that cover various aspects and goals that include mathematical modelling?

Within this strand, Ärlebäck and Albarracín proposed an analysis of various definitions of Fermi problems from a modelling perspective. They focused on analysing how the definition and descriptions of Fermi problems in the literature align with different perspectives on modelling. They also discussed how far Fermi problems and modelling are connected and how strongly that connection is influenced by the definition of Fermi problems.

Concluding remarks and perspectives

The overarching themes tackled in TWG 6 show the variety of research questions the papers dealt with, for example concerning the theoretical frameworks used or the underlying perspectives on the teaching and learning of mathematical modelling [START_REF] Kaiser | A global survey of international perspectives on modeling in mathematics education[END_REF][START_REF] Kaiser | Report from the working group modeling and applications -Differentiating perspectives and delineating commonalities[END_REF]. Furthermore, the educational levels involved span from primary to tertiary education.

In their analysis of the development of the teaching and learning of mathematical modelling over the last decades, Cai and colleagues (2014) introduce five perspectives, which can be helpful in order to identify the progress made during ERME conferences and potential future development.

The first perspective, the mathematical perspective, describes the differences between modelling at practitioner level and at school level. However, industrial examples can play a powerful role in education, because they are authentic and of varying complexity. There is a long tradition of discussing examples from engineering education in TWG 6 at various ERME conferences. In addition, the relation to other disciplines has always played an important role and needs to be mentioned under this perspective.

The second perspective, the cognitive perspective, focuses on students' cognitive processes when modelling, and cognitive barriers, when students work through the modelling cycle. Cognitively oriented analyses have played a prominent role in many sessions of the modelling group at previous conferences. At CERME 10, this aspect can be found within two thematic strands, that on developing modelling strategies and competences and also that on the analysis of modelling problems. The discussions there broadened our focus on the cognitive perspective and helped us develop further our thinking.

The third perspective, the curricular perspective, refers to the inclusion of mathematical modelling in the curricula. This question was addressed in all meetings at previous conferences and is addressed in nearly all thematic strands of TWG 6 at CERME 10 and highlights the significance and urgency of this theme.

The fourth perspective, the instructional perspective, claims the necessity of high quality modelling education in order to promote effective learning. The question on how to implement effective modelling environments is a hot topic that has persisted for decades and was addressed at CERME 10 within various strands of TWG 6, especially within the themes that addressed experimental materials and technology in modelling and teacher interventions.

As a final and fifth perspective teacher education and teachers' activities in school are addressed, because of the obvious necessity of preparing pre-service teachers for the teaching of mathematical modelling, although the importance of this topic at the various sessions of the Applications and Modelling TWG at previous ERME conferences was less prominent than at this. At CERME 10, teachers and their role in teaching as well as teacher education played a prominent role within the work of the Applications and Modelling TWG. This was integrated into various themes such as modelling strategies, teacher interventions and teachers' beliefs. This shift shows a clear further development of the discussions and the work of TWG 6 and needs to be fostered and broadened. Teachers and their education are the key for the effective and efficient integration of mathematical modelling into mathematical education at various levels.

Proceedings of CERME10