ABSTRACTS
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Theoretical and empirical differentiations of phases in the modelling process
Rita Borromeo Ferri (Germany)

The reconstruction of pupils modelling processes can be found in many empirical studies within the literature on modelling. The empirical differentiations of the phases, which includes putting statements and actions of the pupils in the right phase, has not been reconstructed from a cognitive psychological point of view on a micro level thus far. In this article different modelling cycles are discussed with attention to distinctions in the various phases. The “modelling cycle under cognitive psychological aspects” is specifically emphasized in contrast to the other cycles. On the basis of the results of the COM²-project (Cognitive psychological analysis of modelling processes in mathematics lessons, Borromeo Ferri) the phases of the modelling process are described empirically. Some difficulties in the process of distinguishing the various phases are also pointed out.
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Faces of mathematical modelling
Thomas Lingefjärd (Sweden)

In this paper I will discuss and exemplify my perspectives on how to teach mathematical modelling, as well as discuss quite different faces of mathematical modelling. The field of mathematical modeling is so enormous and vastly outspread and just not possible to comprehend in one single paper, or in one single book, or even in one single book shelf. Nevertheless, I have found that the more I can illuminate some of the various interpretations and perceptions of mathematical modelling which exists in the world around us when introducing and starting a course in mathematical modelling, the more benefit I will have during the course when discussing the need and purpose of mathematical modelling with the students. The fact that only some models fit within the practical teaching and assessing of a course in mathematical modelling, does not exclude the importance to illustrate that the world of today cannot go on without mathematical modelling. Students are nevertheless much more charmed with some models of reality than others.
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What are modelling competencies?
Katja Maaß (Germany)

Modelling and application are seen as a highly important topic for maths lessons. But so far the concept “modelling competencies” has not been described in a comprehensive manner. The aim of this paper is to supplement former descriptions of modelling competencies based on empirical data. An empirical study was carried out which aimed at showing the effects of the integration of modelling tasks into day-to-day math classes. Central questions of this study were – among others: How far do math lessons with focus on modelling enable students to carry out modelling processes on their own? What are modelling competencies? Within the theoretical approach, definitions of modelling processes as a basis for definitions of modelling competencies and important views of modelling competencies are discussed. Based on this theoretical approach the transfer into practice is described. Finally we will look at the results of the study. An analysis of the students' abilities and their mistakes lead to more insight concerning the concept of modelling competencies.
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A Framework for identifying student blockages during transitions in the modelling process
Peter Galbraith (Australia) &Gloria Stillman (Australia) 

In this article we present, illustrate, test and refine a framework developed by Galbraith, Stillman, Brown and Edwards (2006) for identifying student blockages whilst undertaking modelling tasks during transitions in the modelling process. The framework was developed with 14-15 year old students who were engaging in their first experiences of modelling at the secondary level.  .
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Teaching mathematical modelling through project work -experiences from an in-service course for upper secondary teachers
Morten Blomhøj (Denmark) & Tinne Hoff Kjeldsen (Denmark)  

The paper presents and analyses experiences from developing and running an in-service course in project work and mathematical modelling for mathematics teachers in the Danish gymnasium, e.g. upper secondary level, grade 10-12. The course objective is to support the teachers to develop, try out in their own classes, evaluate and report a project based problem oriented course in mathematical modelling. The in-service course runs over one semester and includes three seminars of 3, 1 and 2 days. Experiences show that the course objectives in general are fulfilled and that the course projects are reported in manners suitable for internet publication for colleagues. The reports and the related discussions reveal interesting dilemmas concerning the teaching of mathematical modelling and how to cope with these through “setting the scene” for the students modelling projects and through dialogues supporting and challenging the students during their work. This is illustrated and analysed on the basis of two course projects.    
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Modelling in mathematics classrooms: reflections on past developments and the future
Hugh Burkhardt (USA), with contributions by Henry Pollak (USA) 

This paper describes the development of mathematical modelling as an element in school mathematics curricula and assessments.  After an account of what has been achieved over the last forty years, illustrated by the experiences of two mathematician-modellers who were involved, I discuss the implications for the future – for what remains to be done to enable modelling to make its essential contribution to the "functional mathematics", the mathematical literacy, of future citizens and professionals.  What changes in curriculum are likely to be needed? What do we know about achieving these changes, and what more do we need to know?  What resources will be needed?  How far have they already been developed?  How can mathematics teachers be enabled to handle this challenge which, scandalously, is new to most of them?  These are the overall questions addressed. The lessons from past experience on the challenges of large-scale of implementation of profound changes, such as teaching modelling in school mathematics, are discussed. Though there are major obstacles still to overcome, the situation is encouraging.
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Mathematical modelling as bridge between school and university 
Gabriele Kaiser (Germany) & Björn Schwarz (Germany)

The paper reports on university seminars of mathematical modelling at school, that were held together by the departments of mathematics and mathematics education and some schools in Hamburg. Prospective teachers together with students in upper secondary level carried out modelling examples either in ordinary lessons or special afternoon groups. They tackled authentic problems proposed by applied mathematicians working in industry. In this paper we consider three of these examples in more detail and describe students’ attempts at solving the problem.
After a description of the theoretical framework of modelling in schools and the framework and the structure of the course diverse modelling attempts by students to solve two modelling examples are presented. Finally an evaluation of the series of seminars is presented
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