ABSTRACTS
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Enhancing the image of mathematics by association with simple pleasures from real
world contexts
Robyn Pierce (Australia) and Kaye Stacey (Australia)

Those who market people or products choose their images very carefully.  They create positive associations in the public’s mind by photographing their clients with sporting heroes or national icons.  In this paper we present a variety of evidence to show that a major and overlooked reason for teachers’ use and choice of real world problems is to take advantage of this ‘halo effect’ to improve students’ attitude towards learning mathematics.  Analysis of interviews, reports, and results of a brief survey from teachers of middle secondary school classes indicate that they place a very high priority on positive attitudes and hence both choose and enhance real world problems to promote students’ affective engagement through simple pleasures.  Pleasant sensory stimuli, generally non-cognitive and peripheral to the situation to be modelled, are used to promote a positive view of mathematics. This is a good strategy for creating enjoyable and memorable lessons, but there is a danger that it may override more substantive learning goals.
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Mathematical modelling as a tool for the connection of school mathematics
Fco. Javier García (Spain), Josep Gascón (Spain), Luisa Ruiz Higueras (Spain)
Marianna Bosch(Spain)

We start introducing some aspects of the theoretical framework: the Anthropological Theory of Didactics (ATD). Then, we consider on the research domain commonly known as “modelling and applications” and briefly describe its evolution using the ATD as an analytical tool. We propose a reformulation of the modelling processes from the point of view of the ATD, which is useful to identify new educational phenomena and to propose and tackle new research problems. Finally, we focus on the problem of the connection of school mathematics. The reformulation of the modelling processes emerges as a didactic tool to tackle this research problem. We work on the problem of the articulation of the study of functional relationships in Secondary Education and present a teaching proposal designed to reduce the disconnection in the study of functional relationships in Spanish Secondary Education.
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Modeling conceptions revisited
Bharath Sriraman (USA) and Richard Lesh (USA)  

The previous issue of ZDM raised several fundamental issues on the role of modeling in the school curricula at micro and macro levels In this paper we complement the approaches described there by discussing some of the issues and the barriers to the implementation of mathematical modeling in school curricula raised there  from the perspective of the on going work of the models and modeling research group. In doing so we stress the need for critical literacy as well as the need to initiate a new research agenda based on the fact that we are now living in a fundamentally different world in which reality is characterized by complex systems. This may very well require us to go beyond conventional notions of modeling.
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Teachers’ ways of listening and responding to students’ emerging mathematical models
Helen M. Doerr (USA)

In this paper, I present the results of case study of practice of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth.  The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students’ thinking in their classroom practice.  Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students.  The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models.  Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth.  Implications for the knowledge needed to teach mathematics through modeling are discussed.
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Functions: a modelling tool in mathematics and science
Claus Michelsen (Denmark) 

It is difficult for the students to transfer concepts, ideas and procedures learned in mathematics to a new and unanticipated situation in science. An alternative to this traditional transfer method stresses the importance of modelling activities in an interdisciplinary context between mathematics and science. In the paper we introduce a modelling approach to the concept of function in upper secondary school is introduced. We discuss pedagogical and didactical issues concerning the interplay between mathematics and science. The discussion is crystallized into a didactical model for interdisciplinary instruction in mathematics and science. The model is considered as an iterative movement with two phases: (1) the horizontal linking of the subjects:  Situations from science are embedded in contexts which are mathematized by the students, (2) the vertical structuring in the subjects: The conceptual anchoring of the students’ constructs from the horizontal linking in the systematic and framework of mathematics and science respectively.
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Simple thinking using complex math vs. complex thinking using simple math
Steffen M. Iversen (Denmark) and Christine J. Larson (USA) 

Traditional mathematics assessments often fail to identify students who can powerfully and effectively apply mathematics to real-world problems, and many students who excel on traditional assessments often struggle to implement their mathematical knowledge in real-world settings (Lesh & Sriraman, 2005a).  This study employs multi-tier design-based research methodologies to explore this phenomenon from a models and modeling perspective.  At the researcher level, a Model Eliciting Activity (MEA) was developed as a means to measure student performance on a complex real-world task.  Student performance data on this activity and on traditional pre- and post-tests were collected from approximately 200 students enrolled in a second semester calculus course in the Science and Engineering department of the University of Southern Denmark during the winter of 2005.  The researchers then used the student solutions to the MEA to develop tools for capturing and assessing the strengths and weaknesses of the mathematical models present in these solutions.  Performance on the MEA, pre- and post-test were then analyzed both quantitatively and qualitatively to identify trends in the subgroups corresponding to those described by Lesh and Sriraman.
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Mathematical modelling in classroom: a critical and discursive perspective
Jonei Cerqueira Barbosa (Brazil) 

In this paper, I outline a socio-critical perspective of modelling in mathematics education and discuss implications for the analysis of students’ activities at the micro level. In particular, a discursive perspective is presented with contributions from discursive psychology. Recent studies and classroom examples are taken into consideration.
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A global survey of international perspectives on modelling in mathematics education
Gabriele Kaiser (Germany), Bharath Sriraman (USA)

In this article we survey the current debate on modelling and describe different perspectives on this debate. We relate these perspectives with earlier perspectives and show similarities and differences between these different approaches.
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