ABSTRACTS
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Materialization and organization. Towards a cultural anthropology of mathematics

Roland Fischer, University of Klagenfurt in Vienna (Austria)

Abstract: This summary of six articles which have been written in the past fifteen years focus on the question of the social relevance of mathematics on a principal level. The main theses are: Mathematics provides materializa­tions of abstract issues, thereby it supports mass communication. The principles of mathematics are basic for our social organization. The limits of mathematics are limits of organization. But they can be overcome by emphasizing the reflexive potential of mathematics.
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Reflections as a challenge
Ole Skovsmose, Aalborg (Denmark)

Abstract: Reflections on mathematics-based actions and practices bring an ethical dimension to the notion of reflection, and this is the aspect I consider and develop in this essay.
I elaborate on the notion of reflection by addressing eight different issues. (1) The necessity of reflection emerges from the observation that mathematics-based actions do not have any intrinsic link to progress by virtue of being mathe­matics-based. Such actions can be as complex and as questionable as any other actions. (2) Although reflections, from this perspective, are believed to be necessary, one could cite a functionality of non-reflection. For example, non-reflection enables the school mathematics tradition to continue to ensure that the future labour force has particular competencies in the right measures to match the social order for which they are destined. (3) Reflections often presuppose speci­ficity, as they include general as well as specific reconsiderations with respect to some knowledge, actions and practices. (4) I use collectivity of reflections to refer to the observation that ethical considerations can be facilitated through inter­action and communication. Often this pre­supposes that challenging questions be formu­lated in order to open up the ethical dimension with respect to mathematics in action. (5) Reflections presuppose directedness and involve­ment, and this brings me to analyse the inten­tionality of reflections. (6) Reflections can address very many different issues, which leads me to recognise the diversity of reflections. (7) It is easy to ignore or to obstruct reflections, and when reflections emerge, they can easily be elimi­nated from an educational context. We should never ignore the fragility of reflections. (8) This brings me to recognise the uncertainty of reflection. Reflections cannot rely on any solid foundation. Still, I find that reflections are necessary.
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Social reflection in mathematics classes: Cooperation or denial
Franz Picher, Neunkirchen (Austria)

Abstract: Is it possible to reflect reasonably with pupils on social behaviour by means of mathe­matics? Which importance can this subject have for those who learn as well as for society? Within an educational project, and with the help of games like the Prisoners’ Dilemma and texts, situations were discussed in which cooperation of all parties involved would show an optimal result but which also have a great appeal to denial for each party involved. Mathematics can help to distance oneself from consternation and also creates the possibility of abstraction and helps to precise possibilities of reflection. After the end of this project the first question can be answered with “yes”.
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Reflected acting in mathematical learning processes
Katja Lengnink, Technical University of Darmstadt (Germany)

Abstract: Acting and thinking are strongly interconnected activities. This paper proposes an approach to mathematical concepts from the angle of hands-on acting. In the process of learning, special emphasis is put on the reflection of the own actions, enabling learners to act consciously. An illustration is presented in the area number representation and extensions of number fields. Using didactical materials, processes of mathematical acting are stimulated and reflected. Mathematical concepts are jointly developed with the learners, trying to address shortcomings from own experiences. This is accompanied by reflection processes that make conscious to learners the rationale of mathematical approaches and the creation of mathematical concepts. Teaching mathematics following this approach does intent to contribute to the development of decision-making and responsibility capabilities of learners.
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Reflection and metacognition in mathematics education – tools for the improvement of teaching quality
Christa Kaune, University of Osnabrück (Germany)

Abstract: On the basis of a category system that classifies metacognitive activities, the first part of this paper shows to what extent reflection can be understood as one of several metacognitive activities. It is then demonstrated that it proved to be useful to consider different nuances of reflection.
Illustrated by examples taken from math classes on grammar school level, the second part of the essay shows what assignments look like that cause pupils to reflect, and how pupils face up to the demands to reflect on different matters in mathematics education. 
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