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Sensitivity to complexity – an important prerequisite of problem solving mathematics teaching
Thorsten Fritzlar (Germany) 

Teaching is deciding and acting in a complex system. If a teacher attempts to fulfil demands to teach mathematics with a stronger problem solving orientation, it becomes even more complex. This complexity must not be reduced arbitrarily. Instead, a sufficient degree of sensitivity is necessary to competently and flexibly deal with emerging demands on the teacher.     
In this article I provide an introduction to the concept of sensitivity to complexity of mathematics teaching and report on specific realistic and interactive diagnostic instruments. A particular focus is placed on a diagnostic interview about decision-making situations which could occur in a mathematics lesson. A first pilot study with student teachers from different German universities – briefly outlined in the last part of this article – suggests its suitability for gaining important indications of the agent’s degree of sensitivity to complexity of problem solving mathematics teaching..
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Experiments with diagrams – a semiotic approach
Gert Kadunz (Germany) 

A challenging task when doing research in mathematics education is the comprehensible description of activities shown by students and their construction of new knowledge as well when doing mathematics. Charles S. Peirce’s semiotics seems to be a well promising tool for fulfilling this task. Since several years, Peirce’s semiotics is well known and extensively discussed in the scientific community of mathematics education. Among the numerous research reports several papers dealing with Peirce’s semiotics concentrate on the meaning of diagrams as a tool for gaining new knowledge. The aim of the following paper, where a case study will be presented, is to offer the usefulness of such a view on diagrams. In this study two students, which have to solve a problem from elementary geometry, are introduced. The question presented to them asked for a mathematical description of the movement of a rigid body. To answer this question they started experimenting with this rigid body and afterwards invented and used diagrams in manifold ways. Video-based data show these diagrams to be the source of new mathematical knowledge for these students. Therefore, this paper offers Ch. S. Peirce’s semiotics as a successful theoretic frame for describing and interpreting the learning activities of students and their use of diagrams to solve a given mathematical task.
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Teachers’ beliefs on teacher training contents and related characteristics of implementation – the example of introducing the topic study method in mathematics classrooms.
Sebastian Kuntze (Germany)

So-called “bottom-up” strategies for implementation based on mathematics teachers’ own developmental activities are considered to be a powerful approach when encouraging teachers to introduce alternative instructional practices. For evaluational research of in-service teacher training programs using “bottom-up“ implementation strategies, the way how teachers implement contents of the teacher training is at the centre of interest. As the teachers’ active role in the implementation process is necessary, their individual beliefs on the contents of the teacher training and their expectancies might influence the teachers’ implementational activities. These beliefs can be considered as components of professional knowledge and pedagogical content knowledge (Shulman, 1986) in particular.
For this reason, the study focuses on the development of beliefs on contents of a teacher training program throughout the training on the one hand and relationships with characteristics of implementation on the other hand. We consider the example of introducing a student-centred learning environment, the so-called topic study method, in the teachers’ classrooms.
The results indicate that there are interdependencies between beliefs on the teacher training contents and characteristics of implementation.
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An academic experiment on the use of computers in elementary school math classrooms
Silke Ladel (Germany) 

The role of computers in elementary school math classrooms is still being determined. Although computers are promised effective visual tools to promote independent work and study, many educators neglect to use them. Since there are varying points of view, individual teachers generally decide whether to incorporate computers into their methods. Purpose: My experiment analyzes and quantifies the value of computers in elementary school math classrooms. Method: Over a course of 11 weeks, my first grade class worked with the teaching software “Mathematikus 1” (Lorenz, 2000). Using both interpersonal and video observation, I completed written evaluations of each pair of my students’ will and ability to cooperate, communicate and independently solve mathematical problems. Conclusion: My results show that it is generally beneficial to use computers in elementary school math lessons. However, some elements of said software leave room for improvement.
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LeActiveMath’ – a new innovative European eLearning system for calculus contents.
Marianne Moormann, Christian Groß (Germany) 

This contribution gives an overview of the project "LeActiveMath". Within this project a new mathematics learning software has been develop. LeActiveMath is an innovative eLearning system for high school and college or university level classrooms which can also be used in informal contexts for self-learning, since it is adaptive to the learner and his or her learning context in many respects. Topics cover elements of basic knowledge like `linear equations´ as well as more sophisticated contents like `differential calculus´.  This article describes some of the innovative components of the software that are meant to support the students' self-regulated learning. We conclude by reporting on the first evaluations in math classrooms in fall 2005.
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The structure of German mathematics textbooks.
Sebastian Rezat (Germany)

From a socio-cultural perspective it is argued that the modality of artefacts has structuring effects on the activities in which the artefact is involved. The mathematics textbook is an artefact that has a major influence on the activity of learning mathematics. Against this setting, the structures of the units in German mathematics textbooks for different grades and ability levels have been analysed. Firstly, the different structural elements have been examined with regard to: characteristics in terms of content; linguistic characteristics; visual character­istics; their pedagogical functions within the learning process; and situative conditions. Secondly, the orders of the structural elements within the units of the different textbooks have been compared. The findings reveal that the structure of the units is very similar in different mathematics textbooks. The units are not only composed of analogous structural elements, but these elements are also arranged in almost the same sequence. In order to develop a deeper understanding of these findings the structure of the units has been compared to the influential learning theories of J. F. Herbart and H. Roth. On this basis it is argued that the structure of the units seems to reflect the phases of idealised learning processes in general. The issue is raised if this is an appropriate structure in order to provide opportunities to learn mathematics.
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Solving methods of combinatorial geometric problems
Ingrida Veilande (Latvia)

There is considerable experience of organization and management of mathematical contests and interest groups in Latvia. It is necessary to analyse solutions of different mathematical challenges in out-door activities for to develop students’ skills of solving non-standard problems. For this reason collections of thematically related problems with references to applicable methods are useful as valuable manuals for teachers and also as a source of original ideas for the students’ independent work.
Frequently students’ attention is attracted by problems which do not require complicated mathematical formulae. For instance the combinatorial geometric problems deal with systems of geometric objects requiring to estimate dimensional quantities, or to investigate the features of decomposition as well as covering or colouring of geometric figures. Although these problems seem rather simple, various methods are used in their solutions, where general geometric regularities are supplemented with results from different fields of mathematics in combination with general thinking methods such as mathematical induction, the method of invariants and the mean value method. An effective auxiliary in problem solution is the mean value method, which allows making qualitative estimates of given objects by dealing with their quantitative properties.
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Reconstructing basic ideas in geometry – an empirical approach
Andreas Vohns (Germany) 

“Basic ideas” (or “fundamental ideas” etc.) have been discussed in mathematical curriculum theory for about forty years. This paper will centre on the hypothesis that this concept can only be applied successfully by using it as a category for the analysis of concrete mathematical problems. This hypothesis will be illustrated by means of a sample problem from the Austrian Standards for Mathematics Education (“Bildungsstandards”). In this example, basic ideas are used in a content matter analysis which takes students’ solutions to the problem as a starting point for the creation of a potentially substantial learning environment in trigonometry.
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