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• Pupils' meta-discursive reflection on their cooperation in mathematics: a case study

  147-169

  Petros Chaviaris

  Sonia Kafoussi

  Francois Kalavassis

  Views:

  33

  This article addresses the issue of how 10–11 year old pupils in pairs can actively get involved in reforming their behavior as they reflect on their interaction in order to solve mathematical problems. We studied the opportunities offered for the development of meta-discursive reflection in a pair of pupils in two alternative environments: (1) pupils' observations and discussions on their video-recorded cooperation and (2) pupils' participation in playing and acting in a drama. The results of the research revealed three levels of the pupils' meta-discursive reflection on their interaction: (1) focusing on the achievement of personal goals, (2) focusing on partners' responsibility and (3) focusing on mutual responsibility. Both environments helped the pupils to improve their socio-mathematical interaction.

• On four-dimensional crystallographic groups

  391-404

  Eszter Horváth

  Views:

  10

  In his paper [12] S. S. Ryshkov gave the group of integral automorphisms of some quadratic forms (according to Dade [6]). These groups can be considered as maximal point groups of some four-dimensional translation lattices in E^4. The maximal reflection group of each point group, its fundamental domain, then the reflection group in the whole symmetry group of the lattice and its fundamental domain will be discussed. This program will be carried out first on group T. G. Maxwell [9] raised the question whether group T was a reflection group. He conjectured that it was not. We proved that he had been right. We shall answer this question for other groups as well. Finally we shall give the location of the considered groups in the tables of monograph [4]. We hope that our elementary method will be useful in studying linear algebra and analytic geometry. Futhermore, 4-dimensional geometry with some visualisation helps in better understanding important concepts in higher-dimensional mathematics, in general.

• CAS as a didactical challenge

  379-393

  Edith Schneider

  Views:

  33

  The paper starts with the discussion of a concept of general mathematics education (mathematics education for everyone). This concept views the focus of teaching mathematics in the reduction of the demands in the field of operative knowledge and skills as well as in an increase of the demands in the fields of basic knowledge and reflection. The consequences of this concept are didactically challenging for the use of Computer Algebra Systems (CAS) in the teaching of mathematics. By reducing the operative work we reduce exactly that field in which the original potential of CAS lies. It is shown that in such maths classes the main focus of CAS is on their use as a pedagogical tool, namely as support for the development of basic knowledge and reflection as well as a model of communication with mathematical experts.

• Looking back on Pólya’s teaching of problem solving

  207-217

  Kaye Stacey

  Views:

  229

  This article is a personal reflection on Pólya's work on problem solving, supported by a re-reading of some of his books and viewing his film Let Us Teach Guessing. Pólya's work has had lasting impact on the goals of school mathematics, especially in establishing solving problems (including non-routine problems) as a major goal and in establishing the elements of how to teach for problem solving. His work demonstrated the importance of choosing rich problems for students to explore, equipping them with some heuristic strategies and metacognitive awareness of the problem solving process, and promoting 'looking back' as a way of learning from the problem solving experience. The ideas are all still influential. What has changed most is the nature of classrooms, with the subsequent appreciation of a supporting yet challenging classroom where students work collaboratively and play an active role in classroom discussion.

  Subject Classification: 97D50, 97A30

• Why do we complicate the solution of the problem? reflection of Finnish students and teachers on a mathematical summer camp

  405-415

  András Ambrus

  István Hortobágyi

  Marjatta Näätänen

  Riitta Liira

  Maija Salmela

  Views:

  32

  This paper deals with reactions and reflections of Finnish secondary school students and teachers on Hungarian mathematics teaching culture. The experiences were collected at a mathematics summer camp in Hungary.

• Mathematische Bildung im Klagenfurter Doktorand(inn)enkolleg

  67-84

  Werner Peschek

  Edith Schneider

  Views:

  30

  In 2003 we set up a programme for PhD-studies ("Doktorand(inn)enkolleg") at the University of Klagenfurt which should promote and support PhD-studies in the field of mathematics education.Within this programme it is worked on the topic "general mathematics education" from different perspectives.
  In the first part of this paper intentions, the fields of work and the form of organisation are briefly demonstrated. The second and main part considers in detail the work in one of the four fields of work, and finally, the third and last part presents some experiences with regard to the contents as well as general ones.

• Beweise von Sätzen mit Hilfe der Modelle der hyperbolischen Geometrie

  159-167

  Jenő Horváth

  Views:

  23

  We give simple proofs for some problems of elemental hyperbolic geometry using the Poincare's half-sphere model. Our method is that a point of a figure is transformed to a special point of the model.

• Reflecting upon reflections

  1-12

  H. Zeitler

  D. Camp

  Views:

  8

  This paper considers many applications of reflections in geometry. It begins with a few motivational problems for the classroom and goes on to consider the formal application to cases involving reflections across one line, two lines and three lines. It wraps up with a summary of results for reflections in higher orders.
  All this stuff was treated in German and American schools too – so the paper is a typical example of German-American didactics.
  "Thinking is one of the greatest pleasure of mankind." – Galileo Galilei

• Consequences of a virtual encounter with George Pólya

  173-182

  John Mason

  Views:

  107

  The consequences of a virtual encounter with George Pólya as a teacher are recorded. An instance of his influence on my mathematical thinking is recounted through work on one of the problems in one of his books.

  Subject Classification: 01A99, 11A05, 97-03, 97D50