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Pupils' meta-discursive reflection on their cooperation in mathematics: a case study
147-169Views:212This 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
391-404Views:199In 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
379-393Views:202The 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. -
Error analysis in teaching combinatorics: the development of prospective teachers’ confidence and problem-solving skills
103-125Views:76This study investigates the pedagogical potential of error analysis in the teaching of combinatorics within mathematics teacher education. Building on previous research that highlights the role of incorrectly worked sample solutions in cognitive, metacognitive, and affective learning processes, we conducted a mixed-methods study with prospective mathematics teachers at Eötvös Loránd University. Quantitative results from Likert-scale questionnaires (n = 26) indicate that regular analysis of incorrectly worked solutions substantially enhanced participants’ self-confidence, strengthened their problem-solving skills, and positively shaped their attitudes toward future teaching practice. Complementary qualitative data, analyzed through grounded theory, revealed five interrelated categories – self-reflection and confidence, discernment, deeper understanding, methodological surplus, and combinatorial surplus – that together explain the mechanisms through which error analysis supports professional growth. The findings suggest that systematic analysis of conceptual errors not only improves problem-solving competence but also fosters self-confidence, self-reflection, and teaching-related attitudes. By comparing our emergent model of error-analysis thinking with Schoenfeld’s problem-solving framework, we argue that “discernment” constitutes a distinctive and central dimension of error-based learning. The study contributes both theoretically, by refining models of mathematical problem solving, and practically, by offering concrete recommendations for integrating error analysis into mathematics teacher education curricula.
Subject Classification: 97C30, 97K20, 97D40, 97C70, 97C99
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Looking back on Pólya’s teaching of problem solving
207-217Views:637This 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
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Mathematische Bildung im Klagenfurter Doktorand(inn)enkolleg
67-84Views:167In 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. -
Challenges that a teacher-researcher faces during an action research – a case study
89-99Views:336This paper explores the dual role of the teacher-researcher in a four-year action research project focused on problem-based learning in mathematics. It highlights the challenges faced during the phases of planning, implementation, analysis, and reflection. Drawing on insights from the author’s experiences and observations based on both qualitative and quantitative data collection methods, the study identifies distinct challenges linked to the dual role, like differing design goals or subjective-objective voices. The author also proposes solutions to the identified challenges, such as collaboration with university experts and using reflective practices. Furthermore, the research underscores the beneficial impact of action research on enhancing teachers’ awareness and bridging the theory-practice gap, calling for further studies in this area.
Subject Classification: 97D99
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Beweise von Sätzen mit Hilfe der Modelle der hyperbolischen Geometrie
159-167Views:133We 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
1-12Views:173This 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
173-182Views:265The 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
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Why do we complicate the solution of the problem? reflection of Finnish students and teachers on a mathematical summer camp
405-415Views:259This 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.
