Volume 37 (October 2005) Number 5


Zentralblatt für Didaktik der Mathematik

International Reviews on Mathematical  Education

Articles • Electronic-Only Publication • ISSN 1615-679X

(Full texts are available to subscribers only)

“Open learning? Computeralgebra?... No time left for that...“
Bärbel Barzel, Essen (Germany)

Nowadays mathematics teachers have to deal with two challenges concerning their classroom-arrangements: include new teaching methods and integrate computers.  The title expresses the fear of many teachers when following those trends, that realizing both makes curricular prescriptions even more difficult to achieve. In contrast to this other teachers perceive those trends not as an impediment, but as a special opportunity to achieve aims in terms of contents and processes. It was intended to investigate the question whether impediment or opportunity by a research project at the University of Duisburg-Essen. Teaching material was developed to introduce investigating polynomial functions in an open classroom-arrangement integrating CAS.
According to the multi-faceted arrangement a complementary research design was chosen which collects qualitative and quantitative datas. The qualitative part is an interpretive study based on video tapes. The quantitative part is an experimental large-scale study. The material was used in 45 classes (about 1200 students) from different schools in order to check if general conclusions can be drawn. The large-scale study also includes a post-survey and a comparative post-test. To understand the aims of the project it is necessary to grasp the idea of the material. Therefore chapter 1 points out the main ideas of the material, chapter 2 explains the focus of the research project and in chapter 3 you will find first results
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Appropriate Problems for Learning and for Performing – an Issue for Teacher Training
Andreas Büchter, Soest (Germany), Timo Leuders, Freiburg (Germany)

Selecting, modifying or creating appropriate problems for mathematics class has become an activity of increasing importance in the professional development of German mathematics teachers. But rather than asking in general: “What is a good problem?” there should be a stronger emphasis on considering the specific goal of a problem, e.g.: “What are the ingredients that make a problem appropriate for initiating a learning process”  or “What are the characteristics that make a problem appropriate for its use in a central test?” We propose a guiding scheme for teachers that turns out to be especially helpful, since the newly introduced orientation on outcome standards a) leads to a critical predominance of test items and b) expects teachers to design adequate problems for specific learning processes (e.g. problem solving, reasoning and modelling activities).
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Working with tasks for the learning of problem solving in maths teaching as an issue of the first teacher training phase
Regina Bruder, Darmstadt (Germany)

This article describes learning goals of teacher training for the working with tasks in maths lessons. Selected common and different features of tasks intended for the learning and performing are especially referred to.
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Individual ways of dealing with the context of realistic tasks – first steps
towards a typology
Andreas Busse, Hamburg (Germany)

In this paper interim findings of an empirical study on the effects of context are presented. The study focusses on the question how upper secondary students deal individually with contextual aspects. This research project is based on a qualita­tive approach. Triangulation of methods is applied in order to get a broader access to the field. It becomes clear that the con­text is neither an objective nor an invariable feature of the task. Students deal very individually with the context, and it can be an object of change during the solving process. Four types of dealing with the context are gained from the analysis of the empirical data. These types can be embedded in and ex­plained by the concept of sociomathematical norms and the theory of situated learning.
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Connecting arguments to actions – dynamic geometry as means for the attainment of higher van Hiele levels
Thomas Gawlick, Landau, Germany

New technology requires as well as supports the necessity to raise the level of geometric thinking. Freudenthals view of van Hiele’s theory corroborates a dynamic multi-level curriculum that offers material support for higher levels. For levels higher than 2, the dynamic locus capability of Dynamic Geometry software is crucial, e.g. in the study of loci of orthocentres and incentres. Discrepancies between their algebraic and geometric descriptions can motivate a deeper involvement with basic curve theory on the side of the teacher, who thereby can predict in which cases the students may succeed in restructuring the construction to overcome the discordance.
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Motivations and meanings of students’ actions in six classrooms from Germany, Hong Kong and the United States
Eva Jablonka, Berlin (Germany)

This article presents an analysis of about 100 interviews with students from eight-grade classrooms in Berlin, Hong Kong and San Diego that reconstructs student motivations and the meanings they attribute to classroom activities. The data of the six classrooms were produced in the Learner’s Perspective Study (LPS). The LPS is an international collaboration of researchers investigating practices in eighth-grade mathematics classrooms in 13 countries. Although not the central focus of the research, the case study of six classrooms revealed a variety of students’ beliefs and perceptions, which are the focus of this article. These correspond to the possibilities the classroom practices offer. The study also reveals some similarities among student motives and concerns across classrooms. The findings are an important reminder that basing a curriculum upon an alternative vision calls for changing mathematical content as well as the social relations that are established through teaching methods and principles of evaluation.
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The computer as “an exercise and repetition” medium in mathematics lessons: Educational Effectiveness of Tablet PCs
Andreas Kittel, Astrid Beckmann, Volker Hole, Silke Ladel, (Schwäbisch Gmünd) Translated and Edited by Bharath Sriraman (USA) 

The request of a new educational culture within the classroom goes hand in hand with the introduction of the new Educational Standards. That is they are essentially connected with a paradigm shift. The Project supports this aim via different tasks administered through Tablet PC’s within the scope of exercise and repetition phase of learning. The central concern is to find appropriate lesson approaches through computer use in everyday life at school, which are conducive for the math learners and are at the same time easy to effectively implement in other classrooms. In the summer of 2004 the use of Tablet-PCs in school took place in two 9th classes of an Ostalbkreis secondary school in Baden-Württemberg. We report on the effectiveness of this new technology in the classroom.
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Visage – Visualization of Algorithms in Discrete Mathematics
Anne Geschke, Brigitte Lutz-Westphal, Ulrich Kortenkamp, Dirk Materlik, Berlin (Germany)

Which route should the garbage collectors' truck take? This is just a simple question, but also the starting point of an exciting mathematics class. The only “hardware” you need is a city map, given on a sheet of paper or on the computer screen. Then lively discussions will take place in the classroom on how to find an optimal routing for the truck. The aim of this activity is to develop an algorithm that constructs Eulerian tours in graphs and to learn about graphs and their properties. This teaching sequence, and those stemming from discrete mathematics, in particular combinatorial optimization, are ideal for training problem solving skills and modeling – general competencies that, influenced by the German National Standards, are finding their way into curricula. In this article, we investigate how computers can help in providing individual teaching tools for students. Within the Visage project we focus on electronic activities that can enhance explorations with graphs and guide students even if the teacher is not available – without taking the freedom and creativity away from them. The software  ackage is embedded into a standard DGS, and it offers many pre-built and teacher-customizable tools in the area of graph algorithms. Until now, there are no complete didactical concepts for teaching graph algorithms, in particular using new media. We see a huge potential in our methods, and the topic is highly requested on part of the teachers, as it introduces a modern and highly relevant part of mathematics into the curriculum.
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Is the definition of mathematics as used in the PISA Assessment Framework applicable to the HarmoS Project?
Helmut Linneweber-Lammerskitten (Switzerland), Beat Wälti (Switzerland)

The project known as the “Harmonisation of the Obligatory School”, or in its shortened form as “HarmoS”, has a high priority for Switzerland’s educational policy in the coming years. Its purpose is to determine levels of competency, valid throughout Switzerland, for specific areas of study and including the subject of mathematics. The general theoretical basis of the overall HarmoS Project is constituted by the expertise written under the direction of Eckhard Klieme and entitled “Zur Entwicklung nationaler Bildungsstandards” (Klieme 2003) [i.e. "On the Development of National Education Standards"]. The proposal announcing the HarmoS partial project devoted to Mathematics includes references to the results and subsequent analysis of PISA 2003. It thus seems appropriate for us to begin our work on HarmoS with a critical consideration of the definition of mathematics and mathematical literacy as they are used in the PISA Study.  In a first part, we want to describe the core ideas of HarmoS. In a second part, we will address the meaning of general educational goals for the development of competency models and education standards to the extent that it is necessary to properly locate our problem. In a third part we will analyse the concept of mathematics which is at the basis of the PISA Study (OECD 2004) and more precisely defined in the publication “Assessment Framework”.(OECD 2003) In the fourth and last part, we will try to provide a differentiated answer to the question posed in the title of this paper.
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On the way to open standards for education
Fritz Nestle, Ulm (Germany), Nikolaus Nestle, Darmstadt (Germany)

The newest answer of German education authorities to the problems of learning, the so called „Bildungsstandards“, is far away from any suitability. The more than forty years old dream of R. Mager in „Preparing Objectives for Programmed Instruction“ has got a great resonance in Germany too in that time. It was to early. Today we can realize this dream of modern learning environments in the world wide web: E-Testing as the base of successful E-Learning. The „Dortmunder Manifest“ presents the requirements. An adaptive basic program system in HTML/PHP with the respective properties to offer such checks will be given. A new form of evaluation is suggested.
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Picturing Student Beliefs in Statistics
Katrin Rolka, Duisburg (Germany), Michael Bulmer, Brisbane (Australia)

Statistical skills and statistical literacy have emerged as important areas in education. While it has a rich mathematical basis, successful understanding and application of statistics incorporates other types of knowledge. In a similar way, beliefs about statistics can be described using the same framework as beliefs about mathematics, but statistical beliefs bring other aspects as well. This article describes a project for investigating student beliefs in statistics through the creation of pictures of understanding. It presents a classification of statistical concepts and attitudes which are motivated by research in statistical literacy and then shows how these can be refined to be more reliably applied in practice.
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CAS integration into learning environment
Csaba Sárvári, Pécs, (Hungary)

In order to successfully integrate computers into education, it is necessary to organize effectively the prerequisites of human-machine interaction. In the organization of competence-centred education computers could provide valuable assistance for both personal- and group- learning activities. In this paper, we will examine various applications of Computer Algebra Systems (CAS) in classroom settings. The elements of the learning environments are CAS, E-Learning-portal, and Tight VNC remote control system. CAS assisted teaching can become genuinely effective in a complex learning environment if students’ instrumental-genesis evolve into instrumental-orchestration. We will demonstrate the evolution of this process by using one of our developed applications. As an example, we developed, tested, and evaluated our model in the Department of Engineering at the University of Pecs. The study took place during the 2004-2005 academic year with computer science and computer engineering participants.
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Thinking wants to be Organized
Empirical Studies to the Complexity of Mathematical Thinking
Johann Sjuts, Leer (Germany)

Can the describable complexity of test problems concerning mathematical thinking and the empirical results of their dealing with be put into a relation? Can graded test prob­lems be constructed which lead to results which can basically be predicted? Empirical studies give interesting and helpful answers which lead to didactically important consequences, just like the evaluation of the PISA results.
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Balancing Mathematics Education Research and the NCTM Standards
Bharath Sriraman (USA), Michelle Pizzulli (USA)

The release of the Principles and Standards for School Mathematics in the United States by the National Council of Teachers of Mathematics (NCTM) brought to the forefront the debate of whether research should determine the validity of the espoused Standards? Or conversely whether the Standards should influence the research agenda of the mathematics education community? How should university teacher educators address this issue? Should pre-service and practicing teachers blindly accept the Standards as well as the research, or do we cultivate the critical thinking skills that will allow preparing teachers to resolve this dilemma? In this article a university mathematics educator and an idealistic pre-service elementary teacher try to resolve the dilemma of balancing the Standards with research and personal beliefs about the teaching and learning of mathematics.
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High-performing students in the 'Hauptschule' – A comparison of different groups of students in secondary education within Germany
Alexander Wynands, Bonn (Germany), Gerd Möller, Düsseldorf (Germany)

We take a look at mathematical achievement of high-performing students in the Hauptschule, the low track of the German educational system in secondary education. Furthermore, we compare this group with students from other systems in Germany (Gesamtschule, Realschule and Gymnasiums). Our interest is to find out differences and characteristics between the different groups. The results from the national test of PISA 2000 are the empirical basis of our analysis.
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Patterns – a fundamental idea of mathematical thinking and learning
Rose Vogel, Ludwigsburg (Germany)

Taking advantage of patterns is typical of our everyday experience as well as our mathematical thinking and learning. For example a working day or a morning at school displays a certain structure, which can be described in terms of patterns. On the one hand regular structures give us the feeling of permanence and enable us to make predictions. On the other hand they also provide a chance to be creative and to vary common procedures. School students usually encounter patterns in math classes either as number patterns or geometric patterns. There are also patterns that teachers can find in analyzing the errors students make during their calculations (error patterns) as well as patterns that are inherent to mathematical problems. One could even go so far as to say that identifying and describing patterns is elementary for mathematics (cf. Devlin 2003). Practising good interacting with patterns supports not only the active learning of mathematics but also a deeper understanding of the world in general. Patterns can be explored, identified, extended, reproduced, compared, varied, represented, described and created. This paper provides some examples of pattern utilization and detailed analyses thereof. These ideas serve as “hooks” to encourage the good use of patterns to facilitate active learning processes in mathematics.
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