Volume 37 (October 2005) Number 5
Zentralblatt für Didaktik der Mathematik
International Reviews on Mathematical Education
“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.
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).
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.
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.
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.
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.
The computer as
“an exercise and repetition” medium in mathematics lessons: Educational
Effectiveness of Tablet PCs
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
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
Is the definition of mathematics as used in the PISA Assessment Framework applicable to the HarmoS Project?
Helmut Linneweber-Lammerskitten (Switzerland), Beat Wälti (Switzerland)
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.
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
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.
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.
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
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.
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.
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.