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Didactics of Mathematics as a Scientific Discipline, 55-60.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.
edite
d and introduced
by
Rolf Biehler
Bielefeld

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INTRODUCTION TO CHAPTER 2
sented in the other chapters of this book are relevant to teacher education
and to teachers' knowledge.
However, teacher education has its own constraints, and the variation be-
tween and within countries seems to be much larger than in mathematics
education itself. Different systems are in action: The relative function of
university studies in mathematics and in mathematics education, institu-
tionalized training on the job, and in-service education of experienced
teachers varies. The process of giving life to research results and innovative
curricula in everyday classroom practice through communication with
teachers is itself a complex process whose success has often proved to be
fairly limited. That is why the following three topics have become domains
of research and reflection within the didactics of mathematics:
1. teachers' cognitions and behavior;
2. the relation between theory and practice;
3. models and programs of teacher education.
In other words, these three problem domains have shifted from being
merely practical problems to problems at a theoretical level. The four papers
in this chapter discuss all three problem domains from different perspectives
and with different emphases. However, the major concern of all papers is
teachers' knowledge: its structure and its function in teaching practice, de-
scriptive models of teachers' knowledge, normative requirements based on
theoretical analyses, and possibilities and failures to influence and develop
teachers' knowledge.
Teachers' beliefs and teachers' knowledge are increasingly considered as
research topics in didactics of mathematics. Two chapters of the Handbook
of Research on Mathematics Teaching and Learning (Grouws, 1992) are de-
voted to this topic and provide a review of research mainly from a North
American perspective. Hoyles (1992) analyzes how research on teachers has
developed from isolated papers to a new major direction at the international
conferences of the group of Psychology in Mathematics Education (PME).
One of the recent conferences on Theory of Mathematics Education (TME)
organized by Hans-Georg Steiner was devoted to the topic of Bridging the
gap between research on learning and research on teaching (Steiner &
Vermandel, 1988).
Compared with other professions, the special structural problem of the teaching
profession is that it does not have one "basic science" such as law for the lawyer,
medicine for the physician ... scientific theory is related in two utterly different
ways to the practical work of mathematics teachers: first, scientific knowledge
and methods are the subject matter of teaching; second, the conditions and forms
of its transmission must be scientifically founded. (Otte & Reiss, 1979, p. 114-
115)
These two kinds of scientific knowledge have always played different roles
with regard to teacher education for different school levels. Whereas, in
primary teacher education, the mathematical content knowledge was often
56

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regarded as trivial compared to the emphasis on educational knowledge, the
situation for high school teacher education was the reversed. Although this
sharp distinction has become blurred, the different emphases still exist and
can be explained partly by the complexity of the knowledge on the respec-
tive level.
Didactics of mathematics in its relation to teachers can be viewed in two
ways: First, as an endeavor to bridge the gap between theoretical knowledge
(mathematics, educational theories, psychology, etc.) and the practice of
mathematics teaching. However, second, didactics of mathematics as a dis-
cipline sometimes regards itself as the "basic science" for the mathematics
teaching profession. In this sense, didactics of mathematics itself creates a
theory-practice problem insofar as it has developed scholarly knowledge of
its
own.
Teachers' knowledge related to mathematics is crucial. The question what
kind of knowledge, experience, and understanding of mathematics a mathe-
matics teacher should have has turned into a research question for the di-
dactics of mathematics. A symposium of ICMI at the ICM in Helsinki, 1978
(Steiner, 1979), offered a perspective on this topic based on the assumption
that mathematics has to be interpreted within its larger cultural role and in
relation to other subjects, and not only as an academic subject. For primary
teacher education, Wittmann (1989) argued for a type of course on elemen-
tary mathematics that should have a quite different character than usual aca-
demic mathematics courses, for instance, it should be rich in relationships to
history, culture, and the real world; it should be organized in a problem and
process-oriented way; it should involve a variety of representations (con-
crete materials, diagrams, symbolic language, etc.); and it should allow for a
variety of teaching/learning formats. Dörfler and McLone (1986) provide a
differentiated analysis on relations between academic mathematics, school
mathematics, and applied mathematics with regard to the knowledge teach-
ers should have about the different characteristics and natures of mathemat-
ics, (see, also, Niss, this volume).