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Hans-Joachim Vollrath describes a course in pre-service teacher educa-
tion for high school teachers in mathematics that should enable teacher stu-
dents to reanalyze, restructure, and evaluate the academic mathematical
knowledge they have already learned from a didactical point of view.
Reflections on mathematical concepts as starting points for didactical
thinking are taken as a focus, because problems of mathematical concept
definition and meaning can be related to psychological aspects of concept
learning, principles of teaching concepts, and the historical development of
concepts. The examples are taken from calculus, a field of advanced math-
ematical thinking that recently has received more attention from researchers
in mathematics education (Tall, 1991). Vollrath discusses the possibilities
and needs for integrating historical and epistemological aspects of mathe-
matics (see chapter 7, this volume) in teacher education. His contribution
ROLF BIEHLER
57

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relates to a tradition in German didactical thinking of trying to get teachers
to reflect on the relation between school mathematics and university math-
ematics in order to enable them to make conscious choices instead of simply
reproducing either of them. In other words, teachers should be enabled to
reflect on, understand, and actively shape the process "of preparing mathe-
matics for students" as it is also analyzed in chapter 1 of this volume.
Vollrath's paper can be read an as example of how teachers' knowledge
related to mathematics should be extended, enriched, and transformed from
a didactical point of view, even if teachers have had a high-quality academic
mathematics education. Complementary to this normative conception is a
descriptive-empirical orientation toward the question how the knowledge of
practicing teachers can be modeled and whether and how their knowledge
does affect their classroom behavior. Empirical research on this question
should, at least in the long run, inform teacher educators with reliable
knowledge on how to overcome mere intuitive priorities and content selec-
tion in their courses.
Rainer Bromme develops a psychological topology of teachers' profes-
sional knowledge that distinguishes between several kinds of knowledge re-
lated to mathematics, namely, mathematical content knowledge, school
mathematical knowledge, philosophy of school mathematics, and subject-
matter-specific pedagogical knowledge. On the basis of this model, he re-
views and reinterprets empirical research concerned with identifying and
analyzing the function of teachers' knowledge and beliefs for teaching prac-
tice. The paper is situated in an increasingly important research tradition
concerned with modeling teachers' knowledge and beliefs (Fennema &
Franke, 1992; Thompson, 1992). Instead of the notion of teachers' beliefs,
the author prefers the notion of philosophy of school mathematics, similar to
Ernest (this volume) who theoretically extends this topic. By this, the in-
terindividual aspects of this knowledge and its interwovenness with subject-
matter aspects as compared to mere subjective belief systems should be
stressed. Bromme reinterprets research results that have found deficiencies
concerning teachers' knowledge about individual students' understanding
and concerning the subject-matter-specific pedagogical knowledge of teach-
ers, showing that, nonetheless, teachers' have shown practical competence to
cope with the demands of the classroom that indicates the richness in intu-
itive knowledge that teachers have developed during their professional life.
Bromme's approach of considering teachers as experts from the perspec-
tive of an educational psychologist establishes a certain tension to those re-
flections in didactics of mathematics that criticize teachers too easily but do
not take sufficient account of their concrete working conditions, the limits to
rationality in everyday acting.