Book · January 994 citations 110 reads 2,264 authors

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INTRODUCTION TO CHAPTER 2
60
a shared situation in the sense of Steinbring that may foster the dialogue be-
tween theory and practice and develop the teacher in the direction of an in-
tellectual leader rather than the determiner of mathematical truth.
The papers in this chapter elaborate the complex demands on teachers
spanning from the teacher's role of being a representative of the mathemati-
cal culture outside school to being a confident manager of classroom inter-
action. In doing this, the papers have analyzed the teacher's role as a subsys-
tem of the complex system of mathematics education, which is elaborated in
the other chapters of this book.

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REFLECTIONS ON MATHEMATICAL CONCEPTS AS
STARTING POINTS FOR DIDACTICAL THINKING
1.1 Mathematics Didactics in Teacher Education for Gymnasium
In Germany, the Gymnasium comprises Grades 5 to 13 and is oriented
toward preparing students for university studies. Nowadays, about 20% to
40% of an age group attend the Gymnasium. Students preparing to teach
mathematics at the Gymnasium (see Weidig, 1992) traditionally have to
master a complete university education in mathematics. This means that
they are introduced to calculus, linear algebra, analytical geometry, theory
of functions, algebra, number theory, differential geometry, differential
equations, probability and statistics, numerical mathematics, and so forth.
This mathematics is far beyond the elementary mathematics they will have
to teach as future teachers. But the idea of this type of education is that
teachers can only present elementary mathematics at the Gymnasium in a
valid manner if they are familiar with the higher mathematics behind it.
Elementary Mathematics from an Advanced Standpoint by F. Klein (1968)
made this notion explicit: A mathematics education of this type should
make the future teachers think mathematically.
But F. Klein also saw the need for lectures about the didactics of mathe-
matics in teacher education to help student teachers to think didactically.
This was supported by other university mathematicians such as A.
Pringsheim. As a result, lectures in didactics of mathematics were offered at
some universities (Griesel & Steiner, 1992). This development was contin-
ued in the 1960s by mathematicians such as H. Behnke, H. Kunle, D.
Laugwitz, and G. Pickert, who invited experienced teachers to offer lectures
in didactics of mathematics. It turned out that these lectures stimulated re-
search in didactics of mathematics, and that the growing didactical research
helped to improve these lectures. Very typical were H.-G. Steiner's lectures
at Münster. His lecture on the foundations of geometry from a didactical
point of view was published in 1966 (Steiner, 1966a). During the following
decades, didactical theories for most of the mathematical subject areas of
the Gymnasium in Germany were developed, for example, algebra
Hans-Joachim Vollrath
Würzburg
1. INTRODUCTION
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),