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Michèle Artigue illustrates the concept of didactical engineering and its
theoretical background. This systemic approach is connected to theoretical
ideas prevalent in the French didactics of mathematics but also introduces
many "engineering" elements. These are decisionist and practical elements
that are based on scientific research and theories but necessarily have to
extend to more complex, concrete objects than the simplified objects of the
theories. The author describes the concrete studies and developments a
curriculum reformer has to undertake in order to cope constructively with a
specific perceived teaching problem; her concrete case is the inadequateness
of a traditional part of university mathematics teaching (differential
equations) due to modern developments in mathematics, sciences, technol-
ogy, and society. She clearly and explicitly elaborates the tension between
the theoretical ideals of the researcher, whose teaching aims at researchable
results in strictly controlling as many variables as possible, and the practical
needs of the constructive developer, whose measure of success is a sound,
accepted, and adaptable teaching sequence. The systemic approach consists
in a careful analysis of the teaching situation to be acted upon, of the epis-
temological, cognitive, and didactical obstacles against change, and of the
possibilities for global (macrodidactic) and local (microdidactic) choices.
The complexity of the object requires repeated application of the design -
experimental teaching - redesign cycle on increasingly higher levels, and
also consideration of the obstacles when the product of the engineering is to
be distributed – obstacles not only in the students but also in the teachers
who tend to adapt new ideas to their old teaching styles and thereby to
destroy them.
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BERNARD WINKELMANN
In the course of reforming mathematics teaching in connection with the
new-math movement, the question of justification became very virulent; it
had to be dealt with in a scientific debate that, to a certain extent, was inde-
pendent from the question of realization in practical mathematical teaching.
This is the theme of Uwe-Peter Tietze's paper. He describes the historic de-
velopment in the efforts of the community of mathematical educators in
Western Germany and Austria to cope with the problem of defining and
justifying mathematical curricula and the underlying goals. How can we
decide which part of mathematics, which insights, applications, and meth-
ods of mathematics are worth being taught and learned? The author explains
the logical difficulties of argumentations about normative aspects. In a tour
de force on the German didactical discussion about the problems of elemen-
tarization and justification, he describes and criticizes many constructive
concepts dealing with the problem, such as the formulation of didactic prin-
ciples, the development of general objectives, the efforts to identify funda-
mental ideas in mathematics as a whole or in specific domains, the idea of
exactifying as teaching goal and teaching process, and the role of applica-
tions in justifying goals of mathematics teaching. (The historical introduc-
tion to his section on applications should be compared to the more detailed
account in Jahnke's article, this volume.) The survey is very condensed and
rich in content, arguments, criticisms, and even constructive examples,
mostly taken from the debate on calculus teaching in German upper sec-
ondary schools (Gymnasium).
All three authors mark in different ways the tension exerted on curricu-
lum designers between the practical question "what can be taught and what
can be done to make it happen?" and the connected but somehow indepen-
dent theoretical question "what should be taught, and why, how, to whom?"
It is the tension between the ideal of knowing and taking into account the
real possibilities and constraints as described in other chapters of this book,
and the necessity to develop argumentations and theories of an applied sci-
entific or engineering character in order to prepare for the necessary deci-
sions in domains that are only partly known.
REFERENCES
Chevallard, Y. (1992). A theoretical approach to curricula. Journal für Mathematikdidaktik,
13(2/3), 215-230.
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ECLECTIC APPROACHES TO ELEMENTARIZATION:
CASES OF CURRICULUM CONSTRUCTION IN THE
UNITED STATES
James T. Fey
Maryland
1. INTRODUCTION
Translation of mathematical concepts, principles, techniques, and reasoning
methods from the forms in which they are discovered and verified to forms
that can be learned readily by a broad audience of students involves at least
two fundamental tasks: (a) choosing the mathematical ideas that are most
important for young people to learn, and (b) finding ways to embed those
ideas in learning experiences that are engaging and effective.
At first glance, it would seem that, for a highly structured discipline like
mathematics, design of curricula and instructional strategies would be
straightforward tasks that are dealt with routinely by experts in mathematics
and its teaching. But American school mathematics programs are developed
in a complex and loosely structured process involving a wide variety of
people with different values, expertise, interests, and experiences. While
there are mathematics educators and educational policymakers who attempt
to guide curriculum development and implementation through application of
thoughtful content analyses and coherent research-based theories of learning
and teaching, it seems fair to say that American school mathematics is actu-
ally the result of compromises that emerge from informal competition
among many opinions. Furthermore, the competing opinions are usually
formed by intuitive reflection on personal experiences with mathematics
and teaching, not by systematic didactical analysis.
Over the past decade, curriculum advisory reports for American mathe-
matics education have been offered from groups representing classroom
teachers (NCTM, 1989, 1991), research mathematicians (Pollak, 1982;
Steen, 1990), scientists and science educators (AAAS, 1989), educational
psychologists (Linn, 1986), and political groups without any special exper-
tise in education (Bush, 1991). Those recommendations, and the changes in
school mathematics programs to which they have led, have been widely de-
bated in a variety of professional and public political forums. Analysis of
this lively but eclectic process shows something of the effects of curriculum
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),