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tegrative trend.
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These trends differ from each other mainly with respect to the aims asso-
ciated with applied mathematics and mathematical modeling. Representa-
tives of the first trend plead for an emancipatory education. They demand
the use of mathematical methods in realistic situations, where this use serves
to elucidate situations that are really important to the student. This concep-
tion can be illustrated by teaching units such as analyzing unemployment
and the effect of a reduction of weekly working hours, comparing special
train fares for young people, and discussing the effects of speed limits in
cities and on highways. In calculus courses, one can treat problems dealing
with the planning of freeways (e.g., the alignment of crossings) and the eco-
logical implications. This is not only to develop problem-solving qualifica-
tions, but primarily to enhance the students' general political abilities (cf.
Böer & Volk, 1982).
The second trend in the argument aims at developing the central ideas of
mathematics and its epistemology. Students should gain basic epistemologi-
cal and methodological experiences and insights, so that they acquire a
broad and flexible understanding of mathematics (cf. e.g., Steiner, 1976).
Calculus seems to be too complex to meet the requirements for these objec-
tives in school.
The integrative trend demands a balanced relation between utilitarian,
methodological, epistemological, and internal mathematical objectives. This
trend is strongly influenced by the pedagogical aims of mathematics teach-
ing formulated by Winter (see section 2). Blum (1988) illustrates how such
objectives can be reached in applied calculus by analyzing the problem of
constructing functions for income tax as a teaching example.
The natural sciences provide numerous opportunities for teaching applied
calculus. Physics yields a great variety of examples appropriate for teaching
purposes in senior high school. In the 1970s, several applied problems from
biology were developed as teaching units, especially those problems con-
cerning processes of growth. Other important fields for the teaching of ap-
plied calculus are the social sciences and economics (e.g., relations between
cost, profits, prices, supply, and demand; the modeling of markets).
While the textbooks of so-called traditional mathematics contained a
great variety of applied problems and exercises from physics that could be
solved by calculus, and that were actually covered in class, applied prob-
lems were avoided in the textbooks of the new math period. But during the
last 5 years, many examples of mathematical modeling in the fields of eco-
nomics, the social sciences, and biology have been incorporated into calcu-
lus textbooks. Economic problems are especially stressed in special senior
high schools for economics ("Wirtschaftsgymnasium"). The importance of
physics in applied mathematics teaching has faded, since today's students,
especially in basic courses, lack knowledge and interest. Before the school
reform, physics was a compulsory subject in senior high school; now it is
optional and very few students take it, an exception being students in tech-
UWE-PETER TIETZE
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52
nical senior high schools. Another reason lies in the diminished number of
teachers who teach both subjects.
Kaiser-Messmer (1986) investigated the question of whether and to what
extent the general objectives of an application-oriented mathematics teach-
ing can be realized. She carried out extensive case studies on classes ex-
posed to application-oriented calculus teaching. Most students in her sample
improved considerably their ability to understand and cope with everyday
situations; they acquired simple abilities of applying mathematics. But there
were only a few students who gained or improved their general abilities to
cope with mathematical modeling problems. The development of compo-
nent skills was more easily achieved. The students' motivation and attitude
with regard to mathematics improved in nearly all cases.
5. CONCLUSION
New empirical research shows the limits of curriculum development in
principle. The teacher alone determines the effectiveness of curriculum by
his or her decisions, behavior, attitudes, and cognitive processes, no matter
how carefully the curriculum has been developed. The high expectations
educators once had about the benefits of scientifically developed curricula
have been supplanted by a more modest assessment. Recent research has
placed more emphasis on everyday curriculum in the classroom, on teach-
ers' ideas and subjective theories concerning their quotidian preparation of
classes, their subjective learning theories, implicit and explicit objectives,
philosophy of mathematics, and the influence of these cognitions on their
teaching.
6. REFERENCES
Blum, W. (1988). Analysis in der Fachoberschule. In P. Bardy, F. Kath, & H.-J. Zebisch
(Eds.), Umsetzen von Aussagen und Inhalten. Mathematik in der beruflichen Bildung.
Alsbach: Leuchtturm (Technic didact Bd. 3).
Blum, W., & Kirsch, A. (1979). Zur Konzeption des Analysisunterrichts in Grundkursen.