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Didactics of Mathematics as a Scientific Discipline, 171-175.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.
TECHNOLOGY AND MATHEMATICS TEACHING
edited and introduced
by
Bernard Winkelmann
Bielefeld

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INTRODUCTION TO CHAPTER 4
and learning mathematics, since this has had the most dramatic effect on
discussions on the goals and methods of mathematics education at all levels
in the last decade and will continue to be one focus of didactical research
and development. The short history of the struggle of didactics with
software relevant for mathematics education may be sketched as follows:
Ideas, considerations, reflections, and concrete suggestions for the use of
computers in teaching mathematics depend on the knowledge about and ex-
perience with such instruments shared by mathematical educators and
teachers. Fifteen years ago, these people had access to computers mostly as
programmers in numerically oriented languages. Thus computing power
was mainly used for numerical algorithms, for instance, in the form of short
BASIC programs. Ten years ago, another step – but again in the algorithmic
spririt – was taken with the availability of Logo on various personal
computers. Logo introduced its underlying philosophy of exploring
mathematics in specially designed microworlds and of learning mathematics
by teaching it to the computer; it also included the use of geometry and
symbolic manipulations. The proliferation of so-called standard software on
personal computers in the last decade led to new considerations and
experiments, especially with spreadsheets, programs for data representation,
statistical and numerical packages, databases, CAD (Computer Aided
Design)-software, and computer algebra systems. But such software was at
first not very user-friendly, and became too complex afterwards. The need
for special school adaptations soon became obvious; these ideally allowed
easy specializations, employed mathematical notations similar to those used
at school, and used powerful and helpful metaphors, so that even users with
little training and only occasional practice (as is typical of school users)
could handle them successfully. This led to the creation of general and
didactical software tools that sometimes also had a tutorial component,
thereby integrating some traditions of computer-aided instruction (CAI). All
these forms of using the computer came into being in sequence, but can now
be found simultaneously in discussions about teaching mathematics (cf.
Graf, Fraser, Klingen, Stewart, & Winkelmann, 1992, pp. 57-58).
Those developments impact on the different actions in curriculum devel-
opment, such as discussions on content/process goals, on teaching/learning
styles, and on means of assessing not only specific mathematical/
computational activities such as numerical, graphical, and symbolic compu-
tations but also multiple representations of information (cf. Fey, 1989).
In accordance with the postulated changing demands of a computerized
society (cf. Niss, this volume), increasingly less attention is being given to
those aspects of mathematical work that are readily done by machines,
while increasing emphasis is being placed on the conceptual thinking and
planning required in any tool environment. In addition, students should
know not only which mathematical activities could be given to machines to
solve and which not but also, for example, which kind of preparations and
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answers could be expected by using numerical or symbolic computations
(cf. Graf, Fraser, Klingen, Stewart, & Winkelmann, 1992, p. 58).
There is also a certain shift toward mathematical ideas and applications
of greater complexity than those accessible to most students via traditional
methods, such as system dynamics, data analysis, simulations, and a general
trend toward more experimental mathematics (cf. Cornu & Ralston, 1992).
While these considerations belong to the domain of context/process
goals, the papers in this chapter are generally more concerned with the new
possibilities to enhance the teaching-learning process in mathematics
opened up by computers with modern software. The first three papers throw
a specific light on the issue of preparing mathematics for students (cf.
chapter 1): They describe impacts not only on possibilities and
implementations of mathematical teaching methods but also on the
problems of justification of certain contents. This is most explicit in the
paper by Dreyfus.
The activity of programming is not just present in the use of special pro-
gramming languages but also in the use of most other mathematical soft-
ware. Most Computer Algebra Systems include a programming possibility –
normally on a higher level than general-purpose programming languages.
Other mathematical software environments allow for the relatively uncon-
strained creation or definition of certain objects such as functions,
geometric figures, geometric constructions, simulations of data sets,
calculation and drawing procedures, and sometimes also of transformations
regarding these objects. Such activities are normally subsumed under the
general concept of programming, especially if the algorithmic character of
the activity is evident.
In her paper on the role of programming in mathematical education,