Book · January 994 citations 110 reads 2,264 authors

Intelligent tutoring systems (pp. 109-142). Hillsdale, NJ: Erlbaum.
Wertheimer, R. J. (1990). The geometry proof tutor: An intelligent computer-based tutor in
the classroom. Mathematics Teacher, 83, 308 - 317.
Wenger, E. (1987). Artificial intelligence and tutoring systems. Los Altos, CA: Morgan
Kaufmannn.
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CHAPTER 5
Psychological research on mathematical learning, thinking, and instruction
has accompanied the rise of didactics of mathematics as a scientific disci-
pline since its very beginnings. In 1910, the German experimental psychol-
ogist David Katz (1913) produced the volume Psychologie und mathematis-
cher Unterricht (Psychology and Mathematical Instruction) commissioned
by the ICME. Obviously, this research project had been initiated by Felix
Klein. Chapters of Katz's book deal with topics like the development of the
concept of space and number.
The interest of mathematics teachers both in the nature of mathematical
thinking, learning, and instruction and the methods psychologists use is also
reflected by the Leipziger Lehrerverein (Leipzig Teacher Association) who
founded and financed the "Institut für experimentelle Pädagogik and
Psychologie" in 1906. One of the main outcomes of this institute is
Freeman’s (1910) volume on children's and adults' conception of numbers.
Note that Freeman's studies used rigorous laboratory and experimental pro-
cedures.
As is well-known, many mathematicians also theorized on mathematics
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),
Didactics of Mathematics as a Scientific Discipline, 225-230.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.
PSYCHOLOGY OF MATHEMATICAL THINKING
edited and introduced
by
Roland W. Scholz
Bielefeld / Zürich

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INTRODUCTION TO CHAPTER 5
as a human activity. Using the method of introspection and referring to his
own experience of creating mathematics, Poincaré (1910, 1914) and many
other mathematicians dealt with psychological questions like insight or
modes of thought in mathematical thinking. Thus, traditionally, one may
find a close relation between epistemology and the theory of mathematical
cognitions.
The work of the psychologist who is most strongly associated with re-
search on mathematical thinking, that is, Jean Piaget, was strongly influ-
enced by Klein and Poincaré. For instance, when dealing with the question
"What exactly is meant by geometrical intuition?" (Piaget, 1948/1963, p.
447), he discusses various definitions of intuition and intuitive thinking
made by mathematicians.
In some respects, research on mathematical thinking attained a new qual-
ity through the constitution of the "International Group of Psychology in
Mathematics Education" during ICME 3 in 1976 at Karlsruhe. Psychology
in Mathematics Education (PME) was predominantely initiated by Ephraim
Fischbein, Richard Skemp, and Hans Freudenthal in order to promote the
exchange of scientific information in the field. Through approaching mathe-
matical thinking from different perspectives, the research work of all three
founders of PME was concerned with understanding qualities of mathemat-
ical thinking.
The object of understanding qualities of mathematical thinking and their
dependance on types of (contextual) framing and representations, is still a
main issue of current reseach in the PME community (see Goldin, 1992;
Vergnaud, 1990). The relation between external and internal representation
is, in some respects, the core linkage that brings cognitive psychology into
mathematics education. Historically, there is a close relationship between
the psychology of thinking and epistemology; thus one will find many cog-
nitive issues being addressed in chapter 8 on history and epistemology of