Book · January 994 citations 110 reads 2,264 authors

Parts of this contribution are based on Bromme, R. (1992). Der Lehrer als
Experte. Zur Psychologie des professionellen Wissens. Bern: Huber.
88

---

DIALOGUE BETWEEN THEORY AND PRACTICE IN
MATHEMATICS EDUCATION
Heinz Steinbring
Bielefeld
1. NEW PERSPECTIVES ON THE RELATION BETWEEN THEORY
AND PRACTICE
Traditionally, the central task of mathematics education has been to con-
tribute in a more or less direct manner to improving the practice of teaching
mathematics and to solve teaching problems. Accordingly, the didactics of
mathematics is mainly conceived of as an auxiliary science, which has to
transform the scientific mathematical knowledge into a suitable form of
knowledge for teachers and students and which has to provide well-tested
methodological procedures to teach this knowledge effectively. Mathe-
matics education often is taken as a methodology for elementarizing,
simplifying, and adapting scientific subject matter to the abilities of stu-
dents.
Additionally, the role of the referential sciences, such as pedagogics, psy-
chology, or the social sciences, is mostly understood as a further support for
this central task of didactics: to improve everyday teaching practice. In par-
ticular, these sciences should help solve those educational, psychological,
and social problems that go beyond the actual field of teaching
mathematics.
Also with regard to the mathematics teacher and his or her pre- and in-
service training, the didactics of mathematics primarily has the role of a ser-
vant: Didactics should prepare teacher students methodically for their future
teaching practice and endow them with useful teaching strategies. And, in
in-service seminars, experienced teachers expect more or less direct support
for their everyday teaching practice from confirmed research results and re-
liable teaching materials.
Such an expectation toward didactics of mathematics seems to be domi-
nant in the beliefs of many mathematics teachers and researchers: Useful re-
search in mathematics education is characterised by a straightforward appli-
cability of research findings to the problems of teaching practice. This ought
to bring about direct improvements of practice. But, contrary to this
widespread opinion about didactics of mathematics, there is agreement that
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),
Didactics of Mathematics as a Scientific Discipline, 89-102.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.

---

THEORY-PRACTICE DIALOGUE
most teachers simply do not refer to research findings at all and do not use
them in their professional activity. "... if teachers needed information to
solve a problem, it is unlikely that they would search the research literature
or ask the researcher to find an answer" (Romberg, 1985a, p. 2).
Are the results of didactical research much too far removed from the ac-
tual problems of teaching practice? Is it necessary to adjust scientific results
even more strongly to the conditions of teaching practice? Or are teachers,
for different reasons, unable to make professional use of research findings
in their teaching profession (Romberg, 1985b, 1988)? Or is it even
impossible to meet these implicit expectations addressed by practitioners to
didactical theory and, vice versa, the expectations of educators addressed to
practitioners, because they are unfounded and must be reconsidered? Could
it be that scientific results cannot be applied to teaching practice in a direct
and immediate way, on principle, but that the application of theory to prac-
tice is always very complex and depends on many premises (Kilpatrick,
1981)?
The dominant structure that is believed to control the relation between
theory and practice could be described as a linear follow-up: Theory fur-
nishes results that gain direct access to practice, improving and developing
it. This linear pattern is not just found between didactical research and the
practice of teaching; the relation between teacher and student in teach-
ing/learning-processes is often interpreted as a linear connection, too: The
teacher is the conveyor of the mathematical knowledge that he or she must
prepare methodically and then hand over to the students in order to extend
their comprehension and insights into mathematics.
This view is based on an interpretation of mathematical knowledge, as criti-
cized by, for example, D. Wheeler (1985):
In this model, the subject matter to be taught is already determined in content and
form, the teacher knows this subject matter and passes it on, "as it is," to the stu-
dents, and the students rehearse it until they can show they know it as well as, or
nearly as well, as their teacher. What place can there possibly be for research if
this is the state of affairs? (p. 10)
According to this model, research, at best, has to determine content and
form of new mathematical subject matter for mathematics teaching.
This comparative analogy of the relation between research and practice of
teaching to the relation between teacher and student seems to be helpful for
many reasons. The assumed interpretation of the organizational structure of
one of these relations implies a similar conception of the other relation (cf.
90

---

HEINZ STEINBRING
91
AG Mathematiklehrerbildung, 1981, p. 205; Rouchier & Steinbring, 1988).
A linear model of the connection between theory and practice often is based
on a similar linear model of the teaching/learning process of mathematics.
Many research studies have criticized the perspective of the teacher as the
conveyor of mathematical knowledge and the student as the receiver
(Cooney, 1988; Mason, 1987). The teacher is viewed as providing learning
situations in which students have to contribute their own potential for ac-
tively reconstructing knowledge, for establishing a personal relationship
toward this knowledge.
The central perspective on the relation between theory and practice in the
following is the forms of cooperation between didactical research and the
mathematics teachers who already possess some professional experience;
that is, an in-service training perspective and not university training. The
reality of everyday teaching cannot be influenced in a direct way by didacti-
cal research, nor is it arbitrarily changeable and restructurable. In the
framework of its socioinstitutional conditions and with regard to the specific
epistemology of school mathematics, teaching practice is relatively au-
tonomous of external influences; indeed, it has produced very effective
provisions for maintaining this autonomy. Changing interventions into this
complex practice have to reflect more carefully the hidden preconditions
and mechanisms that are relevant in teaching practice.
This leads to consequences for both parts of the theory-practice relation:
Didactical science has no direct possibility of controlling the everyday prac-
tice of the mathematics teacher, and the teacher has no straightforward pos-
sibility of controlling the students' process of either learning or comprehen-
sion. The partners participating in this process of mediation (necessarily) act
relatively autonomously within the framework of the socioinstitutional
conditions, a fact due to the difficult epistemological character of the
knowledge under discussion, which can ultimately only be understood by
means of personal reconstructions.

---

This requires a modified interpretation of the role and perspective of didac-
tical theory in relation to practice. This could be expressed in the model
shown in Figure 2.
This model tries to display the new fundamental paradigm shift in the
theory-practice relation: There are no direct influences or hierarchical de-
pendencies, but exchange and feedback between two relatively independent
social domains of reflecting upon and mediating mathematical knowledge.
Only such a structure could enhance a real dialogue: between teacher and
students and between theory and practice, with all its ways of sharing,
jointly observing, reflecting, and discussing, and its modes of communica-
tion that enable positive feedback that supports the subjective construction
of mathematical meaning by means of integrating the fruitful ideas of dif-
ferent partners. The realization of such a dialogue can probably be estab-
lished between researchers and teachers more easily if the teacher is not
subjected to a "didactical contract" with the researcher. A dialogue between
teachers and students under the usual conditions of the didactical contract is
more difficult to establish. This model of cooperation between theory and
practice must take into account the following three dimensions:
1. Knowledge (in very general terms about mathematics in teach-
ing/learning situations): the relation between theoretical/scientific knowl-
edge and practical/useful knowledge.
2. The professional practice and social role of persons involved in the
theory-practice relationship, and the education of teachers.
3. Forms and models of cooperation between theory and practice in
mathematics education.
Obviously, it is necessary for these three dimensions to overlap, but this
analytic separation helps to get an adequate idea of the complex factors in-
volved in the theory-practice relation. For 10 years, the international re-
search project "Systematic Cooperation Between Theory and Practice in
Mathematics Education (SCTP)" has been analyzing the problem of relating
theory to practice from a broad perspective. A main basis has been a
number of case studies from different countries reporting on diverse
projects trying to improve the relation between didactical research and
mathematics teaching practice (see Christiansen, 1985; Seeger &
Steinbring, 1992a; Verstappen, 1988). Despite their examplary character,
these cases in principle cover all the three dimensions developed here; some
of the research papers reported below might be taken as an example of
emphasis on some important aspect of the 3-dimensional network.
1. Knowledge. This is a complex dimension, because it not only contains
the mathematical knowledge (the subject matter) to be learned by students
or by teachers; it also refers to the related scientific and practical knowledge
domains necessary to improve teachers' professional standards (epis-
temology, history of mathematics, psychology, pedagogics, curricular ques-
tions, etc.) and it has to deal with the difficult problems of mathematical
THEORY-PRACTICE DIALOGUE
92

---

meaning and understanding (at the university and at school; cf. Bazzini,
1991; Ernest, 1992; Seeger & Steinbring, 1992b; Wittmann, 1989).
2. Professional practice and social role. This relates to the social framing
factors influencing and supporting endeavors to mediate knowledge, be they
in the classroom or in cooperation between researchers and teachers. The
indirect ways of relating theory to practice require forms of social participa-
tion and sharing common experiences that belong to different professional
practices and communicative situations (cf. Andelfinger, 1992; Brown &
Cooney, 1991; Mason, 1992; Voigt, 1991; Wittmann, 1991).
3. Forms and models of cooperation. Cooperative efforts to implement
this changed intention often take the form of case studies and applied pro-
jects, implicitly or explicity using attributes to describe the role of the part-
ners involved and the status of the mathematical knowledge. Such practical
case studies necessarily have their own "history," but a fruitful connection
between the complex knowledge involved and the social embedment of co-
operation between theory and practice can be organized only in concrete
frameworks that then have to be investigated for general and universal in-
sights. (cf. Bartolini Bussi, 1992; Bell, 1992; Burton, 1991; von Harten &
Steinbring, 1991; Verstappen, 1991).
A major fundamental insight discussed and explored in the SCTP group
is to more thoughtfully analyze the conditions of the "dialogical structure"
of communication, cooperation, and materials (textbooks, reports, research
papers) in the relation between theory and practice. Unlike a hierarchically
structured conveyance of "context-free," absolute knowledge, a dialogical
structure aims to be particularly aware of the specific contexts and condi-
tions of application and interpretation for the mediated knowledge in which
the partner of cooperation is involved. Scientific knowledge for mathemat-
ics teachers essentially has to refer to the circumstances of everyday teach-
ing practice. A consequence is that neither a separate change of research nor
of practice could improve cooperation, but that the relation between theory
and practice has itself become a problem of research.
2. THE THEORETICAL NATURE OF MATHEMATICAL
KNOWLEDGE: COMMUNICATING KNOWLEDGE
AND CONSTRUCTING MEANING
In the framework of the range of important topics in the theory-practice
relation, I shall concentrate on certain aspects of the mathematical
knowledge negotiated and mediated in this relationship. The theoretical
perspective will not be curricular, historical, or mathematical, but an attempt
to use the epistemological basis of mathematics. If it is accepted that
epistemology is the scientific enterprise of investigating the status,
structure, and meaning of knowledge, then this perspective becomes
indispensable for the analysis of such indirect modes of cooperation
between scientific didactics and everyday teaching practice that aim at
HEINZ STEINBRING
93

---

94
THEORY-PRACTICE DIALOGUE
communication as a reciprocal dialogue searching for possibilities of
constructing and enhancing meaning and not simply conveying knowledge
matter. The intention is not to describe the mediation of a coherent
didactical theory named "mathematical epistemology" to the practice of
mathematics teaching, but to stress and to use epistemological
considerations of mathematical knowledge, because this is an essential char-
acteristic of every process of mediating knowledge between teacher and
students as well as between researcher and teacher. This section presents an
epistemological analysis. The next section discusses how classroom
episodes can be interpreted along these lines and discussed with teachers as