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participants to become aware of the conditions of this teaching and of the oppor-
tunities of change. Interpretation and evaluation of the actual immediate class-
room reality indeed requires us to adopt a theoretical view. Insofar, the seemingly
immediately empirical and real lesson transcripts are highly theoretical con-
structs. They must be understood as individual cases of a varying scope of possi-
ble classroom situations, (von Harten & Steinbring, 1991, p. 175)
Such cooperative work between teachers and researchers serves a twofold
purpose: It is a means for researchers to communicate their theoretical ideas
in a context of shared perspectives and it is used to explore exemplarily the
teacher's practice, or better, to obtain feedback and to learn from the teach-
ers.
3.2 A Classroom Episode
An example may illustrate the development of the two epistemological lev-
els (contextualized and decontextualized) for the teacher's professional
knowledge within the framework of a fruitful dialogue between theory and
practice (for more details, see Steinbring, 1991b).
A short grade-6 teaching episode contains a sequence of exercises that the
teacher poses for training the translation of fractions into decimal numbers.
Despite this intended character of a phase of exercise, a shift to conceptual
problems occurs very soon, which the teacher does not notice at all. The
teacher starts with the first problem: to translate 
into the correct deci-
mal. The solution comes immediately: 0.3. The three following problems
are also solved more or less quickly, with the help of a brief reminder on the
rules of the fraction calculus:
The next problem causes some productive confusion: What is the decimal
for The 
students 
can no longer simply follow the teacher's explicit me-
thodological intention to first enlarge the fraction given to one of the form:
When trying to solve the problem, the students propose
the following transformation: The 
teacher 
rejects this re-
sult, because it ignores the formal method he has proposed. In a second at-
tempt, the students come up with a similar solution:
Now the methodological rule is fulfilled, but still the teacher is unsatisfied.
There is a decimal number as numerator in this fraction, a nonadmitted
combination of signs! In a kind of funnel pattern (Bauersfeld, 1978), the
teacher forces the correct solution by first calculating the number of en-
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largement to the fraction that 
is, 
125; the necessary arithmetical
division of 1,000 : 8 =?, is more complex than the division of 5 : 8 =?,
which would have given the solution directly.
Different intentions were interacting during this student-teacher episode:
The teacher simply followed his methodological aim of training the fraction
translation into decimals; and he relied on one rule, which he thought of as
easy and universal: "Transform the fraction given into one of the form:
and so forth, and then read off the correct decimal number!"
The students still have to cope with the unfamiliar new mathematical
knowledge. They try to uncover the teacher's expectations and to follow his
methodological rule as far as possible. The first four problems are solved;
for the teacher, the fifth problem seems to be only technically more com-
plex, but the students really encounter a new conceptual problem. In their
attempts to give a solution, they offer (still unknowingly) an interesting
conceptual generalization and, at the same time, an improved understanding
of the connection between fractions and decimals. However, the teacher is
not aware of this, because he is keeping strictly to his methodological plan.
Because of his strict goal of performing only some exercises, the teacher
is not open to the conceptual ideas hidden in the students' proposals. He
simply rejects the two fractions and 
for 
reasons of method and
definition. The interpretation from our perspective is that the teacher was
not sensitive to the epistemological dilemma of the mathematical symbols.
He could not understand or accept the possible new meaning of these signs,
the combination of decimals and fractions, which reflects the fundamental
conceptual relation of decimals in a new way: the variable choice of the unit
of measurement as a fraction with a denominator as a power of 10. Ac-
cepting the fraction would 
lead immediately to the answer
or or 
0.625 
by 
using the already known rule of shifting the position
of the point. But being able to agree with this interpretation would require
an epistemological vigilance toward the changing meaning of mathematical
signs and their combinations, which is regulated within the framework of
the epistemological triangle of object, symbol, and concept.
3.3 Analysis of Lesson Transcripts in a Dialogue Between Teachers
and Researchers
This episode, and some of the epistemological issues presented here, can
and have been taken as the common referential situation in a dialogue with
a group of teachers together with the teacher of this episode. This common
object served as a reference context to explain general epistemological ideas
(i.e., the epistemological triangle, the epistemological dilemma, etc.) and, at
the same time, to try to detect general constraints of the given concrete
teaching situation.
The exemplary dialogue between theory and practice in this case included
general and specific aspects. The discussion of the transcribed episode of-
HEINZ STEINBRING
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THEORY-PRACTICE DIALOGUE
fered means for the teacher to detach himself from his subjective immersion
in the teaching episode. This opened perspectives for a better comprehen-
sion of the students' remarks and intentions and for seeing some general fea-
tures in the specific and particular teaching situation; a view that was sup-
ported by the different interpretations given by colleagues. Specific aspects
concerned the interference of the teacher's methodological intentions with
the epistemological constraints of the mathematical knowledge and its
meaning as constituted in this interaction with the students. The seemingly
unique mathematical signs and operations developed by the teacher entered
a different context of interpretation in the students' understanding. How can
the teacher become sensitive to such epistemological shifts of meaning?
Here again, the very fundamental problem of the nature of (school) mathe-
matical knowledge is questioned: The new knowledge cannot be "given" to
the students; the teacher has to be aware of the way the students are trying
to reconstruct the meaning of the mathematical signs and operations he has
presented to the students. The shared discussion and dialogue between dif-
ferent practices enhanced the possibilities of becoming aware of underlying
complementary perceptions and ways of integrating them.
This social situation of dialogue and sharing between theory and practice
displayed the different paradigm of the theory-practice relation: to recon-
struct from a common object one's own conceptual ideas and practical con-
sequences by seeing the variable and general in the concrete, singular situa-
tion with the help of critics and the different perspectives of the participants.
4. CONCLUSIONS
Every productive dialogue between theory and practice in mathematics edu-
cation has to unfold the dialectic between the concrete context and abstract-
ing decontextualizations. This is not simply for reasons of presenting an il-
lustrative example for abstract theoretical considerations. The concrete con-
text has to play a basic role in the sense that it serves common and distinct
roles for the different partners: It links different views, which are based on
different professional activities, and it offers the establishment of referential
connections and referential meaning with particular and comparable aspects.
In this respect, communication and mediating materials in the relation
between theory and practice need to reveal different conceptual compo-
nents:
1. a common referential object;
2. specific generalizations of the knowledge (mathematical, epistemologi-
cal, professional) bound to the particular domain of experience;
3. means of social sharing, participating, and exchanging in commu-
nicative situations.
The dialogue between theory and practice in mathematics education can-
not aim at a direct conveyance of ready knowledge, but can offer only oc-
casions for a self-referential reconstructing of all aspects of professional
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REFERENCES
A. G. Mathematiklehrerbildung. (1981). Perspektiven für die Ausbildung des