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formation of the teaching-learning process. I recognize the importance of
the first kind of study to make both teachers and researchers aware of the
existence of an implicit ideology of teaching as well as of the power of
some hidden interaction rules. The above studies act, so to speak, as demol-
ishers of illusion (ICMI, 1993) and are both a backdrop and an incentive for
other studies. Yet, in my paper, I shall consider other kinds of study that are
supposed to be more pragmatic (yet not at all atheoretical, as I shall argue in
the following), because they are based on designing, implementing, and
analyzing teaching experiments, in which the traditional implicit rules of
interaction and the underlying ideology are voluntarily and systematically
substituted by different explicit ones.
I shall be concerned with two issues, which need to be discussed before
any tentative overview of literature: the function of theoretical assumptions
(section 2) and the effects of choosing among different theoretical elabora-
tions (section 3). The former is prior to any choice, while the latter concerns
just the choice of a theory of learning. The aim of this paper is to elaborate
Steiner's (1985) claim for complementarity on both issues from the
perspective of my research on the relationship between social interaction
and knowledge in the mathematics classroom (Bartolini Bussi, 1991).
R. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),
Didactics of Mathematics as a Scientific Discipline, 121-132.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.

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2. PRIOR TO FRAMEWORKS: THEORY AND PRACTICE
IN THE STUDY OF SOCIAL INTERACTION
2.1 Research For Knowing Versus Research For Acting Purposes
Two contrasting perspectives are represented by the so-called Recherches
en didactique des mathématiques (Douady & Mercier, 1992), which are pe-
culiar to the French community (referred to in the following as RDM), and
by research on innovation (RI) developed in different countries (e.g., the
Purdue Problem Centered Mathematics Project, Cobb, Wood, & Yackel, in
press; the Genoa Project, Boero, 1988, 1992; the Mathematical Discussion
in Primary School Project, Bartolini Bussi, 1991).
The purpose of RDM, at least as regards its core (the theory of didactical
situations by Brousseau, 1986), is to describe the functioning of didactical
situations. The researcher acts as a detached observer of the didactical sys-
tem and looks for conditions of reproducibility in the teaching experiments.
The possibility of falsification is a criterion to judge the acceptability of re-
sults.
Research for innovation (RI) is not framed (it cannot be framed, as I shall
argue in the following) by such a coherent theoretical approach as RDM. Its
main purpose is to introduce examples of good didactical transpositions and
to analyze the resulting processes. As reproducibility cannot be assured by
the mere description of the teaching setting, it is substituted by gradual ex-
pansion to larger and larger groups of teachers. The possibility of verifica-
tion is a criterion for the relevance of results.
The main difference is in the underlying motive for research. RDM aims
at building a coherent theory of phenomena of mathematics teaching; RI
aims at producing tools (either adapting them or constructing by itself) to
transform directly the reality of mathematics teaching. RDM is oriented to-
ward knowledge of classroom processes, while RI is oriented toward action
in classroom processes. RDM is supposed to ignore the results of the latter,
as they usually do not meet its criteria, while RI can borrow results from the
former, because of its intrinsic eclecticism.
2.2. Action and Knowledge Reconciled
The development of different conceptions of didactics of mathematics is
surely dependent on social and historical factors. The analysis of this issue
could be the subject matter of comparative studies in the social history of
didactics of mathematics. References to some documents (e.g., Barra,
Ferrari, Furinghetti, Malara, & Speranza, 1992; Douady & Mercier, 1992;
Schupp, Blum, Keitel, Steiner, Straesser, & Vollrath, 1992) reveals that na-
tional conditions of development are very different. The image of didactics
of mathematics seems to suffer from local conditioning (Boero, 1988).
However, when an image is built or in construction, criteria to judge the rel-
APPROACHES TO CLASSROOM INTERACTION
122

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evance of problems and acceptance of methodologies within a scientific
community are given.
Balacheff (1990a) calls for a confrontation and discussion of theoretical
research and research for innovation. In my opinion, this sounds difficult:
What is in question is not only the nonexistence of a universal language in
which to execute the critical comparison (which is involved whenever com-
peting theories are confronted) but also the existence of different meanings
of didactical research. I shall adopt Raeithel's (1990) description of three
models of relationships between actor and observer in the enquiring
activity: (a) the naive problem solver who considers the symbolic structure
inseparable from the perceived reality; (b) the detached observer, who
represents reality by means of symbolic models, and (3) the participant