Commentary on Theories of Mathematics Education

530 H. Jungwirth
obvious by inspecting the concept of interaction. The idea that an interaction is determined,
for example, by the roles of the participants (which is assumed in stabilityoriented
paradigms), and the idea that an interaction is a negotiation process from
which (also) roles emerge (which is the understanding in transformation-oriented
ones) cannot be combined to an integrated view on interaction serving as a base for
analysis. Thus, a difference in the solution of the dualism precluded a connection of
the respective theories.
The general issue arising from the discussion above is which elements of their respective
grounds theories have to share in order that networking on the level of some
synthesis of theories, or of an integrated analysis, can take place. The answer, those
being essential for the object of research, is obvious but the meaning of “essential”
always remains to be worked out related to the respective object.
A concordance of paradigms in regard to that essence can be called a fundamental
criterion for compatibility. Without a shared ground in that respect, concepts will
become contradictory and thus cannot provide a starting-point for data analysis and
theory development.
Neighbourhood of Phenomena’s Sites
The above aspect has been about positioning of theories, this one will address the
“sites” of the phenomena research is interested in. In my case attention has been
directed to classroom processes. On the one hand, I have been interested in face-toface
relationships, and on the other hand, in larger, partly discontinuous, purposive
networks of activities. Thus, in comparison the first kind of phenomena (having been
thought of as interactions according to symbolic interactionism) are temporally and
spatially smaller units than the second kind (i.e. activity complexes according to
linguistic activity theory). The resulting notion of mathematics teaching combines
both by the idea of “activity complexes in interactive step-by-step development”.
This works because activity complexes can be seen as covering interaction. The
initial concepts are neighbouring concepts, or, by taking up the grain size metaphor
again: The theories use different grain sizes to grasp the phenomena.
For a better understanding of the issue it may be helpful to refer to the notion
of a “layered” social space in the life-world; elaborated by Schütz (1981), Schütz
and Luckmann (1973), and used later for a classification of communicative events
and contexts (Knoblauch 1995). For my argument here it will be sufficient to note
that the social entities members of society may encounter, and the social practices
in which they may involve can be used for building layers. As for the first aspect,
groups, networks of individuals, organizations, or institutions of different sizes, and
still larger collectivities, like social classes, or nations, can be separated. As for social
practices, layers are arranged according to the anonymity of the procedures. The
“lowest” layer is that of face-to-face relationships (happening between individuals
“here and now” in their full concreteness), a “higher” layer is that one encompassing
social processes basing more or less upon role behaviours, and rather at the