Commentary on Theories of Mathematics Education

516 S. Prediger and A. Bikner-Ahsbahs
2. The local theory building level: Jungwirth and Gellert go beyond the empirical
analysis developing an empirically grounded synthesized or integrated local theory.
3. The meta-level: Because they are empirically based, both authors reflect their
strategies of connecting theories and the theory building on a meta-level. They
contribute to the general discourse on the networking of theories, its benefits and
limits, its preconditions and different ways to carry it out.
More concretely, Helga Jungwirth regards a qualitative study on the role of computers
in mathematics classrooms as a case of locally synthesizing theories. For
many years, she has used the interpretative approach situated within the microsociological
perspective of symbolic interactionism and ethnomethodology. As
there is a long tradition of research practices integrating these two approaches, they
are nowadays handled as one theoretical framework. In the current study, she connects
this framework with linguistic activity theory. Both theoretical frameworks are
briefly presented with respect to their role in her research study; their coordinated
expression in research practices is shown by a concrete example of interpreting a
transcript.
On the local theory building level, she can synthesize both theories by specifying
different types of activity complexes as the “outcome of multitudes of similar negotiations
among participants”. She distinguishes types of computer-based (mathematics)
teaching “ranging from a highly verbal teaching emphasizing subject matter
aspects to a teaching that is totally devoted to carrying out manipulations at a computer”.
Helga Jungwirth is very explicit on the role of both theories in this process
of building a local grounded theory: “Through the micro-sociological theories the
formation of an activity complex becomes visible, and through linguistic activity
theory a multitude of interactions can be spoken of and treated as an entity.”
On the meta-level, Helga Jungwirth raises two important issues: three aspects
that enhance compatibility of theories and the interplay between the networking
strategies of synthesizing and coordinating in research practices that follow the program
of grounded theory (Glaser and Strauss 1967). Both are absolutely worth being
taken into account carefully in the further discussion of networking strategies.
Gellert’s considerations are embedded in a framework that has been presented by
Radford (2008) regarding theories as triplets (P, M, Q) with principles P, methodology
M and paradigmatic questions Q as constitutive parts of a theory. Gellert
also connects theories of two different grain sizes for analyzing a transcript from
a mathematics classroom: Ernest’s semiotic theory about rules, describing interactions
on a micro level, is coordinated with Bernstein’s more global structuralist view
of pedagogical rules. This connection is executed focusing on the dialectic between
explicitness and implicitness.
On the level of local theory building, this dichotomy is overcome by questioning
what kind of balance between explicitness and implicitness in teaching mathematics
supports learning mathematics for all.
On the meta-level, he starts criticizing the metaphor of “bricolage” for theorizing
and presents an alternative view, the metaphor of “metaphorical structuring”
illustrating its strength by discussing an empirical example. Gellert stresses that the