Commentary on Theories of Mathematics Education

On Networking Strategies and Theories’ Compatibility 531
“high-end” there is a layer of domains of social practice showing regular patterns
and serving their well-defined functions.
As a consequence, different sites for phenomena are provided, and, thus, positions
theories can assign to those they deal with. To give some examples: Phenomena
can be regarded to exist distributed in small groups, or in their face-to-face
interactions; in networks of individuals, or, accordingly, in their more extended interlinkages;
or, as a further option, in standardized social practices related to some
larger collectivity. If individuals are taken into account as a further site of phenomena
(of cognitions, for instance, or of individual behaviours) a wide range of positions
is provided that seems to cover a good deal of those needed in mathematics
education. Certainly, the position of a phenomenon can vary; for instance, emotions
can be treated as features of persons and as interactive performances as well (Gergen
1991); knowledge can be placed in societies, or in local relationships between
individuals (because it is assumed to be negotiated in face-to-face interaction). Furthermore,
a theory may relate to more than a single layer only; like, for instance,
symbolic interactionism. It is not restricted to those small, short interactions I am
interested in. It refers also to large joint actions that are usually spoken of as given
entities having their specific character; examples include even war (Blumer 1969,
p. 17). However, in symbolic interactionism even such joint actions are always addressed
under the aspect of interactive formation.
In my case, I have combined events on neighbouring layers: face-to-face interactions,
belonging to the lowest layer, and activity complexes that can be assigned to
the next one. For a generalized version, reflections suggest that networking of theories
requires sites being “close” enough to each other. All theories involved should
have positioned their phenomena on rather neighbouring layers, if they do not have
used the same at all (for instance, that all theories figuring in a combination focus
on features of individuals). A predominance of neighbourhood is plausible as
in case that the original positions of the phenomena are too distant it will not be
possible to find some sort of shared phenomenon that can be approached from all
theories. Thus, the development a networked conceptual element that can become a
component of the combined theoretical framework will fail.
Regarding compatibility, sufficiently neighbouring sites of phenomena are not a
condition sine qua non like the concordance of paradigms. Its violation does not
result in a contradictory concept. But it is a condition being necessary for the very
feasibility of networking of theories. For the grain size metaphor mentioned above,
the notion of a layered social space provides a certain meaning: Grain size refers to
the layer being addressed by a theory.
Theories’ Differences in Empirical Load
The third aspect is the “empirical load” of a theory (Kelle and Kluge 1999). Accordingly,
theories can be classed by the risk of empirical failure: whether or not
they comprise concepts and statements that provide properties and hypotheses that