Commentary on Theories of Mathematics Education

528 H. Jungwirth
Besides, in technologically shaped classrooms manipulation affairs are given priority
over mathematical ones in the sense that they can always interrupt mathematical
negotiations in case that manipulation-related complications occur. The connection
of the theories has proven fruitful for a detailed reconstruction of all those features
just in the way I have illustrated by the small example above.
The overall procedure to elaborate that synthesized view on computer-based
mathematics teaching has been presented in several publication of grounded theory.
Thus, I refer to it just to an extent that is necessary to understand the methodological
remarks given below. A researcher starts with a certain unit for analysis (to take a
particular striking episode is a usual way) and proceeds by including more and more
units in the interpretation process. The selection of subsequent units is guided by the
interpretations developed so far (“theoretical sampling”; Glaser and Strauss 1967).
Interpretations are elaborated, enriched, and detailed in a comparative process that
results from the integration of further units for analysis. Hypotheses and their relationships
emerge from that “constant comparative analysis” (ibid). Two strategies
support that process. Including units that are similar to the one(s) already involved
(in regard to contextual conditions) helps to confirm the relevance of interpretations.
By selecting ones that differ in context characteristics, interpretations can be generalized
or tailored more exactly for the respective units. When inclusion of further
cases does not change interpretations any more, their final versions have been developed
(“theoretical saturation”; ibid). That way of reconstructing empirical data is a
general procedure that is not tailored in specific for research that applies, and combines
different theories. In my case both theoretical approaches have contributed
to the process in their specific ways. Micro-sociological theories were a means to
elaborate meanings of tasks and the development of respective interactions, and,
as a consequence, of activity complexes, too. Linguistic activity theory helped in
working out additional features of interactions, that is, the interplay of the modes
of activities, and objectives that refer to larger activity complexes reaching beyond,
and to conceive the latter as entities in the end.
Compatibility of Theories
In discussions about networking of theories preconditions for fruitful connections
play an important role. A deeper insight into the respective qualities of theories
helps in deciding in which respects theories fit together, and thus to appraise their
suitability for a networked enterprise of this or that kind, according to the interesting
phenomena and the research question.
I want to discuss that issue of preconditions in the following chapters. As my
study indicates in accordance with previously presented examples (Prediger et al.
2008), combining theories of different grain sizes seems to be rather a successful
strategy for co-ordinated data analysis and theory development. But this is just one
supportive aspect, there can be further ones as well; and, besides, “grain size” is
just a good metaphor in which the very meaning of that difference is not expressed.
Altogether, I will address three aspects of the theories featuring in my research that