Commentary on Theories of Mathematics Education

On Networking Strategies and Theories’ Compatibility 533
out the rectangle with the largest area) was expected to show no manipulations, as
it is such a case of introduction. But that hypothesis was refuted by the data (the
transcript and the video record of that scene); the introduction of that mathematical
method was interwoven with a use of Derive.
A use of empirically rich theories is characteristic, or even necessary, for quantitative
research as the hypotheses to be formulated need a ground they can be deduced
from. Within qualitative research drawing upon such theories may go beyond
expectations. Literature on methodology for this branch of research (Kelle
and Kluge 1999) even points to the risk that categories having well-defined properties
and refutable hypotheses involving them could dominate and interfere with the
intended “inclusive” reconstruction of reality. However, it is not necessary to use
empirically rich theories as it is done in quantitative research (Hempel 1965); a researcher
is not obliged to restrict her/himself to an examination of certain, refutable
hypotheses formulated in advance. Apart from that, in my case this would never
have done. For instance, the reconstruction of the way of addressing and clarifying
manipulation issues that contributed much to my small, grounded theory about
computer-based mathematics classrooms could not be accomplished just by checking
hypotheses about the mere modes of activities.
My study shows that empirically empty and empirically rich theories can be connected,
and, moreover, that this constellation has led to a satisfying result. But what
is the specific benefit of combining theories of both kinds?
To begin with, qualitative studies prefer theories that lack empirical substance.
This reflects the idea that the reality they can analyze is a reality already interpreted
by the members of society, and, accordingly, research should take those interpretations
in account as a starting-point for analysis (Schwandt 2000). Empirically empty
theories correspond with that position because they just provide perspectives from
which data can be looked at and do not postulate the properties of the phenomena
that will be perceived from those perspectives. These have the role of “sensitizing
concepts” (Blumer 1954). Literature (including that in mathematics education) gives
evidence that a synthesis of solely empirically empty theories has proven very fruitful.
To mention simply one study that draws upon the micro-sociological theories
I have used: Voigt’s (1989) reconstruction of the social constitution of a patterned
mathematical practice in classrooms is a prominent example. So it would be presumptuous
to assert a general superiority of a mixed combination. The respective
empirical loads of theories do not matter in their synthesis in the sense that the
loads are decisive for the quality of the result.
However, connecting theories that differ in that respect has a positive effect on
the very development of a grounded theory; on its “verification”, to be more precise.
This term refers to the check procedure for hypotheses and their relationships within
their application to further data. It is a moot point among the authors of grounded
theory whether or not this has to be done in a deliberately carried out examination:
by predicating certain events and checking whether or not they occur in the data.
According to Glaser (1978), such an explicit verification of statements is not necessary
because deductions from the evolving theory are always checked by further
data, and, consequently, a verification already takes place within the ordinary procedure.
From Strauss’ point of view (1987), however, an explicit verification is a