Commentary on Theories of Mathematics Education

532 H. Jungwirth
can be examined, and thus refuted through data. In the first case a theory has an
empirical substance, in the second one a theory has no empirical substance. That
difference does not mark a basic difference in the value of theories; in particular, a
lack of empirical substance does not indicate a deficiency that should be avoided.
Both states are equally valuable though not equally relevant for all research methodologies.
More precisely spoken, they are just the poles of a whole spectrum of states.
As for the theories figuring in my research, symbolic interactionism is at the
second pole. It is a philosophy, a stance towards the world that can be hold, or
rejected. It is not possible to formulate refutable hypotheses for the position that
objects get their meanings in the course of interaction. Furthermore, that theory
does not provide any information about meanings that will be established in certain,
given interactions, or about the development of interactions in certain cases. For
instance, I could not suppose from the beginning that introductory mathematical
negotiations would be argumentative processes in some contexts at least, and thus
not set out to check that assumption. Ethnomethodology, too is a theory that lacks
empirical substance. There is no empirical decision-making whether or not people’s
methods to settle their everyday affairs make these everyday affairs. This is also a
position that may be hold, or not. As a further parallel with symbolic interactionism,
methods as such are not specified in advance either.
As for linguistic activity theory, the decision is not as clear-cut. Linguistic activity
theory does not provide any information about the objectives, or outcomes
activity complexes try to achieve. Thus it is not possible to formulate in advance a
hypothesis about an orientation that can be confronted with a maybe contradicting,
empirical statement. However, the assumption that activity complexes are not mere
encounters but serve their respective purposes provides a perspective that can be
elaborated within data analysis. For instance, that puzzling switch to the completion
of the table in the second part of my example made me develop the hypothesis of an
orientation towards full technological solutions. First it referred to that introduction
to maximum-minimum problems only but it could be applied to further episodes.
In that application it was modified. For instance, the irrelevance for the very mathematical
processes being a feature in the original version of the hypothesis had to be
abandoned in the end. There were also episodes in which improvements remained
related to advantages for students’ development of mathematical understanding; for
example, a teacher enlarged the dots of a graphical representation of a function but
did not play around with that diagram any more.
On the other hand, linguistic activity theory has some empirical substance. The
category of activity has two distinctive properties, verbal, and practical. Thus, from
the very beginning it is possible to examine episodes in respect to those qualities.
In particular, it is possible to find out whether or not practical activities at a computer
occur and thus to formulate hypotheses about the modes used for task solving.
To give an example related to the transcript above: Basing upon its first part that
is about a mathematical introduction and does not show any practical activities at
a computer, a hypothesis about mathematical introductions in that lesson could be
developed: They take place without any use of a computer. Hence, the first application
of calculus for the solution of the given maximum-minimum problem (to find