Commentary on Theories of Mathematics Education

<strong>Commentary</strong> on Networking of Theories—An Approach 511
students’ mathematical thinking). With the interpretative empirical research on everyday
mathematical teaching–learning situations (for example, the work of the research group
around Bauersfeld) a new research paradigm for mathematics education was constituted:
the cultural system of mathematical interaction (for instance, in the classroom) between
teacher and students. (Steinbring 2008, p. 303)
Within this new plot I think it is possible to develop an idea of networking of theories,
which concerns specifically mathematical education and not education in general.
Many examples in the ZDM special issue are mathematically driven, but unfortunately
this aspect fades in the landscape pictured in the paper (perhaps it is
implicitly present in the five aspects of “theory guiding research” illustrated in the
paper: see its Fig. 1).
A second remark is a consequence of the first one. The point has been raised by
Radford in his paper in the special issue of ZDM. There he observes that
a condition for the implementation of a network of theories is the creation of a new conceptual
space where the theories and their connections become objects of discourse and
research. This space is one of networking practice and its language, or better still, its metalanguage.
In particular, the meta-language has to make possible the objectification of and
reference to new conceptual “connecting” entities, such as “combining” or “synthetizing”
theories (Prediger et al. 2008b). (Radford 2008, p. 317)
Hence he suggests
to look at this social networking practice and its meta-language as located in a conceptual
semiotic space that cultural theorist Yuri Lotman, in the context of the encounter of various
languages and intellectual traditions, called a semiosphere (Lotman 1990), i.e., an uneven
multi-cultural space of meaning-making processes and understandings generated by individuals
as they come to know and interact with each other. (ibid., p. 318)
Radford comments further his suggestion writing:
I prefer to see the network of theories as a dynamic set of connections subsumed in the
semiosphere where integration is only one of the possible themes or “plots” (to use another
of Lotman’s terms) of the metalanguage of the semiosphere. (ibid., p. 319)
A possible plot in the semiosphere could be that of nonlinearity sketched above or
others: from his side, Radford suggests the plot of identity:
...by reacting to our own theory and our claims about it, the other theories make visible
some elements that may have remained in the background of our theories. This is why, in
dialoguing, we enter into a process of extraction: we pull out things from the brackets of
common sense (the brackets of things that we take for granted in our theory to the extent
that we no longer even notice them) and, with the help of other theories and in the course
of dialoguing, we subject these things to renewed scrutiny. As a result of this connection
(that may fit into Prediger et al.’s category of “comparing/contrasting”), the connection may
result in a better self-understanding of one’s own theory. (ibid., p. 319)
Possibly the wonderful plot described by the authors is only one of the many which
are possible in the semiosphere. In a sense, the idea of semiosphere is a more flexible
tool, which can give account of the different lenses through which the Networking
of theories can be looked at. Hence at the metatheoretical level we find a landscape
which is similar to the one that we are trying to describe at the theoretical level.
That must not be seen in the negative. As at the theoretical level the existence of