Commentary on Theories of Mathematics Education

548 U. Gellert
classroom practice, practice is seen rather as a building and the static aspects of the
practice of mathematics instruction are emphasized. ‘Static’ and ‘dynamic’ are orientational
metaphors and emerge from our constant spatial experience: things move
or do not move; they are static or dynamic. Metaphorical orientations have a basis
in our physical and cultural experience. They are at the very grounds of our conceptual
systems. By the local integration of semiotic and structuralist perspectives,
these grounding metaphors are forced into interaction. The more static perspective
on mathematics instruction is challenged by the more dynamic perspective, and vice
versa. These mutual challenges help acknowledge, and partially overcome, the principled
restrictions of theoretical perspectives. The emergence of “new” paradigmatic
research questions Qnew, such as ‘What is an appropriate balance of explicitness
and implicitness in mathematics instruction? Is it the same balance for all groups of
students?’, which have not been included in Qi and Qj , supports the view that the
restricting effects of the principles Pi and Pj of the theories τi and τj can actually
be mitigated by mutual metaphorical structuring.
Conclusion
Local theory integration is not aiming at complementary accounts. Semiotics as
well as structuralist theory have their origins outside the field of mathematics education.
On the grounds of both theories, when ‘applied’ or ‘adapted’ to the study of
the teaching and learning of institutionalized school mathematics, knowledge about
issues that have long been disregarded might be generated. Still, they are not theories
of mathematics education. However, the strategy of integrating them locally,
that is in the region of mathematics teaching and learning, results in an escalated
approximation to the field of mathematics education. This process is essentially a
development of theory.
Lerman (2006) is drawing heavily on Bernstein’s (1999) differentiation between
hierarchical knowledge structures (with science as the paradigmatic example) and
horizontal knowledge structures (such as sociology or education), where mathematics
education knowledge is considered as horizontally structured. Horizontal knowledge
structures are to some extent incommensurable (Bergsten and Jablonka 2009;
Gellert 2008). According to this perspective, knowledge growth in mathematics education
can evolve from two distinct processes: First, by developing knowledge
within (theory) discourses; second, by the insertion of new discourses (theories)
alongside already existing ones. It needs to be observed that the local integration of
the semiotic and the structuralist perspective on explicitness escapes this dichotomy.
Both theories offer rather new discourses for research in mathematics education, as
their paradigmatic research questions Q do not focus on the teaching and learning
of mathematics. However, what counts as ‘new’ and ‘old’ theories in mathematics
education might be a contested terrain, as is what counts as research in mathematics
education. Apparently, there is no consensus among researchers in mathematics
education about the boundaries of the research domain, making classifications of
old/new and within/outside problematic. Similarly, whether local theory integration