Commentary on Theories of Mathematics Education

<strong>Commentary</strong> on Theories of Mathematics Education: Is Plurality a Problem? 113
Communication and ‘Translation’ Between Discourses
Another problématique linked to the horizontal structure and weak grammar relates
to the possibilities of ‘translation’ between the different discourses. Plurality might
indeed be a problem when research outcomes are to be integrated. Moore (2006,
p. 40) argues, while a horizontal knowledge structure with a strong grammar makes
translation possible, one with a weak grammar does not. In the first, there is no break
in the language, because strong grammar aims at the integration of language rather
than the accumulation of languages. “Horizontal discourses with a weak grammar
are established through a process of reinterpretations of the empirical more than
through theoretization of objects” (ibid., p. 40). This way of knowledge formation
(in the sociology of education) was by Bernstein described as being due to a lack of
proper theorising. A theory, in his conception, as Moore points out, is a generating
principle for a range of possibilities not necessarily yet observed in the empirical.
Moore uses the invention of the periodic table as an example. Mendeleev invented
a principle that generated a matrix of positions for the elements (in terms of similar
characteristics and of atomic weight). The point is that this matrix generated theoretical
possibilities that were not yet empirically realised (only 63 of the 92 elements
were recognised). A theory proper, according to Bernstein, has to include the description
of such a generative principle for objects yet to be observed, in addition to
helping to make sense of already given empirical instances. Such theorising would
provide the possibility of more than only redescribing the empirical by different languages
based on incompatible approaches. However, such a theory need to include
an external language for making sense of how a distinct combination of features of
a phenomenon under study would look like if encountered in the empirical.
The horizontal knowledge structure amounts to a plurality of theories and eventually
leads to incommensurable parallel discourses, as Lerman points out. This is
indeed a consequence in case of a dogmatic reading of ‘imported’ theories. When
incommensurable theories are used to mediate between a phenomenon and outcomes
of research studies, the results can only be juxtaposed (see Bergsten and
Jablonka in print); the interpretations of the outcomes might even be contradictory.
This is, for example, the case in studies about the difficulties students face when
confronted with contextualised tasks. Some see the problem in the students’ lack
of using their common sense (mostly within an individual cognitive framework),
while others point to the students’ problem of using too much everyday knowledge
when solving these tasks (mostly from a discursive or sociological perspective) (cf.
Gellert and Jablonka 2009). Incommensurable or contradictory interpretations are
less likely to occur in case of undogmatic reading of theories or of selective pitching
on some concepts to guide data analysis. The theoretical hybrids produced by
such approaches might indeed ease the communication across discourses. There are
also examples of theory reception that in the long run amounts to increased use of
some key concepts without full recognition of the basic principles of a theory. As
an example, embodied cognition found its way into mathematics education during
the 90ies in two ‘waves’, though the remaining impact on the field seems to be an
increased acknowledgement only of the roles of bodily based metaphor and gesture
for mathematical cognition (Bergsten 2008).