Commentary on Theories of Mathematics Education

556 T. Wedege
on action—when working on local theory integration, is made “as transparent and
grounded as possible”. I find that this explicitness demonstrates what professionalism
can be in the field of connecting theories and conclude that the purpose of
the work on strategies can be seen as developing a language for scientific metacognition
in mathematics education.
Theoretical Approach Versus Theoretical Perspective
The terminological context for Gellert’s discussion of connecting theories in mathematics
education and for my comment is developed by the CERME Working Group.
Thus, I have adapted the notion of theory—or theoretical approach—proposed by
Prediger, Bikner-Ahsbahs and Arzarello as a dynamic concept where a theory “is
shaped by its core ideas, concepts and norms on the one hand and the practices
of researchers—and mathematics educators in practice—on the other hand” (2008,
p. 176; see chapter ‘Networking of Theories—An Approach for Exploiting the Diversity
of Theoretical Approaches’ in this volume). According to this dynamic understanding,
theories and theoretical approaches are constructions in a state of flux
and theoretical approaches guide and are influenced by observation (p. 169). A first
consequence of “theory” being synonymous with “theoretical approach” is that theory
is not only a guide for thinking but also for acting—for methodology. In line
with this conception, I follow Radford (2008)—together with Gellert—when he
suggests to consider theories in mathematics education as triples τ = (P,M,Q),
where P is a system of basic principles “which includes implicit views and explicit
statements that delineate the frontier of what will be the universe of discourse and
the adopted research perspective” (p. 320); M is a methodology supported by P ;
and Q is a set of paradigmatic research questions.
In the examination of the diversity of theories, Lerman (2006) does not define
“theory” but by looking at the examples and the proposed categorization of social
theories within the mathematics education research community (1. Cultural psychology;
2. Ethnomathematics; 3. Sociology; 4. Discourse) it is obvious that his’s
understanding of “theory” encompasses methodology. When Lester (2005), assigns
to theory the role as guiding research activities through theoretical research frameworks
which provide a structure for conceptualizing and designing research studies,
he includes methodology as well.
Starting from the understanding of theory as theoretical approach, I have had the
ambition to bring a terminological clarification of differences between “perspective”
and “approach” into the work on strategies for connecting theories, or at least to
bring attention to some terminological problems (Wedege 2009b). Looking at the
syntax and semantics of the two English nouns, one observes that “approach” is
a verbal noun meaning the act of approaching (begin to tackle a task, a problem
etc.). “Perspective” means a view on something from a specific point of view (seen
through a filter) (Latin: perspicere = looking through). In the context of the debate
in the Theory Working Group, this noun does not have a verbal counterpart. Thus,
in order to distinguish the two terms, I proposed the following clarification: