Commentary on Theories of Mathematics Education

Modalities of a Local Integration of Theories
in Mathematics Education
Uwe Gellert
The development of a meta-language for the connection of theoretical perspectives
in mathematics education is still in its infancy. This volume is a substantial contribution
to this developing language. In order to support the institutionalization of
this meta-language in the field of mathematics education, my meta-theoretical discussions
affiliates to the conceptual languages proposed by Lester (2005), Radford
(2008) and Prediger et al. (2008). The aim of this chapter is threefold. First, it scrutinizes
the notion of bricolage as a guiding principle for the connection of theories
in research in mathematics education. It argues that bricolage is appropriate for the
tackling of practical problems. But it shows fundamental weaknesses as long as theorizing
and theory development is concerned. Second, the chapter provides a case
of local theory integration as one of the strategies for connecting theories, discussed
elaborately in the Theory Working Groups at CERME 4, CERME 5 and CERME 6.
Third, the example serves as an empirical footing for an analysis of how theory
development advances by integrating theoretical perspectives locally.
Radford (2008) develops a conceptual language for talking about connectivity of
theories in mathematics education. He takes theories as triples τ = (P,M,Q) of
principles, methodologies and paradigmatic research questions. P denotes a hierarchical
structure of implicit and explicit principles that “delineate the frontier of what
will be the universe of the discourse and the adopted research perspective” (p. 320).
The methodology M includes the techniques for data generation and interpretation
and justifies the coherence and operability of the techniques in the frame of P .The
set of paradigmatic research questions Q is stated within the conceptual apparatus
of the theory and in relation to P .
A similar classification of the principles, rules and ideas that serve as a means for
producing research action and understanding has been generated by Lester (2005).
The function of a research framework, according to Lester (2005, p. 458), is to determine
“the way the concepts, constructs, and processes of research are defined”;
the “acceptable research methods” and “the principles of discovery and justification”;
and the nature of research questions and the manner these are formulated.
The difference between Radford’s ‘theories’ and Lester’s ‘research frameworks’
can be seen in the organisation of the first entry of the triple: Radford is emphasizing
the hierarchical structure of the system of principles whereas Lester is more
U. Gellert ()
Department of Education and Psychology, Free University of Berlin, Berlin, Germany
e-mail: ugellert@zedat.fu-berlin.de
B. Sriraman, L. English (eds.), Theories of Mathematics Education,
Advances in Mathematics Education,
DOI 10.1007/978-3-642-00742-2_50, © Springer-Verlag Berlin Heidelberg 2010
537