Commentary on Theories of Mathematics Education

112 E. Jablonka and C. Bergsten
new domains alongside existing ones without replacing these. The general weakness
of the grammar of mathematics education brings about that the growth, within
each domain, does not result in a set of hierarchically organised concepts along
with unambiguous empirical descriptions, as is the case with some areas of science.
The knowledge in mathematics education does not grow by progressive integration
within theoretical systems of increasing generality. But we also find examples of
theories, ‘home grown’ within the field, which resemble more of a vertical knowledge
structure. These theories also differ in the strength of the grammar, especially
in terms of the extent to which the generation of unambiguous empirical descriptions
is concerned. The anthropological theory of didactics (see Bosch and Gascon
2006) may serve as a prominent example of a theory developed from within
mathematics education, given the implicitness of its intellectual sources. This theory
outlines hierarchically organised key concepts. In relation to the empirical, these
concepts leave much interpretive space. The theory suggests to move from the theoretical
to the empirical and then backwards for the purpose of generating empirical
descriptions, which cannot easily be developed from the outset. However, parts of
the theory simulate aspects of a strong grammar by the use of a seemingly highly
classified language (such as a specific symbolic notation for different dimensions
of a praxeology). Another example of a local theory that reflects more of a vertical
knowledge structure is the interpretation of embodied cognition in mathematics education.
It allows for the generation of less ambiguous empirical descriptions and
thus suggests to move from the empirical to the theoretical. We also find examples
of ‘botany’, that is, developing hierarchically organised descriptors of the empirical,
as for example a classification of students’ errors in written arithmetic or of different
types of ‘word-problems’. Even though such classificatory systems resemble some
features of a strong grammar, their development cannot be called theorising.
When looking into recent issues of ESM and JRME, we found it not easy to
locate some papers in the categories provided by Lerman. This is not because the
categories are insufficient, but because of the different relationships to (or ways
of ‘recontextualising’ of) the theories employed. The modes include adopting the
whole (dogmatic reading), selective pitching on concepts for better organising empirical
material (but not adopting the basic principles or the problématique), misreading
or deliberately re-interpreting key concepts, as well as referring to a theory
as a general background to a study (often without any further evidence of a relationship).
The very low percentages reported by Lerman of papers moving from the
theoretical to the empirical perhaps indicate an undogmatic reading of theories. We
also found a number of papers that can be characterised as common sense redescriptions
of empirical material and previous research outcomes under the dominance of
a specialised research question. Examples of this mode include studies with observations
of different phenomena in relation to children’s strategies when working on
arithmetical tasks.