Commentary on Theories of Mathematics Education

Modalities of a Local Integration of Theories in Mathematics Education 545
solving and argumentation, a form of mathematical activity that is said to be reminiscent
of what mathematicians do, in a Gymnasium class. The students, however,
remain completely unaware of the curricular conception behind the instructional
practice. Since the teacher just introduces the race-to-20 activity, without framing it
as (for the students) mathematically sophisticated, he apparently takes for granted
that the students understand his introductory comment “you think you can count
till 20” as an irony pointing to a more advanced mathematical activity to follow.
Hannes’ behaviour suggests that this is not the case. As in Bernstein’s theory the
organising principles of pedagogic practice are hierarchically related, dubiety in
terms of the distributive rules entails ambiguity in terms of the recontextualizing
rules: Most students take the instructional activity as a game and not as an introduction
to problem solving. They do not recognize that the activity of the game is
subordinated to the principles of school mathematics. Accordingly, it is extremely
difficult for the students to access the teacher’s criteria which make the evaluative
rules, that is, what counts, and is marked, as a legitimate participation to his mathematics
class. We know from a broad range of research studies, that invisible instructional
practices, in which the distributive, recontextualizing and evaluative rules remain
implicit, disadvantage all those children who have not been socialized into
the very forms of these practices (Cooper and Dunne 2000; Gorgorió et al. 2002;
Hasan 2001; Theule Lubienski 2000). Only visible pedagogic practices facilitate
that students collectively access, and participate in, academically valued social practices
and the discourses by which these practices are constituted (Bourne 2004;
Gellert and Jablonka 2009).
Local Integration of Semiotic and Structuralist Assumptions
The contrasting views on explicitness reveal that the rules and criteria of mathematics
education practice remain—in part as a matter of principle—implicit. On the one
hand, the need for implicitness is due to the very character of the learning process:
whoever strives for whatever insight cannot say ex ante what this insight exactly
will be. Ernest’s “General-Specific paradox” is an interpretation of this issue. On
the other hand, the principles that structure the practice of mathematics education
remain implicit to the participants of this practice, without any imperative to do so
for facilitating successful learning processes.
However, for that the general can be fully acquired, the students indeed need
to understand that the specific examples and applications have to be interpreted
as the teacher’s means to organise the learning of the general. Successful learning
in school requires the capacity to decode some of the implicit principles of
the teacher’s practice. The structuralist perspective supports the argument that the
students actually benefit more from teaching-the-general-by-teaching-the-specific
if they are conscious about the organising principle that is behind this instructional
practice. By making the organisational and hierarchical rules and the criteria of the
teaching and learning practice explicit, the teacher would provide the basis for that
all students can participate successfully in the learning process.