Commentary on Theories of Mathematics Education

Surveying Theories and Philosophies of Mathematics Education 23
student extracts what he needs. It facilitates the determination of the objects to be studied,
the role of the actors, and the division of the study of teaching among sundry disciplines.
For example, mathematics is responsible for the content, the science of communication for
the translation into appropriate messages, pedagogy and cognitive psychology for understanding
and organizing the acquisitions and learnings of the student.
At this juncture, we will also point out the fact that Brousseau developed TDS
with some practical ends in mind, that is, to ultimately be able to help teachers redesign/engineer
mathematical situations and classroom practice so as to facilitate
understanding. Again, in Brousseau’s (1999b) ownwords:
The systematic description of didactical situations is a more direct means of discussing with
teachers what they are doing or what they could be doing and of considering a practical
means for them to take into account the results of research in other domains. A theory of
situations thus appeared as a privileged means not only of understanding what teachers and
students are doing, but also of producing problems or exercises adapted to knowledge and
to students, and finally a means of communication between researchers and with teachers.
TDS is very much a constructivist approach to the study of teaching situations
(Artigue 1994) and “founded on the constructivist thesis from Piaget’s genetic epistemology”
(Balacheff 1999, p. 23). It could be thought of as a special science complete
with theoretical considerations and methodological examples for a detailed
study of mathematics teaching within an institutional setting. TDS includes a specific
grammar with specific meanings for terms such as didactical situation, adidactical
situation, milieu, didactical contract etc. Taken in its entirety TDS comprises
all the elements of what is today called situated cognition. The only difference is that
TDS is particularly aimed at the analysis of teaching and learning occurring within
an institutional setting. The most significant contribution of TDS to mathematics education
research is that it allows researchers from different theoretical traditions to
utilize a uniform grammar to research, analyze and describe teaching situations. One
example of this possibility is seen in the recent special volume of Educational Studies
in Mathematics (2005, vol. 59, nos. 1–3) in which 9 empirical studies conducted
in Europe used the “classroom situation” (in its entirety) as the unit of analysis.
Such a uniform approach was made possible largely because of the utilization of
Brousseau’s TDS and Chevallard’s ATD (next section) as the common theoretical
framework. However the research sites at which these studies were conducted were
predominantly in France, and Spain, which have historically used these frameworks.
Anthropological theory of Didactics (ATD): The Anthropological theory of didactics
(ATD) is the extension of Brousseau’s ideas from within the institutional setting to
the wider “Institutional” setting. Artigue (2002) clarifies this subtlety by saying that:
The anthropological approach shares with “socio-cultural” approaches in the educational
field (Sierpinska and Lerman 1996) the vision that mathematics is seen as the product of a
human activity. Mathematical productions and thinking modes are thus seen as dependent
on the social and cultural contexts where they develop. As a consequence, mathematical
objects are not absolute objects, but are entities which arise from the practices of given
institutions. The word “institution” has to be understood in this theory in a very broad sense
. . . [a]ny social or cultural practice takes place within an institution. Didactic institutions
are those devoted to the intentional apprenticeship of specific contents of knowledge. As
regards the objects of knowledge it takes in charge, any didactic institution develops specific