reification

David Kirshner (CIKIRS who-is-at LSUVM.SNCC.LSU.EDU)
Fri, 27 Sep 96 15:38:45 CDT

Hello xmca.

I've been following the discussion of reification in xmca
coincident to reading and thinking about it in the mathematics
education literature. It turns out that reification is emerging
as a central organizing principle for understanding the development
of advanced mathematical thinking, both historically and ontogenetically.
For instance Anna Sfard (1995) makes the case that the historical
development in algebra from the rhetorical to syncopated to symbolic
phases reflects psychological processes of reification, conceived of
as encapsulation of mathematical processes into mathematical objects.
The slow historical processes are seen as indicative of the
cognitive difficulties that students experience as they make
(or fail to make) the transition to the symbolic symbol system of algebra.

Jere Confrey and Shelley Costa (in press) argue that the psychological
(intra-individual) orientation of this approach tends to justify the
Platonist and hierarchical conception of mathematics that
mathematicians tend to hold as the basis for their non student-
centered instructional methods. Her concern for a more responsive
pedagogy, and a more socially informed model of cognition seems
relevant to the marxist definition of reification discussed here; for
instance in Paul Dillon's concern that

The notion of a "universal psychological process" is flawed insofar
as it does not recognize that psychological processes are determined
by the social relations that produce them and that these relations
evolve, or ... are historically determined."

I'm sure that Jere would be willing to share her paper with anyone
interested. (Her email address is given on the CC line, above.)

Confrey, J., & Costa, S. (In press). A critique of the selection of
"mathematical objects" as central metaphor for advanced mathematical
thinking. _International Journal of Computers for Mathematical Learning_.

Sfard, A. (1995). The development of algebra: Confronting historical
and psychological perspectives. _Journal of Mathematical Behavior,
14_(1), 15-39.

Thanks, as always, for a stimulating discussion.

David Kirshner
Louisiana State University
cikirs who-is-at lsuvm.sncc.lsu.edu