Re: depth and breadth: The Calculus

nate (schmolze who-is-at students.wisc.edu)
Thu, 20 May 1999 06:48:30 -0500

----- Original Message -----
From: Martin Ryder <mryder who-is-at carbon.cudenver.edu>
To: <xmca who-is-at weber.ucsd.edu>
Sent: Wednesday, May 19, 1999 3:32 PM
Subject: depth and breadth: The Calculus

> Another significant impact to me in those times was
> Jerome Bruner's influence on my career as a young
> teacher. In The__Process_of_Education (1960), Bruner
> corageously challenged traditional curriculur structures
> on the basis of intellectual honesty. He said that "the
> foundations of any subject may be taught to anybody
> at any age in some form." Second-grade children can
> engage in absorbing games governed by the principles
> of integration, differentiation, set theory or
> approximation without immersing them in the formalisms
> associated with college-level Calculus texts. Bruner
> advocated a "spiral curriculum" in which all subjects
> that are worth knowing should be introduced to all
> students at all levels. Calculus need not be the
> exclusive domain of career-bound scientists and
> engineers to be introduced in high school or college.
>

Martin and others,

Ironically, I was recently reading an article in which Bruner's spiral and
"honest curriculum" was interpreted in rather traditional terms. Rather
than the way you, and myself, see Bruner the author took it as a validation
of what Freire would call a banking approach. The assertion of being to
teach any concept at any age was seen as in a scaffolding way in which the
content was broken down into manageable parts. This of course is not
simply by coincidence many textbooks, curriculum etc. that are more banking
use the spiraling concept to legitimate it. In science there is said to be
some real distorted interpretations of Latour's work.

With that said, I have always found something "revoltionary" about Bruner's
assertion of any can be taught at any age as long as its honest. I think
this is often ignored and has some very important implications for "special
education" along with Vygotsky's reference of sign systems. If in
approaching some particular concept we begin with the "concept" it allows
many different approaches. I find Davydov's work very interesting in this
regard because it puts the concept at the center of attention. Renshaw has
an interesting overview of Davydov's work in relation to the Austalian
educational context.

http://www.geocities.com/~nschmolze/renshaw.html

Renshaw refers to Cole as Follows,

This reversal of the common-sense assumption that higher-order thinking
must be built up piecemeal by mastering lower order procedures, is
reflected also in the work of Newman Griffin and Cole (1989). They worked
with children on division and multiplication problems. The children who
experienced most difficulty with division seemed to lack an understanding
of the functional significance of the multiplication facts. Confronting the
division algorithm organised the multiplication facts, according to Newman
et al, giving the facts for the first time a clear functional significance
for some children. They suggest a re-ordering of curriculum content where
higher level actions (concepts) are taught prior to lower level operations
(Newman Griffin and Cole, 1989, p.155).

Only time will tell if the ideas of the cultural-historical school which as
we can see are quite inconsistant with the banking approach will be used to
legitimate it.

Nate