Interesting discussion on abstraction!
At AERA 2005, there was an interesting symposium on abstraction in mathematics learning. A few lines from the overview:
"Situated theories of cognition emerged in part as a reaction to fundamental assumptions regarding knowing and learning in mainstream cognitive theory. The notions of abstraction and decontextualization have been particularly problematic for situated theorists, and as a result are often rejected from situated analyses. Rather than dismissing abstraction, a few researchers have begun to reformulate abstraction so that it is compatible with situated theories."
In her paper, Jurow lists a few approaches on generalisation, but unfortunately for me she does not critique or integrate them. One of the question I have is: how are abstracting and generalising related?
By focusing of the activity of generalising we do not really get an answer to the issues that maths educators are wrestling with - as Ini Haket pointed out. How do we account for the 'objectivitiy' of mathematical abstractions? Or for their wide applicability? Or for the power of using mathematics, if we only focus on generalizing as a product of the outcome of processes distributed across students, tasks, embodied activity, and modelling tools?
To me this sounds Jurow lives in a participation metaphor (Sfard, 1998, Educ. Res.) whereas in maths education we probably need an acquisition metaphor as well.
In our project on Techno-mathematical Literacies in the workplace we use the notion of situated abstraction. Not that putting the adjective 'situated' in front of abstraction does the trick! Noss and Hoyles (1996) use that term for abstractions that get their meanings largely from the context, and less so from the conventional maths. Often these less formal abstractions are more effective and easier to use (though sometimes mathematically seen incorrect) than more general mathematical abstractions. By the way, our findings are in many ways similar to what Michael Roth emailed about his experiences on graphs in the workplace.
From: Ini Haket [mailto:I.Haket@ppsw.rug.nl]
Sent: 03 May 2005 12:35
Subject: Re: abstraction
Michael, I'm not sure I understand your comment.
Do you mean? It is not important, if students generalize in their own way, because
being the object of activity, the generalizations are inherently objective?
If so, doesn't this cause problems in the school situation?
Or do you mean? Students participate in the same activity and, as a result of that, have
the same objective generalizations.
If so, what is the influence of their past - different - experiences?
Or none of these, but ...?
On 2 May 2005 at 7:12, Wolff-Michael Roth wrote:
> When I look at it dialectically, generalizations are objective because
> they are shaped by who states them, they are concrete realizations of
> possibilities that exist at the cultural-historical level in this
> group, they are inherently objective otherwise they could not be the
> object of the activity.
> If you look at the general in genetic way, it makes little sense to
> ask how general is the general, because the general always contains ALL
> the possibilities that are realized concretely. Like all possibilities
> realized in the F1 generation are already contained in the P (parent)
> generation, ALL concrete appearances in the F2 generation are already
> contained potentially in the P generation. This is why P is a true
> On 2-May-05, at 6:15 AM, Ini Haket wrote:
> > The article poses an interesting problem for me. Is the result of the
> > abstraction
> > process one and the same generalizations for all the participants? How
> > common, how
> > general are certain generalizations? Jurow cites approvingly Latour:
> > "....generalizations
> > are not objective, but are shaped by who states them, how they are
> > connected to other
> > claims ..." (page 282). How does this relate to the aims of education?
> > Math teachers
> > wants their students to work with the same general structure, I
> > suppose? Teachers
> > offer guidance to make sure that everybody generalizes and comes in
> > the end to the
> > same abstraction. To reach the first aim the object of the activity in
> > the lessons is
> > generalizing. And inscriptions and the questions teachers provide for
> > the process of
> > conjecturing form an instrument for guidance in the "right direction".
> > But what about the individual(??), situational(??) influences on
> > generalizations, that
> > Latour mentions?
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