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Re: What difficulties encountered by beginners in abstract algebra?
Posted:
Jan 10, 2013 6:24 PM
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In article <c546d56d-b2a1-417d-baa6-7e1e2721244e@googlegroups.com>,
Dan Christensen <Dan_Christensen@sympatico.ca> wrote:
> On Monday, January 7, 2013 4:48:55 AM UTC-5, William Elliot wrote:
> > On Sun, 6 Jan 2013, porky_pig_jr@my-deja.com wrote:
> >
> > > On Sunday, January 6, 2013 11:46:37 PM UTC-5, Dan Christensen wrote:
> >
> >
> >
> > > > I am working on some introductory notes for group theory. What
> >
> > > > difficulties are typically encountered by math or science undergrads
> >
> > > > in an introductory course on abstract algebra?
> >
> > >
> >
> > > The same kind of difficulties as moving from Calculus to Analysis. Need
> >
> > > set theory, need some intro to logic and proofs.
> >
> > >
> >
> > What do you mean need an introduction to logic and proofs?
> >
> > I learned logic and proofs during my high school sophomore year
> >
> > in the Euclidean geometry class. Where are they these days?
> >
>
> Studies have shown that proof-writing skills learned in one branch of mathematics such as geometry may not be easily transferred to other branches such as abstract algebra and analysis.
>
> F. A. Ersoz (2009) suggests that the many informal "axioms" of Euclidean geometry, as usually taught, are based largely on personal intuition and imagination (p. 163). While this may serve as a productive basis for some discussion, it can blur the boundary between the formal and informal, and lead to confusion as to what constitutes a legitimate proof in other domains (branches) of mathematics.
>
> Ersoz also suggests that introductory geometry courses seldom present many of the methods of proof used in more abstract courses Ñü methods like proofs by induction, contrapositive or contradiction (p. 164). http://140.122.140.1/~icmi19/files/Volume_1.pdf
This fails to mention that the proof writing skills
learned in plane geometry are 100% transferrable to
more advanced subjects. _Of course_ plane geometry
takes some liberties, but they are _warranted_
liberties just as in all teaching a little bit of
liberty with the way the subject actually works is
warranted by the student getting a good start on it.
--
Michael Press
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