Date: Jan 9, 2013 10:21 PM
Author: William Elliot
Subject: Re: What difficulties encountered by beginners in abstract algebra?

On Mon, 7 Jan 2013, Dan Christensen wrote:
> On Monday, January 7, 2013 4:48:55 AM UTC-5, William Elliot 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.

Oh sure, logic and axiomatic system learned in Euclidean geemetery
is so much different than logic and axiomatic system using in abstract
alegbra, that it doesn't transfer from one to the other.

> 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.

Personal intuition isn't needed for algebra?

> Ersoz also suggests that introductory geometry courses seldom present
> many of the methods of proof used in more abstract courses ─ methods
> like proofs by induction, contrapositive or contradiction (p. 164).

Yes, geometry is mostly constructive while algebra or caluulus is an
opportunity to extend logically skills learned in Eucidean geometry.