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Re: What difficulties encountered by beginners in abstract algebra?
Posted:
Jan 11, 2013 11:36 PM
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In article <slrnkf0p8c.ovq.hrubin@skew.stat.purdue.edu>,
Herman Rubin <hrubin@skew.stat.purdue.edu> wrote:
> On 2013-01-10, Michael Press <rubrum@pacbell.net> wrote:
> > 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
>
> I agree that proofs by induction are not present in the
> classical Euclidean geometry course. However, contrapossitive
> and contradiction are present, although not to a large extent.
>
> The real problem is that many students do not even have an
> opportunity to take a proof-type geometry course, and I have
> been told by many colleagues that even the "college algebra"
> taught in universities may well not cover induction. Nobody
> who does not understand induction understands the integers;
> it belongs in first grade, and to be used afterward.
>
> Here is how algebra can and should be taught early, and then
> USED:
>
> A variable is a temporary name for something.
> The same operation on equal erntities gives equal results.
>
> The rest is application of this.
>
> > 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.
>
> Plane geometry texts have been written with these liberties
> removed. But it makes little difference; learning how to
> compute answers is of little value in understanding mathematics,
> or just about anything else. Machines can do that well.
I do not see how what you say relates to what I wrote.
--
Michael Press
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