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Re: What difficulties encountered by beginners in abstract algebra?
Posted:
Jan 11, 2013 2:16 PM
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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.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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