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Topic: Linear Algebra Question
Replies: 28   Last Post: Apr 25, 2014 6:10 AM

 Messages: [ Previous | Next ]
 Paul Posts: 780 Registered: 7/12/10
Re: Linear Algebra Question
Posted: Apr 15, 2014 5:51 PM

On Tuesday, April 15, 2014 8:27:23 PM UTC+1, Bart Goddard wrote:
> quasi <quasi@null.set> wrote in news:h5uqk9pe6k135orf8ns3m86bqsfqmgm57v@
>
> 4ax.com:
>
>
>

> > If not, what has changed -- where is the blame?
>
>
>
> Leaving blame aside, one thing that has changed
>
> is that we almost universally teach mathematics
>
> backwards. Somehow we've gotten to the point
>
> where we think we should teach complete mastery
>
> of the tool before we have a something to use
>
> the tool on.
>
>
>
> Example 1: Abstract algebra. Yes, a reasonably
>
> intellegent person can learn the group and ring
>
> axioms and learn to prove facts about groups and
>
> rings. But the whole course is generally
>
> disconnected from the mathematics previously
>
> learned. I think it would be better to state
>
> a problem or two, say, Fermat's Last Theorem,
>
> and try to prove the n=3 case. This creates a
>
> need for extentions of Q, and then the abstractions
>
> follow.
>
>
>
> Example2: We begin calculus with limits. (Well,
>
> after reviewing all the stuff they didn't learn in
>
> pre-calc.) We spend a 1/4 of the course learning
>
> how to compute limits, but really have nothing to
>
> take the limit of. I think it would be better to
>
> start the course with the slope formula and work
>
> our way to the need for limits.
>
>
>
> This would be teaching mathematics in the order
>
> in which it was discovered, rather than the exact
>
> opposite. Yes, we love our well-polished theories,
>
> and, gosh, it's great fun to work really hard on
>
> this seemingly useless tool, only to get to show
>
> that _now_ we can easily work all these sorts of
>
> problems. Great fun, I say, for us. But it's just
>
> a pony show for a student. It would be more fun
>
> for him if he got to re-live the discovery, and,
>
> I'm quite sure, the discovery would make a lot more
>
> sense to him.
>
>
>
> I really think that we're bringing our students up
>
> to believe that mathematicians sit in their offices
>
> cooking up tools and then go looking for problems
>
> to solve with them. No wonder no one wants to be
>
> a math major.

Agreed. I don't do academic research but I do try and read maths papers. I follow the argument line by line, checking that each deduction follows from what has been shown previously, without concerning myself with getting a general overview. I really enjoy the process of moving from axiom to deduction to a complete proof.
Because I have this overly axiomatic approach where I don't develop intuitions, I'm often stuck when it comes to research problems, and that's why I'm not an academic mathematician, but I do use maths a lot in my work. (More precise details of my job are not publicly available).
I basically think Bart's approach is far more productive than my own but I enjoy an axiomatic approach much more.

I'm sure it's an easy google to find a proof of FLT for n = 3 but I'll ask on this thread anyway. Does anyone have a suggested reference for this and other easy cases? I've seen a proof of FLT only for n = 4.

You say that teaching has "changed". So when was it better, and when did it deteriorate?

Paul Epstein

Date Subject Author
4/13/14 David C. Ullrich
4/13/14 Math Lover
4/13/14 Bart Goddard
4/13/14 magidin@math.berkeley.edu
4/13/14 George Cornelius
4/14/14 David C. Ullrich
4/14/14 George Cornelius
4/15/14 David C. Ullrich
4/15/14 Paul
4/15/14 quasi
4/15/14 David C. Ullrich
4/15/14 quasi
4/15/14 Bart Goddard
4/15/14 Peter Percival
4/15/14 Paul
4/15/14 Bart Goddard
4/20/14 Shmuel (Seymour J.) Metz
4/21/14 Bart Goddard
4/21/14 Shmuel (Seymour J.) Metz
4/23/14 Bart Goddard
4/24/14 Shmuel (Seymour J.) Metz
4/24/14 Bart Goddard
4/25/14 Shmuel (Seymour J.) Metz
4/15/14 Roland Franzius
4/15/14 Peter Percival
4/15/14 David C. Ullrich
4/15/14 Rock Brentwood
4/15/14 Rock Brentwood
4/16/14 David C. Ullrich