On Tuesday, April 15, 2014 8:27:23 PM UTC+1, Bart Goddard wrote: > quasi <email@example.com> 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?