|
|
|
|
|
|
|
|
|
 Posted: Thu Aug 23, 2007 1:24 pm
So I am taking real analysis and abstract algebra this semester and I was wondering what the best way to deal with the classes... I've only taken one major proofs class, but these two seem to be quite a bit tougher. I already keep up with important theorems and definitions with index cards, but I was just wondering what ya'll would suggest.
(I'm taking math stats too... It is funderfull.)
|
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Fri Aug 24, 2007 8:40 am
I have a method when it comes to proof-based classes. It may or may not work for you, though:
1) Memorize all the basic definitions and assumptions (you can use flash cards or whatnot if necessary, but there usually aren't that many definitions). Make sure you know these definitions precisely - down to the letter - as they'll be important for making sense later on.
2) When you see a proof, construct a flowchart for it. Know what definitions are necessary, what previous theorems, the results of all lemmas used for it, etc. Oftentimes I've found that if a theorem doesn't make sense, it's 'cos I don't know all the preliminaries. Alternately, making such flowcharts make theorems with seemingly long proofs fabulously transparent. The best example I can think of is Stokes' theorem on manifolds and chains. Ridiculous amounts of preliminary definitions and lemmas and other theorems, but that *actual* proof is ridiculously short.
3) Like with definitions, theorems need to be known to the letter. But; don't spend your time memorizing the theorems. Spend your time understanding the proofs inside and out. Know the definitions, lemmas, etc. that go into it. You should be able to construct quick, dirty proofs for *all* theorems, and full, accurate proofs for the important ones. I've found that knowing the proof of a theorem makes memorizing the theorem a non-issue.
4) Do all of this waaaaay ahead of time. Before a theorem comes up in class, you should know all the lemmas and definitions leading up to it. You should know what the theorem says, and you should have seen the proof. The classtime should be used for asking questions about the proof and understanding it. All of this stuff - especially being able to prove the theorem - should be done *before* you ever use the theorem on a homework assignment.
5) Math is rarely understood the first time around. Read ahead, and reread often. If you can, keep track of four pieces of information:
a) The farthest you've read in the book
b) The farthest you've read and are working on learning the proofs for
c) The farthest you've read and understand completely and absolutely
d) Where the class is
These should be in descending order, with (c) and (d) around the same place.
Like I said, my method might not work for you, but I hope it helps somewhat.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Fri Aug 24, 2007 12:58 pm
thanks.
I usually read the book before class... In reals the prof likes us to explain the proofs, rather than him explain them... Unfortunately the book skips many steps (almost to the point that you can't follow the proof). But the prof is patient... and with only 3 people in the class he doesn't expect us to figure it out too quick.
With abstract I figure I'm going to have to work harder... We work in groups and I just happen to be in a group with someone that really struggles with proofs. >.< I guess I might want to practice explaining things. lol
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Tue Aug 28, 2007 9:07 am
If you know a proof well enough to explain it to someone, you know it inside and out. As far as the steps go...I guess just do your best to fill in the blanks. Or; try to do the proof from scratch, without looking at the steps in the book, and see if you hit the same "landmarks" that the book does. Good luck!
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Sun Sep 23, 2007 8:22 am
A couple weeks in and I like proof classes a lot... Much better than the normal math classes! Unfortunately the other kids in reals are starting to act like they might drop. I don't want to be a in one person class! eek However, I have enough time to study for Putnum now, so I'm not doing so bad!
I feel bad for my profs though, I always seem to do proof different from everyone else. They like the creativity, but sometimes I take the much more difficult road. Oh well! mrgreen We have come up with much better proofs that the profs have decided to use as opposed to the one they usually teach, so I guess being different isn't a bad thing!
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Posted: Wed Dec 12, 2007 9:35 pm
My proofs are also stylistically different due to the fact that I learned proofs by just taking a class that included proofs. I sometimes take the less efficient way or end out proving things in general and then showing that this case is a specific part of the more general and complicated case. Don't worry about it. As long as you figure out a way to prove it, then it's fine. Make sure you understand the logic behind the other ways as well though.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Reply
