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April 2nd, 2010, 05:34 AM   #1
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How to study maths well?

Alright... It's a straightforward question.
How to study maths well?
Is it good to practise more by doing tons of exercises?
or there are some other ways?
I used to study maths quite well,
but recently, I got into troubles again,
so I just want to see if there is any method to study maths well
Thank you
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April 3rd, 2010, 03:18 PM   #2
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Re: How to study maths well?

Exercises are great, but the point is not being able to do the exercise, it's the understanding of the structures that you develop as you complete exercises.

The best way to get a good grounding in a subject is really to try to complete the proofs for the theorems that are in the book with as little help from the book as possible, and to do any exercise which looks like it may be instructive.

Anyway, the key is to look at what is going on:

*You need to learn to see the same structure in different ways. Normal subgroups are my favorite example: a normal subgroup is a subgroup for which the left and right cosets are the same, for which the cosets form a group, and which is the kernel of some homomorphism. Each of these is important, and the trick is to learn when each one of these is relevant-- that's where exercises are helpful. They help you pick up on when certain properties are the important ones.

*A good collection of examples and counter-examples is always helpful. But more helpful is knowing when these are instructive and when they are not.

*Knowing how to create counter-examples is always helpful... it saves a lot of time with bad conjectures.

*Learning when to back out is crucial. Look at the situation: Does the "proof" you're trying to write use all the hypotheses? Does the same proof "go through" in a simpler case where it shouldn't? How does this situation differ from a simpler one?
I can't count the number of times (in the last month!) I've started working on an idea and realized that my conjecture would imply something clearly false, or that if the idea I was trying would work in my case, it would also work in a simpler case where it was clearly false.

*Look at the big picture! I'm not sure how best to say this in words, so I'll give a recent example from something I'm working on. I'm working on the problem of coloring a graphs with only dominating sets. (I.e., we partition a graph G into independent dominating sets), and I've been hitting road-block after road-block. I finally stopped the other day, and looked at the class of graphs I'm working on: I need a certain color to show up too few times in a certain subgraph, and I need it to also show up too few times in the neighbors of this subgraph. I quickly came up with names for the places I don't want this color to be, and found a formula for how big each partition can be in terms of these named sets. Now, hopefully, I can calculate the parts I'm unsure of, and get a better bound than the one I have right now.
By stepping back and looking at it from a wider perspective, I was able to make some progress, and get a better understanding of what it is I'm looking for in the first place.

Man... I'm full of philosophical rambles today... I guess I need to stop reading the n-category cafe.
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April 7th, 2010, 11:36 PM   #3
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Re: How to study maths well?

Thanks for your reply and I got a few questions after reading it.
Firstly, for the first key, do you mean that I should do more exercises and observe the properties(or structures) at the same time?
Secondly, what is the meaning of the term 'counter-examples' ?
Thirdly (and lastly), I don't know what do you mean for the forth key you stated. Would you mind explaining it in a simpler way?
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April 8th, 2010, 09:07 AM   #4
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Re: How to study maths well?

Quote:
Originally Posted by tomorrow
Firstly, for the first key, do you mean that I should do more exercises and observe the properties(or structures) at the same time?
I'm saying you should do the exercises with the goal of understanding the structures you're working with. The fundaments will get practiced by doing the exercises, but these are useless without a deeper understanding of what's going on.

Quote:
Secondly, what is the meaning of the term 'counter-examples' ?
I don't know why I always hyphenate that... it should be counterexample. It's an example which proves a claim wrong. E.g., the Petersen graph.

Quote:
Thirdly (and lastly), I don't know what do you mean for the forth key you stated. Would you mind explaining it in a simpler way?
Sure. One thing that's difficult about math is sometimes you'll get taken pretty far down some road-- trying to use some method-- before you figure out that it doesn't work. So what I gave are hints that it's going the wrong way.

For example, suppose you are trying to prove that if certain conditions hold for an object O, then O has some property P. (And suppose, that in fact, this is not true, but you don't yet know it.) A couple things could suggest that this is false: You find a counterexample; You show that if it were, true, you can prove something that's clearly false; or... a little more subtle-- look at your proof method, it could be that your proof will hinge on a step, and in a simpler case (which fails to satisfy P), this step is "easy"... which means that you're going to hit a flaw at some point in your proof. There are other things that could tip you off as well.

I know that last idea is hazy... I can't give a good example off the top of my head; I know I ran into this situation a few weeks ago. I thought I could prove that certain graphs satisfy a colorability property. I realized that my idea for a "proof" would carry through in a case wher eit was clearly false, so I either had to get more subtle, or look for a different method. (The claim ended up being false, anyway...)
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April 8th, 2010, 09:45 PM   #5
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Re: How to study maths well?

Once again, thank you so much.
I shall put it into practice and hope it works.
For the last key, I hope I'll understand it when I overcome such a problem
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