How to prepare for and study pure math?

[PieceOfPi]

Hi all,

The title is kind of straight forward, but let me add you a few background.

I am majoring in mathematics and minoring (or possibly double majoring) in undecided (i.e. I'm still looking for the second concentration... right now I'm thinking of either physics or computer science). This is my second year as an undergrad, and I'm still not so sure what I want to do after the graduation. I don't necessarily go to a best school in math (I go to a regular public school), but I do think we have a pretty standard math department here.

Last year, I took number theory to make myself familiar with mathematical proofs, and I thought I did well. Then, I took Elementary Analysis (Text: Elementary Analysis: The Theory of Calculus by Ross), and I thought I did really well in that class, so I thought I were sharp enough to take the upper-level analysis sequence (Text: Rudin). But it took me about 3-4 weeks that my thought was wrong-- I tried to keep up with it, but I could not keep up with the pace of the class and the amount of homework. I was also frustrated because everytime I asked a question to my professor, he would just tell me "Oh that's easy; just read the definition!", even though I've already read the definition a few times before seeing him. I felt only ones who were doing well in the class were grad students and some really bright undergrads. My midterm score was 72, whereas the class average was 70, and this probably wasn't as bad as I expected, but I still decided to take the class Pass/No Pass because I couldn't really handle the frustration. I did pass the class at the end, but I decided not to continue with the sequence.

Now here's my concern: So I tried analysis, but I didn't do so well or enjoyed it well. But I'd still like to try out pure math (whether algebra, analysis, or topology) --as a math major, I feel like I should know good amount of these. But this time, what should I do to prepare myself for these courses, and how do I actually study for these courses? Or should I not bother with these courses anymore as the stereotype I have toward this subject (i.e. only geniuses can do pure math) is true? I certainly think I have some interest in pure math, as I enjoyed elementary analysis and number theory. My interests may change as I decide my second concentration, but please let me know if you have any advice.

Thanks,

PieceOfPi

 

[Unknot]

I do believe that being able to understand and intuitively "feel" definitions in something like Rudin is crucial for someone who wants to do pure mathematics. I suppose you are talking about Rudin's PMA. Then it really is basic for further study in math.

I don't think only geniuses can do pure math - I do think it requires some other kind of intelligence than let's say, biology, but as long as you have SOME talent it should be ok. The hard thing is to find out whether you have enough of this talent. If you are going to go to graduate school in math - I would think you would need to work harder, because being able to read PMA in your sleep would be a requirement. But since you have 72 in math, I think you should encourage yourself to try more. I think ANY math major should know some topology (and I suppose you have seen these in Rudin) and even more important, some algebra.

 

[csprof2000]

Perhaps you just don't like analysis? Not all mathematicians are great at every sort of mathematics...

If it's not just an issue with interest, however, I don't know what to tell you. Practice makes perfect?

 

[PieceOfPi]

Thank you for your responses.

I do believe that being able to understand and intuitively "feel" definitions in something like Rudin is crucial for someone who wants to do pure mathematics. I suppose you are talking about Rudin's PMA. Then it really is basic for further study in math.

I agree. In fact I think that's what threw me off the most in that course. I read the definition a few times, and yet I couldn't get a complete understanding of the definition, so I decided to move to the next page, and then by the end of the chapter, I had so many "holes" that prevented me from completing assignments.

I also had a hard time with Rudin's PMA because it was so concise that I felt like there was something missing in proofs when I read it, which made me think "... Wait, why is that obvious?" From reading Amazon.com's feedback, I realize it has a very high rating, and I'm supposed to fill in those gaps by myself. But I think I just didn't have that maturity when I was studying it. So is this maturity something I can gain from more practice? Is there any strategy from that practice?

I don't think only geniuses can do pure math - I do think it requires some other kind of intelligence than let's say, biology, but as long as you have SOME talent it should be ok. The hard thing is to find out whether you have enough of this talent. If you are going to go to graduate school in math - I would think you would need to work harder, because being able to read PMA in your sleep would be a requirement. But since you have 72 in math, I think you should encourage yourself to try more. I think ANY math major should know some topology (and I suppose you have seen these in Rudin) and even more important, some algebra.

I sometimes think that maybe I should have tried harder in that class, as getting an average score in analysis probably isn't as same as getting an average score in some introductory courses.

Perhaps you just don't like analysis? Not all mathematicians are great at every sort of mathematics...

If it's not just an issue with interest, however, I don't know what to tell you. Practice makes perfect?

It is possible that I don't like analysis, as I find myself more interested in courses like linear algebra or combinatorics. I personally want to try algebra next year, but it all depends on schedule and the professors who are teaching algebra or analysis.

And what would you suggest me to do for the "practice"?

Again, thanks for your wonderful comments, and I'd be happy to read more.

 

[owlpride]

I personally did not like Rudin. He is very rigorous and very concise but I always felt that something was missing... Historically, all of math has been motivated by some problem or another, but Rudin just throws all of the definitions and proofs at me without any background or motivation whatsoever. I do use him as a reference, but I don't like the book as the primary text for a course. (At least not if the professor is just copying the proofs from the book on the board.)

And in case someone's interested, I have been doing fine in all of the graduate courses I have taken so far.

 

[csprof2000]

"It is possible that I don't like analysis, as I find myself more interested in courses like linear algebra or combinatorics. I personally want to try algebra next year, but it all depends on schedule and the professors who are teaching algebra or analysis."

You're sort of in the same boat as I was when I was an undergraduate. I started off as a math major with a bunch of AP credit, and this meant that I took the undergraduate real analysis sequence when I was in my sophomore year. I did well in it, but I had to struggle in that class... I found the subject not to my liking.

I subsequently changed my major to Computer Science, and I haven't had a course in analysis of any kind since. The only other math courses I took were some advanced algebra courses, one on graph theory and a couple on combinatorics (enumeration and combinatorial designs), and a few more statistics and probability.

Oh well.


"And what would you suggest me to do for the "practice"?""

Check out a few books on analysis from your library, and read through them. Don't do any exercises. Just read the text and the proofs. Look for good books that motivate you to understand the stuff AND have clear, elaborated proofs. Don't do any exercises until you've read at least two full books. Think of it as immersion training.

 

[quantumdude]

A couple of years ago I decided to get an MS in Math (still working on it--trying to go to school and work full time is a chore!). My prior education consists of a BS in engineering and an MS in physics. So I am not coming from a "pure math" background.

I've found that what I need to do is get the books ahead of time and start working the exercises in the summer. I do the same thing over winter break. I'm usually 2 or 3 chapters ahead of the class when the course starts. It's worked out well, my lowest grade is an A-. Try it!

 

[mrb]

Check out a few books on analysis from your library, and read through them. Don't do any exercises. Just read the text and the proofs. Look for good books that motivate you to understand the stuff AND have clear, elaborated proofs. Don't do any exercises until you've read at least two full books. Think of it as immersion training.

Sorry to be disagreeable, but I just have to make a quick comment that I don't think this is good advice. We learn math by doing math. The exercises are the most important part. In fact it might be better to get another elementary real analysis book and ONLY do the exercises; just reading the exposition when your previous knowledge is lacking.

And in fact that would be my advice to the OP on how to study for math in general. Just do math. Some students seem to try to study math the way they would study for a history test... Memorize facts and things. This is almost useless. You must develop an intuitive understanding of the material, and you do this by working with it. A lot. A good first approach is, for example, to do all or almost all the exercises in a textbook instead of just the handful assigned. You may not have time to do this for every class, but it really helps when you can.

I'm a senior math major, but I've got a 4.0 in math classes.

 

[Unknot]

Personally, I think there's no need for other books. You had problem with Rudin. Then go read more Rudin! If you were coming from let's say Stewart's calculus, it's understandable - there is some gap between that calculus and Rudin. But you had Ross's great book and from that Rudin is not that much of a jump.

It would be better if we knew what kind of definitions/theorems/chapters you are mainly having problem with, but I agree with mrb. You need to do exercises. Have you done exercises for Ross? That may be a good place to start. Then I would move onto reading Rudin, and then doing all the exercises. Exercises in these two books are not that difficult.

My theory is that students cannot intuitively "fill" the gaps between Rudin's proofs because they haven't had that much preparation in actually writing out proofs. Of course, you might have done plenty of these already, but I am just guessing.

 

[symbolipoint]

Tom Mattson said/wrote this:

I've found that what I need to do is get the books ahead of time and start working the exercises in the summer. I do the same thing over winter break. I'm usually 2 or 3 chapters ahead of the class when the course starts. It's worked out well, my lowest grade is an A-. Try it!

Who has comments about what Tom Mattson said? The idea makes excellent sense; in fact that method would seem to prevent academic problems. Does anyone discourage that practice? If so, why?

Imagine what you would have to do if you studied a course as a student, did poorly and then needed to retake the course. Compare this to what Mattson described. One method seems more effective than the other. Maybe some students cannot learn effectively the first time through a course. See that point? Why should students be intentionally discouraged from studying a course BEFORE actually enrolling in it?

 

[owlpride]

I try to play with the most important definitions and theorems from a course before the beginning of the semester. This way I can pay attention to details while others are struggling to grasp the concepts, and I have a framework in which I can sort all the stuff I learn into. When I don't have time to prepare for the entire semester in advance, just working one or two classes ahead works as well.

I don't like reading the entire text ahead of time though. I would know enough to get bored in class and lose focus, but not quite enough to get through the course without paying attention.

But I would definitely second Tom's suggestion to work ahead! Just find a way that works for you.

 

[PieceOfPi]

Again, I thank you ALL for responding to my questions! All of your replies are fantastic, and I'm certainly encouraged to challenge myself to take more pure math courses!

Personally, I think there's no need for other books. You had problem with Rudin. Then go read more Rudin! If you were coming from let's say Stewart's calculus, it's understandable - there is some gap between that calculus and Rudin. But you had Ross's great book and from that Rudin is not that much of a jump.

Ross is amazing! In fact I feel like this is where I should start before I even read Rudin again. My elementary analysis course didn't do all the chapters (we skipped chapters on topology as well as entire sections on differentiation and integration, as it was only a 10-week course), so it might be helpful for me to read Ross, have a complete understanding of Ross, and then hit Rudin. I still own both Ross and Rudin, so only thing I need is some time to crack those texts.

It would be better if we knew what kind of definitions/theorems/chapters you are mainly having problem with, but I agree with mrb. You need to do exercises. Have you done exercises for Ross? That may be a good place to start. Then I would move onto reading Rudin, and then doing all the exercises. Exercises in these two books are not that difficult.

I think Ch. 2 of Rudin (topology) threw me off the most. I did my best to understand basic set theory and metric spaces, but once I got to the definition of compact sets and a few associated theorem, my brain had a really hard time understanding them. The exercises from Ch. 2 were pretty challenging for me as well.

 

[PieceOfPi]

A couple of years ago I decided to get an MS in Math (still working on it--trying to go to school and work full time is a chore!). My prior education consists of a BS in engineering and an MS in physics. So I am not coming from a "pure math" background.

I've found that what I need to do is get the books ahead of time and start working the exercises in the summer. I do the same thing over winter break. I'm usually 2 or 3 chapters ahead of the class when the course starts. It's worked out well, my lowest grade is an A-. Try it!

Very interesting point. So far I've never done this before, so I don't know how this would work out (it might turn out I'm not very good at self-studying), but this certainly sounds like I should try out, as reading a math textbook seems like something I need to practice more often.

 

[khemix]

I have found myself in a similar situation to yours. Pure math is extremely difficult and often not motivated that well. I do not know how anyone can succeed in an analysis course while doing full time study. To me, it required a lot of time outside class. I had to play with the theorems and assumptions until they made sense... something I would not have time for in school. The reading of the textbook alone took more than most other courses, and to do problems was even harder.

I found doing a lot of extra exercises helps the concepts sink in and reinforces the definitions and makes you appretiate the theorems.