I want to get A's in analysis and abstract algebra

[PieceOfPi]

I just finished my first quarter of analysis (Text: Rudin's PMA) and abstract algebra (Text: Beachy and Blair) courses. I must say I really enjoyed these courses, and I feel like I learned a lot from them. However, I still ended up getting B+'s from both of these courses. While I'm not disappointed about this, and in fact I'm glad that I'm not doing anything terribly wrong in these classes, but I still would like to improve myself so that I can get better grades in the future (or just perform better in these courses in general). So I was wondering if you have any tip to how to improve myself. A few remarks...

• I already feel like I spend quite a bit of time for these courses, but maybe I need to spend more time. Or perhaps, I need to come up with more efficient way of studying these materials.
• Both the professors who taught this course last quarter will be teaching again in the next quarter. I think they are fantastic professors, but they do have reputations for being pretty rigorous. On the other hand, I already know their styles (HW, test questions, etc), so I guess I am in a little bit of advantage for that.
• I would certainly like to know how people prepare for the exams. This seems really time consuming, since I feel like there are tons of definitions, theorems, and proofs to understand well enough to answer questions on exams. What I'm basically doing right now is just to re-read the textbook and notes to review myself (or re-teach anything that I did not understand), and look over some homework questions.

Let me know if you have any suggestion/comment. Thanks.

 

[logickills]

One of my favorite approaches, study groups. This only works if you are with a good group of people. Some of the major advantages to studying in groups are:

• Compare notes - You can never get everything, comparing notes solves this!
• Discussion - Being able to have a discussion applies your knowledge.
• Teaching - Someone in your group won't understand something, just like discussion, teaching or helping can further your conceptual understanding.
• Answers - Comparing answers is a big help and can raise your confidence or cue you to get more help. Student explanations can be easier to understand.

Now don't get me wrong, not everyone is able to study in groups.

Preparing for exams, this really depends on the instructor. Ask students who have taken the class already if the instructor has any emphasis lines. Figuring out verbal cues is an important part of note taking. Lines such as "now remember" or "don't forget" maybe "This will be on your exam" :p.That is all the have, hope it helps some!

LogicKills

 

[owlpride]

Studying for a math exam should not take all that long if you stay on top of the material during the term.

I found that I have to study math actively to understand what I am doing. When I see a new definition, I have to come up with a couple of examples and non-examples. When I see a theorem, I have to think of an application and ask myself why anyone would think of a theorem like that. Is it obvious? Surprising? Useful? How does it fit in with other things I have learned in this course? Can I think of applications outside the scope of the course? It helps me to reprove all the theorems from class myself. That shows me if I understand the gist of each proof. I have also heard of students using Rudin as a do-it-yourself analysis course: they read the definitions and theorems, but try to prove the theorems themselves instead of reading the proofs.

Learning math takes a lot of hard work!

 

[snipez90]

Rereading the text passively is not going to help at all. Especially with a text like Rudin, you could reread a paragraph of a proof (which sometimes is the whole proof) and still miss the subtleties and nuances involved. As owlpride said, you need to try reproving the theorems on your own (this should seem automatically much harder unless you've truly understood the theorems) and more importantly, how each step is motivated. Consolidating proof techniques should be a consequence of this approach.

Lastly, but probably most importantly, you should work on more exercises. Rudin's text has quite a few nontrivial exercises/results and unless your teacher hates you, some rewarding but difficult problems will not appear on your homework. For instance, the chapter on metric topology includes the nontrivial result that for metric spaces, sequential compactness implies (open cover definition) of compactness as an exercise. The next chapter includes the Baire category theorem as well as the analytic completion of any metric space as exercises. If you don't do these exercises and the ones related to them, you not only lose practice, but also some rather deep results that comprise core analysis material.

 

[owlpride]

^ Seems like my analysis professor hated us... I knew it!

 

[PieceOfPi]

Thanks for your replies! I would like to comment back on some of the things you have said so far.

Preparing for exams, this really depends on the instructor. Ask students who have taken the class already if the instructor has any emphasis lines. Figuring out verbal cues is an important part of note taking. Lines such as "now remember" or "don't forget" maybe "This will be on your exam" :p.

I am starting to get sense of how my analysis professor likes to writes his exam--If he seems to highlight something a lot in class, then that's very likely that he will put them on the exam. I still haven't figured out that for my algebra prof though... but it seems like she likes to make half of the problems related to something done in class or in homework problems, and other half reserved for "something new" (i.e. definitions/theorems not done in class, but only requires the knowledge of what we have done in class to understand/prove them).

Studying for a math exam should not take all that long if you stay on top of the material during the term.

And I think that is the hard part :P But I will try harder to stay on top of things next term.

I have also heard of students using Rudin as a do-it-yourself analysis course: they read the definitions and theorems, but try to prove the theorems themselves instead of reading the proofs.

I have done it once around week 7, and that actually seemed like a really good way to read this textbook, even though it took me a lot of time. I think I will try to work hard on that next term (and that should force me to stay on top of everything!).

Lastly, but probably most importantly, you should work on more exercises. Rudin's text has quite a few nontrivial exercises/results and unless your teacher hates you, some rewarding but difficult problems will not appear on your homework. For instance, the chapter on metric topology includes the nontrivial result that for metric spaces, sequential compactness implies (open cover definition) of compactness as an exercise. The next chapter includes the Baire category theorem as well as the analytic completion of any metric space as exercises. If you don't do these exercises and the ones related to them, you not only lose practice, but also some rather deep results that comprise core analysis material.

Actually, in my opinion, my analysis professor is pretty good about assigning good amount of problems from Rudin that actually gives us important results of analysis. In fact, we even discussed about sequentially compact iff compact in class on top of proving them on the homework. But nevertheless, he doesn't assign everything, and I don't think we have talked about Baire category theorem and etc, so I will keep in mind and try to read all the problems from each chapter, and hopefully try them on my own (same goes for algebra).

 

[snipez90]

^ Seems like my analysis professor hated us... I knew it!

Heh this was tongue in cheek of course. Our teacher never actually mentioned Rudin ever (our main text was Kolmogorov and Fomin), but Rudin's problems did appear on one exam.

 

[hitmeoff]

One of my favorite approaches, study groups. This only works if you are with a good group of people. Some of the major advantages to studying in groups are:

• Compare notes - You can never get everything, comparing notes solves this!
• Discussion - Being able to have a discussion applies your knowledge.
• Teaching - Someone in your group won't understand something, just like discussion, teaching or helping can further your conceptual understanding.
• Answers - Comparing answers is a big help and can raise your confidence or cue you to get more help. Student explanations can be easier to understand.

Now don't get me wrong, not everyone is able to study in groups.

Preparing for exams, this really depends on the instructor. Ask students who have taken the class already if the instructor has any emphasis lines. Figuring out verbal cues is an important part of note taking. Lines such as "now remember" or "don't forget" maybe "This will be on your exam" :p.


That is all the have, hope it helps some!

LogicKills

Agree with this 100%

 

[zpconn]

I do not believe there is much one can do to prepare for an exam covering proofs. Proofs require at the same time an intuitive and a rigorous understanding of the subject, and both of these come slowly with time over the course of the semester. The key to success is to regularly check whether you are actually gaining this understanding and, if not, to do something about it sooner rather than later.

It's very beneficial to read both the textbook from the class and other well-known textbooks on the subject. For example, while I haven't read the book by Beachy and Blair, I would be very surprised if you wouldn't have learned a lot by reading Artin's book (for example).

My algebra course used Herstein, but during the course I periodically checked in on Hungerford's graduate text, Martin Isaacs' graduate text, and Artin's text. I found Hungerford's the least useful by far, but I still profited from seeing the subject developed from a (slightly) categorical viewpoint. Artin's book is filled to the brim with big jumps of logic masked in simple, friendly language; I found the sections on polynomial rings particularly interesting in comparison to standard treatments. And the book by Martin Isaacs had the encyclopedic nature of Hungerford, but the explanations were significantly better and the proofs very well-developed.

[I gave Lang's textbook a chance, but couldn't stand the horrible grammar, random italicized conclusions in the middle of paragraphs, and countless comma splices.]

As an interesting example, out of those four books the only one that satisfactorily stated the various isomorphism and correspondence theorems was the one by Martin Isaacs. These theorems aren't particularly difficult, but there are a number of subtleties. Herstein left out almost all the subtleties. Artin didn't explicitly state the subtleties but used them anyways (as is his style :D). Hungerford just, well, I don't even remember what he did. And Martin Isaacs had very clear, very complete, very rigorous statements of all of them. Great for learning and great for reference!

In the case of analysis, Rudin is a hotly debated book. This is a great example of a circumstance where you should read both Rudin and another book (and maybe even another).

Of course, you don't have to read every page of every book. It's just important to jump around rather often and get different viewpoints, different levels of rigor and difficulty, and to see the material used in different ways.