Effective Methods of Studying in different areas of Math

Question

I apologize if this question isn't appropriate for this site, but I am looking for advice that I think other math students might be better able to give me.

I am an undergraduate math major entering into my second year.

What I have found is that I love theoretical discussions of mathematics and proving theorems, though the education system focuses on doing exercises and applications, and that is what grades are determined on. I have a hard time doing exercises mainly because I don't find them motivating, although, I admit, doing the exercises does cultivate one's intuition to a certain degree.

Up until now my main technique of learning was to take meticulous notes from textbooks(not in class as all they do is mundane exercises), think about those concepts and then practice exercises before exams, and it worked fine for discrete math, precalculus, and even the first course of calculus, which was mainly focused on derivatives. Further. I as able to maintain A's in all my classes with enough energy left over to even do very well in other classes.

However, this hasn't gone too well in the second course of calculus, which is mostly about integrals of various functions and their applications and techniques. The instructor usually does problems on the board step by step without any discussion for what we are doing. For example, both the fundamental theorems were introduced and given about 15 minutes of time to discuss. But, of course, this could be bad teaching style.

My question is this, should I learn math theoretically and understand the material well even though on tests this would mean getting about B's that I have gotten or should I only care about the grades and just do what I am told, like a human calculator, practice exercises and just focus on getting A's? What is the better for me in the long run if I want to get into a decent grad school? And have any of you ever had to decide between getting an A or spending more energy and time actually learning the theory, which is more satisfying? I could be looking at this completely wrong and perhaps there is a way to do both? Thank you and your input will be appreciated.

Answer 1

I think it's mostly a cultural problem of math teachers. Nowadays it's more or less an habit to avoid to explain almost anything and just copy on the blackboard theorems and formulas. I'm not saying that everyone is doing that, but that is bad habit that is quite widespread.

That's mostly because since math became really rigorous, many times you don't have the time to effectively prepare the lesson, a lot of professor don't want to get errors and they just copy the notes they've done years ago on the blackboard. More of that it's just an habit. 15 minutes of discussion on a theorem was much more that I ever got when I was undergraduate.

My suggestion would be to talk directly to have an insight picture. Talk directly to someone who knows things and which is friendly enough and has the time to explain them to you. You can study alone, but that's the hard and often logorating way. The easiest one is to: talk with friend, talk with professors, find a nice online community of people who knows and can give you an intuitive picture of what's going on! When you know really the subject then difference between theory and practice will slowly disappear.

After you got the idea, work out the detail rigorously. After all this preparatory stuff you will probably find a good professor for your thesis and talk with him as much as you can to have right pictures of what's going on. Usually behind even complicated theorem there's always a practical need or a simple idea, try to catch it and then work out details.

When you'll become a professor try remember the problems you had at the beginning because is quite common to forget them :)

Answer 2

I feel your pain, performing meticulous calculations is not fun and can often make the subject appear dull when you are asked to do them over and over again.

What you can do though, is either attempt to learn the theory that enables you to do those calculations (if it is within your understanding), or more importantly, gain an INTUITIVE understanding of why it works. This might not have the rigor that you're looking for, but will benefit you greatly in the long run when you eventually do enter into the more theoretical classes when it is necessary to formalize your intuitive understanding in proofs.

Answer 3

when I took calculus ii, most of the theorems in the book covered in the course had proofs that, I believe, the author deemed the reader should be able to understand. Proving propositions are fun, and plenty of proof techniques can be birthed and garnished in the mind of the student in courses like discrete math or intro to proofs, but I would suggest spending more time doing what you are told to better secure a decent grade in the course, and then whatever study time you have left for that particular math course to study a theorem and its proof, if it has one. If you can balance your study time well with your math and muggle courses, then you should be able to achieve spare time for fun, namely doing the proofs. For example, when studying sequences in calculus, know how to apply the alternating series test, and then understand its proof later.

Further, the course is designed to ensure that your well suited in understanding the material presented in the course. If you spend the majority of your time understanding the proofs, which does not necessarily mean you'll be better equipped to tackle the problem sets, your problem solving skills will diminish,Problem solving skills extend beyond simply tackling applied problem, for there are instances in pure mathematics, or just proofs in general, where you must simplify an integrand in order to proceed with evaluating an integral necessary to assist in a proof, knowing algebraic tricks in inductive proofs, etc.

You must find a balance between doing the problems assigned to you and reading/doing the proofs. If the course is has a heavy emphasis in doing problems, don't despair: Do what you're told, develop intuition and mathematical maturity, and soon you will undergo courses where the demand is on proving propositions, like real analysis.

Lastly, take charge in your class or your professor's office hours to acknowledge or discuss the theoretical. I'm sure your professors are just as bored as you are in the grim ennui that often accompanies mechanical problem solving.