How to succeed in upper-level math

[bobbobbob]

This question will essentially be more of a how-to plea or general help request.

I'm currently studying math and I'm at the point where I've transitioned into upper-division classes, most if not all of which are proof based.

To be blunt, I currently feel discouraged at the prospect of being able to succeed in "upper-level" classes. Last quarter I was able to get decent grades in my classes, but nothing like the success I enjoyed in lower-level math. My discouragement stems from the fact that in the past if I studied I did well. Now I feel as if I'm studying and not doing well at all.

Obviously this is a reflection of either my study methods or frequency with which I review the required material, but either way, I'm starting to become disillusioned with the good ol' saying that if you "practice, practice, practice" then you'll get better.

My question to whoever cares to comment, and believe me I greatly and truly appreciate all advice, is how did you transition into proof based classes and succeed in them? Obviously the material is different for each course, but in general how did you approach absorbing the material? While everyone is different, what are review methods that you found helped you to best understand the material and also retain your understanding?

I don't know if others have this same issue but I find that for proof based classes, since problems tend to all be different, I don't quite retain a sense of how to tackle problems even after completing an assignment whereas for classes such as calculus in which e.g. I had to learn integration, after so many integrals I had a general sense of how to solve them. Any advice on how to remedy this deficiency?

I apologize if this question is inappropriate in any way, but I'd love some perspective from those that have gone through a similar process.

Edited April 21, 2016 by bobbobbob

[ajb]

First I wish you luck and hope that you stick with it.

 

The general situation is that mathematics is hard, it takes lots of effort and you may have to allow yourself time to absorb the material. You have to accept this and not allow it to bring you down.

 

Sometimes it can help to talk with your class mates. Try to explain problems and solutions to them and vice versa. Getting other peoples thoughts on a problem can really help you understand what you understand and what you don't. Also, feel free to ask questions in this forum, however we will not do all your homework!

 

Once again, best of luck.

[RuthlessOptimism]

I have gone through the exact same thing before.

 

My advice sounds bad, but for me it worked. If you want to get better at doing and understanding proofs / more abstract mathematics the best way for me was to simply do more of it and read more about it. Simply seeing many correctly solved examples and memorizing the basic strategy for solving them is very helpful

 

There is a math competition in North America called the Putnam Mathematics contest. It is not the hardest one in the world but all of the questions are proof based and are marked as either 100% correct, or wrong. Joining / or forming a club for studying for it and competing in it is a good way to practice doing proofs. Even if you are very stuck initially and don't even know how to start simply looking at past solutions is very helpful. Like I said it sounds bad, but the more proofs you've seen and basically memorized the easier they will be to do on your own because you will have many similar ideas / situations to draw experience from basically moments like: "ah this is very similar to that problem that I've already seen done, I will try something similar and see what results".

 

Memorization seems to always be looked down upon in academia and creativity is favored with regards to problem solving. But in my opinion memorization is the foundation for creativity. You can only be creative with things that you understand so intuitively that you don't even need to think about why it works / is true. And like ajb said a good test for understanding is if you can explain in simplistic terms to someone who is not really a mathematician how / why something works.

[TransientResponse]

High level familiarance, on not only basics, but also on theories based on basics.

This comes by concentrated exercise of the knowledge.
Useful sources, and the methods of using them.
Experimenting/visualizing the theories.

Moreover, It is essential to relate phenomenons, facts eachother. And distingusihing among similarities, when needed.

Edited May 7, 2016 by TransientResponse

[StringJunky]

I have gone through the exact same thing before.

 

My advice sounds bad, but for me it worked. If you want to get better at doing and understanding proofs / more abstract mathematics the best way for me was to simply do more of it and read more about it. Simply seeing many correctly solved examples and memorizing the basic strategy for solving them is very helpful

 

There is a math competition in North America called the Putnam Mathematics contest. It is not the hardest one in the world but all of the questions are proof based and are marked as either 100% correct, or wrong. Joining / or forming a club for studying for it and competing in it is a good way to practice doing proofs. Even if you are very stuck initially and don't even know how to start simply looking at past solutions is very helpful. Like I said it sounds bad, but the more proofs you've seen and basically memorized the easier they will be to do on your own because you will have many similar ideas / situations to draw experience from basically moments like: "ah this is very similar to that problem that I've already seen done, I will try something similar and see what results".

 

Memorization seems to always be looked down upon in academia and creativity is favored with regards to problem solving. But in my opinion memorization is the foundation for creativity. You can only be creative with things that you understand so intuitively that you don't even need to think about why it works / is true. And like ajb said a good test for understanding is if you can explain in simplistic terms to someone who is not really a mathematician how / why something works.

That's a good strategy for getting things done but I think you will agree that knowing something is not necessarily understanding it. if you understand something you can actually manipulate it better to apply to more situations and even find novel applications for it..