How do you shorten the feedback loop when learning abstract math?

Question

I have been grappling with a problem in the process of trying to learn abstract math, and was wondering how other people deal with this.

In short, the feedback loop between learning a concept, and knowing whether you are applying it correctly is somewhat long. I can memorize all the definitions and theorems, but I lack the ability to self-regulate in the sense that I don't really know where the gaps in my knowledge are, and if I'm interpreting it correctly. This is particularly problematic when writing proofs, and not being able to see the logical issues you introduce.

When using this material in an academic context, there is the explicit feedback from an instructor or some such, but it usually takes too long for the feedback to really help in the learning process.

Do you have any pointers as to how to go about making learning math a bit more responsive in context of this feedback loop?

Thank you very much.

Answer 1

One advice i can give is yes you need to know the definitions but please dont memorize the proofs, there are 2 things u can do , one is you read the statement and by going through the definitions try to prove it yourself, that will help a lot with your proof writting, if you are not able to do it alone wich is normal after trying some certain amount of time read the proof of the author but rather then memorize it take ideas from it see where he used each condition that he had to use to prove the theorem, and see if you take some certains conditions where would the proof fail to be true or if you really need all the conditions that are given.This is just something i did when started to learn topology form Munkre's, when i was in the basic initial part where the proofs where basically just going through the definitions i always tried to make them alone first to see if i understood the concepts.

Another thing you can do is try to explain the concept your learning to other people, because only if your understand it correctly you will be able to explain it in a way that makes sense.

Answer 2

Work through as many practice problems as you can.

Working through practice problems or textbook examples is one of the best ways to see if you mastered a particular math topic or theorem. If you get it wrong, you get instant feedback. Practice problems do not need to be very applied, but can of course be theoretical as well, such as "how do you get form step A to step B in proof X".

Condense proofs to core points

Do not memorize proofs. Math is not about memorizing, but about being able to reconstitute a great deal of mathematical conclusions from simple beginnings.

Instead, understand what the prerequisite conditions are and why, and then understand, conceptually, the main steps and then practice on how to reconstitute the proof from that conceptual understanding.

For instance, "state $f(x)$ as a tautology by adding and subtracting $f(a)$, make it into an integral, then integrate by parts..." is the start of a conceptual understanding of one famous theorem.

Once you know that, practice reconstituting the proof from this conceptual foundation until you get it.

Join a relevant math community

Find an appropriate math community to discuss the material. There are many of them online. If all else fail, start your own workshop.

Answer 3

I'm of the belief that you need to learn by watching before you learn by doing. I'm an undergrad, btw, so I think I understand where you're coming from.

What has worked for me is to find an author who actually writes thorough proofs and study their proofs like the bible. This is not easy. For one thing, most authors choose to omit certain technical details from their proofs, presumably for environmental reasons, to save pages. Here are some authors I recommend though.

• For algebra, see https://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf or, slightly more advanced, https://www.cip.ifi.lmu.de/~grinberg/primes2015/sols.pdf. This guy was, to my fortune, the professor of my first junior level mathematics course. Although algebra is really tangential to my areas of study, his writing has greatly influenced my own style. You can pick up some good habits by reading his stuff carefully. He's also active on this website (hi Darij!).
• For analysis, Tao is pretty good (google "Tao Analysis I").
• For probability theory, just hold on for dear life. That stuff cannot really be discussed in true rigor at the undergraduate level.

What subjects are you studying rn? I might know a good source. Also, as I write this, it is winter break. I recommend of taking advantage of the break from classes to spend time reading a source such as these (I, for one, am doing exactly that).

Edit: The others who have answered have both recommended to not memorize proofs. I think one recommended focusing on the key points. I do not mean to attempt to declaim this advice, but in my opinion, it is necessary to first understand technical details, before glossing over them. As a friend of mine used to say (with reference to something completely difference), if you are simply bad at painting, you can't just call it abstract art. You first have to learn to draw very well in the classical sense before branching out.