Effective learning techniques for abstract mathematics?

[JohnKeats]

When you're studying abstract mathematics what are some effective techniques to use? When studying something like abstract algebra do you think something like this sounds reasonable?

1. First learn the relevant section until you reach the exercises.
3. While learning that write down the main definitions and proofs.
5. Then try to reproduce the definitions and proofs on your own.
7. After you're able to reproduce the definitions and proofs, start the exercises.

Then repeat and rinse that for the next section after you do the exercises. Does that sound reasonable? The reason I ask this is because I've realized that sometimes I fully follow an argument in the textbook but then I forget it. Like I understand it, but I'm unable to reproduce it. Is there a remedy for this, and do others experience it or is it just me? I guess the other question I've is that the list I've made is sort of like memorising the definitions and proofs. Is this really so bad? I suppose memorising stuff is bad when you don't understand it, but it can't hurt to memorise stuff you fully understand, right?

 

[I like Serena]

Hey JohnKeats!

Here's my 2 cents.

As I see it there are 4 steps to learning something new.
This is not limited to abstract algebra - it applies to any field.

1. Learn the jargon: the new terms, symbols, and definitions.
   These are best memorized, which will speed up understanding.
   I recommend making a list on no more than 1 sheet of paper as early as possible.
2. Reproduce examples/proofs. You can memorize them as an intermediate step, but ultimately that will not help.
3. Apply the theory to a different case for which no example/proof is available.
4. Apply the theory in an out-of-the-box fashion.

These steps usually correspond exactly to the exercises that are roughly divided into subsequent sections.
It means you can jump up and down to the exercises.
If they are easy you can skip stuff. If you get stuck you have to go back.

The first series of exercises will check if you know the terms, symbols, and definitions.
If you think you know them, you can use them to verify that it's the case.
And you can stop if you feel they are just too obvious - you know the terms.

The next series of exercises is to check if you can reproduce the examples/proofs.
You can memorize them, but remember that it will just be an intermediate step to the next phase.
If you can do them, you can probably get a grade on an exam that is sufficient to pass.

Next are the exercises that force you to actually understand what the theory is about.
Otherwise you can't do them.
This is where memorization breaks down - you really have to understand.
If you can do them you can be comfortable in the knowledge that you'll get a high grade.

Finally, often mixed with other exercises, are the 'star' exercises.
You have to be brilliant to do them and/or google a lot and/or ask for help and/or spend an enormous amount of effort on them.
These are optional. You can stop if you run out of time.
Get far enough with them and you'll get a perfect score on an exam.

This is how it works in the undergraduate level.
However, in more advanced texts there are often no exercises at all.
You are supposed to come up with them yourself! ;)

 

[JohnKeats]

Thank you for your invaluable input, Klaas van Aarsen. :D

I'm glad to see that my initial plan at least sort of resembles like yours.

Can you suggest any additional tips on speeding up the process of learning? I suppose learning the right way is always going to take time, but I'm somewhat impatient. Also I realize that you have already suggested some tips that would speed up the process like skipping stuff if they are too easy. I talk about speed because I'm reading Chapters 7, 8, 9 of Abstract Algebra by Dummit and Foote and I've estimated that finishing those three chapters would take a minimum of about 3-4 weeks, leaving no time to learn anything else.

 

[I like Serena]

Thank you for your invaluable input, Klaas van Aarsen. :D

I'm glad to see that my initial plan at least sort of resembles like yours.

Can you suggest any additional tips on speeding up the process of learning? I suppose learning the right way is always going to take time, but I'm somewhat impatient. Also I realize that you have already suggested some tips that would speed up the process like skipping stuff if they are too easy. I talk about speed because I'm reading Chapters 7, 8, 9 of Abstract Algebra by Dummit and Foote and I've estimated that finishing those three chapters would take a minimum of about 3-4 weeks, leaving no time to learn anything else.

A lot of time is usually lost on trying to figure out what is meant.
This is even worse when following a lecture.
While you are still trying to figure out what he said, he has already moved on, and you're missing the next part.
It's like listening to someone talking in Chinese (assuming you don't know the language) - it doesn't make sense - and we're just wasting time.

The key to get the jump on what's going on, is what I already mentioned:

1. Learn the jargon: the new terms, symbols, and definitions.
   These are best memorized, which will speed up understanding.
   I recommend making a list on no more than 1 sheet of paper as early as possible.

Spend 10 minutes before a lecture to scan the italic words in the text and read the sentence just before or after that says what it means.
Or even better, make a list of all italic words in the text, and write the corresponding sentence next to it (may still fit in 10 minutes).
Then, when a lecturer mentions a word and uses its subsequently, instead of being lost and missing out on everything, you can think, ah, that's one of those jargon words, what was it again? And things will start to fall into place.

When doing exercises the same thing applies. At first the exercise just doesn't make sense. And you can get stuck on something like that for a long time. But if you know the words already, and/or can quickly look them up, many exercises become clear and may even seem trivial implying that understanding has been increased.

Finally, before an exam, check the list again and see if you can instantly remember what each term or symbol meant.
If you can, you are ready. If you can't, spend the time to figure out what it was again.
As trivial as it seems, it tends to 'jump' your grade.