Forgotten old results break my motivation

Question

I'll begin graduate school next year and I am very impatient to learn new things such as theories, ways of thinking and so on (I enjoyed reading about category theory on my own and I find Galois theory very interesting for example).

However, during my undergraduate - and I am worried that it would do that again in graduate school - I was often stuck because I had to remember old results and/or had to figure out basic things - that I didn't study - (and exemple of this "basic things" can be found in my previous post where I was stuck with a simple set theory equality involving complements) and I find this very hard to manage (in terms of energy and time consumption). When I finally understand what I missed, I am discouraged (it breaks my motivation and I am very tired). Who knows how many of these things I forgot ? What if I have to use one of these tricks during an exam / while learning new material ? How can I better learn to avoid this situation in the future ?

I would like to know how do you react while dealing with similar problem (those who have good - or fresh - memory are not in this position). Do you have a way to quickly find the good reference (except SE) ? Does that break your motivation like me ? If yes, what do you do to "get back on the rails"?

Thanks for your answers.

Answer 1

In my experience, any material that you do not regularly brush up on, you forget. Sometimes partially, sometimes completely. You don't learn things once, then know them forever. You learn things once, then you remember the stuff that you need again and again. So don't get discouraged when you have to brush up on old material in order to understand new material, think of it as a chance to brush up on material in order to keep that material within your grasp. At some point, when you have had to brush up on certain material enough times, it will stick. This is part of a natural process where you forget material that you don't need, whereas the stuff you need is set in stone.

Always remember that the stuff you need is probably different from the stuff some other mathematician needs. So the process is unique to you, and in the long run defines the type of mathematician you become.

Answer 2

Yes, been there a hundred times myself, the "Oh God, do I need to go through all that stuff AGAIN!" feeling. What helps me is to start a different section of a course, with new material to provide motivation, but to choose a section that depends for its understanding on the old material you have already covered.

So you are forced to go back a bit, then you dig out your old notes and find that after an hour or two, it's a lot easier to remember because, I think, its always still somewhere in your unconscious and I find myself saying, hey yeah, this is easier that I anticipated. I have the difficult parts already explained to myself in the notes.

I also try to keep an good index of my notes, it helps because part of the lack of motivation is the feeling you don't know what exactly you covered and you usually assume the worst. Also depends on your moood that day, I don't over think it or force it , I just let the motivation return which it does naturally when you see some new stuff or stuff you know that you already understand.

Like a golfer tired on practicing his swing a million time, gets fed up then the next day , sun is out, and he is invited to a new course with his friends. Will his motivation return, I reckon it would.

Answer 3

This also has to do with how you used to study. I always forced myself to be able to reproduce everything from scratch. Deriving everything from first principles with all books closed on a blank piece of paper is a bit more time consuming, but this yields superior results.

To this date I can reproduce most of what I learned decades ago without needing to look up anything. The only reason I do look up things is because that saves me time. When I have to teach and I want to refresh my skills, I'll first reproduce the entire theory on a black piece of paper just like I did when I was a student. Only then will I look at the books. I'll then know what I should pay attention to.

What helped me here was the fact that I started studying from advanced math and physics topics from university books when I was still in high school. I therefore had plenty of time on my hands, not under any pressure to submit homework of to be prepared for exams. And then when I went to unversity I was so far ahead that I could just go on with that routine.

Answer 4

Some very good points are made in previous answers. I would just add that it's also good when you learn new definitions and results to dissect and interpret them to the maximum degree possible. It helps to remember theorems if you have a strong intuition for why they are true, what basic/familiar results they generalize (e.g. many things in functional analysis are generalizations from familiar results in linear algebra), what sorts of pathological counterexamples a definition is meant to exclude, etc. It's always good to try to see what would happen if certain hypotheses were dropped from a theorem - why would it fail?

Personally, I find that this kind of reasoning both sticks better in my mind and is more fun if I talk it out with a colleague for whom the idea is also interesting. A colleague can challenge your understanding by asking questions you wouldn't have thought to ask, and somehow the act of verbalizing always forces me to make my ideas more precise than if I just ran through them in my head. Sometimes thinking by myself, I have an over-inflated impression of how concrete my understanding is. Partly this is because I'm always impatient to learn things fast so may try to rush forward, and partly because I have a bad habit of agreeing with myself. This means that a good colleague to talk with is one whose intrinsic nature combats these tendencies. These might or might not be similar to your personal patterns, but maybe the principle generalizes even if not the particulars!