How do people 'master' a subject? What is the difference with 'memorizing' (not purely memorizing, but rather having it drilled inside the brain) it?

Question

I'm in my 3rd year of pure math bachelors, and a thorny issue keeps on reappearing for me. I understand the theory, for a day, maybe two. I've reached the point where our lecturers no longer detail every single step, but leave some details to be filled out myself. Most of the time I have no issue filling in the blanks myself, I understand everything, I can follow the line of thought. But after a few days I just forget everything, conditions for a theorem, etc. Even what I studied yesterday can be forgotten today. I feel like there is a finite box inside my head, and after its full if I want to put something in I have to push something out (involuntarily).

Are there ways to solve my problem? It's very worrying for me I can debate with my friends about the theory when I have access to my notes, but keep on failing theoretical exams (written or oral).

Answer 1

Create lessons, blog posts or the like, explaining what you want to learn to an audience of an appropriate level - could be just yourself, or your friends, students of a similar level, etc. You often learn what you teach, even when not literally teaching as a paid tutor or instructor.

Answer 2

This is purely anecdotal, but it helped for me so perhaps it's useful for you too.

Instead of trying to memorise definitions, theorems, proofs etc. I think it is important for your brain to do some active exploring. By this I mean, try to understand why the definitions are the way they are, what "philosophical" idea they try to capture, and how they connect to other concepts. Try to see how the conditions imposed in the definitions allow for the key theorems to be proven, and what happens when you strengthen/weaken these conditions. Or, at the very least, try to come up with the proofs for a theorem or understand the definition on your own, independently of the textbook first, rather than reading through and then trying to recall it. This active discovery process is imo essential for doing mathematics.

For example, when learning linear algebra, ask yourself: why do we need to consider vector spaces defined over a field? Why do we need the ability to divide scalars, what happens if we replace it by a commutative ring? Well, it turns out that you get modules and some key results about vector spaces don't carry over, e.g. uniqueness of the size of a basis, which makes vector spaces (especially those of finite dimension) significantly simpler. This gives you some deep understanding that is very difficult to forget; moreover, you more easily see the connections between different fields that will be helpful in the future.

I think actively trying to memorise things in mathematics, including the definitions, is totally against the spirit of the subject. It won't even be effective anyway, especially not in the long run. Don't do that, you won't enjoy it anyway.

Answer 3

Perhaps as much an extended comment as an "answer": yes, sheer memory inevitably plays a role in almost any endeavor. But the comparison I'd want to make is that we don't "memorize" how to walk across a room, or throw or catch a ball, or many other things-we-do. We do not formalize them: we do not attempt to describe what's going on, nor the goals, by a list of rules. Rather, just as with a driver's license test in the U.S., the formal test itself is quite disconnected from one's skills as a driver.

As @JochenGlueck commented, thinking to memorize things (with "flash cards", etc.), is at best a shallow "fix" to your situation. Rather, I'd recommend thinking that (contrary to the tradition of many textbooks) the "definitions" are convenient naming conventions that (by this year) have proven very relevant to talking about real phenomena. So (good) examples are the longer-lasting point... Those are the reasons the terminology was invented.

Also, for me, at least, thinking that the objects and their properties are real, rather than made-up formal abstract non-existent things, helps me take them seriously, and remember them.

Answer 4

I realize the work is harder and more abstract, but you need to try to use conventional methods of pedagogy, nonetheless.

1. Make a set of flashcards for key definitions. Drill them. You can carry a pack and go over them when stuck in a line or waiting for class to start or the like (little lost time parts of the day). I know this sounds juvenile, but it will help.

2. Install a system of rolling review. Note, rolling review does NOT mean re-reading stuff. It means retesting yourself. By redoing proofs, doing problems.

3. Try to figure out a way to do some simpler drill problems for each lesson, not just super hard ones. Look at alternate books, etc. Or even just doing the proof several times. Maybe even hokey things, like creating two examples and two counterexamples.

4. Do all the homework problems for each lesson, when you get it. You say you can handle the material at the time. But don't recall it later. Well, do all the problems then. Not just the assignment. But everything in the book. [For rolling review, use other books with more problems, or just repeat the old ones...at least it proves you're not forgetting stuff!] The point is that you need to "overtrain". Don't say "I get it". Drill the content into your brain like Umbridge's pen into Harry Potter's hand. I see students in all kinds of coaching, testing, etc. (not pure math, but many subjects) who think "I got it", but they really haven't firmed up the knowledge enough. And it shows under pressure.

5. Given you are failing, figure out some plan to allocate more time to the coursework. Ask advice from an advisor, but make it your responsibility in the end. Can you slim down the number of courses? Eschew work or clubs? In the end, if it's bad enough, you may want to drop to an easier track (applied?) if possible. But try to make this work first with best efforts.

6. Meet regularly with your academic advisor for discussion of your situation. It can be a little monastic to do so much math homework, so if you have some human connection understanding your effort, maybe that helps you put in the work. Also, he may have advice for specific classes (yes, different/additional to your course teacher).

7. See your teachers regularly with questions and to review homework issues. For one thing it's also human connection. And it may help on the content also. Not to brown nose. But with specific things (keep a list in your notebook...I use the back and go towards the front, since notes and homework drill goes front to back.) A lot of teachers will see you outside of "office hours" if you just make an appointment (can try dropins also, but be prepared for a no...but then they'll usually specify something later).

8. This is controversial, but I have mixed feelings (mostly negative) on study groups. The human interaction is nice. But you can get distracted with socializing. Also, you are not going to have the study group on the tests. And then if others are much stronger, you may just sort of nod your head and make a few remarks. But not really be training yourself to know the material. I would probably moderate the use of study groups (lower time spent there) and try to do prework on your own and have specific questions for your colleagues, when you do go to them.

Answer 5

I had the same problem when I was a student, and it took me a while to learn and accept that forgetting is actually a good thing because (vastly simplified) because it gives you a way to recode the material in a different way and that offers fresh perspective. There are neuroscience books on memory going into details.

Here’s a quote from Andrew Wiles that I found on the internet:

• I really think it's bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it's a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that's what actually enables you to get round [the problem]. So if you remembered all the failed attempts before, you wouldn't try them again. But because I have a slightly bad memory I'll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do.*

I was also surprised to learn that very accomplished researchers tell me that the remember nothing about papers they wrote ten years ago…of course if they went and reread them they would fairly quickly be able to refresh their memory.

In terms of concrete things one could do to help, I would say talking/explaining the material to others via e.g a study group, a student seminar, or a YouTube channel.