Question

How do you actually retain math conceptual understanding?

Answer 1

Repetition. Once you stop using it it slowly oozes out of your brain.

Answer 2

My dude there is no stopping it. You WILL lose it if you dont use it. It definitely WILL be fuzzy unless you keep repeating it over and over and over. Anyone who says otherwise is high on copium.

Answer 3

I forget lots of things. But at this point, I know where I need to start looking if I need to (re-)learn or review something, and I am confident that I have the skills to re-learn anything I've forgotten. It's totally normal to have to go back sometimes.

To add to your list, I would say that doing problems, lots and lots of problems, is a great way to really understand and learn the concepts you are working with. Try to make sure you understand all the proofs and fill in any details for yourself that textbook authors have left out. Read different textbooks and watch videos about the subject to get different ways of looking at things. Talk about it with your friends/classmates, professors, or strangers on the internet. Also think about why these concepts were ever defined in the first place, understand the motivation behind it.

Answer 4

An advice I keep giving people but I rarely follow myself is to write down things in your own terms. Like really let loose and dont try to be elegant or concise or even too organized. Explain to yourself the concept you just got many times be redundant draw very clear pictures with lots of comments as if you were talking to yourself. I cannot count the number of times I go back to my notes and I find some cryptic one line that at the moment I thought was enough to summarize many hours of struggle.

I think you inevitably forget things when you dont use them often and thats fine, its just you need to figure out the key points to connect it all once again.

Answer 5

If you intend to stay on the academic path, at some point you *will* have to specialize in one narrow area of research, to be able to make significant contributions to the field. That means you won't have time to go back and practise some of the things you did as an undergraduate, and those topics will be lost to you eventually.

But such is the way of things. There are just too many things for any one average person to know within mathematics today, and the amount of knowledge simply keeps growing.

Answer 6

I use Anki.

It's not for everyone, but I have, among other types, cards asking me to prove things, especially important results. I think this is nice because it forces me to remember definitions and the gist of proof tricks and techniques as well as being able to make it all work. But ain't it mere memorization? To some degree, yes, inevitably so, but the best way to remember highly structured information such as mathematics is to understand it really well.

I also find the first couple of months I am repping on new cards to be invaluable in helping me grok the ideas and contextualize them. I do very few problems and exercises but when I do tackle them, I am usually able to perform well just from this practice.

This method would be particularly well-suited for fields with lots of details and neat proofs that connect well with other things I learned and that I go deeply into. It sucks when I am dealing with unintuitive and long proofs or with fields in which I am only dipping my toes. Sometimes I break long proofs down. Also, there still is stuff I don't remember well and this gets worse the longer since I have used or studied them.

In my experience though, my retention with Anki is night and day better and I enjoy studying with it. First, it is highly structured so there is little guesswork and I trust the system. Second, it forces me to be disciplined and helps me focus.

Answer 7

For me, I find as long as I can visualize the situation and form a mental picture, that this cements the conceptual understanding.

Having to teach others helps because I tend to teach the picture. This forces me to make sure the picture is there and is correct.

Answer 8

the best thing you can do is just practice. go back and do problems in old textbooks. approach gaps as gaps and fill them when needed.

Answer 9

I studied math about 20y ago and also programming.
I have worked as a self employed business owner in a completely different field since then.

I can't calculate stuff by hand as I used to, the craft is kind of lost. But, even though I don't remember the exact formulas or techniques in math, I haven't forgotten the gist of it, the implications.

 I could, if needed program basically anything (well..) with a few days or weeks of start up. I am pretty out of touch with any modern API's but once you can program you can sort of always ride that bike.

Answer 10

Imagine in your head you're talking to a kid who is just annoying enough to keep asking questions, but reasonable enough to not stray away from the topic. Now imagine you explain to that kid.

To save time, use this technique whenever you have nothing else to think about. For example, instead of singing in a shower, or ruminating about past relationship while you try to sleep, why not do this?

Programmer had a related method: rubber-ducking. Explaining to an imaginary object really help.

But you will lose it if you don't use it, that's unavoidable. But you won't really lose it completely. Relearning will be much faster. And usually I will vaguely remember that I used to know how to do this and where to learn it again, when the situation comes up.