What are effective techniques for retaining knowledge in mathematics?

[The Learner]

Hi everyone,

I'm pretty new to the PF and have rarely posted on forums but looking through the threads, I see that you guys give some great advice so I'd love to hear some responses.

My current situation: I've just started my third year and am majoring in physics. I intend to get an M.S. in physics and be something like a H.S. teacher and eventually hope to improve the education system here and abroad.

The problem: I think what it comes down to is the math. I've taken Calc II & III, I've taken Physics I - IV, Linear Algebra, Diff Eq, and Math Methods in the Physical Sciences but I can't say I've retained much knowledge of the techniques. Familiarity perhaps, but not mastery. (I recognize there's a difference between mathematics and the cookie cutter techniques but I feel like mastery of those techniques/confidence with them is needed to teach)

Not to be too desperate but I think what I'm looking for is reassurance that I'm not alone...is it normal that I don't remember everything I've 'learned'? Is it normal to have to keep returning back to my notes, and basically relearning things after a semester or two, does it just take time?... I'll admit, I haven't done too much repetition and practicing, so that might be the giant problem.

The main thing I'm looking for is what are learning techniques that you guys (you know, you all being the best of the best :) ) find effective? There's a difference b/w learning and memorizing, but I find it hard to say I've actually learned something when I can't remember it. And to what extent people generally remember material with these techniques.

I'm not sure if it's my methods that need work, if I just need to slow down and solidify my math foundations, or if there's something off with my ideals (i.e. a great physics teacher doesn't need to be a master of integrating functions even though that'd be so cool haha). Really, any insights offered are super appreciated. My apologies also for the long post!

 

[chiro]

Hey The Learner and welcome to the forums.

One thing you should be aware of is that the contextual relationship of knowledge when learning something for the first time is not the same as revisiting that same information.

When people start learning, they do not actually have the memory and interpretation of something that some-one who has already learned something, is simply revisiting.

For example when you first see an integral without seeing the ideas behind it, it would look like some kind of weird alien symbol. Then you might look at the identities regarding integrating functions like x^n and you see a pattern, but still you don't get it.

But if you realize how integration relates to addition of changes and differentiation finds those changes, when you see an integral in this context you think of area, volume, length, total amount of change, flux or something else and you have context of what the integral means in another language that you wouldn't have if you are looking at an integral like you are trying to decipher or reverse engineer an ancient language without an instruction manual.

Losing some of the fine details is not going to hinder you, if you remember the stuff that's the most important even if you forget most of the details.

So with this line of reasoning, the best thing I think for you do to in terms of learning is to get the core ideas of mathematics (or anything else for that matter) and understand these things from as many perspectives as you can.

This means that if you do forget formulas, identities and so on, you take the core ideas and look at what they actually mean in any perspective rather than remembering details that have no context and are simply isolated.

Memory is a contextual thing, so it's important to remember that things aren't explicit like 1's and 0's: everything is relative to something else.

I'll leave you with a final thought on mathematics: in mathematics, the descriptions and definitions are written in such a way that they are able to describe a great deal of information in a compact form.

This has its benefits and its shortcomings: the benefits is that you don't have to remember as much, but the shortcoming is that most of the context is lost and it's something that most people discover after doing math for a long time (which includes discovering what other pieces of connected math have to do with these compressed definitions).

If you can find a way to, in your own mental language and organized methodology, to take what you have learned and to write what the core ideas mean to you, then when you forget something, you can pull out this resource and get yourself to remember as quickly as possible.