Are there any memorization techniques that exist for math students?

I just watched this video on Ted.com entitled:

Joshua Foer: Feats of memory anyone can do

and it got me thinking about memory from a programmers perspective, and since programming and mathematics are so similar I figured I post here as well. There are so many abstract concepts and syntactic nuances that are constantly encountered, and yet we still manage to retain that information.

The memory palace may help in remembering someone's name, a sequence of numbers, or a random story, but are there any memorization techniques that can better aid those learning new math concepts?

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Are you asking how to be able to recall theorems/axioms etc... faster? That's a different question. –  sidht Oct 13 '12 at 2:01
I once heard an interview with a memorization champion on NPR. He was asked if his techniques for memorizing long lists of random numbers and names and such could help doctors, scientists, etc. His answer was simply "No". Memory tricks are just that...tricks. He went on to say that doctors and other experts can remember vast quantities of information because they understand the connections between facts and concepts. He went on to say that there really is no short-cut to true understanding. I agree. –  Bill Cook Oct 13 '12 at 2:03
Another example: I was discussing a topic with my PhD adviser and he recalled that there was a great paper by so-and-so which addressed the issue at hand. He went after a moments thought to recall the date of the paper (15+ years old) and picked it mysteriously out of a stack of stuff in his office. At the time I thought that he must have a most singularly amazing memory. Now years later as I study and read more and more, I find myself able to do the same (and trust me, my memory leaves much to be desired). Expertise and time builds these things. –  Bill Cook Oct 13 '12 at 2:07
One more thing...now that I think of it, the NPR interview was with Joshua Foer. I guess my memory is better than I thought. :) –  Bill Cook Oct 13 '12 at 2:09
It does help to know the names of the students. I used to draw up a diagram of their usual seats, and call on different students just to have occasion to associate their names with their faces. I did eventually learn their names. I'm not so sure what they learned. –  Will Jagy Oct 13 '12 at 3:47

You shouldn't try to learn mathematics through memorization at all. It will get you nowhere: anything that can be memorized can be looked up these days. What you should try to learn is the underlying concepts and the way they relate to each other. If you understand those well enough, you won't need to memorize anything.

Think of learning mathematics as being like learning, say, chess. Would you learn how to play chess by memorizing openings? Well, maybe that could work, but it's probably a better idea to learn how to play chess by, y'know, playing a lot of chess.

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This doesn't the answer the question. It questions the validity of the question and basically moralizes towards the person who asked the question. Also, people DO learn how to play chess better in part by memorizing openings. So, by analogy the question has validity. –  Doug Spoonwood Oct 13 '12 at 2:07
@Doug: I think that questioning the validity of a question is a valid answer to a question. If someone asks you "have you stopped beating your wife?" then I would prefer that "that question is predicated on a false assumption" be a valid answer. –  Qiaochu Yuan Oct 13 '12 at 2:10
You told me about what you prefer NOT what actually consists of a valid answer to such a question. Thus, you don't have an actual answer. Also, it does sometimes come as possible to answer a question while also questioning it's validity, "no, I haven't stopped beating my wife, because I never started beating my wife." –  Doug Spoonwood Oct 13 '12 at 2:14
You can’t understand a poem fully until you have it in your memory. Consider the case of a first-year student trying to understand the definition of continuity. She can look it up again and again, but it will still be mysterious and unreproducible until she comes to understanding. If one memorizes the definition, even before understanding, one may turn it over and over in the mind until it becomes internalized. The ultimate aim is for the understanding to be deep enough that memory is no longer required. –  Lubin Oct 13 '12 at 2:45
@QiaochuYuan, from my personal experience I was able to pick up math quickly simply because I memorized the multiplication table when I was a child. Now, something like that is almost expected but isn't much of the rest of math the same thing. Simply memorizing the properties of exponents or real numbers can have a similar benefit to that of remembering the multiplication table. –  Ein Doofus Oct 13 '12 at 3:23

For math there is no better way to remember than to just understand. Though the time required to reach that point may be too difficult to forget.

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This doesn't answer the question and basically just consists of criticism of the question similar to Qiaochu Yuan's "answer". –  Doug Spoonwood Oct 13 '12 at 2:10
True, I just couldn't resist the quip, memorization techniques really are not the best help in mathematics... –  adam W Oct 13 '12 at 2:14

If you say that the techniques that exist to help memory are amazing, I would not disagree. Visualization, they say, is the best way to memorize. I have not seen or heard of the "memory palace" other than what you wrote here and maybe in passing on an infomercial one time or another (am I correct in understanding it as a visualization technique?) I can say that I believe in the power of any visualization that helps in understanding or even just remembering mathematical concepts. The difficulty is in finding the personal imagery that works. Only after much meditation have I ever found such imagery that works for me on any particular problem I am considering, but it would not be something I could translate in order to benefit the random person.

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It is a visual technique, however, from what the speaker describes it's about creating the craziest association you mind gravitates to on a subject. For example, don't remember the name "Ed," instead think about the talking horse Mr. Ed after you enter your bedroom in your mind. Anyways, I think you might be on to something. I do remember that my computer networking class was easier when I visualized packets of information moving about. I'm curious, when you think about math concepts is any movement (animation) involved, or are they static visualizations?Could you please describe, if possible? –  Ein Doofus Oct 13 '12 at 3:55
As I said, description might be difficult. And especially considering we live in a 3-D world, where math is of many dimensions. One simple visualization could be that of shadows (good one for Halloween maybe) when considering projections in vector spaces. But to your movement question, yes my visualizations are usually very animated in one way or another. Paths taken through a tree structure, rotations about an axis, and shifts and scalings of topological shapes are a few easy examples. –  adam W Oct 13 '12 at 4:05
@Ein: in mathematical logic, "$p$ implies $q$" doesn't mean "$p$ has an effect on $q$." (Ordinary propositional logic doesn't talk about effects at all.) It means nothing more and nothing less than "if $p$ is true, then $q$ is also true" (so the only way it can be false is if $p$ is true but $q$ is false). The notion you're looking for is called strict implication (en.wikipedia.org/wiki/Strict_conditional). –  Qiaochu Yuan Oct 13 '12 at 2:58