Not the intended target, but I'll throw in my two c from doing a lot of applied math.

Math readily has two components. The first is a formulaic, formal component that can be readily overcome by rote. The second is the more freeform conceptual understanding that motivates and directs the first. I feel confident that if you ask anyone familiar with advanced math if they understand concept/theorem/tool X, they'll say yes if they know it in the second form and are confident that they can reconstruct the important parts of it in the first.

I think a lot of why people rebel against rote memorization then is that it, as a method, is very likely to prevent you from encountering the second side there. If you honestly use it to improve your fluency with the formal manipulations, it can be a great tool for learning more math. It's just easy to lose that honesty.

To really understand math, you need to recognize that it's a language you must both read and write. I suggest that if you do get strong benefits from rote memorization, then you should complimen t your reading by attempting to synthesize mathematical concepts you've not seen before. Read the claim of a theorem and then prove it yourself without knowing the answer. If you can honestly complete mathematical synthesis at that level as well, then rote memorization isn't hurting you in the least.

To elaborate on this, a working knowledge of some area of mathematics is not like a set of historical facts to be familiar with, or a list of fundamental particles and their properties, or a group of plays or novels to be quoted from, or a set of pigments and their interactions with brushes and paper, or even a code library’s API.

Mathematics is, fundamentally, about model building. The study of mathematics is about learning how to make maps even more than it is about the specific territory being mapped. In my opinion the largest part of mathematical fluency is the constant willingness to test mathematical structures and ideas against each other and against new data, to figure out how parts work at their deepest levels and then to go back and try to see how each one fits with all those known before. What matters in understanding a mathematical concept is not whether you can repeat a witnessed proof step by step or write down a formula, but whether you have an intuitive grasp of the abstraction(s) in question, whether you can explain them to yourself (an ability to explain them to others also recommended), and whether you can apply them to new problems which arise.

It is my belief that this kind of deep understanding and fluency can only be obtained by repeatedly interacting with these abstractions in a wide variety of problems and contexts, writing down the patterns and working through the proofs, questioning the axioms underlying them, asking how they generalize or how they apply to specific cases, and so on. Very little of this work can be done on flash cards, at least for me personally. Indeed, I believe it is precisely the teaching of mathematics as something which can be learned from flash cards which most impedes mathematical education and understanding.

See http://www.maa.org/devlin/LockhartsLament.pdf

Thanks.

Thanks. That's a great response.

Everything I remember about maths I remember because I understood the "why" of it. The stuff I learned rote - just to pass the exams - I have long since forgotten. I'm now frustrated that I was expected to learn anything by rote, as it's all lost to the sands of time now.

Rote memorization might help you with exams and book problems, but it won't help you develop long-term mathematical problem-solving skills. If you can't explain it from first principles, you don't truly understand it.

I'm not saying that problem solving and understanding aren't important (see Bloom's taxonomy).

I'm saying that you still need memory to justify whatever domain your knowledge is in, and that memory seems to be the bedrock of further knowledge (understanding, creativity, problem solving etc.).

Am I missing something here? Do you not need some element of memory to explain things from first principles, and to have an understanding of something?

edit: I've just realized from reading wtallis' comment that I might be confusing two different forms of knowledge: memory and reasoning, insofar as reasoning can produce further truths. Is that what you are getting at?

You can easily manipulate symbols without understanding what's happening. The problem is it's hard to progress significantly past your understanding using this approach. Not that Math really has levels as such, but if you don't get Calculus at a fairly deep level DiffEQ ends up being fairly meaningless.

PS: It's a common thing for most calculus classes to redirive old formulas. Not because they stoped working, but rather because you really should understand why it's 4/3 pi * r^3.

It is similar to the difference between say svn and got: you should not remember the various revisions, but how to get there, because that will knowledge will be reusable in different branches.

As an example, I had a maths professor who derived the formula for solving a quadratic equation in class (in 30 seconds or so) because he did not remember it.

Depends, I had that opinion too until I took university maths. The proofs they give you to learn by rote are actually extremely useful, as they add techniques to your repertoire that you otherwise wouldn't have..and the difference between 'learning the proof by rote' and 'learning the technique' is usually extremely small.

To add to enneff's comment: procedural knowledge is at least as important as the end result of a derivation, and in the long run is more useful.

I had several mathematical classes that weren't actually math classes (eg. statistics and physics) where other students would cram to memorize a page full of formulas, and I wouldn't even know the names of most of them going in to the exams - any formula that I could derive in under three minutes wasn't worth memorizing.

When a major outstanding problem in mathematics is finally resolved, it's surprisingly rare for people to care much about the result. Generally, people have checked enough cases or used other methods to be fairly sure what the answer really is. What gets mathematicians excited is the fact that a new proof brings with it new techniques (because if a problem can be solved using existing techniques, it doesn't withstand attack long enough to become legendary).

Ok, I think I might understand what you are getting at. I seem to be confusing two different forms of knowledge: reasoning and memory (where reasoning can be used to produce other truths).

There are already some great answers to this question.

Math begins with intuition. We don't memorize facts in order to build intuition -- we explore, discover, and synthesize. Formalism is an impediment to the early stages of this process. The most blatant example is set theory. Nearly everyone has a basic grasp of naive set theory and hardly any layperson has a basic grasp of formal set theory.

Just as an aside, many of the most interesting things in mathematics are extremely counterintuitive. It also turns out that our intuition is broken and leads us to say crazy things. Again, set theory has clear examples.

Back to the point, the things you memorize help to organize your untamed mathematical data. I seriously doubt that you'd have any good results from memorizing something about which you do not have intuition.