Memory is crucial. I got much better at learning mathematics when I started memorizing the statements and proofs of key theorems.

You also need the understanding, of course. The largest mistake in mathematics education is to think that memorizing a rote process is the same as learning mathematics (see: US high schools). The second largest mistake in mathematics education is to swing too hard in the other direction.

Absolutely. Being able to instantly recall concepts, definitions, syntax, etc. is incredibly empowering when solving problems, regardless of the subject (mathematics, programming, writing, etc.).

Nobody would suggest that you try to learn a new language without memorizing words, phrases, idioms, etc. Communication is the foundation of any field of interest, so why do we think we can get away with "learning" them without memorizing the sub-language of the field?

Just as memorization is only part of process of learning a new language, so it is for learning in other areas.

If all you have is some rote memorization about mathmatics you can use those memorized items to study in your own head. So while you shower or drive you can think about the math you've memorized and develope a deeper understanding.

As you say, memorization can be a useful part of a deeper understanding.

Learning without thought is labor lost, Confucius said.

At any rate, symbolic representations of knowledge should be committed to memory (or pad and pencil) at the time of understanding (Hebb's rule), or else your "memory" will be isomorphic to random symbols and therefore inaccessible to future thoughts.

That said, some of the greatest mathematicians (e.g., Gauss, von Neumann, Dieudonné) had phenomenal memory, so I suppose rote memorization isn't a terrible hack to try to approximate the advantages this kind of gift confers, but only in a pinch, since these mathematicians were keeping much more than definitions and statements of theorems in memory.

I had an absolute beast of an algebra professor who impressed upon us the fact that things are defined they way they are for a reason, and we, as mathematicians must know them. It is not enough to be able to remember the definition of a normal subgroup, for instance. We must know why we define it the way we do, and what happens if the definition is altered in the slightest, weakened, strengthened, negated etc. It is not enough to know that quotient groups are defined modulo normal subgroups only. You should know why defining a quotient group modulo a non-normal subgroup is meaningless. Even for much more complicated categories, like manifolds, one must absolutely know why all the bits and pieces are needed. Why do we require the transition functions for real n-manifolds to be C^r? What if r=0? What happens if two charts on some underlying topological space are not compatible? Why do we need the atlas to be maximal? Why on earth do we need a countable base? Hausdorff? What is this business about needing manifolds to be Hausdorff? [0]. Why is it that bijective homomorphisms are the isomorphisms in the category of groups - i.e. why is it that its inverse is not required to be a homomorphism as part of the definition? Learning definitions by rote is the surest route to mathematical mediocrity.

"Don't just read it - fight it!" - P.R. Halmos.

[0] Yes, I know these can be relaxed, but this is the standard presentation in Kobayashi-Nomizu/Spivak/Lee/Tu/Barden-Thomas/Do Carmo/Petersen etc etc.

Surprisingly - as stated by others - memory is crucial to math and problem solving. Most problem solving is just memory on a higher domain anyway. Break down a problem with first principles (hard part), and solve with a mental model gained from memory.

Euler - probably the greatest mathematician of all time - had an amazing memory. He memorized the entire Aeneid and could recite any line at will.

In that case you're learning foreign vocabulary, something Anki is more traditionally associated with.

The Greek roots that appears in English words aren't really foreign, they're part of the English language and have been since forever.

That may be the case, but most people don't know what they are and what their original meanings are, so they might as well be foreign words.

So they haven't heard of epiphenomena, epithelium, epigenetics, epigraphs, epiderm, epithets? Or monosyllabic words, monoski, monochromes, monogamy, monograms or monologues. It's not hard to guess the meaning of the roots.

The only word of these that I would assume any given english speaker knows is monogamy, and maybe monologue. I myself know those, epithet and monogram. Perhaps I could guess on the others, but I certainly wouldn't know what epi- meant.

I used handmade cards and it made all the difference. A good foundation was necesarry for me to make things click. I tried anki later on, but it was too cumbersome to create and work with digital cards for me.

Memorization and definitions go hand in hand. After all, you need a certain amount of memory to understand that epi means 'upon' or that mono means 'alone'. If I retain that, is it my understanding, or my memory?

That leads me to my next point, which is Anki is best used when you do what's called rote-memorization, but memorize simpler aspects -- so have one card that asks what does 'upon' mean in Greek, and another asking what 'mono' means in Greek - in addition to making more general questions regarding this subject.

Perhaps easier: if you can remember that "mono" means "one" (as in monophonic, monosyllable, monomania, monopoly, ...) then that should make it easy to remember it's for (the analogue of) one-to-one functions -- same as injective.

(But the OP is French and I don't know whether French mathematics terminology includes anything corresponding to English "one-to-one".)

Came here to say the same thing.

This won't give you an understanding of mathematics, just perhaps a memory of it.

A memory is also crucial. Things like standard derivatives/integrals or standard probability laws need to be memorised be it by this method or by experience.

I agree to the extent that once understood, things need (presumably!) to then be remembered.

I suppose people's methods differ, but personally I don't think I could have an effective memory without understanding, and that my memory follows understanding.

By "effective memory", I mean that I could actually apply some definition, not merely remember the pattern of notation on the page. I fancy if I learnt it from a flash card I'd struggle when variables were named differently, or with the slightest variation in form.

I used to be in your camp but now I think they go hand in hand. If you insist on understanding before remembering, how do you start?

When we start a lot of things are interrelated and unless you a bare minimum is committed to memory, it's difficult to start.

It is akin to learning a language where one need to cram the basics building blocks i.e. words and grammar rules before applying them. I think idea of trying to understand before remembering ends up impairing learning at least in my case.

I have a two year old and I see this process in action. She will first cram things like counting and then start attaching meaning to it slowly.

There are some things we must remember, of course. A child has no choice but to memorise the digits. Axioms, essentially.

When it comes to addition however, any attempt to learn addition by tabulation and memory would clearly be futile.

I certainly don't mean to suggest that one should study a theorem in a particular corner of mathematics from fundamentals up; rather understand it "one level down", to something previously interned in a similar manner (and since forgotten because one no longer had need of it) or to somemething axiomatic.

For example, say I come across an equation for the area of a right triangle for the first time. I've previously understood and accepted some other problem that I can reduce it to - area of a rectangle, say, or integration - so I can do so and actually understand why, and be able to reproduce it. Easier to learn, recall, and apply than rote memorisation in my opinion.

I see it more like freeing up you active working memory to do other things, similar to how you can only become a better driver once you aren't constantly fretting over exactly how much to lift the clutch as you accelerate. You can (and must!) understand what/why you do what you want to do, but you can begin to abstract over it and find other patterns.