Best learning techniques for maths.

[Quixot]

Is it true that the people who succeed the most in academia are best at rote learning and self-discipline?

If not, what are the best ways at learning new material for you?

 

[e.bar.goum]

Rote learning is almost certainly totally the wrong way to learn. True learning comes through understanding rather than memorization.

The way understanding comes varies between people, but the stock answer is by practice.

 

[chiro]

Is it true that the people who succeed the most in academia are best at rote learning and self-discipline?

If not, what are the best ways at learning new material for you?

Hey Quixot and welcome to the forums.

My advice for learning math is to get the big ideas first and then fill in the gaps by doing the exercises.

When I say big idea, I am referring to finding out the whole point of a particular subject. For example in calculus, you are concerned with modelling change and by understanding what is actually changing, how that relates to some mathematical representation, and how that is calculated, you can put all of it together and then reinforce your understanding by doing specific things that are covered in your problems.

If your lecturer doesn't tell you the main ideas early, then I suggest you ask them. By keeping these in the back of your mind while you do any work you have to do, be it assignments, exams or other assessments, you will be able to help your understanding of both the broad and the specific concepts in your course.

 

[homeomorphic]

Rote learning has a bad rap. Maybe it would have better if I said that procedural knowledge is what will get you through maths? Thanks for the input!

No, rote learning is really not a good strategy. Maybe there are a few procedures you should know, like maybe Gaussian elimination. But it's mostly about understanding concepts and just practicing.

One of my big strategies is to rehearse arguments for why some theorem is true in my mind until it becomes obvious that the theorem is true. This doesn't mean I memorize the proof, exactly. I try to see the idea behind the proof. To look at it in just the right way, so that it makes it obvious. That can be tricky. Usually, I just keep trying different ways of thinking of it until one of them just clicks and is completely satisfying.

 

[scottbekerham]

firstly read the material to get the big picture then work out the details . The only way to be good at some math subject is to do a lot of exercises.Reading the material and understanding the proofs is not sufficient

 

[Obis]

It's better to deeply understand 3 theorems than to rote learn 100 theorems. You would forget them anyways.

 

[sweetpotato]

Rote learning is important for the most basic math facts that you will use over and over again. You should remember things like the following by rote learning:

multiplication tables
basic algebra formulas: slope of a line, quadratic equation formula, (a+b)^2, etc.
constants like e and pi
calculus formulas like integration by parts

These are just some examples.

Generally speaking, the more advanced you get in math the less you have to learn by rote, but yes, especially in pre-college math, rote learning still has its place.

 

[homeomorphic]

My point about rote learning was mainly about situations where the student and/or the teacher doesn't have the time to go through the theorems. Yes, it is best if you can derive them yourself; but, if under time constraints then it comes down to rote learning by default.

I think you have it the wrong way around. In the less desirable circumstances, maybe you don't have time to learn a proof, so you can delay understanding the proof for later and just focus on how to use the theorem. That can work sometimes. A lot of times, doing that will be effective because you wait until you are pretty good at the subject to understand the proof and by then the proof is easy.

But, I think the default should be that you understand it. It's only when it is prohibitively difficult and will slow you way down that you should delay understanding it. There are also some proofs that are unenlightening and not worth learning. Typically, you hope there is a better proof out there, but you kind of play it by ear.

Maybe the important thing is not to let yourself get stuck on things too much, whether it's proofs or problems. Getting stuck on problems is good because that means it is challenging you, but you don't want to stay stuck.

 

[homeomorphic]

Rote learning is important for the most basic math facts that you will use over and over again. You should remember things like the following by rote learning:

multiplication tables
basic algebra formulas: slope of a line, quadratic equation formula, (a+b)^2, etc.
constants like e and pi
calculus formulas like integration by parts

These are just some examples.

I have to disagree with most of this. Multiplication is tables are okay to memorize by rote. It worked for me. Plus, a lot of practice doing arithmetic. However, I think the right way to do it is to practice counting like 2 4 6 8 10, 3 6 9 12, etc. I heard of a Russian teacher who got kids to come up with their own multiplication tables that way, and a lot of them were very successful when they grew up.

The slope of a line is very understandable and should not be learned by rote. Also, constants like e and pi are very understandable. Integration by parts is really just the product rule in reverse. Again, understandable. It's okay if some things kind of go over your head the first time you learn, but it's not desirable.

I use the quadratic formula by rote, but I also know you can get it by completing the square, which isn't very difficult, so if, for some reason, I forgot it, I could easily derive it again.

The fact that things like that are learned by rote is just an indication of the math being dumbed down for a "lower-level" audience.

Generally speaking, the more advanced you get in math the less you have to learn by rote, but yes, especially in pre-college math, rote learning still has its place.

Maybe that is the way it tends to be because things are taught badly, but not the way it should be. Actually, in advanced math, there are a lot of proofs that are just impenetrable and it's better not to go through the proof, at least as things stand. And there is just too much to learn when you get to research level stuff, so you always have to decide if it's worth your time. There are a lot of proofs that I know of that people working in the field typically won't know. It's my personal mission to try to make a lot of these proofs more understandable so that it will be possible to understand more proofs. But, as I said, it's important not to get stuck, and if you try to learn all the proofs in topology (and I mean graduate level or research level), let's say, you'll just end up getting stuck too much. I am the type of person who really wants to know why, but a lot of times, I skip proofs, and tell myself I will come back later and try to come up with my own crystal clear proof when I have time.

 

[Obis]

Depends on what do you call "easier".

Memorization of theorems is boring and useless - it's very hard.

 

[demonelite123]

i think it is good to memorize certain important theorems (and their hypotheses as well) so you can whip them out whenever you need them. Of course, it is even better to try to understand the derivations or proofs which will make remembering the theorem easier. While some proofs are enlightening and show interesting techniques that you may also want to remember well, others may not be as important, such as those that involve only tedious computations, and these are not as important to commit to memory.

 

[homeomorphic]

homeomorphic, but isn't it easier to just memorise a theorem that let's you know the integral of (sinx)^n? I wouldn't know how to derive it.

Well, if n is odd, it's a pretty easy integral. If n is even, I think it gets ugly, but the strategy is fairly easy.

There's another option besides memorization. Just looking things up. Some facts are just not that important to know, so you don't need to know them. Another option is to use a program like Maple or Mathematica (although I find sometimes the answers they give you will be too ugly).

 

[Sankaku]

Anything that can be done by 'rote' memorization can be done better by a computer. The whole point is to understand what is going on, which is something a machine cannot do.

It is beneath the dignity of excellent men to waste their time in calculation when any peasant could do the work just as accurately with the aid of a machine.

—Gottfried Leibniz (1646-1716)