How to actually learn math (not just memorizing and rote)?

[MissSilvy]

Almost everyone agrees that learning math by rote or superficially gets one nowhere fast. I am currently starting my calculus sequence and I know it's a tad early to be asking this question, but how can I learn so that I actually understand the meanings and relationships behind the numbers? I've been told that drilling away at example problems and memorizing equations is not good enough at some point: the math eventually gets too difficult and the creativity required to solve advanced problems is too much for people who haven't internalized the actual ideas behind math properly.

I become quite intimidated when I look at textbooks that my senior friends are using and I want to make sure that I can actually do make the transition from computation to more advanced concepts. I know my explanation is vague but I suppose it can be best summed up by Terence Tao's http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/" post. I'm concerned about being ready to make the transition from stage one to stage two. Any help would be appreciated.

(I know I post on this forum quite often. I'm terribly sorry! I did a search and nothing on this topic seemed to come up)

 

[Vanadium 50]

Proofs. Lots of proofs.

I have not found a shortcut, I'm afraid.

 

[symbolipoint]

MissSilvy fears Calculus:

I am currently starting my calculus sequence and I know it's a tad early to be asking this question, but how can I learn so that I actually understand the meanings and relationships behind the numbers? I've been told that drilling away at example problems and memorizing equations is not good enough at some point: the math eventually gets too difficult and the creativity required to solve advanced problems is too much for people who haven't internalized the actual ideas behind math properly.

You might be worrying too much without justification. How is your conceptual understanding and skill with Intermediate Algebra and Trigonometry? If those were adequate, then the only big difficulty to expect is the epsilon-delta limit proofs during the first two semesters of Calculus. Almost everything else should basically be well learnable. Difficult, maybe very much so, yes; but still, learnable.

 

[chiro]

Almost everyone agrees that learning math by rote or superficially gets one nowhere fast. I am currently starting my calculus sequence and I know it's a tad early to be asking this question, but how can I learn so that I actually understand the meanings and relationships behind the numbers? I've been told that drilling away at example problems and memorizing equations is not good enough at some point: the math eventually gets too difficult and the creativity required to solve advanced problems is too much for people who haven't internalized the actual ideas behind math properly.

I become quite intimidated when I look at textbooks that my senior friends are using and I want to make sure that I can actually do make the transition from computation to more advanced concepts. I know my explanation is vague but I suppose it can be best summed up by Terence Tao's http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/" post. I'm concerned about being ready to make the transition from stage one to stage two. Any help would be appreciated.

(I know I post on this forum quite often. I'm terribly sorry! I did a search and nothing on this topic seemed to come up)

Apart from the proofs I think a bit a curiosity can go a long way in helping you learn and understanding things. You can get yourself started on a problem and you can discover so many hidden relationships as a result. It may not be something so rigorous either.

I've been doing some research in compression for example and a lot of what I do is work with symmetries of some sort. Its not the standard huffman type compression that comp sci students study. Anyways just by working with these I have found some relationships and results of using symmetry that can help aid data compression.

Calculus would be no different. You don't necessarily have to always do problem sets. Just be curious about something and follow it up. If you think you don't understand something try and use a different perspective. Maths is not a linear science it is more non-linear than anything. You can analyze mathematics by its linguistic structure, by change, by symmetry, by topology, by measure.

If you are thinking that a calculus proof doesn't make sense try and map it out using some sort of mechanical resimulation. Usually I find that humans learn when they can make an association with something that can be represented in the physical world. So if you're having a problem try and simulate it somehow. If its to do with deltas and vanishing epsilons picture them over time. If its to do with a linear algebra problem with rotations, derive the results using normal equations using trigonometry. A lot of things usually have a few core main points to them and usually when you can mechanically simulate the problem in your head it becomes a lot easier once you can picture what your trying to simulate.

If your reading a tonne of proofs and you have to follow a lot of them take note of how everything is transformed at each stage to arrive at the proof. Try and generalize the behaviour of proofs so that you can in your mind make a note of common types of transformations both linguistic and mathematical. Also try and simulate it if possible.

If your working with problems dealing with constraints again one by one start with an unconstrained thing and add constraints one at a time.

If you understand how things are transformed in mathematics you can excel in fields such as algebra, proofs and a lot of mathematics. Understand what makes a powerful transformation as applied to a mathematical object and by doing so you'll realize what the transformations are actually doing.

There's probably a tonne more but I can't think of it at the moment. Good luck!

 

[mbisCool]

Since you are just learning calculus i would assume the main focus of your class will be computational. Often the homework will simply be a lot of computational problems which are good practice to some degree. I would suggest when doing homework don't concentrate on the intuitive computational probelms but try to do a lot of the applied or abstract problems. This will ensure you understand instead of you just being able to compute.