Advice to how to be better at mathematics

[tgt]

What general advice would you give to a keen young (upper high school and undergrads) student of mathematics?

I would say most important is to develop your mathematical intuition by doing and reading as many examples and problems as possible. Anything else?

 

[Orodruin]

This is something for which there is likely as many answers as there are students. How you approach the subject is individual as is what is going to work the best way for building experience and intuition.

 

[tgt]

This is something for which there is likely as many answers as there are students. How you approach the subject is individual as is what is going to work the best way for building experience and intuition.

Yes, I know the approach taken by students are different. But what are some over arching goals that students should aim for (apart from the obvious of aiming for the highest marks).

I am looking for general advice that everyone can take on board - as goals to achieve.

 

[DrSuage]

Pick mathematics that you think is unlearnable and then learn it. No mathematics is beyond you.

Don't read mathematics books - read very small sections in mathematics books with a large stack of blank paper and a pencil whilst making sure you can fill in all the gaps between each line in the derivations. Then do the exercises and problems.

If you think something sounds interesting - try it out for yourself.
Come up with your own examples. If a book states a theorem, immediately try to construct counter-examples - this will help you understand why the theorem has to be true on an intuitive level. Then look through the proof, checking each line. Then close the book and try to prove the theorem yourself from memory - this will show if you really understand what happened.

 

[tgt]

Pick mathematics that you think is unlearnable and then learn it. No mathematics is beyond you.

Don't read mathematics books - read very small sections in mathematics books with a large stack of blank paper and a pencil whilst making sure you can fill in all the gaps between each line in the derivations. Then do the exercises and problems.

If you think something sounds interesting - try it out for yourself.
Come up with your own examples. If a book states a theorem, immediately try to construct counter-examples - this will help you understand why the theorem has to be true on an intuitive level. Then look through the proof, checking each line. Then close the book and try to prove the theorem yourself from memory - this will show if you really understand what happened.

These are good advice but are more directed at the older student. Also what about some general principles?

 

[micromass]

These are good advice but are more directed at the older student.

They're not directed at the older student. The methods mentioned by DrSuage are so very important and it is crucial that even young beginners with mathematics get to know this style of reading math. One can not read math like a novel!

 

[tgt]

From my experience, even if you were very pedantic with reading and understanding proofs, without a solid "mathematical background" and mathematical intuition, it doesn't leave a large imprint in your brain. To me the mathematical background and intuition is gotten from doing lots and lots of problems starting at a basic level.

For an extreme example, it is not good to get someone to start learning set theory before they have a good grounding of algebra, analysis etc.

 

[IGU]

What general advice would you give to a keen young (upper high school and undergrads) student of mathematics?

Study mathematics. Most high school curricula include almost none, just Euclidean geometry. Mathematics is proofs. The usual high school algebra, trigonometry, and calculus classes avoid proofs, teaching instead just lots of rules and techniques. Completing these classes, even with excellent grades, tells you almost nothing about whether you like or are good at actual math.

I would say most important is to develop your mathematical intuition by doing and reading as many examples and problems as possible.

Depends on the problems. I would say that problems that take at least fifteen minutes of hard thinking are worth doing, better if they take longer. The improvement of your mathematical intuition comes from thinking about how to solve such problems, especially the wrong paths you take. This is why people who give up and look at answers too soon don't learn as much. If you are susceptible to that temptation, don't use materials that have answer keys.

Also, recognize that generic advice may not apply to you. Some people love pictures and examples, which others find them irrelevant and unenlightening. Some people find proofs uninteresting -- these people are destined to be something other than mathematicians. It's best to understand yourself as well as and as soon as possible.

From my experience, even if you were very pedantic with reading and understanding proofs, without a solid "mathematical background" and mathematical intuition, it doesn't leave a large imprint in your brain. To me the mathematical background and intuition is gotten from doing lots and lots of problems starting at a basic level.

Not to put too fine a point on it, if you have no understanding of proofs then you have no "mathematical background". You develop one by understanding proofs, one tool for which is working difficult problems.

For an extreme example, it is not good to get someone to start learning set theory before they have a good grounding of algebra, analysis etc.

On the contrary, learning elementary set theory and logic and other foundational topics is an excellent way to start.

You might find it useful to read Measurement, a fairly recent book by Paul Lockhart.

But what are some over arching goals that students should aim for (apart from the obvious of aiming for the highest marks).

This too is quite wrong. Aiming for the highest marks is a useless waste of time. You should aim for understanding the subject you are studying, and good enough marks will come from that. There are many ways to go about getting high marks, and most of them are of no use other than for getting high marks. Such knowledge is of little value once you are done with being a student. The understanding of the subject should be your main goal, as its utility is long term.

 

[tgt]

Not to put too fine a point on it, if you have no understanding of proofs then you have no "mathematical background". You develop one by understanding proofs, one tool for which is working difficult problems.

I'd disagree. Look at Ramanujan. Didn't know much about proofs, tremendous "mathematical background". Shows Mathematics is "bigger" than just the technical constructs smart people have put in place. Don't get me wrong, proofs are very important and with a mere human brain, we need proofs in order to investigate and understand better mathematical systems and relationships in them.

Look at logic, turned out there are many different types of logics. Also there are many problems that cannot be proved nor disproved in the standard mathematical foundation (ZFC).

On the contrary, learning elementary set theory and logic and other foundational topics is an excellent way to start.

I was talking about set theory as in zermelo frankel set theory. Offcourse maths students should know basic set theory early on.

This too is quite wrong. Aiming for the highest marks is a useless waste of time. You should aim for understanding the subject you are studying, and good enough marks will come from that. There are many ways to go about getting high marks, and most of them are of no use other than for getting high marks. Such knowledge is of little value once you are done with being a student. The understanding of the subject should be your main goal, as its utility is long term.

High marks usually equates to understanding especially in more difficult maths subjects. There are freaks who are excellent mathematicians but never got high marks but they are definitely the minority.

 

[epenguin]

Given that the book has sold 1 million copies and has been in print without interruption for 70 years, it cannot be too much of a waste of your time to read the slim book 'How to solve it' by George Polya.

Do not throw it away after you read it. In fact you might think that most of it is perfectly obvious and you will know how to handle anything. It's later, when you have forgotten, but are stuck.

Maybe, I don't know, this and a lot of the advice is wasted on students. That is, they are already spoonfed. They are given an exercise on the chapter, the methods to use will already have been illustrated in that chapter and so on. Unless they themselves vary the problems

I didn't need that sort of help when going through maths books. But I needed it (and would have saved time if I had known about it earlier) when I tried to solve problems arising in biophysics, with nobody to tell me what method to use. Nobody to tell me what conjecture to make - just a feeling that underneath that this system there is something there. I could almost say, you want to get good at maths, for many people best idea might be study some specific physics, engineering, circuit theory, biophysics, population genetics, DNA Evolution, mathematical economics,... lots of things, all open-ended. Try to see if you can have conjectures and solve problems there where nobody is telling you all that much what method to use. It would also make you more useful in the end - after all why do you want to get good at maths?

Just a brainstorming idea - maybe it would be too ideal and time-consuming for average students.