How to become good in Mathematics?

Question

I don't know if this is the right place to ask this question...but let's try...

I have seen lots of people all around me, interested in mathematics-ranging from teachers to friends.But,I have seen some have different approach towards the subject.Their thinking is different from the rest.Both groups of people may have a good reputation among others about being good in Mathematics but approach of some of them is completely different and simple.Let us take an example.

See this question.The answer provided is good enough and effort is appreciable.But,there's another hint provided to the solution in the comments by Andre Nicolas.Using his method,the problem can be solved in a few lines but the answer provided extends the problem to about $2-3$ pages and makes things quite complicated.

So,I want to know what has helped in this difference in approach?Is it only practice or do some people have special inborn aptitude towards mathematics?

I think I fall in the first group.I have to improve myself through practice.Can I ever become so much proficient in mathematics only through mathematics and no inborn abilities?

Thanks for any response!!

Answer 1

In short:

There's probably some inborn aptitude, but not as much as people think. Especially not as much as all those people saying "oh math just isn't for me" would have you believe. Practice is a much bigger factor if you ask me.

Longer answer:

It also matters which level you want to get to.

• The level required to be considered "good at math" in modern society is a level that is accessible to anyone with a semi-working brain, as long as they work hard enough. Note, however, that even this "basic" level is not reachable with zero work, unless you are an absolute born genious.
• The level required to graduate in mathematics requires a very small amount of inborn aptitude and quite a lot of hard work. I would say that of people who study math in university and fail, most fail because they don't work hard enough, but some actually work very hard but simply don't have the spark needed for some of the more abstract subjects
• The level required to be insert brilliant mathematitian here is not reachable to most of us. It requires a large amount of inborn aptitude that is probably only bestowed to one in a million, and then a simply enormous amount of drive and stamina to actually acchieve greatness.

Answer 2

Like any other ability, it's mostly a combination of inherent talent and obsessive focus and interest in the subject. Some particular ideas that are unique to success in mathematics or a mathematical career:

• A particular kind of laziness: trying to find a clever, deep idea rather than just relying on brute force.
• Comfort with abstraction. There are a lot of questions here along the lines of 'What does [definition] really mean?' or 'Draw me a picture to explain [thing]'. Being a mathematician means being able to handle abstract concepts abstractly. You don't have to limit yourself to, say, the most abstract areas of algebraic geometry (or whatever), but you do have to be comfortable dealing with things that have no real-world analogue or for which familiar intuition is misleading.
• Working on single problems for long periods of time. Now, it's generally a good idea to have multiple topics at once. Compare the publication rate of mathematicians versus academics in other theoretical fields, though.
• An early start to one's career. Mathematicians have a short shelf life, and there's a huge amount to learn to be prepared to be a professional mathematician. It's not a career you can jump into at a later date, and there's really no entree into the subject except through climbing the academic ladder. (There are vanishingly few exceptions of people who've gone from industry or non-mathematical subjects into mathematics--- Raoul Bott is the only name that comes to mind--- especially compared to the opposite direction.) Now, there's certainly a difference between being good at mathematics and being good at being a mathematician; my point is that the former is not a particularly useful or relevant ability without the latter. If you want to do real mathematics, or even get enough practice in order to do real mathematics, you'll have to start out strong in your career in academia and remain there.
• Sheer luck in having the right contacts. As mentioned in the previous point, it is extraordinarily rare to be a mathematician except by going through a certain series of steps. It's not like, say, computer science, where it's not particularly difficult to jump between academia and industry in either direction; and hardly anyone is working on nontrivial mathematics research in industry. There are gatekeepers at every step of the path, and if you get stuck with a bad advisor, department chair, etc., you're screwed.

One other thing I'll mention is that while mathematics competitions are certainly not a bad thing, they're not really indicative of what professional mathematicians do; they're more about using elementary methods in clever ways to solve contrived problems in a short amount of time. (That having been said, feel free to practice math with them, and you should legitimately feel proud if you do well in them.)