Naturalness in mathematics Why is math so natural to some people? Even trying hard to study with dedication, worrying about ideas and at the same time the technique, nothing for me happens naturally.
 A: That's a very good question!
I'm inclined to say that (to some degree) we're all math people. We're all born with an aptitude for learning. Some aptitudes are stronger than others, but that's okay as long as one keeps learning!
I think reading up on the psychological views of "nature versus nurture" and on the theory of personality. I'm sure a lot of it can be chalked up in that.
If I can encourage you, I would say writing a post on this forum site shows you are a math person. Just because it doesn't come at the first attempt doesn't mean it won't happen at all. We all have our strong points and our weak points in math.
For example, when I started my masters in math, I dropped partway through the first semester because I felt I couldn't do it. I returned the next semester and now I am on the honor role and holding an assistantship with teaching responsibilities. I would definitely be considered a natural-born mathematician, but that doesn't mean I don't have to work at it. For example, statistics comes the most naturally to me. This year I'm focusing my studies on analysis and abstract algebra and am sometimes having my posterior served to me on a plate. As my sister would say, I'm riding the struggle bus.
Don't give up. Eventually you'll get there! As I read on a workout app, progress is progress no matter how small. Math is a workout for your brain. If you're the gym type, you're not going to bench press 500 lbs. on Day 1. Same thing with math: you have to be able to "work" just the curl bar before you can add weight.
I believe in you!
A: Here is just my thought. Math is a tool that we all used to communicate. Thus, it is a language. So here is a question: English (or whatever language you are native in) is a language as well, so are some (English-speaking) people 'natural' in English and some just don't?
Here is a deeper thought: what does 'natural' in something even mean? Assume 'natural' in English means fluent in using it. Then everyone using English well is 'natural' in English. The same thing holds with math. Some people might seem 'natural' in math by birth (i.e. having some sort of mathematical gene), but it doesn't mean anybody else cannot be 'natural' in math. If you don;t speak English first, you can be fluent in using it by means of experience. So if you want to be 'natural' in math, you need experience. If you feel the math you are doing is not 'natural', you should acquire more experience. A sudden click will come if you persist enough. After that, things that you thought as unusual will become natural
At the end, I would like to say this as well: nothing should be deemed as 'natural' at first. If everything is natural, then there is no science at all, because we will understand everything already by naturality. To learn math, you need to learn the way that math people are thinking. For example, an integral is often understood as a signed area under the curve. If someone told you this in the first day learning integral, you (and I) would think of this is unnatural (and insane, probably). But if I tell you that an integral is computed by 'chopping the region into many tiny rectangles and add up the areas', then the notion of 'area under the curve' is clear now, and it becomes a natural concept. Similarly, every math concept should be learned with its underlying motivation. Often the definitions/theorems seems nonsense, but the underlying motivation is natural
A: This is generic advice I would give to an undergraduate.  You may already be at a more advanced level, but as my research publications are competent rather than stellar, I am not qualified to advise on this.  As mentioned in comments, getting top grades at world leading universities is child's play compared to doing publishable research, let alone truly groundbreaking work.
There are different kinds of hard work.  In my experience of teaching, certain kinds of hard work are not helpful at all for mathematics:

*

*Diligently memorizing whatever the lecturer says,


*Doing similar exercises over and over again so that you are trained to quickly do certain calculations.
Unfortunately when most students think of hard work these and similar activities are what come to mind.  I assume these methods are useful in other subjects (and what passes for high school "mathematics"), but they will not get you very far in undergraduate mathematics.
On the other hand there are some kinds of hard work that you might not call work at all (so you may make the mistake of thinking someone is not putting in any effort and is just a natural), but really do build you into a successful mathematician:

*

*Challenge everything the lecturer says.  If the lecturer says "The definition of ... is ....", your reaction should be no I would define it another way.  Then after thinking about it you might come to realise the definitions are the same after all, or maybe you will find that your one does not quite capture the idea.  Then in the end you will appreciate the lecturer's definition all the more.


*Ask your own questions - what you think might be important.  You will often find that those end up being the things that get addressed, but it will mean more to you if you wondered about it naturally, rather than being told "this is a thing that you are going to learn".  Make a sincere effort to answer your own questions.  If you can't then that is what this stackexchange is for.


*Do not look at proofs until you have tried to prove the theorems yourself.  This way you will appreciate how clever they are, and why the complicated steps are needed.


*Talk to your friends about mathematics.  Argue - make claims and justify them.  Laugh and joke about it.
These may not look like hard work, but to do them properly takes more time and effort than reading notes and doing exercises.  It may feel like you are taking time away from focussing on the things you will need in the exam, but in the long run this is how to become one of those people that others call "a natural".
