Lot's of good answers here, thought I'd just add a personal story here - I was not that brilliant at maths at school. Partly because - I'm not that great at arithmetic - have to check calculations carefully - many mathematicians are the same.
Are some that are lightning calculators like Von Neumann, but others who are not so good, and can easily be average or below average at arithmetic.
Like Emma King, cosmologist, "The Mathematician who can't add up".
And I was also not brilliant at remembering formulae and methods for solving problems.
Good at deriving them from scratch, good at inventing a new idea to solve a problem I'd never seen before - but not so good at remembering formulae and remembering methods for solving problems that the teachers had taught me.
So with the way school exams particularly tend to emphasize ability to remember formulae and procedures in maths, and to do arithmetic well and quickly - well I didn't do too badly but got a B in my A levels in maths instead of an A.
Still greatly loved my subject and - also went on and studied after my A levels - and I was especially attracted to the abstract side of maths.
So just to take an example, a mathematician might puzzle over details of how exactly the number of primes gets less and less as you get to higher and higher numbers. Might spend his or her life studying that very problem in intricate detail.
But might not even be especially quick at testing to see if a particular number is prime, or factorizing, for the ordinary small numbers we use in arithmetic.
Might, might not be, is largely irrelevant at that stage.
So anyway - turned out that I was pretty good at maths - but not arithmetic, and not school level maths (was okay at it, could learn it eventually - no slouch really, but compared to many mathematicians, I was a little slow at it).
However, I was good at university level maths and went on to get a good first class degree in my subject.
So - is a case of finding what your subject is. Is not at all a black and white thing, that some people are universally good or whatever.
Within maths also - there were particular topics that I enjoyed much more than others and were also much better at than others.
So similarly, within your subject area whatever it is also, you might find some particular area that particularly suits you, often happens.
Your first year or two may be partly a case of finding out what that is, what part of your subject particularly interests and suits you.
If they had taught maths in uni similarly to the way it is taught in school - or even worse (for me) - taught lightning calculation and arithmetic at uni, I'd still be quite a way down the list and would have thought that I was a rather mediocre mathematician - might well have given up despite my love of the subject.
So - that's my kind of extra take on it - that it's not at all a single scale like you are either good at maths, or not so good, but can be that you are good at say fourier analysis, or group theory or whatever - some special topic. While in other areas may be not so good or even hopeless, in the same subject.
Just in the same way that some people are brilliant at maths but hopeless at languages and some people are brilliant at languages but hopeless at maths, or whatever.
And - most important thing really is if you enjoy it, then you are doing something you enjoy and eventually it may be your life's work, or lead you to a job that has satisfaction or whatever - what can be better than that?
That others may be lightning calculators or whatever - well that is something they need to sort out and work with - and may be asset, may be a problem for them, either way - it is not much to do with what you do with your life :).
And - one of the great things about being at college is - all the other students you meet. I learnt a lot from the other students as well as the teachers.
But then the other answers here have already answered that part pretty well :).
Are some that are lightning calculators like Von Neumann, but others who are not so good, and can easily be average or below average at arithmetic.
Like Emma King, cosmologist, "The Mathematician who can't add up".
And I was also not brilliant at remembering formulae and methods for solving problems.
Good at deriving them from scratch, good at inventing a new idea to solve a problem I'd never seen before - but not so good at remembering formulae and remembering methods for solving problems that the teachers had taught me.
So with the way school exams particularly tend to emphasize ability to remember formulae and procedures in maths, and to do arithmetic well and quickly - well I didn't do too badly but got a B in my A levels in maths instead of an A.
Still greatly loved my subject and - also went on and studied after my A levels - and I was especially attracted to the abstract side of maths.
So just to take an example, a mathematician might puzzle over details of how exactly the number of primes gets less and less as you get to higher and higher numbers. Might spend his or her life studying that very problem in intricate detail.
But might not even be especially quick at testing to see if a particular number is prime, or factorizing, for the ordinary small numbers we use in arithmetic.
Might, might not be, is largely irrelevant at that stage.
So anyway - turned out that I was pretty good at maths - but not arithmetic, and not school level maths (was okay at it, could learn it eventually - no slouch really, but compared to many mathematicians, I was a little slow at it).
However, I was good at university level maths and went on to get a good first class degree in my subject.
So - is a case of finding what your subject is. Is not at all a black and white thing, that some people are universally good or whatever.
Within maths also - there were particular topics that I enjoyed much more than others and were also much better at than others.
So similarly, within your subject area whatever it is also, you might find some particular area that particularly suits you, often happens.
Your first year or two may be partly a case of finding out what that is, what part of your subject particularly interests and suits you.
If they had taught maths in uni similarly to the way it is taught in school - or even worse (for me) - taught lightning calculation and arithmetic at uni, I'd still be quite a way down the list and would have thought that I was a rather mediocre mathematician - might well have given up despite my love of the subject.
So - that's my kind of extra take on it - that it's not at all a single scale like you are either good at maths, or not so good, but can be that you are good at say fourier analysis, or group theory or whatever - some special topic. While in other areas may be not so good or even hopeless, in the same subject.
Just in the same way that some people are brilliant at maths but hopeless at languages and some people are brilliant at languages but hopeless at maths, or whatever.
And - most important thing really is if you enjoy it, then you are doing something you enjoy and eventually it may be your life's work, or lead you to a job that has satisfaction or whatever - what can be better than that?
That others may be lightning calculators or whatever - well that is something they need to sort out and work with - and may be asset, may be a problem for them, either way - it is not much to do with what you do with your life :).
And - one of the great things about being at college is - all the other students you meet. I learnt a lot from the other students as well as the teachers.
But then the other answers here have already answered that part pretty well :).
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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