Of course, faster calculations help solve problems quickly. But does that also mean that faster calculations open more opportunities for a career in mathematics (like a researcher)? I like mathematics and can spend weeks trying to solve any problem or understanding any concept. But nowadays, there are many contests that focus on faster calculations rather than problem solving. I am very slow at calculations due to which I end up doing badly in these types of contests. Does that mean I am lagging somewhere? Can this cause hindrance in pursuing a career in mathematics?
closed as primarily opinion-based by Adam Hughes, Semiclassical, Dominik, saz, Leucippus Dec 7 '16 at 0:06
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This is a vague concept. True mathematics does not deal with numerical calculations, let alone faster calculations. Science has devised machines called calculators to perform this task of faster calculations. What real mathematics deals with are concepts, mathematical subtleties, reasoning, logic and new lines of thought to imagine things that we cannot see. Strictly speaking, mathematics is the only science which ventures into things that cannot be seen or felt in reality nor properly visualised. A classic example of this notion is the n-dimensional space, generalising everything possible theory, theorem, law and equation to n-dimensions, assuming some kind of a universal symmetry, perfectness and order in this generalisation. If you really want to do research in mathematics, leave all calculations to the calculator and start to develop the art of thinking. That's what's most required to carry on in this line.
In general I would say that problem solving skills are more important than being 'skilled' in arithmetic. The contests you mention, often test both problem solving and speed as well.
However, being fast at doing calculations often means that you understand theorems and proofs faster, and thus tend to get stuck less often. This, in turn, might mean that you are just able to read and learn more during a shorter period of time. This is of course not true for everyone.
I don't think that you should feel hindered by the fact that you feel that you are slow at calculations. I know several mathematicians (myself included) who sometimes feel inferior to others when it comes to calculating things fast. Often I feel that this has to do more with self confidence. If I try to do calculations fast in a somewhat stressful situation (especially if someone goes "Your the mathematician, calculate this!"), I'm often blocked by the stress, and tend to do silly mistakes. Because of this, I often double check my calculations, in my head, even really easy ones, and this slows down calculations substantially.
In conclusion, when it comes to pursuing a career in mathematics, don't feel that this is too much of an obstacle, you're not alone. Get an app that trains basic arithmetic in your head and don't let this get in the way of your studies. It is far more important that you have an ability to connect different areas of mathematics and have an ability to be creative, and most important, that you really think that mathematics is interesting, fun and important in its own right. Get away from the thought that mathematics is a tool that has to be applied to some other problem (often physical). Being fast at calculations is handy but, unless you are really slow, it is more important in social situations than doing research in mathematics.
I don't think being slow at computations will make pursuing research math impossible, but it might make it harder. Getting an undergraduate degree in math (at an American university) still requires a fair amount of test taking and problem sets which involve computation. If you're really slow it may depress your grades, which is a barrier you'd have to overcome. However, this becomes less and less important as you progress through a course of study.
Secondly, there are many areas of research mathematics where 'computation' play a large roll. There is quite a lot of algebraic computation in discrete mathematics for example, so it might be the case that it impacts what field you choose to study.
Being faster at recognizing when/how to apply basic theorems and techniques will make you more efficient, but in the end it's inventiveness that is necessary for substantial breakthroughs.
After reading you question I remembered an interview given by Scott flansburg who is worldwide accepted as the fastest calculating hymen in the world ( I can't provide that video to you as I found that one luckily). He said in the video that calculation is purely a strong consequence of strong logic and ability to find the pattern.I thought starting the answer with this US a good idea. Since now I have given the definition of calculation to you now so let's come to your question now.
The pure mathematics or generally called higher mathematics is not so related with calculation. It is based on higher order thinking , strong reasoning ability and knowledge application ( since it deals with problem solving to a large extent). The mathematics contest held worldwide test you knowledge in higher mathematics and hence test those things which are essentials of higher mathematics ( which I have already mentioned earlier).
So in my opinion, you don't need to worry about the role of calculations in international/higher level mathematics contest.