Is solving problems quickly an important trait for a mathematician to have? Is solving textbook/olympiad style problems quickly necessary to succeed in math? To make an analogy, is it better to be a sprinter or a marathon runner in mathematics?
To some degree speed matters. If you are going to sit there thinking "hmmmm ... what is a subspace?" every single time you see the word "subspace" you may never get through even the simplest of linear algebra proofs. So, some knowledge must be there automatically, that's why we all practice so much. That's why we don't just learn definitions but memorize them. That's the point of knowing a proof by heart (and I do mean knowing not just memorizing it in that case) so that it falls off of your tongue with ease. That way, when you need to prove something similar you aren't struggling with the mechanics. You have room in your brain to think about the big picture.
In some sense, the process of doing math is all about doing things more quickly. Think of how long it took you to solve a quadratic when you were in grade school. Now you just look at it and you know the solutions are there, and you can probably see the factors, or write the quadratic formula without thinking about it. What once was a big problem is now just a tiny step in much larger problems.
That all said, I do think that at times lay people mistake being fast with being good at mathematics. Some of the mathematicians I admire most are careful... even plodding in their thinking, going over each step in a way that seems almost naive. --yet they never seem to miss anything and in their careful way of reasoning they make many small discoveries that someone who was rushing to the answer might miss.
These mathematicians whom I admire stop now and then and think carefully about even the solutions to quadratics, they are looking for something deeper in each detail. And they are not self-conscious about the speed at which they work. Rushing to impress others is a great source of errors.
I'm as guilty as anyone else of this. But, I will try to emulate my mentors and not care if it takes me a long time. As long as in the end what I say is logical... and correct.
Depth of thought is much more important than speed. A good illustration of this is the work of Grigori Perelman, who proved the Poincaré Conjecture. In their article about him published in The New Yorker, Sylvia Nasar and David Gruber wrote:
'At Leningrad University, which Perelman entered in 1982, at the age of sixteen, he took advanced classes in geometry and solved a problem posed by Yuri Burago, a mathematician at the Steklov Institute, who later became his Ph.D. adviser. “There are a lot of students of high ability who speak before thinking,” Burago said. “Grisha was different. He thought deeply. His answers were always correct. He always checked very, very carefully.” Burago added, “He was not fast. Speed means nothing. Math doesn’t depend on speed. It is about depth.”'
You can find all the quotes here and I recommend you to read it. But for the completeness of the answer I quote some of them which I find most important.
In The Map of My Life mathematician Goro Shimura said,
I discovered that many of the exam problems were artificial and required some clever tricks. I avoided such types, and chose more straightforward problems, which one could solve with standard techniques and basic knowledge. There is a competition called the Mathematical Olympic, in which a competitor is asked to solve some problems, which are difficult and of the type I avoided. Though such a competition may have its raison d'être, I think those younger people who are seriously interested in mathematics will lose nothing by ignoring it.
In his lecture at the 2001 International Mathematics Olympiad, Andrew Wiles gave further description of how math competitions are unrepresentative of mathematical practice,
[...]The two principal differences I believe are of scale and novelty. First of scale: in a mathematics contest such as the one you have just entered, you are competing against time and against each other. While there have been periods, notably in the thirteenth, fourteenth and fifteenth centuries when mathematicians would engage in timed duels with each other, nowadays this is not the custom. In fact time is very much on your side. However the transition from a sprint to a marathon requires a new kind of stamina and a profoundly different test of character. We admire someone who can win a gold medal in five successive Olympics games not so much for the raw talent as for the strength of will and determination to pursue a goal over such a sustained period of time. Real mathematical theorems will require the same stamina whether you measure the effort in months or in years [...]
In his Mathematical Education essay, Fields Medalist William Thurston said,
Quickness is helpful in mathematics, but it is only one of the qualities which is helpful. For people who do not become mathematicians, the skills of contest math are probably even less relevant. These contests are a bit like spelling bees. There is some connection between good spelling and good writing, but the winner of the state spelling bee does not necessarily have the talent to become a good writer, and some fine writers are not good spellers. If there was a popular confusion between good spelling and good writing, many potential writers would be unnecessarily discouraged.
In his book Mathematics: A Very Short Introduction, Fields Medalist Timothy Gowers writes,
While the negative portrayal of mathematicians may be damaging, by putting off people who would otherwise enjoy the subject and be good at it, the damage done by the word genius is more insidious and possibly greater. Here is a rough and ready definition of genius: somebody who can do easily, and at a young age, something that almost nobody else can do except after years of practice, if at all. The achievements of geniuses have some sort of magic quality about them - it is as if their brains work not just more efficiently than ours, but in a completely different way. Every year or two a mathematics undergraduate arrives at Cambridge who regularly manages to solve a in a few minutes problems that take most people, including those who are supposed to be teaching them, several hours or more. When faced with such a person, all one can do is stand back and admire.
And yet, these extraordinary people are not always the most successful research mathematicians. If you want to solve a problem that other professional mathematicians have tried and failed to solve before you, then, of the many qualities you will need, genius as I have defined it is neither necessary nor sufficient. To illustrate with an extreme example, Andrew Wiles, who (at the age of just over forty) proved Fermat's Last Theorem (which states that if $x$, $y$, $z$, and $n$ are all positive integers and $n$ is greater than $2$, then $x^n + y^n$ cannot equal $z^n$) and thereby solved the world's most famous unsolved mathematics problem, is undoubtedly very clever, but he is not a genius in my sense.
How, you might ask, could he possibly have done what he did without some sort of mysterious extra brainpower? The answer is that, remarkable though his achievement was, it is not so remarkable as to defy explanation. I do not know precisely what enabled him to succeed, but he would have needed great courage, determination, and patience, a wide knowledge of some very difficult work done by others, the good fortune to be in the right mathematical area at the right time, and an exceptional strategic ability.
This last quality is, ultimately, more important than freakish mental speed: the most profound contributions to mathematics are often made by tortoises rather than hares. As mathematicians develop, they learn various tricks of the trade, partly from the work of other mathematicians and partly as a result of many hours spent thinking about mathematics. What determines whether they can use their expertise to solve notorious problems is, in large measure, a matter of careful planning: attempting problems that are likely to be fruitful, knowing when to give up a line of thought (a difficult judgement to make), being able to sketch broad outlines of arguments before, just occasionally, managing to fill in the details. This demands a level of maturity which is by no means incompatible with genius but which does not always accompany it.
Fields Medalist Alexander Grothendieck describes his own relevant experience in Récoltes et Semailles,
Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.
What do YOU think?
For mathematics, intelligence is needed. And intelligence is instantaneous. And something that is instantaneous has no notion of speed; speed requires time.
Speed as a measure cannot be applied.
For arithmetic, speed can be applied.
But there is an almost universal confusion between arithmetic and mathematics. Mathematics is the process of digging into the unknown. Arithmetic is the process of playing with the artefacts that the process of mathematics has uncovered.
What is termed mathematics in school is usually nothing more than arithmetic. Mathematics depends not upon the material but upon the teaching. If the student is actively enquiring, with the teacher guiding this enquiry only when needed, then mathematics will ensue. If the teacher is intent upon stuffing material into the head of the poor student, the best that can be hoped for is arithmetic.
At first, I believed speed was a necessary attribute in order to become a mathematician because classes like AP Calculus require for the student to answer questions in a short limited amount of time. Speed is important to let your mind think of other hard and abstract ideas but it really doesn't matter because if you truly love and understand mathematics to its core (even if you are a middle school student taking Algebra 1), than you are already a mathematician. I did learn something important thought. Even if you are great at mathematics, don't let your overconfidence take over because without practice, your mind will struggle in remembering how to do a problem (although you might get it in a few minutes but it is better to now it right away right?).