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 deep.”'
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.
Speed is great for brute force calculations and computer simulations.
For proofs that rely on abstract ideas and arguments, speed is inconsequential.