I have an analogy that I use with my students, and it is applicable here.
Mathematics is like literature. Things like elementary arithmetics (grade school $+$,$-$,$\cdot$, etc. on real numbers) are like the "abc"s. Things like algebraic manipulation (but not Algebra) (e.g. $log_2 (4 x)=9$, solve for $x$) are like words. (A lot of Americans reach only this stage and proclaim they hate math; it's as preposterous as saying you hate books when you can't even read the word the). Things like elementary calculus and real analysis are like sentences, and by the time you have an undergraduate degree, you can probably read a board book or two (without help, gasp!).
The whole rest of mathematics lies beyond.
The application is this: your "computations" are as fundamental to mathematics itself as the printed word is to literacy. It's great that you're excited about math, but I think it's extremely important to not just understand, but grok the basics before you try to build anything on it.
A lot of the stuff grade school kids learn is crap--like about not-dividing-by-zero, like about fractions, like about conics. And a lot of it is a dumbed-down-version or otherwise irrelevant once you get to higher math. But, a lot of it provides useful intuition. The analogy is: do you need to know what the letters' names are to read a book? No. Do you need to know that "?" is called "question mark"? No. As long as you understand that a is different from b and that ? is interrogative, then you're in good shape for understanding--but knowing those facts is useful too. If you know that a and e represent vowels, which all words in English phonology require, for example, then you automatically know why sdfslkjhrwfbv is an ill-formed English word.
The main points here are:
- You do not have to understand useless dogma educators teach kids to simplify lessons. I'm of the opinion that such lies should not be taught in the first place, since they screw up peoples' ideas later. I know. I've fixed a lot of it.
- However, by not being familiar with basic methods, you are missing part of the fundamental essence of math. If "just don't think about math like that", then you're missing part of what math actually is.
- You do not technically need to be competent at executing computations, so long as you do thoroughly understand them. It is difficult, however, to have the latter without the former--and, as one commenter noted, it is easy to delude yourself.
- Being incompetent at computation will harm your ability to interact with other mathematicians.
- Being incompetent at computation will minorly harm your ability to understand higher math.
Under these circumstances, I recommend becoming competent. If you really do understand computations, this should be relatively straightforward. It will be well, well worth your increased understanding and increased ability to interact. You don't have to aim for complete fluency at first, but I'd still hit the grammar monograph before I tackle Tolstoy.
Best Answer
Here's the short version: When the award was proposed, those involved were specifically worried about lingering rivalry between mathematicians from opposite sides of World War I (especially French v German), a rivalry which at that point had basically sunk the first attempt at an International Mathematical Union. Fields's proposal (adopted after his death) tried to sidestep the problem of "invidious comparisons" by inserting fuzzy language about future achievement (which is hard to measure and debate) as vs just past accomplishments (which can be debated more easily and thus lead to conflict). Early Fields Medal committees interpreted this as saying that the Fields Medal should be something like an early/mid-career award for someone not already decorated with other prizes and career milestones. A big motive for doing this was to avoid getting into debates over who was really the top mathematician, especially since they had so many top mathematicians (including many they knew personally and who might be offended) to choose from. Before 1966, when the limit was codified, there were people under 40 who were ruled out as being too advanced in their careers, and also people over 40 who were seriously considered even past the stage of initial nomination.
Working on an article that explains the bigger story, but if you're around Cambridge, MA on March 6, 2017, you can see me give a preview at MIT: http://events.mit.edu/event.html?id=16503559
Also important is that the Fields Medal wasn't really in the same conversation as the Nobel Prize until 1966, and at the beginning few would have taken the comparison seriously (seeing it as more of the early-mid-career award that it was). See my article: http://www.ams.org/notices/201501/rnoti-p15.pdf