Bad at computations... but not math?

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.

When working on my bachelor's I didn't like computations much and what appealed to me about math was the concepts. Senior year that changed. I'm not sure what to attribute it to, but my perspective shifted. Computations, done well, build the intuition that motivates all the nice concepts. The nasty technical stuff of cutting edge research gets sifted, the key pieces get names (hence definitions) and then have concepts built around them so they can be pushed to their limits.

I've heard many students complain that analysis seems like a "bag of tricks". You could claim the same about an elementary group theory course. But the truth is that there is an interplay between technique and concept. It is not a one way relationship. Even category theory has techniques (as in symbol shifting to reach 'nice' arrangement) and sometimes very technical criterial in order to manifest its concepts. Remember that the abstract stuff came from abstracting the nitty gritty. We had nasty integrals to work with before we had functionals on Banach spaces. You can't do much with a Banach space if you cannot actually apply its structure to nasty problems. A smooth manifold is really nice intuition, but at the end of the day you are locally working with concrete functions in $\mathbb{R}^n$.

I would suggest that you attempt to solve problems more concretely than you currently are - for a season. It will force you to develop better technique. Use your abstract intuition to motivate the concrete.

When I'm having trouble grasping something abstract, I make it concrete. If I am having trouble with something concrete, I abstract it. The dialectic between these improves my understanding of the material.

Finally, and someone else will have to do the details on this, many important open and solved problems involve very specific objects in their statement or proof.

You could understand mathematical concepts without fighting with exercises, but the knowledge become more superficial. In philosophical subjects as category theory it is more possible but in disciplines as combinatorics it would hardly work at all.

In each discipline the class should be divided into theory and practice, almost like two different subjects, and it would be a mistake taking too easy on any of those parts. Struggle with the exercises improves the ability to prove theorems.

The ability to cope with the calculations is much about the techniques you learn as you go. Lazy students risk to end up like amateur philosophers like me.

But you might not be as bad as you think? It may be that other students just have more experience? Which might have evened out by next year?