"It is a profoundly erroneous truism, repeated by copybooks and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them." ---Alfred North Whitehead
"All math instruction problems are primarily insufficient speed at a previous step. If you can't run multiplication in your head, you can't do long division and have it make sense. If you can't factor numbers in your head, you can't do decent fraction problems. If math facts aren't instant, basic algebraic manipulations (2x + 3 = 17) take too long to do ... and you never learn the patterns. Algebraic fractions, quadratic factoring, the fundamental theorem of calculus, integration over the complex half-plane ... it's all the same thing. How fast are you at the prior step? Can you do it in your head?" ---Aretae
What specific exercises are helpful in internalizing and automatizing computational skills? I realize that the obvious answer is "drill and repetition," but I was hoping the community would have some something more specific to add, so I have a number of related subquestions.
Say, are timed drills helpful? Should one try to do simpler exercises in one's head? Does anyone have tips on how to develop the mental toughness to persist through computations that might be boring or painful in the moment even as one understands that they are ultimately necessary for building understanding? Is there some esoteric metacognitive art of debugging one's own mental algorithms: figuring out the most efficient way of doing something, and training oneself to do that thing without thinking?
After two years of pure autodidacticism in mathematics, I'm taking a differential equations course at the local community college, and it's been more difficult than I expected. Humiliating as it is to admit, I don't seem to yet have the patience or mental toughness for large problem sets, and I worry that I may have developed a few bad study habits. I have a tendency to leisurely examine proofs and casually attempt a few exercises, without (I fear) taking care to establish the firm base of quick and reliable skills needed for higher understanding. To take one example, I can do integration by parts, but only after a noticeable hesitation; the operation is not introspectively obvious to me in the way that (say) the distributivity of multiplication over addition is introspectively obvious and hardly even feels like a step. Integration by parts still feels like a step---and I wonder if perhaps it shouldn't, supposing one really wants a deep understanding. So while I'm proud of myself for (say) having picked up a lot of cool complex analysis insights from my few months with Mathews and Howell and my quick skim of Tristan Needham, I also feel as if I am missing some fundamental skills that are needed to achieve deeper insight, and I was hoping maybe this community would have some ideas.
Obviously this is a soft question (perhaps too soft?---I dearly apologize), and should therefore be community wiki; however, I don't have the reputation to enable that. Perhaps a moderator could be so kind? I thank you-all in advance for any advice you have to offer, and remain yours.