Problems versus Exercises

Why do we need to do simple arithmetic efficiently in our minds? We already have readily accessible computers for this, let's focus on something computers can't do for us yet.

> but a competent soldier needs to exercise that capability until it's second nature

I'd prefer a robot to fight in a war(instead of me).

What best improves reading comprehension, or conversation in a foreign language, or mathematical problem-solving, or understanding of history, or eye for photography, or anything else, is experience doing the thing itself: reading novels, having conversations, solving puzzles, reading primary sources and analyzing them, taking and printing pictures. The thing itself is the most interesting/fun activity, and the most genuine: it allows many more and more subtle connections to be made between ideas/fields/actions. It aligns with the ways our minds naturally work. It inflames curiosity and creativity.

Any “exercise” is, at best, a limited model of the skill to be learned to do the real thing. It is one tiny piece of a longer process, split apart semi-arbitrarily.

It’s important to understand when an “exercise” can be helpful. Namely, when there is a very clear way to break down a skill into parts, and then analyze just one part, looking for specific weaknesses to eliminate. So for example, a soccer player might want to take many corner kicks to practice placing them in just the right place, or a chess player might want to repeat just the opening phase of the game, or a student of mathematics might want to practice multiplying numbers, or a foreign language learner might want to try to pronounce a tricky word over and over.

But! These exercises should not simply be done for their own sake. They should be done with careful attention to making specific improvements to skill or technique. Ideally, they should be overseen by an expert mentor who can give useful advice about what to fix. And more emphasis should be put on doing the real activity itself. A soccer player might have a perfect corner kick in the abstract, but without understanding where the players are on the field and when to time the kick in the context of the game, it’s not very useful. A chess player might be able to find every possible fork on the board, but unless he understands the rest of the current position, just knowing the forks alone won’t save him.

For someone who expects to become a world-class performer, where the performance must be flawless, I think there’s probably more value in doing exercises and focusing on minutiae, since such a person is dedicated and understands what the purpose is, and usually has a skilled coach for advice, often in a 1-on-1 setting.

But swamping beginners in exercises usually does more harm than good: exercises are boring, they lack context or purpose. They suck all the life out of a subject, and they retard real learning.

Trivial usually denotes solutions that are obvious and not very interesting such as 0, 1 or the empty set. It's more of a property of the solution than the difficulty of the problem.

By extension sometimes you can say a proof is trivial, when it's very easy, but I think the primary use is the one I mentioned above.