How to retain the key points of an exercise? I do not know if this question is relevant to this site or not. But I'm going to ask it anyway as this site has helped me answer my maths questions hitherto.
I've experienced that the problems I do as exercises have a predisposition to not stay in my mind. For example if a problem I did in the last month is given to me today I have to go through my thinking process anew. I don't know if it is normal or not. I don't know if this issue is rather psychological either. But I believe it is not. How would you remedy this? What are the tips that one could propose so that I may appraise and retain the important or the key points of an exercise? Please help.
Thank you.
 A: Well, in my experience it's not unusual to have to re-do the thinking - it would be a waste of mental space to remember every problem you've ever solved. But what should happen is that the second time you do it should be much easier. If that's not happening for you, I do have a little advice: when you're doing a problem the first time, don't just try to solve it. Look for what the exercise was intended to teach you, what techniques you used. Look for what you could have done better - try to find not just a solution, but the best one. And do the same thing when you finish a series of exercises from the same section of a textbook or the same assignment in a class: what strategies did you use repeatedly? What connections did you see between the type of problem and the technique you used?
To take a simplified example: say we were asked to solve the equation $x^2 + 3x + 1 = 0$. We could try to solve it by factoring (won't work) or the Rational Zeroes Theorem (won't work either); or we could do it by completing the square, or by using the quadratic formula. Of these four approaches, only two will be successful; personally, I find the quadratic formula the easiest, so I'd do that. We get the solution $x = -\frac 32 \pm \frac 12\sqrt{5}$. What do we take away from this exercise? Well, first, we can notice that of the four techniques we have, two are unreliable; but completing the square and the quadratic formula always work, so we can default to those. Next time, we might not even have to think about which technique to use. Second, we can notice that our solution has a $\pm$ and a radical in it; if we don't see those in our answers to similar problems in the future, we might want to check our work.
But then suppose we're asked to solve $x^3 - x = 0$. Now, the quadratic formula isn't available, and neither is completing the square - both of those require us to start with a quadratic expression. We could try either factoring or the Rational Zeroes Theorem - either one will give us the answers $x = 0, 1, -1$. If you try both, you'll probably find that factoring is easier.
What have we learned from this section? Not just that the solution to $x^2 + 3x + 1 = 0$ is $x = -\frac 32 \pm \frac 12\sqrt{5}$ and the solution to $x^3 - x = 0$ is $x = -1, 0, 1$. We've learned that the quadratic formula is a magic-bullet technique for quadratic expressions, which produces messy results; we've learned that for equations that aren't quadratic, the quadratic formula is irrelevant, and so we should try first factoring, and then Rational Zeroes. Here's my point: the next time we see a polynomial of any degree, we can use this general-purpose knowledge to make the solving process much easier. This will work even if we forget these specific polynomials - in five years, if someone asks you to solve $x^2 + 3x + 1 = 0$, you won't remember this post, but you'll know to go straight for the quadratic formula.
I'm of the opinion that this advice is accurate no matter what level of math you're at. If you're doing proofs, look at what the shape of each proof is and what techniques it uses - for example, proofs about integers usually wind up using induction, and often have to do with divisibility by $2$, so I usually start by setting up an induction and asking myself "what happens if $n$ is even? What if $n$ is odd?"
