How to retain the key points of an exercise?

I'm a student majoring in maths. The sole reason I chose mathematics is that I like mathematics. I'm sure I do not belong to that "gifted" category. I haven't participated in maths competitions either. I don't know if the question I'm about to ask is a consequence of lack of formal training when it comes to problem solving and the like. The following is my question.

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.

• Not to take away from the answers below, but I suggest you obtain and read a copy of "How to Solve It" by George Polya. – G Tony Jacobs Jun 2 '17 at 16:46

For me, the process is as follows:

1. Do the exercise.
2. Do the exercise again. This is probably faster than the first time, since I have a vague feeling of what I should be doing, and maybe remember some dead ends that I can avoid.
3. Repeat step number 2. At some point, if I have to do the exercise often enough, I learn it by heart. This is rare.

In practice, this rarely happens unless I am teaching and encounter the problem often there, or I do research on a given subject and have to think through the same lemma several times.

Instead, the problem solving heuristics and patterns of thought that are repeated often are what stick to mind. When encountering a familiar problem, a problem of familiar type, I have a feel for the problem and know which common technique to try first, and I know what the likely steps in solving the problem are. I still have to do the work to be sure the attempted proof works (unless I have done the same argument several times recently).

It is, in general, okay to not remember solutions by heart. If you can repeat the argument, even if it feels like fumbling, that is already good.

The most elementary context in which I can observe this is multiplication; say, the powers of two. How much is $2^7$? I can remember $2^6 = 64$ (because I have calculated this so frequently) so $2^7 = 128$. If I want to be sure this is the right answer, I have to check it. If I do this calculation often enough, then I can guess the right answer immediately, but must still check it to be certain. Before I remembered $2^6$, I had to start with $2^5 = 32$, and before that, I had to start from $2^3$, and so on.

A similar process happens with less elementary problems, though the steps that happen by heart or from strong intuition are more interesting.

Any exercise is just one example of a family of similar exercises. You can explore those similar exercises by writing and solving your own variations of the problem.

To be more concrete, I assume that you have had at least first semester calculus so have seen something like the sliding ladder problem as an exercise. Suppose that you had the following exercise:

A 10 ft. ladder is leaning against a wall. The bottom of the ladder is being pulled away from the wall at the rate of 2 fps. How fast is the top of the ladder sliding down when it is 8 ft. high?

After you have solved the problem, start by simply changing the numbers. For example change 10 to 12 or 8 to 5. You will quickly notice that the original problem used a Pythagorean triple ($6,8,10$ satisfy $6^2 + 8^2 = 10^2$) which is convenient for a numerically nice answer but not essential -- radicals present little problem. Once you solve the same problem numerous times with different numbers, see if you can solve it once and for all with symbols like $L$ replacing specific numbers like $10$.

Then, start making more substantial changes. Push the ladder towards the wall rather than away from it. Change the given information so that it gives the rate that the top of the ladder sliding down the wall and asks about the rate at which the bottom of the ladder is moving. Maybe even get a bit creative -- what if it is an extension ladder which is extending (or contracting) at a given rate in the course of the sliding? Or -- what if the wall isn't perpendicular to the floor (an implicit assumption)? Convince yourself that you can still do it with the law of cosines replacing the Pythagorean theorem.

Once you get to the point that you not only know how to solve a given problem but are able to create similar problems on your own, then you will have a deeper understanding of the relevant key points, an understanding which is unlikely to be so quickly forgotten.