I still forget concepts even after answering numerous math problems

Question

Note: this is particularly aimed at high-school/entry level college problems

When I'm learning a new topic:

1) I read the theory given in the textbook at the start of each topic

2) proceed to read the solved example problems which the textbook provides (usually 3-5 with full solutions)

3) I then proceed onto answering every question within each exercise.

My problem is that I still forget concepts, for instance, let's say a week later (or even a few days later sometimes).

What am I doing wrong?

Answer 1

You don't say anything about writing notes or summaries. For each type of problem, while it is still fresh in your mind, write a detailed reminder to yourself on how to do it -- maybe a particularly good example. Include key definitions, theorems with examples and non-examples. Keep these summary notes separate from your lecture notes so you can quickly review them.

Answer 2

Usually the things that we remember are those things that we both understand completely and have found useful.

Understanding requires an intuitive grasp of why in the first place you want to consider certain mathematical objects, why a theorem should be true, and how intermediate objects were conceived. If you don't understand why someone constructed some strange object in the middle of a proof, it's impossible to have a full grasp of the underlying structure. You can know whether you fully understand a topic if you can re-derive all the theorems on your own without knowing how it was done in the textbook. Typically that means that you'll have to try to prove each theorem yourself before reading the given proof. If you get stuck, you can glance quickly through the given proof to find roughly where you have reached, and then read one or two more lines as a kind of hint so that you can try to continue by yourself. At the same time you should try to figure out what aspect of the problem you had missed in being unable to proceed. This may not be possible with some textbook proofs that pull intermediate objects out from thin air without explanation of how they got it, in which case you should ask someone familiar with the topic.

Secondly you need to find out what it is that what you're learning is useful for. If you cannot apply it to the real world, you can still try to extract the core structure and perhaps that might appear in other more applicable areas. If you can't use what you learn, it's quite natural that you'll forget it sooner or later, even if you practise numerous times using exercises in the book.