How to understand the book and the material to the deepest possible level?

Question

I'm a first year mathematics major and I have a problem with my learning process. In my university, I only have books and questions that the university published, so I have to learn the most of the material by myself. The books are books that my university has written. I'm having problem with getting a deep understanding the material. My learning algorithm goes like this most of the time:

1. Opening the book, starting to read and most of the time reading fast.
2. After a not very long time, starting to skip the proofs that I didn't get.
3. Starting to skip some exercises.

So at the end of the day, I know the definitions, I can solve some exercises, but I don't understand the material deeply. When I'm reading I usually don't have questions and that's not a good thing. I want to read the book, ask questions all the time, prove things on my own, etc. How can I solve this? I don't have motivation to understand the proofs. I thought about proving to myself the theorems, but I don't know if that's a good idea. The courses that I'm taking now are Linear algebra 2 and Real anaylsis 2.

Answer 1

1. Buy some alternate texts, that are more pedagogically useful for self studying. There is cheap stuff available on Amazon from Dover, etc. Check reviews on the web to see what is best suited for you. (May not be the hardest ones, may be the easier ones.) Also, check out what is in your library. Yes, the actual brick building. If you are too poor for buying books, then just look at the web and steal pdfs.

2. When you read, some degree of speed (not too much, but not too painstaking either) is good. Just realize, it will never be as immersive as reading a novel or a history book. BUT, when you hit examples or derivations, you need to slow down and "read with a pencil" (in your notebook, see 4 below). Recreate the example/derivation or if material is easy, just attempt it yourself and then see how the book did it. But treat each example/derivation as an en passant exercise.

3. Do all the exercises. Yes, including the easy ones (hopefully there are some...if not, find some). They can build familiarity in a way that "discover the world" ball-buster problems do not. Part of the struggle in these abstract classes is just getting used to a new language (e.g. in physics, few people "get" quantum mechanics at first...not the way kinematics is a blinding revelation, but over time they just get used to it.)

4. Work with a notebook (bound or spiral). One active one per class. From the front, you have study/drill. I usually have no icon at top of page for exercise. Stars for lecture. And hashes for book study. From the back (IMPORTANT), keep a running list of questions (with space for answers). Just this process of writing your questions down will help you study, help you feel less lost. Often you will find yourself answering them...but in some cases, not. But they are great questions for followup with the instructor.

5. Read and do drill work prior to lecture. I know this sounds hard, but paradoxically it will make the work easier. And the lecture will be extremely comprehensible! This takes discipline. But pays off.

P.s. There is no royal road to geometry. Buckle down and study. Good luck!

Answer 2

Your main concern is "but I dont understand the material deeply". Especially in mathematics, a deep understanding is very important, so you should do your best to reach that level. By just passively reading a book chapter, you rarely get that deep understanding.

Maybe the Rubber duck technique can help you.

After reading some chapter, explain it to some patient audience. In the process of re-phrasing your newly-acquired knowledge in your own words, you often find your weak points. It isn't even necessary that someone from your audience actively asks about aspects they didn't understand - you'll often feel that yourself. And that's why the audience can be a rubber duck if a group of fellow students isn't available.