Importance of Exercises in Mathematics for Self-Studying

I am a high school student wanting to major in Mathematics in the future. I started to like Mathematics recently, starting a year ago and I watched some interesting math videos on YouTube for fun (ex: History of Mathematics Prof. Wildberger, Intro to Real Analysis Prof. Su, Linear Algebra MIT, Multivariable Calculus Prof. Shifrin) and I was studying these topics/subjects only from videos. I still have to finish the Linear Algebra videos though.

While watching videos was not very in-depth or rigorous, it was fun for me and that's all it mattered until recently. Recently, I got a local University Library Card so that I could actually access the textbooks instead of taking the information solely from video lectures. So I borrowed Rudin's Principles of Mathematical Analysis, Munkres' Analysis on Manifolds, Lang's Linear Algebra, and Herstein's Topics in Algebra for the subjects I was studying through videos so that I could have some mathematical maturity and problem solving skills that I currently lack.

The problem I have is that I'm used to having the professors on screen solving problems and proving proofs for me :( So it is difficult for me to do the exercises by myself or prove the theorems myself. Even as a novice with not much experience with Mathematics, I just seem so lacking as I can't even do the exercises on the textbooks. I did not expect/plan for this to happen because I falsely assumed that watching the video lectures would further my understanding of concepts in Mathematics and my mathematical maturity.

When I read the textbooks (I haven't got far into any of the texts, just a few chapters), I read every word but I do not do the exercises. When the author goes through proving some result step-by-step, I enjoy it because I usually follow along and it seems very clean and logical and when I read the author's insights or background information, that interests me as well. While reading these books are fun and entertaining, it is only fun because I skip the exercises; I read the exercises, I think about how to solve them, and if I get an idea, I handwave and say that's obvious but I never write it out with paper and pen and when I do, while the idea is there, I cannot formulate it well. And the questions that I don't have an idea of how to tackle, I skip them, too.

So basically I read but skip all the problems which I know is not beneficial for me. After a month or two of almost no progress with solving exercises, I am bothered because I feel like I'm getting nowhere and reading further seems pointless without understanding how to do the questions from previous chapters since it feels like I didn't learn/get the fundamentals right. I get frustrated and sometimes even reading the books do not interest me.

Is there a systematic way to fix this problem? Also, I just keep rationalizing my actions of not doing the exercises by telling myself that I'll have to learn these topics/subjects again when I get into a university so that I can just relax for now and just read the materials/concepts without solving the problems. It feels very wrong not to do the exercises though because then I'm not really learning how to do mathematics, I'm just reciting information and concepts. If anyone has advice, I would appreciate it a lot if you commented. I'm interested to know the methods of doing exercises and formulating solutions without skipping them all. I also want the solutions to the exercises but there is a lack of solutions for these books which isn't helpful to me as well. I'm also curious as to how many exercises I should do from the book. Should I do all of it, most of it, or some of it? Thank you very much. I probably want to make this a community-wiki but I do not know how to.

P.S. Does it matter how much time I spend studying a day if I want to be successful? Is there or should there be a minimum amount of time I'm dedicating per day for studying Mathematics? I know that this is personal and can depend on different people but I was just curious. I would like to spend 2~3 hours doing Mathematics each day but it seems like a lot of people spend more time and are more dedicated. If this P.S. seems pointless/useless, feel free to ignore it as I'm more interested in the main question.

Honestly, for the most part you can even skip reading most of the book. Problems are millions of times more important.

When you get a new book, don't even read it. It doesn't matter. Go directly to the problems, skipping everything in the book. Don't even glance at chapter titles. Go directly to the first problem. Obviously the problem probably won't make sense to you. THEN go back and briefly look for just enough information to do the problem. What does ___ mean? What theorems look somewhat relevant? Are there example problems similar to your problem? After you do a few problems, maybe read a little bit more if it is interesting and to get a more general idea of what's going on. But problems should always always always be your number 1 priority.

For the most part, other than reading the main theorems and definitions, and maybe a couple example problems, reading the prose of a text is a time sink and not that useful. This might not sound like much to you now, but when your time stops being so free it will matter a lot.

Edit, I don't mean to never read the book. Just go to the problems FIRST. The problems should always be the first thing you look at in a textbook. Then read the book with the purpose of solving those problems. If you're reading the book just to read the book, you're wasting a lot of time.

In learning there are two goals, one is obtaining knowledge and the other is obtaining skill.

• Having knowledge without skill means you can talk about it, but when it comes time to act, you lack the tools to produce results.

• Having skill without knowledge means you can generally produce results, but your limited understanding of the world has you relying on trial and error, or has you being unaware of techniques better suited to the task at hand.

Consider building a table.

• A knowledgeable person might know how to calculate the sag of a table, but lack the ability to build one. To do so, he must learn about selecting wood, safe use of shop tools, etc.
• A skilled person can build a table, and perhaps in the building discover that his table sags unacceptably, forcing him to add stiffening elements. He won't know how much or little to add, and under some circumstances (extremely long tables), his initial attempts at stiffening might add more weight exasperating the problem.

The right knowledge can save you a lot of effort, but it can't make you learn the patterns needed to do things. Being able to do things is good, but if you lack study, you will have to learn it all the hard way (and there's only so much time on the planet to do so).

So I think you are doing a fine job of expanding your knowledge of Mathematics, but it sounds like you need to sit down and do "just enough" problems to develop your skills.

By the way, double checking your work is a skill too, and since you lack a professor to grade for you, it sounds like you're going to be developing that skill early (which is a very good thing).

As a counter to other answers, I think of mathematics as a kind of language to learn. You can measure language proficiency by understanding, speaking, and writing. Reading books is by no means a waste of time, since if you can follow what it's saying, you are improving your ability to understand, and hopefully by extension to speak and to write at some point. One way to measure your progress is to see if you can't write out a complete treatment of some key theorem. An overlooked part of mathematical proficiency, I've found, is the ability to say grammatically correct mathematical utterances.

But: the thing about math is that you cannot fully understand what the math is about until you struggle with problems. The form of math (the definitions, theorems, etc.) follow from the function of math (solving problems). You will see what issues arise when you solve a problem, and you will gain an appreciation for the structure set out by the author of the textbook. You will also see what isn't in the textbook. Authors of mathematical textbooks tend to leave it to you to figure out what all of it is for. Sometimes authors are lazy about constructing good exercises, and it is up to you to come up with your own!

Where mathematics is different from usual languages is that the objects it talks about are not the everyday familiar objects like apples or chairs. A math textbook is like a guided tour where the guide points out objects and facts on a fixed route. Can you really understand an apple without picking it up, turning it around, cutting it open, or tasting it?

I found Liebeck's A concise introduction to pure mathematics useful for seeing a cross section of problems from various fields. Also, don't worry if it takes days to solve a problem (I've spent weeks on many). Take a look at Polya's How to solve it for a bunch of heuristic questions to ask yourself when you get stuck. I once heard something like "a mathematician's happiest state is confusion."

Things to think about when reading a mathematical text:

• For a definition, what are examples, non-examples, gotchas?
• For a theorem, are the hypotheses necessary, can they be strengthened? Is there an interesting special case of the theorem? (Math texts tend to go for maximal generality on the theory that you can figure out the specifics.) Can you come up with a different proof of the theorem?
• After solving an exercise, what might be interesting related exercises? What insight did you need in solving it? Can you turn it into a more general theorem?