Some Confusing Experiences in Self-learning Maths

Don’t ask me why I would self-learn mathematics because sometimes I feel like it is a waste of time especially when I found the materials difficult. I feel that maths is the foundation of today’s science, and therefore, important. I feel that I lack training in maths. Not that I don’t know how to do computations in Calculus or linear algebra, but those rigorous constructions of proofs and theorems. And don’t ask me whether Maths is useful to me or not. It is not. In my daily work, I don’t need a very high level of maths. I will not earn a lot of money because of Maths. Should I spend my time elsewhere? Probably. But I would like to waste some time, just like watching Netflix and Youtube.

One very confusing experience when learning maths alone is I don’t know whether I understand the materials or not Or whether I understand it well enough. For example, I am reading Calculus written by Michael Spivak. When I read the text, I feel quite comfortable. It doesn’t mean easy but at least I can move on. But whenever I tried to work on the problems, I struggle. There can be 30-40 questions in each chapters. For some questions, I am able to complete it at ease. For some, I can spent hours or days but still have no ideas what’s going on. I ended up spending a lot of time in just one chapter by working on as many problems as I can. Then, I tried to search online to find out how the materials are usually taught. I read syllubus from different schools and find that for a normal homework set, it only covers 3-5 problems in each chapter. It looks like I am doing too many problems LOL. But even I get the 3-5 problems correct, does it mean I understand it well enough? There are many more problems in the chapters that I am not comfortable in solving.

Another confusing point is whether I should look at the answer, especially when help is not as accessible as in University. I found that to learn Maths better, the struggling process is actually very important. I tend to not read the solutions until I finish all the problems in a chapter. But without help (like someone gives you a hint), I end up spending a lot of time struggling and struggling. This is discouraging sometimes. It makes you feel dump. Maths always make me feel dump lol. And I will be impatient because of it.

It is like a balance between learning/struggling and time. I have only that much of time. Sometimes, I just need to move on to the next topics but not untill I can at least understand the prequiste of the next topics. I think that can be the value of teachers to guide you through the learning curve.