To teach people to think like a mathematician is merely to teach them to think properly.
Precisely.
If I can see how the relationships work then I can process math to solve problems. I try to work it out in my head but I can't handle symbols I don't know what they mean. I need to see examples and read explanations on what is happening. That is what handouts did for me, the books confused me. The handouts were step by step. The books just expected you to know what was going on. There was no structure.
I know what you mean, I've observed this as rather normal. Your problem is not as much with math as it is with lack of understandable information about math.
I guess the ones who
really understand math grow up and get well paid jobs. Teachers are picked from the rest. That is to say, you're on your own. It would be easy to say it's all someone else's fault, but we all have to go through this, this is just the reality of learning math on planet earth.
Handouts compile the most important information, leaving out as much of the fluff as possible, which is why everyone understands them. Such "refined information" is very valuable, but is only available on the beginner levels of math, because a teacher has to write those handouts, and that teacher needs to have a refined understanding of math first. Same problem as before.
Therefore, learning math relies heavily on your own skill of "information mining". Which is a skill of "extracting the essence out of a heap of bullshit". As a great counterexample: Wikipedia. It's a great reference, but it relies too heavily on foreknowledge of various concepts/symbols. Your information mining skill must be already extremely high to be able to extract the essence of the concept you are researching.
Information mining is like a game where you connect new concepts to the ones you already understood.
New concepts are learned by finding/mining an explanation. A good example of such an explanation is the "handout", which illustrates a new concept by using concepts you are already familiar with. This converts an unknown concept into an understood one, a red into a blue one.
But what if you find an explanation that uses concepts you are not yet familiar with? This is the situation you described. The books that are confusing.
You need to execute a recursive algorithm like this: For every red-concept that you need to understand the first explanation, you have to find yet another explanation.
Hopefully, if you're lucky, the second explanation that you find is useful = an explanation that relies of blue concepts as in this image:
If not, you might have to execute this algorithm recursively many many times. It all depends on how complicated the concept is that you are trying to understand. The higher the red dot is above your blue dots, the more steps you will need to reach it.
But finding/mining those explanations is still all up to you. Like any mining job, an arduous process. Some of this you probably already know, it's all part of the "learning how to learn" process.... maybe I just wanted to draw some pretty pictures