This only answers a small piece of your question, but it's important to realize that the mathematical notation you see is only how mathematics is recorded and c

This only answers a small piece of your question, but it's important to realize that the mathematical notation you see is only how mathematics is recorded and communicated. Most of what goes into doing mathematics happens in your head. "Doing" mathematics requires you to know what the object of study is, and what tools have historically been brought to bear on it, and these are communicated and taught via notation. But once you've internalized those concepts, the notation doesn't usually play a significant role. Yes, you may trial your ideas with some calculations, and once you're convinced of something you need to prove it with some degree of rigor. But you don't usually set out to prove something you don't already have some reason to believe in -- it's not like the math workbooks you may have encountered in school. (I second the sibling reply's mention of Lockhart's Lament!)