During my online course study, sometimes the instructor would say, intuitively, the reason go such as such. We spend a lot of time doing proofs that are the primary insight and knowledge. But the instructors also speak of developing an intuition on the subject.
It dawn on me that these intuitions are the superpower that separate people who "just know" from the rest. This is not some short cut but the critical skill of mastery. Some really simple algebraic intuition are x2 grow faster than x; 1/x decrease as x increase, etc. This is obvious to those who know them. But to others, they may not see this immediately and may need to think or make calculations. These students don't have the intuition. They may have figured out at the end, but they do not have the mastery.
I was fairly good at math at school. Now I am learning a lot about probabilities, a somewhat a new territory to me. A lot times my intuitions plainly fail. I stare at some simple symbols or equations and cannot make out what they are. I may have to slowly go thought definitions and try to work them out. But after working on these topics for many months, I begin to be able to see more things intuitively. Mastery in mathematics is not just knowledge in formal proofs. A lot of times it is to develop the intuition so that you just know.