There are "Levels" to Understanding Mathematics

The way it was described to me when I was in high school was in terms of 'levels'. Sometimes, in your mathematics career, you find that your slow progress, and careful accumulation of tools and ideas, has suddenly allowed you to do a bunch of new things that you couldn't possibly do before. Even though you were learning things that were useless by themselves, when they've all become second nature, a whole new world of possibility appears. You have "leveled up", if you will. Something clicks,...