I've grown to see math as far more pattern matching than I remember it as a kid. I think that explains why "rote memorization" works more than folks want to admit for it.
Mostly because they have rote memorised it (and partly because much of the education system is a game to be played e.g. speaking with professors during office hours can give very strong hints to exams).
I suspect this is a lot like being able to recognize a piece of art to the artist by sight. Strictly, not required. But a lot of great artists can do it.
For real fun, I saw an interview with Magnus Carlsen where someone was quizzing him on famous games. He was able to say the match on the first 2-3 moves a remarkable number of times.
John von Neumann