Memorization in mathematics

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Broadly speaking, there seem to be two "schools of thought" regarding memorization in mathematics, as well as a third "default position":

Name Description People who hold this position
Default position/status quo (often implicitly held) Memorization is not just necessary to learn mathematics, but it is the primary means through which to learn mathematics. Mathematics is the study of memorizing algorithms for solving problems. Mathematics pretty much consists of rote memorization. School teachers
Reactionary position The point of mathematics isn't to just memorize things; the point is the understand why things are true and to appreciate the beauty of mathematics. One should eschew memorization in favor of trying to deeply understand the concepts. Richard Feynman, Paul Lockhart
Revised status quo position Memorization is pretty important for learning mathematics, and understanding comes from having fluency/competency. Understanding is the real goal, but memorization for the purpose of gaining fluency and storing facts in long-term memory (so as to free up working memory) is actually an essential part of the learning process. Tim Gowers, Michael Nielsen