That point is very early. Math finally really took off for me when I realized that, no, it's not enough to understand an equation, or a set of requirements, etc., when I'm introduced to it. I have to actually memorize it. Literally with a flashcard app that I open every day.
By rote memorization, it is available in my head immediately, and I can use it to reason through things, without having to derive anything. It helps tremendously.
Then, at some point, it will become so familiar and understood through working with it, that the rote memorization becomes superfluous. For example, I've worked with the Laplace Transforms and the various Fourier Transforms (FS, FT, DFT, DTFT) so intensely by now, that I can absolutely just recite them by concept and understanding of why they are what they are.
But until then, rote memorization is basically a necessary tool to work with math.
The moment that made me realize that was when I saw a math genius at university, one who clearly understood the topics extremely well, going through his flashcards.
About three weeks into my first undergraduate class on abstract algebra, it dawned on me that the instructor wasn't giving me math tests. He was giving me vocabulary tests. In that class, most of the answers to questions flow straight from the definitions. Once I broke out the flashcards and started memorizing definitions, that class became almost trivial.
I used flashcards in all my classes after that to memorize terms, definitions, and concepts. Math and engineering are, for me anyway, like a foreign language. To converse in that language fluently, one must be very comfortable with the vocabulary. It just makes sense.
Well put, and I like that analogy. You have to learn the vocabulary, and only then are you well equipped to discuss grammar and finer subtleties of the language. It goes hand in hand.
By rote memorization, it is available in my head immediately, and I can use it to reason through things, without having to derive anything. It helps tremendously.
Then, at some point, it will become so familiar and understood through working with it, that the rote memorization becomes superfluous. For example, I've worked with the Laplace Transforms and the various Fourier Transforms (FS, FT, DFT, DTFT) so intensely by now, that I can absolutely just recite them by concept and understanding of why they are what they are.
But until then, rote memorization is basically a necessary tool to work with math.
The moment that made me realize that was when I saw a math genius at university, one who clearly understood the topics extremely well, going through his flashcards.