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Does one learn mathematics by memorizing the rules?

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Take a look at 3Blue1Brown on YouTube. His videos are really quite fantastic at helping one visualise what is meant by seemingly "pure" maths. A few things like Taylor Expansion I was able to "do" but didn't get a feel for it until I watched his videos. Although I would still recommend getting a good book from a library or online and just practising lots of problems which use the maths you're stuck on. That really is the key to understanding maths - it's in the doing.
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To put it simply, you learn math by solving problems.  If you solve enough problems then you will end up storing the rules in your memory; however, this does not me the most effective way of studying math is memorization!  The best thing to do to learn math is do the problems, check your answers, if you get one wrong go back and correct it, and repeat until you are finished with your homework or you have used up all the time you've allocated to studying math that day.

Good day!  And good math!
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no, this is exactly the wrong way to learn mathematics, although I believe it is what most people below an undergraduate level are doing, and also that it is one of the main reasons why so many people are proud to hate math (or rather, what they think math is).

the point of learning math is so that you can *understand* the concepts and apply them in new situations that you haven't seen before. if you "learn" by just memorizing the procedures to get the answers to certain types of problems, then you are not understanding concepts and you gain no knowledge that can be applied more generally, defeating the entire purpose of studying math. memorizing the procedures is not a useful skill to have, because most likely someone has already implemented the procedures on a computer, which can execute them faster and more reliably than you ever could by hand.

mathematicians, and more generally, just people who actually understand the concepts, do not dedicate any time to memorization. if you understand the concepts, and you use or think about them regularly, then you will automatically remember them (at least, the most important parts) with no conscious effort required. if you don't regularly use the concepts then you will eventually forget them, but this is not a problem because they can always be looked up when needed. if you feel that dedicating time to memorization is *necessary*, then this is a sign that something has gone wrong.

relying on memorization will only get you so far. people who rely on memorization typically get stuck around the end of high school level, often in calculus. the only reliable way to fix the issues caused by memorization without understanding is to go back to the beginning and relearn everything, but without relying on memorization.
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Strictly speaking, the answer is unequivocally yes. However, that's extremely misleading if you don't know what a "rule" is.

Rules are things that are handed down to you. Some rules are chosen but make sense (like distributive property). Some rules are chosen because we just had to pick something (like x-coordinate points to the right). Some rules are actually just result of other rules (doing the same thing to both sides of an equation is just inherent to what "=" means, since otherwise the two sides wouldn't be equal).

Many things are NOT rules. Like "Solve 2x-7=0 by adding 7, THEN dividing by 2". You could also divide by 2 first, then add 7/2. There are basically no procedures from algebra that you should memorize. But the above *is* a Strategy: "usually solving by reverse order of operations is easiest." There are also Tricks, like recognizing a difference of squares. You learn a list of tools. The understanding you need there is just when to apply them (which usually comes from experience).

It's important to draw distinctions between dumb tricks, interesting rules, and rules that are just there, because then you know where to spend mental resources. To a degree, it's a matter of taste. But most things in math before proofs classes are not deep. There is a reason behind it, but that reason is usually "that's the definition". Or something like "you use the quadratic formula on quadratic equations because that is the solution to quadratic equations" is all the depth of understanding you need. My personal general rule is to try to feel out anything that's visual/spatial, and to just categorize everything else as "that's the definition" or "that's something that works".

So, like everything in life, don't fall into a camp. Whatever situation you're in, do the thing that applies to that situation. If something is deep, learn more; if it's not, don't. There is no magic reason others seemed to be doing better than you. Some people just act like it's fine, some people learned what I said above from an early age and it snowballs, some people memorize a bunch of stuff that isn't rules and do fine in basic algebra but then bomb in college. Overall: learn the fundamentals, take things for what they are, and don't worry too much. If something's not working, talk to people and try something else (like you are doing now).

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