There are many ways to “learn math”. For some people, learning math is just about memorizing some formulas and learning to apply them. I will explain what I think is a good way to learn math properly and well.

Think, don’t memorize.

Be skeptical.

Try to understand so well that you can explain it to yourself and to others.

Think deeply of simple things.

Find a good teacher/mentor if possible.

Think, don’t memorize: If someone like a teacher repeats a phrase like “d=rt, distance = rate times time”, don’t just passively listen and believe it. Think about it logically each time, and rederive this each time you need it. I’ve never liked teachers who constantly repeat an acronym for a formula in the hopes you will remember it, when you should seek to understand the underlying idea, and “rederive” (re-come up with) the formula each time.

Be skeptical: Don’t blindly trust any formula. Always wonder: Is it true? If it seems true after some testing, then: why is it true? Maybe the formula is true for all the integers from 1 to 12. Be skeptical. Maybe it breaks down later.

If you’re ever given a formula sheet, try to understand why each formula is true. Formulas aren’t things to memorize. They are things to understand. They were developed by someone very very human. Humans make mistakes, sometimes even in the copy-writing stage. Perhaps someone mistranscribed an otherwise correct formula. Perhaps everyone else in the world who checked the formula was wrong. Don’t blindly trust the work of previous mathematicians, if you want to learn math well.

I was on math team in middle and high school, and sometimes we were given formulas without them being explained to us. This-the lack of understanding of a formula, or deep understanding of a formula, always bugged me. If I didn’t understand why the formula was true, or if I understood, but not well enough to the point where I could explain it to someone else who doesn’t understand it in a way they can understand it, as well as convince them it is indeed true/correct, then I am not satisfied that I understand it on a deep enough level, and I will struggle to understand it better. Even if it is a “simple” fact like why (−2)×(−2)=4(−2)×(−2)=4 (why two negatives multiplied together make a positive, and why is that positive number exactly 2×2?2×2?) (this ties into “think deeply of simple things”)

Once you understand something, practice explaining to others or yourself (or your imaginary younger self who still does not understand), if you can’t do it, then, while you may understand completely yourself, you can still improve your understanding to a level where you can in fact help someone else understand too. For instance, if someone learning math uses reasoning of spoken words to understand math, can you help this type of learner understand using only English words? This is a challenge, converting math conceptual understanding to people who want to gain math conceptual understanding from trying to understand in English words instead of, let’s say, symbolically, for algebra.

This process of trying to understand, understanding, trying to understand better, convincing yourself of the correctness of some mathematical fact, explain to your imaginary younger self, helping others understand, is a constant struggle.

I would go so far as to say, be skeptical of everything you are told in math class. If you are not, and you fall into the trap of blindly believing and accepting a bunch of formulas, algorithms, theorems, that some math teacher told you, you miss the whole point of mathematics, which is to think, understand, and help others understand.

Find a good teacher/mentor: This can be tremendously helpful if you are struggling to understand at any point in your math-learning career. It can be helpful even if you can do well at some local math tournaments but don’t understand so well that you can communicate to and help others understand. It’s hard to find a good mentor, but if you do, don’t miss your opportunity to learn from them! Learning to learn from others is also an important skill to develop (and is one that often isn’t exercised in a standard U.S. math class) At some point, you have to go past struggling to understand K-12 maths and struggle to understand other people’s work in mathematics, if you want to become a mathematics professor.

Once you can go through the process of learning and explaining that I described above, you can tackle the challenge of learning to solve problems you haven’t seen before and learning to write proofs. (I’ll skip mentioning this phase too much)

Once you’ve learned to write a variety of proofs, you can finally get past the rigorous stage of your math-learning career and hand-waive some things that you know you can go back and prove later if you are doing a long argument for a proof.

The most important piece of advice I listed above is to Be Skeptical. A healthy skepticism is good for a mathematician, and it is good in real life too, when listening to facts discovered in science and stories people tell.

Mathematics is a very social endeavor, and I believe learning math should be a very social endeavor too. It’s not easy to learn math properly, and I’m not sure the extensive process I described above is for everyone.

Good luck and have fun in understanding math to your personal level of satisfaction!

Thanks a lot

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