If you feel the need to close the post, please at least read the whole post, I really think it will be very useful for other people.
I am a programmer, and my personal believe is that math math should not be hard to "understand." by understand I mean you can absorb the concept. So with my definition of "understanding," it is not required you to be able to proof, or derived math equations from certain theorem, or apply them.
think of it like you are a chess newbie and know all the basic stuff, and you are watching, says, video of chess match between professional. In this case, you understand what is happening on the board after a move is made (eg. checkmate), but you are incapable of making the "correct" move before the move is made.
Similarly in math, i think, it should be possible to just "understand" the concept of certain equation, even though you are incapable of using it in anyway. I guess its like knowing just intuition but not the math. okay, let me get to the point, with this minimum requirement, I am wondering is it possible to learn math like learning to write a program.
program is similar to math concept in that "concept" build upon each other. (program build upon "class", "variable", "function", etc. and math concept build "premise", "theorem", "definition", etc)
Example of step to learn to write a program.
- I want to understand how to use method A
- I read method A and but I don't understand what variable B is used for
- I check how variable B is used
- Now I understand how method A is used.
- List item
step provide above is pretty obvious in programming and it should applied to learning math concept too, but the problem is programming "concept" is searchable, but math "concept" is not required to be searchable ([ ] may mean different things in different math field).
another reason that above step is difficult to apply in math compared to programming is because math lack "validation steps" (a step to validate degree in which you understand the concept.) In programming, if function don't run, you don't get it. if function doesn't do what you want, you don't get it, and you don't even need to read the function line by line neither because it simply compute for you. However, in math, I is hard to even tell what you don't understand and also hard to test if you are not comfortable working with math (No way to just run the "concept" to test you understanding)
so I give myself a challenge to understand a research paper that I have absolutely no clue about, and see if I can understand it. This implies that I can skip things that doesn't help me to understand the higher level, such as theorem's proof. (in case concept that I care is built upon this theorem.)
My current difficulty is searching for certain math notation given a context. (eg. even if I know what operation A is, but if i don't know its symbol, it wouldn't know that it is operation A; aka, operation A is searable, but its symbol is not searchable) here is the paper for my challenge: Universal Invariant and Equivariant Graph Neural Networks https://papers.nips.cc/paper/2019/file/ea9268cb43f55d1d12380fb6ea5bf572-Paper.pdf
so far i am doing okay, I understand what the paper is saying in higher level, I will pass the test of "understand" if I can do the following
- write essay with minimum equation explaining what the paper is trying to solve, what is the usecase of it, what the concept is about?
- I can still recall the concept with in the paper, and "reuse" them in the future. ("reuse" means I understand general concept rather than per use case basis)
- at any point in time in the future if I need to learn "lower" level concept, I can simply fill out the blank like pluging in lower level concept into a bigger one. (nothing is changed; same shape but less hole)
It is worth noting that I do have note taking system to allow for point 2, 3, but what I am doing here is to create a framework to learn math in such a way that my current note taking system can be applied.
why am I doing this?
- no need to learn math from book. like reading chapter 1, 2, 3... next book,..
- learning is more fun. (project based learning (eg. understand certain paper) . rather than "just learn")
- learning can be "paused and resumed"
- learning is adhoc. (only learn when i need to)