You don't need to memorize rules when studying math. Just like you don't need to spend any time to memorize syntax for programming languages. You automatically remember things you use a lot.

Once you have spent countless hours doing exercises to the extent that you understand the math, you already remember the rules. If you have not spent countless hours doing exercises, you don't understand anything at this level.

You don't hire a programmer who has read all the books and 'understands' programming but has never programmed. It's the same with math. You don't just read a math book from start to finish. You can use wolfram alpha for visualizing functions, not for learning math.

+ 100. I can't upvote this enough.

Programmers have spent countless hours practising programming to the point where they have forgotten how difficult it was in the beginning. A non programmer might think of programming as "memorizing hundreds of rules" to get anything done, but one doesn't learn programming by sitting around explicitly memorizing hundreds of rules and then begin to program.

Actually writing programs with a minimal set of 'rules' memorized and then adding more as needed is how one typically learns programming.

I've been teaching programming for five or six years now. I always start with HTML, then add CSS, and then add JavaScript. That way they experience mastery all the way, and see how they can be creative with the code. It's so great to see a pupil "get it" -- and sometimes even pupils that "suck at math" or even pupils who have problems spelling the most basic sentence correctly. In fact I've found that there's a strange correlation where pupils who have dyslexia often seem to be better than others at programming.

> I always start with HTML, then add CSS, and then add JavaScript

So when do you get around to teaching programming, then? ; P

/ducks

Starting with HTML and JavaScript is one of the most efficient ways to get children interested in programming.

I wrote something similar to what you are saying. For instance, I said: "understanding integrals and derivatives doesn't require one to memorize all the mechanical rules".

I'm not against memorization though. Memory is very useful when studying math or programming or any other subject. You don't want to have to "reason your way through" every time, shortcuts are very important!. I think of this like brain-memoization. Without it, it would be very inefficient to make progress. A lot has been said about this relationship [1]. Also, I think this is how some breakthroughs happen, "connecting the dots", so to speak.

Maybe when you say: "...spending any time memorizing syntax...", you are thinking flashcards or something like that? Sure, you don't need flashcards, anything you do often enough is gonna be easier to remember.

My comment was more in line with the fact that, with 2000 pages, maybe the author elaborates a lot on things that are very mechanical in nature and maybe require a few pages to describe (and are very inefficient for humans to compute? Just use a computer! :-). Say, Gaussian elimination; couldn't one be told: this is a matrix, this is a determinant, this is the relationship between them, this is what it means to invert the matrix, etc. and skip the full description of Gaussian elimination? (put in an appendix? on a second book? less pages!). I don' think is super helpful to, say, spend a lot of time inverting matrices with pen and paper in order to get proficient in linear algebra.

1: https://www.google.com/search?q=intelligence+memory+relation...

I'm coaching my son through high school math. One of the things I'm trying to impress upon him is focusing on understanding _why_ these formulas work rather than just memorizing the formulas themselves - if you understand why they work, you can always re-derive them if you need to, and you may forget the details of what they do, but you'll never forget the details of why they work once you understand them.

The best strategy for this in my opinion is to always ask “what’s the picture” (as in visualization)? I can’t remember the mechanical rules for say Newton’s method, but I have the picture of what Newton’s method does in my head. From that, I can quickly work out the algorithm.

No, you don't need rules to understand or use language. You need rules to explain why.

> No one should go with pen an paper double checking if that polynomial integral is correct or not!

Hm - maybe I'm fortunate that I studied calculus before there were (accessible) software packages that could just do this stuff for you, because back then, the only way to solve these was to do them on paper. I'm sure I would have been tempted to just "skip ahead" to letting the computer do it for me, but I definitely learned a lot more going through all of the steps myself than I would have if I had just gotten a high-level understanding of what was going on and plugged the rest into a computer. Because, honestly, integrating polynomials is really, really easy - if you know how to do it, you can do it on paper faster than you can load up wolfram-alpha, type it in, and wait for an answer.

This one does not. it's proof heavy, and there isn't really a mechanical way to do proofs that's efficient. The understanding comes first and often all in bunches when "the light turns on," then the proof follows.