So you want to do calculus ? You need algebra. What parts of algebra ? Go figure !
this is one big hurdle in learning math backwards. You discover new missing pieces at every corner. Each missing piece leading to another missing piece.
Learning math from the basics to advanced (as recommended by most) is very frustrating at how slowly you actually develop the math muscle.
At a deeper level, conceptual grasp does not make you good at math, its not enough. You may fool yourself into thinking you "get it" till you try to solve a few exercises. You need to repeat the lower levels enough to make it into muscle memory (which some people refer to as math intuition or groundwork) before embarking onto higher levels that build on it.
So working your way bottom up is slow and frustrating, top down is slow and frustrating. What do you do?
Just keep at it. One key observation for me was that at some point the misery and rabbit hole nature diminishes, quite rapidly. The groundwork of solving all those exercises repeatedly pays off and the next set becomes a little easier. Getting to calculus after spending ridiculous amount of time on algebra is the only way I have known to work.
And this is true for learning progamming too. knowing the concept of loops is essential but, you still can't write efficient code to sort an array. You need to get the syntax and write enough loops and then progress to exercising writing specific sorting algorithms repeatedly to get them into muscle memory.
But there is an inflection point beyond which the same concepts repeat but in different variations and they take progressively lesser time to get a grasp on.
thats just how I've learned math and programming. Also why a large percentage of people just give up hope and accept they just don't have the math gene. Meh.
> this is one big hurdle in learning math backwards. You discover new missing pieces at every corner. Each missing piece leading to another missing piece.
Yes this is exactly what happening to me.
E.g. I got up to Week 2 of the course and suddenly made the big (to me) discovery that sqrt(a/b) = sqrt(a)/sqrt(b).
It seems trivial I know when you see it written like that, but the problem is to recognise and apply that principle in the context of a broader problem such as factoring.
> Just keep at it.
Thanks, this gives me confidence that I'm not wasting my time haha
I am beginning to get better at it, to the point that I can often work out why I got a question wrong on my own without referring to the answer.
It's really frustrating how every single person I know who got (really) good at math or programming got there the same way, but never even hinted about it to me till I saw them use the same techniques. The clever ones figured out the important parts faster and spent more time on repeating the common idioms, theorems and required prior knowledge (e.g. the sqrt(a/b) = sqrt(a)/sqrt(b) piece for you) instead of the problem or spending too much time on conceptual understanding
The really important part for me was to rip these small but critical parts out and form somewhat like mental workout routine that I kept repeating multiple times per week. By week 5/6 I could solve the same/similar/related problems which weeks ago took me several minutes with ease and I had more brain power left to think about higher level and related concepts and techniques that formed more connections, making the experience a lot more fruitful, productive and faster. Without that mindless, disciplined mental routine to get the basic and critical stuff in muscle memory, I do not believe I could have made it through.
this is one big hurdle in learning math backwards. You discover new missing pieces at every corner. Each missing piece leading to another missing piece.
Learning math from the basics to advanced (as recommended by most) is very frustrating at how slowly you actually develop the math muscle.
At a deeper level, conceptual grasp does not make you good at math, its not enough. You may fool yourself into thinking you "get it" till you try to solve a few exercises. You need to repeat the lower levels enough to make it into muscle memory (which some people refer to as math intuition or groundwork) before embarking onto higher levels that build on it.
So working your way bottom up is slow and frustrating, top down is slow and frustrating. What do you do?
Just keep at it. One key observation for me was that at some point the misery and rabbit hole nature diminishes, quite rapidly. The groundwork of solving all those exercises repeatedly pays off and the next set becomes a little easier. Getting to calculus after spending ridiculous amount of time on algebra is the only way I have known to work.
And this is true for learning progamming too. knowing the concept of loops is essential but, you still can't write efficient code to sort an array. You need to get the syntax and write enough loops and then progress to exercising writing specific sorting algorithms repeatedly to get them into muscle memory.
But there is an inflection point beyond which the same concepts repeat but in different variations and they take progressively lesser time to get a grasp on.
thats just how I've learned math and programming. Also why a large percentage of people just give up hope and accept they just don't have the math gene. Meh.