It’s not enough to just understand what is written. You have to understand why it is like described. Why it can’t be different - or could it? You need to build an intuition for both the physics and the math describing it.
You have to push until it clicks.
It’s hard work.
Two mathematicians are discussing a theorem. The first mathematician says that the theorem is trivial. He then proceeds with two hours of exposition. At the end of the explanation, the second mathematician agrees that the theorem is indeed trivial.
A C programmer with no experience in concatenative or, especially, functional languages will have little chance of understanding code written in Forth or Haskell. They will have to start at the bottom of the learning curve.
The differences between languages aren't just skin-deep. Same goes for the design patterns in those languages. Haskell's applicatives have no analog in C, for instance.
You're not wrong, but those don't crack the top-10 of in-demand languages. Enough functional paradigms have also crept into more popular languages that you wouldn't be completely lost.
I went through a similar phase with math, struggled with basic algebra for years, then at some point it all clicked into place and I unintentionally ended up skipping all high school algebra to get to early calculus.
It makes it really hard when helping other people through those early concepts (like pointers), because "keep working and at some point it'll just click for you" isn't very convincing.
I’ve started to call it paying the “time tax” or “staring tax”, I have to just review tough concepts (code, concept, leetcode solution, math problem) for a long while before it eventually clicks.
Haha, monads still haven't clicked for me despite knowing F# and Clojure pretty well. I think of them as "functors that you can flatten" (and functors are "things you can `map` over", and `map` is "a structure-preserving transformation"). I know monads syntactically, but not intuitively. Despite this I still manage to work with monads pretty well - sometimes rote knowledge is enough. Maybe if my work involved discovering new monads I'd feel like I'd know them better. (I just see the same option/list/result(ish)/generator-style monads over and over.)
What’s strange to me is how people seem to resist this fact. Like it’s obvious to pretty much everyone that you need to practice an instrument to be able to play music on it. But for math and physics at least they seem to think it’s somehow different? That they ought to be able to solve the problems just by picking up a pencil? And when they don’t get a decent grade on the exam they look over at the person who got 100 and say “that’s why! I can’t do it because I’m not a genius like they are!”
Maybe that's the difference between people who are good at it and people who aren't. I found solving math problems (the ones you get in a practice course at a university) hugely rewarding. Finally figuring it out was so satisfying.
Coincidentally, my endeavors playing an instrument were never particularly fruitful.
The feedback is only fast if you're already skilled. It's pretty common for new students to have no proper sense of what sounds good (particularly since it's familiar to them), such that they struggle to figure out what they need to improve. Similar to digital artists flipping the image often to make it easier to identify issues by making the image somewhat unfamiliar to them, forcing their brain to actually look at the image rather than just using its previous idealized mental representation of what the image looked like.
Saw plenty of this in school. Not appreciated.
I feel you missed the main point of my comment, though. Even if abstract reasoning, or certain forms of abstract reasoning, were not natural to you (something that I can impossibly judge), it wouldn't mean, as you initially claimed, that it would be impossible to teach you mathematics.
Just got a little upset about the rest. I wouldn't have reacted as I did had you phrased it a bit differently, like in the post I am currently replying to.
Either way I recommend reflecting on it.
Quoted sentence does a lot of heavy lifting in your reasoning. Being intrinsically interested in something is integral part of learning. And, yes, obviously for people who have zero interest in subject it will act as "specific learning disability". I know this because I tend to be interested in programming-related subjects and, consecutively, capable of learning them, but I could never muster any interest trigonometry or calculus homework. I've also known multiple people who felt the same way about debugging their software development homework - they weren't interested and hated every second of it.
Ironically, this is one of the first things they teach you in college level math (or at least they did in my case).
There are a lot of different reasons why students feel the way you do and (unfortunately) give up on math. It’s a shame because I believe most of those reasons are specific to the high school setting and from what I’ve seen, students have a much better time if they come back to math later in life and study without that environment.
Interest also plays a role, of course. And not everyone will have the interest to go back and study again, later in life.
Which brings us to my original point - not everyone can do it (well).
He would intuit solutions to problems based on extrapolating from the textbook. The undergraduate stuff all clicked right away for him.
Even when we scored the same on an exam, the difference in effort expended was dramatic
Some people "practice" by trying to play the same thing over and over, and it is equivalent to "hearing" vs "listening". Learning how to play an instrument (and also a particular piece of music) is a very complex process. Just like you need to learn the skills to "learn from a book", you need to learn the skills to "practice an instrument".
In most intellectual fields, you can take your sweet time to remember what you learned so long as you remember it eventually. Music requires practice to recall everything in real time.
Those students, they do not get FAANG jobs, or make big money. The few that want to understand are much more like the articles description, and they end up five years on far ahead of their peers at work.
Just my $0.02.
That sort of initial seed would give me so much information...
On that topic... Does anyone have any clue how and why magnetism works? I have a A Level Physics test next week :)
Like, 2-D, sure, seems easy enough. But once you go to 4-D, things get really hard to grok.
Passionate science educators online have really filled in a lot of my education.
of course these are rarely static (after all one can learn proofs of theorems without being able to solve any problem), hence the need for the "muscle memory" of how to apply these techniques to the particular patterns.