Why do some people struggle with basic algebra and generalizing

you should stop discuting with monkeys then

>>8240734
I think the thing that surprised me most when trying to teach calculus to people is that they get completely hung up on function notation (y=f(x)) and start multiplying crap out and everything. Idk maybe programming since I was 11 got me used to it? It never seemed hard or even noticeable to me (I honestly thought they always spent way too much time explaining it) but some people just really struggle and I can't understand why.

>>8240737
>discuting

>>8240739

>hung up on function notation (y=f(x))

What does that mean?

Y is a function of X?

>and start multiplying crap out and everything

What does that mean?

>>8240749
I think he's saying that when he showed people the notation f(x) for the first time
to mean f(x) is some function where you give it an x value and it spits a value out, to mean something like 2(3) = 6, or f(x) = fx

>>8240734
You hear people talk about a learning curve; but I really do think there's a "learning wall", a point which you come up against something new and detached from your experience thus far. At which point you have to work quite hard to jump over the wall, after which it's back to the nice smooth curve. Take your example, everyone by the time they're at least 8 can multiply with a high degree of competency, but then you first see algebra and it's devoiced from everything you've seen up to that point, so you have to work harder to understand the concept of abstraction, and all it takes is some decent exposition and maybe a couple of concrete examples. But then you run up against the human condition: when things get hard people get depressed, the more they get depressed, the harder it gets to work on something, at the end you either stop or keep going and make a breakthrough; it end's with you just not being able to pick up the skill.

But once you have it, once you understand that algebra is just using letters as a stand in for numbers (and then you assign the letters numbers when you need to) you hope over the wall and carry on up the nice, smooth curve, applying and expanding this new skill and understanding.

At least that's what I always thought.

I think it is the education system, I just finished 12th grade and I can tell you out of my 6-7 math teachers I had only 1 was truly good. when I say "good" I mean he explained the 'why' and not only the 'who'.
I think lots of kids are not taught what the 'Letters' mean in algebra properly, I noticed while teaching my little brother algebra that he treated the letters as letter and not numbers, so I had to explain to him that the letters are only substitutions for unknown numbers and that they are essentially number.

>>8240756

Right ok, so people find it difficult to just accept the underlying logical framework of the relationship y = f(x) and want to immediately apply to something.

It's kind of like saying that people have trouble with abstract thought/concepts, which seems to be the case.

>>8240734
I think I can relate. For example, when I was teaching myself category theory, I've thought I have a good understanding of fundamentals up to the point when I tried to wrap my head around Yoneda lemma, when I realized I don't actually see beyond the diagrams and arrows, and it's all nonsense to me. Then something "clicked" and now I don't even know what I couldn't understand.

tl;dr I think these people could make the jump, but just they neither want nor need to.

>>8240734
Generalization through representation is a difficult skill. It's probably the main skill that separates us from the other animals. Representation leads to abstraction but the problem is that to abstract you must have a solid base and variety to abstract from. People are pattern recognition machines, they need a shitload of examples in order to learn.
>>8240786
Exactly

I think the biggest problem is the way people are taught nowadays. I mean really, I'd say 95℅ of people who have passed a calculus course don't really understand it at its core and the reason why is really simple. When I want to understand something and I mean REALLY understand it, the first thing I ask myself is why? Then comes the how. If I ask an engineering student to perform a simple integration, you bet they'd be able to perform the operation, because they know how to. But if I were to ask that same engineering student to put themselves in the shoes of Isaac Newton and explain the process of integration in the same way he did, would they be able to? Hell no. What I'm trying to say is that students see mathematics as a means to an end nowadays, not a venture in and of itself. If you want people to really internalize the concepts of calculus, give them the tools to come up with the principles on their own, then if they can't come up with these principles show them how to do so step by step (in essence, walk them through the thought process of those that invented calculus).

>>8240897
This is a big problem in schools. They always tell us " thats how it is, just memorize it" and a lot of people seem to do fine with that. But there are people like me, that need to understand it to learn it. It's comletely useless to me if I can't tell how to use it properly. If you understand a concept then you can draw analogies to the real world and the concept becomes usefull. I was completelt lost in calculus until i tried to understand a simple NN backpropagation. I went through the derivations step by step and i got that 'click' moment.