You mean sitting in a classroom and listening/talking to people? Because that is all it takes.
Edit: It is really irritating to me how so many people thinks it is impossible to grasp math by just sitting there. Look, there are so extremely few concepts to learn and you have many hundreds of hours of class time to learn it, how can you not learn during that time? Nobody told me to do work during class, so I didn't. Nobody told me to do homework, so I didn't. I just sat there, when they gave me a test I did it, the tests was the first time I ever tried to solve a problem and I solved them because I had seen how to solve similar problems during class. This strategy worked up to a masters level in physics/maths, so not just high school.
For high school, you're basically right, there isn't that much content. But it's not like you're in a math class particularly long either. Few hours a week for a few dozen weeks a year, for a few years, and a fair bit of each class is getting everyone seated, telling people the homework, and reviewing the previous lesson. If you look at a math/science high school curriculum, you tend to just follow some book that's a few hundred pages, and it takes the whole year to do it. An engaged kid could read it all over a few evenings, per class.
> It is really irritating to me how so many people thinks it is impossible to grasp math by just sitting there
It's very easy to think you really understand something when someone's explaining it to you. Kinda like watching a show on Discovery. Then when it comes to solving problems, there's a fair chance that you're exposed. I found this a lot at university, there'd only be 10 questions but they were good ones that necessitated looking at a variety of sources and getting a real understanding. Often nobody I knew would be confident on all the answers, and this is kids that got into a world famous university.
I didn't think about it though, I didn't assume I understood anything. Just that one day there is a test, I went to the test and answered everything as well I could. I didn't get everything right, I did a lot of stupid mistakes, but I got enough things right.
I think that if you need to practice to do well on the tests then you don't understand the subject well. There comes a point where you feel you have perfect clarity of a topic, I just reach that level very fast. I am not always correct when I feel that way, but it was good enough.
> Did people really explain things to you at university? Where I went you were expected to read a lot on your own, and then you had a few hours of tutorials where you could ask some questions.
Right, higher level courses makes you do more independent work. But the courses people take for physics, CS or engineering are still very well explained. The courses you take to prepare for math research didn't go as well for the reasons you explained, I still passed them but with worse grades and started doing math research, but after I solved my first research problem and wrote about it I realized I didn't really like math that much and learned programming and got into Google instead. I figured that if I have to work on something I don't care about I can just as well earn a lot of money doing it.
Just explaining things here to make my position more understandable. I always hated teachers who grades after how hard you work or how many boring practice problems you solve etc, so I have a strong distaste for any argument along those lines. "But you need to work hard to learn!", no you really don't. That is my point. Don't make me do boring stuff because you had to.
But most people do need to do something other than just show up to class to learn things. That's got to somehow be obvious, based on everyday observation of your peers? Certainly my main experience of university was that kids who were diligent understood things, and those who weren't didn't. Plenty of kids thought they could either quickly catch up, or get an overview at the lectures and just be smart and figure it out. It might have worked for one or two, but for the large mass of kids, they needed to sit and look at things themselves, and take some time.
Also I don't understand how you think courses were well explained. Where did you study? I never thought anything was explained much at all, lectures were mostly a summary of major ideas, with a lot of glossing over the details by professors who were long past the point where it mattered how well they understood the stuff.
My point is that to me they were well explained. To most others they are not, but that wasn't what we were discussing. A guy above said that nature is completely dwarfed by nature even today. I disagree with that. A bottom 10% kid with respect to nurture can still get great results if their nature just is good enough.
Most of my professors in college called me a genius. I got some research projects after my first year, but I didn't have any work ethics so I didn't do anything with it, I mostly just sat there and didn't do anything just like how I spend most of my time in class. I solved the original problem quickly, but all time after that was just sitting. I know I am not normal, I just want to say that people like me exists, and I get angry when others argue that people like me doesn't exist, that we just fake it or are privileged etc. It is just nonsense and it irritates me a lot.
I'm a total mental wreck, I could barely do anything, arguing that the reason I did so much better at math than median persons was because I worked hard, got tutoring, had a better home environment, had better focus etc, all of that is just nonsense. I'm a bit better today, there is no way I could even hold a job back then as just sitting and not doing shit doesn't fly at work, but I still lived through all that shit.
I'm not saying really smart people don't exist, obviously we've all met mentally disabled people who cannot be taught higher math so why wouldn't there be people who get it really easily?
But I do think most people by far are in the middle group. Teach them well, encourage them, and they do well. Fail to do so and they do badly.
But if you agree with such a model you must see that in aggregate, nurture wins. There's a bunch of people who wouldn't know any calculus or whatever if they hadn't been set to do certain problems while being educated.
If you look at something like private education in the UK, you see this very thing. Kids are coached and they do well, on average. They're confident and they're well drilled. Not every kid in those schools does well, and not every kid who does well went to one of those schools. Yet stats show that the 7% of kids who went to private schools get about 50% of the Oxbridge places.
It’s disappointing, but also amusing. When I was in high school Harrison Bergeron was assigned reading. And it’s a great loss too, but one that most people won’t even notice. Observe how much skepticism your personal testimony gets from those who cannot even imagine what it’s like to have extremely high intelligence.