> you don't learn math by pouring over the text and reading page after page. You read something and try to understand it. You play with it, you stand up, take a walk and try to work out why something is the way it is, think of examples, try to clear up what you don't quite get.
Absolutely this. Math is a not a spectator sport, you can't acquire the skill and knowledge by reading. You must engage with it, fence with it, wrestle with it, make it your friend and spend time with it.
I remember not so long ago playing with some equations and accidentally proving Pythagoras' Theorem. It wasn't new, it wasn't deep, but it was satisfying, in much the same way as getting code to perform as you want, accomplishing some task in an elegant way.
But if you don't spend the time, you'll never get the insight(s).
Yes, I think some people falsely learn early on that stumbling upon something they don't quite understand is a sign that they are failing.
I sometimes even think (though I'm not so sure) that having too good textbooks can be an impediment for above-average students as they have less to wrestle with and things are laid out and pre-disgested too much. I may be wrong though, because you can then just get to the next levels until you hit the wall.
But I feel like anecdotally I got good value from deriving some basic things and reorganizing the somewhat badly presented materials in some of my classes. Later on I found very good American textbooks (although widely out of the reach of my wallet at the time), from MIT etc. which were crystal clear in some of the intuition that I had to sweat out for myself. Maybe I could have spent that energy at higher levels, though.
In my country (Hungary, but I saw similar attitude in Germany as well) its not unusual to learn entirely from the lecture material (own notes or provided script and assignments) without any textbook. Some people do get a book from the library, but it's not really expected from you.
my original comment before editing said book/teacher. I use them interchangeably, but I hope the message is still there. Namely: Is it a difficult subject or do you need a better teacher?
More people should play with math like they play with new programming languages or tools. I've learned a lot by just trying to describe in geometric terms the relationships among shapes I see around me or use in my design work.
I learned more about trig working out what angle I needed to input in some design software to make a line cross two exact points than I ever did in high school.
I'm very much a visual learner so I find geometry in particular very satisfying since I work out relationships visually (and then mathematically to prove it).
I find that I learn immensely more in even tiny little tasks where I care about finding the result, compared to any assignment or exercise. Suddenly you look at the resources in a totally different light, critically evaluating what they are saying. Like, I'd pull some lecture slides or papers from the internet and read them in a very different way, compared to actual lectures. The roles are changed, I suddenly see the math (or whatever the topic is), and evaluate what it can do for me in the particular case. Not like what I need to do for the math to conquer it and learn it to the satisfaction of a teacher.
This also applies to computers. Just get your hands dirty, try to build something and you'll find yourself learning about so many things and tools incidentally while trying to fix something.
I can learn much better this way than going chapter by chapter in a book.
But interestingly, even graded, project-style assignments don't accomplish the same effect, only if I'm really onboard with the topic. I seem to need a spark of excitement of "this is not exactly the way you're supposed to be doing things" and just breaking things and playing.
Couldn’t have said it any better. Whenever someone is asking me if I have been introduced to a tool needed to do a job, I tell them “no, but only because I never had a reason to learn it. I’ll pick it up.”
I can pick something up very quickly when I have an actual reason to know it, and getting a good grade in a course is not a strong enough reason compared to “Learning this will help me save the world”. I’m not going to be passionate about it or inquisitive or asking myself the extra questions I normally ask myself when I’m learning something in order to apply it to a problem. And real problems are better than made up problems.
Absolutely this. Math is a not a spectator sport, you can't acquire the skill and knowledge by reading. You must engage with it, fence with it, wrestle with it, make it your friend and spend time with it.
I remember not so long ago playing with some equations and accidentally proving Pythagoras' Theorem. It wasn't new, it wasn't deep, but it was satisfying, in much the same way as getting code to perform as you want, accomplishing some task in an elegant way.
But if you don't spend the time, you'll never get the insight(s).