This might be difficult for me to put into words, but bear with me because I think it's an important question.
Among the many people who study math, I am one of them. I'm not particularly advanced but I enjoy it. I'm working through calculus 2 in my free time. I do it for fun.
It occurred to me: I have a certain amount of ideas in my head: calculus, trig, geometry. These are all tools that required hundreds of years of development. I've developed my ability to reason as best as I can so far.
However, I find that I'm not capable of very much, other than I can work through a textbook, I can solve the problems, maybe even sometimes in clever ways that surprise me. But what can I do? If I am capable of more, I don't know it.
The question, more specifically, is this: if someone has studied up to, say, calc 2, what should they be capable of?
Mathematicians and physicists over the years have achieved much more with much less. So what I'm saying is, with what I know, I feel like I should be capable of more and I'm sure there must be students out there that feel the same.
I'd like to think that with knowledge comes more ability than just regurgitating their studies or passing contrived tests. Insight, intuition should lead them to discover new-found abilities.
Well I hope I've made my question clear and, if I have not, feel free to delete it. Hopefully if I've left any gaps, then the reader can read between the lines and understand what I'm getting at.
Okay think of it this way. Let's say someone has a powerful computer and they're using it as a paperweight. How could they use this resource to the fullest?
I'm expecting question of a format like, "well, if you have trig under your belt, I'd expect you to be able to give me a good estimate of the size of the Earth. You have the tools for it."
Thanks for reading, it's a bit long-winded.