I posted my K-12 school experience in one of my first posts.
I tested out of most of the lower division classes and my first taste of college classes was Engineering Thermodynamics with an older German professor who used to teach at Caltech. It was a very non-trivial class. He got my number and I learned how to study for the first time.
I also took a lot of classics classes, ie Greek and Latin, and math classes. I liked upper level math better than upper division engineering due to the way it was taught - and with professors' urging switched my major to math.
The math program was very rigorous, small, collegial, and Socratic with students and professors having to daily defend solutions in front of the class. This approach put a premium on solving problems quickly and then being able to explain the solution or being able to follow a train of thought and then point out the errors immediately. It was a lot of fun and required an immense amount of preparation. It was like getting a Law degree in Math. I formed very close relationships with everyone in the program.
Later when I took upper division engineering and physics, I was clearly superior to the students ( and many professors) in those classes. I and other math majors were able to solve problems the professors could not. I ended up with a math degree heavy in engineering and classics that included 3 graduate level courses in math.
I took the GRE and got accepted at a number of places, but decided to go into industry as the job offers were very lucrative for those who could teach themselves and who could work hard. I later went back to MBA school which was trivial academically.
As far as schools go, it comes down to the programs and the professors. Based on my math experience, I would prefer smaller, niche, intense, Socratic programs at large state schools where there are 8-10 students max per upper division class. The large state schools will attract top professors and the students in the smaller programs will be self-selected for motivation and high intrinsic ability. Or smaller engineering programs where you can get a lot more of the professors' time.
The key is to end up obsessively working on something so you get very, very good at it within an environment that will support you academically and emotionally.
I second what Beckee said about grad programs. I took a few more graduate math courses at a different school a few years ago. I felt very different at that school as it was much more formal. I did very well with the stats/numeric methods profs but the algebra/topology professors were a different story.
I think intelligence is not fixed. I think that beyond a certain point, it comes down to hard work. In my math program we had a handful of bright people from Eastern Europe and China. They were clearly talented mathematically and had been singled out back home for extra preparation. They were clearly ahead when we started the Real Analysis/Abtract Algebra classes, but after the first few months, everyone caught up. The pace of work was so hard and we all worked so hard and lived and breathed math that they were soon in the middle of the pack. In later classes where you had to do original thinking and grunt stuff out, they were no better than anyone else.
After two years in the math program I went back to take some engineering and physics electives. Quantum physics was easy. Thermo was easy. It was all easy where it had been moderately difficult before.
I've also posted on here about my HS classmates who were not National Merit nor tested as I high as I did, yet they are all MDs leading departments, research PHDs with dozens of papers, etc.
The Polgars' life story really begs your question.
Polgar No substitute for hard work!!!
