Gap year to study math

You're self-motivated; the kids you mention as a general rule pushed by parents trying to compensate for what they feel is a lack in their lives. Some will excel; some will flame out as a lot that drives them is 'being a wonder kid,' and being admired for it. At a good college, things equalize fast; and this can turn into an emotional barrier for them. I recently was at a coding bootcamp with an extreme such case: conversing at 10 with MIT faculty; lost at 22.

If you study hard, and are truly driven, things equalize fast. Take what's new and hard for you, and some in your comfort zone...and don't listen to your class-mates telling you homework was easy for them. Except for the occasional true math genius where it's no lie - and this student is unlikely to harp on it -, a strangely arrogant attitude is common among students in certain technical fields; so expect it, and shrug. I like to share advice given to me by my first hw buddy: if someone has a more elegant solution than you, it was copied from a nicer book than you had access to. Or from the senior mathematician at my undergrad alma mater: "it is easily seen means that you see it after a sleepless night agonizing over it." Math isn't easy for anyone.

In my grad studies, some of my classmates already had Ph.D.'s in physics prior to starting this new Ph.D. (and I have great respect for physics), and I was worried about being inadequate. As long as you are willing to work really hard, doing hw in the structured environment of a school will help you much more. Try to find a friend who to team up with, and to have friendly competitions with how to solve things more elegantly, and who to meet when you're stuck (and you will be). Working on your own is very hard (it's what I'm doing right now), and nowhere near as productive.

I started my math degree without having ever taken past Algebra 2 in high school. I studied and took it seriously, the kids who thought they already knew everything didn't. Your success will be determined by your energy, commitment, and readiness to learn. Head starts don't matter. I think you'll be just fine.

Some things to keep in mind:

• No matter what, don't lose your self-driven attitude. This is is probably your greatest strength, and is rare to come by. Always engage in some form of self-study.
• Will your offer of acceptance to your top-5 institution still be there next year if you don't go this fall? If not, are you willing to risk losing that acceptance? (This is just a question to consider.)
• Self-study is certainly the best way to learn math (IMHO). You will learn an incredible amount through the "struggle" (not the best word, but I think it approximates my meaning) to understand a topic.
• I am taking a non-traditional route to college education, by spending an extra year at a community college before continuing to a four-year school. There are plenty of people who take off a year between high school and college--so long as you are purposeful with your time, it won't cause many problems.
• One of my classmates who stands out as a "great math person," (who is actually majoring in math, whereas a lot of my other friends are engineering students) took precalculus for the first time (I believe) at the college level. You certainly don't have to finish Spivak before entering college to do well in mathematics.