Open a map, close your eyes and point randomly.
Pack you backpack, buy tickets and go there without any reservation. Spend at least 1 month away from home, comfort and routine.
When you are back you will know exactly what to do.
When you are back you will know exactly what to do.
This sounds like an interesting and exciting life-changing plan and everything, but I don't think it's particularly actionable advice for most sixteen-year-olds in today's world, and I don't think it's likely to help with the question at hand. The submitter isn't asking /what to do/; the submitter is asking /how can I be more motivated in what I'm doing/!
I think better advice is to focus on how the things you're learning relate to the things you like to do. So try to focus on how math can improve your programming, research, and other interests. Also, keep in mind that you're building a foundation--things are more interesting once you get to the stage where you're building on top of it. Even if you decide not to use it in a few years, it's a really nice thing to have that will give you a lot of flexibility.
I see your point that it's not a very actionable bit of real advice. Pragmatically, if you told my 16 year old to pack a bag and leave, I'd want to smack you :D
1) The plan takes more than a trivial amount of money. Most sixteen-year-olds cannot afford this sort of plan.
2) Many parents, if not most parents, are unlikely to support this idea. There's of course a spectrum of what parents might do to stop it if they're not supportive, but it's easier to just save up money and wait until you're done with school.
3) It can only be done in the summer without causing a lot of problems, and perhaps not even then.
It's a potentially helpful suggestion for someone who's over eighteen, but for many sixteen-year-olds it's probably a pipe dream or something that may bring more problems than benefits. The suggestion assumes a certain type of parents and a certain level of privilege that I would estimate are the exception and not the norm. I would have been extremely frustrated if someone gave me advice that presented this as a serious option when I was sixteen.
Advice that makes these sort of assumptions may end up decreasing motivation. I think more universal advice that addresses the actual question is a lot more helpful.
I can agree that distance creates perspective. All I am saying is that if my 16 year old daughter told me she was going to set out for some random spot for a month, I'd probably nix the idea and not feel like I was being unreasonable.