How can a curious high-schooler to go about doing mathematical research? I hope you are having a wonderful day. My main question is how/what are the best ways for a high school student to go about working on, or at least attempting to do mathematical research. 
I am currently a senior in high school, and have a working understanding of integral and differential calculus (Im in AP calc BC right now if that helps) and all preceding subjects. I have already been dabbling in self-study of vector/multivariate calc, basic differential equations, and basic linear algebra, but I would not yet trust myself to work with these concepts yet. I can look at work and understand it, but am not sufficiently skilled yet to tackle these problems myself. I could study these more thoroughly if it is essential, and I am honestly very interested in all of these subjects.. but I digress
I am really just asking all of you (much more advanced mathematicians than I will probably ever be) what your experience researching math tells you would be the most effective way for me to start. My career plan is to major in physics, get a PhD, and work in some form of theoretical or research physics. 
Thank you for any time or thoughts you can share with me. It really means a lot, and I apologize for any inconvenience. 
 A: Good for you!! One way to begin is to look around you for questions that you find interesting. This will give you practice 


*

*modeling situations mathematically, 

*looking for and playing around with new problems you like, 

*trying to solve problems where you don't know exactly what tools you're "supposed" to be using, but instead you have a bunch of things you've learned that you can try applying, 

*expressing your thoughts and conclusions carefully, and looking to prove that they're correct.


An example problem might be: you notice that water bottles fill up faster at the top where they round off, which seems interesting, so you think about it and decide to model it in a simpler way that you can probably solve: you have a 2D triangle (narrower at the top) filling up with water at a constant rate---how does the height of the water level change as it's filled? Does it fill faster toward the top, and if so by how much? What about other triangles, other shapes, or if the triangle is tilted over (seems harder)?
Or you might see a documentary about people walking across hot coals unharmed, and ask how you might model the problem physically to see how they do it.
Or you might learn that osmium is the densest naturally-occurring element, and try to get better intuition for density, so you decide to imagine a 1cm cube of osmium and a 1cm cube of some other metal like aluminum: if you balanced them on a seesaw, how far apart would you have to put the aluminum cube from the osmium cube? how long would the seesaw have to be? what happens if you increase the size of the cubes? 

This is my way of thinking about getting into mathematics, with a selection of more physics-based problems. For you, you may have other problems you think are interesting: they may be more abstract, more like games, more practical, more based on engineering, and so on---playing around will help you discover what kinds of problems you like.
