Let me comment on the opinions that "many high school students do research". Again those statements referred to lab science, and I am talking about math. Still there may be comparisons.
There are certainly some high school students today who also claim to have done "research" in math, but this claim is highly susceptible to interpretation, at least if you understand the situation as I think i do. I know of two examples.
1) for years there have been high school math competitions in which students entered by proposing certain projects they supposedly carried out, in math research.
I have been advisor for such projects. Even some of my students who got admitted to Ivy league schools, did essentially nothing on their own project. I.e. I came up with all the ideas, and all the questions, and the student just wrote down what I had said and added nothing and did not even try to add anything.
Some more dedicated advisees at least worked out examples of what I suggested, in addition to writing up the basic theory, but usually it was impressive enough to the judges that they were even familiar with the terminology i had taught them from the college curriculum.
I recall one exception of a student who actually learned some abstract algebra on his own, and then generalized a small result about modular arithmetic. The result he proved was well known, but he did not know that and ddiscovered and proved it on his own. This impressed me. He is now a junior level mathematics professor at Columbia I believe.
Even his high school work was not "research" in the sense of new information, but it is research in the sense of self discovery and verification. I.e. it did show his potential.
Oh yes, I remember several more students who did their own excellent work as well. (As I learned the game, I stayed out of it more, and also some of the students were just really gifted, and their high school teacher was outstanding as an advisor as well, much better than me.)
I have also seen hugely impressive lab
science projects at math/science fairs which were obviously carried out with the benefit of a large engineering lab at a university, and hardly likely to have been the work of the student at all, except possibly as an assistant to his father or uncle. Even if not, it would have been impossible for a student without access to a major lab. Quite unfairly, these projects won major gold ribbons.
2) There is a current funding initiative from congress and NSF called "VIGRE", or vertical integration of research and education. This mandates that universities involve undergraduate students in research early in their careers. The fact that there is a lot of money available for this means it is happening and many people can claim participation.
It does not mean that what is happening there is actually significant research. Faculty are pressed to use this money as congress intended, and also challenged to respond to the need to involve more American students in math and science (the money is restricted to US citizens), but we are very challenged to come up with "research" experiences that undergraduates can actually participate in.
So there are students claiming participation in VIGRE research experiences who will have not done any research that I, or maybe anyone, would consider real research.
So I stand by my original statements on this topic, in spite of rumors people may have heard that "research" is common among high school students and undergrads, and a necessary ingredient of an application. If the reader of the application knows as much as I do, and all university professors know this, those claims are highly suspect. Or at the very least the meaning of the word "research" has to be understood carefully.
I hope this helps clear the mystery up some. Of course as always, I could be wrong, even though this statement is based on long experience. Also be aware that college admissions officials normally do NOT know as much as faculty about these things, and they are sometimes impressed by things that do not impress scientists.
For example, although sad to me, it may be true that the high school student who actually did his own research may have been benefited more on his college application by the less meaningful fact that he took calculus from a local college while in high school. I.e. almost anyone trying to enhance his resume can do that, but not anyone can discover and prove a theorem, even a small previously known one.
Indeed the students who did the least on their projects with me got into the most famous schools, since that was their goal, and they did what it "took" to get into those schools throughout their careers. Nonetheless the more talented students probably did get in where they wanted to. No actually the best of them was turned down at one top place, or at least got less scholarship than needed to go there.
So there is a tension between the truth about a candidate, and the BS on a vita that may help get him in a certain school. Still I advocate being as straightforward as possible, without being self destructive, and not becoming cynical. Sometimes you will not be treated fairly, but eventually you will be likely to come out where you want.
I.e. take calculus at the local college if that will impress the admissions official, but also do the real work to learn as much as you can, to impress a professor who may interview you.
good luck.
