What I described is what I consider to be "doing mathematics". As distinct from applying mathematics to other problems (also a very worthwhile endeavor!) I'm c

What I described is what I consider to be "doing mathematics". As distinct from applying mathematics to other problems (also a very worthwhile endeavor!) I'm certainly not encouraging everyone to pursue a career as a math professor; the job market is poor, and is getting bleaker. But I feel like prospective math high school teachers should have an experience like the one I described. Much more important, in my opinion, than learning how to row reduce matrices, do integration by parts, or prove the intermediate value theorem.