This semester I am teaching a course for freshman mathematics majors. The course is called Introduction to Formal Mathematics. One purpose of the course is to develop a tight-knit cohort of mathematics majors and another purpose is show them that mathematics is about more than “solve for $x$.” We do some problem solving, a little proof writing, and introduce them to a few topics they may or may not see in future courses. The course is a lot of fun. Feel free to check out the course webpage.
The last couple weeks we have been doing a little combinatorics and some graph theory. Today one of the students remarked something to the effect, “I liked math before, but this stuff is just so cool.” I agree. There is no reason why we couldn’t teach these topics to high school students or even middle school students. Unfortunately, we are too obsessed with trying to make sure students pass standardized tests and can take calculus in high school.