Many other commenters have pointed out how great examples and ties to the real world are; the love of applications is well-represented in this discussion. But I

Many other commenters have pointed out how great examples and ties to the real world are; the love of applications is well-represented in this discussion. But I'll throw out beauty as an underrated motivator! I love math because it's beautiful, primarily, and this isn't represented often in elementary or high school education. The idea of "function" or the idea of "constructive proof of bijection" are beautiful concepts. The patterns of math are beautiful (so beautiful I created an adult coloring book called Math with Crayons that focuses on Aztec diamonds, fully packed loops, lozenge tilings of hexagons known as totally symmetric self-complementary plane partitions -- you know, the stuff people are actually doing research on today!). Numbers to me are kind of boring. But how many kids get to draw fractals in class and then talk about the concept of dimension, and then argue about how one could rigorously define dimension, with the example of trying to define dimension for fractals as an edge case? How many kids get to draw non-crossing partitions and Dyck (lattice) paths and then try to prove that they are both enumerated by the Catalan numbers? How many kids get to think about the mathematical structures you allude to, and play with the fundamentals of category theory? Well, very few. And why? Their teachers don't know any of that stuff. If teachers only know computation, that's all they can teach. If we had teachers who really had breadth and depth in math teaching math, I think we could start teaching concepts, structures, and beauty very early. I've had conversations about the basic ideas of category theory (relations not objects!) with many kinds of people. This is not out of the grasp of a kid. Set theory -- not out of the grasp of kids. The very idea of what a number is: in high school I got to take a seminar with other high school students on this, looking at John Stuart Mills and Frege and Russell and Whitehead and all this thinking about what a number is. But this is extraordinarily unusual material for a high school or even college student. Of course it's difficult to deal with these notions later, then -- people have been trained out of their natural curiosity and into a view of math as computation at worst and manipulation to find x at best, rather than seeing it as a way of dealing with beautiful structures.