Hi All! My name is Dan Finkel. I'm the founder of Math for Love. It's a pleasure to be on this panel with all you fine people!
While I always enjoyed math, my moment of real math conversion came when I attended the Hampshire College Summer Studies in Mathematics program when I was fifteen. That was my first taste of "higher math," and the beginning of the passion for mathematics that still guides my work. Since then I went on to teach math to high, middle and elementary school students, get a PhD in algebraic geometry, found Math for Love, and design games, puzzles and lessons that I hope share the same excitement that I found in those first moments of genuine mathematical thinking.
There are, for me two reasons to share the ideas from higher math with students at a young age. The first is that mathematics is much, much bigger than arithmetic. It's bigger than all the math that students see in school. Consider that calculus, the endpoint of the tradition high school math sequence, is about 350 years old. Ignoring more modern results would be like never teaching anything in English more current that Shakespeare. (Are his plays great? Sure. But a LOT has happened since then!)
But there's another reason too. The central point for me is not necessarily the content you share (though there's an argument for sharing beautiful ideas with kids at any age, as long as they are ready for it); it's about sharing the process of mathematical thinking. Accelerating kids into higher grade material isn't worth doing if the process of genuine mathematical thinking isn't part of the deal. On the other hand, when you ask students to consider interesting patterns, ask questions, make conjectures, and try to break them with counterexamples—another "advanced" idea that very young kids can play with—the students may take you into territory you weren't expecting.
Many years ago I was working with a fantastically gifted second grader. One of his great gifts was the ability to ask natural, fascinating questions. He once came in and asked whether you can always cut a collection of identical squares into a single square. For any n, that is, can you dissect n identical squares into one single square? For four squares you don't need to cut them: just arrange them in an array. For two squares it's relatively simple. Five is trickier, but still doable. Three seemed impossible.
I didn't know anything about this problem at the time. It came from the student. But I ended up finding it discussed in an old Martin Gardner book, with the three square case solved, gorgeously, by a ninth century Persian astronomer. The problem, in fact, was part of the inspiration for one of Hilbert's 23 problems (the third). Did we go looking for "higher math?" No. But by letting the act of asking questions about novel ideas and situations play a part in our process, we played the game of mathematics that gives rise to higher math.
There are so many beautiful areas of mathematics, from games, to knots, to topology, to modular arithmetic, that are accessible to young kids. But the point is not just to convey the knowledge. It's to participate in the thinking.
Even arithmetic comes alive when you attack it this way. A student I knew once asked his mom if 3^(-2) had any meaning. When she started to answer, he stopped her, saying, "Don't tell me. I want to figure it out."