Hi all, glad to join you here. (This discussion already has so much meat in it, it might take me a while to digest it all!)
I teach math at a community college in California. I've also worked with homeschoolers, and run some math circles, and math "salons" at my home. I put together a book showcasing some of the amazing work being done by math enthusiasts with kids of all ages, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. I have a blog at Math Mama Writes, but I haven't been writing as often there lately. (I'll definitely link to this fascinating conversation!) Basically, I love playing with math, I want to share that love, and I look for ways to bring that playful spirit into my college classrooms.
I'm not sure what the term "higher math" means. To me, all of math can be approached playfully, and I'm not sure what makes something 'higher'. If we simply mean the topics that aren't arithmetic, there is so very much!
I hope I'm not being too much of a curmudgeon by questioning each word (I think that's a math skill, perhaps...), but I also know that 'teaching' can be problematic. I want to offer delights to young children, and then follow their lead. Does this game intrigue them? No? How about this puzzle? Or maybe this art project? So much math can be achieved this way.
My department asked us to wear something math-related on Thursday. I was grumbling to myself for a bit, and then I found a t-shirt I love in the back of my t-shirt drawer. It has the dragon curve on the front, and on the back are images that show the stages you'd get when folding a strip of paper (this site might be a good way to see how it works). So I pulled it on and went to work, where I chopped up some paper, and started showing people the coolness of the pattern you get when you keep folding and opening it back up the same way. (We looked at the pattern of "valley folds" and "mountain folds".) It only took a few minutes of classtime, and maybe, just maybe, one or two students will be intrigued enough to keep playing.