I bought this book by Robert Osserman years ago at a used book store and finally started reading it on the plane to Cleveland last week. It is wonderful. For the first time, I feel I am starting to "get" spherical geometry. Not that I could do any calculations or anything, but Osserman explains it by starting with the basic problem of map-making: there's no way to make a two-dimensional map that is not distorted; the issue is how best to manage the distortion for the purposes of the particular map. He then translates this problem into mapping four-dimensional space using three dimensions. Apparently Dante's "map" of the earthly vs. heavenly spheres uses pretty much the same idea as the mathematical "hypersphere." If Osserman can even begin to get this across to a spatial illiterate like me, he must be onto something.
I wonder what would happen if math teachers taught this book or a similar one, along with the usual calculations and story problems. The students could read a chapter and they could discuss it for a half hour once a week or something. I understand there's no time in the high school classroom for this sort of thing, but it would be great. I never cared about figuring out when two trains would pass each other--but learning that I could use math to understand how the universe works would have hooked me a lot more.