Consider what we teach in high-school geometry:
- There’s a lot of content from various two-dimensional topics, such as congruence, similarity, angles, polygons, and area. Much of this rehashes what was already done in middle school, though (we hope) in greater depth.
- In some schools there’s some right-triangle trigonometry.
- Usually there’s a lot of time devoted to proof. In honors classes this becomes the principal focus of the course. Some students love proof; it gives them a real sense of power. Many students hate it and can’t make sense of it.
- There’s a bit of content from the third dimension, such as volume. It’s a pity that there’s so little, since we live (apparently) in three dimensions, as Tom Banchoff has eloquently pointed out.
- And what about non-Euclidean geometry? What is the real shape of space, anyway? For some reason this topic is treated as appropriate for honors classes but too abstract for everyone else.
There has to be some way to sell proof to more students. It doesn’t have to be done in geometry class — it could be tackled somewhere else — but experience has shown that geometry is where it’s most effectively learned, probably because one can draw pictures and because there’s a long tradition of postulates and theorems. Proof is important, since it teaches students to justify an argument and not merely take their teacher’s word for it. But that’s not the subject of this post. Here I want to consider the shape of space. Is it Euclidean? If not, why do we pretend that it is?
And…why is this topic considered so abstract? Probably it’s because seeing isn’t believing. The principal advantage of geometry — that one can draw and examine figures — seems to go away. And yet MacArthur-award winner Jeff Weeks has produced exciting and concrete mathematics materials that can be explored by kids as young as ten. Shouldn’t we be including this sort of content in regular high-school geometry? Let’s spend less time on rehashing middle-school material and more time challenging our students to think about the universe in new ways!