UPDATE: Wow, thank you everyone for the great insights!
A couple of months ago I stumbled across Paul Lockhart's essay A Mathematician's Lament and it made perfect sense to me. I'm not meaning to argue this essay one way or the other, except to say that 12 years of what I did in math class really isn't mathematics as you — and I, as an "enthusiastic amateur" — enjoy it.
We're probably going to homeschool our daughter, who will be kindergarten age next fall. I feel there's a place for knowing your times tables and the like, but there's also a place for knowing that mathematics is more than arithmetic and formulas: it's discovery, playing with ideas, etc.
Where I'm going is that I know enough to understand the difference, but I'm not quite confident enough to teach (or lead) this process effectively, since I'm not a professional mathematician.
Are there any curricula that would help provide some structure to facilitate this kind of learning, so that there is one less student who has a shortchanged opinion of the mathematics profession?
Or, alternatively, if you felt that your elementary school math education hit the mark, how was it done?
Thanks for your time!
Best Answer
As far as a full curriculum goes, I don't believe there is one that does exactly what you want. Books (in the United States, at least) divide into two camps:
"Constructivist" (e.g. Everyday Math, Connected Math)
"Traditional" (e.g. Saxon, Singapore)
Now, any search you make that even has a whiff of these terms will summon up loud and angry missives (try this article from the New York Times for an idea).
Constructivist curriculum is an attempt to catch the "joy of mathematics" approach to learning; for example rather than a worksheet with addition problems there might be a question about all the different possible sets of numbers that add up to 20.
The downside (as pointed out by the article above) is that (especially when taught by teachers who aren't themselves strong in mathematics) it can lead to basic skills being missed.
This is a problem Lockheart's Lament acknowledges. He seems to think students won't miss anything important. This can be true if the person steering the education is a mathematician, but with a non-specialist (i.e. most elementary school educators and homeschoolers) things can go horribly wrong.
Now, it's possible to balance to pull off a fantastic curriculum, but the ones I know about (say, at the Russian School of Mathematics in Boston) are, as self-described by the teachers, not following a curriculum at all. That's great if the teachers are experts, but put homeschoolers in a quandry.
I think the world is still waiting for an inquiry-type elementary curriculum that can be followed by non-experts and doesn't shortchange basic skills. So for now I'd suggest:
a.) Pick a traditional curriculum (Singapore is fine, although do shop around).
b.) Supplement. This very question is filling with lots of suggestions.