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!