I don't think you can "support" public education while sending your child to private school. The hypocrisy is clear. Sure, as a politician you can vote other people's money for public schools, but by sending your own kids somewhere else you're sending a message about "good enough for thee, but not for me".
Of course the NEA supports no one but Democrats for national office, and all of them send their kids to private schools. "Do as I say, not as I do." Obama is yet another in a long line.
Here's a defense of Obama's sending his kids to private school. In my opinion it's a weak defense, but I post it here anyway as others may disagree with me. (You may have to scroll down a few posts to get to the one under discussion.)