So I bought the California Common Core books for 3rd grade to understand the requirements. In our 10 minute car rides to and from camp every day, I was able to teach her all of it, without books or writing anything down.
The way they teach this in schools is just so far from optimial it's heartbreaking. To give one concrete example, a huge part of 3rd grade here is memorizing the times and divide tables. After some quick experiments, it was obvious that all she'd need to do was learn to count each number series, which is MUCH easier than memorizing flashcards, and can be done anywhere. Once you can count the number series, times and divide problems fall out naturally with practically no memorization needed.
The second big thing this year is word problems for those math facts. These word problems are so easy if you give the child the vocabulary and concepts to break them down in into their key parts. Instead, schools just leave it up to the courage and interest of the student to keep hitting their heads against the wall.
We're on to 4th grade math now and she's a week into 4th grade. I'm keen to check the Alan Kay link. :)