One math teacher that I follow is Christopher Danielson. Christopher makes regular posts on his blog about talking with his own kids at home about math. I've found these posts to be very interesting, and have extended some of those same conversations with my kids. I have three boys, ages seven, five, and two. It is very interesting to talk through mathematical thinking with them and get a sense of how they see the world through numbers.
Here's a conversation my wife had with Aaron, the 7-year old, this morning while processing some summer practice subtraction he was working on. He had been taught "borrowing" and "carrying" this year in school, and we wanted to see if 1)did he retain anything, and 2)what sense, if any, did he make from those procedural skills.
Mom: Okay, so let's try out this one here: 32 minus 19. How would you start this problem?
Aaron: [after some pause for thinking, he starts counting on fingers] two, one, zero, negative one, negative two, negative three, negative four, negative five, negative six, negative seven.
He writes -7 under the problem.
Mom: Okay, so negative seven is your final answer?
Aaron: I don't know
Mom: What do you do with the three and the one?
Aaron: Three minus one is 2.
Mom: Oh, is it just two? What do the 3 and the 1 represent?
Aaron: Oh, it's twenty
He writes 20 under the problem next to the -7.
Mom: Nice, so now what?
Aaron: [pauses to think it through] That makes it 13.
It was interesting to see his problem solving process here. She went on to look at the "borrowing" method he learned last year, but really focused on the relationship between the tens and ones places and what it actually means to borrow (we busted out some Cheerios to help visualize it)
But it was very cool to see him work through this process. Did he start subtracting 9 from 2 on the right hand side because that's where he's been told to start, or did he know he was comparing units? When he had 20 and -7 on his paper, did it make sense for him to combine those values the way he did. If the 7 had been a positive value, would he have added them?
When we teach kids procedural skills based on memorizing steps without attached meaning and understanding of numerical relationships, there's definitely a disconnect there. It was interesting to see his wheels turning after 2 months of summer vacation.