A few months back, Tabitha (5) wanted me to ask her some math class questions. This led to some disagreement between the two of us on what constituted a math class question.
On our way back from our fall camping trip last weekend, Tabitha announced that she knows what 8+8 is.
Me: Ok. What is 8+8?
Tabitha: [longish thinking pause, eyes gazing up and to the right] 16!
Me: Wow. I could really see you thinking that through. How did you know that?
Tabitha: I counted.
Me: OK. Good. But how did you keep track of how much to count?
Tabitha: huh?
Me: How did you know when to stop counting?
Tabitha: When I got to 8. That was 16.
Me: Right. But how did you know when you had counted eight times?
Tabitha: I just counted to 8. That was 16.
Me: Oh! I know! Was it like this? Eight…nine (that’s one), ten (that’s two)…like that until you got to sixteen (that’s eight)?
Tabitha: Yes.
Me: How did you know to do that? Did you learn to do that in Kindergarten? Or was it your own idea?
Tabitha: [long pause] OK. I admit it; it was my idea.
I have written before about how sticking with this stuff pays dividends eventually. Tabitha announced that she knows what 8+8 is, but what she really meant was that she knows how to find 8+8. Tabitha knows that talking math is not about telling answers, but about actively thinking things through. That doesn’t come for free in classrooms, and it doesn’t come for free at home. It is a result of lots of prep work and lots of encouragement and lots of conversations.
Returning to the theme of the knowledge required for me to have this conversation with Tabitha, though, I needed to know something about possible strategies, and I needed to have thought some of those strategies through for myself-by trying them out.
If the sum in question had been 8+2, I wouldn’t have questioned her claim that “I counted”. You can subitize two, so you can know that you have counted twice without having a separate system for keeping track.
But counting 8 times without having a way to ensure that you have done so? This seems pretty much impossible. As an adult, I know to stop at 16, but that’s only because I already know the sum. As I count 9, 10, 11, 12, I am conscious that the only way I know I have counted four times is because I know that 8+4 is 12. I’m not able to keep track of the number of counts past three.
I had watched her fingers; no motion there.
So there had to be something else going on. I was only able to press because I knew what she knew (how to count) and what she didn’t know (addition facts) and some possible ways that this knowledge could be used to find the sum of 8 and 8.
The more I talk with these kids, the more impressed I am with the work of CGI. Much of what I know in this area is the result of studying their work.
UPDATE: The original CGI link in that last paragraph went to a tag search of CGI on my blog. That was confusing. It has been replaced by a particular post. Here’s the original link. And here’s a link to CGI on Wikipedia.
Overthinking my teaching 
(CGI link leads right back here.)
Sue, it actually only appears to. The link is to CGI-tagged posts on OMT. Of which the present one is latest and so appears at the top. I can see I did not think this through.
I’ll be changing it so people can actually learn something new if they click on it. Thanks for the critique.
OK. (Didn’t know it was a critique. I think CGI must stand for cognitively guided instruction, but I know/remember nothing more.)
You are correct, Sue, on meaning of CGI. Critiques are good.
You must insist that your college library purchase this book: Children’s Mathematics. I checked; they don’t have it. It’s an easy read and super important for anyone working with young children, or with those responsible for the mathematical development of young children.
You got me chuckling. I have duly passed along your recommendation. (I told the librarian it’s from a blogger I admire.)
I can remember my son having the same observation, and I kept note of what strategies he was using. At first he would just count, and he would move his fingers while counting. Next he counted without moving his fingers but sub-vocalizing the numbers. Then he counted and had a little nod of his head as he counted, and now he counts without noticeable change that I have spotted. He actually has started using the count up to an even number strategy, ie “I know it takes 2 to get to 10 from 8, and then I have 6 more to use and 10 and 6 is 16.”
Given that I have never taught him these strategies directly, nor has his teacher, I guess he must have figured them out on his own. It’s an interesting gradient between counting with one’s fingers and doing addition in one’s head.