Griffin and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.
In Minnesota, this means we get to study integers.
Me: Griff, guess the temperature.
Griffin (eight years old): Two below zero.
Me: It’s three degrees above.
G: So I was off.
Me: Not by much, though. How much were you off by?
G: [muttering to himself, then loudly] Five degrees!
Me: How did you know that?
G: It’s two degrees up to zero, then three more.
Let’s pause for a moment here. You know how I just won’t shut up about CGI (Cognitively Guided Instruction)? It’s because they’re right. Children know mathematics before it is formally taught.
Consider the grade 6 (for 11-year olds) Common Core Standard 6.NS.C.5
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
Griff pretty much has this nailed down and is making progress on grade 7. But no one has formally taught him how to subtract integers. He reasons his way through a problem by making sense of the relationships in the context. He can find 3-(-2) without knowing keep-change-change.
But it’s not just Griffin. CGI demonstrated that children—all children—develop mathematical models of their worlds that precede instruction, and that instruction sensitive to these mathematical models is better than instruction that ignores them.
Back to our conversation.
Me: So what if it 10 degrees out, and you guessed 3?
G: [quickly] I’d be seven off.
Me: Right. How do you know that?
G: Ten minus three is seven.
Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?
So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.
So three minus negative 2 is five.
G: [silent]
By this time we were nearing the bus stop. I had offered this tidbit as an intellectual nugget to chew on, rather than a lesson I expected him to absorb. But that is what it means to have instruction be sensitive to children’s mathematical models.
Overthinking my teaching 
While we don’t have as many negative temperatures to talk about here in Vancouver, I find a link between integers and playing ball. See http://davidwees.com/content/number-line-activity for a description of how that worked.
I have noticed my own son solve problems like this all over the place without my help. He has the advantage of strong number sense, which is certainly not an accident. It has been nurtured by myself and my wife as much as we can by including him in activities that involve numbers on a daily basis.
When you say something like “So three minutes negative two is five,” do you also write it down on paper? Or can G follow that in his head?
Although you site compelling evidence for making sense of integers, I still don’t think the pedagogical opportunities provided is a good enough reason to move from temperate San Francisco to Minnesota.
🙂
Yeah, Bree. Guess the temperature would be a pretty crappy game in San Francisco, wouldn’t it?
Me: Griff, guess the temperature.
G: 58.
Me: Nope. 59.
G: I was off.
Me: By how much?
Repeat exactly like this all year long.
Karim, no I do not. I am not opposed to it, but usually—as in this example—we’re just walking to the bus stop or sitting on the couch. Also, I have absolutely no evidence that he followed that line of reasoning. I was just planting a seed. I’ll report back on whether it germinates.
@Breeden – Advantage of living in MN is that it will help Griffey develop understanding of functions earlier. For instance, “what is the relationship between temperature and thoughts of madness?”
@Triangle – The old Mark 4:5 approach, eh?
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Bless you, bless you, bless you. Can we please teach all parents to do this with their children? I teach 8th grade remedial math and have very few students who can accurately do integer addition and subtraction 90% of the time. The majority of those who *can* do it accurately do it using the stupid mnemonics they’ve been taught. I hate hearing “keep-change-change” simply because there’s no understanding behind it. This becomes clear when they try to KCC on multiplication or division…
I’m working on it. Do your part and spread the word about my Talking Math with Your Kids posts, OK? Share the love and I’ll keep it coming. Lots more fun to be had in the coming months.
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