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.
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.