My 6th- and 7th-grade students are pretty effective at calculating with negative numbers. They all know, for example, that 5 – (-2) = 7. Ask them why, and you’ll hear this:
“Because two negatives make a positive!”
Then, if you listen carefully, you will hear something else: the low rumble of my teeth grinding together with tectonic force.
“Two negatives make a positive” is one of those math slogans that drives me crazy, because it is so pithy, so memorable, so easy to apply… while also being so vague and non-mathematical that I’m amazed students find it useful at all.
We can all think of many, many cases where two negatives don’t make a positive. Rain on your wedding day plus grand larceny on your wedding day does not make for a winning combination, despite what “two negatives make a positive” would suggest.
It’s not even true with negative numbers, where -10 + -30 does NOT equal +40 (although I have seen students claim that it does, citing “two negatives make a positive” as their justification).
In fact, that’s one of my major complaints with “two negatives make a positive”: it is such a swift, over-arching generalization that students wind up applying it in places where it doesn’t make much sense.
In fact, “two negatives make a positive” doesn’t really make much sense anywhere.
What does make sense is a slight variant, less catchy but far more true: “The opposite of the opposite is just the thing itself.”
What’s the opposite of “the opposite of happy”?
Well, “the opposite of happy” is sad.
So the opposite of that is “happy” again.
For adding and subtracting with negatives, I tend to favor a debt model.
For multiplying and dividing with negatives, I think a slightly more abstract approach is necessary – it’s all about the properties of multiplication.
Good mental models are more effective than mantras like “two negatives make a positive,” I believe. But even if they weren’t – even if the use of mantras led to error-free computation with negatives – I’d still favor the “mental model” approach. Learning new models engenders the kind of rich thinking that math class is supposed to be about; learning new mantras engenders the uncritical thinking of the cult-follower.