So today as a starter I thought I’d do a recap on negative numbers. Students have starter books (see: lazy teacher) choc full of puzzles and basics revision. The one in question asked students to circle the correct answer for a set of about 20 negative number problems. Each one had four choices, and each choice was essentially calculated as so:
So above, the student should circle the bottom left box, etc.
I gave them around five minutes. During that time, I expected the occasional “aah I can’t remember how to do these…” to which I would reply “that’s why we’re doing them”. Sadly, I also frequently heard students desperately trying to recall the ‘rule’ they’d memorised (or rather, not memorised). “oh what is it again? A minus and a minus makes a plus. No wait, that’s for multiplying, a minus and a plus make a plus” “But what if it’s a minus minus a minus? That must be a minus then”.
This was going on all around the class. I’d love to say these were young students. They weren’t. These were high ability students at the twilight of their schooling. We had a brief discussion, and all sorts of random answers came flying in at me, with baffling explanations that weren’t really explanations at all.
So we started again. They were really annoyed at me for trying to wipe the slate clean. They had a method and they remembered the method and it WORKED.
I pointed out they didn’t have a method, they had a kind of rhyme, that didn’t work, nor had it been correctly remembered. They were still annoyed with me.
First I asked that they stopped using the word minus in any way shape or form. It was too confusing. If it’s a negative number, then say negative, not minus. Second, if it’s a subtraction, say subtract. That way at least we’re all talking about the same thing.
“But a minus and a minus”
I snapped a little. “Look, the reason you can’t do this has nothing to do with your maths ability. It’s because you’re trying to remember a phrase and that’s all you’re focusing on. Not one of you is even TRYING to think about it, you’re just trying to recall that phrase. So let’s please just stop, and think about what’s going on in the question.”
And it was absolutely true. They are such talented mathematicians, yet they became utterly fixated on a phrase, and lost all sight of trying to rationalise what they were doing or whether it made sense. Never has the detrimental effect of memorising a ‘trick’ been so crystal clear in front of me before.
So we drew a number line with just one negative on it, a zero, and one positive. We talked through which direction you go when subtracting a positive number, and therefore logically which direction would you go if you were subtracting a negative number?, then again for addition.
Sadly I’m utterly unconvinced that they were happy with this new ‘rationalising’ approach, and I fear I’ve confused them even more for challenging and dismissing their trick, which none of them got to work anyway. So, next lesson we’ll relent. We will get there. But what annoys me is that I’m the bad guy. I’m the guy who in trying to get them to see why minus minus blurg doesn’t work ( – 3 – 4 is not 1, yet it is often incorrectly recalled as ,a minus and a minus make a plus) is NOT that memorable, and negates all thinking and creates nonsense answers that nobody is even questioning.
I’m the bad guy!
I’m also faced with the horrible task of undoing bad learning, and trying to re-do it, in the face of students who are convinced they had it right.
And worse still, I stop myself and think “maybe I should just go through “a minus and a minus makes a plus” again, then they’ll remember it… for now. And we can move on.