It took a while to dawn on me: I’d obviously missed something.
I got frustrated when the kids couldn’t cancel pairs. All they had to do was match up a red with a yellow, move them to the side, and count what was left over. They could add more pairs, and the number could stay the same. Easy, right?
Not so much. So I tried a lesson from AIMS.
Here is my version of the basic manipulative1:
I’d hoped that since the +s & -s are already paired, that the number would be more obvious.
It turns out to be some odd conceptual problem: the same 1/3 that didn’t get it with the algebra tiles didn’t get it here either. As I probed further, I realized that they had a bad understanding of the word “more”.
At first I thought it might be a language issue. But translating the question (with the help of my kids that did get it) made no difference. It was the same as the algebra tiles problem: they could tell me which one was more, but when I asked “How much more” I’d get completely wrong answer. (As an example, for the image up there, I’d get answers like 8, 5, or even 13 before I’d get a correct answer of “3 more red”).
The problem was “more”?
I was stunned.
So I ran a mini tutorial – I had the 1/3 who got it come up and explain how they did it. Their explanations weren’t that great – the biggest difference to their stumped classmates was seeing a peer proclaim that this was easy and obvious. That way, when I finally did give them a useful methodology, they were eager to jump on it.
I spent the rest of the day without notes, without writing anything down, with no procedures, just having them play “stump your buddy” with the charge cards. Once a pair got good at it, I’d suggest variations (like not matching up the plus & minus signs, or using irregular groupings).
The next day, all the stuff I’d expected to be hard stuff was a piece of cake.
I still can’t believe the problem was that they didn’t understand “more”.
1 The biggest physical problem with this manipulative is keeping the tabs folded to the right place. Unless you crease the paper close to the point of failure, they tend to want to stand up, making it confusing as to whether a particular tab is showing or not. I solved this by digging out my old pile of transparencies, and having them lay one on top of the card. This held the tabs down, and made it much easier for them to count.