This afternoon, I had two goals for my hungry student: 1) find missing lengths of right triangles and 2) find various areas of composite figures. We were working on the latter, and came across an interesting problem; instead of being given the triangle’s lengths, you were given other measurements that could lead you to deduce their lengths.
We’re looking at this problem, and it’s been established that “this whole length is 25m, and this part says that 5m of that is already taken up. So what part is left for the base of the triangle?”
She couldn’t tell me. She started fidgeting and doing this nervous rocking thing she does, and I knew I needed to say it a different way. So I say, “let’s say I’m standing here, and you’re standing 25 meters away from me. If I walk 5 meters toward you, shortening that distance, how far away am I now?”
She narrowed her eyes and looked at me, and said, “isn’t it just… 5?”
“Well, if it was 5, then that would tell me that this 5 meters plus the 5 meters I just walked would add up to the total 25 meters. But I know 5 + 5 isn’t 25…”
Fidgeting, rocking more, and a murmur: “… that’s tricky…” (adorable)
Sigh. I gave in.
“So if I have 25 meters and I take away 5 of them, how many do I have left?” She immediately typed 25-5 into her calculator (yes, you read that right) and said, “20?”
The vocabulary of math is crucial. And crippling. She heard those key words “take-away” and knew that meant “push the minus button.” She could connect “take-away” to “subtract” but couldn’t connect either of those words to the actual concept of a number of meters being reduced by a certain amount. Even “take-away” has become a meaningless memorized term, something that obviously started out to ignite real understanding, but memorized to the point that it’s now just another math word. So when I’m trying to scaffold down to meet them so they actually get it, I run into these land-mine key words that clue them in to some prior math-class knowledge and screw everything up. It’s really weird to have to come up with ways to explain the concept of subtraction without saying the words “take” and “away” to close together. Crazy. What do they think “Take. Away.” means?