I've taken to putting these ( )conceptions front and center. Put them out there for kids to wrestle with. Plug in some numbers. Argue. Get frustrated. And then walk away with a little more understanding than they did before.
Today's episode centered on the equation:
(2/3) (3x + 14) = 7x + 6 and students were asked to multiply both sides by 3.
And, of course, they came up with 2(9x + 42) = 21x + 18.
Well because, naturally, a(bc) = (ab) (bc).
So what do you do?
The younger me would have said something profound like, "You don't distribute multiplication over multiplication. I'll say it again slower for those of you taking notes. You. Don't. Distribute. Multiplication. Over. Multiplication."
There ya go. My finest pedagogical moments summed up by slowly repeating a negative definition of a property they obviously don't fully understand.
The older, wiser, 5-kid-having self is a bit more patient.
Up on the board goes:
True or False
2(3 · 4) = 2(3) · 2(4)
2( 3 + 4 ) = 2(3) + 2(4)
Most kids said that their gut told them that both equations were true. In fact, many said, "true" before the virtual ink had dried.
"But what does your head tell you? Verify that both equations are true."
Oh, no they're not both true.
"Ok, good. So now you have a conflict. What you think should be true is different than what you know is true. Why?"
This is why I have been calling these things preconceptions. Students bring something to the task. Always. They never come empty handed. These responses that #needaredstamp are usually a right idea used at the wrong time. It's like a kid who has never played sports before goes from learning basketball to soccer. Coach says dribble and the kid picks up the ball and bounces it as he runs down the pitch. Right rule; wrong application.
I've had kids tell me that they do certain operations on a problem because "it just felt right." I'm not sure how to address that other than to put them in a position for their feelings to betray them and help them deal with the disappointment in a constructive way.
Next weeks episode: Why Love Isn't an Emotion