I want to write about my response to an (incorrect) student approach today that I forgot to make big deal about–by that I mean I failed to get excited about the mistake, failed to celebrate the discovery of a very reasonable strategy that doesn’t work, and failed to share that students’ discovery with the class.
A little context: We have be doing conservation of energy problems this week and turned today toward momentum problems. We worked a lot of momentum problems, and then I gave them a more challenging problem. The problem they were working on near the end of class today involved a 3kg object sliding along with a speed of 10 m/s into a 7kg object, and then the two objects (now stuck together) fall off an edge through a height of 4 m. The question asks about the speed of the two masses. All of the students correctly used conservation of momentum to solve for the speed of the mass after the collision (3 m/s), but about half of the students failed to use that speed to calculate the total energy after the collision. These groups simply set the potential energy at the top equal to the kinetic energy at the bottom, forgetting to include the kinetic energy it initially had.
One group however, realized they made this mistake and went to correct it–not by going back and re-working the problem, but by simply adding the 3 m/s to their final answer. The strategy (while wrong) makes sense. They saw themselves as calculating how much speed it would have gained if it didn’t start with any speed, so why shouldn’t they just be able to add the initial speed back in? I’m mean we’ve spent the entire semester doing that essentially– with equations like xf = xi + vt, and vf = vi + at, etc. Figure out the change and add that to what was there to begin with.
It would have been a cool thing for me to share with everyone what the approach was, why it seems like a reasonable thing to do, helped to point out that it led to a different answer, and leave it as an open puzzle about why. In the moment of teaching, however, I merely gave quick lip service to strategy as being reasonable, and gave some silly explanation about why it won’t work. Believe it or not, I said something about how it had to do with the fact that √( a² + b²) ≠ a + b. I quickly reared them back to the correct strategy. Now granted, there were only 4 minutes left in class. I didn’t have a lot of time, but I know that wasn’t the reason I didn’t do it. It’s that I really didn’t recognize how cool of an idea it is, and I didn’t think about the impact that my tone of voice and response would have on that student, their group, etc. I just sort of reacted. And what weird about me not recognizing it as a reasonable approach, is that I’ve even talked about this mistake before.
So tomorrow, I get to try to redeem myself. It’s going to be the first thing we talk about.