Sep 15, 2007
My Algebra II students and I pose with problem that stumped us.
One of a teacher's greatest fears is the fear of being stumped, of standing before your credulous students--they who trust that you know All, because you are the teacher--and having the spigot of knowledge suddenly dwindle to a thin stream before drying up all together. There's nothing worse than struggling in real time to understand a concept you're supposed to be teaching your students while they look on expectantly. Your heart pounds, you break into a cold sweat, you clear your throat and go "hmmmmmm" a lot. Eventually you start talking to yourself, turn your back on the students to face the board and start scrawling, desperate to figure out how the heck to do this, before finally in hasty shame suggesting that "we're running short on time so we'll get back to this one later"
I used to run into these nightmare moments more frequently when I first began teaching, but as the years have gone by and my confidence and knowledge of the subject matter has grown, they've become fewer and further between.
But this year, I've started teaching Algebra II, a subject I haven't taught in about seven years. I had the distinct sense that sooner or later I was going to be stumped. And this past Thursday, it finally happened.
We were taking questions before grading the previous days homework, when:
"Mister, how did they get the answer for number 15?"
The odd numbers have the answers in the back of the book. Smart students use those answers to be sure they are doing the work correctly. Not-so-smart students copy the answers down and move on without a second thought. These were smart students. So they already knew what the answer should be. They just couldn't figure out how to get that answer. The very fact that they cared about the "how" should have given me a clue that this was going to be no ordinary "stumped session."
I glanced at the problem--"Solve for S," it said--and then quickly solved it on the board. The answer was "S is equal to 2P over R."
Except it wasn't.
I studied the problem. How on earth did they get that? I puzzled, feeling my skin growing warm. Now I'm sure the math whizzess that are looking at this problem see the solution clear as day, but for some reason, I couldn't see it. I cleared my throat. I "hmmmmed." I turned my back to the students and scrawled away on the whiteboard, momentarily insensible to their presence in the room.
But this time, I didn't suggest we move on. And here, one of the worst experiences a teacher can have turned into one of the best moments any teacher can ask for.
The students rose out of their seats and gathered with me at the whiteboard. One, a young lady who I call "The Vice President" snatched the marker out of my hand and started writing. Her classmates "The Treasurer" and our newest student who I'll call "Violet" called out suggestions and started scrawling themselves. Their minds were whirring and they weren't just hazarding wild guesses--they were using their Algebra skills, applying the principles they'd learned. These kids were 100% engaged--something every teacher longs for in their students. I was so proud of them, I thought I'd burst. It was worth being stumped, just to see them jump into the deep end and really wrestle with this equation. That was the first great reward of being stumped that day--seeing my students joyfully engaged in challenging themselves and learning, using the skills they'd acquired. A teacher can't ask for more than that.
"Violet" was the first to find the solution, using a cross-multiplying method I'd always used for equal ratios but had never seen used to balance an equation like this. More amazing than the fact that she solved it, was the fact that I wasn't embarrassed that she solved it before I did. I was pleased and gratified to learn something new as well. But I wasn't satisfied, and neither were the other students. If it could be solved through cross multiplication, then it could be solved using the traditonal means of balancing an equation that I'd been teaching. We continued to tangle with the problem until at last we saw where our error had been, and how the problem could be solved. But there was still one more question--we'd gotten our answer, partially by clearing the decimal. But what if we left the decimal there? Why wasn't the answer turning out the same? I stared at the problem for a few moments and then the light came on. The answer was the same. I practically shouted this news to my students and began to enthusastically explain it. At first they didn't get it, but then they did, and we ended the class on a literal learning high. The second great reward of being stumped was learning something new, understanding a familiar principle in a whole new light. Ideally, a teacher never stops learning, and I was thrilled to see my own education continue that day.
Often we think that as teachers, as adults, the most important thing is to know everything. But sometimes it's when you're stumped that the real teaching--and the real learning--begin.
Here's a recreation of Thursday's problem. I didn't have the presence of mind to photograph the actual class, but on Friday I asked the Algebra II students to rewrite the problem and the solution we found, so that I could post it here. (Unfortunately, they neglected to show all the steps in solving the equation, but you get the idea).
And in other news. . .another amazing class! Here's "T" and a classmate acting as "news anchors" for RNN (Revolution News Network). On Friday, the class filmed a news report segment on the Battles of Trenton and Princeton for their 7th/8th American History class. The set has a sort of hyper-patriotic, Fox News type vibe, doesn't it. But hey, this is the American Revolution after all!