by Lois Burke
How I found POWs, used them, lost them and eventually came back to them!
I actually found the Math Forum POW’s many, many years ago (we won’t talk about how long). When I first started using them I taught mostly Geometry and I really loved finding non-traditional problems for my kids to try. I started by working a few in class, in small groups. I was trying to get my kids to take some risks and to start diving in to problems that weren’t quite as “direct” as the ones often offered in textbooks. I was looking to stretch their thinking and increase their confidence. Over time my kids become great at taking those risks and really did jump in whenever I gave them a POW. They competed with one another as to who could come up with the most unique solution and whose solution might be the most elegant – and what that meant. As a student myself, I always did the problem the “hard” way — I was the brute force mathematician — so it was always fun to see how they would work the problem. The ideas they would come up with were amazing!
Then came state testing and suddenly, I didn’t feel like I had time for the PoWs. Had to prepare for those state tests. My kids did well but the ones who didn’t do well were struggling. Why? What weren’t they getting? After looking at test questions and working with and talking to kids for a while, it seemed that the kids who were struggling just didn’t seem to be able to handle any problem that didn’t look like the five examples we had done in class. Anything new threw them for a loop. They just didn’t know how to attack it. Enter the PoW’s. I decided that the only way my kids could build their problem solving muscles was to do exactly that – problem-solve!
The love affair starts again. Now I make a habit of including PoW’s as often as possible. We take time to notice and wonder and I’ve noticed (no pun intended) that noticing and wondering has made the transition over to the regular ‘ol math lessons on the typical stuff too!
“Ms. Burke, I noticed that when you multiplied (x + 3)(x – 3) the middle canceled out. I wonder if that always happens when multiplying two binomials when one is positive and one is negative?”
“I noticed that when you have one root at 4 + i then you have another at 4 – i. I wonder why that is?”
Now that being said…. It didn’t happen overnight. The first one went “ok” – and that’s about as enthusiastic as I can be about it. I was still getting used to teaching with the PoWs; they were still getting used to learning with them. The noticing and wondering we did in class was crummy. For example, when we did The Function Challenge (#628), they noticed that there were five functions; and they noticed that two of them were quadratics — and that was it. Come on… really… that’s it? Nothing about the lines? And I was at a loss as to how to get them to think more deeply. The solutions weren’t much better – they lacked detail and were often just an answer, even though I had given them a rubric and gone over it in class. Again… really…Yuck… what to do next?
Enter the importance of feedback and revision! I started having my kids submit their answers to me in a Google doc. I told them that I would give them feedback, provided they worked on the PoW early – before it was due (this was a mistake, by the way; I’ll explain shortly). I provided lots of comments in their documents, pushing them to think more deeply and explain more thoroughly. It was hard to come up with comments that didn’t give them answers. I decided, though, that this might be a good place to model for them some noticing and wondering.
“I noticed that you looked at the graphs and which was higher. Good thinking! I wonder if you thought to compare the compositions? What function is created by B(D(x))? What about D(B(x))? Which is larger algebraically or graphically? “
The students who worked on it ahead of the due date got lots of feedback. Be prepared: giving quality feedback takes time but it is so worth it! The kids who took the time, read the feedback and even asked questions back did much better than the ones who left it until the last minute. Lesson learned….
Enter next PoW and the importance of the scenario vs. the actual problem we did Don’t be Square #736. The first time I had used the scenario but I wasn’t as prepared as I should have been. This time I was ready! We put up EVERY single thing they noticed – EVERYTHING! We put up EVERY single thing they wondered – EVERYTHING! Crazy stuff went up there! It was fun! Then we started analyzing our thinking. What seemed important? What did we think the question might be? This bugged them at first – no question was there….
“I can’t do this … there isn’t a question? Why are you putting up a problem that’s not a problem?”
They also hated that I wouldn’t give any value one way or another to their responses. I just asked them to repeat for the class and the class decided what they thought we should think about and how one idea could connect to another. It was probably one of the best classes I’ve had in a long time! They got it.
Their responses were so much better too! Everyone had to submit a rough draft this time. No exceptions. Two due dates: one for the rough draft that I gave them feedback on, and one for the final. This way I got to comment on ALL of them. No one got to wait until the last minute. It was a beautiful sight to behold. Even parents got into it! One parent helped his daughter with the assignment rough draft and actually thought they needed Calculus. She took great pleasure in going home and explaining it to him “the easy way!”
My students were thinking! Hallelujah! Loads of things improved: those in-class “noticings and wonderings” I mentioned earlier; their explanations to each other — and their willingness to to get up and explain in front of the whole class.They were taking risks and enjoying getting that right answer and really understanding how they got that answer. They didn’t balk at a problem that they didn’t get right away. They dove right in! Success! I didn’t feel like I was just teaching math but teaching how to solve problems – whether they were math or not. Even when the kids hadn’t seen that type of problem before, they still felt they could try it . Any problem seemed doable — even those on the state tests.
To give kids confidence … the ability to persist … the ability to communicate — that’s huge!!