At Twitter Math Camp, one of the highlights for me was the “My Favorites” presentations. They started off with a bang… or maybe a pufffft…..whap! That’s the sound of Hedge @approx_normal shooting me with the marshmallow gun she taught us all how to make. And I’m still pondering what Glenn (@gwaddellnvhs) showed us on the last day about geometric interpretations of imaginary roots of quadratics.

Somewhere in the middle I got to talk about one of my favorite things: pausing the student/teacher interaction (by doing it online) and then practicing diagnosing students and asking really good questions. I was asked to blog about it, so here goes…The basic structure we use for looking at student work together as teachers is:

• Work on the problem students will be working on. Some tips:
• Use noticing and wondering to see some of the relationships you might not have noticed.
• Try to solve the problem they way your students might, not just your favorite way.
• Try to solve the problem multiple ways — are there any pieces that all solutions have to have?
• Anticipate where students might get stuck.
• Wonder “what math does this connect to?” — how could you stretch the student?
• Read the student’s work. Some tips:
• Assume the student is working to make sense of the problem, even when they don’t succeed. Can you understand the reasoning behind their steps?
• Start with a focus on what you have evidence the student DOES understand.
• Use “I notice,” and “I wonder,” to focus yourself on the data, and be clear when you’re making assumptions (wondering).

Here is the student work we looked at at #TMC12:

First I tried to divide 756 (the area) by 7 because there are 7 small rectangles inside the big rectangle. I got a total of 108. Then I divided 108 by 4 and got 27. Right away I knew that was not the answer because it was to easy and to small of a number. Then I tried to divide 756 by 2 because we tried to find the measure of the oppisite sides and got 378. After I found out that answer I also divided 378 by 2 and got 189. I realized it could be a possibility of one of the sides. Then I used guess and cheek to find out what the side was and got 4. Because 189*4=756 which is the area. So the perimeter we have to add 189+189+4+4=386 which equals 386 which is the answer of the perimeter.

Here are some things I noticed and wondered about the student’s work:

• The student correctly calculates perimeter from two side length, and correctly “undoes” the area formula to find a missing side length given the area and one side length of a rectangle.
• If the student had been given one side length, she could have solved the problem completely.
• The student guessed side lengths. Her strategy for checking the side lengths seemed to be based on gut feeling “too easy and too small of a number.”
• The student didn’t end up using the small rectangles, though she began focusing on them.
• I wonder if the student would say the problem was hard because there were too many possibilities or she didn’t know how to narrow down the possibilities.
• Think of one or two things you appreciated about the student’s work and one or two questions you could ask the student to move their thinking forward. Some tips:
• “I notice… ” and “I wonder… ” are great, non-threatening feedback starters.
• Be specific!
• Think about what the student might be interested in doing next, as well as what you want them to do.
• Prompts like, “what would happen if…” or “I noticed you said this about… could that apply here?” that ask students to make connections and predictions are powerful.
• What problem-solving strategies is the student using? Could some of their conceptual or procedural difficulties be supported with strategy tips — like organizing their work, looking for patterns, checking their guess, looking for constraints, etc?

Here are some things I could think of asking the student to help her move her thinking forward, and for both of us to learn more about her process. I don’t know which is my favorite or would be the most useful.

• How many answers do you think this problem can have? How do you know? Are they all correct?
• What makes this problem hard? Is there anything you would change or any hint you could get that would make it simpler?
• Do you think the small rectangles matter? If they did matter, how might they matter?
• For the answer you gave, have you checked it? Have you checked it with the original story/picture? What happened?
• What did you notice about the problem at first? Have you used all of your noticings in solving the problem/checking the problem?

If you thought this process of doing math together, looking at our students’ work, and zooming in on the best possible questions was cool, that’s great. If you’d like to be able to do the process slowed down, online, with lots of feedback, that’s even better. If you’d like to do it using your actual students’ work, and have them write back to you, and analyze your questions based on other teachers’ feedback and your own students’ responses, then have I got a proposal for you!

The Math Forum is getting together a group of teachers to do this kind of “feedback study” together this year. All online, all asynchronous (but we could have twitter chats, too!). The plan is to use a collection of Math Forum PoWs around different problem-solving topics (and whatever topics are important to you and your curriculum), get student work through the online system, and study it (and our responses) together. Wanna play? Comment, tweet @maxmathforum, or email me: max at mathforum dot org.