Sunday, January 26, 2014

What Would Happen If We Took a Problem Apart and Put It Back Together?

Ben Blum-Smith's challenge to, "describe and (jargonlessly) name concepts for thinking about task design" got me thinking about an activity my colleague Shannon and I tried last year with our fourth graders.
  After watching Dan Meyer's TED Talk  Math Class Needs a Make-Over, I couldn't get out of my mind the part where he shows a problem from a text book and then visually strips away all the elements until he's down to the graphic.
  What would happen, we wondered, if we took apart a problem like that and then asked the kids to put it back together?  We decided to take a grade 4 PARCC prototype, remove the question and some of the information, and present it to the kids as, well, just what it was: an incomplete question.

     We asked them to imagine what the question might be.  They brainstormed some ideas in their notebooks, and then got together in groups to put down some of their questions on chart paper.  Some were basic and pretty much what we expected.  But many were complex and needed additional information, supplied by the kids, in order to solve.  This was something we did not anticipate, but once it started the floodgates opened and all this math just started pouring out!

Some of the original questions. They  were first posted in the room; later they went up on a bulletin board.
Next, we gave them the same problem, but with the rest of the information.  Still no question.

Again, they were asked to take a guess as to what the question might be.  They started getting closer.

Here's an example from the second go-around.  
And another.  
Again, we were surprised at the complexity of the questions.  There was a high level of engagement; as long as they didn't have to figure anything out they thought they were getting away with something!
     Finally we gave them the whole thing:

They went to work immediately.  They had spent so much time with the task they had internalized its parameters.
  Then we realized we had this whole pile of really interesting questions, and that it would be a shame for them all to go to waste.   And even better, they were kid questions!  We did some culling, made sure there was variation in the problem content, and that there were some easy, medium, and hard ones.  We taped them all around the room, and told the kids to spend a few days looking them over and thinking about which ones most interested them.  We then put big sheets of construction paper and some pencils and markers under each question and let them have at it.  Some kids wanted to solve the problems they had written, others were interested in solving ones their friends had written.  Interestingly, some questions could not be solved because the authors had not included enough information.  Those had to be revised.
     When all was said and done, we agreed that it was a very productive problem-solving experience.  It had a low barrier to entry, it scaled both horizontally and vertically, and had a high engagement level.  Shannon and I agreed that one of the reasons for that was because of the student-centered nature of the project.  The questions were created by the kids, and they had freedom to decide which ones they wanted to tackle.  Kudos to Shannon for turning her class into a "math lab" and for using the lesson again this year with a new group of fourth graders.  Any help with a name to describe this activity?


  1. This is really cool.

    Do you think you would've gotten the same reaction out of kids if you had just tasked them with asking a mathematical question about that context? Or was it crucial that they were trying to guess the actual problem?

  2. Good question and one I hadn't thought about. I wouldn't say that it was crucial that they were trying to guess the actual problem; I think it added a little extra excitement, maybe in the kind of way the kids get excited when they do the estimation 180 activities. As more information is revealed you can re-evaluate your prediction and then see how close you get. I'm working now on a post that will describe a similar problem-solving model that I used in 5th grade this year that did not involve guessing the actual problem, and the engagement and excitement level was still really high. I had a notion that it would be valuable for the kids to see how those kinds of problems were built, that maybe then they wouldn't seem so intimidating.

  3. Hi Joe,

    I'd like to reference this blog post in an upcoming webinar for teachers in Georgia. May I?

  4. Ooops- forgot to say why. Because I love it.

  5. Thanks so much. Go right ahead and please let me know how it goes.

  6. I guess I am late to this party.
    This is interesting. I had a book club meeting with some other 6th grade math teachers. The leader discussed chapter 4 of a book called. Powerful Problem Solving by Max Ray of the Math Forum at Drexel. It was the same idea. Giving kids some information and asking them what they wondered about the data and what they saw.

    1. Thanks for stopping by Michael. Fortunately for us the party never ends. I'm sure this is an idea that has been circulating around for years. Let me know how it works in your classroom.

  7. Seems like Powerful Problem Solving might be a good name for the process. Thanks for sharing.

  8. Thanks for the suggestion Karen. It is very powerful. Right now we're in the middle of another round with our fourth graders with a different prompt. If you try it out, let me know how it goes.

  9. I would love to implement something like this in my classroom, but am having a hard time getting my head around how to still address all the Common Core standards while doing it. (High school algebra and geometry)


    1. Steve,
      Thanks for stopping by and commenting. Most of what's happening in the "Math twitter blogosphere" is centered around middle and high school content, and those folks (Dan Meyer, Fawn Nguyen, et. al) are much more qualified than I am to suggest how this approach could be used there. You might want to start with one of Dan's or Andrew Stadel's 3-acts. Robert Kaplinsky also has a great collection of interesting ideas and lessons on his site, and they're all connected to common core standards.
      But I will say that we struggle with the same tension. We attempt to embed content standards in our projects, and try whenever possible to let the math arise from the problems. We also turn our attention to the practice standards, and I understand that even common core critics agree that the practice standards are valuable; some say the best thing that came out of the entire enterprise.
      I am glad you are interested in experimenting. Good luck and let me know how it goes.

  10. Great blog. Thanks for sharing. Trying to think what that might look like in kindergarten. "Our class has 3 caterpillars,2 chrysalises, and 1 butterfly. What questions/problems could we be trying to solve? " Perhaps it could be a FAQ assignment where the students need to come up with the FAQ for a presentation. Just thinking aloud. This could cross over into science easily. Thanks again.

    1. Thanks for your comment. I think it's awesome that you're thinking about how to apply some of these ideas in a kindergarten classroom. Just getting them used to asking questions instead of answering questions would be a big step. Maybe even starting with picture prompts. We have amazing kindergarten teachers at my school who are eager to experiment and explore. Next year my goal is to focus a little more on our K-2 classrooms.