Saturday, May 5, 2012

If Lincoln and Douglas Had Been Math Teachers

"Punches are no longer being pulled." -MSNBC
"At each other throats." -CNN
"Ugly as a wedding dress inside a slaughterhouse inside a divorced vegetarian." -Fox News

As you can see, the 2012 debate season is heating up. And by debate season, I am obviously referring to the debate of how best to pose the following math problem to students:
Draw a rectangle on a square grid. An example 9 by 3 rectangle is drawn for you below. Draw one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size.
First, the back story.
The players
On August 12, 2011 Kate Nowak posted a seemingly innocent post on her blog sharing this problem. On par with my general google reader habits, I read this post about six weeks later. I liked the problem. Liked it very much. When I got the call from Gallup, it got my vote for being a good problem.

I worked on the problem myself. A pattern eluded me. Worked some more. Had a particularly helpful insight. Now a pattern emerged. Proving this pattern now eluded me. A few false steps and insights later, I was satisfied with my argument for why this pattern would in fact continue.

The set-up
On October 29, 2011 I used this problem in an Escape from the Textbook workshop as the context for a meta-discussion about mathematical habits of mind. From some perspectives, it was a great discussion. From other perspectives, it was a lovely way to spend a Saturday morning. Alas, history is complicated.

The tale
The following week I posed the below "problem" to my sixth graders.

But where's the problem, you ask? Exactly. By now my kiddos are used to seeing things like this. They know that the first step is to come up with an interesting mathematical question, what some might call a #anyqs or 100qs or "what can you do with this" or WCYDWT or WWJDWT. For now I'll ignore the barks of "Who cares!" My students bought in and came up with a great list of questions. Here's a selection (edited for mathematical fanciness) for your viewing pleasure:
  1. What is the length of the diagonal?
  2. If I change the size of the rectangle, how will the length of the diagonal change?
  3. What's the area of the rectangle?
  4. How many squares are in the rectangle?
  5. What's the area of the triangles?
  6. What's the perimeter of the rectangle?
  7. Does the red line cut the rectangle in half?
  8. Can I walk around the blue and red lines without crossing my path?
  9. How many times does the diagonal go though an intersection of the grid (a vertex)?
  10. How many squares does the diagonal pass through?
I gave them 20 minutes to work on whatever problem(s) interested them. I then shared with them that I found #10 to be particularly intriguing and had everyone work on this (if they weren't already) for the remainder of the period. Oh, and again the systems were already in place, but I didn't have to remind anyone that they weren't finished after counting the nine squares in the above shape. They were all over trying to find the general solution. Anyway, conjectures were made, simpler problems were solved, but no one solved the problem completely.

The wire
On November 5, 2011--a warm and rainy day in Palm Springs, CA--I presented this problem as part of my workshop Making the Process Standards More than an Afterthought at CMC-S. As I am known to do, I blogrecapped and blogflected on this presentation soon after. Dan Meyer was at this workshop and, being the Brad Pitt of the math ed world, picked up on a subtle but important choice (actually, I would call it a mistake) I made in my presentation.

In short, I posed the same problem I had earlier posed to my students to a group of teachers and asked them what questions they had. The problem is that I didn't explicitly tell them that while I appreciated their answers, we were going to focus on one question in particular. Another problem is no one asked the problem I had set up on the next slide. Anyway, you can read the details on Dan's blog if this recap wasn't enough.

I am indebted to Dan for pushing me to be more careful about how I talk about posing problems with both students and teachers. That said, the piece that fascinated me the most was the assumptions in the comments about why this #epicfail occurred.

Part of the problem, of course, is grounded in teachers’ fear of what would happen if a truly open question led to something that exposes teacher ignorance.
Expecting students to play mind-reading games and then disrespecting their ideas when they fail to come up with exactly the “correct” question/answer seems like an error to me. 
I don't think this is just me being defensive, but I think these comments are way off the mark (at least in my particular case). I have zero fear of exposing my own ignorance in front of my students. Feel free to ask them if you don't believe me. :) And I did not have a "correct" answer in mind, I just had a next slide. A later comment by Maria Droujkova actually did an excellent job of describing what was in fact going on and how this could easily be resolved.
[Y]ou have to tell people, explicitly, what it is you are doing. Here is how I would plan it:
1. What questions would you ask about this diagram?
2. Thank you! If we had an extra hour or five, students could solve these fine questions and more. Hopefully, forming the questions helped you to get to know the diagram more personally. The next task: ask just questions about things you can count.
3. Thank you! So, again, we could pursue these questions with students. I prepared a question of this type ahead of time to demonstrate the next stage of the process. I like it because it recently inspired a good discussion in math teacher blogs, and has some history in Olympiads.
The sting
Fast forward to today. I posed this problem again, this time using the updated version of:

I made sure to preface with the fact that I valued their questions and would love to have the time to explore each and every question, but had a particular question in mind. Once again, teachers came up with some questions (nowhere near the number or diversity of questions my sixth graders came up with, but my sixth graders had more time and more training). Once again, no one asked the question about counting squares (which, I will remind you, was a question one of my sixth graders asked).

So I'll leave you with this question. Was this another #epicfail? Should I give up with this particular problem and just include the question? Does it really matter? Anyone at today's session reading this feel cheated? Maybe I'll see what 101qs thinks.


  1. I missed your session, but agree that there is value in both worlds. Opening up the floor gives students the chance to pose problems that may or may not be as interesting as the one you want to pursue. It also generates a list of problems based in a variety of different math topics. It is powerful, regardless of future intent, for students to see the interconnectedness of Math.
    I don't see the giant offense in asking for questions, dwelling (shortly) on their solutions, and then moving into a question of particular importance. Opening up closed problems, even for a short time, can be beneficial. Not everyone has the luxury of a large piloting pool.

  2. Avery, this is a spectacular post! I was hooked, and I love the section headings.

    This is a really important point. Dan is saying “Make the prompt scream the question you are looking for." This makes perfect sense, if we have explicit objectives in terms of content, technique, and mastery at the heart of math instruction. There is no doubt value in Dan's work, but it is completely predicated on a sort of inception of the teacher's question into the brains of the students. It's easy to see this as a teacher's central objective.

    If, instead, we're interested in the questions that our students want to explore, then we have no reason to assess the work this way. I think this is a really provocative math scenario. What if a primary goal of a mathematical education was simply for students to improve their ability to sense "mathyness" in the world and digest it?

    Thanks again for a really great read.

    1. If it's all about getting MY question asked, then maybe we shouldn't call it "anyqs."

  3. Given your goals (and your awareness that your session audience hasn't been trained) why ask for their questions at all? Do you want them to practice mathematical habits of mind or do you want them to see how it might look in a classroom? If it is the former, then get rid of the "next slide" and let them explore their own questions. If it is the later, get rid of sharing their questions out loud. Instead have them do a quick "Turn-and-Talk" and then move onto the sixth graders' questions.

    As someone who tries to pack way too much into my presentations, I can say that focusing on my purpose has helped to streamline my sessions (mostly).

  4. It seems like when you move on to a particular question it signals importance or relevance. If you are trained to think of right and wrong answers, focusing on one of many questions seems to imply importance or "rightness."

    I'm left thinking, why did you move on the that particular question from all the questions submitted. Do explore why people thought of certain questions at all?

  5. First of all, revising the image can help narrow people in to your question even more. I would have never guessed it from your first image. I might have from your second. If you were to have two or three rectangles with the diagonal squares highlighted, I think I definitely would have. This is where feedback of 101qs is helpful (except I wish there was the ability to modify the image to see if you can get the questions to converge even more.)

    I strongly agree with the point of having a particular question in mind though. With my children at home, it can be unguided exploration, but in school you need to take the class somewhere they don't know to go (for the CCSS at bare minimum, as well as many other practical reasons). By default, it seems most people (and my students) will ask questions that they know the answer to. Maybe it's a confidence thing, or limited vision, but that's only marginally helpful. I think that's where your culture of inquiry seems to be working (and where it fails in groups of teachers). Perhaps a gentler way to move people to your question would be to ask "Tell me the strategies you would use to answer each of your questions", which will let you weed out the easy ones, and focus on the ones no one already knows how to do (yours).