Grand Forks, ND

I added an #anyqs component to the workshop I facilitated in Grand Forks, ND, last week. This was new for me. At the end of the first day, I assigned homework:

Give yourself one photo or one minute of video to tell a mathematical story so perplexing that all of your students will want to know the ending, without you saying a word or lifting a finger.

I received e-mails all the way through the night and into the morning before the second day’s session. I loaded each entry into a slidedeck and reeled them off to the group over 25 minutes.

At the same time, I had the participants working in a Google Form. For each entry, they’d write up a) the name of its designer, b) the question it provoked, c) the perplexity of that question (ie. how bad they wanted to know the answer).

The yield on that investment of 25 minutes was incredible. We spent the next hour mining and interpreting data and drawing conclusions about effective curriculum design with digital media. It was some of the productive professional development I’ve ever been a part of.

I posed several rounds of questions for table discussion. The first round began after they had just submitted all of their questions:

  1. Was the exercise fun? Was the exercise useful?
  2. What will be the most common question for your entry?
  3. How curious will people be about it?
  4. Which ones were you most curious about?

Then I sent out a link to the Google Form results and posed the second round:

  1. How effective was your entry at provoking a common question?
  2. How could you have made your entry more perplexing?
  3. Whose entry was the “best”?
  4. What do we mean by “best,” anyway?

While they batted those questions around, I dug into the spreadsheet and found the median response for each entry’s perplexity rating. Six of them tied for the highest median rating:

We reviewed those six, briefly, and then I posed a third round of questions:

  1. What’s special about these entries?
  2. If you had a favorite that didn’t make the cut, what isn’t captured by this measurement (the median perplexity rating)?
  3. Dan and Nancy both feature leaky faucets filling up a container. In what ways are those two entries different?

Selected Answers To Those Questions

  • The unanimous consensus was that the exercise was fun. Fun isn’t necessarily a prerequisite for an effective PD exercise, but man does it ever help.
  • One participant said the #anyqs exercise was useful, mainly, for “training my eyes.” She elaborated that after just one pass at #anyqs the day before, she was already more alert to the applications of math in her life outside the math classroom.
  • The issue of subjectivity has been one of the most fascinating conversations about #anyqs online, and so it was in Grand Forks also. Are the questions of math teachers about this image a useful proxy for the questions of students? Will a student from North Dakota have a different question about a video of a wheat thresher than a student from California? One participant noted that the high school and middle school teachers in the workshop asked questions that were linked closely to their content areas? (Strong data mining, right?) The sum of my thinking to date? Yes, the process is subjective. In spite of its subjectivity, field testing my curriculum with teachers has improved it immensely for students. Perplexity can transcend our demographic differences.
  • I forgot to mention this in Grand Forks but the easiest, best way to make your video-based problems more perplexing is to use a tripod. Or to simply put your camera down on something sturdy. The reason being is that it’s so much harder to gauge so many different measurements (speed, for instance, or height) when the camera is wobbling back and forth and up and down, throwing your subject around in the frame.

General Remarks

  • I have designed a lot of different constructs to explain to myself (and others) perplexing curriculum design. None has been as effective as mathematical storytelling. I’m particularly chagrined to think back on all the times I’ve broken a problem down into the four tasks of “verbalization, visualization, abstraction, and decomposition.” That construct resonates with my grad school peers, but it’s terrible vocabulary for teacher professional development. (ie. “Okay, so where do you find the decomposition of this task. How would you help the student abstract the problem space?” etc. Gross.) I’ve never heard table groups reference “decomposition” in one of my design activities. The language of storytelling, in contrast, was a constant feature of their conversations.
  • One participant: “We need a website for sharing these.” Yes.
  • Another participant: “Kids should bring in their own photos and videos.” Maybe.
  • This is the dy/dan drinking game: every time I put my readers to work to make me smarter or more effective in my studies or at my job, drink. I was legally unsafe to drive after receiving hundreds of pages of student work for my Michael Benson experiment. I was black-out drunk after using the work of @salmathguy, @reimerpaul, @eduz8, @techsavvyed, @fnoschese, and @wpeacock202, for fodder in my #anyqs workshop. Y’all should be so lucky to have readers like y’all.
  • There was a horrible moment in the early morning of the last day when I planned to hand each participant 28 strips of paper (one for each person’s #anyqs entry) on which she’d write her question and the name of the designer. Once we finished, I figured we’d trade the slips back to each other, a process that would probably take forty-five minutes on its own, right? So take a shot for Google Forms as long as you have the bottle out.

Your Homework

  • How are the two leaky faucet videos [Dan, Nancy] different? Which one is better? Define “better.”
  • One participant submitted this video of Carl Lewis’ 1984 Olympics long jump. He was perplexed by the parabolic motion of Lewis’ jump. Instead, nearly all of his colleagues (and yours truly) wondered how fast Lewis was running at lift-off. Given infinite resources, tools, the ability to travel anywhere in time and space, how would you capture Lewis’ long jump in a way that highlights the perplexity of his parabolic motion?

And Now A Word From Our Sponsor

If any of this seems interesting to you, let me recommend my Perplexity Session, which I’ll be hosting in Mountain View on 9/10/11.

Here’s John Scammell with a celebrity endorsement:

As someone who was fortunate enough to see one of the early incarnations of this workshop, I can tell you that it is incredibly valuable professional learning. Dan is a skilled facilitator, and more importantly a great teacher. I highly recommend it.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Joshua Schmidt

    June 22, 2011 - 4:23 pm -

    Dan’s video requires outside information to be given in order to solve the question “How long until the bucket fills up?” Since we haven’t seen inside the bucket, then we don’t know what the volume is, and more importantly, the rate at which the water is filling up the bucket.

    Nancy’s video a) gives us a timer and b) gives us a measurement to answer the same question. Therefore, one could solve this problem without any type of outside information.

    I think Nancy’s video is better for 2 main reasons:

    1) The volume in the problem is given, suiting a structure that almost frames the question. I think it’s more likely that students would be able to figure out what was going on much quicker.

    2) Since no extra information is needed outside of the video, students are empowered to solve this problem with no teacher interaction. It that case, it’s hard to do any less.

    Both videos would have been greatly sorted by having the camera sit still. Nancy’s would have been great if we could have had a shot of the cup the entire time without having to zoom up the video at the end. However, great job to both. I think there are very good problems to give a class.

  2. Hey Dan,

    Against your better judgement, I’m planning to invite students to shoot photos of things that are mathematically perplexing to them this summer.

    I’m curious how much you primed the teachers in Grand Forks… did you show a bunch of #anyqs before giving them their homework? How many? What did y’all talk about?

    One thing to note: the kids’ job is to capture the scenario (we’re suggesting signs & prices & discounts in stores as one good place to look). They’ll work together in groups led by college student volunteers to think of mathematical wonderings. I will, of course, let you know how it goes.


  3. To make parabolic motion leap* out from the Carl Lewis video, I wonder about:

    1) slow motion
    2) that neat effect where you leave the image from the previous frame in each subsequent frame, like Dan’s basketball video
    3) pausing Act 1 with him near the vertex of his jump (and maybe with some previous record jump or current leader’s jump marked in the sand)

    I guess I’m suggesting that we use what worked in Dan’s basketball video, where I definitely got to thinking about parabolic motion, and apply that here.


    *Honestly, there was no pun intended.

  4. I like the production of Dan’s better. I think I am more drawn into a question, based on the artistry, BUT I have no intuitive mathematical thought about it. I have no beginning information with which I can guess about the things I want to know, like volume, and rate of fill, or amount of waste due to a leaky faucet. A clear bucket would have made this awesome IMO.

  5. Re the “we need a website to store all these” I completely agree. If there is sufficient interest I am prepared to put time and effort into setting something up. Sorry to hijack Dan’s comments thread here but if people would like to comment if they are interested in helping out in any small way then I will get organized.


  6. I actually think Dan’s is more effective, Nancy’s discreetly tells the student to figure out how long it will take to fill the container you need to know (amongst other things) the time already passes, current fill level and the volume of container. Dan’s on the other hand leaves the entire project up to the imagination of the student: nothings given, what is it that we actually need?

  7. Joshua Schmidt

    June 23, 2011 - 2:32 pm -

    Going back to the videos again, I still like Nancy’s better for one specific reason.

    I am assuming that there is one question that you want the students to ask when the video is being made. I think it’s safe to say that Dan’s video is more inherently interesting, but I think that Nancy’s is much more likely to get the students to the same question every time.

  8. @J Edwards and anyone else interested. Yes, I am considerign a wiki. I am currently investigating either MediaWiki (which runs wikipedia) or TWiki and leaning towards TWiki.

    If anyone wants to discuss you can get me on Twitter as @numbat63

  9. Max: I’m curious how much you primed the teachers in Grand Forks… did you show a bunch of #anyqs before giving them their homework? How many? What did y’all talk about?

    We checked out four photos and four videos beforehand. I asked them to come up with a question and a rating for how bad they wanted to know that question. They over the next day they made their own.

    I’ve done this now in three one-day workshops using the #anyqs materials developed on Twitter and Grand Forks. It isn’t quite the same thing having people interrogate other people’s entries. There’s much more of a charge when the community interrogates its own.

    As for the Dan v. Nancy thing, there are obviously a lot of different variables working for and against us here, but I’m going to come down in favor of Dan for the single reason (alluded to by J. Edwards) that Nancy already has her students in act two when they should still be enjoying, speculating about, and arguing over act one. The students are keenly aware they’re in a math problem in Nancy’s. She’s presenting information (with the timer, with the measuring cup) that should be given only once students ask for it in the second act of the problem.

  10. For the parabolic motion thing, in a perfect world, the best way to do so would be to have the video cut to a different shot just before the jump, where the camera stops panning and is far enough back that you can see the entire jump from start to finish from the side.

    The panning camera emphasizes motion and speed, but de-emphasizes distance. Also, the angle of the camera (shot from slightly above) de-emphasizes vertical movement. A stationary camera with a side view low to the ground would emphasize the parabolic arc the runner makes.

    A follow-up video with the multiple freeze-frame trick would be nice, but having that in the Act 1 video might make it too obvious that that’s what the teacher wants you to see.

  11. @1:

    My question is, why should the TEACHER tell them the volume of that container? To me, the obvious question is “I’ve seen those buckets all over the place, so why don’t I know how much water one of them holds?” Not to mention “how big is that drop of water?”. The key to problem solving is to ask yourself the many little questions that the authors put in those mealy mushy textbook bits.


    Maybe, maybe not. The nice thing about Nancy’s video is that there is an obvious question that is as pointless as many textbook problems. I’m wondering how much my water bill goes up if I have a drippy faucet, and how much my electric bill goes up if that happens to be the hot water faucet, and how long it would take for the sink to overflow. Then I need to go figure out a whole bunch of things, including whether I can use that video to solve the problem in Dan’s video!