Dan Meyer developed a lovely challenge for math teachers and curriculum writers last week:

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.

He used it to start a Twitter discussion [#anyqs] that has yielded some really interesting discussion both on Twitter and on Dan’s blog.

I want to follow up on one of my entries in this fun and challenging game.

### Griffy counts

The following question have all come via Twitter.

- How many times will Griffy make it around?
- How many times would he make it around if he counting at the correct speed?
- How close will his counting be to 2 minutes?
- Which will occur first; 2 minutes or his counting to 120?
- What number will he be on when the 2 minutes is up?
- How many numbers/minute is Griffy counting?

Mission accomplished. Each of these is relevant to my intention; they overlap and interact in ways that would make exploration of any one of them relevant to any of the others.

These questions get at mathematical modeling. What assumptions should we build into the model? Should we assume that his counting continues at the same pace up to 120? Should we assume that his running continues at the same pace? If we answer *no *to either of these, do we think he’ll speed up or slow down? Etc.

Answers in video below. Note that he starts counting 10 seconds into the video.

Worth watching for the very end, most of all! Hi to Griffy and you. :)

Like I said on Twitter, really fun stuff here, Christopher. I’m trying to sort out in my head the effect of

showingsome kind of answer at the end of a problem. I wonder if I’m overestimating the catharsis. My understanding of CMP is that it pays close attention to the “resolution” of a math problem, with the teacher leading a conversation to summarize all the disparate student methods. Do you see a place for this technique within that structure?I have to say I was captivated. I really did need to know the answer to my question…

Two extra things I appreciated: proper counting by saying one hundred-one rather than one-hundred and one, and “Can I see the video now?” Classic!

Dan:

Absolutely.

And now that I have your attention on the matter, let’s dig in.

Connected Mathematicshas for a long time been concerned with narrative structure-both in a lesson and across the lessons in a unit. Prior to meeting you and Karim Ani, I wouldn’t have used the language ofnarrativeto describe it, but we talk aboutstorylinesof units for sure, and we ask teachers-both implicitly and explicitly-to tell stories to their students in getting them started solving problems.We have a basic structure to our lessons: Launch, Explore, Summary. Your work with multimedia has been helpful in our thinking about the Launch part of this. I am seeing it as being helpful with: (1) how to increase the power of our launches (to make them more “hooky”), (2) how to decrease the language load on (especially but not exclusively) second-language learners, and (3) how to encourage students to see their lived world mathematically.

But this is the #anyqs part of wcydwt, right? Your #anyqs is our Launch, with some serious constraints. Launch encompasses lots of ways to engage students-#anyqs has a really useful role as one of these.

Reading comments on your blog, you’ve got lots of people wanting to know what happens after you get kids intrigued with a good mathematical question. My observation is you, Dan, tend to dismiss this question with “do whatever makes sense”. I don’t mean

dismissin a pejorative way; I mean to say that it’s not where you’re focused. You are more interested in the initial engagement than in changing other aspects of teacher practice right now. InConnected Mathematics, we refer to this next phase of the lesson as Explore and we are quite explicit about the kinds of practice we intend to promote there. Furthermore, because we intend to encourage certain kinds of practice in Explore, teachers have to do more in the Launch than engage kids with a mathematical question. Teachers have to give students enough direction so that they can resolve (or make substantial progress in resolving) the situation for themselves. And teachers can’t give students too much direction so that there’s nothing left to resolve.At the end of the lesson-the Summary-teachers are pulling together students’ strategies in order to compare them, to connect them and to generalize. Checking answers is an important part of this, but it’s not really the major focus.

Psychologically, though, there is something very very nice about

seeingyour solution play out (or not). The multimedia can even increase the conflict in the situation (see e.g. 50-50). Therevealcan push the narrative along.But at heart, the Summary of a CMP lesson is really about process. How do we arrive at our answers? What can we learn from what we did? etc. Whether the answer is shown through multimedia or arrived at through consensus is not quite as important to the mathematical storyline.

I wish that were true! Most people seem content to point to some interesting webpage, photo, or video they see online and say “math!” without paying much respect to how students learn from it.

It’s true, though, that coming to a shared understanding of the context (act one) is much harder than developing solution strategies (act two). It requires that mixture of content and pedagogical knowledge that is pretty difficult, if not totally impossible, to convey online.

So I read this and tried to show a little video in 8th grade CMP while teaching The Shapes of Algebra. Usually when my students walk in there is a warm up or launch question on the board. This time I didn’t have a problem on the board and didn’t say a word, as instructed :) I had a clip from YouTube on the screen and after the bell rang pressed “play” and the students immediately got quiet and watched. It was of two airplanes that had a near collision. I stopped it just a few seconds before their paths would intersect.. when you couldn’t quite tell whether they would collide or not. Then I asked my students to explain one way the video related to math and one mathematical question they had. I got some less thought provoking questions (“What will happen next? Are they really gonna crash?”) but I also got some comments like. .”I don’t know if they’ll crash, but I knew their paths would have to intersect because they weren’t flying in parallel lines” and “So maybe the sky is like a giant coordinate grid with planes making the lines on the grid” and “How can someone graph this to prevent a crash?” This led into our lesson on solving a system of equations. I think (hope) it went well. . I hope this counts as more meaningful and not one of the problems in contrived contexts that you posted about. The CMP book has a problem about flight patterns but I decided to go with the video and discussion first rather than just have the kids do the problem and assume they “get” the big picture real life meaning behind the math. I wanted to see what level of mathematical inquiry they would produce without being prompted. Thanks for the sharing the idea. I would like to try something like this again. Could you respond to this posting when you have a chance to let me know if this is along the lines of what you were thinking? Thanks, Alex (PS My middle school kids were relieved – though a bit disappointed – that there was no dramatic crash when I showed the remainder of the clip :)

Students should ask questions that honestly perplex them. If mathematics isn’t useful for answering the questions that perplex them, we need to find better ways of representing perplexing mathematical situations or just admit that they’ve figured us out, that math isn’t useful for resolving any of life’s perplexing questions.

That isn’t true, of course. But asking your students “what

mathematicalquestions do you have?” about a video is a bit like asking “what do youlikeabout me?” on a first date.Hi Dan. I’m Alexandra Otto. I teach middle school math in Alaska. I was trying to get my students to think on a deeper level beyond just asking about the planes, “Are they gonna crash?” But are you suggesting that it would have been better for them to ask, “Are they gonna crash?” and then talk about how math can answer the question? I was excited to try this and now I feel like I got an F on your experiment :( . . I did think we had a good discussion and students were interested. Any suggestions for how to redo this in the future – from picking film clips to framing questions? I thought it seemed like a fun idea.

Alex:

Dan:

Alex, you don’t fail this experiment at all! Not by a long shot!

I will say that I was so, so excited by your story. And I think Dan is expressing a useful subtlety in quite strong language. But it really is quite subtle, and much less important in terms of teacher practice than the step you took trying the experiment in the first place.

Dan, I think, would be happy having you quit once you get kids to

Are they gonna crash?At that point, you have them interested in the narrative of the situation. They want to know what happens next. Then you can show them how mathematics resolves the situation.Dan’s subtlety comes in when you ask students

What mathematical question do you have?I’m less troubled by this. He argues that you implicitly communicate to students that the only way to get math into the situation is to shoehorn it in-to demand that it be there. But he would also say that after you toAre they gonna crash?you proceed to Act 2, where you tell them how mathematics can answer the question.So the real difference between what he advocates and what you did is the word

mathematicalin your first question. Ask instead,What questions do you have?Get them toAre they gonna crash?Get some gut instinct guesses about that from kids. Then push onwards by either (1) asking how we might resolve this situation, or (2) telling. I, of course, advocate asking. I want them to get as far as they can in ways that make sense to them.Forgive Dan. He’s coming off a week-and-a-half Twitter experiment in which he has seen teachers attempt to shoehorn mathematics into some quite phony situations (and he’s seen some really well-produced images and videos that do prompt good mathematical questions quite naturally-but these are harder to come by).

Yeah, let me apologize for my tone, Alex. I think I know what video you’re referring to and I’m kicking myself for not thinking about the classroom uses when I first saw it. A lot of us in this experiment are struggling to get students to come up with

anyquestion, much less a fairly focused question like you managed with “Are they gonna crash?” You seemed kind of down on that question in your comment, though, and I’d hasten to point out that that question is thenaturalone anyone — student, parent, teacher, math teacher — is going to have. And we math teachers have a tool to offer students to help them resolve their own question. That’s great.I just wouldn’t start from the premise that your students’ questions need to be mathematical. Some students who don’t like math at all will be completely perplexed by the near plane crash. Their perplexity is an invitation — whether they know it or not — for learning mathematics.

Thanks for your comments. I see what you mean now. I guess it is sort of like when I tell my two year old daughter to click her seatbelt on and she grabs her seatbelt and announces “click” without actually fastening it. Just because I announce that something is “mathematical” doesn’t necessarily make it so – for myself or my students.

I have to confess that I didn’t read the details of the original experiment on Dan’s blog – just Christopher’s discussion of it. But it sounds like the questions posted by Christopher in response to the Griffy video were generated by math teachers – after all, who follows you on Twitter? Certainly not middle school students, I would think. If I had shown the Griffy video (cute by the way!) to my students and asked them to come up with a question, I think I would have gotten some of these questions:

1. Is that your son?

2. Why is he running?

3. What does it say on the fridge?

4. How old is he?

So by nature your audience is geared toward mathematical thinking and I think mine is more geared toward themselves and their social lives. (I realize this is a gross generalization and this is not to say that mathematical thinking cannot occur in my students – if I believed that I wouldn’t be in my profession.) My use of the word “mathematical” was to avoid certain questions like “Who is on the plane?” “Are they leaving Kodiak?” And I thought the statement about the crash was almost too obvious to mention – after all, that was the point in showing it! But I appreciate your comments and realize the experience is meant to be more organic for students than contrived and that trying to force a mathematical epiphany to take place may take away from a true learning experience.

Is there a place where I can see video clips that other teachers effectively used?

Thanks,

Alex

You are spot on, Alex. Great critique, and deeply thoughtful as always.

Dan writes a lot about the mathematical narrative of a lesson. Kids are interested in that mathematical narrative only insofar as it resolves or moves along the larger life narrative in which they are interested. If I wanted to use

Griffy countswith middle school kids (and I’m not sure I do), I would want to tell a story that led up to the video. How this is my son, how I have been making math videos involving him since he was three, how on this day I had told him that he needed to wait two minutes before we could do something he wanted to do, how this had led him on this day to count the two minutes off by himself, inexplicably by running in circles around the first floor of our house. That story answers nearly every question in the list you cite; those questions are part of the larger life narrative in which middle school kids are rightly interested.Then we would watch the video. Middle school kids can identify with having to wait for something, and with the unfairness of an adult claiming “two minutes” and really taking much longer. I imagine someone in the class would think to ask the questions that got asked on Twitter.

I see the issues you raise as being fundamental to what we try to do in

Connected Math. We use the story line to engage kids with mathematical questions. I am learning from Dan how to be more thoughtful, elegant and smooth in setting up the mathematical investigation; how to make the mathematics arise more naturally. So that the video (or the photograph) isn’t just window dressing; it’s got the mathematical information we need to pose and solve the problem.You ask:

I’ll put this on my to-do list-a small curated collection of effective prompts produced by others.

In the meantime, you’ve got to check out Karim’s work over on Mathalicious. He’s less pure than Dan in the use of multimedia for problem posing, but he’s doing really interesting stuff with the life narrative discussed above. Note especially the Pimp My Feet lesson in which combinatorics quickly morphs into larger life lessons about choice. Right up your alley, I would think, Alex.

I’m not sure how to respond to this but it seems important.

Maybe this: I taught the kids on campus who hated math. No group could’ve been less geared towards mathematical thinking. But when I put up this video a) they were wondering the same question and b) that question was mathematical.

My readers are probably sick of me citing that video but it’s an existence proof: it’s possible to generate mathematical questions off the fuel of student curiosity alone, provided we give them something rich enough to be curious about. And there are others.

Aha – it’s much easier to get students to ask the same question if it is a yes/no question. “Will he make the shot?” Then everyone is on the same page and you can plan a math lesson to answer the question. I was trying to get my students to CREATE higher order questions on Bloom’s taxonomy – at the levels of synthesis, analysis, & evaluation (hence my use of the word “mathematical”) assuming that higher level questions would yield higher level thinking skills. Creating good, challenging questions is itself a challenge for anyone – even adults. But what you are suggesting, I think, is that a more basic question that requires a simple yes/no response can effectively generate higher level thinking skills as long as it plays upon an innate sense of curiosity.

I taught language arts for years before switching over to math just a few years ago. When I taught the concept of “foreshadowing” in language arts classes, I used a similar idea – I showed a film clip of an imminent event. (I think it was the tornado just about to hit in Twister.) Just as the outcome would be revealed, I stopped the film to the students’ protests. Then we talked about what clues we had seen in the film that foreshadowed what would happen next. I think the timing here is crucial – stopping the clip just before the expected outcome leaves students with a sense of discomfort and leaves them wanting to know how it will all turn out.

Thanks for a good dialogue and may I have permission to use your basketball clip in class, Dan? I would be happy to post about the outcome. I will also check out your blog and maybe jump in to some discussions there. I teach 7th and 8th grade math in Kodiak, Alaska.

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