Mathematical storytelling

I tend to tell stories when I teach. Most of the time, they are relevant to the topic at hand. As a seventh-grade teacher, I specialized in telling a story that ended with a mathematical twist, and I was particularly pleased with myself when students:

  1. didn’t see see it coming, and
  2. wanted to argue with each other right away about how the story was going to end

Ask my college students and they would likely recognize that aspect of my teaching, although I tone it down a bit for adults.

In the past six months, I have found a wealth of examples of amazing storytellers in mathematics. I met Dan Meyer and Karim Ani. Each of them uses the language of acts for describing lessons (although they don’t quite agree with each other on the meaning). And I (along with several hundred thousand others) discovered Vi Hart. Fabulous, smart mathematical storytellers; each and every one.

Out of this rich stew of stories comes the following. It’s the first of several I’ve got brewing. Help me make them better, will you?

UPDATE: In response to initial feedback in the comments, I revised the original video and posted it above. Two changes: (1) Sloppy audio levels were fixed (this is embarrassingly easy to do in iMovie), and (2) The video ends with the two pricetags sharing screen space-for easy reference. I’m unsure whether the second change is an improvement. See comments below.

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7 responses to “Mathematical storytelling

  1. My first thought is, do you really need to state the question at the end, at least so explicitly? I’m curious how often a significantly different question would arise. Sounds like a test for #anyqns .

    I’m also curious about your thinking about the value of the front story. I don’t doubt there is value, I just don’t know what specific value you see it bringing for the viewer/student.

  2. These are both questions that I’ve given quite a bit of consideration in the past couple of weeks. They are important ones. I’ll give quick replies here, but I’ll save the full brain dump for a later post, after I’ve gotten a couple more of these made.

    Do you really need to state the question? Perhaps not, but I’m not sure it’s a problem that I do. I appreciate the value of the great #anyqs experiment, and I’ve learned a lot from it. But I’m not sure I always need to leave the question asking to students. As my dear friend Karim once wrote, I think that a prohibition on exogenous questions may be a bit too strict.

    As indicated by my series of posts on Five Practices, I am primarily concerned with student problem-solving strategies. In posing the question, I ensure we’re all working on the same thing. But I don’t give up the strategy. And indeed, my College Algebra students were doing a wide variety of things with the problem.

    I’m also curious about your thinking about the value of the front story. Versions of this problem are standard fare in a College Algebra course-there’s nothing new in the mathematics I’m suggesting students should work on here. I’m thinking about the front story in a couple of ways. The main one is empathy with my students. They’re in College Algebra mostly viewing themselves as non-mathematicians. I imagine that they quite reasonably ask who in the world thinks up these problems and how. I am making quite clear how this problem arose, and I am hoping that it makes it plausible to them that they can ask these sorts of questions in their own lives.

    Another aspect of the front story is modeling where information comes from. In past semesters, I have had students work on an extended problem about the shenanigans you go through these days with gasoline when renting a car. They have repeatedly asked me to specify such things as the mileage of the car, the amount of driving that is assumed, etc. But that’s not how math gets done. Real problems don’t come so neatly packaged. They come in a messy form and we need to gather information and make additional assumptions. I want to model what that looks like. Where did the information about price and mileage come from in the video?

    I’ll have much more to say, and I’ll try to do it in an organized fashion over the next few weeks. Keep the critical questions coming. It was helpful to know that you see in the example some of the things I’ve been struggling with in my own mind.

  3. Very interested in this. . told my kids the legend of Descartes staring at the fly and envisioning the Cartesian plane – whether myth or reality the story captivated them. Though I think you mean not just storytelling for attention getting purposes but to procure deeper mathematical thought. Sometimes I wonder how far we have really evolved from primitive man sitting around a fire, listening to tales of a time gone by. . kids (and adults) just love a good story. I’m wondering too what role students’ own storytelling could play here too.

  4. My sense is that the problem could use an image like this one, which I ripped off Toyota’s website. This doesn’t abstract the problem for the students fully. It doesn’t say, “Gas prices and the price difference between cars are all you need here.” If we want students to learn to abstract and problematize their own lives, it’d be good to give them some practice at it in math class, right?

    Christopher:

    “But I’m not sure I always need to leave the question asking to students.”

    I’ve created something of a monster with #anyqs where some teachers believe asking any kind of question is verboten while others ask their students “what questions do you have?” while coercing them towards a single question.

    When a problem space lends itself to multiple worthwhile questions, it’s great to ask “what questions do you have?” and be prepared to run with the possibilities. If it really only lends itself to one question that’ll be worth the class’ while, best just to ask it rather than faking it.

    The joy, for me, of #anyqs, is that it lets me know if the question I’m about to ask my students is one that they will find (for lack of a better word) natural. Which is to say, if sixty teachers on Twitter wondered a certain question without any prompting from me, safe to say that’ll be a good question to ask my students.

    • The inverse of that last claim may be the more powerful one, Dan. If sixty teachers on Twitter can’t agree on the question, safe to say students won’t find it natural either.

      Either way, point taken (I think). Which is that the value of #anyqs is in the feedback it generates for your own problem posing.

  5. The pacing of the front story was a bit slow. The news clip was at a different audio level from the rest—I had to quickly adjust the volume up so I could hear it, then back down to avoid getting blasted.
    You leave the kids with a question, but there was no indication that they were supposed to have memorized all the numbers (except one) along the way.

    This is one problem with videos as a problem statement—it is damned hard to glance back to get the needed data, which is trivial with a written problem.

    Quite frankly, while the problem is a good one, the video detracts more than it adds to the problem. Giving the same story and putting the numbers up on a whiteboard would e better.

  6. Hmmm….yes. Audio levels were pretty sloppy. That has been fixed.

    gasstation and Dan each point to getting the necessary data into students’ hands. I’ve improved that in the updated version as well. I’m not sure it’s for the better, though. I’ll have to think about that. The file Dan links to solves the problem in a different way, I think. That file becomes the handout.

    Why might the revised video be for the better? One way to characterize a student’s experience with this video is that they were supposed to have memorized all the numbers (except one) along the way. Another way to characterize it is that the first question we should ask is, What information will we need to figure this out? The revised video telegraphs this information to students. The original video presents that information as it really appears to people in their lives; in the flow of their experience. It’s video; we can go back and extract the information we think we need. Or a teacher can give out the Toyota handout that has all same information (and more!)

    I’ll take the video detracts feedback seriously. I’ll think hard about what I want it to add, and for whom.

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