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a/k/a Annuli Follow-up

Is there any advantage to these images over the analogous problem in a textbook?

I vote "definitely, yes." The first four of these questions offer two enormous bonuses on top of the fifth while assessing the same skills.

The first is that you can guess them intuitively before you answer them mathematically.

What do you think? 500 tickets? 5,000 tickets? 50,000 tickets?! Give me a wrong answer. Give me an answer you know is too high. Give me an answer you know is too low.

I spent six years looking for high-yield techniques to draw students who hate and fear math into conversations and then calculations about math. Given another six, I'm sure I'd find something more effective but that right there is the best I have. It costs you nothing and it gets them talking. It gets them interested in an outcome. It gets them interested in the tools to determine that outcome.

The other advantage to this curriculum is that the student doesn't need the teacher to verify the answer.

I usually envy all the fun ELA instructors get to have with their students. Not here, though. ELA instructors have to grade essays using subjective measures of form and content. "Was my thesis coherent?" the student wonders. "Was my essay persuasive?" The student waits for the instructor to render judgment. This is necessary, I suppose, but it's also adversarial and it forces the teacher to double down elsewhere to restore a spirit of collaboration to the relationship between teacher and student.

Meanwhile, my math student wonders, "Was my original guess correct? Is my math right?" to which I can respond, "Beats me, man. Let's find out." And we count up the tickets. Or I show them the playlist from which I burned that CD. Or we measure the toilet paper. Or we look at the front of the dental floss container.

Every answer but the last one disposes the student to see that math makes sense on its own terms, that math coheres to the world, that math exists apart from her teacher's say-so. Her teacher doesn't determine the correctness of her answers.

This did wonderful things for my relationship to my students. At our very best, we became peers, collaborators, and co-conspirators in the creative exercise of mathematics.

2012 Mar 12: "It’s Killing Me. I Gotta Know."

2012 Dec 12: Watch Students Watch The Answer To Their Math Problem

2013 Feb 14. Mr. Ward has another illustration.

2013 Feb 27. The conclusion of the Barbie Bungee activity has students testing out their predictions, making sure their bungee cord is long enough for Barbie's head to come close to the ground but short enough that it doesn't touch the ground. Kids flip for this, apparently. Here are examples from different teachers' classrooms:

2013 Feb 27. Brian Miller's class solves The Bone Collector challenge and watches the answer.

2013 May 11. Nat Banting's students "gave a round of applause" when they saw the end of Toothpicks.

23 Responses to “You Don’t Have To Be The Answer Key”

  1. on 17 Jun 2010 at 12:59 pmAmy

    Love this. Sending it to my math teacher mother.

  2. on 17 Jun 2010 at 5:55 pmmark vasicek

    Dan,

    I love that you bring ideas to the classroom that help us connect to the students. Thanks for this entry in once again explaining why. I hope you will continue to post useful stuff for my classroom.

    Mark

  3. on 17 Jun 2010 at 6:38 pmAnne

    I’ve been having that same issue with the IMP curriculum in my classroom. Is it project based? yep. Do we have a big central problem we’re trying to solve? sure. Do the students care/ Are they invested? Not really.

    How do I take this amazing cirruculum and make it relevant? Is it really as simple as changing textbook picture to google images?

  4. on 18 Jun 2010 at 5:26 amMr. K

    >Is it really as simple as changing textbook picture to google images?

    I don’t think so.

    What it takes is finding a question that students can make a reasonable guess at, but knowing for sure takes some math. It also takes not framing it as a “we need to solve this” problem, but more as a “I wonder” problem. Get the kids wondering too, and they eventually want to solve it.

  5. on 18 Jun 2010 at 8:29 amSteven Kimmi

    While my grade-level change this coming year has left me without circles for the most part (I mean anything beyond that they are flat and round), I still am compelled to comment…on your musical choice. Bon Iver, woo-hoo, good choice. Do students in your neck of the woods even know who he is?

  6. on 18 Jun 2010 at 9:48 amDan Meyer
    Mr. K: What it takes is finding a question that students can make a reasonable guess at, but knowing for sure takes some math. It also takes not framing it as a “we need to solve this” problem, but more as a “I wonder” problem. Get the kids wondering too, and they eventually want to solve it.

    Good word.

    Cultivating that genuine posture of wonder towards answers and methods that are already known to you was one of the most fun challenges this job ever offered me.

    Steven Kimmi: Do students in your neck of the woods even know who he [Bon Iver] is?

    The awesome ones do!

  7. on 18 Jun 2010 at 2:49 pmAlex

    –>How do I take this amazing cirruculum and make it relevant? Is it really as simple as changing textbook picture to google images?

    Anne, I’ve faced the same dilemma. I took cookies (from IMP, I think year 2) and made it into a bakery that my wife and I operate. I create group competitions to see who can earn me the most money based on the constraints. I allow the students to create their own businesses with their own constraints. I brought in a guest speaker who works for a fashion company and can tell the kids that optimization of sales is a real thing, and then we adjust the lesson to work with his company. Kids are pretty smart, they can sense the genuine relevance versus the fabricated one (really, when are you going to hide in an orchard and need to know the radius of the trees that blocks the line of sight???).

  8. on 19 Jun 2010 at 3:02 amChris R

    Am I missing something here?

  9. on 21 Jun 2010 at 9:07 pmJulie

    As a 6th grade ELA teacher in an inner-city charter school, I’m often jealous of math teachers in my school. For one thing, you can make math gains so much more quickly and directly (i.e., in a linear fashion) than reading level gains, which are often all over the place. For another, math problems have actual answers that can’t be disputed, at least at the sixth-grade level. That’s why teaching grammar acan be totally rewarding–there are right answers and kids are psyched when they can figure it out. Teaching vocab can be the same way–it’s fun to try out new words and see how they fit together.

    We teach our ELA kids good “active reading” techniques, like asking questions and making comments as they read, things that good readers do automatically. The more compelling the text, the better the questions and comments and make kids want to read on and care about literature and discuss the issues in the novels we read, which helps put teacher and student on the same team.

  10. on 26 Jun 2010 at 9:48 pmTouzel Hansuvadha

    @Anne
    >>”I’ve been having that same issue with the IMP curriculum in my classroom. Is it project based? yep. Do we have a big central problem we’re trying to solve? sure. Do the students care/ Are they invested? Not really.”

    @Alex
    >>”Kids are pretty smart, they can sense the genuine relevance versus the fabricated one (really, when are you going to hide in an orchard and need to know the radius of the trees that blocks the line of sight???)”

    Can you explain how the Orchard Hideout problem and other problem-based curricula (I’m only familiar with IMP) are different than Dan’s dental floss problem? Both are miles better than traditional problems. Both allow kids to guess intuitively (I’m thinking especially of IMP units like Pit and the Pendulum and The Game of Pig).

    I suppose it differs according to Dan’s second criterion (“the student doesn’t need the teacher to verify the answer”), but by how much? When Dan teaches that lesson, he’s not just going to sit back at his desk and wait for the kids to verify the answer. He’s going to ask questions, prod for explanations, ask for generalizations, and scaffold for students that need help (without lowering the cognitive demand). The same happens in IMP. Both are student-centered. Both offer rich problems, with multiple-entry points.

    I agree with Anne in that the IMP curriculum is amazing. I also agree with you in that it’s not directly relevant to their
    everyday lives. But does it need to be? I think there’s as much rich math that is completely irrelevant to their daily lives (like the orchard hideout problem) as their is contrived math that *tries* to be relevant.

  11. on 06 Jul 2010 at 8:04 amMimi

    By the way, a cute picture for the WCYDWT category: http://wildammo.com/wp-content/uploads/2010/04/4349098831_aa94049cf5_b-675×450.jpg

    From http://wildammo.com/2010/04/06/what-stormtroopers-do-on-their-day-off-part-2/ .

  12. on 05 Aug 2010 at 4:17 pmMWZ

    I have been teaching IMP for 5 years and I like how Mike Schmocker describes this type of curriclulum…”The Crayola Curriculum.” It dumbs down some concepts and teaches other concepts that are beyond the students’ training. It wastes valuable class time with projects and POWs that are not relevant to the the unit that it’s in. Traditional math also is lame. There must be something that balances the rigors of the traditional approach with the ideas of that students need to be engaged.

  13. on 05 Sep 2010 at 7:53 pmAlex

    @ Touzel – the difference, as I see it, is that kids will NEVER hideout in an orchard waiting to see how long until it becomes a true hideout.

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