## You Don’t Have To Be The Answer Key

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.”

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.

2014 May 19. Reader Amy Hughes writes in:

After some work on rectangular prisms with 6th graders, we worked on the file cabinet problem. It took all week to weave the videos into our work/homework, etc but on the final day, when the last post-its are being place on the cabinet, the bell rang and students would not leave the room until their calculations were verified – it was awesome to see them care that much.

2014 Aug 1. Kate Nerdypoo:

They would literally CHEER and high five when they discovered they had the right answer.

2014 Dec 28. Nat Highstein:

… in my experience, this is not your typical reaction to getting the right answer on a math problem!

2015 Sep 9

2015 Sep 29

2015 Oct 2

At this point one student was about to leave the room and said “I’ll wait because I need to know how this turns out!” – how cool is that?

2016 Feb 07.

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

1. #### mark vasicek

June 17, 2010 - 5:55 pm -

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

2. #### Anne

June 17, 2010 - 6:38 pm -

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?

3. #### Mr. K

June 18, 2010 - 5:26 am -

>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.

4. #### Steven Kimmi

June 18, 2010 - 8:29 am -

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?

5. #### Dan Meyer

June 18, 2010 - 9:48 am -

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!

6. #### Alex

June 18, 2010 - 2:49 pm -

–>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???).

7. #### Julie

June 21, 2010 - 9:07 pm -

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.

June 26, 2010 - 9:48 pm -

@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.

9. #### MWZ

August 5, 2010 - 4:17 pm -