Dan Meyer

Total 1615 Posts
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.

The Teaching Muscle I Want to Strengthen in 2018.

[a/k/a 3-Act Task: Suitcase Circle]

It’s the muscle that connects my capacity for noticing the world to my capacity for creating mathematical experiences for children. (I should also take some time in 2018 to learn how muscles work.)

By way of illustration, this was my favorite tweet of 2017.

Right there you have an image created by Brittany Wright, a chef, and shared with the 200,000 people who follow her on Instagram. Loads of people before Ilona had noticed it, but she connected that noticing to her capacity for creating mathematical experiences for children. She surveyed her Twitter followers, asking them to name their favorite banana, receiving over one thousand responses. Then on her blog she posed all kinds of avenues for her students’ investigation – distributions, probability, survey design, factor analysis, etc.

That skill – taking an interesting thing and turning it into a challenging thing – is one of teaching’s “unnatural acts.” Who does that? Not civilians. Teachers do. And I want to get awesome at it.

But Ilona ran a marathon and I want to run some wind sprints. I need quick exercises for strengthening that muscle. So here are my exercises for 2018:

I’m going to pause when I notice mathematical structures in the world. Like flying out of the United terminal in San Antonio at last year’s NCTM where I (and I’m sure a bunch of other math teachers) noticed this “Suitcase Circle.”

Then I’ll capture my question in a picture or a video. Kind of like the one above, except pictures like that one exist in abundance online.

Civilians capture scenes in order to preserve as much information as possible. That’s natural. But I’ll excerpt the scene, removing some information in order to provoke curiosity. Perhaps this photo, which makes me wonder, “How many suitcases are there?”

In order to gauge the curiosity potential of the image, I’ll share the media I captured with my community. Maybe with my question attached, like Ilona did. Maybe without a question so I can see the interesting questions other people wonder. You may find my photos on Twitter. You may find them at my pet website, 101questions.

I want to get to a place where that muscle is so strong that I’m hyper-observant of math in the world around me, and turning those observations into curious mathematical experiences for children is like a reflexive twitch.

(Plus, that muscle will be more fun to strengthen in 2018 than literally any other muscle in my body.)

BTW. Check out the 3-Act Task I created for the Suitcase Circle. It includes the following reveal, which I’m pretty proud of.

BTW. The suitcase circle later turned into Complete the Arch, a Desmos activity, which has some really nice math going on.

[Suitcase Circle photo by Scott Ball]

Featured Comment

Ilona Vashchyshyn :

I would just add that we shouldn’t forget that the classroom is a world within a world for us to notice, and that while many great, unforgettable tasks are based on interesting phenomena that we’ve observed or collected outside of school, on a day-to-day basis, high-impact tasks are probably more likely to be rooted in our observations and interactions with our students (in fact, even the banana tweet and post were sparked by a conversation with a student who was eating what was, to me, an exceptionally green banana). They tend not to be as flashy, but can have just as much impact because they’re tailored to the kids, norms, relationships, and histories in our classrooms.

I Have Big Reservations About Chalkbeat’s Teaching Competition

At SXSW, Chalkbeat is hosting The Great American Teach Off:

Top Chef. Project Runway. The Voice. Live competition shows have introduced audiences to the worlds of cooking, fashion, and singing — and opened a window into the intricate craftsmanship that these industries demand. Now it’s time for one of America’s most under-recognized professions to get the same treatment. Hi, teachers!!

Two teams of math teachers will teach a lesson to a live audience and receive judgment from a panel of “teacher celebrities.”

I linked to that description on Twitter and people were unsparing in their criticism:

I agree with the spirit of those criticisms, and David Coffey’s in particular:

Good teaching requires complicated decision-making based on a teacher’s long-range knowledge of a student and of mathematics. We should reach for any opportunity to make those decisions transparent to the public, who will always benefit from more education about good education. But a live event with an audience you don’t know and can’t interact with individually will necessarily flatten “teaching” down to its most presentational aspects, down to teachers dressing up in costumes, down to Robin Williams standing on desks in Dead Poets Society.

Not good teaching.

I asked teachers what kind of TV show would do justice to the complexity of teaching, if The Voice and Top Chef were the wrong models. Jamie Garner and James Cleveland both suggested The Real World, which seems dead on to me.

The Real World a) isn’t a competition, b) allows for characters to develop over time, and crucially, c) isn’t a live event. It is edited. You don’t watch the cast members do anything mundane. In the case of teaching, we’d love for the public to understand that good teachers assess what students know and adjust their instruction in response. But no one wants to watch a class work quietly on a five-minute exit ticket in real time. So the show would edit quickly past students completing the assessment and straight to the teacher trying to make sense of a student’s thinking, involving the audience in that process.

The challenge I’d like to see the folks at Chalkbeat take up is how to make those invisible aspects of teaching – the work that takes place after the bell – visible to the public. The work of presenting is already teaching’s most visible aspect.

BTW. Jamie Garner expands on The Real World: Math Class.

2018 Jan 1. Chalkbeat’s Editor-in-Chief, Elizabeth Green, clarifies her rationale for launching the competition and responds to some concerns raised here and on Twitter. She describes lesson study as the touchstone for her Teach Off and how she’s had to alter that format to fit SXSW.

It’s a really interesting article, full of references to the education scholars who have inspired her work for a decade. But I still tend to think she and the members of her design team have underestimated the magnitude of those compromises and how they’ll distort the approximation of good instruction her audience will encounter.

2018 Jan 8. In a revised contest page the organizers have eliminated the competition and clarified other aspects.

Featured Comment

Organizer Elizabeth Green weighs in:

I’m weighing in late here, but in response to one of the above threads, we never intended to have the whole audience serve as the students. As we’ve clarified in our revised page, which has more specific language, we’ll have 7-10 adult audience volunteers serve as students. Imperfect as a representation, for sure, but we still think everyone will get something important out of the 20-minute instructional activity + the followup discussion — that “something important” being better understanding about the nature of teaching and math teaching in particular. And for the record, Dan, at the 1,000-person “Iron Chef”-style teach off in Japan that Akihiko described, the students were the teacher’s actual students, and they all sat onstage.

This Episode of “Arthur” Gets Basically Everything Right About Math

Depictions of mathematics in TV and film generally lack nuance. When Hollywood doesn’t hate math, it reveres it, genuflecting before the eccentric, generally white male weirdos taking up space in its highest echelon – your Will Huntings, your John Nashes, etc. – with little in between.

But Arthur nails the nuance in “Sue Ellen Adds It Up,” and reports three important truths about math in ten minutes.

We are all math people. (And art people!)

Sue Ellen says No one in my family is a math person.

Sue Ellen is convinced she isn’t a math person while her friend Prunella is convinced there’s no such thing as “math people.” You may have this poster on your wall already, but it’s nice to see it on children’s television. Meanwhile, Prunella is convinced that, while she and her friend are both “math people,” only Sue Ellen is an “art person.” Kudos to the show for challenging that idea also.

Informal mathematical skills complement and support formal mathematical skills.

Prunella says You were using math and just didn't realize it. It's called estimating!

Sue Ellen says that she and her family get along fine without math everywhere “except in math class.” They rely on estimating, eyeballing, and guessing-and-checking when they’re cooking, driving, shopping, and hanging pictures. Prunella tells Sue Ellen, accurately, that when Sue Ellen estimates, eyeballs, and guess-and-checks, she is doing math. Sue Ellen is unconvinced, possibly because the only math we see her do in math class involves formal calculation. (Math teachers: emphasize informal mathematical thinking!)

We need to create a need for formal mathematical skills.

Prunella says Now let's measure the space on the wall.

Sue Ellen resents her math class. She has to learn formal mathematics (like calculation) while she and her family get along great with informal mathematics (like estimation). Then she encounters a scenario that reveals the limits of her informal skills and creates the need for the formal ones.

She’s made a painting for one area of a wall and then she’s assigned a smaller area than she anticipated. She encounters the need for computation, measurement, and calculation, as she attempts to crop her painting for the given area while preserving its most important elements.

Nice! Our work as teachers and curriculum designers is to bottle those scenarios and offer them to students in ways that support their development of formal mathematical ideas and skills.

[h/t Jacob Mehr]

[Presentation] Math Is Power, Not Punishment

I’m happy to release video of the talk I gave throughout the 2016-2017 school year, including at the NCTM Annual Convention in San Antonio, TX.

This is my best attempt to tie together and illustrate terms like “intellectual need” and expressions like “if math is aspirin, how do we create the headache.” If you’re looking for an elaboration on those ideas, or for illustrations you haven’t seen on this blog, check out the video.

The Directory of Mathematical Headaches

This approach to instruction seriously taxes me. That’s because answering the question, “Why did mathematicians invent this skill or idea?” requires a depth of content knowledge that, on my best days, I only have in algebra and geometry. So I’ve been very grateful these last few years to work with so many groups of teachers whose content knowledge supplements and exceeds my own, particularly at primary and tertiary levels. Together we created the Directory of Mathematical Headaches, a collaborative document that adapts the ideas in this talk from primary grades up through calculus.

It isn’t close to complete, so feel free to add your own contributions in the comments here, by email, or in the contact form.

Watch an Expert Math Teacher Put Three Kinds of Knowledge to Work in the Same Class

Lisa Bejarano’s post Two Kinds of Simplicity offers a useful idea about teaching complex fractions, but much more interesting to me are the three kinds of knowledge she puts to work in her class.

Knowledge About Teaching

Lisa has read widely from sources online and offline and has a great memory. So when she asks herself, “How am I going to teach [x]?” she can quickly summon up all kinds of helpful posts, essays, books – even the mental recording of previous classes she’s taught on [x].

Knowledge About Students

I stopped to think about how this would work with my class.

Lisa has taught long enough and knows her students well enough that she can test each of those resources out in her head, all during the lunch break before class. You can see her swiping right and left on each of them – “Yeah, maybe this idea. Definitely not that one.” – as she sees her students in her imagination. I’m sure Lisa is open to the possibility that her flesh-and-blood students will differ in surprising and awesome ways from her mental model of those students. I wouldn’t bet against her intuition, though.

Knowledge About Math

She ultimates decides to start her precalculus students with the elementary school analog of their lesson, turning an abstract fraction division problem into a more concrete one.

Then, as her students acquaint themselves again (or in some cases for the first time) with helpful models for that division, she builds back up to the abstract version of her task.

Lisa is only able to move up and down the ladder of abstraction like this because she knows a lot of math – specifically where it builds from and towards. If she doesn’t know that math, her options for helping her students basically shrink down to “let’s solve a few together.”


I don’t know if it’s possible to practice what Lisa is doing here. It’s knowledge, the tightly connected kind you get when you spend thousands of hours in math classes, reflect on those observations, write about them, talk with other people about them, and then use them to inform what you do in another math class.

It’s possible, even easy, to spend the same number of hours without acquiring that tightly connected knowledge.

It’s something special to see it all put to use.

BTW. My guess is a lot of those knowledge connections were tightened because Lisa is a dynamite blogger. On that theme, let me recommend The Positive Effects of Blogging on Teachers, an article which does a great job describing ten reasons why teachers should think about blogging.