Dan Meyer

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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.

Rough-Draft Thinking & Bucky the Badger

Student work on the Bucky Badger problem.

Many thanks to Ben Spencer and his fifth-grade students at Beach Elementary for letting me learn with them on Friday.

In most of my classroom visits lately, I am trying to identify moments where the class and I are drafting our thinking, where we aren’t looking to reach an answer but to grow more sophisticated and more precise in our thinking. Your classmates are an asset rather than an impediment to you in those moments because the questions they ask you and the observations they make about your work can elevate your thinking into its next draft. (Amanda Jansen’s descriptions of Rough-Draft Thinking are extremely helpful here.)

From my limited experience, the preconditions for those moments are a) a productive set of teacher beliefs, b) a productive set of teacher moves, and c) a productive mathematical task – in that order of importance. For example, I’d rather give a dreary task to a teacher who believes one can never master mathematical understanding, only develop it, than give a richer task to a teacher who believes that a successful mathematical experience is one in which the number on the student’s paper matches the number in the answer key.

A productive task certainly helps though. So today, we worked with Bucky the Badger, a task I’d never taught with students before.

Bucky doing pushups.

We learned that Bucky the Badger has to do push-ups every time his football team scores. His push-ups are always the same as the number of points on the board after the score. That’s unfortunate because push-ups are the worst and we should hope to do fewer of them rather than more.

Maybe you have a strong understanding of the relationship between points and push-ups right now but the class and I needed to draft our own understanding of that relationship several times.

The scoreboard for the game. Wisconsin scored 83 points. Indiana scored 20 points.

I asked students to predict how many push-ups Bucky had to perform in total. Some students decided he performed 83, the total score of Bucky’s team at the end of the game. Several other students were mortified at that suggestion. It conflicted intensely with their own understanding of the situation.

I wanted to ask a question here that was interpretive rather than evaluative in order to help us draft our understanding. So I asked, “What would need to be true about Bucky’s world if he performed 83 push-ups in total?” The conversation that followed helped different students draft and redraft their understanding of the context.

They knew from the video that the final score was 83-20. I told them, “If you have everything you need to know about the situation, get to work, otherwise call me over and let me know what you need.”

Not every pair of students wondered these next two questions, but enough students wondered them that I brought them to the entire class’s attention as Very Important Thoughts We Should All Think About:

  • Does the kind of scores matter?
  • Does the order of those scores matter?

I told the students that if the answer to either question was “yes,” that I could definitely get them that information. But I am very lazy, I said, and would very much rather not. So I asked them to help me understand why they needed it.

Do not misunderstand what we’re up to here. The point of the Bucky Badger activity is not calculating the number of push-ups Bucky performed, rather it’s devising experiments to test our hypotheses for both of those two questions above, drafting and re-drafting our understanding of the relationship between points and push-ups. Those two questions both seemed to emerge by chance during the activity, but they contain the activity’s entire point and were planned for in advance.

To test whether or not the kind of scores mattered, we found the total push-ups for a score of 21 points made up of seven 3-point scores versus three 7-point scores. The push-ups were different, so the kind of scores mattered! I acted disappointed here and made a big show of rummaging through my backpack for that information. (For the sake of this lesson, I am still very lazy.) I told them Bucky’s 83 points were composed of 11 touchdowns and 2 field goals.

Again, I said, “If you have everything you need to know about the situation to figure out how many push-ups Bucky did in the game, get on it, otherwise call me over and let me know what you need.” The matter was still not settled for many students.

To test whether or not the order of the scores mattered, one student wanted to find out the number of push-ups for 2 field goals followed by 11 touchdowns and then for 11 touchdowns followed by 2 field goals. Amazing! “That will definitely help us understand if order matters,” I said. “But what is the one fact you know about me?” (Lazy.) “So is there a quicker experiment we could try?” We tried a field goal followed by a touchdown and then a touchdown followed by a field goal. The push-ups were different, so now we knew the order of the scores mattered.

I passed out the listing of the kinds of scores in order and students worked on the least interesting part of the problem: turning given numbers into another number.

I looked at the clock and realized we were quickly running out of time. We discussed final answers. I asked students what they had learned about mathematics today. That’s when a student volunteered this comment, which has etched itself permanently in my brain:

A problem can change while we’re figuring it out. Our ideas changed and they changed the question we were asking.

We had worked on the same problem for ninety minutes. Rather, we worked on three different drafts of the same problem for ninety minutes. As students’ ideas changed about the relationship between push-ups and points, the problem changed, gaining new life and becoming interesting all over again.

Many math problems don’t change while we’re figuring them out. The goal of their authors, though maybe not stated explicitly, is to prevent the problem from changing. The problem establishes all of its constraints, all of its given information, comprehensively and in advance. It tries to account for all possible interpretations, doing its best not to allow any room for any misinterpretation.

But that room for interpretation is exactly the room students need to ask each other questions, make conjectures, and generate hypotheses – actions that will help them create the next draft of their understanding about mathematics.

We need more tasks that include that room, more teacher moves that help students step into it, and more teacher beliefs that prepare us to learn from whatever students do there.

[NCTM18] Why Good Activities Go Bad

My talk from the annual convention of the National Council of Teachers of Mathematics last week was called “Why Good Activities Go Bad.” I hope a) you’ll have a look, b) you’ll forgive my voice, which as it happens I left at the Desmos Happy Hour the night before.

The talk is a deep dive on a single activity: Barbie Bungee.

But the goal of the talk isn’t that participants would walk away having experienced Barbie Bungee or that they’d use Barbie Bungee in their own classes later. Phil Daro has said that the point of solving math problems isn’t to get answers but to understand math. In the same way, the point of discussing math tasks with teachers isn’t to get more tasks but to understand teaching.

So during the first few minutes, I give a summary of the task relying exclusively on your tweets for photo and video documentation. Then I interview three skilled educators on their use of the task – Julie Reulbach, Fawn Nguyen, and John Golden. Two teachers saw students engaged and productive, while the third saw students bored and learning little.

What accounts for the difference?

My talk makes some claims about why good activities go bad.

BTW:

Here are the previous five addresses I have given at the NCTM Annual Convention.

2017: Math is Power, Not Punishment
2016: Beyond Relevance & Real World: Stronger Strategies for Student Engagement
2015: Fake-World Math: When Mathematical Modeling Goes Wrong and How to Get it Right
2014: Video Games & Making Math More Like Things Students Like
2013: Why Students Hate Word Problems

Watch the Four ShadowCon Talks from #NCTMAnnual and Sign up for the Follow-Up Conversation

Image of the ShadowCon Auditorium

On Thursday at #NCTMAnnual, four speakers urged teachers to reflect on their power not just to help students encounter mathematical knowledge but to change how students define themselves in relationship to math and to each other.

Video of each of their talks is online right now and each presenter invites you to join them in a follow-up conversation about their ideas over the next month. (More below.)

  • Lauren Lamb told us about her experience learning mathematics as a young woman of color, how she often felt invisible in her classrooms and unrepresented in her textbooks. She described the ways her teachers did and didn’t involve her in her own mathematics education.
  • Javier Garcia contrasted the ways we talk about students (as though they’re incomplete, fallible) and mathematics (as though it’s complete, infallible) and made a case that teachers should reverse those two descriptions.
  • Nanette Johnson impressed upon the audience the fact that each of them will leave behind a legacy for their students, an indelible imprinting of their efforts, either positive or negative.
  • Andrew Gael revealed the potency of our presumptions about student competence, and how students often live up and down to those presumptions. What we believe about student competence affects how we work with those students, which affects their opportunities to develop competence.

Great talks each one. Each one well worth your time.

But what happens to talks like these after they’re over? The ShadowCon Hypothesis is that the ideas from even great talks rarely survive contact with the reality of classroom instruction; that absent any kind of conversation or community organized around their implementation, those ideas are too easily put in a box labeled “Nice to Think About” or “Maybe Later.”

Each of our speakers agree with that hypothesis and each one wants to participate in a conversation with you over the next month. Sign up for a course you’d like to think and talk more about. We’ll place you on an email thread with a couple of random, interesting colleagues. Then you’ll receive a new discussion prompt once per week for the next four weeks, starting May 7. On a weekly basis, the speakers will summarize the most interesting ideas and answer the most perplexing questions from across all the groups.

It’s going to be a very interesting month.

[image via Cassie Sisemore]

Where You’ll Find Me at #NCTMAnnual

The icon on my airplane’s wifi signal indicates I’m somewhere over Wyoming right now, en route to Washington, D.C., for collaboration and conviviality with thousands of math teachers from all around the United States. I’m looking forward to reconnecting with old colleagues and meeting new ones so let me tell you where we’ll find each other. If we’ve met, let’s catch up. If we’re just meeting, let me know what you’re working on or wondering about.

Wednesday

Desmos Preconference Workshop

A picture of the Desmos preconference at TMC in 2017

My team will be running a morning workshop and an afternoon workshop on our newest, hottest technology and activities.

Also: Emdin’s opening session; NCTM Game Night.

Thursday

ShadowCon

ShadowCon will be at 6PM on Thursday in Ballroom B.

Zak Champagne, Mike Flynn, and I have recruited four interesting speakers – Lauren Lamb, Javier Garcia, Nanette Johnson, Andrew Gael – each offering their own variation on a similar theme. The presenters and the organizers collaborated on these ten-minute talks for the last several months. The process was a joy and the resulting talks are really exceptional. We’ll also introduce a new way to continue the call to action of those talks long after they have ended.

Desmos Happy Hour & Trivia

The Desmos Math Trivia Happy Hour is Thursday April 26 from 6:30-9:30PM at Clyde's of Gallery Place in the Piedmont Room 707 7th St NW, Washington DC.

Beep beep! Right after ShadowCon, I’m speed-walking straight to NCTM’s Top Rated Happy Hour Event. Then we’ll commence NCTM’s Highest Grade Trivia Competition. I can’t divulge any of the categories but if you were to brush up on your naughty words that rhyme with math vocabulary, I don’t think you’ll regret the effort.

Also: Sessions with Stiff, Rosen, Briars, Usiskin, Cirillo, Pelesko, and time at the Desmos booth, trying to convince people to buy our free calculators and free activities. (Pretend you don’t see me, friend. Pretend you’re on the phone with your mother. Pretend I didn’t invent those exhibitor-dodging moves. We are having this conversation, friend.)

Friday

Full Stack Lessons

I know my talk is at 8 AM. I know that. But I’ll be bringing coffee for at least me and one other person in the room. Maybe three more people if I can find one of those coffee carrier trays.

Here’s the description:

Two teachers can take the same idea for a lesson and experience vastly different results in class. This is often because one teacher taught from the full “stack” of questions and the other taught from just part of it. We’ll look at the contents of that stack and learn how to put the full stack of questions to work in your classes.

Also: Zager, Martin, United Airlines.

Remainders

Here are three items that crawled into my head some time during the last few months and didn’t find their way out yet. This post is brain surgery.

Lesson Exploder

I have thought about this tweet from David Coffey at least once per week for the last five months.

The Song Exploder podcast interviews artists about the craft of songwriting. The artists describe their motivations for creating their songs, what they were trying to accomplish, and how they tried to accomplish it, all while the Song Exploder team teases out key elements of the song for illustration. I feel smarter about the craft of songwriting whenever I listen to it. Maybe not as smart as if I had spent a year at the Oberlin Conservatory of Music, but for how much smarter I’m feeling, it’s hard to argue with Song Exploder’s cost (free) and scale (internet-sized).

Now swap “teacher” for “artist” and “lesson” for “song.” I know what we can swap in for “Oberlin Conservatory of Music.” Classroom visits. Lesson studies. Problem solving cycles. Professional learning communities. Those are all very effective and also very expensive. I don’t know what to swap in for “Song Exploder,” though – an option that is less effective but basically free and scales with the internet.

What kind of digital media could we use to a) highlight something significant and useful about the craft of teaching b) as quickly as possible c) distributed as widely as possible d) in a form that’s replicable and episodic? (Song Exploder is up to 133 episodes right now.)

What current examples can we find? Teaching Channel videos? Blog posts? Lesson plans? Unedited classroom video? Marilyn Burns distills classroom anecdotes into really popular tweets.

What inspiration can we take from other fields? Delish videos? NFL Red Zone / Mic’d Up? Mystery Science Theater 3000? Twitch streaming?

New Jersey Turnpike

I can’t figure out the tolls on the New Jersey Turnpike.

If you don’t come from turnpike territory, how it works is you enter the turnpike somewhere and you exit the turnpike somewhere else. You pay depending on where you entered and exited.

My assumption is that the pricing would look pretty linear as a function of the miles traveled. Like this:

But it doesn’t. It looks like two linear functions with the second piece starting maybe at the Garden State Parkway. (Why?) And the Pennsylvania Turnpike exit is also way more expensive than a linear function would predict. (Why again?)

Here is the website that tells you the cost of different trips on the turnpike. Eric Berger, our CTO at Desmos, helped me type code into my browser’s Javascript console that returned all the data. Feel free to dig in. I’m looking for answers to my questions about pricing and I’m also interested in possible classroom applications of the data.

Cape Town’s Zero Day

Cape Town has a water crisis and a website that until recently calculated a “Zero Day” for their water reserves, a day when faucets will run dry and people will collect a daily allotment of water from central locations throughout the city.

That’s either terrifying or mathematically interesting, depending on which part of my brain I subdue while I’m thinking about it. How do they calculate that zero day? How can we put students in a position to appreciate, replicate, and even adapt those calculations for their own contexts?