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Designing for Mathematical Surprise

“Surprising” probably isn’t in the top ten list of adjectives students would use to describe math class, which is too bad since surprise lends itself to learning.

Surprise occurs when the world reveals itself as more orderly or disorderly than we expected. When we’re surprised, we relax assumptions about the world we previously held tightly. When we’re surprised, we’re interested in resolving the difference between our expectations and reality.

In short, when we’re surprised we’re ready to learn.

We can design for surprise too, increasing the likelihood students experience that readiness for learning. But the Intermediate Value Theorem does not, at first glance, look like a likely site for mathematical surprise. I mean read it:

If a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

[I slam several nails through the door and the floor so you’re stuck here with me for a second.]

Nitsa Movshovits-Hadar argues in a fantastic essay that “every mathematics theorem is surprising.” She continues, “If the claim stated in the theorem were trivial it would be of no interest to establish it.”

What surprised Cauchy so much that he figured he should take a minute to write the Intermediate Value Theorem down? How can we excavate that moment of surprise from the antiseptic language of the theorem? Check out our activity and watch how it takes that formal mathematical language and converts it to a moment of surprise.

We ask students, which of these circles must cross the horizontal axis? Which of them might cross the horizontal axis? Which of them must not cross the horizontal axis?

They formulate and defend their conjectures and then we invite them to inspect the graph.

In the next round, we throw them their first surprise: functions are fickle. Do not trust them.

And then finally we throw them the surprise that led Cauchy to establish the theorem:

But you can’t expect me to spoil it. Check it out, and then let us know in the comments how you’ve integrated surprise into your own classrooms.

Related: Recipes for Surprising Mathematics

Rough-Draft Talk in Front of Hundreds of Math Teachers

This was new. I was on a raised platform with seven middle school students to my left and six to my right and several hundred math teachers surrounding us on all sides.

This wasn’t a dream. The MidSchoolMath conference organizers had proposed the idea months ago. “Why don’t you do some actual teaching instead of just talking about teaching?” basically. They’d find the kids. I was game.

But what kind of math should we do together? I needed math with two properties:

  • The math should involve the real world in some way, by request of the organizers.
  • The math should ask students to think at different levels of formality, in concrete and abstract ways. Because these students would be working in front of hundreds of math teachers, I wanted to increase the likelihood they’d all find a comfortable access point somewhere in the math.

So we worked through a Graphing Stories vignette. We watched Adam Poetzel climb a playground structure and slide down it.

I asked the students to tell each other, and then me, some quantities in the video that were changing and some that were unchanging. I asked them to describe in words Adam’s height above the ground over time. Then I asked them to trace that relationship with their finger in the air. Only then did I ask them to graph it.

I asked the students to “take a couple of minutes and create a first draft.” The rest of this post is about that teaching move.

I want to report that asking students for a “first draft” had a number of really positive effects on me, and I think on us.

First, for me, I became less evaluative. I wasn’t looking for a correct graph. That isn’t the point of a rough draft. I was trying to interpret the sense students were making of the situation at an early stage.

Second, I wasn’t worried about finding a really precise graph so we (meaning the class, the audience, and I) could feel successful. I wanted to find a really interesting graph so we could enjoy a conversation about mathematics. I could feel a lot of my usual preoccupations melt away.

After a few minutes, I asked a pair of students if I could share their graph with everybody. I’m hesitant to speculate about students I don’t know, but my guess is that they were more willing to share their work because we had explicitly labeled it “a first draft.”

I asked other students to tell that pair “three aspects of their graph that you appreciate” and later to offer them “three questions or three pieces of advice for their next draft.”

  • I like how they show he took longer to go up than come down.
  • I like how they show he reached the bottom of the slide a little before the video ended.
  • I think they should show that he sped up on the slide.
  • Etc.

If you’ve ever participated in a writing workshop, you know that workshopping one author’s rough draft benefits everyone’s rough draft. We offered advice to two students, but every student had the opportunity to make use of that advice as well.

And then I gave everybody time for a second and final draft. Our pair of students produced this:

Notice here that correctness is a continuous variable, not a discrete one. It wasn’t as though some students had correct graphs and others had incorrect ones. (A discrete variable.) Rather, our goal was to become more correct, which is to say more observant and more precise through our drafting. (A continuous variable.)

And then the question hit me pretty hard:

Why should I limit “rough-draft talk” (as Amanda Jansen calls it – paywalled article; free video) to experiences where students are learning in front of hundreds of math teachers?

My students were likely anxious doing math in front of that audience. Naming their work a first draft, and then a second draft, seemed to ease that anxiety. But students feel anxious in math class all the time! That’s reason enough to find ways to explicitly name student work a rough draft.

That question now cascades onto my curriculum and my instruction.

How should I transform my instruction to see the benefits of “rough-draft talk”?

If I ask for a first draft but don’t make time for a second draft, students will know I really wanted a final draft.

If I ask for a first draft, I need to make sure I’m looking for work that is interesting, that will advance all of our work, rather than work that is formally correct.

How should I transform my curriculum to see the benefits of “rough-draft talk”?

“Create a first draft!” isn’t some kind of spell I can cast over just any kind of mathematical work and see student anxiety diminish and find students workshopping their thinking in productive ways.

Summative exams? Exercises? Problems with a single, correct numerical answer? I don’t think so.

What kind of mathematical work lends itself to creating and sharing rough drafts? My reflex answer is, “Well, it’s gotta be rich, low-floor-high-ceiling tasks,” the sprawling kind of experience you have time for only once every few weeks. However I suspect it’s possible to convert much more concise classroom experiences into opportunities for rough-draft talk.

To fully wrestle my question to the ground, how would you convert each of these questions to an opportunity for rough-draft talk, to a situation where you could plausibly say, “take a couple of minutes for a first draft,” then center a conversation on one of those drafts, then use that conversation to advance all of our drafts.

I think the questions each have to change.




[photo by Devin Rossiter]

Lonely Math Teachers

Check out this lonely math teacher on Twitter:

Taylor registered her Twitter account this month. She’s brand new. She’s posted this one tweet alone. In this tweet, she’s basically tapping the Math Teacher Twitter microphone asking, “Is this thing on?” and so far the answer is “Nope.” She’s lonely. That’s bad for her and bad for us.

It’s bad for her because we could be great for her. For the right teacher, Twitter is the best ambient, low-intensity professional development and community you’ll find. Maybe Twitter isn’t as good for development or community as a high-intensity, three-year program located at your school site. But if you want to get your brain spinning on an interesting problem of practice in the amount of time it takes you to tap an app, Twitter is the only game in town. And Taylor is missing out on it.

It’s bad for us because she could be great for us. Our online communities on Twitter are as susceptible to groupthink as any other. No one knows how many interesting ways Taylor could challenge and provoke us, how many interesting ideas she has for teaching place value. We would have lost some of your favorite math teachers on Twitter if they hadn’t pushed through lengthy stretches of loneliness. Presumably, others didn’t persevere.

So we’d love to see fewer lonely math teachers on Twitter, for our sake and for theirs.

Last year, Matt Stoodle Baker invited people to volunteer every day of the month to check the #mtbos hashtag (one route into this community) and make sure people weren’t lonely there. Great idea. I’m signed up for the 13th day of every month, but ideally, we could distribute the work across more people and across time. Ideally, we could easily distinguish the lonely math teachers from the ones who already experience community and development on Twitter, and welcome them.

I’m not the first person to want this.

So here is a website I spent a little time designing that can help you identify and welcome lonely math teachers on Twitter:

It does three things:

  • It searches several math teaching hashtags for tweets that a) haven’t yet received any replies, b) aren’t replies themselves, and c) aren’t retweets. Those people are lonely! Reply to them!
  • It puts an icon next to teachers who have fewer than 100 tweets or who registered their account in the last month. These people are especially lonely.
  • It creates a weekly tally of the five “best” welcomers on Math Teacher Twitter, where “best” is defined kind of murkily.

That’s it! As with everything else I’m up to in my life, I have no idea if this idea will work. But I love this place and the idea was actually going to bore a hole right out of my dang head if I didn’t do something with it.

BTW. Thanks to Sam Shah, Grace Chen, Matt Stoodle, and Jackie Stone for test driving the page and offering their feedback. Thanks to Denis Lantsman for help with the code.


18 Jan 22:

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]

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.