Mathematical Surprise

I gave a talk at the Wisconsin state math conference earlier this month and this woman was the best part.

I don’t know her name. I’ll call her Jan. Jan is about to testify to the power of surprise.

I asked the crowd to give me three numbers between 1 and 6, numbers you might get from a roll of the dice. They said 2, 3, and 5. Then I asked all of them to evaluate those numbers in this expression.

Most of the crowd started working on that task, but Jan didn’t. She laughed and said, “I teach second grade,” excusing herself.

I encouraged her to show off whatever she remembered from the last time she worked with expressions like this. She scribbled on the notebook in her lap and we managed to evaluate x = 2 in the time we had, but not 3 or 5.

I asked the crowd to call out the result for 2, 3, and 5. They called out 2, 6, and 20, one after the other.

Then I asked the crowd to evaluate those same three numbers in this expression.

Jan tossed her notepad on the desk, a reaction of “no way, no thank you” to the length of that expression. I decided not to press her at that exact moment, because I had a secret everyone in the crowd would come to understand at different times, Jan last of all and perhaps best of all.

I asked for their result for 2.


“Okay, what about 3?”


“Okay, that’s weird. What about 5?”


I played up my surprise, acting like I didn’t know all of those terms would simplify to 0.

That’s when I noticed Jan. Out of the corner of my eye, Jan straighted up in her chair and then picked up her notebook to sort out what just happened.

I wish I had a sharper vocabulary to describe this transformation, as well as more strategies for provoking it. By showing Jan a situation where order arose from apparent disorder, she felt something in the neighborhood of … cognitive conflict? Intellectual need? “Surprise” feels closest.

I don’t know all the words and I don’t know all the strategies, but I know there are few gifts a teacher can give a student more satisfying than helping her transform from “no way, no thank you” to “okay, let’s sort this out.”


  • I don’t think this experience has much to do with Jan’s growth mindset about herself, or mine about her, but I’m willing to be proven wrong. How was this experience distinct (or similar) to a mindset experience?
  • Think about the design of this activity, all of its different permutations, and how each one might have affected Jan. What if, for instance, I had given given the class those three numbers instead of soliciting them from the class? What if I had only solicited one number? What if all three numbers didn’t evaluate to the same number? How would these permutations have affected Jan’s interest in picking up her notebook?

Desmos Now Embedded in Year-End Assessments Across the United States

Amy X. Wang, writing in Quartz:

Enter Desmos, a San Francisco-based company that offers a free online version of TI’s graphing calculator. Users across 146 countries, most of them teachers or students, are currently logging 300,000 hours a day on the platform—and today, Desmos announced a major partnership with testing consortium Smarter Balanced, which administers academic exams in 15 US states. Beginning this spring, students in those areas will use the online tool in math classrooms and on statewide performance assessments.

When students take their year-end assessment in 15 states, they’ll see the same free calculator they’ve been using at school and at home the rest of the year. That assessment will more closely reflect what they know, rather than what they were able to express through unfamiliar or costly technology.

USA Today has the reaction quote from Texas Instruments president Peter Balyta:

Peter Balyta, president of TI Education Technology and a former math teacher, defended the purchases, saying a TI calculator “is a one-time investment in a student’s future, taking students through math and science classes from middle school through high school and into college and career.” He said TI’s technology is evolving, but that models like the TI-84 Plus come with “only the features that students need in the classroom, without the many distractions that come with smartphones, tablets and internet access.”

This is interesting. In a world where more and more assessments are delivered digitally (and pre-loaded with digital calculators) the sales pitch for hardware calculators is their lack of features, rather than their abundance.

There is clearly a market today for a calculator that lacks internet access. Around 20% of teachers in my survey said they wouldn’t let students use mobile devices on exams for reasons of “test security” and another 10% cited “distraction.”

Open, interesting questions:

  • Are those figures trending upwards or downwards?
  • Will schools and parents continue to pay Texas Instruments an estimated 50% profit margin for more test security and fewer distractions?
  • How do math coaches and instructional technologists help teachers harness the advantages of the internet while also managing concerns about security and distraction?

2017 May 12. Peter Balyta makes a longer case for hardware calculators, one which won’t surprise anyone who has followed this discussion. He mentions (1) lower cost, (2) fewer distractions, (3) greater test security, (4) more features, and (5) availability on tests.

RIP Malcolm Swan

In trying to explain to family and friends what Malcolm Swan meant to the field of math education, I’ve been putting him in the same category as Michael Jordan – talents that come along once in a generation in disciplines that are as much art as science. In Swan’s case, he designed experiences that endeared students to mathematics, and endeared teachers to students, more effectively than anyone I know. You can pick up his The Language of Functions and Graphs, now thirty years old, and wonder, “What have we been doing all this time?” Swan drew math out of the world and thought out of our students in ways that feel challenging and new even today.

Malcolm was uncommonly humble and generous for someone of his talent. He was willing to spend time and trade ideas with me long before I had anybody’s name to drop, or any name of my own. He was also uncommonly dedicated to the field of math education, writing articles, giving talks, and hosting workshops, and all throughout you knew he believed completely that you too can do what I do, that math education isn’t art or science so much as it’s design. And he believed that design could be taught and learned.

That’s why I’m sad for everyone who knew Malcolm personally, for his family and his colleagues at the Shell Centre, but I’m not as sad for our profession as I thought I would be. Malcolm’s talent was generational and unique, but he did more than any of us could have hoped to explain it. Over his career, he added to our profession in permanent ways far more than his death now subtracts. I know we will still be learning from Malcolm for decades. And throughout those decades, the best day of my week will be any day I get to introduce a new teacher to his work, and pass along his conviction that “you too can do what he did.”

Every Handout from NCTM 2017

tl;dr –

  • There were 740 total sessions at NCTM’s 2017 annual conference. I wrote a script to find and extract the 279 sessions that included handouts, slides, or other attachments. You can also download the entire mess of attachments with a single click. (1.28 GB. Not small.)
  • The process of writing and running that script maps almost perfectly onto the process of mathematical modeling. If the same defective wiring runs through your brain as mine, you’ll understand how that was a total rush.
  • I hope NCTM will make these resources easier to find in the future, especially for non-members.

This is just like mathematical modeling!

I’d been using the same script for this task for the last two years, but NCTM switched website vendors this year and I had to create a new one. On the one hand, accessing handouts from the conference probably shouldn’t be so challenging. On the other, this process is such a fun puzzle for me, and maps almost perfectly onto the process of mathematical modeling.

Here is what I mean. The third step in the modeling cycle is to “perform operations.” I’m not here to tell you that people (old or young) should never perform operations, just that computers are generally much faster at it. When I thought about the task of poking my head into all 740 NCTM session websites and asking, “Hello. Any handouts in here?” I admitted defeat immediately to some software.

So the human’s job is the first two steps: identifying essential variables and describing the relationships between them. Computers are much, much worse at this than humans. That meant looking at cryptic computer chicken-scratch like this and asking, “What do I need to tell the computer so it knows where to look for the session handouts?

If you notice that each session has its own four-digit “id” and that each handout has been tagged with “viewDocument,” you can tell the fast machine where to look.

But the modeling cycle doesn’t end there. Just like you shouldn’t paste the results from your calculator to your answer sheet without thinking about it, you shouldn’t paste the results of the fast machine’s search to your blog without thinking about it. You have to “validate the results,” which in my case meant poking around in different sessions, making sure I hadn’t missed anything, and then revising steps one and two when I realized I had.

I hope NCTM will make these easier to find.

These handouts are basically advertisements for the conference without substituting for attendance. (No prospective attendee will say, “I was this close to attending but then I found out some of the handouts would be online.”) If they’re easier to find, not only will existing attendees be happier but non-attendees will have a nice preview of the intellectual activity they can expect at the conferences, making them more inclined to attend the next year. Nothing but upside!


Presentation Advice from 14 of My Favorite Presenters

As I prepared for a presentation on presentations, I asked fourteen of my favorite math education presenters, “What is your best advice for preparing and delivering a presentation?” Here is their advice, edited for brevity and sequenced for story.

Michael Serra, author of Discovering Geometry:

Watch others and take note of what you want to see and do not want to see in your presentation.

Matt Larson, President of NCTM:

Whose style do you like and want to emulate? You can’t be someone else but what elements from other presenters can you make your own and integrate into your own style.

Patrick Callahan, Callahan Consulting:

I prepare the talk that I would like to see. I like to be surprised and challenged. I don’t like things tied up in a neat bow. I like talks that I have to keep thinking about.

Jo Boaler, author of Mathematical Mindsets:

Relax! Find some space to think. We now know that there are different modes of thinking and that speed and pressure block creative and innovative thoughts. I always like to plan talks when I feel like I have some good time to think expansively and creatively. I don’t do it in in-between moments, I do it when I have a good block of time and I feel as though I can think deeply and well.

Preparing a talk is like writing a book or a paper – think through the key ideas. How do they flow to and from each other? Don’t overwhelm your audience with too many ideas. Keep them all connected.

Cathy Yenca, Apple Distinguished Educator and awesome edtech blogger:

Ideas come at weird times – intentionally jot down every idea as it comes to you. I tend to have Post-its on my nightstand, in my kitchen, on my teacher desk, on my laptop (yes, on the flat spots to the left and right of my trackpad). I transfer these to a Google Doc, giving myself permission to brainstorm freely. Not every Post-it will make it to the final talk. That’s okay.

Matt Larson:

A good presentation is like a good classroom lesson – it is based on extensive planning and preparation. I have been working on the “versions” of my annual talk since last August. I have literally spent hundreds of hours doing research, putting together hundreds of “potential” slides, double checking citations, messing with the order, eliminating slides, etc.

Tracy Zager, author of Becoming the Math Teacher You Wish You’d Had:

Story first, slides later. The most important thing is to plan the storyline of the talk first. Weekend Language is great on this, as is the one-minute video you made a few years ago. Both helped me break the habit of how we were all taught to start preparing your presentation by picking a theme for slide one in PowerPoint. Instead, I start with my big idea. Every talk needs to have a central big idea, and I spend a lot of time clarifying that idea before I get anywhere close to Keynote.

Elham Kazemi, author of Intentional Talk:

Don’t just build a talk by cutting and pasting slides together. Build the experience and then the slides.

Fawn Nguyen, middle school math teacher and awesome blogger:

A good image is better than text. For example, I have a picture of Robert Kaplinsky hiding his face under his [shameful] UCLA sweatshirt. That picture is more powerful than a slide that reads “The Bruins are getting pummeled on their own turf by the Oregon Ducks.”

Matt Larson:

I try to find a place to “test drive” the presentation, perhaps at a smaller conference. In some cases, I’ve simply given the presentation to 3-5 people in a small room.

Jose Vilson, author of This Is Not A Test: A New Narrative on Race, Class, and Education:

Preparation is really similar to the classroom. Eat breakfast. Go through normal routines. Do a few vocal exercises to get your vocals right. Make sure you’re just full enough and that you haven’t had too much to drink.

Tracy Zager:

Create contingency plans. What if the wifi doesn’t work? Be prepared for that. It’s frustratingly common. If you’re playing videos and they’re embedded in your slides, also open the files themselves and minimize them. Sometimes they don’t play nicely and you need to find the originals quickly. (I learned that one the hard way.) What if you’re running long? What will you cut? What if you’re running short? Where can you go deeper? What if the mic sucks and you’re tethered to the lectern area by a wire? That’s my least favorite scenario. If the room is small enough in that situation, I toss the mic in a heartbeat, use my teacher voice, and get out in the crowd.

Elham Kazemi:

Arrive early and greet people that took the time to come to your talk. learn as much as you can about whose in the room in the 10-15 minutes before the talk begins. Use that knowledge during the talk.

Barb Dougherty & Karen Karp, authors, editors, and co-presenters:

We attempt to make a personal connection with participants by going through the room and meeting people prior to the presentation.

Cathy Humphreys, author of Making Number Talks Matter:

Most of us get nervous before a large talk, and once early on when I was in that state a mentor said to me something like, “This talk is not about you. You aren’t the point. It’s the ideas you are sharing that matter.”

Patrick Callahan:

Delivery is about reading the audience and being flexible. You can tell when the room is with you and when it isn’t. Based on that, it’s important to be flexible: skip slides, slow down, speed up, give time for folks to turn and talk, ask questions.

Michael Serra:

Be enthusiastic. Your audience needs to feel that love you have for what you are presenting. Know your stuff but don’t be a know it all.

Jo Boaler:

I never like to talk for more than 15 minutes without asking people to watch a video, do a maths task, or discuss with each other.

Michael Serra:

Your first presentation will probably not be to hundreds of listeners but to a small group of participants. So make sure they are participating and not just listening.

Jose Vilson:

If you’re doing PowerPoint, you’re doing something more image-based, but the images should remind you of what you’re saying to your audience.

Marilyn Burns, founder Math Solutions:

I use PowerPoint slides as guides for my session, to trigger my thoughts and keep me on course. I use the notes section to remind myself about what I was planning to say. I try to avoid presenting lots of information to be read, and I work hard to avoid ever reading what’s on a slide. If there’s something for people to read, I stop talking and give them time.

Barb Dougherty & Karen Karp:

Tell stories, including those that show your own foibles along the way. For example, with our 13 Rules That Expire, we share that we know these rules because at one time we taught them!

Jo Boaler:

Try and tell a connected story, talk about the personal connections to you.

Steve Leinwand, author of Accessible Mathematics:

A good talk or presentation is like a good lesson. You start with classroom action and later summarize the purpose of those tasks or activities. In other words, I think that a good presentation models good instruction

Tracy Zager:

If possible, do just what you do in a classroom, where you listen in on conversations and then ask those people to share out in the larger crowd when you come back together. Those organic moments are to be treasured.

Fawn Nguyen:

Go along with an unintended diversion.

Matt Larson:

Think about how you can be “inspiring” or have a “call to action” at the end. I’ve attended this presentation, now what?

Fawn Nguyen:

Is there a take-away message or a call-to-action from your talk?

Great, right? Test all of this and keep what’s good for you. It’s time to propose your session. Everyone: your NCTM proposals are due May 1. Californians: your proposals (North, South) are due the same date. Share what your ideas and questions.