Intuition, nudges, and other thoughts after Asilomar

I got to make my first trip to Asilomar this weekend. The first of many, I hope. I found CMC-N to be just as warm and retreat-like as people have described. It’s a special conference in a beautiful place with great people. What more can you ask for?

I launched a new talk yesterday, and I get a kick of seeing which ideas or images or phrases resonate or cause reactions, and which fall flat. There are always a few surprises, which is part of the fun of this work.

The talk was about mathematical intuition, and it’s a big sprawly topic I love a whole lot. I wrote a long chapter about it and excerpted a few sections of it for this talk. Here are the slides and handouts part 1 part 2 if you’re interested in the larger arguments and context for this post. I’m going to focus in on one aspect here.

In a little section of the talk, I gave what I hoped would be a gentle but provocative nudge to my colleagues. Based on the response I heard in person yesterday and online, I think I succeeded somewhat in that effort, but I think I need to add a little clarity. This morning, at 3:30AM, as I lay in my hotel room wide awake and jet-lagged, I found myself unable to remember which of the many possible ways I’d planned to say things I’d actually said. (Does that happen to other people besides me? Practice and reality blur together for me in the aftermath, especially the first time I give a talk.) So I thought I’d take to the blog here and lay down what I meant to say in pixels. While I’m at it I’ll take a deeper dive because, well, I’m spending the day stuck on planes on snowy tarmacs and in O’Hare and it’s a good time to make sense through writing. Get comfy.

Over the past several years, I have watched and listened and learned a ton from my colleagues who have been thinking hard about engagement and sense-making and problem-solving. I’m thinking about Phil Daro teaching us all to delay answer-getting. I’m thinking of my friends Max and Annie at the Math Forum, who have taught me so much about slowing down in order to emphasize sense-making when we launch problems. I’m thinking about the idea of low-floor tasks, of removing barriers to access, of the brilliant desmos articulation of making problems easy to start but difficult to finish.

And I’m thinking mostly about Dan Meyer‘s years of work on co-developing the question with students. Nobody has pushed my thinking about this topic more. If you haven’t dug into his work yet, start with these essentials:

Tangent. Bear with me. Last week, I was thinking about Dan’s influence on me and the teachers I support. I met with the 4th/5th grade teachers who had developed a lovely lesson around the open, clear-yet-ambiguous, informal question, “How big is New Hampshire?”

They started with an image like this.


The kids had a great conversation about how big New Hampshire is, and how they might measure it. They dug into the difference between the perimeter and area. They had suggestions for ways to measure the size of New Hampshire using a map that involved string or bendy rulers or graph paper or cubes. Their conversation was gathering steam when one student said, “We could take a string and measure all the way around New Hampshire by putting the string on the line. Then we could take the string off New Hampshire and measure the string with a ruler, and that’s how big New Hampshire’s perimeter is.”


Becky Wright was leading this lesson, and she did a masterful job questioning this student until the student realized that, while she had dome some great thinking, something about her plan didn’t make sense. The length of that string wasn’t big enough to be the length of New Hampshire’s perimeter. I could see light bulbs going off over kids’ heads, one by one, as they realized they needed to know something they don’t yet know about the relationship between the length of the lines on the paper and the distances on earth. A student raised his hand and said something like, “We need one of those maps that has a line that tells you how long the lines really are. So, like, if an inch on the map is 20 miles or something.” Becky revealed the google map she’d queued up for them, and they said, “Yeah! That’s what we need!”


It was a lovely, student-generated lead-in to the huge concepts of scale and proportional reasoning. I stood in Becky’s room, leaning against the wall, thinking about the presentation I saw Dan give in Boston in 2015 in which he talked about the ways technology makes what Becky did possible. Check out this visual that Dan was kind enough to send over. It’s stayed with me.


Because Becky’s maps weren’t constrained by the cost and weight and permanence of textbooks, she could space them out and control when they appeared. Rather than present the entire problem typed up and printed in one big jumble of overstimulating text and images and given information, she could start with the cleanest image and build the question in layers with her kids. She could hold back the map with a scale until the kids had time to figure out what information mattered, and then ask her for it.

Aside from the great mathematical modeling work that was happening here, I was busy noticing how the kids were working. As Dan has argued, starting with informality before formality, sketches before measurements, and so on, has a tangible impact on the way the class participates. The kids started out sleepy, but as the discussion progressed, more and more students joined in, rather than the more typical pattern of losing kids the longer the discussion goes. Engagement went up. Access was great. Students felt ownership. Everyone could contribute. Nobody was boxed out or shut down or denied meaningful mathematics because of an excessively formal problem. All the kids in the class understood the question and had started making sense.

I smiled, ducked out, and walked down the hall to Kitri‘s classroom, where kids were a day or so ahead. They were deep in the work of calculating the area of New Hampshire. A few examples:

imag0711      imag0710

imag0713  imag0712

So fascinating. Kitri had given students no constraints on which tools they could use. She was looking forward to a discussion about which of these tools made life easier by yielding a decent estimate of the area of New Hampshire without a ton of agonizing calculations. What would the students who had chosen centimeter cubes think when they tried to work with the scale based on inches?

This piece of work stood out to me most of all, though.


There’s so much good thinking here, right? The decomposition, the space, the thinking about arrays, the care not to double-count. There’s tons to build on. But what is happening with those numbers? If we put aside the assumption this student made that the graph paper dimensions matched the scale of the map (they didn’t) and focus on what he did here, what’s going on?

I asked him to tell me about his work. He showed me how he counted up the arrays, and how he was now working on putting together all the fractional bits. I asked him, “What does that 20 inside the rectangle mean?”

He said, “There are 20 squares there because it’s 4 x 5.”

I asked, “But what does that 20 mean in New Hampshire?”

He was stumped. “Twenty inches? Twenty miles? No, twenty meters. I think?”

This is the kind of moment that gives me pause. That led me to make the picture I shared at CMC.


Kitri and Becky both started the kids out in this informal, intuitive, sense-making way. In Kitri’s class, before they got the cubes out, they ballparked the numbers some, so they’d dabbled in magnitude. But somehow, once the cubes and graph paper and numbers came out and this student was in the midst of the problem, he got calculating. He got doing. He got executing. And he stopped asking himself, “Does this make sense?”

What I attempted to argue at CMC is that this powerful work to engage students through intuition at the beginning of a problem is absolutely necessary, but not sufficient. We need to keep going with it, to build off it, and to develop our strategies for teaching students how to stay in touch with their intuitions throughout the problem, even in the midst of calculations. I want students to work a little, then check back in with their guts. To think, “What’s going on here? Does this make sense? How does this jive with where I thought this was going?” I want them to lift themselves out of the details periodically and think, “Where am I?” To jump from street view to birds-eye view and get re-oriented. If they’re where they thought they should be, they can get back to work. If they realize they’re lost, they need to figure out what’s going on, readjust, and recalibrate before diving down into the details again. They need to take time to refine their techniques based on their experiences. These little cycles of do a little work, gut check, reflect, do a little work, gut check, reflect, are what mathematicians do, and what I was representing with the blue dream line.


How to teach students to do this is the kind of question that jazzes me up. I offered a few practical strategies yesterday, and there are more in my book. I am definitely about the how, not just theory, and I hope the classroom teachers and coaches in the crowd found useful ideas yesterday. They’re my first priority, always.

My subterranean agenda yesterday, though, was to suggest to my colleagues who think about and work on these ideas at scale that they increase their focus on supporting intuitive thinking during the problem. I wanted to nudge it toward the fronts of their minds when they design tasks and think about pedagogy and teach pre-service teachers and run professional development, so we can build more ideas and routines and instructional techniques that encourage the synergistic relationship between intuition and logic. For example:

  • In a 3-Act task (a brilliant structure from Dan), how do we build on the beautiful, intuitive thinking in Act 1 when we move into the doing of Act 2? What can the teacher do? What’s part of the task design? What’s the pedagogy here?
  • When we activate sense-making and delay answer-getting by withholding some information (the question, the numbers, the given information, the details), how do we keep the students making sense even after they have that information?
  • If we “dial down the math” at the beginning by spending time in the informal world of sketches and estimates and descriptions, how do we maintain and build on that intuitive thinking even as we dial up the formal and abstract and symbolic?
  • If we’re working with beautiful technology that makes it easy to offer iterative feedback to kids, (cough, desmos, cough), how do we use that technology to its fullest potential here? I want my des-friends pondering how they can flex their tech muscles to promote intuition all the way through problems, even after the variables and gridlines and labels come out. Iterative feedback is essential to developing intuition, so they’ve got a big advantage right from the start. Let’s make it count.

I think these questions are worth thinking about. It seems a shame to do all that work to get kids engaged and invested and making sense in act 1 of a problem, but let that intuitive thinking fall off when we start figuring in act 2. I’m in no way criticizing the intuitive work at the beginning of the problem. I’m saying “Yes to all of that, and…”

Dan teased me relentlessly for most of yesterday afternoon and evening, saying my little nudge to him (and others!) was a knife to the ribs. But I also know that he and I agree that this exchange of ideas–this learning from and reacting to and building on each other’s work–is why we have professional guilds. It’s why I crossed the country all day Friday and came back again all day Sunday. It’s why math teachers read and listen and watch and talk and go to sessions. It’s why we travel to conferences and present and blog and tweet and write books and publish articles and teach courses and do podcasts. We listen to other people’s ideas, and we put our own ideas out into the world. We take serious ideas seriously and think about them and react to them. We tell each other when we’re off-base, redirect each other when we’re close, and encourage each other to keep going when we’re on to something good. All this is to say, if you’re not putting your ideas out there yet, you’re missing out on some pretty great feedback.

In our post-session conversation yesterday, Dan asked questions about my talk that I’ve never thought of. His perspective as a designer of tasks and problems and routines is different than my perspective as a coach and author. He pressed and made me think and I appreciated it.

Dan’s work on co-developing the question with students is some of the most provocative and best work I’ve seen. It has led to long and deep thinking in me, and substantial changes in my practice and the practice of my collaborating teachers. That’s the second-highest compliment I can give him.

What’s the highest, you ask?

A nudge to keep going.


I had a great time at the California Math Council (South) conference in Palm Springs a few weeks ago. I went to lots of super talks, but the two that have stayed with me the most were Ruth Parker’s and Megan Franke’s. I’m still mulling over both, and want to start by posting about Megan’s talk because it has made an immediate impact on my practice.

Megan discussed the relationship between children’s counting and children’s problem solving. She made a compelling argument against viewing them sequentially, or thinking that one is a prerequisite to the other, and instead talked about how they can develop in an intertwined, mutually reinforcing way. She argued that children can use what they know about counting to think about problem solving. And she argued strongly that children’s partial understandings about counting are incredibly valuable. Amen to that, sister. The CGI researchers have been leaders in focusing us on what students do know, rather than taking a deficit approach, and it was so gratifying to hear Megan make this argument in forceful terms, in person. I’m a total fan.

Megan showed several videos of students counting collections of objects (rocks, teddy bears, etc.) with partial understanding. For example:

  • perhaps they organized what they counted and had one-to-one correspondence, but didn’t have cardinality, so they didn’t know the last number they said represented the quantity of the group. Or,
  • perhaps they didn’t have one-to-one correspondence and didn’t count accurately, but knew their last number signified the total. Or,
  • they had a lot of things going for them, but didn’t know the number sequence in the tweens (because they’re a nightmare and make no sense). And so on.

Megan showed a video in which a girl was counting a group of teddy bears (I think 15). She did pretty well in the lower numbers, but got lost as the numbers got bigger. The questioner then asked the student, “What if all the green bears walked away? How many bears would be left?” The girl giggled at the thought, collected up all the green bears, shoved them across the table, and counted the remaining 9 bears accurately.

My jaw dropped.

Megan made a powerful case that we can springboard off counting collections into problem solving, even if the counting is partial. She argued that students are already invested and engaged in the collection, so we might as well convert the opportunity. Some of her reasons:


I sat at my table thinking about how much work I’ve asked teachers to do. (If you don’t know what the counting collections routine looks like, take a peek at Stephanie’s kindergarten in this video:)

Teachers have gathered all these little objects, bagged them up, collected muffin tins and cups and plates, created representation sheets, taught the routine of counting collections. And yet, after the kids count and represent their collection, we just clean up.

It’s like we’re leaving players on base at the end of an inning. We’ve done all that work to get the hits and load the bases, but then we don’t bring them home. We don’t make full use of the opportunity we have designed. It was suddenly all so plain.

I tweeted about this idea of bouncing right from counting collections into problem solving, and my friend and colleague Debbie Nichols got the idea right away. She didn’t even wait for a counting collection. She started springboarding off an image-based number talk.

I visited yesterday, and Debbie’s K-1 kids were counting these cupcakes. They had all sorts of beautiful ways to count them.


  • “I see two sixes, one on each side.”
  • “I see four in each row, and there are three rows.”
  • “I counted by ones. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.”
  • “I counted by twos. 2, 4, 6, 8 10, 12.”
  • “I counted by fours. 4, 8, 12.”

Counting by twos, of course.

Students stood up and marked up the image on the smartboard, showing how they counted. They played with strategies to keep track and record. It was all great.

This is normally where we would have stopped, proud of ourselves for a worthy counting exploration. Not this time.

Debbie asked, “If we were going to write a story problem about this picture, what might we write?”

Arms went up right away. A student suggested, “Somebody had 12 cupcakes and took away 10. How many are left?”

Another student said, “Six cupcakes have raspberries. Two have rulers. Two have apples. Two have papers. How many are there altogether?”

Another student said, “I have 12 cupcakes. Evelyn gave me 10 more. Now I have 22.”

Debbie asked students to go get their notebooks and write a story. Solving it was optional.

Remember, this was a K-1. So, some kids who are very young K’s drew a picture and then talked about cupcakes:


Ned has one cupcake. How many Neds have cupcakes?

Most kids were able to write a story problem and read it to us. And most of those kids wanted to solve it and were able to do so successfully:


Davin had 12 cupcakes. Arik brought 12 more. How many does Davin have now?

One precocious student made the context a little silly so he could work with the kinds of numbers he likes. He explained to me how he found the answer of 450 cupcakes: “1000-500 would be 500, so another 50 more would leave 450”:


David had 1000 cupcakes. Aric took 550. How many does David have now?

But my favorite conversation grew out of this piece of work:


Bella had 5 cupcakes. Morgan gave her 35 more. Now how many does Bella have?

I asked her how she’d figure out how many cupcakes she had now. She said, “I’d think 5 in my mind, and then come over to these cupcakes and start counting 6, 7, 8, 9…” A few minutes later, she told me, “It’s 40!”

After admiring her careful work, I said, “If I were going to count this, I would have thought 35 in my mind, and then come over to these cupcakes and counted 36, 37, 38, 39, 40.”

Now it was her turn to drop her jaw. She was so excited that we both found the same answer. I asked her if I could do that–can I switch the numbers around like that? I could literally see the gears turning and the wheels spinning. She is off to the races, starting to abstract and decontextualize and play with numbers.

I especially loved Debbie’s work because kids got to pose problems. We are way, way too stingy with opportunities for kids to pose problems. Most kids think math class is a place where the teacher asks questions or the book asks questions, and kids answer them. If we want students to understand that math is a way to ask and answer your own questions, we need to give them some chances to do the asking.

I hasten to add, during morning meeting, these same children were asking questions about infinity. “What’s half of infinity?” “What’s the biggest number?” “Is infinity a number?” Students who have their thinking honored–who are used to generating questions in math–will ask specific math-problem-type questions, but also large, important, relational, lofty questions. Problem-posing and question-asking in mathematics has a wide range. Kids need practice asking at all the different grain sizes.

Circling back to springboarding, today’s lesson drove home what Megan was saying. These kids had already spent 15 minutes studying these cupcakes, counting the cupcakes, listening to their classmates count the cupcakes. They were already invested, had thought about how they were organized, and were certain they were starting with 12. It hardly took any nudging at all to bounce them from their counting investigation to a problem-solving one. In fact, the two investigations were seamless.

My friends at different grade levels, can this idea transfer? When do you do loads of work, get students invested in a context, and then walk away too soon? I’d love to know.

In Debbie’s room yesterday, I kept thinking how lovely it was for students to see sensemaking as integral to counting and problem-solving, right from the start. How lovely it was for Deb to double the bangs for her bucks with this scenario. How lovely for her students to go deeper into a context that they could already visualize and understand.

Springboarding. I’m a fan.

How Not To Start Math Class in the Fall

My girls started school yesterday. Fourth and second grade. No idea how that happened! IMAG0343_1_1

Today, on the second day of school, each kid had her first day of math, which she spent taking a math test. By their descriptions, the tests were typical, elementary school, beginning-of-year-diagnostics: lots of questions, a whole random collection of content, multiple choice. Each child was told:

  • There will be no talking.
  • You may not work together.
  • I can not help you.

I’m sure the district or school requires this test be given. I’m sure the curriculum starts out with this beginning-of-year-assessment. I’m not criticizing the individual teachers here.

But I don’t get this tradition. NOT ONE BIT.

Teachers have two different dominant needs at the start of a school year:

  1. Teachers need to set a tone and a climate for mathematics. They need to build community and trust and relationships and an atmosphere conducive to collaboration and risk taking and inquiry and learning. They need to establish routines and expectations.
  2. Teachers need to begin gathering useful formative assessment about their new students so they can plan effectively.

The stock beginning-of-year-assessments fail on both counts. I think the ways they fail the first one are obvious. The key word in the second point is useful. On day one, I really don’t care if my students know the vocabulary word for a five-sided polygon, can tell time to the half hour, and can calculate perimeter accurately. I’d much rather know how they attack a worthy problem, how they work with one another, and how they feel about the subject of mathematics. I am much more interested in the mathematical practice standards than the content standards in the fall.

There are many wonderful ways to kick off math. I’ll say it again to give room for a second collection: there are many wonderful ways to kick off math. You can do math autobiographies. You can do Talking Points and tackle some math myths. You can establish essential routines as efficiently as possible and then launch into a great problem. You can teach expectations in a mathy way. You can get kids counting or solving or working or playing a game or talking about math and observe how they work together and how they think. You can ask questions and listen in. You can get to know them.

Above all else, you can make it clear what math class will feel like this year. And please tell me it won’t feel like this:

  • There will be no talking.
  • You may not work together.
  • I can not help you.


Twitter Math Camp Keynote

I was honored to be one of the keynotes for this year’s Twitter Math Camp (#TMC16). For those of you who are unfamiliar, TMC is a conference organized by (secondary) math teachers for math teachers, and it is a truly remarkable and inspiring thing. I have a lot to process about my time here and what I learned. I hope to find time to blog about it. For the moment, I wanted to post the talk and slides somewhere. This seemed like a friendly spot. I’d love to hear your thoughts, especially if you disagree with me. We have so much to learn together.

Video: “What Do We Have to Learn From Each Other?”

Slides (high quality): Zager TMC16 keynote Minneapolis

Slides (lower res): Zager TMC16 keynote Minneapolis smaller file size

Postscript. David Butler blogged about the Lunes of Alhazen and this talk and what it all means to both of us. Please check it out. It’s beautiful.

Extending the Book Experience?

By now you’ve probably heard about ShadowCon, the mini-conference hosted by Zak Champagne, Mike Flynn, and Dan Meyer. One of the governing principles of ShadowCon is that the organizers want to “extend the conference experience.” To this end, talks are videoed and put on a website where people can watch them and have conversations with other people, including the person who gave the talk. The session doesn’t live and die in a convention center in another city, but goes back home with attendees and connects to their work in schools.

I was thinking about ShadowCon the other day, and then about books. Which got me wondering, what would it mean to extend the book experience? In the interests of disclosure, I’ll tell you I’m asking that question as both an author and a publisher. I want to experiment with ways to increase interaction and discussion around books so 1) it’s a better reading experience for readers, and 2) authors would get smarter because they’d listen to people’s reactions and stories and perspective about what they wrote.

I’m starting to mull over ways to use my own book as a test case. I already have lots of online additional content to share. 13 blog posts–one for each chapter–are sitting in my drafts folder, waiting for me to press publish when we get close to book publication date. These blogs are full of videos and articles and resources and related blogs and all kinds of good stuff. But what I’m wondering about is how to turn those blogs into two-way spaces, where I share content, yes, but I also hear from readers. If someone reads something in the book and tries it in a classroom, I’d love to know about it. I’d love to hear what worked and didn’t. I’d love to give feedback, if desired, and get feedback (always desired).

So I’m hoping you can help me think about how to do that? When we read books, we usually don’t have access to the author. What I’m wondering is how could access to the author enrich the experience of reading a book? If I open up a forum (here or elsewhere) and make it so readers can talk to each other and to me, and I’d both moderate and be an active participant in the conversation, how would that deepen and extend the experience for all of us?

This internet thing is pretty marvelous, and I have come to treasure the ethos we have in the Math Twitter Blog-o-Sphere (#MTBoS). At the same time, books are marvelous. I love them. I love the thoughtfulness, the depth, the level of argumentation, the pace, the quality.

I wonder how to bring what I love about books to the MTBoS? And what I love about the MTBoS to books?

If you feel like sharing ideas, I’d be much obliged. If you could talk to an author during and after reading a book, what would that do for you? How would you like to do that? Comments sections, webinars, uploading video and discussing it, book clubs? Other ideas?



Straight but Wiggled

I visited a first grade last week, and the teacher asked me to take over an already-in-progress Which One Doesn’t Belong? (#wodb) conversation with a small group. She’d chosen this image, shape 2 from Mary Bourassa at


I’d heard the last couple of comments, and noticed kids were referring to these shapes rectangle, square, diamond, and pentagon. I know that children are usually describing shape and orientation when they use the word diamond, so the first thing I did was turn the page 45° to see what would happen.


Abby said, “Now you turned the gray one into a diamond and the white one into a square.” The other kids nodded their agreement.

I have learned so much about this moment from Christopher Danielson. In his brilliant book, Which One Doesn’t Belong? A Teacher’s Guidehe digs into the mathematics of diamonds and rhombuses, children’s informal and formal language, and how we might teach in a moment like this. My favorite sentence, which is pinned above my desk at work:

I have come to understand that talking about this difference is more important than defining it away.

Earlier in my career, I would have been tempted to define it away. With Christopher on my shoulder, I engaged the kids in a #diamondchat instead. I drew a quick #wodb with a rhombus, a kite, a diamond-cut gem, and a baseball diamond. I asked, Which of these shapes are diamonds? We played around with the word and I learned a lot about their thinking. Mario looked unsettled and said, “Now I’m not so sure what a diamond is.” He turned to me and asked, “What’s a diamond? Which one is right?”

I said, “I don’t know. It’s up to you.”

The kids gasped.

I smiled and went on, “Diamond is a great word. We can use it to talk to each other so people understand what we mean. But it doesn’t have a strict definition in math, like some other words do. For example, the word ‘square’ has a meaning in mathematics, and we can all agree on that meaning. But diamond isn’t like that. It’s meaning is really up to you and what you’re talking about. If you think this is a diamond, it’s a diamond.”

Students liked this idea.

It was time to move on to a new #wodb, so I looked through the ones the teacher had printed out, and went for something different:


This one is from Cathy Yenca. It produced the desired effect right away:

Rianna: “I thought we were talking about math! Why’d you put this up! There are no letters in math!”

Mario: “Well, I guess you could think about them as shapes. Like U doesn’t belong because it has two lines and then a curvy handle thing at the bottom. And A doesn’t belong because it has an inside space. And T doesn’t belong because it’s the only one made out of 2 straight lines…”

Abby interrupted, “T only has 1 straight line.”

This one caught me off guard. “What do you mean?”

She gestured up-and-down with her hand and said, “That’s a straight line, and then it has a bar across the top.”

I asked, “How many straight lines does the N have?”

Abby: “Two.”

“What about now?”


Abby: “Now the N doesn’t have any straight lines.”

“What about the T now that it’s turned?”

“It still has one straight line, but now it’s that one.” She pointed to the now-vertical part of the T.

I pulled out a marker. “Is this a straight line?”


Abby: “It’s pretty close.”

“Right. Pretty close. Pretend I’d made it perfect.”

“Then yeah, it’s a straight line.”

“What about this one?”

“That’s a laying-down line.”


What about this one?


“That’s straight but wiggled.”


Oh man. How awesome is that?

The kids started arguing, in the best sense of the word.

Willy: “I don’t think that’s straight. Straight means it doesn’t have any wiggles or curves.”

Julie: “No, straight means it goes up and down. It doesn’t matter if it’s curved.”

Those were the clear terms of the debate. So now what do I do? I have Christopher on my shoulder still.

I have come to understand that talking about this difference is more important than defining it away.

I also have the knowledge that defining it away isn’t going to convince Abby or Julie or Mario. They might say “OK” to please the teacher, but I’m not going to change any minds that way.

At the same time, my mind is reeling, wondering how we use this word “straight” informally? Why have kids inferred that straight means up-and-down?

“Kids, line up in a straight line.”

“Sit up straight.”

“That picture is crooked. Can you straighten it?”

“Put the books on the shelf so they’re straight.”

In our everyday language, we sometimes do use “straight” to mean upright or vertical or aligned. I didn’t think of all these examples on the spot, but in the moment, I was confident I would think of them later. In the moment, I knew Abby had a good reason for her argument, based on her lived experience, even if I hadn’t figured it out yet. That faith in her sense-making matters a lot in this interaction. Her understanding of “straight” isn’t wrong; it’s incomplete. My job is to help her layer in nuance and context to her understanding of “straight,” so she knows what it means in an informal sense and what it means in geometry.

With the kids, I told them it might help us communicate if we added a few other words to the mix. I asked if they knew the words vertical or horizontal? They did, Abby included. They were able to correctly match the lines to the word.

Abby said, “I think there should be more words for lines.”

“Like what?”

“Like, vertiwiggle. That would be up-and-down and wiggled. And horizontawiggle for lying-down lines that are wiggly.”

This was the moment to stop pressing. That much I knew.

By coining these new words, Abby had let me know she was now thinking about two different attributes at the same time: the orientation and the wiggliness. I wasn’t about to resolve that complexity or define it away. I wanted her to think about it. I’ll check in with her next time and see what she thinks.

The classroom teacher said her mind was blown by this conversation. She had the whole class join us, and I erased the board. I used a straightedge and drew only the straight, vertical line. The whole class agreed it was straight.


I drew the straight, horizontal line, and asked if it was a straight line. All hands went up, even Abby’s. I looked at her and said, “Tell me what you’re thinking now.”

“I’m thinking that, even if it’s lying down, it’s still a straight line.”


I drew the diagonal line. Now about half the hands went up. I called on a student who said it wasn’t a straight line:

“Well, it’s almost a straight line, but it’s slanted.” Lots of nods at this.

Nate said, “I don’t think it matters if it’s slanted. Straight means it’s not curved. That line is straight.”

Emma said, “No, straight means it goes like this.” She gestured up and down.

There were lots of furrowed brows.

I drew the squiggly lines and asked who thought they were straight. A few hands went up. Abby raised her hand halfway, then put it down. She said, “I made up a new word for that kind of line. It’s vertiwiggly.”

The class laughed. I smiled and capped my marker. Time was up.

The teacher wants me to come back Monday and pick this up again. So, where should I go from here?

Whatever I do, I’m in no rush to define this loveliness away.




I gave my first Ignite! talk in San Francisco. I was super honored to be asked, especially given what a Math Forum fangirl I am. The lineup was AMAZING. I hope you’ll watch all the talks.

An Ignite! talk has a unique format: 5 minutes, 20 slides, 15 seconds per slide. The slides auto-advance whether you’re ready or not. Ten of us presented in an hour.

In my mind, I wanted to really nail this talk. I wanted to script it and practice it and polish it and rehearse it for a month so I could give it in my sleep. In my real life, I had so much work piled up between the time I finished my book and the time I set sail for San Francisco that I didn’t get to start drafting the talk until seven days before I was to give it and I hardly had time to practice during conference week.

I found the process of choosing a topic interesting. I gave a pretty political talk at ShadowCon last year, and decided to go for something more substantive this time. I chose to talk about the last big chapter in my book. It’s the biggest chapter, actually, which makes it a strange choice for a 5-minute talk. But it’s my favorite, (shh! don’t tell the other chapters), and I liked the challenge of seeing if I could get the framework into 5 minutes, even if I couldn’t even start on the classroom stories and specifics.

I learned that 15 seconds is enough time for me to say 3 sentences. I tightened my script until I was pretty happy with every word. I started practicing in the odd 5 minutes here or there in my hotel room in San Francisco. Everything was smooth with the script in my hand. Then the night before Ignite!, my awesome roommate, Jenny Jorgensen, made me put the script down and give the talk to her, and the shit hit the fan. It was a hot mess.

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I always say that the first time you give a talk to an audience is like cooking your first pancake. This analogy comes from Scott Hamilton talking about ice skating in the 80s. I wish I could find the reference, but I’ve looked and failed. I swear, he said it on TV and it stuck with me. No matter how much you’ve practiced, there’s nothing like getting out in front of the crowd to make you realize where you need to turn up the heat or thin the batter. Practicing in front of Jenny made me realize I was about to give a first pancake to 1,000 people. Not my favorite feeling.

The morning of the Ignite!, my dear friends Graham Fletcher and Kristin Gray met me in at 7:30 in an empty room to let me screw it up in front of them a few times so I’d give a second pancake to 1,000 people. I felt better after. Not great, not confident, but better. Mostly, I felt lucky to have Graham, Kristin, and Jenny as friends. Such support.

I think there are two plausible ways to do an Ignite! One is to script it out and then practice the hell out of it until you really do know it cold. The other is to bullet point it and have it feel a little more improvisational.

What I did this time was find the no-man’s land in the middle. I scripted it out enough that I lost the normal, extemporaneous flexibility I have to change my words and keep going. When I said the wrong word, what was happening in my mind was, “No! I decided to say dispute here, not debate!” I was tied too tightly to my script. But I didn’t have the time to practice enough to deliver the smooth performance I would have liked.

I’m not sure if there will be a next time. The main thing I noticed in San Francisco was how relaxed and happy I was the next day, in a normal 60-minute session. I could walk around, check in on people, add tangents, be funny, go with the flow. I never would have thought 60 minutes would feel long and relaxed, but compared to a 5-minute Ignite!, it was joyful. I felt so much more in my element. I’m really more of a long-form girl.

I am grateful for the opportunity, though. It pushed my skills as a presenter and forced me to tighten up my thinking. I learned a lot. Mostly, I learned to always have a turn-and-talk or something for the crowd to read at slide 3 so I can take a drink of water when my inevitable cotton mouth shows up. That’s the one thing I can count on.

Well, that and my friends.