Welcoming Dissent

It’s been so gratifying to hear from people enjoying Becoming the Math Teacher You Wish You’d Had all around the world. After five years of work on it, I’m over the moon that teachers are finding it useful, approachable, and inspiring (their words, not mine, but oh boy do I love those three words).

I’m worried, though.

I’m worried about the normal, human tendency to not want to hurt my feelings. I’m worried I’m not hearing from people who disagree, or think, “OK, maybe. But what about…” or “In my experience, actually…” I’m worried I’m missing out on my chance to learn from your critiques.

So I wanted to make it explicit. I’d love to hear how, when, and why you disagree or are unconvinced. You can tell me in the comments, on the forums, on twitter using #BecomingMath, via email, on the (nascent) facebook discussion page, or in person next time you see me.

Of course, please keep it civil and constructive. No need to tag in or poke a stick at the ideologues and name-callers from the math wars. I’m not that in need of dissent.


You’ve probably noticed some changes here at the blog. I decided to go through the hassle of moving platforms for one big reason: forums. WordPress.com doesn’t support the use of forums, and I wanted forums very, very badly. I think it’s worth telling you why.

The value of any professional resource–book, conference, course, PLC–lies in the thinking it sparks in you, and how that thinking impacts your practice. That’s why I wrote a thought-provoking book, filled with a mix of meaty ideas and practical details: I’m hoping to stimulate thinking that leads to positive actions in classrooms. There’s something that comes in between reading and actions, though, and it’s something I can’t put in a book because that’s not where it happens. That thing is processing, and it happens within each reader and among readers in conversations, writing, and personal reflection.

I love the image of teachers gathering together in coffee shops on Saturday mornings or in classrooms after school to talk about their thinking as they process this book. It’s a hoot to imagine where those conversations might go because I know the book will be a totally different read for different readers, depending on which questions each reader is currently asking and which problems of practice each reader is wrestling with right now. Reading is a always a partnership between an author and a reader, and I’m about to partner with thousands of different people who each bring individual experiences, contexts, curiosities, and personalities to our interactions.

So my first motivator is it would be a crying shame for me to miss out on all of that!

I have a second motivator. There has been a fair amount of angst in publishing in recent years about what will stay and what will change with the advent of new technology. “Will people still want to read books in the age of the internet?” has been the dominant question. The answer is undoubtedly yes. Despite all the fears and worrying, most people feel like Kristin.

Changes in technology have made publishing a beautiful paper book easier (wait until you see the vivid color throughout this book OMG), but they haven’t led to books being replaced by screens, as once imagined. As a book-lover who lives in a house full of books and works for a publishing house situated over an independent book store and across the street from a public library, you can imagine I’m happy about the durability of paper books in the 21st century.

That’s not the end of the story of technology and books, though. I find myself wondering what changes in technology could mean for the reading of a book–rather than the form of a book–in the 21st century. And that’s where I see a lot of promise.

Over the past several years, I have wholeheartedly joined the community of teachers in the Math-Twitter-Blog-o-Sphere. I learn daily through conversations on twitter, longer conversations via our blogs, in-person conversations at conferences (even at a conference of our own), conversations in chat rooms at our webinars, and conversations via video chat, phone call, email, and direct message with colleagues all around the world. In short, we have learned how to use technology to further our conversations–to think individually and together in a supportive, welcoming, professional guild. My in-person PLCs have benefited from my larger community because we have a constant influx of new ideas to process together. In short, it’s the greatest place ever.

I wondered about creating a similar place to host conversations about the book. In-person book clubs and staff-wide PLCs are fantastic, but not everyone has access to one of those. Twitter is great, but most teachers aren’t on twitter. It’s also easy to miss conversations or bits of threaded tweets because it flies by. Blogs are essential, but they’re scattered across the internet. Facebook book clubs are–well, we’ll see about that one. I honestly have no idea what to expect, but I’m going to give it a try for sure because that’s where most teachers are. That said, I know lots of people who either don’t use facebook or keep it solely for their non-professional life.

I’m sure we’ll talk in all those places because we tend to talk where we are, but I wanted an additional place, a designated place with no sifting needed where people can come together and think. I wanted it to be a place with the culture of the MTBoS: where thinking-in-the-raw is welcomed and treated with respect. I certainly plan to moderate toward that culture, but I’m counting on you to create and support it with me.

And, selfishly, I wanted it to be a place where I could listen in on and participate in the conversations sparked by the book. I can anticipate what lots of them will be about, but I also know I’ll be surprised and that’s a delightful idea. I’ve built the forum with this spontaneity in mind. I’ve started seeding some topics to get us going. They’re focused on the Calls to Action in the study guide, which are suggested places to put your book down, go try something specific in your teaching, and then report back. But I’ve also structured the forum so that participants can start topics of their own and take the conversation where they need it to go.

I’m hoping some of you early readers will help out by starting or joining conversations soon. As you read or finish a chapter, pop by the forum and process out loud with the rest of us. Tell us what your in-person book chat talked about, or what idea keeps rattling around in your mind. Tell me what you disagree with, what resonated, and what the implications of my arguments are. If you wrote something up on your blog or had a conversation on twitter, link it in the forum so people can check it out. I’ll do my best to curate conversations so people can follow them, and I certainly see myself as a participant, learning alongside and from everyone else.

The forum is an experiment. We’ll see how it goes. But I’m optimistic that creating a place to ground and capture the rich conversation people around the world are having may improve many readers’ experiences with this book. It will help them talk through their thinking, learn from other people’s thinking, and reflect meaningfully on which changes they want to make in their professional practice. And it will help them grow their own professional communities, bringing in new colleagues who listen respectfully and share thinking generously.

I think it’s worth a try. That’s why I built it. Now we’ll see if they–you–come.



The Book

You can now take a look at what I’ve been up to since 2011. btmtywyh-jk-final-jpg The preview for
Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms is live. You can flip through the whole book, if you like, before deciding whether it’s a good fit for you and your colleagues.

The book’s imminence has me feeling reflective. Couple that with the regularity with which I’m asked, “So what was it like, writing a book?” and you have the makings of a long, navel-gazey blogpost. Here we go, for those who are curious. Pour some tea.

[Disclaimers. I have two of them. One, from an author’s point of view, I’m only speaking to my experience. Two, I now have the good fortune to see this process unfold from the other side of the editor’s desk, as a Math and Science Editor for Stenhouse Publishers. You should know I work there.]

What do I have to say?

Five years ago, I ended up on the phone with my editor, Toby Gordon, kind of by accident. I was working for a university here in Portland, supervising and coaching student teachers at a wild range of schools. I loved that work, and had done it for quite some time in Boston, mostly with Boston University, and a bit with Tufts University and the Boston Teacher Residency. After we moved to Maine, I thought I’d continue that work while my kids were little, and then return to full-time teaching. Politics and state budgets kinda blew my plans, though. My department at the University of Southern Maine (USM) was hemorrhaging high quality faculty and not replacing them. Adjuncts were brought in to teach methods for $3000 a class, which is downright insulting. I loved my colleagues and my students, but the writing was on the wall, and I started looking around for new work.

I reached out to Toby at Stenhouse to see if I could work for her as an outside reader or freelance editor. Toby came highly recommended by, well, pretty much everyone and most especially my math hero, Elham Kazemi. We hit it off on the phone, and eventually she said, “Yeah, you could do those things, but what about you writing a book?”

I asked, “About what?”

She said, “Well, that’s up to you; it’s your book. What do you have to say?”

Inarguably the second-best question I’ve ever been asked. (“Will you marry me?” is tough to beat.)

She sent me off to brainstorm. I created lists of ideas and sent them to her. Toby told me which ones had legs. Some had been done before. Some wouldn’t sell. Some were unfocused. And one stood out. I just went and found my email to her, dated 11/29/11:

I was just looking at my shelf and my eyes fell on the first edition of Mosaic of Thought by Ellin Oliver Keene and Susan Zimmerman. That book was so path-breaking in that it taught teachers to view themselves as readers, first; to explore their own reading and think about what great readers do before delving into how we might start to teach children those same skills and strategies. I started wondering if anything like it had been attempted for math?….Has anyone tackled teachers’ affects about math in a cozy, curl up by the fire, more intimate and friendly way like Keene and Zimmerman did for teachers’ affects about reading? Attempted to help wary teachers embrace the discipline of mathematics as adults, not by telling them I’ll hand them a bunch of blackline masters so they don’t need to be scared (or think), but by guiding them on an internal journey about their own thinking and math?

It evolved, but that was the genesis, right there.

Toby sent me off to read. I think new authors often don’t know how important it is that they read. I had so much to read. Marilyn Burns’s book about math phobia, everything Deborah Schifter ever wrote, especially the pair of What’s Happening in Math Class? Reconstructing Professional Identities books, everything from Ball, Lampert, Fennema and Nelson, Hersh, Gresham, Beilock. Journal articles, books, blogs. I read.

And I started writing. Toby would check in on me periodically to make sure I wasn’t just reading; I was also writing. Four months later, in March of 2012, I submitted a proposal. I nervously waited, hoping they’d say yes!

They said…maybe.

It stung, but I received useful feedback from the editorial board and three outside readers. I saw what they were saying, for sure. I needed to work out the balance between practicality and big ideas. I needed to figure out exactly who my reader was. I needed to be more positive, and not spend so much time stating the current problem. They asked for an additional chapter from classrooms, based on fieldwork I hadn’t done yet. In fact, I wasn’t really sure what my chapters were yet. I knew my Table of Contents would grow out of my work in classrooms and my research, and I had a tough time trying to nail it down before doing that work. I started to sweat committing to anything before I’d done the research, research I had to do with an open mind.

In other words, I was afraid I was losing the book for the sake of the proposal.

So I did something kind of ballsy. I told Toby I was going to pretend they’d said yes. I was going to stop saying, “If I write this book” and start saying, “I’m writing this book.” I’d begin fieldwork and mucking around and gathering information and literature reviews. When I’d done more of that thinking and legwork, I’d know where I was headed, and I’d be ready to resubmit a proposal with a full chapter.

She was tickled. And supportive, as always.

Your shift in thinking makes absolute sense to me. Rejigging the proposal and chapter now is like putting the horse before the cart. You’ve done a lot of thinking and reading about this project, but clearly there’s a lot more to come—plus you want to start tracking down and working w/ teachers. So, I’m all for it! I have no doubt this shift will serve you—and the book—well. Everyone has different ways of going after a book. You’re well acquainted w/ your non-linear approach—so just trust it and see where you land. I’m here whenever you need me.

The research

This is where my project looks radically different from most projects. Most of the time, people already have some idea what they’re going to say. Most of the time, their books grow out of the work they are already doing.

Not me. I didn’t have answers at that point, just questions. My main ones:

  • What’s essential to the discipline of mathematics? What ways of knowing, coming to know, changing our minds, thinking, agreeing, and disagreeing are unique to mathematics?
  • What would it take to make math class more like mathematics?
  • How can we close the gap between math class and mathematics when teachers have never really experienced mathematics as it’s actually practiced?
  • How can we improve the system when we’re the products of the system? What supports do teachers need? What could I offer teachers that would be useful?
  • In The Process of Education, Jerome Bruner said, “We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development” (1960). This statement has been a core belief throughout my career in education, and I treasure my copy of that little red book. But what does “intellectually honest” math look like to students at different stages of development? And how do we facilitate that? (Curious? Ball dug into this question a bit in her With an Eye on the Mathematical Horizon article.)
  • What’s replicable? When I find teachers who are teaching math class so students are doing the real work of mathematics, will I be able to help other teachers learn from them? How?

I read a ton. I also hit the road in the fall of 2012, working around my teacher ed schedule at USM. I started visiting classrooms, asking for recommendations of great math teachers in a variety of settings. I started with 40 names, and observed and interviewed them all. I saw a lot of very skilled teachers, but I quickly came to realize that what I meant by the phrase “great math teacher” wasn’t what other people meant by “great math teacher.” For me, that phrase wasn’t code for good management. Or high test scores. Or clear explanations. Or “fun” activities.

I meant something bigger. Different. Something about how the kids were interacting with the mathematics, and each other, and the teacher. Something about the combination of joy and intensity, curiosity and passion, safe exploration and high standards. Something about the transparency of the raw thinking, and the way math felt empowering. Something about who owned the thinking, who owned the truth, who owned the authority to decide if an argument was convincing.

Heidi’s second graders

I first found what I was looking for in Heidi Fessenden‘s 2nd grade classroom in the Mattapan/Dorchester neighborhood of Boston, MA, and in Shawn Towle‘s 8th grade classroom in Falmouth, ME. I couldn’t get enough of these two. They teach in totally different grades, different settings, different student populations, with different curricula. They’d never met or taken a course together and have taught for different lengths of time; yet, somehow, when Shawn opened his mouth, he said just what I’d heard Heidi say. When Heidi looked at a piece of student work, I saw the same wheels turning I’d seen in Shawn. When a student made a mistake with lots of potential to dig down into understanding, the same delight came into their eyes. I started spending as much time as I could with them.

It was through months of conversations, observations, and interviews with Heidi, Shawn, and their students that I started to figure out what my book really was about. I grounded my ideas in classroom realities and started identifying these teachers’ most practical, replicable, effective techniques. I started seeing what “intellectually honest” mathematics teaching and learning looked like in different content areas, with different aged students. Mostly, I started putting to words what made me so darn happy in their classrooms.

Shawn and one of his eighth graders

Around that time, I broke my ass. I’m not kidding. I was walking our two big dogs on an icy morning and they pulled at the wrong moment and I looked very much like Charlie Brown when Lucy snatches the football. I fractured my tailbone and had to call off my observations for a bit because I couldn’t sit in the car or on the train.

That was early December, 2012, and I was suddenly stuck at home with my laptop. I let Toby know I was starting a new proposal. I’d learned so much since the last one. This time it flowed. This time I knew where I was headed. By February, I had a chapter and revised proposal ready to go. I submitted, and this time the editorial board and reviewers were unanimous in accepting it. I was under contract to write a book in March of 2013.

Jen’s fourth-grade student

In the meantime, I’d kept researching, reading, and looking for other teachers. That’s how I found Jennifer Clerkin Muhammad that May. She taught fourth grade in a two-way bilingual school in the South End of Boston, and she rocked my world. The level of discourse in Jen’s room was astounding. My regular trips to Boston to visit Heidi now became trips to visit Heidi and Jen. Jen taught in the morning; Heidi in the afternoon. Their schools were across town from each other, but it was doable. I’d take the 5AM train out of Portland, visit both teachers, crash with my friends Shoma and Josh, visit both schools again the next day, and catch the nighttime train back to Portland. I loved those trips so much. Riding back on Amtrak, I’d have dialogue and images and questions and student work whirring in my head and on my computer. I’d play back the video for a bit, and then stare out the window at New England going by, and think. What, exactly, were the instructional decisions the teachers made and techniques the teachers used to engender such amazing mathematical communities?

Me, with Debbie’s K-1s

I still felt like I needed a small-town setting, because so many of my examples were urban and suburban. I kept looking, and that’s how I found Debbie Nichols, in rural NH, in August of 2013. From my first day in Debbie’s room, I knew I’d found a home. I started spending every Tuesday with Deb and her young students, and I learned something every time I visited. Still do.

Heidi, Shawn, Jen, and Deb are the four anchors of the book. They were the teachers who helped me figure out the core of what I was trying to say. I also visited dozens of additional teachers who were fantastic and added new voices and stories and teaching challenges and opportunities. You’ll meet lots of them in the book as well.


This header is a little misleading. I didn’t do all the research and then do all the writing. Both were happening all the time. But I did frontload the research and then really dig down into the writing.

I loved writing this book. I mean, I loved it. It was my trusted companion for years. I thought about sections in the shower, chapters when I was falling asleep at night, a better way to close that story while walking the dogs. With every revision, I got clearer about what I was saying, about what mattered. It was hard. It was a worthy problem, though, and I enjoyed the challenge. Some days I struggled all day and still barely filled my one-inch picture frame. Other days, I found some kind of flow, and I could hardly sleep because I was reworking and moving and dreaming about passages.

Throughout, I counted on Toby. I sent her chapters and bits of things as they were ready, when I needed feedback. She’d tell me what she loved, and I tried to do more of that. She’d tell me when I was dropping in too much research, and I’d cut. Some. Ha! She’d tell me if she got bogged down in the details or if something didn’t make sense, and I’d clean it up and prune it back. Throughout, it was clear it was my book, and I could take or leave her suggestions. I took most of them. Sometimes I argued back, and she would come around to seeing why I felt I had to do something a certain way. Sometimes, months later, I’d realize she’d been right all along and change it. She never gloated about those moments, which was gracious.

Something I’ve learned since becoming an editor: there’s a duality at the core of our work. We take tremendous pride in the books we publish, but we’re egoless during their publication. It’s our job to make the authors look as good as possible, to help them say just exactly what they want to say in the best way possible, to usher their books out into the world as effectively as possible. But we erase our fingerprints from that work. The author is the one who shines; the editor works behind the scenes. The book is the author’s baby. The editor is the midwife. Both roles are immensely satisfying.

I took a master class in editing by being edited by Toby. I apprenticed with the best.

We emailed, we texted, we talked on the phone, and we met for lunch sometimes (because we live in the same place, lucky me!) to discuss my progress over fish sandwiches. I always walked away clearer on what I needed to do, or at least what the terms of the struggle were. Often I’d be halfway through an email to her and I’d realize the solution to the problem I’d been describing.

I like feedback a whole lot, so I also sent chunks of the manuscript out to other people. I sent every chapter to all the teachers named therein, and I wouldn’t go forward without their approval. I also sent chapters to experts on specific sub-topics. Reuben Hersh read my work on intuition. Danny Bernard Martin read a section on equity. Virginia Bastable helped me with representation-based proofs. In each of these cases and many more, people were generous with their time, expertise, and insight, and I am indescribably grateful.

I was interrupted during the writing of the book. Cancer is rude that way. I thought I’d be able to write during my mom’s treatment and mine, but I wasn’t. I made it through that year watching back episodes of Comedians in Cars Getting Coffee, walking the dogs in the woods, staring out the window, and reading The Princess Bride on a loop. You do what you gotta do. Toby told me to write if it helped me cope, and not to write if it didn’t. I couldn’t. All I could manage to write during that time were tweets. Thank god for twitter, and my friends there.

When I was able to come back around to the book, I fell in love with the process of writing something long all over again. I love building a larger argument, thinking about how to guide the reader through it, crafting a larger story. I wrote the last word on the last page in December of 2015. I celebrated for an evening, and then I turned back to the first word on the first page and started revising the next morning.


Again with the misleading header. I’d revised all the way through. Every time I worked on a chapter, I’d start at the beginning and work my way toward where I’d left off. Some days, I never got past the first page. Revision-during-writing is essential.

But once the whole thing was drafted, I got to focus exclusively on revision, and that was extremely satisfying. Time had made it much easier to cut unnecessary words, quotes, and examples. Writers say “Murder your darlings” for good reason. Some of my darlings had become less dear over time, and knocking them off was much easier after a bit of distance.

I’d also discovered I’d gotten better at writing the book as the book went on, so I went back and revised the early chapters to match the later ones. Interestingly, this often meant writing with more confidence than I had at first. First-time authors are often uncomfortable saying, flat-out, what they think. That got a lot easier for me with practice.

A big part of revision was working on the artwork. Throughout the writing of the book, I’d collected all the high-resolution photographs and original student work I thought I might use, along with permission slips from parents. I accumulated a giant box, and then sloIMG_20160216_195318_472wly trimmed them down to the most essential figures. No fluff! At the end, it was time to number those figures, insert design notes in the manuscript to show where they should go, build the art chart for the production team, make sure all my permissions were in order, put sticky notes on student work saying when names needed to be removed or ancillary problems needed to be cropped out, and so on. I made an efile and file (the yellow folders in that picture) for each chapter, and organized all the figures carefully.

Getting Ready for Turnover–The Team Grows

When everything was as done as I could get it, I handed the whole pile and files to Toby. She started back at the beginning and read the entire manuscript again. When a change was needed or something could be cut, she flagged it. When she was ready, we met again and went through, sticky note by sticky note, making decisions together. We pulled some figures, which meant renumbering again. Most of the changes were quite manageable, though.

Once I cleaned up the files, the manuscript was ready to be turned over to the production department. Turnover is a slightly misleading term, because the editorial work is not over. Toby stays with the project from that first nervous phone call until well after the book is out in the world. But at turnover, the team of people working on this project grew from the two of us to a whole group of people. Just the briefest of descriptions:

  • Production Editor Louisa Irele carefully worked through every figure and picture, scanning original work, correcting my errors, and checking permissions. She went through the text files and put them in mansucript form.
  • Editorial Production Manager Chris Downey is responsible for everything that happens with the text from turnover to publication. She handles the words. She is incredibly kind, professional, and knowledgeable–I swear, the complete Chicago Manual of Style resides in her head somewhere. She has enviable attention to detail. The designer of my book told me he can pick out Chris’s manuscripts from the ones he receives from other publishers. The quality of her editing is remarkable. Chris guides each manuscript through copy editing, typesetting, multiple rounds of proofreading, and indexing with care.
  • Obviously, that means we have professional copy editors, typesetters, proofreaders, and indexers on this team as well. I am so grateful for everyone’s eagle eyes! When I got nervous about mistakes, Chris reassured me. “Don’t worry. We have six pairs of eyes looking at these proofs right now.” It helped.
  • Jay Kilburn is the Senior Production Manager, and he handles the design of the books. We never use templates at Stenhouse; rather, each book is individually designed. Jay is so gifted at pairing authors with the right designer. Take a look at Which One Doesn’t Belong? It’s so Christopher. And my book is so me. It’s amazing! Jay works with the designer on the overall feel of the book. They decide on the trim size, the covers, the fonts, the headers, all the design elements in the interiors (boxes, sidebars, pullquotes, tables, chapter openers, running heads, dialogue, etc.). Jay also manages the actual production of the books at printers, binderies, and warehouses. He has this huge realm of knowledge about glues and sigs and press runs and where the readers’ eyes need places to rest and widows and orphans and kerning and spot varnish. (Look at the covers of Which One Doesn’t Belong?. See how the shapes have a bit of shine that makes them pop? That’s spot varnish.) I’ve been at Stenhouse just about a year, and I’m at a place where I can begin to grasp the deep knowledge of bookmaking that Jay has, or at least get most of the vocabulary.
  • Lucian Burg designed my book. He also happens to be a lovely person and one of my neighbors! We hire local whenever possible, and Lucian is really local. A few weeks ago, the girls and I stopped by his studio to drop off page proofs. Lucian told me, “I’m proud of your work and I’m proud of my work,” which made my day. He loves converting these ugly word processing files into a beautiful, visual, satisfying experience for the reader. His studio might be the prettiest workspace I’ve ever seen, largely because its filled with the beautiful books and covers he’s designed. Throughout this project, he understood the feel and vision I was after, and I couldn’t be happier with what he did. It’s a perfect marriage of content and design, which is the whole goal.

So, at the turnover meeting, the editor helps the production team get a feel for the book and its style, as well as any particular themes or ideas that might translate visually. Production then gets to work.

People kept thinking I was done at this stage. Nope. The manuscript came back to me two more times: once after copy editing and once after proofreading. Jay also sent initial design files to see if I liked the way the designer was approaching the book. Did I like the way he handled the different elements of the text, like dialogue, block quotes, captions, etc.? You bet.

While I missed writing the book, I mostly loved this phase. What had been my project was now our project. Every person was adding his or her expertise and knowledge and skill to this effort, and together they took my pile of files and folders and documents and turned it into a book. I am in awe of my production colleagues and the work they do.

Getting Ready for Publication–The Team Grows Again

As publication appears on the horizon, the marketing, sales, operations, and customer service departments get involved with the book. Without giving away any trade secrets, I’ll say the marketing team puts together a specific plan for this book, based on its strengths, the author’s desires, and their robust knowledge of the market. They get the catalog ready, the product page launched, and the advertising lined up. The sales team starts educating our regional representatives about this book so they can begin talking it up in schools. The operations team works with Jay to make sure the book moves from the bindery to the warehouse on time and properly packaged, so it’s ready to ship out on day one. Customer service, sales, and operations together work with readers, teachers, school districts, booksellers, and international distributors. From preorders to purchase orders, they are experts in the business of publishing, and they are the ones who get books into your hands.

Out It Goes, Into the World

The team is about to get a lot bigger, and change again. We’re about to add readers.

This is the mind-blowing part. Toby has always told me that books take on lives of their own once they’re published. You plan and you try and you hope and then you send it out there and see what happens.

I had the best time writing this book. Selfishly, it was worth every minute. But I didn’t just write it for me. I wrote it for you. I expect some of you will like it, some will not, some will agree, some will disagree, some will be put off by the length, and some will want more. All good. What I hope above all else, though, is that it will make you think. Hard. It’s jam-packed with ideas, and I hope you’ll find yourself reflecting on them as you take a shower or chop tomatoes into your salad. Whatever you decide about the arguments I make, I hope this book helps you teach with intentionality and joy.

I have one last bit of work to do before it sets sail. Over the next few weeks, I’ll be adding some dedicated spaces here and on social media about the book. Here, there will be a site for each chapter, along with a robust study guide, supplemental resources, calls to action, and all that good stuff. I haven’t worked out the tech yet (#growthmindset), but I will be building forums for discussion throughout those sites. I’m most eager for those forums, because I’m jazzed to hear your reactions. I wrote this book to move the conversation forward, not have the final word. I can only get wiser if you educate me.

I must really want that dialogue, because I’m even willing to establish a book study page on facebook. Oh my.

So once I get the spaces built, please stop by and tell me what you’re thinking about in response to the book. What new questions are you asking? What are you trying in your teaching? What did I get wrong? What should I be thinking about next?

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 wodb.ca.


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. Its 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.