Representations and Manipulatives and Tools and Things

One of the things I think about is the relationship between teaching math and the physical stuff that goes along with teaching math. This relationship gets distorted sometimes.

For a while, the elementary world got all kinds of swept up in manipulatives. All lessons became “hands-on” because somehow “hands-on” led to “minds-on.” Deborah Ball’s classic Magical Hopes article does the best job I know exposing the flaws in this stance. If you haven’t read it, by all means, click that link and read it right this second.

People sometimes get hung up on tech in the same way. Recently, I had the chance to share some outstanding work Kristin Gray got from her students when she asked them to take out their notebooks and write down what they were wondering about doubling and halving. They’d been working on 14 x 25 = 7 x 50. Check out these conjectures:

I mean, so great.

Both times I shared this work, people oohed and aahed, and then asked the same question: “Could you use tech to do this? Maybe a google doc?”

I have to confess, I don’t understand this question. With paper and a pencil, students were able to shift back and forth between words and numbers effortlessly, much faster than 11-year-old kids can type. If they’d wanted to make a quick sketch or doodle (perhaps an area model, in this situation), they could. They didn’t have to lose their train of thought while hunting through their device’s symbols for ÷ (an obelus, for fellow #wordnerds). The only apps I know that allow students to think and write so freely are apps that turn tablets into $800 notebooks by letting you write on a screen with a stylus.

I kept wondering, what’s the value added there? What’s the rationale for adding tech? What can it do for you that cheapo paper notebooks can’t?

That’s the question I ask myself about tools, in general, whether they require charging or storage in a plastic tote. What will they do for the mathematical teaching and learning here? Sometimes, the answer is not much. Other times, A LOT.

This year, I did something new in my school. My principal and I made it a priority for me to work with our paraprofessionals. These colleagues are overworked, underpaid, undertrained, and almost never supported to go to PD. Yet they’re responsible for educating about 20% of our students–the neediest 20%. So, this past year, my principal worked out a schedule where, one week per month, I had all the paras in two different shifts on Tuesday and Thursday afternoons. Over the year, we did a book study of the second edition of Children’s Mathematics. I chose it because it’s the foundation of everything in elementary math. The book and its embedded videos teach educators how students think about the operations, and how that thinking progresses. They also teach educators to listen closely–and quietly–to students while they think through problems. Given that the paras had been jumping in too quickly to tell or spoonfeed or direct kids, I wanted them to start learning how to tolerate wait time, how to trust that their students have mathematical ideas, and how to listen as a core part of teaching mathematics.

I also knew that digging into students’ mathematical ideas would allow me to get the paras digging into the mathematics itself. One para loves math and I knew she’d be game, but many were reluctant and some were downright hostile to the idea of this year-long focus on mathematics. I had my work cut out for me last fall.

Fast forward to June, and I gotta tell you, I loved my time with the paras. I think it’s some of the most important work I’ve done in the building and I can’t believe it took me so long to get there. We built a safe space and strong relationships and, most of the time, they got more and more willing to try new ideas, wonder about why things worked, and make sense for themselves.

One observation I’d made over the year was that several paras openly despised both the area model and the related partial products strategy for multiplication. They didn’t understand why anyone would do that, and were resistant to multiplying any other way than the standard algorithm. As our year drew to a close, I wanted to devote time to multiplication of double-digit numbers and see if I could get anywhere with this animosity toward these two essential multiplication strategies. I knew that if I just drew or recorded the strategies with equations on chart paper, I would lose them. I’d learned that lesson the hard way, and wanted to avoid the shut down I’d caused before. So I needed something. I needed a tool that would unsettle our typical patterns.

In this case, I reached for graph paper. I handed them each a rectangle of graph paper and asked them, “How many squares are there?” Note, I did not use the word “multiply.”

Nobody shut down. Everybody got to work, and I got a great range of strategies, including the ones above. (The graph paper was 14 x 21, if that helps.)

Man I love a variety of strategies. It’s just the best. Now I had a whole range of decisions to make about where to go from here. (If you want a thoughtful discussion about that decision-making process, you need to read Intentional Talk. It’s had a huge impact on me.) There were a number of things I could do, and a number of competing goals in my head. A few of them:

  • I needed to explicitly connect what they did with the graph paper to multiplication.
  • I needed to get them more comfortable with representations of what they did with the graph paper, both in pictures and numbers. Optimally, they’d be the ones recording, not me.
  • I needed to expose some rich mathematics by digging down into one of these, or by drawing connections among a few of them. Which ones?
  • I needed to take this opportunity to highlight the fresh mathematical thinking from some paras who have had negative histories in math, who started out the year reluctant but dove into this problem bravely, and who still needed support to see themselves “as math people.”
  • I needed them to explain their strategies to one another so they could put words to their own thinking, and listen to and try to follow their peer’s thinking. (This was an ongoing goal all year and we’d made a lot of progress.)

So here’s what we did. Each person explained their strategy. While they did, I asked for volunteers to come up and record on paper what their colleague had done on our chart paper.

I stayed quiet while the person with the marker recorded, and then naturally turned to their colleagues and said, “Is this what you did?” or “What was your equation again?” We were getting somewhere.

In each group, someone had surprised me with their strategy. In the first group, J said she looked at her rectangle and thought, “Way too many to count.” So she folded her rectangle in half:

She thought, “Still too big,” and folded it in half again:

She looked at that and thought, “I can count that and then multiply by four.” The thing is, one of the factors was odd, so quartering led to a fractional result. She didn’t bat an eyelash:

She had a 7 x 10 rectangle, which yielded 70 squares. She then combined 6 half-squares to make 3, and had 1/2 left over. Each quarter had 73 1/2 squares. To figure out the total number of squares, she pinched together the 4 half-squares into 2 whole squares, multiplied 73 x 4 to get 292, and added them together to find 294 squares. She told us that the folding grew out of her comfort with sewing, and she was completely in command of her strategy.

I did the recording on J’s work. Not beautiful, but everyone agreed that what I drew matched what she did.

In the next section, L also began by folding her rectangle in half, but the other way:

Then she groused at me, “Oh, you made it not work out evenly!” A moment later, she said, “That’s OK, I love thirds.” Look how pretty:


A perfect square!

When she unfolded it, she had this:

Now we see one power of a tool. If I had explicitly asked them to solve the multiplication problem 21 x 14, I would have had almost all identical column multiplication solutions, which aren’t ripe for rich discussions. But this little graph paper rectangle yielded a wide range of approaches, including two strategies that made beautiful sense, visually, but had almost no chance of emerging if we’d only worked with numbers:

21 x 14 = 4 x (10.5 x 7) and

21 x 14 = 6 x (7 x 7)

Not only that, both J and L were in the spotlight for innovative math, when both J and L have historically not been so keen on the subject. They were the experts on these strategies, teaching me and the rest of their curious peers. If I could have bottled that moment and given it to them for safe keeping, I would have. It was a highlight of my year.

That brought us to the end of Tuesday. I knew I’d start with these on Thursday. Because J and L were in different sections, they hadn’t seen each other’s solutions, although word spread quickly throughout the staff and I heard them comparing notes after school (yeah baby). We began Thursday by marveling over the two strategies, comparing and contrasting the difference it made to want to avoid fractions or not be bothered by them. I then focused us on L’s strategy, written numerically. When we looked at the piece of paper, we could all see that there were six 7 x 7 squares. But when I wrote the equation:

21 x 14 = 6 x (7 x 7)

there was a lot of wondering about it. They all agreed that, if they’d been working with numbers only, they never ever would have transformed 21 x 14 into 6 x (7 x 7). They wondered, where did those numbers come from? Especially that 6? How could a 6 come out of 21 x 14? Eventually, there was recognition that 7 was a common factor of both 21 and 14. That insight led us to write:

21 x 14 = (3 x 7) x (2 x 7).

The 6 was in there somewhere, starting to become more visible, but this is a place where nobody was sure about the rules they’d learned once. What could you do with that 3 and 2? Add or multiply? Five or 6? We went back to the paper:

Do you see a 2 x 3 array there? A 2 x 3 array of 7 x 7 squares? Six 7 x 7 squares? Holy smokes, there it is. They saw where the 6 came from.

We played a little with the quartering strategy in the same way:

21 x 14 = (10.5 x 2) x ( 7 x 2) = (10.5 x 7) x 4

We wrote them up, talked some more about the associative property and what happens when you break factors up by multiplication. Now, to be clear, I am not saying everyone in the group would be able to recreate this logical flow independently yet. But I am saying everyone in the group was following along. They didn’t shut down at the formal math vocabulary, at the symbolic representation, at the diagram.

And that’s why I was glad to have a tool. In this case, the tool made it possible for everyone to access the mathematics here. It helped me gather a variety of solutions so we could make connections among them. It made tangible what had been abstract. It allowed my colleagues to bridge from something intuitive to something a little out of reach. And it made us talk with each other about the mathematics more, not less.

I’d call that added value.

I don’t get fussed over whether tools are high tech or low tech. I love and use them all. But I do take care to use them thoughtfully, not for the sake of using tools or tech, but for the sake of the mathematical learning and conversation they’ll allow me to engineer.

In early drafts of my book, there was a chapter called Mathematicians Use Tools. I was planning to get into all of this stuff. I cut it for the sake of length–it was already a huge book–and I thought tools had been written about a lot elsewhere. They have. I decided, instead, to showcase thoughtful use of tools throughout the book, which wasn’t hard because effective lessons often involved the strategic use of tools. Probably the right call.

There are times I regret not taking the deep dive into tools, though. I see so much tech-for-the-sake-of-tech, tools-for-the-sake-of-tools. I also see teachers still afraid to use tools for fear of mess or noise or lack of control or time or organization. I’d love to explore when and how and why we reach for them–or don’t.

Maybe this is the space.


I’ve been having fun playing around with lesson closes lately. It’s something I’ve been meaning to write up or present on for a few years now, and one of Mike Flynn‘s awesome Mt. Holyoke classes gave me the reason to do it. I’ll be giving a condensed version of that webinar tonight at the Global Math Department. Register and join us if you’re able. If not, I’ll come back to this space and post the video of the webinar once it’s up, in the next day or two.

Update! Here it is:

Some resources:

Slides from the webinar

The list of purposes and formats the Mt. Holyoke crew brainstormed

The planning templates for Open Strategy Shares and the six different Targeted Discussions from Intentional Talk

#BecomingMath 318-321 is an excerpt from my book that shows a close with a different pedagogical purpose

Kristin Gray‘s blogpost, “Number Talks Inspire Wonder

Christopher Danielson‘s blogpost, “What Did You Learn?

Max Ray-Riek‘s googledoc gathering the goodies from MTBoS folks playing with #lessonclose

Someday, I would like to explore these ideas further, in writing or a full-day workshop or something. There’s a lot here! But for now, I hope this helps you get started. I know I’m having fun chewing on #lessonclose.


Every year, handouts edition

Every year, I plan to get my slides and handouts done well ahead of time so they can be uploaded.

And then, every year, I think well, at least I’ll find time to upload them during conference week.

And then, every year, I think I’ll upload them as soon as I get home.

And then, every year, I think I’ll upload them before Dan scrubs the conference websites for the files.

And then, every year, I say screw it and just post them here.

Slides: Gut Instincts NCTM

Handout: Gut Instincts handout San Antonio NCSM NCTM

Questions that press intuition: Printable Intuition Questions


Finally, every year, I promise myself to get ’em done on time. Next year.


NCSM/NCTM My Favorite

I’m starting to bounce back from conference week. I love it every year and this one was no exception. I’m still processing (and digging out from emails), but there’s one thought I need to share now.

Every year there’s a certain amount of soul searching among the folks in the Math Twitter Blog-o-Sphere (#MTBoS) about our role and relationship to our professional guilds. People talk about conferences and communication and membership and journals and the future. All understandable, and a good conversation to have. I agree with some, disagree with others, enjoy listening in. No sweat. Something Michael Pershan said on Dylan’s blog has stayed with me, however, and made me really uncomfortable:

I’ve started wondering if being in the MTBoS is sort of like being a fan. To be in the MTBoS means that you love Three Acts, Which One Doesn’t Belong, Talking Math With Your Kids, Estimation 180, Problem-Based Stuff, Max Ray-Riek, Tracy Zager, Desmos or something.

To be fair to Michael, the next sentence was:

[Is this a good time to remind you that I’m making stuff up and have no reason to believe ANY of this with much confidence? I think it is.]

Ha! I appreciate that Michael tossed his idea out there so we can think about it. Because if he’s thinking it, others probably are too. And I get where it comes from, I think. Mostly. Maybe? I mean, there’s this “cool kids” thing that happens that I can’t stand. So that’s part. And then there are also the amazing resources made by people in the community, (WODB, visual patterns, #3Actmath, Estimation 180, etc.), which many people use and promote. True.

But for me, the MTBoS is not at all about fandom. It’s a community where people who love teaching math and thinking about teaching math can congregate. It’s a place to find like-minded folks, but also dissent. It’s a place where we can make our ideas better by listening to others, putting our work out there, and asking for feedback. It’s a supportive space where some genuine, deep friendships and collegial relationships are born and maintained. It has norms that matter–openness, inclusiveness, camaraderie–even though nobody is officially moderating for those norms. (That’s amazing, if you think about it.) For me, MTBoS is the only place on the internet where the comments are overwhelmingly constructive, and worth reading. It’s not about any one person or group of people. It’s a community, not a cult of personality.

What’s my evidence? There are so many choices (Twitter Math Camp, Global Math Department, every day online), but my recent focus was #MTBoSGameNight. It’s a zany idea, something a few of us hatched before Boston. The idea was to create a way to meet our online community members face-to-face at the close of NCSM and the kickoff of NCTM. We have no budget, no sponsor, no organizing committee. All I do is organize a few logistics and make a slide inviting anyone who wants to come.

Our first year, we had something like 40-50 people? I don’t know. Our second year, we were over 100. This year, 200-250. Matt Larson, as President of NCTM last year, asked how NCTM could help. He asked how could we make sure people could go both to the MET gala and Game Night? Given that Game Night is something the members want to do, what institutional support do we need? This year, NCTM provided the venue and a cash bar. So appreciated.

The night before Game Night, I was talking to Graham Fletcher in a bar and he asked what he could do. I told him I’d love a clever way to give away door prizes. He texted me the next morning. “Where’s the box of books? I’m on it.” He, Zak Champagne, and Mike Flynn set up a series of estimation challenges about them. Now we needed A/V, so Stenhouse helped out there. I arrived at the space 15 minutes before. It was a brightly lit hotel conference ballroom. Muzak was playing. The bartender was drifting around aimlessly. I thought, “This is not going to work.” Nothing about it felt like a party.

And then the people came. Christopher Danielson put his wonderful mathy playthings on every table. People started introducing themselves to each other. Drinks started flowing and the bartender started grinning.

My friend and fellow Mainer, Sarah Caban, came up to me and told me she had an idea. The game she brought was a Rock, Paper, Scissors tournament. Would it be OK if she explained it to everyone? Of course. She got a gaggle of demonstrators, grabbed the mic, and off she went:

Play escalated quickly:

Until the final, glorious climax:

To me, this is the wonder and worth of MTBoS. Somebody has an idea, other people are game to try it, everyone else has a good attitude, and we make stuff (moments, memories, play, thinking, resources) together. Honestly, I felt such an outpouring of love for all these people at this moment. It was the highlight of my conference week.

I don’t see fandom here. Rock, paper, scissors was Sarah’s brainchild, and it was her first game night. She joined twitter about a year ago. The rock, paper, scissors finalists were not Dan Meyer and Andrew Stadel. They were two women I’d never met before. I hope our winner will remember what it felt like to have all those math teachers at her back, chanting, “Dana! Dana! Dana!” Because to me, that’s what we do every day. We have each other’s backs. We cheer each other on. We share our passion for this work, and our ideas, and our energy.

Sign me up to be a fan of that.

I’m pleased to say Matt and I have institutionalized this event going forward. #MTBoSGameNight will continue to be hosted by NCTM. That may sound like a small thing, but it’s not. From where I sit, the merger with The Math Forum, the emphasis on #MTBoS representation on all NCTM committees, the MTBoS keynote in Nashville, and the support of our fringe events like Game Night and ShadowCon are meaningful. I’m all for pressing NCTM to be what we need it to be. But I’m also all for recognizing the big shifts made over the last few years. Change comes faster within the (unstructured, unregulated) #MTBoS than it can within the (highly structured, institutionalized) NCTM, just by the nature of the beasts. But good change is happening nonetheless.


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. 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.