How Not To Start Math Class in the Fall

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

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

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

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

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

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

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

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

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

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

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


Twitter Math Camp Keynote

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

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

Slides (high quality): Zager TMC16 keynote Minneapolis

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

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

Extending the Book Experience?

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

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

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

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

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

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

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



Straight but Wiggled

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


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


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

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

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

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

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

The kids gasped.

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

Students liked this idea.

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


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

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

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

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

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

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

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

Abby: “Two.”

“What about now?”


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

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

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

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


Abby: “It’s pretty close.”

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

“Then yeah, it’s a straight line.”

“What about this one?”

“That’s a laying-down line.”


What about this one?


“That’s straight but wiggled.”


Oh man. How awesome is that?

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

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

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

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

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

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

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

“Kids, line up in a straight line.”

“Sit up straight.”

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

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

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

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

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

“Like what?”

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

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

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

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


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

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


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

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

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

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

There were lots of furrowed brows.

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

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

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

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




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

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

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

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

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

ignite script 1.jpg

ignite script 2.jpg

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.

Talking Math in Ghirardelli Square

My stellar week of NCSM/NCTM fun ended with my family flying to San Francisco for a weeklong holiday. We started out at the Embarcadero, and ended up in Ghirardelli Square. We were sitting on a bench, eating our hot fudge sundaes, when my husband, Sam, started talking about the sale on big bags of Squares® inside.


He said he’d worked out the math, and big bags came out to about three Squares® per dollar. He was thinking about picking up a bag to take back to his office. I thought about grabbing one for Stenhouse, and one for my school. Conversation moved on. Eventually we all fell quiet.

Several minutes later, I was people-watching when my first grader, Daphne, asked, “Is it possible to break ten into three equal pieces?”

I asked why she was asking. (I do this often.)

“This is a question I have a lot. Like, today, when I wanted to buy my new doll, Olivia, it was some amount of money for three of them, and I wondered how much it would be for one of them, because I only wanted one.”

I was a little slow on the uptake. Long conference week. I sat there, blinking, for a minute. And then I said, “Oh! You’re asking about the chocolate, aren’t you?”

“Yeah. Because Daddy said they’re three for a dollar. What if you only wanted to buy one of them? How much would that be?”

“I see. But you didn’t ask if you could break a dollar into three equal parts. You asked if you could break ten into three equal parts. Where’d the ten come from?”

“Well, you know how there are ten 10s in 100?”

“I do.”

“It’s basically the same thing. Like, 6 + 4 = 10, and 60 + 40 = 100.”

I asked where the 100 came from.

“The 100 is because there are 100 pennies in a dollar. So if I figure it out for 10, then I’ll be able to figure it out for 100, because it’s basically the same thing.”

This is where I stopped her for a second so we could high-five. I mean, holy use of mathematical structure, Batman.

Daphne went on, “What I really need are ten things. Oh, rocks! Perfect!”


Daphne worked with the rocks for a long time. She was thinking hard. I kept her sister quiet, which is the challenge in moments like this.


Daphne finally said, “I don’t think I can make three equal pieces with rocks, because I can’t break this last rock apart. It works out to 3 + 3 + 4 or 3 + 3 + 3 + 1. I need a piece of paper to show you.” She grabbed the envelope I had in my bag.


“See? It doesn’t work because I can’t break this rock! I can’t cut rocks into thirds.”

I asked, “What if they were crackers or cookies instead?”

“Then I could break up the last piece into three equal pieces. Then they’d each have three whole ones and one-third.”

“What do you mean, one-third?”

“Well, a third is one of three equal pieces.”

“What if you have three pieces, but they’re different sizes?”

“Then they’re not thirds. They’re just three chunks. I learned this from listening to you and Maya talk, by the way.”

Hmm. +1 for older sisters.

I asked, “So what do you think they do if they want to sell one piece of chocolate? How much should it be?”

She said, “With money, I should be able to to break it up. I can make change for the dollar. So I have 100 cents. So, 30 + 30 + 30 is 90, and that leaves 10 more…”

At this moment, she leapt up. “Wait! It’s the same thing again! It’s going to go on forever! With 100, it was 30 + 30 + 30 with 10 left. With 10, it’s 3 + 3 + 3 with 1 left. I can split up that 1 into three pieces, but there’s going to be a piece left. That one extra piece MAKES IT GO ON FOREVER! There’s always going to be an extra piece! Three, three, three!”

I happened to have a conference schwag calculator in my bag, and she got to see 100 ÷ 3.


“So how much should one piece of chocolate cost?”

“One-third of a dollar.”

“How much is that?”

“About 33 cents. If they charge 33¢, they get pretty close to a dollar.”

“How close?”

“Well, three chocolates would be 33 + 33 + 33, so 30 + 30 + 30 is 90, and 3 + 3 + 3 is 9, so that’s 99¢.”

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I just spent a week thinking about the teaching and learning of mathematics with all kinds of amazing people. So much of the conversation is about how we can create the conditions so students do what Daphne did here:

  • She noticed math in the world around her, and wondered about it.
  • She posed an original (to her) mathematical question.
  • She used structure to think about that question (in this case, the structure of place value).
  • She used the strategy of solving a simpler problem.
  • She looked for patterns and regularity.
  • She stuck with her problem when it was hard for her.
  • She used tools, representations, and models.
  • She decontextualized and recontextualized the problem.
  • She reasoned and justified.

I mean, she was all over the SMPs, right? And naturally, too. I wasn’t pushing any math at that moment, believe me. I was sitting there like a lump, very tired and very happy to be with my family. But I’ve done things at other times. Namely, I’ve made it clear to Daphne that math belongs to her. That her ideas are valuable. That I’m interested in them. That math involves asking questions. That she can figure things out for herself. That she owns the results of her investigations. That math is all around her. That she is a mathematician.

It’s paying off. Now, if only she can hold on to all of that.




Monday, 1:30 – 2:30, OCC 210/211.

“I’m Not Really a Math Person”: Coaching Anxious Elementary Teachers

NCSM Oakland 2016 slides



Friday, 9:30 – 10:30, Ignite! Moscone 134.

Going Beyond Groupwork: Teaching Students to be Mathematical Colleagues.



Saturday, 8:00 – 9:00 Moscone 3003 and then 9:30 – 10:30 Moscone West 2nd floor cove 1.

How do they relate? Teaching Students to Make Mathematical Connections

Zager NCTM SF 2016 presentation