The Zeroth Law: Thoughts after NCTM 2017

“Joy is a zeroth law in mathematics.  When true mathematics is happening, joy is involved.”

Someone’s joyful Cuisenaire rod creation during choice time in my classroom.

Anita Wager and Amy Noelle Parks started off my time at the NCTM Research Conference and Annual Meeting with this proclamation, and as I’m sitting on this airplane, trying to synthesize all of the ideas buzzing around my brain, it strikes me that this is a uniting principle.

It’s certainly not a given, the idea that a national math teachers’ conference would be centered around joy. In fact I think if you asked most people, “What do you think they will be talking about at the national math teachers’ conference?” joy would not be one of the top 5 answers on the Family Feud board. And yet. There it was. Everyone talking about joy.

So many of us want to make mathematics a joyful experience in our classrooms. We want it for our kids, and honestly, we want it for ourselves. You know that feeling when you’re laying in bed, or standing in the shower, or walking up the stairs to your classroom, and you’re playing the day forward in your head, and there’s a lesson that you “have to teach” and you’re dreading it? You’re already thinking about how not fun it’s going to be for you, and for the kids? Well, Anita and Amy’s talk said to me that the feeling I’m describing should be a red flag. A call to action. A moment to absolutely stop in your tracks and NOT. TEACH. THAT. LESSON.

@TracyZager talked about gut instincts and how we can get our students to listen to the “little voice” in their heads rather than telling them to listen to us. I’m thinking that applies to us teachers, too. When the little voice is telling me that what I have planned for my kids that day is not going to be joyful for them, and in turn for me, I need to change something.

But, a few questions, right? What about “covering the material?” What about all that stuff that needs to get taught? Some of it might be a little boring. You know, like … (racking my brain for something that is categorically, inherently boring to teach) (still racking) um, I couldn’t really think of anything. Maybe like “how to fill in the bubbles on this standardized test grid.” So if there is no mathematics topic that is inherently boring to teach (and MTBoS, please, speak back to me if you think there are some), I’m thinking that means that making mathematics a joyful experience for my students is up to me. For the most part, I have the agency to decide what I teach and how I teach it. So in regard to covering the material (a term that I deeply hate because it feels like the complete opposite of teaching for understanding), I think we need to know the math we’re teaching deeply and we need to know ways that we can teach it that promote joy. I’m thinking about when Tracy quoted somebody’s grandmother, in answer to the question, “Doesn’t this type of teaching take too much time? What about the pacing?” Somebody’s grandmother said, “If you don’t have time to do it right, you must have time to do it twice.” I’m arguing that doing it right doesn’t just mean teaching for understanding; it also means teaching for joy.

As Amy said, joy in mathematics teaching and learning does not mean “ ‘wrapping peanut butter around something that tastes bad’ to make it more palatable.” It means revealing to kids the beauty and creativity in math. It means giving them a chance to have the deeply pleasurable feeling of figuring something out, of seeing connections, of seeing something in a new way, of having something someone says to you or shows you mathematically suddenly “click.” It’s hard, sometimes, when we are thinking about the content that we want our kids to understand before they leave our class, to keep a goal around joy in mind. But just like we don’t want to teach kids “rules that expire” or procedures for solving problems that don’t make sense to them, because that wouldn’t serve them well in the long term, teaching kids that math is not inherently joyful or pleasurable doesn’t serve them well in the long term. We owe it to our kids and to mathematics to always be cultivating an image of the discipline that reflects its intrinsic joyousness.

How are we going to do that? At first I thought step one was what I described above: stop doing lessons that lack joy and regularly, committed-ly seek ways to teach that same content joyfully. But then I took one more step back. Because I don’t actually just want joy in my classroom. I want it in every classroom.  Then I came to the crux of the problem, or one of them at least: if teachers don’t view mathematics as joyful, they can’t cultivate classrooms that do. We all know the “math autobiographies” of our colleagues who don’t like math, because of their experiences with math as kids. We know there are many teachers who still think “I’m just not a math person,” some of whom are trying valiantly to steer their children toward a growth mindset, showing Week of Inspirational Math videos and making their kids attach “yet” to any self-disparaging comments, while still holding inside of them their own insecurities about their relationship to math.

So my idea is this: we have to give teachers lots and lots of opportunities to experience joyful feelings when they do math. Beautiful, visual math like Which One Doesn’t Belong? and Estimation 180 and Lusto’s Dots. Things that make teachers gasp and exclaim out loud. That is the feeling that we want people to have. That is the feeling we want people to associate with doing mathematics. There are lots of ways to get that feeling, but those that I just mentioned are great, easy entry points.

Let me also add an important note about the definition of joy: Amy and Anita say that joy and sadness are not opposites.  Joy and disengagement are. Putting that together with the statistic @JoBoaler shared, that 70% of K-12 teachers do not feel engaged in their work, makes me feel somewhat disconsolate about the state of things. But a possible silver lining is this: if you’re leading a PD and teachers cry over a problem they can’t solve, or get mad and argue with you or their colleagues, you don’t have to see those moments as non-joyful. In fact, those strong feelings come with engagement. And if disengagement means no joy, then doesn’t engagement mean … joy?

Leaving NCTM 2017 I feel so terribly engaged. I’m so grateful for the MTBoS and all of the teachers, coaches, teacher educators, and researchers who made my synapses fire like crazy and gave me chances to play with math this week. I’m leaving with a lot to think about, write about, and do. In the spirit of many of the presentations I loved this week, I’m going to leave you, and myself, with a call to action, and I’d love to hear from you if you take it up:

Call to Action

  1. Change one of the lessons that you hate to teach to add more joy. Do a WODB or a unit chat or any one of the things @Trianglemancsd shared on Friday.
  2. Find one of your colleagues whom you know thinks of math as something they have to teach rather than something they love to teach. Do the same thing with them. See if you can get them to gasp.



Tape Diagrams, Big Feelings and other Predicaments of Teaching

This week in math I did something I have deeply mixed feelings about … I introduced the tape diagram to my 2nd graders.  For those of you not familiar, the tape diagram is a model for representing a mathematical situation that is a favorite of the EngageNY/Eureka Math curriculum, which my school started using about four years ago.  While EngageNY loves the tape diagram, I can tell you from experience with both 2nd and 4th graders that kids remain unconvinced of its magical qualities.  It’s pretty common to hear groans when you say, “Can you make a tape diagram to show what’s happening in that problem?”  Many, many 4th graders I taught repeatedly told me, “Tape diagrams don’t help me.”


There is so much confusion around the tape diagram, for kids and teachers alike.  Kids seem to find it extremely onerous to represent the action of a story problem in a tape diagram, even when they understand what’s happening in the problem and can write equations that match the steps needed to solve it.  They don’t WANT to make a model. They just want to get the answer!  

Teachers wonder how much value they should assign to a child’s ability to make a tape diagram that accurately represents the problem.  If a kid can solve the problem but can’t make a tape diagram that matches, what does that mean?  Is a tape diagram a tool for problem solving, or a way to represent what’s happening in the problem after you’ve solved it, or a way to make sense of what’s happening in the problem?  Is there an audience for the tape diagram — like is it something that you use to communicate your understanding of the problem to others, or is it a tool for deconstructing the problem and understanding it yourself?  These are some of the questions my colleagues and I have tossed around during our conflicted relationship with the tape diagram.

A big, big question for me as a 2nd grade teacher is whether or not making tape diagrams is developmentally appropriate for my kids.  At times I’ve thought that it’s an abstraction of the problem that just doesn’t make sense to my students, because they are not yet able to think abstractly.  I worry about what happens when we push kids to do things that they aren’t ready to do.

Enter my classroom on Wednesday morning.  EngageNY has offered my students, on their first foray into making tape diagrams, the following problem:

Shawn and James had a contest to see who could jump farther. Shawn jumped 75 centimeters. James jumped 23 more centimeters than Shawn.  Draw a tape diagram to compare the lengths that Shawn and James jumped.

After introducing the tape diagram to the whole class, I sent students off to work on making diagrams for a set of word problems.  I was working with two students on the rug, and slowly but surely, my group grew and grew as students trickled over for help.  “I don’t get it,” said Veronica, a feisty seven-year-old with a flair for the dramatic.  She rubbed her head and frowned.  

“OK, come join us, we’re all reading the problem together,” I tried.  I read the problem out loud to the group.  I got up and acted it out, jumping Shawn’s jump, having the kids mark where I landed, then working together with them to actually measure out 23 centimeters beyond that to show how far James jumped.  I said, “Do you see it? Do you see each of the jumps here?” I gestured at the space on the carpet where I had just jumped. “Can you try to draw a tape diagram that shows Shawn’s jump and James’ jump?” I asked them, my tone revealing both hope and doubt.

Veronica frowned again.  “I can’t,” she said.  She looked up at me.  “My brain can’t do all of these things.” (Big Feeling #1 – frustration.)  She looked down at the paper.  “It’s too small,” she said, pointing to the space allotted for drawing the dastardly diagram.

“You need more space?  Sure, go get another piece of paper,” I suggested.  But the whole time, I was thinking, “Yup, she’s right. This is just way too hard.”  I hate the feeling when I see the work through my kids’ eyes and I feel how hard it feels and I think, “Is what I am doing totally inappropriate?” and then I slog forward anyway, because that’s what we’re doing that day.  I really hate those moments. (Big Feeling #2 – self-doubt.)

Veronica came back with her new piece of paper, but she never made a tape diagram on it.  Instead, she poked holes in it with her pencil until it resembled Swiss cheese.  Then she put her pencil near her top lip and pushed it up above her teeth to make a funny face, distracting another kid in the group (who was really just attempting to copy the tape diagram I had drawn on the board during the lesson, because she too couldn’t figure out how to do anything else).  I started getting that feeling like, “Now I’m going to lose the whole group and no one even understands this anyway!” (Big Feeling #3 – anxiety.) “Veronica, go back to your table,” I said.  She sighed and stomped off. (Big Feeling #4 – anger.)

Can you believe it? I sent that poor girl back to her table.  I’m ashamed to tell you, world, that I did this, but that is what happens when tape diagrams come out.  Big Feelings ensue.  And sometimes they make you do things you wish later you hadn’t done.

About fifteen minutes later, it was time for lunch and recess.  The students cleaned up their math materials, got jackets, and lined up.  Veronica’s face said it all.  She pulled her hood up over her head.  The tape diagram Big Feelings had not gone away just because math was over.  We started walking down the stairs to the lunchroom.  I heard some commotion in the middle of the line.  When I turned to look, Veronica was coming around the corner, and then I saw her push the boy in front of her.  Pretty hard. Hard enough to make him fall down the stairs (which, thankfully, he didn’t).  I gasped, audibly. (Big Feeling # 5 – terror.)  The rest of the class turned to see what had happened. Veronica froze.  Everyone was OK. But wow.

When we got down to the lunchroom (safely), I pulled Veronica aside.  She immediately started weeping.  And then I apologized to her.  (Big Feeling #6 was the same for both of us – remorse.)  I told her that I was so sorry that I had sent her away from the group during math.  I explained that my Big Feelings had gotten the better of me, just like hers had on the stairs.  We hugged.  She joined the class for lunch, seeming a little bit lighter.

What I thought about after this experience was how much Big Feelings impacted our ability to teach and learn math that day.  I thought about something I had read recently from David Cohen’s Teaching and Its Predicaments:  

“To turn up evidence that students have not learned is one of the most threatening things teachers can do; a student who fails to comprehend is an actual or potential failure for the teacher. The more vivid the evidence that students did not learn, the more troublesome it can be. This is another predicament of teaching: acquaintance with students’ knowledge is full of promise but loaded with problems.”

When we teachers feel that our students don’t understand something we’ve attempted to teach, we have choices.  We can think that we need to try harder, or change the way we taught it.  We can think, “This wasn’t developmentally appropriate.  They’ll get it when they’re ready.”  We can think that we tried our best and if that kid is just going to fool around, it’s on them.  Each of these possible interpretations reflects the teacher’s emotional response to the deeply threatening possibility that a student has not learned.  I think I went through all of those with Veronica on Wednesday.  Teaching is just so hard.

I still don’t know whether tape diagrams are worth the Big Feelings they bring up.  But what happened on Wednesday reminded me that all teaching and learning, math or otherwise, is emotional business.

Saying vs. Doing

img_2469Something I am continually working on as a teacher is saying less.  This became even more important last year, when, after teaching 4th grade for many years, I became a 2nd grade teacher.  The attention span of the 7 year old is pret-ty short.  And I quickly learned that they have little to no interest in long, thoughtful lesson introductions that connect what we’re going to study today to the previous day’s work. I wish I could illustrate for you the bored stares, the poking of the person in front of them, the taking books out of the classroom library baskets next to the rug and thumbing through them, the vehement “can I go to the bathroom?!” crossed-finger silent gesture that my sweet 7 year olds were throwing me last year as I fumbled my way through learning that I needed to get. to. the. point. Fast.  With 2nd graders, it’s all about shock and awe.  You have to jump right into it.

I’ve been thinking about this a lot, this “say less” thing, as I move into the new school year.  The first few days of the year in my classroom have been filled with wordy explanations of classroom routines and procedures. Are you yawning yet? I am. As critical as I know it is to teach my students how to properly unpack and pack their backpacks, it’s extremely uninteresting to me, and to them.  I spent the first few days of school feeling a bit uninspired by all of this procedure-teaching.  The best moments of each day, I noticed, were when we were DOING: singing the Shark Attack song together, playing hospital tag during outdoor play, building patterns with unifix cubes.

I started thinking about math teaching.  When I teach math, I want my kids to DO — I want them to discover the way our number system works, explore the properties of operations, engage with each other, and wrestle with mathematical ideas.  It’s quite active.  It does not look like me telling them, “Here’s how you solve the problem.”  But when it comes to teaching classroom routines, I’m doing the opposite.  I’m telling them, “If we all go to the closet at the same time, we’re going to have a problem.  Here’s how we’re going to solve that problem…”  In my deep boredom with the teaching and practicing of classroom routines, I started wondering about what it would be like to let my students DO more, right from the get go, instead of teaching them how I want them to do it preemptively.

I realize this is classroom management 101 anathema.  Anyone who read The First Days of School by Wong and Wong (shudder) is thinking, “Disaster!”  But I honestly wonder.  Just as in math, I don’t teach my kids procedures because I want them to think about what makes sense, maybe when it comes to classroom routines and procedures, I could let them DO first and discuss what made sense and didn’t make sense later.

Taking an approach like this would definitely involve a higher tolerance for classroom chaos than I presently have.  But even that element is not that different from math teaching.  Over the years, I’ve developed a high tolerance for messiness when it comes to math teaching and learning.  I know I used to feel anxious when it seemed like kids weren’t “getting” what I wanted them to see in the math, but over time I’ve been able to develop the part of my teacher self that sees kids’ struggles as part of the road to making sense, and to not get freaked out by the chaos that sometimes ensues on that road.

I know that content-wise, classroom routines and procedures are not the same as math.  If kids don’t know how to move around the room or get their materials quickly, learning time is wasted.  Maybe it does make sense to teach classroom routines explicitly.  But I think doing so this year has brought out for me the tension around my desire for control versus my desire for exploration and sense making in my classroom.  So this is an important thing that I’m going to keep thinking about:  How can I continue to create a classroom where doing comes first, and saying comes later?

First post: What We Notice

Studying student work is important.  It can tell us so much about what students know.  It’s our jumping off point for future lessons.  It gives us a window into the minds of our kids, especially the quiet ones from whom we don’t hear much in class.  I personally enjoy studying student work in the peace and mental quiet of my own home, in my classroom at 4 pm, or in any place that is not a room filled with children shouting, “Can I go to the bathroom? I’m done.  What are we supposed to do now?  He hit me. Is it almost time for lunch?”

And yet, I’ve noticed that when teachers get together and study student work, it can sometimes be hard for us to see what kids know and can do.  We often see what they don’t know.  Why is this?

Recently I was at a conference where I was working with a group of teachers to study some student work and consider what our next teaching steps might be if these were our students.  To give some context, we had never met the students whose work we were examining, nor did we have access to their teacher.  We just had the work. We put it between the four of us.  We put on our serious teacher faces.  We studied.

I’m always wary of the person who speaks first in one of these situations.  I’m wary to be that person and I’m wary of that person.  But someone’s gotta do it.  So that person (it wasn’t me this time) did it:  “Well, we can see that he really doesn’t understand the meaning of the equals sign,” she said.  I considered this.  It was true that the student seemed to be confused about the equals sign, I thought to myself.  The next teacher chimed in:  “It’s hard to get a sense of anything from this chicken scratch.”  Uh oh. Handwriting commentary gets under my skin.  I waited a little longer.  Next teacher:  “He subtracted where he should have added.  He doesn’t understand negative numbers.”

None of these comments are out of the ordinary for teachers studying student work, and I don’t think any of them are “bad.”  But they made me wonder why it’s often our default to approach students from a deficit perspective.  Until I recognized this in my own teaching (which took a while), I was guilty as well.  Maybe we default to this because it justifies our existence and importance as teachers — if there’s a problem, we have some work to do! If they don’t know something, we can fix it! We can be the Givers of the Knowledge.

The problem with this perspective, for me, is that a) I don’t want to see my students as knowers of nothing or heads stuffed with “misconceptions”* awaiting correction, because I don’t think it is true and I don’t think it honors kids very well; and b) I don’t want my students to see me as knower of everything and provider of correction, because I don’t think it’s true and I don’t think it honors teaching very well.  If I believed my job was just about correcting mistakes and telling kids how to do stuff, I might never have been interested in teaching in the first place.

Since I want to be a teacher who helps kids see their own power to make sense of math (and the world), I have to be a teacher who sees kids as having power and knowing things.  If I want to be that teacher, I have to approach students and their work by thinking, “What does this student know?”

Even with this approach, though, it’s not always easy to understand student thinking.  If you like reading research, this article by Jacobs, Lamb and Phillips nicely highlights how difficult it is for teachers to get good at studying student thinking and noticing what’s important.  They write, “Expertise in attending to children’s strategies is neither something adults routinely know how to do nor is it expertise that teachers generally develop solely from many years of teaching.”  They also say that the skill of attending to children’s strategies is regularly overlooked by professional developers.  In their study, Jacobs, Lamb and Phillips found that it wasn’t just processing capacity (e.g. all the kids in the room yelling your name as you’re trying to look at a kid’s work) that created challenges for teachers in noticing the right things in student work; it was that teachers actually need more instruction in noticing what is mathematically significant and need help developing “skill in finding those mathematically significant indicators in children’s messy, and often incomplete, strategy explanations.”

How can we get better at noticing what kids know mathematically?  Approaching students with the disposition that they have important mathematical ideas, that they are making sense of something, and that their ideas are meaningful to them, is a necessary, but not sufficient, first step.  It’s a goal I’m taking with me into my classroom this year (and every year) as I get ready to teach my 2nd graders.  I offer it up if you’re looking for a goal for yourself this year as a math teacher that doesn’t involve a lot of making copies, putting manipulatives in baggies, or even writing awesome tasks that will get your kids excited about math.  It’s more of an internal math teacher goal, but I like it.


* I can’t stop thinking about Rochelle Gutierrez’s talk at NCTM this April:  “Mathematics Teaching as Subversive Activity: Common Core, Social Justice, Creative Insubordination”  She had a lot to say about the weight of the words we use when we talk about students, and one of her pet peeves is the word “misconceptions.”  She said (paraphrasing from my notes here), “We often tell teachers ‘anticipate your students’ misconceptions.’  Students don’t have misconceptions!  They have conceptions.  They have conceptions until those conceptions bump up against something that causes them to no longer work.”  I loved this reframing.  I still say misconceptions, but this idea tickles my brain every time I do.  Passing that on to you.