Finding Newness

One of my dear friends became a teacher about 10 years after me.  Recently, he reminded me of a conversation we had after the end of his first year of teaching.  He was telling me about all of the things he wanted to do differently the following year (pretty much, everything – isn’t that how your first year of teaching is?) and I said, “Oh yeah. Your first year is all about your second year.”

I strongly remember the feeling of bumbling my way through my first year of teaching, every moment filled with thoughts of HOW I WILL DO THIS DIFFERENTLY NEXT YEAR.  So many goals! I love goals. Goals give me something to do, something to dig into, something to read about and talk about and learn about. They help me know that I’m making progress. In short, I am all about setting goals.

Flash forward: it’s teaching year 19 for me. And coaching year three.  And this early September, I’ve been having some trouble figuring out my goals.

One reason for this is that goals come in many sizes, and the longer I’ve been teaching, the bigger my goals have become.  I miss the goals that were like “revise these lessons to make them better” or “play music in the classroom.” Those goals were so attainable. When I became the math coach at my school, I had some attainable math coach goals, too – like “make a math website for our school” and “revise these lessons to make them better [for every grade]”.  My work was still meaty, in that I was working on a greater scale, albeit with a narrower scope than when I was a classroom teacher. 

But the big goals — the truly big goals, like “align our curriculum and our teaching with our beliefs about math pedagogy” and “improve teachers’ ability to facilitate math discussions” and “deepen and broaden teacher content knowledge” — these are not goals that can be attained in a year, or two, or three.  You can make progress, and I think we have, but progress is hard to measure on a schoolwide scale, and from a coaching perspective. What metrics should we use? Do those actually reflect progress? How will I know when I can check off a goal like any one of those as “done”? These are some of the questions that are plaguing me as I enter year 19/year 3.

Even more, what I was missing a little bit as I started to prepare to talk with staff today and tomorrow, was the excitement of embarking on a new project.  When a goal is new, I have so much energy and momentum to put towards working on it. But what about when a goal is old? How do you find the newness, the inspiration, in an old goal?  And if I can’t find it, how am I going to get the teachers I coach excited, inspired, about our goals?

My mom is selling my childhood home and this past weekend, I was there, sorting through piles and piles of papers and notebooks and artifacts from my school career, preschool through college. What I re-found out about myself from reading through my high school English journal entries and my papers on Othello and Ulysses, is that I might be a math coach now, but I’ve been a reader and a writer my whole life.  Words – my own and those of the authors I love and admire – are everything to me. So when I was feeling a little lost about how to inspire myself and everyone else around our new/old goals, I sought my own inspiration in books, in other people’s words and ideas. Tracy Zager’s chapter on mathematicians using intuition, Deborah Ball’s AERA talk on discretionary spaces and the power of teaching — these texts helped me remember why I’m doing what I’m doing, where I’m hoping to go with it, and they helped me find some newness in places that felt a little old.


The Work Inside of the Work

Finally, fractions season is upon us. I’m sure there are people who dread it, but I’m not one of them.  I love the complexity of teaching fractions, the way you have to choose your words so carefully around them, the way they act like other numbers even when you don’t think they will, the fact that they allow you to talk about the number line, the spaces in between and how they are the thing … the spaces are the THING!  The very appearance of a fraction is just so intense. It looks like two numbers, but it’s one number! You just can’t beat all of that for teaching fun.

Giant cuisenaire rods!
These are my new giant Cuisenaire rods on a masking tape number line. I’m kind of obsessed with them.

For the month of March I’m going to write a series of posts on my classroom experiences with the teaching and learning of fractions.  As the math coach at my school I have the opportunity to work in 3rd, 4th and 5th grade classes simultaneously, so I get to really dig into the CCLS fractions progression with kids and teachers, and see firsthand the ways in which students’ ideas about fractions build, slip backwards, and leap forwards.

In January, I had the chance to see Phil Daro (one of the authors of the CCLS) speak at Teachers College in New York City.  Phil said a number of things that stuck with me, but the one that I’ve been thinking about the most this past month came from a story he told about the authors’ debate over whether or not to include long division in the CCLS (“it’s surprisingly unuseful,” Phil said, “…these days, with calculators…”).  When they consulted with a group of Japanese teachers on why they included long division in their curriculum, the Japanese teachers said something like, “Long division is such a wonderful context for teaching the unfinished business of place value.”

The idea of unfinished learning is a powerful reframing of “learning gaps,” a term I hear often (along with “strugglers,” “kids who can’t access this math,” and a plethora of other names for, well, regular kids who are learning math at a pace that doesn’t seem to match with the pace of the curriculum or the state testing schedule).  It brought me back to a related sentence from Bill McCallum’s blog post of long ago that I still can’t stop thinking about:  Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.”

I’ve been carrying the idea of unfinished learning as an opportunity, not a hindrance, and the idea of using the progressions to understand student thinking about fractions, into a whole bunch of 5th grade classrooms this past month.  One thing I’ve seen is that when you welcome unfinished learning, instead of feeling threatened by it, you can get pretty excited about how much of it there is to explore.

Last week in one of my 5th grade classrooms, the teacher posed this question:  Is 7 x ⅘, and ⅘ x 7, the same as (7 x 4)/5?  Exploring these three related expressions, students drew tape diagrams to match each one and talked animatedly with their partners about their ideas.  When they came together as a whole class to share their thinking, one student seemed particularly unsettled. “I know that 7 x ⅕ is 7/5, so 4 of those is 28/5,” Allison started. (Me: marveling at her application of the unit fraction work she did in 3rd and 4th grade to this new concept for her.)  “But the tape diagram would look like this, and I don’t understand what that has to do with (7 x 4)/5.”  She showed her tape diagram to the class and other students raised hands to try to answer her question.  

“7 x ⅘ is the same as ⅘ x 7 because in multiplication you can switch the numbers around,” a boy said (Me: marveling at his application of the commutative property of multiplication, an idea introduced in 3rd grade, to this new situation.) (Also, this didn’t exactly speak to Allison’s question – have you noticed this is how kids often respond to each other? With something only sort of related?)

Allison still looked unconvinced.  “What do you mean?” she said.

“You know, like 6 groups of 2 is the same as 2 groups of 6,” another girl chimed in.  A little side discussion ensued in which several kids made the motion of turning a rectangular array one way and then the other to illustrate the commutative property.  “And so if you were going to make a tape diagram for this one,” the girl said, pointing to (7 x 4)/5, “first you would make 7 4s, or I guess you could make 4 7s.  And this other one is like 7 four-fifths. So it’s like you have 7 copies of 4 but now the 4 is four-fifths.”

Allison paused.  “So 7 x 4 means 7 copies of 4?” she asked.  

At this moment her teacher and I exchanged a look of genuine surprise.  Here was a bit of unfinished learning, not about fractions, but about the meaning of multiplication (one meaning of multiplication, anyway), bubbling up in this fractions/multiplication/division discussion.  Without this conversation, we never would have known that Allison had unfinished learning around the meaning of 7 x 4. I should also say that Allison is not a “struggler” in math class, and she generally appears to be making sense of grade-level work.  So whatever unfinished learning she had around single-digit multiplication has not been holding her back. What I loved about this moment was the way that Allison was really working to make the math make sense to her, bringing everything that she understood from before, along with all of her unfinished learning, into her math community.  And the other kids were bringing all of that too. And they were just figuring stuff out together, like putting together pieces of a puzzle, a puzzle that doesn’t get finished in one school year.

There are a lot of different joys that come from teaching, and one of them is when you’re really wanting kids to understand something and then they do.  But another kind of joy, that’s newer to me, is the joy that comes from just watching kids wrestle with ideas and NOT reach that moment of understanding.  Watching kids wrestle with ideas and letting the conversation end unsettled, letting the learning be unfinished, and thinking that true honest learning mostly goes that way.

Scenes from a Maybe Understanding

You know when you’re working with kids, and you’re feeling really good about their progress, and about yourself as a teacher, and all of a sudden you ask them something and the answer you get makes you stop dead in your tracks?  The answer makes you wonder if you might have misheard, or if you have actually been deluding yourself all year long?

Yeah. So that happened.

To back up:  This year in my new role as math coach, one of my jobs is working with small groups of students who need extra support in math. Two of my fifth graders have been enthusiastic participants in math group all year.  Fifth grade math, by the way, is hard. These students entered 5th grade on somewhat shaky ground and they’ve been working hard to hold their own, but there’s just so much to hold on to. I admire their tenacity. Both are girls. One is a student for whom English is a new language.  They are often quiet in their whole-class math lessons and discussions, but seem to find math group to be a place where they can share their ideas freely, and they often become joyful and animated in our work together.

About a week and a half ago, these two girls (I’ll call them Ariana and Samara) were given some problems around finding volume with fractional units. They were being asked to multiply mixed numbers.  The dimensions of the first rectangular prism were 4 ½ x 3 x 2 ½ inches . As they eyed 4 ½ x 3, Ariana said, “So do 4 x 3 and ½ x 3…?”

“Does that make sense to you?” I said, trying to stay neutral.  Ariana said that it did, and Samara agreed, but when they went to calculate ½ x 3, they were stumped.

I was a little surprised that ½ x 3 was such a curveball for them.  I prompted a little … “So ½ x 3 … you could also think of that as 3 copies of ½ …” and then I waited.  (I realize that in some senses, ½ x 3 is not the same as 3 x ½, but I intentionally used commutativity here because I thought it would give them more access.)  

“Ohhhh! Yeah! 3 x ½ ! So 3 times 1 is 3, and 3 x 2 is 6, so it’s 3/6,” Ariana said.

Common misconception, right? What’s the big deal? Ariana is certainly not the first kid to begin blindly multiplying stuff because, well, because she can.  But. We had done so much work on understanding the meaning of the numerator and the denominator earlier this year! And they did that work in 3rd grade! And in 4th grade! In fact, 3 x ½ is a 4th grade standard problem!  What was going on here? I was disturbed.

I put 3/6 and 3/2 up on the board as two possible answers to 3 x ½ .  Ariana was certain that it was 3/6, and Samara said she was not sure, but she thought maybe it could be either one.

Wait, WHAT?!

I had them draw a picture to support their ideas.  Look at this. (I had to reconstruct the work later because I didn’t realize I was going to write about it until it was over.)



Can you see how both of them used these models to make an argument for 3/6? I was losing my mind, and at the same time, I was kind of loving the challenge.  I took a deep breath.

“Samara, your picture shows this,” I said, writing “½ 3” next to her model.  “Can you see why?” She said she sort of could, but sort of couldn’t.

Ariana’s model looked more like 3 copies of ½, but she was still convinced that there were 6 equal parts and 3 of them were shaded, so it must be 3/6.  Again, this might all seem normal and understandable except that our curriculum and all of our teaching in 3rd and 4th grade is around developing a deep understanding of unit fractions and the way they work.  The language, for example, “3 copies of ½” is intentionally used to help students think about ½ as a unit, and 3/2 as the result of making 3 copies of that unit. How could all of that work have flown out the window?

Our session was over.  I sent the girls on their way leaving this question unresolved, but knowing that we’d come back to it next time.  After they left, a 4th grader I also work with came into the room and saw the work on the board. I explained what the 5th graders and I had been working on, and that the girls thought 3 x ½ was 3/6.  “That’s not right,” my 4th grade friend said.

I felt somewhat heartened. “Why not?” I asked.

“Because the denominator always stays the same,” she said. I deflated again. Was she just repeating a “rule” she’d heard somewhere? Was this an overgeneralization she’d made on her own?  

“Why does the denominator stay the same?” I asked.

“Because …” she paused. “Because it’s the unit!”

There are a lot of things to unpack in here.  One is the amount of emotional rollercoaster that is involved in teaching for me.  Another is the question, how do you know when kids really understand something? And another is the question, can you really understand something as a 4th grader, like the idea that 3 x ½ is 3 copies of ½ or 3/2, and then lose hold of it as a 5th grader when you are trying to hold on to multiplying fractions by fractions, dividing fractions by fractions, dividing whole numbers by fractions, multiplying mixed numbers, 10 to the power of whatever, whole number multiplication and division with numbers of any size, and more?  How deep does your understanding have to be for it to be unshakeable, and transferable? What I felt in this conversation was that we were standing on a house of cards.

Today I returned to 3 x ½ with Ariana and Samara. I put up 3/6 and 3/2 again and again asked them to make a representation to support one of these two answers.  Both girls started by writing ½ + ½ + ½. This time I asked them to try a number line representation in addition to the other models they had drawn. I asked, “Where does ½ go on this number line?”  Here is Samara’s attempt #1 to answer that question:

fullsizeoutput_4740I was floored, but I tried to show no emotion as I asked, “How did you decide to put ½ there?”  And then she did this:


I realized I had to take a further step back.  I asked, “What does ½ mean?”

Samara responded, “It’s like you split it in half.”  

I took out the pattern blocks. “Can you find a block that represents half of this hexagon?” I asked Samara. She immediately grabbed the red trapezoid.  “How did you know?” I asked.

“Because a hexagon has 6 sides and this trapezoid has 3 sides,” she said. (Her class is currently studying attributes of shapes, so it didn’t totally surprise me to hear her say this. It also made me feel like, ahhhh! There’s too much math swimming in this 5th grader’s head!)  

Ariana jumped in. “No,” she said. “I don’t agree. Because this hexagon has 6 sides but the trapezoid has 4 sides and 4 isn’t half of 6. I don’t think it’s about the sides.”

“So what makes those halves then?” I asked, with full knowledge that we were now in the territory of 2nd grade standards.  

“They’re equal!” Ariana said, and Samara agreed. I put 3 blue rhombuses down on top of the hexagon. “So are these also halves then? They’re equal,” I said.

“No!” Samara said. “Those are thirds.”

“Why?” I asked.

“Because there are three of them,” she said. I put the trapezoids back. “So these are halves because … ?”

Ariana said, “There are two of them and they’re equal.”

So many things had to happen to get to this moment of understanding. First, I had to keep asking questions and trying to draw out the girls’ real, true ideas about fractions.  Second, I had to set aside my big feelings about the fact that they didn’t seem to understand things that I really thought they understood, things I had a personal stake in them understanding from all of the work we did together this year.  I struggled many times during these exchanges to find the right direction to take the discussion in, to find the right model or counter example that would nudge them toward sense-making. It was so hard. In the course of this discussion, we used equations, area models, number lines, and geometric models to re-engage with content these girls “learned” in 2nd, 3rd, 4th and 5th grades.  As their teacher, I had to know all of that content, how it connected, and how it had been taught to them in order to try to pull together a coherent set of experiences for them that would build or re-build their understanding of these ideas.

And after all of that incredibly complex work, where we ended up was: with a maybe understanding of the meaning of one half.

How Long is Too Long?

I recently wrote this post for The Math Collective blog and am re-posting here, because, well, when you only write a blog post every 6 months, you want to put it in as many places as possible!

“Keeping kids on the rug” is kind of a thing in elementary schools.  Since the dawn of, well, rugs in elementary classrooms, teachers, administrators and coaches have been pondering the length of time that is appropriate to keep kids in the meeting area for a lesson.  How long is too long? Should a math lesson be 8, 10, 13 minutes long? We seem to universally agree that 20 minutes is too long. Yet I doubt there’s a teacher I know, myself included, who hasn’t kept kids on the rug for 20 minutes or more. There’s just so much to teach!

A typical day in my classroom could easily look like this:  The kids gather on the rug for the math lesson.  Everyone is settled and attentive.  I start out with a little warm-up, kids are engaged, we’re feeling good.  I move into the big idea for the day.  Mild fidgeting ensues. We’re about 8 minutes in. Maybe a turn and talk will help, I think to myself.   After the turn and talk, I want to show them what one solution path might look like.  But I want to show it with unifix cubes first.  Noah and Eric have started poking each other, and Alysia is fully reading a book from the classroom library at this point.  “Alysia, can you make 18 with these cubes?” I ask, trying to pull her in. I’m sweating a little bit. We’re at like 12 minutes right now and I haven’t even done one example with them yet.

ontherugWe work through one problem with the cubes and I think they can see it. I think like 12 of them can see it. But there are 14 other kids in the class! I’m not sure if they can see it.  Let me try one more with the cubes, I think to myself (now arriving: 15 minute mark).  After the second example, I feel like I’ve gotten maybe four more kids on board.  At the same time, at least five more kids are completely checked out.  Alysia has moved on to Chapter Two of the book after her brief stint as “18 maker.”  I am definitely sweating.  

“OK guys, let’s just take a quick look at the problem set together,” I say.  They take out the workbooks they’ve been dutifully sitting on for 20 minutes.  We read the directions out loud.  About two kids total are paying attention at this point.  I am 100 percent sure that they’re not going to know what to do when they go off.  At the same time, it’s been 22 minutes, which is officially WPTTGOTR (Way Past Time to Get Off the Rug).  “OK, go on back to your seats and get started,” I mutter in defeat.

I know why all of this happened, and happened repeatedly, in my classrooms.  It’s a very simple, very honorable reason:  I wanted to make sure they learned.  

But aren’t there other ways that kids can learn, beyond sitting on the rug for a lesson?  I knew that there were.  I just had to let go of some control over the learning, and have some faith that learning could happen even when I wasn’t right there to see it.  At first, I tried just telling myself something I wanted to believe, but didn’t totally believe yet:  that the lesson is just one opportunity for kids to learn, not THE opportunity for kids to learn.  I experimented with giving kids lots and lots of time to work on an interesting and appropriately problematic task. I walked around. I conferred. Kids were engaged. Kids were learning. I tried structures like partner coaching that provide kids time to work independently and talk about their thinking in small groups and partnerships.  I overheard conversations kids were having when they didn’t know I was listening. They were actually talking about math.  I started to believe it.  I didn’t need to keep them on the rug until they got it.  I needed to give them something to think about, and let them go think about it, with tools and partners and good problems, for a long time.

There is some anxiety that comes with letting go of control.  I admit that it can feel a little scary to send kids off the rug, knowing that they don’t totally “get it” yet.  So much of our self-worth as teachers is tied up in students understanding.  But there is also great freedom and relief in admitting that I, the teacher, am not the sole nexus of learning.  Learning can happen even when we’re not there to witness it.  If we believe this is true, our work as teachers can be directed towards creating opportunities for kids to learn in all of these different ways — from us, yes, but also from rich, meaningful tasks, from studying the work of their peers, from having time to consider new ideas alone, and from having time and space to explore and grow ideas with their math communities.  

Unit Switch

Today I’m presenting with @pcipparone at the CGI Seattle conference.  Our presentation is entitled “Bringing out the Unit Across Mathematical Domains.”  You can check out the slides here.

One of the things we’ll be talking about is a new routine I created this year, called Unit Switch.  The idea behind Unit Switch is to have students repeatedly explore the idea that when you change the size of a unit, you need more or less of them to make the same length.   My interest in creating this routine came from a lot of work with 3rd and 4th graders around unit fractions, and seeing the struggle they had with understanding the idea that 1/4 is smaller than 1/3, because it takes four 1/4ths to make one whole unit, so the pieces are smaller than thirds, which require three to make one whole unit.  In one of my student’s words:

Version 2

I hoped that by giving my kids repeated experiences around changing the size of the unit, I could lay some groundwork for the important work they would need to do in the upper grades and into middle school around fractions, multiplication and division, and proportional reasoning.  What I didn’t realize is that talking about units with my kids would also be a way to connect SO MANY different mathematical domains that we tend to teach discretely in elementary school.  More about that in the presentation.  But here, I wanted to share the Unit Switch routine, and I would love to hear your thoughts if/when you try it out with your own kids.  Could I start my own #unitswitch hashtag? That would be crazy.

Here’s a quick rundown of how to lead a Unit Switch:

  1. Show an object (or a picture of an object).
  2. Show the first unit.
  3. Ask, “How many of these will it take to make this length?”
  4. Collect estimates – too high, too low, just right.
  5. Reveal (or collectively determine) how many – discuss if their estimates were close or not, why or why not.
  6.  Show the second unit.  Ask, “How many of THESE will it take to make this same length?”  Again have kids turn and talk about estimates and share thinking.  This time, when collecting estimates, compare to the original unit … do they think they’ll need more or less of the new unit? Why?
  7.  Reveal how many.  Ask, why did it take (more/less) of these?  Focus discussion around the idea that as the size of the unit gets bigger/smaller, you need more/less of them to make the whole.

Here are a few photos of some Unit Switches we did in my classroom this year, to give you a mental image.

There’s more I’d like to say about this, but it’s time to go present.  Looking forward to hearing your thoughts!

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.