The Time I Was "The Low Kid"

At the start of a recent meeting, a math team I am a part of engaged in a mathematical challenge. The task was to take the following expressions, written on post-its, and place them on a number line: x, 2x, 3x, x/2, x^2, x^3, x+2, x-4, and √x.

As an elementary math coach, I spend all my time thinking about elementary math. I had a partner for this task and, in this partnership, I was the less confident and less competent member. I was, if you will, the "low" kid.*

After a few moments to think on our own, my partner, who I'll call Joel, stood up enthusiastically, moved toward the number line on the wall, and began to place post-its on the number line. I stayed in my seat a few more moments than he did. (What kind of body language do students use to give us the message that they are unsure or unconfident about a task?)

The facilitator, Kyara, asked me to stand up and move closer to my partner and the number line. She prompted me to try to place a post-it on the number line.

I had ideas about where to place some of the expressions, but because Joel was so enthusiastic and jumped right in, I hung back and let him do the beginning work on his own, even though I really wanted to think about the math. (When do students hang back because of an overly enthusiastic classmate? How can we moderate this dynamic?)

Joel and I started to talk about where the expressions might go. Kyara pushed us to be explicit about the assumptions we were making as we placed expressions. "Under what conditions would that work?" she asked a few times, and we elaborated what x would have to equal in order for our placement to be true.

We played around with what would happen if x were a negative number. Joel placed x near the middle of the number line, with x^3 off to the left and x^2 off to the right. "Why did you do that?" Kyara asked. Joel began to explain, easily and quickly. I could hear his explanation, but that didn't mean I really understood it. Kyara asked me to restate it in my own words. This was the perfect facilitation move, giving me the chance to say for myself how I understood it.

"Let's see, if x = -2, for example, then x^2 would be -2 x -2, which equals 4," I said. "And x^3 would be 4 x -2, which equals -8."

(When do we make sure students have the chance to say for themselves what they understand, to explain and re-state and give another example so they understand more deeply?)

As teachers, we sometimes say, "Some of my students really need to use manipulatives," or "My low kids need more time to think through the math," or "Even though they know more efficient strategies, sometimes they keep using the old, inefficient strategies."

When I was in the position of thinking through a task that was somewhat challenging for me, I needed time to think, time to talk through my thoughts, and time to try out sometimes inefficient strategies to be sure about my and my partner's reasoning.

It's not that "some" students or "low" students need those things. Everyone needs to think, to talk about their ideas, and to have space to try things out even if they aren't doing it in the most efficient way. That's how learning happens.

I'm excited to spend time this June doing some hard math with teachers. We all need to be reminded of what it is like to work through a hard problem, to be a little stuck, to have a partner who goes faster than we do, to need extra time, to need to talk in order to solidify our learning. We all need regular reminders of how learning happens.



*I do not use the phrase "low kids," and I cringe when I hear it. I used it in this blog post to emphasize my experience of being the "lower" kid in a partnership and what I learned from that experience.

Discretionary Spaces in 4th Grade

In the 2018 American Educational Research Association Presidential Address entitled “Just Dreams and Imperatives: The Power of Teaching in the Struggle for Public Education,” Deborah Ball describes the many “discretionary spaces” a teacher faces each day in her classroom. As Ball explains, teaching is a “highly idiosyncratic and individual” practice in which teachers must make in-the-moment decisions dozens if not hundreds of times a day -- decisions which cannot be mandated by a curriculum or supervisor.

Ball shares a brief video of one minute and twenty seconds in her classroom, in which she identifies twenty discretionary spaces. That’s how often teachers have to make decisions about seemingly minute things such as who to call on, what to say to a student who calls out, how to respond to an incorrect answer, and so on.

Ball explains that it is through these discretionary spaces that racism, sexism, and all kinds of oppression enter our classrooms. While teachers can reinforce bias through their classroom interactions on a daily basis, often without realizing it, they can also use discretionary spaces to begin to free students from racist and sexist stereotypes and roles. This is not easy to do, however, as all of us are steeped in the culture of white, male supremacy -- it’s the air we breathe every day.

One of Ball’s points is that teachers can use discretionary spaces to begin to build new narratives for children of color, girls, and other marginalized or low-status students, and we can do this by “taking as axiomatic the brilliance of Black children.” This is not enough, however, as teachers also have to have “something else to do,” rather than the traditional teaching moves we’ve absorbed through teacher education, formal and informal mentoring, and the “apprenticeship of observation.”

In the video from her classroom, Ball shares an example (and you really, really should take the time to watch if you haven’t yet -- her speech starts at minute 46:45, and this part happens at 1:23:05) in which one of her students, Toni, a Black girl, calls out a question without raising her hand, laughing and playing with her hair as she speaks. Ball shares the ways most teachers would respond to Toni (chiding her or ignoring her), and then she offers an alternative: she publicly asserts that Toni’s question is mathematically important, and she asks another student to reply to Toni’s question. In this way, Ball gives everyone in the class, including Toni herself, a moment of seeing “Black girl brilliance” (rather than seeing a Black girl get in trouble for calling out).

Recently, I had the pleasure of watching a white fourth grade teacher, Ms. Lynn,* lead her class in a forty-five minute long math discussion in which they asked each other questions, experimented with ideas and explanations, and explored ideas about multiplication and division. A seemingly trivial question from one student set them off on this unplanned conversation, turning what was meant to be a 15 minute number talk into the entire math class.

Here’s what happened. See what connections you can make between how Ms. Lynn handled this student’s question and Ball’s discussion of discretionary spaces.

Math class started in Ms. Lynn’s room with students partnering up and tossing a large, soft ball back and forth, counting by 3s as they tossed. The goal was to set a new record for the class, counting as high as possible before dropping the ball. The vibe in the room was lovely: energetic but calm, focused, and collaborative.

After five minutes of this warm-up, students came to the rug, and Ms. Lynn began their number talk by writing the number 42 on the board. Next to it, she wrote, “Is 42 divisible by 3?”

Student thought quietly and then some began to share ideas. Only a few words had been spoken, though, when Josiah, a Black boy who was sitting, along with two other students, on yoga balls at the edge of the rug, called out, “What does that mean?”

I suggest you pause right here and think for a minute. How would you respond to Josiah’s question? This is the discretionary space, and Ms. Lynn had to decide how to respond.

While you think, I’ll tell you what I observed about Josiah in the few minutes preceding this moment. I had watched him counting by 3s with his partner, and it wasn’t easy for him. He had to stop and think often. His partner would coach him by reminding him of the last number she had said, but she didn’t tell him what his number was. I also observed that he seemed energetic, cheerful, friendly, and easily distracted during the first few minutes of the number talk on the rug.

As Ball does in her address, I’ll list some ways I think teachers might commonly respond to Josiah’s question.

“Josiah, please don’t call out. Raise your hand if you have a question.” 

“Who can help Josiah remember what ‘divisible’ means?”
“We’ve been talking about that word for a while now, Josiah.”

Some teachers, having observers in the room, might be embarrassed by both Josiah’s question and his calling out.

Here’s what Ms. Lynn did.

“Good question!” she responded warmly. “What’s another way I could have asked that question?”

Students began to raise their hands and she started a list on the board. The second question on the list was Josiah’s. “How can you get to 42 by 3?” he asked. Although his wording wasn’t mathematically precise, Ms. Lynn wrote it down on the board, because although his wording wasn’t perfect, his thinking was.



After another student offered, “Is 42 a multiple of 3?” Ms. Lynn asked, “Can anyone use the word ‘factor’ to ask the question?” This sparked a few more ideas.

Next Ms. Lynn asked, “How could I ask this question using just math symbols?” One student wrote 3 x ? = 42. Then Zoe came up and wrote 3 x 42 = ?.

“What did Zoe say?” Ms. Lynn asked. “Do you agree or disagree?” Hands went up all over the room.

“Zoe, look how many people are excited to talk about your thinking!” Ms. Lynn said, and Zoe smiled.

And they were off on a thoughtful conversation about the relationship between multiplication and division in which they drew a variety of models, wrote numerous equations, and thought deeply about the Big Ideas of fourth grade math.

Like Toni, Josiah asked an important question, and by taking him seriously, Ms. Lynn gave him (and other students who might have had trouble remembering what “divisible” means) an entry into the lesson. Josiah’s question showed that he wanted to participate in the math discussion and that he was thinking about the math. That was what mattered, not whether he raised his hand or remembered a mathematical term.

Not only did Ms. Lynn value Josiah’s question, she also leaned into it, figuring out in a matter of seconds how to use it to spark a conversation that was precise in both language and mathematical ideas. She did this again with Zoe’s incorrect thinking, putting the responsibility of finding and explaining the error on the students in her class. Here again, she turned a “mistake” into a springboard for real thinking and discussion.

Here is a picture of what the board looked like at the end of the lesson. (At one point someone asked what would happen if they tried to find out if 43 was divisible by 3, so in this photo the number on the board is 43.)



At one point in the conversation, I was sitting next to Josiah, and he asked, “Can I share the meaning of division?” This is a student who in some classrooms would be in trouble, would be admonished for calling out, and would probably not participate in math class because of a fear of making mistakes. Instead here he was, part of the math conversation, thinking out loud, taking risks, sharing his Black boy brilliance.

*All names in this classroom vignette are pseudonyms.

Keeping Class Agreements Alive

I've started reading Not Light, But Fire: How to Lead Meaningful Race Conversations in the Classroom, by Matthew Kay, and it is a joy. I'm only on Chapter 3, but it's already got me thinking about starting the year in a classroom by purposefully planning for race conversations from Day 1.

Although Kay's book is written from the perspective of a high school teacher, so far I find it very applicable to my work as an elementary educator. It's gotten me interested in talking, writing, thinking, and tweeting about how one could tweak his ideas in elementary school. This is the first of a few blog posts where I'll write about community-building as I've done in it my classroom over the years.

(Because I'm also thinking a lot about facilitating adult learning, I found as I wrote this that I also kept adding notes about how I'd done similar things in groups of adults learning together. That seems important, too.)

These posts are not overtly about race, but Kay makes the (not surprising but perhaps sometimes overlooked) argument for why you must build community in your classroom in order to have race conversations. He is clear that his ways of building community are not the only, or best, ways. So these posts will detail some ways that have worked well for me in elementary classes that talk about race.

Months ago, Kristin Gray was asking about how teachers keep classroom norms alive in the classroom past the first week or two of school, and I've been wanting to write about that too. So this first post is about Class Agreements and how I make our list of agreements a living document throughout the year.

Class Agreements: What are they? 

First of all, a note about what Class Agreements are NOT. Class Agreements are not a list of rules. As I tell students (and adults when I work with them in groups), rules are things we have to do. Agreements are things we are all going to try our best to do. We are all going to make mistakes, because they are hard things to do. None of us will follow the Agreements all the time. But we will try. (Thanks to Tina Blythe for teaching me this distinction.)

Now: What do you call them? I used to call them "Class Promises," but it feels strange to say we promise to do something that we have just said we won't always do. "Norms" feels a little businesslike to me, but other people like it. One class I worked with last year wanted to call them "Vows," like a wedding, which seemed intense, but that was their decision. In general I like the term "Class Agreements" because we all agree on them, but it doesn't matter too much what you call them.

How do you create them?

I begin in September by having students talk, write, and draw about their Hopes and Dreams for the year. We then talk about how we need to act and treat each other in order to make our Hopes and Dreams come true.

This is a well-known process created by Responsive Classroom, which you can read about here, but it's not easy. In particular, coming up with and finalizing the list of our Class Agreements is an arduous process, and I don't think I'm a particularly skilled facilitator of this part. But, in the end, we come up with a list we can all agree to.

Usually I try to keep the list of Class Agreements to four or maybe five items. One year's list was:

1. Treat others the way you want to be treated.
2. Be peaceful and respectful and talk out problems.
3. Try to be positive and think good thoughts.
4. Try your hardest and stay focused.

These are not my ideas -- they come from the students, and are probably not worded how I would word them if I wrote them. I try to make sure they come up with one about how to resolve conflicts, one about treating each other well, and one about working hard. There's often one about taking care of our learning environment as well.

After we create the Agreements and everyone signs them (agreeing to try to meet them), the Agreements don't just hang on the wall and get forgotten. I've done different things with them over the years, but here are a few.

Refer to the Agreements in Morning or Afternoon Circle

One easy way to keep the Class Agreements alive is to include them in Morning or Afternoon Circle in some way. I've often had students look at the Agreements in the morning and choose one they really wanted to focus on that day. They silently put up their fingers to show which one they've chosen (three fingers for Agreement #3, etc.). This just takes a couple seconds, but it keeps the Agreements in our minds. In Afternoon Circle, students can briefly share a way they or someone else followed the Agreements that day.

(I'd like to note that we did this in a class for adults I took this year with Gene Thompson-Grove: revisited our norms and thought about which ones we needed to focus on ourselves and which ones we would like the group to attend to.)

Times of Crisis: Class Meetings

Once, at a moment of crisis, when something went badly wrong in our class community and we called a class meeting, my co-teacher Annie Shah took the Class Agreements off the wall and put them on the floor in the middle of our circle to ground our discussion. She reminded the students how important they were and why we had written them, and then we talked about which ones we needed to do a better job with. This act of physically centering our community around the Agreements was a powerful reminder of the list as a living document, not an ancient artifact of the first weeks of school.

Use Class Agreements for Written Reflections

Finally, Class Agreements are powerful tools for written reflections. Students can reflect at the end of a unit, term, week, month, or year and write about ways they have embodied one or more of the Agreements. I also use them on reflection sheets that students complete when they've done something they shouldn't (usually involving hurting someone else physically or emotionally). These reflection sheets include a list of the Agreements and ask the student to circle which one they did not follow, then write about what happened and how they can fix the problem they caused.

(Written reflections on the norms are also something I've done as an adult student. In Tina Blythe's semester-long class, we did regular reflections about the norms we had created, sometimes verbally, sometimes in writing. Mid-way through the term, we folded a piece of paper into four squares and wrote about one of the following in each square: a norm we had done well with personally, a norm we needed to do better with personally, a norm the group had done well with, and a norm the group needed to do better with. This was useful feedback for the instructor.)

A Note About Compliance

I loved this post from @Jess5th about shifting the narrative away from compliance on the first day of school. She engaged her students in conversations about when it's important to follow the rules and when it's important to break the rules. I have never had that conversation so early in the year, but I think it's an important idea, so don't miss her post.

In my next post on building community, I'll write about a routine I learned in a mindfulness class called Appreciations. 

Rule #1: Provide Think Time

Our second graders are on a field trip. It is quite a good field trip, novel and engaging, making them excited to be writers. But I am paying attention to who participates and why.

The second grade is a diverse group in the true sense of the word: racially, socio-economically, and in terms of learning profiles. Of the 22 students, about 8 to 10 participate often and actively during traditional, teacher-led lessons at school. These 10 tend to be whiter and come from more economically-advantaged families (although not exclusively, of course). 

The field trip is no exception. The man leading the class is energetic, hearty, and jovial. He asks fast questions and wants fast answers. He calls on the first hand up. 

“Where does our story take place? Who should be the main character? What does he look like? What is his name?” The questions come fast and furious, and only a few hands keep up. I am paying close attention to the race, gender, and learning profile of those who are getting called on. White boys are speaking most. White girls are second.

The man leading us is definitely in a hurry. “You’ve gotta have an idea right in your head when I call on you!” he says urgently, after calling on a few kids who didn’t have an answer at the tip of their tongues. 

Some students who try to have fast answers have their responses dismissed.

“Who should be the main character?” he asks.

“You!” says Wilfred, a student for whom English is a second language and who is clearly enamored of our host.
“Me? No!” the host replies.

“Oh. No,” Wilfred says, shaking his head as if the wrong words had just popped out. “I forgot.”

“What will happen next in our story?” our host asks.

“The dog smelled its own butt!” Jaquan says. Jaquan has some holes in his formal education and often needs extra time to process directions and content. 

“No, that’s too gross,” our host replies, and so neither Jaquan nor Wilfred’s ideas make it into our class story. 

“What is the doghouse made of?” is the host’s next question.

“Wood!” Teddy calls out. Teddy is a white boy who can do well in school but is often disengaged. He hasn't been speaking on this field trip until this moment.

“Raise your hand,” the host admonishes. Someone else raises their hand and says “wood,” which he accepts.

These three boys make attempts to participate in the fast-paced lesson. Who hasn’t even raised a hand? A few other quiet boys, and a bunch of the girls. The girls who haven’t raised their hands are primarily girls of color, girls for whom English is their Second Language, or girls with learning disabilities.

I start to feel urgent about the unequal participation, so I talk in whispers to students sitting near me. I repeat the host’s questions, asking them what they can come up with. By the time they think of something, he’s already two or three questions ahead of what we have been working on.

At some point, the other teacher on the field trip also wants to increase participation. She does it by saying, “Abby, we haven’t heard from you. Abby, Evita, do you have any ideas?” But they can’t, or won’t, come up with something when put on the spot like that, with no think time. 

Later, I sit with Roberto as he works on his own story. I ask him what he wants to name one of his characters. I then sit and count the seconds silently as he thinks. Twelve seconds later, he tells me a name. Twelve seconds. At the pace of our host, we would have answered three more questions by the time twelve seconds had passed.

I used to teach in ways that are somewhat reminiscent of how this host runs the field trip (although I hope never quite so fast-paced). Over the years, I’ve learned strategies to increase and equalize participation and access to the conversation. There are very simple routines I use when I teach that provide enough think time for everyone, communicate my expectation that everyone think and prepare to share their thinking, and minimize the domination of the quick, loud voices — voices which are often male, usually native English speakers, nearly always traditionally successful students, and, in a racially diverse school, often white.

Here are some of these simple routines I employ on a daily basis, in large and small groups.

Provide Think Time

The single most common strategy I use to ensure that everyone has time to process a question or direction is to provide Think Time. I explicitly teach what Think Time means and how students should use their Think Time, using anchor charts like this (which I completely stole from Kassia Wedekind and Christy Thompson).




This means that we all get comfortable with long silences of 30-60 seconds, sometimes longer. I usually ask students to indicate where they are in their thinking with their fingers.

“When you have one answer [or idea], put your thumb up in front of your chest. When you have another strategy [or idea], put up a finger.” This lets me know when they’ve had enough Think Time. I make sure everyone has at least one thumb up before I start asking for responses. (This thumb strategy is borrowed from Sherry Parrish and her Number Talks books.)

Choral response

I used to think that having a class respond chorally was old-fashioned and a bad idea. How do you know if students got the right or wrong answer if they all respond together? Won’t they just copy each other and recite mindlessly rather than thinking the question through? I have since learned that there are times when choral responses are preferable to individual ones, especially when paired with Think Time.

If I have asked a question that has a short, factual answer (such as What kind of syllable is this? or What goes with 10 to make 15?), I ask students to think quickly (for just a few seconds), indicate with a thumb when they have their answer ready, and then recite it all together.

“How many syllables are in this word?” I may ask. “Think it. Show me a thumb when you know.” Pause. “Now say it.”

The point of small questions like these is usually to keep students mentally engaged as the lesson moves on to something deeper and more important, or to provide short practice with a basic skill. By providing Think Time and then asking for a choral response, I make sure everyone is caught up and “minds on,” and everyone gets a chance to share their thinking. If they are wrong, they’ll hear that most people said something else and they’ll adjust their thinking. If I hear that a few people are wrong or there is disagreement, I’ll stop and we’ll discuss it.

Think, Pair, Share

There’s no need to write much here since this is a common enough practice, but I’ll mention how important it is to teach a clear structure for this routine and to practice that routine. I use this anchor and I actually have younger students touch their ears or mouths to show who is going to speak and listen first. I count backwards from 5 to 0 to give them time to find a partner, and when I get to one I always say, “Stand up if you don’t have a partner” so solo kids can find each other. We also talk about and practice how to look around the room to make sure everyone has a partner and how to include someone or change partners when it makes sense. We usually precede a Turn and Talk with Think Time.



Student-led discussions

… are not topics for this blog post, as the participation techniques I’ve laid out here are primarily for teacher-led lessons (the kind that was modeled during our field trip and which surely have a place in schools). But they bear mentioning as I have found them to be very effective tools for increasing participation (verbal and mental) among all students. I’ve written a bit about Hands Down Conversations here and you can find more on Kassia Wedekind and Christy Thompson’s blog devoted to the topic.

These structures are so simple and so tiny, yet they define most lessons I lead and they increase access to the conversation tenfold. Boiled down to two words, they can be summarized as Think Time. Well, Think Time and respectful silence so that others can think too. No one has their best ideas instantaneously, and our expectation should always be that students go deeper and think well rather than quickly. If I could give one gift to all teachers and field trip leaders, it would be the gift of Think Time.

Going Back Home

Today I visited the place where I became a teacher.

I haven't been inside that building in 9 years. But for the eight years before that, I went there nearly every day. I sat on the front steps calling people I loved on Sept. 11, 2001. I cried in the bathroom stalls and out on the back steps after hard days as a new teacher (and a not-so-new teacher, too). I had some of my most joyful teaching moments there, with first graders who grew a garden and read stories and explored pattern blocks together, and second graders who wrote poetry and discovered their neighborhoods and did yoga together. There I learned so much about good teaching, from some of the wisest educators I know, about relationships, about justice, about why we teach and how to teach and our responsibility to kids and families.

These halls I haven't walked for 9 years are so familiar to me. The metal grate in the middle of the hallway floor. The sliding closet doors, askew in their tracks. The apple tree outside, whose apples we used to press for cider in the fall. The door handles, the bulletin boards, the fraying blinds that hang over the bookshelves in the hallway so little hands won't disorganize the leveled books.

My school no longer lives there. A new school has called that building home since my school moved into a bigger building a mile down the road. I walked through the hallways as children arrived with their book bags and lunch boxes, hanging their things on the hooks I used to label carefully each September. I could feel the ghosts in those hallways, spirits who inhabit my memory so vividly. Ghosts of the teachers I laughed and cried with. Ghosts of the children who shared my days and, mentally, my nights, with whom I struggled daily as we learned and worked together. Ghosts even of teachers I didn't know well, but whose work shaped lives in ways I know are still remembered, and ghosts of the ideals we lived in that school, the ways we wanted to teach our students to live and think and be.

I am surely a bit melodramatic about the good old days we had there. So many of them were hard days, but now I know that that was probably the best place I will ever work. We were a community that was committed together to a common good -- though a common good that was not easily agreed upon, and often arrived at with great struggle.

Schools are places where adults' and children's joys and struggles are lived. Walking down the hallway, I thought about schools that are closed in the name of saving money and increasing "achievement." I thought how lucky my school had been because instead of being closed down, we had been relocated and merged with another school. Even though those moves changed our school irrevocably, the loss wasn't as great as the losses of communities whose schools are shuttered. 

Eve Ewing writes about the damage to a community that accompanies school closings. This line of Ewing's runs through my mind today: "[T]he decision to shuffle students from one building to another in the name of numbers is shameful. The decision to do so is based on the premise that children, teachers, and schools are indistinguishable widgets, to be distributed as efficiently as possible across the landscape. But the fact is that schools are ecosystems, each with its own history, culture, and intricately woven set of social relationships. Schools are community anchors. They are not interchangeable, nor are they disposable. Schools are home." 

Going back to my old school was as poignant and wistful as if I had gone back to the house where I lived for 18 years as a child. I understood vividly in that moment how true Ewing's words are, and what a loss it is for a community when a school is closed. "Schools are home."