Supporting Instructional Growth in Mathematics (Project SIGMa)

Good news to share: another research grant has been funded by the National Science Foundation. Yay!

For this project, my research team and I will be working with Math for America in Los Angeles to design a video-based coaching method for their Master Teacher Fellow program.

sigma logo

This is what we pitched to the NSF:

This study addresses the need to develop processes for adequate and timely feedback to inform mathematics teachers’ instructional improvement goals. In this study, we propose using design-based implementation research to develop and investigate a process for documenting mathematics teachers’ instruction in a way that is close to classroom practice and contributes to teachers’ ongoing pedagogical sense making. The practical contribution will be a framework for formative feedback for mathematics teachers’ learning in and from practice. The intellectual contribution will be a theory of mathematics teachers’ learning, as they move from typical to more ambitious forms of teaching in the context of urban secondary schools. Both the practical and theoretical products can inform the design of professional development and boost other instructional improvement efforts.

In a recent Spencer study, my team and I investigated how teachers used standardized test data to inform their instruction. (That team was Mollie Appelgate, Jason Brasel, Brette Garner, Britnie Kane, and Jonee Wilson.)

Part of the theory of accountability policies like No Child Left Behind is that students fail to learn because teachers do not always know what they know. By providing teachers with better information, teachers can adjust instruction and reach more students. There are a few ways we saw that theory break down. First, the standardized test data did not always come back to teachers in a timely fashion. It doesn’t really help teachers adjust  instruction when the information arrives in September about students they taught last May. Second, the standardized test data took a lot of translation to apply to what teachers did in their classroom. Most of the time, teachers used data to identify frequently challenging topics and simply re-taught them. So students got basically the same instruction again, instead of instruction that had been modified to address central misunderstandings. We called this “more of the same,” which is not synonymous with better instruction. Finally, there were a lot of issues of alignment. Part of how schools and districts addressed the first problem on this list was by giving interim assessments –– basically mini versions of year end tests. Often, the instruments were designed in-house and thus not psychometrically validated, so they may have not always measured what they purported to measure. Other times, districts bought off-the-shelf interim assessments whose items had been developed in the traditional (and more expensive) manner. However, these tests seldom aligned to the curriculum. You can read the synopsis here.

Accountability theory’s central idea  ––  giving teachers feedback –– seemed important. We saw where that version broke down, so we wanted to figure out a way to give feedback that was closer to what happens in the classroom and doesn’t require so much translation to improve instruction. Data-informed action is a good idea, we just wanted to think about better kinds of data. We plan to use a dual video coaching system — yet to be developed — to help teachers make sharper interpretations of what is happening in their classrooms.

Why did we partner MfA LA? When I reviewed the literature on teachers’ professional learning, they seemed to be hitting all the marks of what we know to be effective professional development. They focus on content knowledge; organize their work around materials that can be used in the classroom; focus on specific instructional practices; they have a coherent and multifaceted professional development program; and they garner the support of teacher communities. Despite hitting all of these marks, the program knows it can do more to support teachers.

This is where I, as a researcher, get to make conjectures. I looked at the professional development literature and compared it to what we know about teacher learning. MfA may hit all the marks in the PD literature, but when we look at what we know about learning, we can start to see some gaps.

*Conjecture 1 Professional learning activities need to address teachers’ existing concepts about and practices for teaching.

 

Conjecture 2 Professional learning activities need to align with teachers’ personal goals for their learning.

 

Conjecture 3 Professional learning activities need to draw on knowledge of accomplished teaching.

 

*Conjecture 4 Professional learning activities need to respond to issues that come up in teachers’ ongoing instruction

 

*Conjecture 5 Professional learning activities need to provide adequate and timely feedback on teachers’ attempts to improve their instructional practice to support their ongoing efforts.

 

Conjecture 6 Professional learning activities should provide teachers with a community of like-minded colleagues to learn with and garner support from as they work through the challenges inevitable in transformative learning.

 

*Conjecture 7 Professional learning activities should provide teachers with rich images of their own classroom teaching.

 

The conjectures with * are the ones we will use to design our two camera coaching method.

We need to work out the details (that’s the research!) but  teacher’s instruction will be recorded with two cameras, one to capture their perspective on significant teaching moments and a second to capture an entire class session. The first self-archiving, point-of-view camera will be mounted on the teacher’s head. When the teacher decides that a moment of classroom discourse illustrates work toward her learning goal, she will press a button on a remote worn around her wrist that will archive video of that interaction, starting 30 seconds prior to her noticing the event. (As weird as it sounds, it has been used successfully by Elizabeth Dyer and Miriam Sherin!)  The act of archiving encodes the moment as significant and worthy of reflection. For example, if a teacher’s learning goal is to incorporate the CCSSM practice of justification into her classroom discourse, she will archive moments that she thinks illustrate her efforts to get students to justify their reasoning. Simultaneously, a second tablet-based camera would record the entire class session using Swivl®. Swivl® is a capture app installed in the tablet. It works with a robot tripod and tracks the teacher as she moves around the room, allowing for a teacher-centered recording of the whole class session. Extending the prior example, the tablet-based recording will allow project team members to review the class session to identify moments where the teacher might support students’ justifying their reasoning but did not do so. The second recording also captures the overall lesson, capturing some of the lesson tone and classroom dynamics that are a critical context for the archived interactions. Through a discussion and comparison of what the teachers capture and what the project team notices, teachers will receive feedback on their work toward their learning goals. We will design this coaching system to address the starred conjectures in the table

Anyway, I am super excited about this project. I am working with amazing graduate students: Grace Chen, Brette Garner, and Samantha Marshall. Plus, my partners at MfA LA: Darryl Yong and Pam Mason.

I will keep you posted!

 

 

 

Renegotiating Classroom Treaties

Many classrooms are governed by tacitly negotiated treaties. That is, students trade in their compliance and cooperation –– student behaviors that alleviate the challenges of crowded classrooms ––  for minimal demands for engagement by the teacher. When I have worked with teachers trying out open-ended tasks for the first time, I will often hear about “pushback” or “resistance” from the students: “I tried using this activity but the kids balked. They complained the whole time and refused to engage.”

These student responses indicate that teachers are violating their part of the treaty by going beyond minimal demands for engagement and increasing intellectual press. Put differently, by using an open-ended task, teachers raise the social risk, leaving students open to judgment since they can not rely on the usual rituals of math class to hide their uncertainty. Treaties may, as their name suggests, keep the peace, but they reflect norms of minimal engagement that interfere with deeper learning.

In my own observations, I see teachers struggle to move students past their initial reluctance to participate and make it clear that active involvement is required in their classrooms. Renegotiating classroom treaties requires a clear vision for what student participation can look like, structures to support that vision, along with the determination to see it through. The teachers I interviewed for my forthcoming book all emphasize how critical the first days are for setting these expectations for their students, particularly since their expectations may differ from what students are used to in math class. “It’s entirely intentional that I begin setting norms and structures on the first day of school,” Fawn explains. By launching the new school year by showing students what it means to do math in her class, Fawn renegotiates the classroom treaty through norms and structures, introducing the Visual Pattern and other discourse routines from the start. She says, “I need to provide students with ample opportunities to experience the culture that we have set up. We need to establish and maintain a culture that’s safe for sharing and discussing mathematics, safe for making mistakes, and a culture that honors each person’s right to contribute. There needs to be a firm belief among everyone that mathematics is a vital social endeavor. Building this culture takes time.”
Starting the school year with clear expectations is important, but guiding individual students’ participation is an ongoing project. The teachers I interviewed have numerous strategies for monitoring and building positive participation throughout the year. Students students who hide or students who dominate make for uneven participation. The teachers describe how they contend with these inevitable situations.
When figuring out how to respond to quiet students, the teachers try to understand the nature of students’ limited participation. Not all quiet students are quiet for the same reasons. At times, quietness is rooted in temperament: some students inclined to hang back until they feel confident about what is going on, but they are tracking everything in class. These students do not contribute frequently, but, when they do, their contributions add a lot to conversations. This kind of quiet is less of a concern and can even be acknowledged: “Raymond, you don’t talk a lot, but when you do, I always love hearing what you have to say.”
Other times, quietness signals students’ lack confidence. That is, students indicate some understanding in their work or small group conversations, but they do not have the confidence to participate in public conversations. With these students, the teachers seek out individual conversations. Chris calls these doorway talks, while Peg calls them sidebars. (“Trying to deal with calculators and rulers at the end of class, I couldn’t make it to the doorway!” Peg tells me when I note the different names.) “I might say to a kid, ‘You know, you had really good ideas today, and I would have loved to have heard more of them in the conversation we had a the end. I think you have a lot more to contribute than you give yourself credit for.’” Sometimes, there are ways of encouraging good ideas to become public that do not directly address the student. Chris explains that he might say something like, “I haven’t heard from this corner of the room.” He then asks other students to hold their ideas while waiting for a contribution from the quiet group.

Of course, some students are quiet because they really do not know what is going on. This could be due to a language issue, in which case, the teacher needs to modify instruction to give them more access to the ideas. If there are other learning issues going on, this might suggest the need to check in with colleagues about the students performance in previous years or in other subjects.

eager-students
Talkative students pose another kind of challenge to the expectation that everyone participates.  On the one hand, they can provide wonderful models of sharing their thinking. They can be the “brave volunteers” who explore their thinking publicly, and teachers can lean on them to get conversations started. On the other hand, they can be domineering, making it difficult for other students to get a word in. The quiet students who temperamentally need to think before they speak have their counterparts in some talkative students: these are the students who think by talking. Asking for their silence sometimes gets heard as asking them not to think. When I have had students like that in my own classes, I make sure to assure them that I value their engagement but that I need them to find other strategies for processing so that other students can be heard. Sometimes, students with impaired executive functioning, like those with ADD, have a hard time with the turn-taking aspect of classroom dialogue, so not only do they talk a lot sometimes, they struggle to take turns. Again, teachers can respond by valuing students’ ideas while helping them participate more effectively: “I know you get excited, but we need to take turns so that we can hear each other.” Finally, domineering behavior can get expressed through a lot of talking: students who are highly confident in their understanding and want to explain to others. Teachers need to judge the extent to which this is altruistic, a sense of trying to share knowledge, and the extent to which it shuts conversations down. In the first case, students can be coached towards asking questions of their classmates, channeling their impulse to talk into something constructive. In the second case, the dominance can be corrosive to the classroom culture and the students might need stronger redirection. For all of this feedback, similar strategies of direct address (via sidebars or doorway talks) and indirect address (“Let’s hear from somebody else”) can help teachers manage participation.

Laying the Groundwork for Logarithms

s

Strangely, I have had occasion to do a few tutoring sessions with different kids recently around exponential and logarithmic functions.

This particular mistake set off a few alarm bells:

logblog

 

Do you see what the student is doing here? She is treating

log a

like a variable that is being divided instead of a function.

I looked at the student’s notes, and all the usual log laws were there. But she did not yet have the unshakable understanding that logs are functions. I realized that there are some foundational ideas that she needed before we could really make sense of all of this.

Here are a couple of essential ideas I want to communicate to students about logarithm functions.

First, functions can be described as actions, so I always make students explain what a function is doing.

The question you should ask about every function is: what are we doing to the input to get to the output? I call it “saying the function in English.”

Since we usually teach logarithms after exponential functions, let me start with them.

I ask, What do exponential functions do? They provide rules based on repeated multiplication. So the function

2x

tells us that “some number (y) equals 2 multiplied by itself any number of (x)  times” to get y. We can do this with different examples, talk about how the function grows, look at the graph, look at tables, compare the growth of exponential functions to linear and quadratic ones. My goal is to get kids to have a feel for what is happening with exponential growth so well that when somebody says, “It was growing exponentially!” they can decide whether that is an accurate statement or not.

This is the first part of the groundwork for understanding logarithms.

Second, remember that anything we do in mathematics, we always find ways to undo.

This is thematic in all of mathematics. It becomes a chant when I teach math.
I say to students:
“Since this is math, anything we learn to do, we need to ….?”
They soon learn to respond with:
UNDO!!!”

Doing-and-undoing is a good mathematical habit of mind to emphasize, because students start to anticipate that when we learn some new funky function or operation, an inverse is coming down the pike. They are not at all surprised to learn that trig functions have an inverse and so on.

In this case, since we have learned to exponentiate, they can guess we need to un-exponentiate.

shrug

That’s just how math works!

I like to show inverses of functions in all of the representations. The idea is the same in tables, graphs and equations: the x’s and y’s switch places.

For tables and graphs, it’s fairly easy for students to figure it out. But the algebra gets tricky. To find the inverse of the previous exponential, for example, we need to derive it from:

inverse 1

This immediately creates a mathematical need to “un-exponentiate.”

So when we want to solve that equation for y, let’s undo exponentiation with a function we call a logarithm. Logarithms undo exponentiation.

logging the inverse
Since the log undoes the exponentiation, we end up isolating the y.

this one!

I also tell them we read this as “log base two of x equals y.”

So when you see an equation like:

fixed

you are asking “2 to what power equals 8?” I have them practice explaining what different equations mean.

Now your students are ready to learn all the details of working with logs!

Tell me your ideas in the comments.

[Before I close, vaguely related Arrested Development reference:

bob loblaw

Because this is a log law blog. But I guess I don’t really want to talk about log laws. Anyways…]

First, Do No Harm

I have often wondered if teachers should have some form of a Hippocratic Oath, reminding themselves each day to first, do no harm.

Since the network of relationships in classrooms is so complex, it is often difficult to discern what we may do that causes children harm. Most of us have experienced the uncertainty of teaching, those dilemmas endemic to the classroom. Was it the right decision to stay firm on an assignment deadline for the child who always seems to misplace things, after giving several extensions? Or was there something more going on outside of the classroom that would alter that decision? Why did a student, who is usually amendable to playful teasing, suddenly storm out of the room today in the wake of such an interaction?

What I have arrived at is that there are levels of harm. The harm I describe in the previous examples can be recovered if teachers have relational competence — that is, the lines of communication are open with their students so that children can share and speak up if a teacher missteps.

What I am coming to realize is that mathematics teachers have a particular responsibility when it comes to doing no harm. Mathematics, for better or worse, is our culture’s stand-in subject for being smart. That is, if you are good at math, you must be smart. If you are not good at math, you are not truly smart.

I am not saying I believe that, but it is a popular message. I meet accomplished adults all the time who confess their insecurities stemming from their poor performance in mathematics classes.

Here is an incomplete list of common instructional practices that, in my view, do harm to students’ sense of competence:

1. Timed math tests

Our assessments communicate to students what we value. Jo Boaler recently wrote about the problems with these in terms of mathematical learning. Students who do well on these tend to see connections across the facts, while students who struggle tend not to. But if timed tests are the primary mode of assessment, then the students who struggle do not get many opportunities to develop those connections.

2. Not giving partial credit

Silly mistakes are par for the course in the course of demanding problem solving. Teachers who only use multiple choice tests or auto-grading do not get an opportunity to see students’ thinking. A wrong answer does not always indicate entirely wrong thinking. Students who are prone to getting the big idea and missing the details are regularly demoralized in mathematics classes.

Even worse, however, is …

3. Arbitrary grading that discounts sensemaking

Recently, a student I know had a construction quiz in a geometry class. The teacher marked her construction as “wrong” because she made her arcs below the line instead of above it, as the teacher had demonstrated. This teacher also counts answers as incorrect if the SAS Theorem is written as the SAS Postulate in proofs. Since different textbooks often name triangle congruence properties differently, this is an arbitrary distinction. This practice harms students by valuing imitation over sensemaking.

4. Moving the lesson along the path of “right answers”

Picture the following interaction:

Teacher:    “Can anyone tell me which is the vertical angle here?”

Layla:        “Angle C?”

Teacher:     “No. Robbie?”
Robbie:       “Angle D?”

Teacher: Yes. So now we know that Angle D also equals 38˚…

That type of interaction, called initiation-response-evaluation, is the most common format of mathematical talk in classrooms. Why is it potentially harmful? Let’s think about what Layla learned. She learned that she was wrong and, if she was listening, she learned that Angle D was the correct answer. However, she never got explicit instruction on why Angle C was incorrect. Over time, students like Layla often withdraw their participation from classroom discussions.

On the other hand, teachers who work with Layla’s incorrect answer –– or even better yet, value it as a good “non-example” to develop the class’s understanding of vertical angles –– increase student participation and mathematical confidence. And, they are doing more to grow everybody’s understanding.

What are other kinds of teaching practices that stand to “harm” students?

Relational Density in the Classroom

Recently, Michael Pershan has been thinking about why it’s so hard for teachers to share knowledge and ideas. He has been playing with building cases to discuss as teachers, wondering about what counts as sufficient description to invite consultation.

In my work, I find that one of the challenges to building shared professional knowledge comes from the irreducible situativity of teaching. If that sounds like an academic mouthful, my apologies. But what I mean is that we can’t escape how much of what works in teaching comes out of nuances of our practice and resources in our context that we may not even be aware of. Just as fish don’t see the water they swim in, so too teachers often miss things like community norms or material resources that shape what is possible in the classroom.

In addition, I think the relational part of teaching has been understudied –– especially in mathematics education. As I have said before, asking students to share their thinking is a socially risky proposition and depends on the relationships in the classroom and the norms for participation.

Here is Courtney Cazden on this:

“In more traditional classrooms, social relationships are extracurricular, potential noise in the instructional system and interference with ‘real’ schoolwork. What counts are relationships between the teacher and each student as an individual, both in whole-class lessons and in individual seat-work assignments. In nontraditional classrooms, the situation has fundamentally changed. Now each student becomes a significant part of the official learning environment for all the others” (2001, p. 131)

So to get students to share their ideas, teachers have to attend not only to their individual relationships with students, but to students’ relationships with each other.

This is decidedly challenging work. Most classroom teaching situations exhibit tremendous relational density. As Philip Jackson observed decades ago, classrooms are among the most crowded of institutional settings. In order to function, they require some degree of cooperation from the students. Teachers often achieve that through setting up systems of compliance, by building relationships with students, or some combination of the two.

Although students who have an instrumental view of schooling are less dependent on a teacher’s relational skills, a teacher’s success often depends on engaging and shaping students’ sense of purpose.

But the relationships in the classroom do not simply exist between the teacher and students; they exist among the students themselves. Once we take this into account, the social complexity of the classroom is stunning. Instead of just seeing the relationship one teacher builds with each student, we must account for the combinations of relationships among the students themselves. As a consequence, the difference between having 16 students or 32 students in a classroom does not simply double the relational density of a classroom: each set of students has potential for harmony or conflict. Just considering the smaller class of 16, there are 120 possible pairings between students. In the larger class of 32, there are 496. The number of students only doubled, but the relational complexity has more than tripled.

CocktailPartyGraph_700

Figure 1. Student pair-wise relationships grow quadratically while the class size grows linearly. The red dots represent students, and the connecting lines represent potential relationships.The last diagram represents the relationships among 5 students and a teacher, illustrating the fast growing relational density with every added student.

Relational density serves as a backdrop of potentialities in classrooms: not all relationships are actively engaged. When I talk to experienced teachers, however, I notice that they are alert to the relational potentials across the classroom social network, usually framing them as classroom dynamics.

Returning to Michael Pershan’s question, how do we adequately capture these dynamics when we describe our teaching situations? Some teachers talk about the kids with “strong personalities” or “the quiet kids.” I have heard teachers talk about students who are hot spots in the classroom relational network: most other students have an active experience of liking or disliking them. These experienced teachers respond by building lessons with their hotspot students in mind, anticipating possibly corrosive behavior or harnessing potential leadership.

Obviously, not all teachers attend to classroom dynamics in this way. Whether or not these dynamics are  on a teacher’s radar, they contribute to the situativity of teaching. That is, we can’t really talk about teaching without addressing some of these particulars. Inattention to details of a teaching situation leads to invisibility in critical aspects of the work. This makes knowledge sharing hard.

So the question is: what sufficiently describes the character and dynamics of one situation to help teachers productively compare it to another? Often, teachers fall into language that relies on stereotyped understandings: an urban school, an honors class, an ADHD kid. These everyday categories stand in for broader dynamics but, in my view, do not adequately describe teaching situations.

Yet leaving critical dimensions of teaching situations underspecified contributes to the lack of consensus around expertise. What constitutes successful teaching remains hotly contested, evidenced by policy debates around standardized testing and value-added models of teaching. Grossly underdescribing teaching situations has led to an overdetermination of desirable, visible outcomes like test scores. In this way, invisibility creates a reliance on other kinds of representations of the work when communicating about instruction.

How do teachers teach responsively?

The idea of responsiveness is one of the biggest challenges of equity-geared teaching approaches.

Responsiveness, by definition, means that lessons cannot entirely be planned without considering the students. What is more, since the students’ input and ideas are actively sought out, it increases the uncertainty of how a lesson will unfold.

This weekend, I have been reading a book by Adam Lefstein and Julia Snell called Better than Best Practice. Like me, these scholars spend a lot of time thinking about good teaching, although their study is in literacy classrooms in the UK, while I spend my time thinking about US mathematics classrooms.

Nonetheless, the premise of their book resonates with me. As the title suggests, they argue against “best practice” language that seeks to “prove” the efficacy of exact teacher moves or curricula. Like me, they are interested in the kind of teaching that seeks out, engages, and responds to students’ ideas.

Lefstein and Snell refer to this as professional teaching, arguing that involves sensitivity, interpretation, judgment and a flexible repertoire of methods. I found this to be a useful framework.

lefstein and snell coverBy sensitivity, the authors refer to teachers’ attentiveness and openness to critical moments in the flow of a class. Did a student raise an important issue? Did another student speak up for the first time? Does a conflict seem to be brewing? Classroom dynamics involve numerous people, all with their own feelings and thoughts and challenges, and a teacher must thoughtfully navigate these while moving lessons in a productive direction.

Once teachers are alert to a critical moment, they must then figure out its significance –– what Lefstein and Snell call interpretation. Was a student’s objection to a teacher’s premise simply an attempt to derail a lesson, or is there an important question that needs to be aired?

What will the broader message to the student and the class be if the teacher pursues the question? What if she shuts it down?

In the latter set of questions, sensitivity and interpretation work together as the teacher figures out which part of her repertoire to engage. By repertoire, the authors refer to a teacher’s flexibility and depth in calling upon a range of possible actions and success in implementing them.

Together, these resources come together to constitute judgments about teaching. Teachers make hundreds of decisions a day, and the demand only increases when they seek out student input.

I like this framework because it positions the teacher not as just “doing” things in the class, but actively responding to and making decisions about students. It also broadens the object of professional learning beyond the usual activities or specific teaching moves to increased sensitivity to student and classroom dynamics and their relation to ongoing judgments.

“What do you think and why?”

Today I got to virtually meet up with the amazing math teachers at the Park City Mathematics Institute. In addition to doing beautiful math problems, they have been involved in daily sessions called “Reflections on Practice.”

Image

21st Century PD. I am beamed into the room. Photo: Suzanne Alejandre.

I knew that they had been talking a lot about the 5 Practices, so I decided to spend my time talking about how hard it was for most students to answer the question:

What do you think and why?

Persuading children to answer this question is a big obstacle to getting rich mathematical discourse off the ground in any classroom.

But think about it. That is a really tricky question to answer, both socially and intellectually.

I asked the teachers to spend some time thinking about why students might be reluctant to participate.
Slide05

They brainstormed a great list:

  • Sometimes students are not able to articulate their thoughts.
  • Students might fear the judgment of their classmates.
  • Students have incomplete thoughts.
  • They are not always sure whether a question is a “right or wrong” question or a “share your thinking” question.
  • There may be social norms that communicate that being smart is bad.
  • They can be in crisis in their outside lives, making the question besides the point.
  • They may not see sharing their thinking as a part of their role as students.
  • They may have a very individual, internal process that makes “sharing” difficult.
  • They may try to share their ideas but find that they are not listened to or valued.
  • Sometimes students would rather not risk trying and failing, so it is safer to just not try.
  • Language barriers can make it difficult to share.

I have seen all of these things as a teacher and an observer of mathematics classrooms. It is really hard to get kids to share their thinking.

I told the teachers about two concepts that I found to help teachers address these challenges and successfully establish rich classroom discourse with their students.

The first one is classroom norms. The second is addressing social status, which I have written about here and here.

I shared a list of norms that I have found to help encourage participation.

Slide08

Then I talked a little bit about status problems and how they can get in the way of productive mathematical conversation. First I defined status…

Slide09

Then I talked about how status problems play out in classroom conversations.

Slide10

My goal was to help teachers think about the things they can actually do to support productive participation in mathematical discussions. I gave the teachers some more time to think about these ideas and brainstorm ways of developing norms that help alleviate status problems.

Another great list was generated. I am adding my commentary to the teachers’ ideas.

  • Frequently vary groupings so people can be exposed to other people. This is important. A lot of times teachers want let students choose groups, which can especially aggravate status problems around social desirability. Other times, teachers use a “high, mid, low” achievement scheme. Students quickly size that up and know where they stand in the pecking order, which reinforces academic status problems.
  • Use “round-robins”: everybody gets 1 minute to speak, whether or not you use all of it. This is not one that I have used, but the teacher who introduced this idea talked about how they let the clock roll for the full minute, even when students only spoke for 15 seconds. The quiet time was usually good thinking time for his students.
  • Randomly call on kids. The teacher who introduced this one explained that she had playing cards taped on students’ desk, with the number representing their group (“the kings”, “the 12s”) and the suit representing an individual student. She could then pull out a card from her deck and call on “the 2 of diamonds.” I asked her what she did when a student didn’t know. She told me that she would sometimes get others to help them or move on then come back to them later, even if only for a summary statement. I added that I think it is really important to have a clear understanding in the class that partial answers count (see the “right and wrong” answer problem above) to successfully use random calling on kids. Otherwise students might shut down and feel on the spot.
  • Making an initiative to make norms school-wide.This was an insight close to my heart. As the teacher who contributed this idea said, it will be much more powerful for students to get the same message about how to participate from more adults in the school.
  • Tension: having students value ideas without getting stuck on ideas. This referred to the way kids can get wedded to particular ideas, even when they are wrong. I talked about how important it was to emphasize the value of changing your mind when you are convinced, not based on who is arguing with you. This is the heart of productive mathematical conversations.
  • Tension: shifting from right/wrong to reasoning. Need to be transparent. The teacher who talked about this saw that emphasizing reasoning can be a game-changer for students who are good at seeing patterns and memorizing methods. They may know how to do things but have no idea why they do things: they suddenly go from “good at math” to “challenged.” I suggested addressing the concerns of these students from the perspective of advocacy: “I love your enthusiasm for math! I know what happens as you go up the curriculum, and you will really need to understand why things work, so I am giving you a chance to build those skills now.”
  • Normalizing conflict through “sentence starters.” Conflict and arguing are usually seen as bad things to students, yet we want to create situations that allow for mathematical disagreements. By using sentence frames  –– and even posting them on the classroom walls –– we can help students learn to civilly disagree. For example, “I disagree because ____” or “How do you know that _____?” This also helps students press each other for justification.
  •  “Everyone listening, everyone speaking, everyone responsible for understanding.”
    This was a norm that could really help encourage participation.
  • Role playing & discussion as a way of (re)establishing norms. This teacher pointed out that norms sometimes need to be talked about explicitly –– and they often need to be revisited over the course of a school year. I added that I notice that certain curriculum topics (e.g., fractions) can bring up status issues, requiring certain norms to be revisited.
  • Celebrating mistakes as opportunities to learn. How is that for normalizing confusion? Normalizing mistakes as a way for everybody to think harder about a topic or idea. I asked for some specific language for this, and the teacher suggested something like, “Thank you for bringing that up. We will all understand this better by discussing this.” (Sorry! This is from memory!)
  • High social status kids as “summarizers,” give them math status. Sometimes students with high social status do not have high academic status. By giving them a mathematical role, we can marshal the fact that others listen to them and help build their understanding by giving them a particular role.
  • Valuing different ways of contributing. Another one close to my heart! There are many ways to be smart at mathematics, and by valuing different ways kids can contribute, we can increase participation.

Thank you to the teachers of PCMI for the great conversation! Please add anything that I forgot to the comments section, and stay in touch!