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.

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.

Why Meaningful Math Learning Matters

What Meaningfulness Means

Learning and schooling are not the same thing. There are children who are great learners but terrible students. These young people are full of ideas and questions, but they have not managed to connect their innate curiosity with their experiences in school. There are many possible reasons for this. Children may find school to be a hard place to inhabit, due to invisible expectations that leave them feeling alienated. Sometimes, school curriculum just seems irrelevant: their personal questions about the world do not find inroads in the work they are asked to do.
Although many parenting books extol children’s natural curiosity and emphasize its importance in their learning and development, schooling too often emphasizes compliance over curiosity. Thus, it is not surprising that children who are great learners and weak students have their antithesis: children who are great students but who are less invested in learning and sense making. Make no mistake: these students hit every mark of good organization, compliance, diligence, and timely work production, but they do not seek deep engagement with ideas. Given the freedom to develop a question or explore an idea, they balk and ask for more explicit directions. I have heard teachers refer to these children as “teacher-dependent.”
Too often, meaningfulness falls through this gap between learning and schooling. There is a fundamental contradiction at play: meaningfulness arises from and connects to children’s curiosity, yet “curious children” is not entirely synonymous with “successful students.” Meaningfulness comes about when students develop an appreciation for mathematical ideas. Rich and meaningful learning happens when students draw on prior knowledge and experiences to make sense of ideas and explore problems, invoke their own strategies, get to ask “what if…?”  In short, meaningful learning happens when students’ activity connects to their own curiosity. To make meaningfulness central to math teaching, then, teachers need to narrow the gap between being curious and being a good student.


Meaningfulness: When students connect their own curiosity and experience to ideas, thereby developing an interest in and appreciation for mathematical content.

Why Meaningfulness Matters

Every math teacher, at one one time or another, has been asked the question, “When are we going to use this?” While this question often gets cast as students’ resistance to learning, it can be productively reinterpreted as a plea for meaningfulness. When the hidden curriculum of math class –– the messages that are inadvertently relayed through classroom organization and activity –– consistently communicates that meaning does not matter, we end up with hordes of students who no longer reason when they are doing math. They instead focus on rituals, such as following the worked example, and cues, such as applying the last learned procedure to the current problem.
As researcher Sheila Tobias explained in her classic exploration of math anxiety, a lack of meaning exacerbates many students’ negative experiences learning mathematics. When math class emphasizes rituals and cues that rely on memorization over sense making, students’ own interpretations become worthless.

For instance, they memorize multiplication facts, and, in a search for meaning, they decide that multiplication makes things bigger. Then, they learn how to multiply numbers between 0 and 1. Their prior understanding of multiplication no longer works, so they might settle on the idea that mulitiplication intensifies numbers since it makes these fractional quantities even smaller. Finally, when they learn how to multiply negative numbers, all their ideas about multiplication become meaningless, leaving them completely at sea in their sense making. The inability to make meaning out of procedures leaves students grasping and anxious, as the procedures seem ever more arbitrary.
In contrast, when classrooms are geared toward supporting mathematical sense making, they reap multiple motivational benefits. First, students’ sense of ownership over their learning increases. Students see that multiplication can be thought of as repeated addition, the dimensions of a rectangle as related to its area, or the inverse of division. When they learn new types of multiplication, the procedures have a conceptual basis to expand on. Relatedly, their learning is more durable. Because they understand the meaning behind the mathematics they are learning, they are more likely to connect it to their own experiences. This, in turn, provides openings for their curiosity and questions. Beyond giving students opportunities for sense making, meaningful mathematics classrooms provide students chances to identify and explore their own problems. Indeed, in a systematic comparison of teacher-guided and student-driven problem solving, educational researchers Tesha Sengupta-Irving and Noel Enyedy found that the ownership, relevance, and opportunities to engage curiosity in student-driven problem solving supported stronger outcomes in student affect and engagement.[1]

The challenge, then, for teachers is how to help students engage in meaningful mathematical learning within the structures of schooling. I would love to hear your ideas about how to achieve this.

[1] Tesha Sengupta-Irving & Noel Enyedy (2014): Why Engaging in Mathematical Practices May Explain Stronger Outcomes in Affect and Engagement: Comparing Student-Driven With Highly Guided Inquiry, Journal of the Learning Sciences, DOI: 10.1080/10508406.2014.928214

Professional Development is Broken, but Be Careful How We Fix It

This morning, Jal Mehta tagged me on a tweet to linking to his recent Education Week blog post, entitled “Let’s End Professional Development as We Know It.”

The following exchange ensued:

He then asked if I could share some of my research to back my perspective. I sent him an email with journal articles and such, but I thought I would share my ideas with y’all too.

Here is my argument about why putting professional development (PD) back in schools may be necessary but not sufficient to improving its impact on teachers’ instruction.

Unlike medicine and other scientific fields, where problems are taken-as-shared and protocols for addressing problems are roughly agreed upon, teaching problems are locally defined. What needs attention in one school may not need attention in another. For instance, some schools’ “best practices” may center on adapting instruction to English learners, while other schools’ might center on the mental health ailments that have become prevalent among affluent teens. Likewise, other professions share language, representations, and goals for critical aspects of their work — these all important resources for learning together. In teaching, we see repeatedly that terms acquire the meaning of their setting more often than they bring new meanings to these places. Take, for instance, Carol Dweck’s ideas about mindset. The various ways that her construct has taken hold in education led her to explain why what she means by mindset is not how the idea is being used. If we leave professional development entirely up to individual school sites, this means that “doing PD” on Topic X probably looks fairly different from place to place, so radically localized professional development will exacerbate this problem.

Leaving professional development to local sites also limits teachers’ access to expertise. When my colleagues and I have studied teachers’ collaborative learning, we found that the learning opportunities are not equally distributed across all teacher groups. Some of this has to do with how teachers spend their time (e.g., focused on logistics or deeper analysis of teaching). But some of it has to do with who is sitting around the table and what they have been tasked to do.

Teachers’ collaborative learning can be described as an accumulated advantage phenomenon, where the rich get richer. That is, teachers who have sophisticated notions of practice are able to identify teaching problems in complex ways and deploy more sophisticated strategies for addressing them. This follows from my previous points, since problem definition is an important part of teachers’ on-the-job learning. For instance, if we have a lot of students failing a course, how do we get to the bottom of this issue? In many places, high failure rates are interpreted as a student quality problem. In others, they are taken as a teaching quality problem. Interpretations depend on how practitioners think this whole teaching and learning business goes down. In other words, problem definition is rooted in teachers’ existing conceptions of their work, which in other professions, are codified and disseminated through standardized use of language and representations.

Unequal access to expertise is only one of many reasons the optimistic premise of teacher community often does not pan out. There is a tendency to valorize practicing teachers’ knowledge, and, no doubt, there is something to be learned in the wisdom of practice. That being said, professions and professionals have blind spots, and with the large-scale patterns of unequal achievement we have in the United States, we can infer that students from historically marginalized groups frequently live in these professional blind spots. For reasons of equity alone, it is imperative to develop even our best practitioners beyond their current level by giving them access to more expert others.

Even in highly collaborative, well-intentioned teacher communities, other institutional pressures (e.g., covering curriculum, planning lessons) pull teachers’ attention to the nuts-and-bolts of their work, rather than broader learning or improvement agendas. Add to this the norms of privacy and non-interference that characterize teachers’ work, you can see why deeper conversations around issues of teaching and learning are difficult to come by.

What about, you might say, bringing in expert coaches? Research shows that expert facilitators or coaches can make a difference. In fact, there is evidence that having expert coaches may matter more than expert colleagues when it comes to teacher development. At the same time, we suspect that expert facilitators are necessary but not sufficient, as coaches often get pulled into other tasks that do not fully utilize their expertise. In our current study, we see accomplished coaches filling in for missing substitute teachers, collating exams, or working on classroom management with struggling teachers. None of these tasks taps into their sophisticated instructional knowledge. Additionally, being an accomplished teacher does not guarantee you have the skill to communicate your teaching to others. In our data, we have numerous examples of really great teachers underexplaining their teaching to others.

Lee Shulman famously called out the missing paradigm of teacher knowledge, giving rise to a lot of research on pedagogical content knowledge (PCK). While PCK gave a very useful way to think about teachers’ specialized knowledge, little progress has been made on understanding how teachers develop this and other forms of knowledge, particularly in the institutional context of schools, which often presses teachers’ practice away from what might be deemed “good teaching.” As long as we don’t have strong frameworks for understanding how teachers learn, PD –– even localized, teacher-led PD –– risks being just another set of activities with little influence on practice.

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.


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.

Teacher Community and Professional Learning

One of the things I study is how teachers learn with colleagues. I focus primarily on urban secondary math teachers. I basically film people working together and analyze it to death. I am interested in this because teacher collaboration is repeatedly shown to support both teacher learning and student achievement, so I am curious about why.

First things first. Strong collaboration is very rare. Very few high school teachers report even simply sharing ideas with colleagues. Productive collaboration goes beyond just sharing ideas or resources into what I have called collaborative pedagogical problem solving. This is really unusual but super cool when I get to see it.

I want to make two points about what I have observed, and then pose some questions to the #MTBoS .

Observation 1: Effective collaboration is hard.

There are a number of challenges to effective collaboration. First of all, it takes an investment of time, energy, and emotional commitment. These are scarce resources, particularly in high turnover schools. Teachers face a lot of structural obstacles to collaborative learning. The typical 50 minutes of daily planning time, for instance, is already overfull with the demands of grading, planning, and home communication.

Second, it’s hard to talk about instruction with colleagues. When teachers talk about instruction, this is almost always asynchronous from the active work of instruction. Unlike scientists, who have standardized ways of representing what happened in the laboratory, teachers do not have standardized ways of representing what happened in a lesson. We can use things like student work, but then we do not have standard ways of interpreting these. Some teachers will look at student work with a right/wrong lens, while others will want to understand a students’ thinking.

At the same time, one of the advantages of working in a school-based teacher community is that your colleagues are close by: they know your administrators, they know your community, they know your students. You don’t have to explain those things to them, which makes the description part of sharing a little easier.

Observation 2: Typical teacher collaborative talk does not support deep professional learning. When I have analyzed the learning opportunities in teachers’ conversations, I have looked at two things:

(1) what conceptual resources are being developed as teachers talk about instructional problems, and

(2) how are these connected to their future work.

Most teacher collaborative talk does not offer much in the way of professional learning.

For example, most teachers plan together by organizing a pacing calendar. They will say things like, “The book says 7.1 will take 1 day, but with our kids we’ll need 2.”

In this case, the opportunities to learn are thin. We don’t know what the math content is, we don’t know why we need two days, and we don’t know how that extra time will be used.

In contrast, if teachers plan by building on students’ thinking, their talk may sound different. They will say things like, “Our kids freeze when we do fractions. Let’s just focus on these problems as rates of change. We can show them on the graph how this is change over time, like, “for every 5 seconds, the car moves 10 feet.'”

In this case, concepts are developed about who the students are, what their experiences of math are, and what instruction might look like to keep them engaged and develop their mathematical understanding. These concepts are directly linked to what the teachers will do next in their classrooms.

MTBoS Challenges:

Regarding Observation 1: In some ways, the bloggy/tweety teachers have overcome some of the limitations of school-based teacher community by finding like-minded folks online. They have found their kindred spirits to share with. This is awesome and overcomes some of the limitations of traditional collaboration. Also, the MTBoS are typically tech savvy. I have been impressed with the ways they manage to represent their classrooms through samples of student work, lesson plans, photos of their classrooms with kids doing things. But, other details of our teaching situations –– the tetchy administrator, the new curriculum policy –– are not as readily available.

Is this an issue? How much does this limit what teachers can learn together online?

Regarding Observation 2: I have seen so many impressive exchanges among teachers in the MTBoS. Most of these have focused on dissecting mathematical content, sharing rich activities, and refining instructional language. It seems harder to share about the particulars of students and their thinking because those are so much more specific to people’s schools.

Is it possible to hit the sweet spot of professional learning –– to develop concepts about the interrelationships among students, teaching, and mathematics  –– through online interactions?

“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.”


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.

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.


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…


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


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!

How do we build math- and kid-positive department cultures?

I was pleased with the responses to my last post. A number of teachers reached out via twitter and comments, asking how they might build math- and kid-positive cultures in their own schools.

I can’t offer any large scale studies of the answer to this question, even though I am currently engaged in a research project that is trying to work with districts on similar issues. But I can share some of the experiences I had working with teachers in the Pacific Northwest toward this goal.

Gather invested colleagues around a common problem.

I always say, I have yet to meet a teacher who goes into the profession for the glamor or the money. Almost everybody who becomes a teacher wants to help kids. Find the folks whose heart is still in that. Find the ones who are willing and able to invest the time in their professional growth and look for a problem to work on.

That is what we did in the partnership project that went on for 6 years with some urban high schools. We started teachers at a school we called Septima Clark High. To get started on our work together, we used a process called “The 5 Whys” to try to get at the root of a problem that was bothering them. Their burning question was: why are so many kids failing 9th grade math?

First we brainstormed the answers to this question. I listened and recorded the brainstorm non-judgmentally and without conversation. This went on for over an hour, and we only got to a second level of “whys.” The result came to be known among us as “The Wall” because, as I wrote all the reasons on giant post it notes, they filled an entire wall. Seeing all the answers to this question was rather overwhelming.

The Wall

The next step in the process was to look at this vast list and identify the things we could actually do something about. I underlined these. A small fraction of the reasons were actionable, but they gave us a way in to make a plan and set goals.

We sorted the actionable reasons into categories. From this, the teachers arrived at two conclusions:

  • that their curriculum wasn’t engaging all students, and
  • they needed to update their teaching practices.

The process was vital to teachers’ sense of ownership over our subsequent work.

Work together on a productive framing of that problem, linked back to math teaching and learning.

It’s one thing to identify a problem, like a high failure rate in 9th grade math. It’s altogether another thing to come up with a productive framing of that problem. Problem framings are how we define the parameters of something.

All too commonly, it is easy to point fingers and play the blame game: prior teachers did not do their job; the promotion policies that pass kids along; a lack of academic role models in kids’ lives. These reasons made the teachers’ brainstormed list. But none of them supported anything actionable on the part of the teachers. On the other hand, things like “kids aren’t engaged in the mathematics” did provide inroads for the teachers. By pressing teachers on what they can do, the framing that came out of this observation was that classroom activities and structures needed to encourage more participation. That was something teachers could work on.

Get support to have the time and space to meet to work on that problem regularly.

The hardest part about this process is that there is no way around how time-intensive it is. We all know teachers’ days are already overly full, despite the myth of teaching as an “easy, kid-friendly schedule.” Time diary studies of teachers’ work show that they work long hours, fitting a lot in on schooldays, weekends, and summers.

This is where administrative support can help. If there is already professional development time designated for your school, see if you can repurpose it for your goals. Even one hour after school weekly can make a difference. The best situation is to have common planning time with your collaborators, but this is a tricky and even expensive investment for schools.

Set short term and long term goals for your work, and find resources.

Too often, educational reform is treated like an appliance that can be brought anywhere and work the same way every time. We expect schools and teachers to “try” something, as if it’s just a matter of flipping a switch and saying yea or nay.

Education, however, is a human endeavor. The specifics of any setting and situation matter a lot for what works and how to get it working. Change takes time, especially ones that press on teachers to examine their core assumptions about teaching, learning, and mathematics.

One of my former doctoral students, Nicole Bannister, studied the Septima Clark teachers for her dissertation. She writes about their learning process and how they found ways to see and support their struggling students in their classrooms.

Celebrate the small victories, because there will always be setbacks and challenges.

School was originally designed on the factory model. Knowledge was thought of as a product that teachers could give to students efficiently on a set schedule. We now know that learning does not work that way — deeper understandings that support retention and fluency with mathematics cannot simply be delivered.

Re-culturing teaching –– reimagining the relations among students, mathematics, and teachers, as well as the activities that happen in the classroom –– to support more effective learning  is challenging work. Fundamentally, it involves working against the institutional grain of schooling, so there will be setbacks and challenges. For this reason, the small triumphs cannot go unnoticed: the student who makes sense of an idea for the first time, the one who participates after a long period of silence, the eagerness students have for a certain problem or project. All of these moments matter and need to be shared with the team. Otherwise, a team risks discouragement and burnout.

Share your work to help build critical mass.

Even before any results came in, the Clark teachers worked hard to communicate what they were doing with colleagues and parents. They held a meeting in the school library one evening to explain their understanding of the failure rate problem and the work they were invested in addressing it. Having the community support mattered. Even skeptical parents were heard saying, “If the teachers are this excited about what they are doing, I won’t stand in their way.”

Eventually, the kind of results administrators care about came in too. In the 2004-05 school year, before the team’s work began, less than half of the students who entered ninth grade at or below grade level were promoted to 10th-grade math. The following academic year, 83%
of those students were promoted.


This was not about dummying down content. In fact, the mathematical depth of classroom activities increased, as did student participation.

During the next state testing cycle, the student gains at the school caught the attention of the district. As the above chart shows, we saw higher achievement among Black* students and low income students, two groups that were of concern in the school and district. Soon, other schools wanted to learn about their work. Clark became a place that administrators visited, as did other teams of teachers.

*   *   *

The ongoing challenge for departments that reculture is how to sustain that work over time. In the last post, I told you about Railside, a place where I studied and taught. They managed to sustain their work for over two decades before policy pressures undid significant aspects of their work. Maybe if we can get critical mass at a national level, we can convince more people that organizing teaching so kids can learn is a worthwhile investment.



* I use the term “Black” because some students were African and some were African American, but they generally referred to themselves as black.


My Opinion on Common Core State Standards in Mathematics

I have had the luxury of taking time to form my opinion on the new Common Core Standards.

There are three issues to consider, all of which get discussed when we talk about them.

1. The content of the standards themselves.

2. The nature of the assessments used to hold schools accountable for them.

3. The implementation of them, from curricular support, professional development and accountability processes.

My take on Issue 1 is that they are a strong first draft. The practice standards are the boldest and most important innovation, since they press on higher order thinking. Nonetheless they have some flaws. For instance, a teacher friend told me one grade asks that students learn to make box-and-whiskers plots while the subsequent grade asks for students to compare them to look at differences in measures of central tendency. Well, making those plots without looking comparatively is a silly exercise since the whole point is that they make measures of central tendency and spread visible. Goofs like this could be tweaked in field testing, but the authors did not have that chance.

2. I had some hope that the ‘second generation’ assessments developed for CCSS-M would be a step up from a lot of what we have seen. The release items I have seen so far have not carried out that promise.

3. The biggest problem, in my mind, is the rush of implementation and the lack of resources to make this ambitious goal feasible. Perhaps the most fatal aspect of implementation is that CCSS-M is getting put into the very flawed infrastructure of NCLB/RTTT. On the ground, it ends up feeling like a turning of the screws in the already problematic accountability pressures schools and teachers are facing.