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!




Playful Mathematics Learning

I have had the great pleasure of spending the last several days at the Minnesota State Fair.

math on a stick

My colleague Melissa Gresalfi and I got a National Science Foundation grant to study a very special exhibit there called Math On-A-Stick. We have an awesome team of graduate students helping us with the research. They are Lara Heiberger, Panchompoo Fai Wisittanawat, Kate Chapman, and Amanda Bell.

PML Logo

Math On-A-Stick is the brainchild of Christopher Danielson,  educator, promoter of talking math with your kids, and mathematical toy maker.

christopher danielson

That is Christopher on the right. The woman in the pink jacket is a former math teacher. She made the beautiful quilt for Math On-A-Stick.

The exhibit is just a delight. Not only is it a lovely respite in a shady, relatively quiet corner of the fairgrounds, it is filled with math play. Here are a few of the stations in the exhibit.

On the left is a tile station. The tiles are half black, half colored, and children can make all kinds of patterns with them. The center image comes from the pentagon station. I could spend all day there myself. I made that creation. On the right are tessalating lizards and turtles.

Everyday there are visiting mathematicians and mathematical artists. The first day I was there, Megan Schmidt brought some of her spiral magic. Yesterday were hexaflexagons.

Today, Christopher was the Visiting Mathematician. He built a giant pattern machine that children could play with. It is made up of little “pattern machines,” and the buttons pop up and down, making a satisfying clicking noise.

Melissa and I are interested in studying two things about children’s encounters with the exhibit. First, we are interested in the design, investigating how the various activities support mathematical interactions between children, the exhibit, the mathematics educators, parents –– and each other. Second, we are interested in children’s engagement. We want to examine how the children engage with different parts of the exhibit, looking for relationships among children’s ideas about mathematics, reported experiences in math class, and the exhibit design.

Our primary data come from recordings the kids make while they are playing. We outfit them with GoPro cameras so we can see how they interact with the exhibit, recording their interactions, their general gaze, and the time they spend at the various stations.


Melissa and Fai set up a stationary camera, while Lara pretends to be a kid at play.

IMG_9942 (1)

A couple of kids getting outfitted with GoPros

This is supplemented by entry and exit surveys, brief interviews, and stationary recordings of the stations (e.g. a camera positioned at the Pentagons so we can see how a cross section of children play with that station and compare that to the activity of our focal children).


Data collection station. It’s a well oiled machine.

We weren’t sure how kids would feel about us approaching them and asking them to wear the cameras on their heads. It turns out that they love it. They are really happy to share, as are the amazing volunteers, who have been very agreeable to getting captured in the video footage as the children play.

Our research findings will help us identify more ways to make mathematics play a part of instruction. Already, many children are telling us that Math On-A-Stick math is different than what they do in school –– even kids who are inventing and problem solving in impressive and novel ways. We are looking forward to analyzing the hundreds of hours of data we collect and sharing what we learn with all of you.


What Does It Mean to Study Teachers’ Learning from a Sociocultural Perspective?

I try to be a plain-talking academic when I engage in the public realm of social media. Sometimes, despite my best efforts, I find myself wanting to use academic jargon. My goal in writing this blog is to have conversations with both educators and researchers, so I think it is okay to have “turns” of conversation that lean a little more on my research voice than my educator voice.

Sociocultural is jargon word that I have wanted to invoke from time to time when talking to my practitioner friends. In particular, the research I do uses sociocultural learning theories as a way of describing both how students and teachers learn.

But what does that mean? In order to understand, you need a little history on how we have come to think about learning the way we do.
In the late 19th and early 20th centuries, U.S. research on learning was dominated by behaviorism. Seeking a rigorous empirical basis for a study of behavior, researchers like E.L. Thorndike and B.F. Skinner sought to explain how learning happened by documenting what they could see empirically.

Out of this theory, we have ideas like operant conditioning, where actions are shaped by stimulus and responses in the environment to ultimately change behavior. Skinner famously made little operant conditioning chambers called “Skinner boxes” that successfully “taught” pigeons to dance. Through the boxes, food was dispensed in response to the pigeon’s movements. If he turned his head to the left –– the stimulus –– he would get a food pellet –– the response. The next time, he had to turn his head a little further to get his food. Eventually, through operant conditioning, the pigeon learned to turn in a full circle –– to “dance” –– to get food.

dancing pigeons

Behaviorism explained some forms of learning, but it couldn’t explain everything. In the 1950s, the cognitive revolution began. Researchers like Jerome Bruner began to critique behaviorism, noting that a sole focus on behavior precluded a study of how people created meaning, a central question in understanding why people do what they do. Researchers realized they could do empirical studies that included a theory of the mind. Using methods like case studies and talk aloud protocols, investigators could examine how people made sense of their activities in the world.

Cognitive science, as it came to be called, led to important insights like schema theory and conceptions. A schema is a general system for understanding how knowledge is represented and how it is used.

Researchers can look for evidence of different schemata (the plural of schema). Like the behaviorists, they observed what people did to understanding learning. However, they augmented this by asking people to explain their thinking through interviews and surveys.

To give an example of a schema, let’s take the word “dog.” When I say “dog” what do you imagine?

You probably think of four-legged animals that bark, are furry, have tails. But how do you know that these are all dogs?

How do you know that these are not?

This is the question that underlies the idea of schemata.

The examination of schemata started to point to the importance of culture. Schemata are closely related to prototypes. So, for example, when I say the word “furniture” what do you think of?

Linguists have found that when you say the word “furniture” to Americans, they think the best examples are chair and sofa.

When you say the word “möbel” to Germans, however, they think the best examples are bed and table. Our schemata and our prototypes –– the building blocks of concepts in the world –– are culturally specific.

By the early 1990s, this increasing recognition of the importance of language, culture, and context shifted our ideas about learning yet again. Language and culture were not just the setting for development and thinking –– some kind of external variable to be controlled for –– they were, in fact, fundamental components of these mental processes. This insight meant that, to explain some learning phenomena, researchers needed to do more than describe mental structures.

This required another broadening of research methods. Using linguistics, anthropology, and sociology, learning researchers wanted to account for how concepts stretched beyond individual minds and into the world. Deeply influenced by Soviet psychologist, Lev Vygotsky, researchers working in this sociocultural tradition examined learning as it happened in interactions in the world, requiring new units of analysis. That is, instead of studying individuals as they learned, researchers sought ways to study individuals in context.

My own research takes up these sociocultural insights to re-think how we study teacher learning. Let me paint a bit of a picture for you about the intellectual traditions that shape my work.

First, when I entered my doctoral program at UC Berkeley in the mid-1990s, debates between cognitive and sociocultural perspectives on learning were quite active in my courses and in research groups. Although most arguments centered on questions of student learning, there was a growing interest in what was often called “out-of-school learning.” Influenced by anthropological researchers like Jean Lave, a small group of scholars studied workplace learning, a particularly pressing topic in our modern information economy, where workers must constantly adapt to a rapidly changing world.

Meanwhile, in educational policy studies, there was a growing recognition that research on school organization, curriculum, and teacher professional development had overlooked a central question: How do teachers’ learn? Since almost all school improvement efforts want to improve instructional quality –– through curricular reform, changes in scheduling or assessment techniques –– they all depend on what happens inside of classrooms. And that, of course, depends on what happens with teachers.

For this reason, educational policy scholars like Judith Warren Little and Mike Knapp were recognizing that teachers’ learning is an underanalyzed component of any efforts at school change or instructional improvement. Yet it was not central to policy designs –– let alone to analyses of their effectiveness.

The moment was ripe for somebody to connect these ideas. My work starts with the policy-based observation that designs for instructional change must consider teacher learning. I then use methods and insights from sociocultural theories of learning to examine how teachers’ learning happens in the school as a workplace. As the sociocultural theorists suggest, what teachers know and learn is not solely a product of what is in their individual heads.

Concepts for teaching draw on culturally specific practices and language in the world. For instance, in the U.S., we often start grouping children by ability levels at a very young age. The concept of a “high ability 6 year old” makes sense for American teachers in a way that it would not to teachers in countries that do not track in the elementary years. There are consequences to that concept having social meaning, as educators make decisions about their schools and classrooms and parents advocate for certain experiences.

By using sociocultural perspectives to explain teachers’ learning, my research is culturally specific and theoretically specific. Although the details of what I find about U.S. teachers may not generalize to other countries, it is my hope that my descriptions of teachers’ learning can be more generalizable.

Structure Can Change Agency

One great privilege of the work I do are the many opportunities I get to share the things I care about with different groups of people. If you do it enough, you get a chance to clarify your own ideas, learn from others, and notice connections.

This past weekend, I had the honor to give a keynote talk at the Carnegie Math Pathways Forum. If you don’t know about their work, it is worth checking out. Briefly, their work addresses the enormous blockage in the math pipeline as students transition from secondary to post-secondary. A staggering number of students get placed in developmental math classes, and often, these courses become a holding bin students cannot get out of. The Carnegie folks have worked primarily with community college instructors to re-think developmental math curricularly and pedagogically. It’s fascinating and important work.

My talk was about the relationship between structure and agency, how both contribute to inequalities in mathematics education. When we are teaching in a classroom, it is easy to see problems of inequality as they look locally: high enrollments in developmental math, over-representation of students coming from poverty and students of color, a sense of student apathy. To make progress, however, instructors can learn by linking the local to broader social processes: the maldistribution of qualified math teachers, STEM classrooms that are hostile environments to minoritized students, a K-12 curriculum that often reflects the institution of schooling more than what it means to do meaningful mathematics. I argued that if we frame these problems through what we see locally, we give ourselves, as teachers, less leverage to make progress on them. I shared two key concepts for linking these social processes to what we see in our classrooms: social risk and status. I have written about both of these (click the links if you are curious), but briefly, social risk refers to the threats people feel are posed to their status in a community while status describes the perception of students’ academic capability and social desirability. Both of these ideas link the social process explanations for inequality to what teachers see in their classrooms locally.

Teachers can then work to design classrooms that reduce social risk by, in part, attending to status dynamics. In other words, to connect structure and agency, we need ways to think across scale and look at the social origins of problems too often narrated as individual issues. Instead of, for example, blaming students for being apathetic about mathematics learning, we need to recognize what their history has likely been in our current system and accept their apparent apathy as a reasonable response. Our task shifts from finger pointing (“My students just aren’t motivated!“) to having the productive challenge of honoring their experience while trying to change their ideas about math and learning.

In the end, then, structure can help us change agency in two ways. First, by recognizing that it is there, along with the social processes it holds in place, we can arrive at more productive framings of the problems we face locally. Second, we can leverage the structural designs in our classroom to invite students’ agency.

I have written about designing structures to promote agency before. If you don’t feel like reading that (I realize it’s summer!), maybe watch this video instead. It is quite a joy.

And don’t we all need more of that right now?


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.

Reinventing Mathematics Symposium at The Willows School

I am honored to be presenting tomorrow at the Reinventing Mathematics Symposium at the Willows School in Culver City, CA.

My workshop is on Playing with Mathematical Ideas: Strategies for Building a Positive Classroom Climate. Students often enter math class with fear and trepidation. Yet we know that effective teaching engages their ideas. How do we lower the social risk of getting students to share to help them understand mathematics more deeply? I will share what I have learned from accomplished mathematics teachers who regularly succeed at getting students to play with mathematical ideas as a way of making sense.

In my workshop, I will develop the concepts of status and smartness, as well as share an example of “playful problem solving.” Here is the Tony De Rose video we watched, with the question: How is Tony De Rose mathematically smart? If he were a 7th grader in your classroom, what chances would he have to show it?

Usually teachers like  resources, so I have compiled some here.


Bellos, A. & Harriss, E. (2015). Snowflake, Seashell, Star: Colouring Adventures in Wonderland. Canongate Books Ltd; Main edition

Childcraft Encyclopedia (1987). Mathemagic. World Book Incorporated.

Jacobs, H. (1982). Mathematics: A Human Endeavor. W.H. Freeman & Co Publishers.

Pappas, T. (1993). The Joy of Mathematics (2nd Edition). World Wide Publishing.

Van Hattum, S. (2015). Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. Natural Math

Weltman, A. (2015). This is Not a Maths Book: A Smart Art Activity Book. Ivy Press.

Blogs that Feature Playful Mathematics

Math in Your Feet Blog

Talking Math With Your Kids

Visual Patterns

Math Munch

Some Inspiring Ignite* Talks that Give Ideas about Teaching Playfully

*Ignite talks are 5 minute long presentation with 20 slides and with the slides advancing automatically every 15 seconds. It’s the presentation equivalent of a haiku or sonnet.

Peg Cagle, What Architecture Taught Me About Teaching

Justin Lanier, The Space Around the Bar

Jasmine Ma, Mathematics on the Move: Re-Placing Bodies in Mathematics

Max Ray, Look Mom! I’m a Mathematician

There are tons more. The Math Forum does a great job of getting outstanding math educators to share their work in this series of talks.

Please feel free to add other good resources in the comments section!


What are the Grand Challenges in Mathematics Education

Back in March, the National Council of Teachers of Mathematics put out a call for Grand Challenges in Mathematics Education.

A Grand Challenge is supposed to spur the field by providing a focus for research. NCTM came up with the following criteria for a Grand Challenge in math education:

Research Commentary-Grand Challenges_1

So I ask you to help the brainstorm. What are the complex yet solvable problems we face in mathematics education that can have a great impact on people’s lives?

Add your thoughts in the comments below or through Twitter (@tchmathculture). Use the hashtag #NCTMGrandChallenge.

Facilitating Conversations about Student Data

Sometimes, you ask and the internet answers. My research team and I (doctoral students Britnie Kane, Jonee Wilson and Jason Brasel and postdoctoral fellow Mollie Appelgate) wrote this a couple of years ago at the request of one of our district partners. We have been studying how teachers learn through collaborative time.

This memo focuses on research-based ideas on how to support common planning time so that it has the greatest potential for teacher learning about ambitious mathematics teaching. To that end, we provide a framework for effective conversations about mathematics teaching and learning. We develop the framework by using vignettes that show examples of stronger and weaker teacher collaboration.

We use vignettes for a couple of reasons. Primarily, we are concerned about the confidentiality agreements we have with study participants, which protect their right to remain unidentified. Additionally, sometimes raw conversational data takes ramping up to understand. Details about particular group histories or the nature of the problems they are looking at that do not communicate well in brief excerpts. The vignettes are clear illustrations of key ideas that also protect our participants’ confidentiality.

Part I: A Brief Conceptual Framework for Understanding Teacher Collaboration

Our research centers on how teachers learn about important instructional issues through their collaborative time. Based on our work in MIST as well as previous research, we have found that teacher workgroups’ discussions are richest when they include rich depictions of connections across students’ thinking, teaching, and mathematics.


The Instructional Triangle

These are three critical aspects of teaching that are frequently represented as Schwab’s Instructional Triangle (see above). Rich collaborative discussions draw upon and make connections among the three elements of the instructional triangle. For instance, teachers can consider how students’ understandings of particular mathematical ideas can be drawn out and developed through the design of a particular lesson. Notice how this example accounts for relationships at each point of the triangle. As we will elaborate, the consideration of multiple dimensions of classroom teaching makes such a conversation richer for learning than, say, one that solely focuses on what mathematics comes next in the curriculum, without accounting for the particulars of students’ thinking or other lesson details.

Sometimes, it is assumed that doing certain types of activities will lead to better learning opportunities for teachers. For example, the very name “common planning time” implies that planning should be a central activity, perhaps with an assumption that co-planning is more important than looking at student work. One important finding in our research is that activities in themselves are not richer in learning opportunities. In other words, there are versions of “planning” that are replete with teacher learning opportunities and there are versions that have few of them. Likewise, there are strong and weak versions of “looking at student data” or “looking at student work.” In the following sections, we provide examples of strong and less-developed collaborative conversations, along with commentary to help you all make sense of these. It is our aim to both illustrate this point and fill out our notion of rich teacher conversations.

Part II: What does rich teacher collaborative talk sound like?

In our data, we often see teacher workgroups participating in three different activities: planning (which teachers seem to find most useful, since they have to plan anyway), looking at student data (which administrators often encourage because of accountability pressures), and looking at student work. We will provide stronger and weaker examples of “looking at student data.”

Vignette 1: Rich Talk about Student Data

In this vignette, three teachers are discussing their students’ interim assessment results. In front of them, they have the test booklet and their school’s distribution of student responses. Using these materials, they have been looking at which questions were frequently missed and then looking at the items to make sense of what their students struggled with. Their conversation included the following discussion about a problem involving supplementary angles.

Maricela: On this one I think our kids are having a hard time with this because it asks for supplementary angles but the angles are next to one another. And that is not what the kids are used to seeing.

Diane: Yeah, that’s how I showed ‘em too.

Marcus: Exactly. I think it was confusing to them because they were looking for angles that butted right up next to each other and obviously, on this one, there is not a straight line at the bottom which would say supplementary to them.

Diane: I don’t think this was so much about not understanding what supplementary meant as…

Maricela: No, I agree… which is frustrating that they would understand what it meant but still miss it just because of the picture.

Diane:  So how can we teach this differently next time?

Maricela: We should probably use different ways to represent the supplementary angles.

Diane: Yeah, not always using the straight line and asking, “What is the supplementary angle?” but also just drawing an angle.

Maricela: Or even stressing that the definition is really adding up to 180 degrees, that the angles don’t all have to be together to be supplementary.

Commentary: This conversation provides teachers with a rich opportunity to learn from the assessment data. Their conversation integrates student understanding, the mathematics, and the implications for teaching. Their discussion of the student understanding of supplementary takes the reason for the error into account. Specifically, the data push them to think about how they have been teaching about supplementary angles in their classrooms. When Diana asks about how they could teach this differently next time, all three suggest ways they could be more versatile in both their representations of supplementary and their use of the definition. In this exchange, they integrate students’ thinking, the mathematics, and how they should adjust their teaching to help the two come together more effectively. This is similar to rich talk about planning but the teachers are making connections between the test results and making sense of what it tells them about these critical aspects of teaching.

Vignette 2: Weak Talk about Student Data

In this vignette a group of teachers are looking at interim assessment scores and they are asked to label each student as commended, passing, bubble or growth based on what the teachers feel the students have the potential for earning on the state-wide test.





After the teachers have labeled each student they review their numbers.


Okay, looking at “commended,” I have 0%. “Passing” I believe I only have about 20%. Bubble kids need that extra little help – that’s 50% And 30% on “growth.” Of those, that 30%, a fifth failed it last year.

I have about 33% “commended,” 17% should pass, 30% that are borderline with a little help could probably be passing, and then one or two students not.

Commentary: The majority of the 45-minute meeting was devoted to this activity and the conversation that followed. While this may be a useful administrative activity, there are few opportunities for the teachers to consider the relationships among student thinking, mathematics, and teaching. Instead, the teachers focus on the distribution of students in the different NCLB categories.

To make this activity richer, it would help to connect the data to the particulars of instruction, student thinking and mathematics. While this may help teachers recognize their students’ progress and which students might need extra support, there is little in this conversation that would help teachers to think more deeply about their teaching or change their instruction. Even looking for patterns in what topics are frequently missed, as the teachers did in Vignette 1, would get closer to this goal.

Summary of Vignettes 1 and 2: What makes for more productive discussions about student data

To move discussions about student data from a weak to a strong activity, with richer opportunities for teacher learning, ensure that data conversations discuss and make explicit connections between student thinking, the mathematics of the questions and the implications for instruction.

Below are some questions that may help to productively discuss data and more clearly make connections between student understandings, mathematics and instruction:

  • Making sense of the data: What did we learn about student understanding on a particular math topic from looking at this data? What trends in student understanding do we notice?
  • Thinking back on previous instruction: What are students thinking about the math to have answered in this way? How might our instruction led them to think this way?
  • Thinking ahead to subsequent instruction: How should we consider adjusting our instruction to address what we found for this group of students? Why would that work? How can we address these issues in student thinking when we teach it next time?

When teachers look at student performance data, learning opportunities will be richer if the teachers have to resources for looking at overall trends alongside the details about mathematical topics, individual students, or both. These details allow teachers to delve more deeply into the connections among what they know about student thinking, the mathematics and their instruction.

Sometimes administrative activities, such as those in Vignette 4 must happen. However, it is important that these take up a minimum amount of time or that the information garnered from such analysis gets taken up later to develop connections across mathematics, student thinking, and instruction.

Part III: Facilitation

As the examples in Part 2 illustrate, conversations that are richer for teachers’ learning build connections across teaching, students’ thinking, and mathematics. Sometimes, we have found that facilitators can help support these kinds of conversations. In other words, the facilitator’s job is to support teachers in connecting the three elements of the instructional triangle –– and to do so as specifically as possible.

One challenge of teacher collaboration is that some critical aspects of teaching –– students and the classroom interaction –– are not available to examine together. Good facilitators come up with strategies to help teacher groups get “on the same page” about some issue in teaching. Sometimes they do this by re-enacting the voices of students and teachers in the classroom to creating shared images of actual classroom talk. Once the teachers have some shared image of the issue, they have to work together to make sense of it together. To this end, good facilitators ask teachers to provide rationales for instructional decisions that they make (e.g. “so, what are students learning in doing this for homework?” “How does this activity help students in thinking about and understanding the idea of what volume is, beyond memorizing the formula?”)

Good facilitators also support teacher engagement. They do this in several ways. First, they build supportive relationships with individual teachers, identifying their strengths and coming up with reasonable next steps for their professional growth. When teachers are engaged, they participate more readily in conversations. Of course, when teachers share their ideas honestly, there is greater potential for conflict. Good facilitators make a safe space for learning, respectfully listening to different ideas while continuing to press for deeper understandings about teaching, students, and mathematics.

In summary, good facilitators:

  • Get teachers on the same page about some important questions in teaching.
  • Press teachers to explain their pedagogical reasoning.
  • Link instructional issues to clear statements that connect teaching, students, and mathematics.
  • Support individual teacher engagement and development.
  • Develop norms for honest but respectful conversations.

As we did with our framework for teacher conversations, we will develop our notion of good facilitation through vignettes that show facilitators of different skills. As with the other vignettes, these are based on our data but have been cleaned up for reasons of clarity and confidentiality.

Vignette 3: Sophisticated facilitation

In this vignette, two teachers, Jack and Soledad, work with and Coach Rachel. The team works to plan a launch for the following day’s lesson. Coach Rachel begins by asking teachers to make connections among instruction, mathematics, and student thinking:

Coach Rachel:          Ok. So, would you look at the book’s lesson on place value, and decide what you think the kids have to know in order to be able to do it?

Soledad:                    They definitely have to know exponents, which is scary, because we haven’t done exponents yet. See how it says “10 to the—”

Jack:                         Yeah. Neither have we. I hate how this book skips around. Like, my kids don’t get exponents yet. Why can’t we stick to place value if 2.3 is about place value?

Coach Rachel:          Ok. I hear ya—you guys are worried about the exponents. Let’s pretend we’re students, and we have a shaky understanding of exponents. How else could we approach this problem?

Jack:                         They could. Um. They could use the idea of multiples of ten—or, you know, like what an exponent actually means. Like ten to the first is ten times one, ten squared is ten times ten, you know…

Soledad:                    Oh! I see what you’re getting at, you sneaky thing. ((laughs)). You’re saying it has to do with the, like, base ten?

Jack:                         Like, 10 times 1 is 10 and 10 times 10 is 100? Ok, so how can we connect that to place value for them? Because that’s tough.

Coach Rachel:          Yeah. You’re right, it is. Um. So, the idea is that place value stands for an order of ten, right. That’s what we need kids to understand in order to be able to do this problem.

Soledad:                    Yep. Especially when we get into decimal numbers. My kids get really freaked out by decimal numbers.

Jack:                         Right, so how can we launch this so kids get that?

Soledad:                    Ok. So, what if we use money. Like, kids get money. Right? Like pennies, dimes, dollars, you know…

Commentary: Coach Rachel’s work in this vignette illustrates some of the important qualities of effective facilitation. To get the teachers on the same page about their lesson planning, the group works together from the textbook and teacher guide. There is a positive, supportive, and honest tone in the conversation. Jack does not hesitate to share his frustration with the curriculum ( “Why can’t we stick to place value if 2.3 is about place value”), and Coach Rachel uses it as a way to connect the topic of place value to their concerns about the exponents. She is respectful of this concern (“I hear ya—you guys are worried about the exponents”) but manages to redirect the group so that they consider students’ perspectives and new thinking about mathematics. This is a critical move: the conversation could easily devolve into a gripe session about the curriculum, but she brings it back to the territory of the Instructional Triangle we introduced in the introduction of this memo. She does this by building an additional representation of the classroom, getting them further on the same page, asking the teachers to pretend like they are students and to think of alternate mathematical approaches to the work. Once teachers begin to reconceptualize the task from students’ perspective, Coach Rachel then marks exactly what students need to be able to see. The summary statement she provides links teaching, students, and mathematics. In the end, Coach Rachel helps teachers arrive at a specific instructional goal, based on a (re)consideration of student thinking and mathematics.

Vignette 4: Weak facilitation

In the following vignette, a group of teachers plans a launch on place value, using money as a jumping off point. They are using the same unit we heard about in Vignette 5, but it is a different teacher team and facilitator.

Coach Melissa:








Coach Melissa:



Coach Melissa:



Coach Melissa:

Ok. We’re going to role-play this launch. Tamara, will you play the teacher, and then, Trent—you’ll be the student.

Ms. White, I’m tired. ((laughs))

((laughs)) Ok. Um. So, what is this ((holds up a dime))?

A dime.

Right. So, how do we write that down?

Like this, “10¢.”

Nooo. Write it down the right way.

Well, that IS the right way ((crosses arms in front of chest)).

How about, “Like you’d see it on a price tag?”

I’ve seen price tags like that.

Ack! You’re just as frustrating as real students ((laughs)).

Ok. Go back to the role-play. Tamara, you’re the teacher.

Um. Ok. Are there any other ways to write it? ((Trent writes 0.10)). What happens if you multiply that by ten?

I don’t know, the decimal moved.

Why do you think the decimal moved?

Commentary: Coach Melissa does a number of things well in this interaction. Clearly, she has strong rapport with the teachers, who joke around and eagerly participate in the activity she has designed. The idea of the role-play has some potential to get the teachers on the same page about some issue in teaching. Nonetheless, we see this facilitation (and the meeting that surrounds it) as providing very few opportunities for teachers to learn about ambitious instruction. Drawing on our framework for rich teacher conversations, we see that few connections are made across teaching, students, and mathematics, and Coach Melissa does very little to press it in that direction.

Weak facilitation may result from focusing on any one of the three points of the instructional triangle, to the exclusion of the other two. This is an example of an over-emphasis on teaching with little consideration for mathematics or students. Strong facilitators often use role-plays, but effective role-plays allow inquiry into the connections between student thinking, teaching, and mathematics. Although Coach Melissa asks Trent to enact a student, the “student” gets very little air time, and “student” contributions are not taken up as meaningful: Coach Melissa revises Tamara’s question to “[Write it down] like you’d see it on a price tag,” but ignores the “student” objection to using decimal numbers or, as the following section points out, the mathematical import of that objection. Tamara playfully tells Trent that he is “as frustrating as real students,” and Coach Melissa urges a return to the role-play.

Less Attention to Student Thinking and Math: Coach Melissa overlooked an important moment for considering student thinking and mathematics in conjunction with teaching. We see this as a missed opportunity for teacher learning. For example, when Trent writes “10¢” and argues it IS the right way to write down the value of a dime (line 19), Coach Melissa could have led a discussion about student thinking: Why do students prefer to think of a dime as “10¢,” rather than 0.10? Perhaps they prefer whole numbers to decimal numbers? What is the relationship between the two ways of writing ten cents, and how does this connect back up to the unit topic of place value? Such a discussion would be relevant to a planning the lesson and would involve a richer consideration of student thinking and of math.  


Strong facilitation involves effectively working to build trust among group members so that teachers feel secure airing their confusions and struggles. It also involves connecting the three elements of the instructional triangle. In order to have rich opportunities to talk about connections between student thinking, teaching, and mathematics, the facilitator needs to help teachers get on the same page about important questions in teaching. To do this, the facilitator can press for pedagogical reasoning and ask teachers to re-enact student and teacher voices in order to create rich representations of students’ thinking. Because facilitating teachers’ opportunities to learn is complex work, it is also important for facilitators to make clear statements that tie together teachers’ representations of student thinking (which are often “impersonations” of students’ voices or examples of student work) with teachers’ understandings about the mathematics involved in particular lessons and, ultimately, the reasons for their instructional decisions.

Suggestions of ways for facilitators to press on teacher learning:

Here are some strong questions facilitators might ask:

  • What do you think students will need to understand in order to do this task?
  • How does (this activity) help students develop their understanding of (a key mathematical idea)?
  • What do you hope students will learn by (doing this activity/worksheet)?
  • What is the big idea that you want students to come away with from this lesson?
  • What happened in your classroom when you tried to do (a new teaching technique, for instance)?
  • What did students say in response?
  • What were students’ misconceptions?
  • Why do you think students had that misconception?
  • What led to students’ misconceptions? (Help teachers to focus on things over which they have control)
  • How can we address that misconception in our class next time? (“Re-teach it” is not a clear enough response—it doesn’t help teachers think about what they did last time or what they need to do differently next time.)

Concluding Thoughts

Over the years, research has repeatedly shown that teachers can benefit from professional interactions with other teachers. At its best, working with other teachers supports teachers to more deeply understand their work and prevents isolation. This is particularly important when teachers are attempting to try new practices, including moving toward more ambitious mathematics teaching. Our research has found that in collaborative conversations that allow teachers to make connections across student thinking, the mathematics being taught, and instruction have a greater potential to move their teaching closer to the ambitious instruction. Given the rigor of the current set of state assessments, students will need to access to this instruction to increase their chances of success.


Smith, Margaret, Victoria Bill, and Elizabeth K. Hughes. “Thinking Through a Lesson: Successfully Implementing High-Level Tasks.” Mathematics Teaching in the Middle School 14 (October 2008): 132–38

Policy and Math Education: A Conference


This past week, I attended a conference at UC Berkeley about policy and math ed organized by Geoff Saxe, Na’ilah Nasir, and their amazing graduate students. The gathering had two main purposes:  to get a group of math ed researchers together to talk about issues related to the Common Core, and to mentor junior researchers in their work. I think the conference met the second goal very strongly and the first one more loosely.

Let me give a brief overview of the main events.

  • Alan Schoenfeld gave a keynote about his work on TRU Math and the Formative Assessment Lessons. He shared his work, some initial findings, and areas of research opened up by these tools.
  • Marty Simon, Jenny Langer-Osuna, and Elham Kazemi led breakout discussion sessions on learning trajectories, equity, and professional development respectively. (Elham is also on Twitter and worth a follow.)
  • Doctoral and postdoctoral students poster sessions
  • A symposium on improving mathematics teaching and learning. Danny Martin talked about researching issues of race in mathematics education. Paul Cobb shared the district partnership work from the MIST project, pointing to gaps in what we study and what school leaders need. Then I talked about how policy operates as a context for teacher learning, sharing some of my findings about math teachers’ encounters with NCLB.
  • A closing session with commentary on the research shared. Carol Lee talked about different challenges in implementing the Common Core in English Language Arts, as well as what it means to teach in ways that consider children’s cognitive, social, emotional, and physiological development. Rogers Hall provided a synthesis of much of the research, talking about the importance of “mutterings” about research (complaints and dissatisfaction) and what it takes to turn mutterings to utterings so that different voices are heard and valued. Anna Sfard challenged researchers to increase their conceptual accountability in their work by making their language clearer and their communications more accessible.

Of course, this summary does not do justice to the richness of the conversations or the key insights gleaned. (Raymond Johnson storified some of the tweets from the conference if you want to check them out.) Of course, the in-between social time was enriching as well. I had some great conversations with Kris Gutiérrez, Tesha Sengupta Irving, Niral Shah, and Nicole Louie.

The question and answer sessions after the main events had a collegial but challenging tone. The conversations gave us a chance to ask our most pressing questions to people who are great to think with.

I, for one, am left with tremendous humility about the complexity of the research and educational enterprise. The doctoral students and postdocs seemed to really appreciate the experience as well. It seems like these small, focused forums have a value we can’t always get at bigger conferences whose aim is to present finished work instead of share in difficult puzzles.