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 researchbased 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.
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 coplanning 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 lessdeveloped 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 statewide test.
5
10 
Jolene:
Austin: 
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 45minute 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 reenacting 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:
Trent:
Tamara:
Trent:
Tamara:
Trent:
Tamara:
Trent:
Coach Melissa:
Trent:
Tamara:
Coach Melissa:
Tamara:
Trent:
Coach Melissa: 
Ok. We’re going to roleplay 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 roleplay. 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 roleplay 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 overemphasis on teaching with little consideration for mathematics or students. Strong facilitators often use roleplays, but effective roleplays 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 roleplay.
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.
Summary
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 reenact 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? (“Reteach 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.
Reference
Smith, Margaret, Victoria Bill, and Elizabeth K. Hughes. “Thinking Through a Lesson: Successfully Implementing HighLevel Tasks.” Mathematics Teaching in the Middle School 14 (October 2008): 132–38