Building Teaching as a Responsive Profession

Those of you who spend real or virtual time with me have heard me talk about how hard it is to talk about teaching.

One frequently mentioned issue is that, unlike other professions, teaching does not have its own technical language. Professions like aviation and medicine have common professional terms that highlight important features of critical situations and guide practice. In aviation, for instance, pilots identify wind patterns to aid in landing planes. Likewise, surgeons have cataloged human anatomy and surgical procedures so the protocol for appendectomies can be named and routinized, with appropriate modifications for anatomical variations such as hemophilia or obesity. But a strong headwind in China is similar to a strong headwind in Denmark; a hemophiliac in Brazil will require more or less the same modifications as a hemophiliac in Egypt.

In contrast, an urban school may not be the same as an urban school a few blocks away, nor an ADHD kid the same as an ADHD kid in the same classroom. Although such terms attempt to invite descriptions about particular teaching situations, the language often relies on stereotyped understandings. Everyday categories like an urban school, an honors class, or an ADHD kid seldom work to describe teaching situations adequately to help teachers address the challenges they face. Words characterizing social spaces and human traits are inherently ambiguous and situated in particular social, cultural and historical arrangements.

The variation teachers encounter cannot always be codified, as they often are in aviation and surgery. In fact, in the United States, when educational situations are codified, they often presume the “neutral” of White, English-speaking, and middle class culture. However, the widespread practice of glossing cultural particulars, or only seeing them as deviants from a norm, reduces teachers’ ability to teach well. From Shirley Brice Heath’s  seminal work comparing home literacy practices in White and African American communities to Annette Lareau’s identification of social class-specific parenting patterns, we see time and again that children from non-dominant groups frequently encounter schooling expectations that are incongruous with their home cultures, often to the detriment of their learning. Conversely, when instructional practices align with children’s home cultures, teachers more are more effective at cultivating students’ learning. (See, for a few well documented examples, this work by Kathryn Au and Alice Kawakami, Gloria Ladson-Billings, and Teresa McCarty.)

Culturally responsive pedagogies are, by definition, highly particular and have been documented to yield better student learning. To communicate sufficiently, professional language for teaching would need to encompass this complexity, avoiding simplistic –– perhaps common sense –– stereotypes about children, classrooms, schools, or communities.

How, then, can we develop shared professional language for teaching and build professionals responsive to the children they serve? I have some ideas I will share in another post.

Making Sense of Student Performance Data

Kim Marshall draws on his 44 years’ experience as a teacher, principal, central office administrator and writer to compile the Marshall Memo, a weekly summary of 64 publications that have articles of interest to busy educators. He shared one of my recent articles, co-authored with doctoral students Britnie Kane and Jonee Wilson, in his latest memo and gave me permission to post her succinct and useful summary.

In this American Educational Research Journal article, Ilana Seidel Horn, Britnie Delinger Kane, and Jonee Wilson (Vanderbilt University) report on their study of how seventh-grade math teams in two urban schools worked with their students’ interim assessment data. The teachers’ district, under pressure to improve test scores, paid teams of teachers and instructional coaches to write interim assessments. These tests, given every six weeks, were designed to measure student achievement and hold teachers accountable. The district also provided time for teacher teams to use the data to inform their instruction. Horn, Kane, and Wilson observed and videotaped seventh-grade data meetings in the two schools, visited classrooms, looked at a range of artifacts, and interviewed and surveyed teachers and district officials. They were struck by how different the team dynamics were in the two schools, which they called Creekside Middle School and Park Falls Middle School. Here’s some of what they found:

  • Creekside’s seventh-grade team operated under what the authors call an instructional management logic, focused primarily on improving the test scores of “bubble” students. The principal, who had been in the building for a number of years, was intensely involved at every level, attending team meetings and pushing hard for improvement on AYP proficiency targets. The school had a full-time data manager who produced displays of interim assessment and state test results. These were displayed (with students’ names) in classrooms and elsewhere around the school. The principal also organized Saturday Math Camps for students who needed improvement. He visited classrooms frequently and had the school’s full-time math coach work with teachers whose students needed improvement. Interestingly, the math coach had a more sophisticated knowledge of math instruction than the principal, but the principal dominated team meetings.

In one data meeting, the principal asked teachers to look at interim assessment data to predict how their African-American students (the school’s biggest subgroup in need of AYP improvement) would do on the upcoming state test. The main focus was on these “bubble” students. “I have 18% passing, 27% bubble, 55% growth,” reported one teacher. The team was urged to motivate the targeted students, especially quiet, borderline kids, to personalize instruction, get marginal students to tutorials, and send them to Math Camp. The meeting spent almost no time looking at item results to diagnose ways in which teaching was effective or ineffective. The outcome: providing attention and resources to identified students. A critique: the team didn’t have at its fingertips the kind of item-by-item analysis of student responses necessary to have a discussion about improving math instruction, and the principal’s priority of improving the scores of the “bubble” students prevented a broader discussion of improving teaching for all seventh graders. “The prospective work of engaging students,” conclude Horn, Kane, and Wilson, “predominantly addressed the problem of improving test scores without substantially re-thinking the work of teaching, thus providing teachers with learning opportunities about redirecting their attention – and very little about the instructional nature of that attention… The summative data scores simply represented whether students had passed: they did not point to troublesome topics… By excluding critical issues of mathematics learning, the majority of the conversation avoided some of the potentially richest sources of supporting African-American bubble kids – and all students… Finally, there was little attention to the underlying reasons that African-American students might be lagging in achievement scores or what it might mean for the mostly white teachers to build motivating rapport, marking this as a colorblind conversation.”

  • The Park Falls seventh-grade team, working in the same district with the same interim assessments and the same pressure to raise test scores, used what the authors call an instructional improvement logic. The school had a brand-new principal, who was rarely in classrooms and team meetings, and an unhelpful math coach who had conflicts with the principal. This meant that teachers were largely on their own when it came to interpreting the interim assessments. In one data meeting, teachers took a diagnostic approach to the test data, using a number of steps that were strikingly different from those at Creekside:
  • Teachers reviewed a spreadsheet of results from the latest interim assessment and identified items that many students missed.
  • One teacher took the test himself to understand what the test was asking of students mathematically.
  • In the meeting, teachers had three things in front of them: the actual test, a data display of students’ correct and incorrect responses, and the marked-up test the teacher had taken.
  • Teachers looked at the low-scoring items one at a time, examined students’ wrong answers, and tried to figure out what students might have been thinking and why they went for certain distractors.
  • The team moved briskly through 18 test items, discussing possible reasons students

missed each one – confusing notation, skipping lengthy questions, mixing up similar-sounding words, etc.

  • Teachers were quite critical of the quality of several test items – rightly so, say Horn, Kane, and Wilson – but this may have distracted them from the practical task of figuring out how to improve their students’ test-taking skills.

The outcome of the meeting: re-teaching topics with attention to sources of confusion. A critique: the team didn’t slow down and spend quality time on a few test items, followed by a more thoughtful discussion about successful and unsuccessful teaching approaches. “The tacit assumption,” conclude Horn, Kane, and Wilson, “seemed to be that understanding student thinking would support more-effective instruction… The Park Falls teachers’ conversation centered squarely on student thinking, with their analysis of frequently missed items and interpretations of student errors. This activity mobilized teachers to modify their instruction in response to identified confusion… Unlike the conversation at Creekside, then, this discussion uncovered many details of students’ mathematical thinking, from their limited grasp of certain topics to miscues resulting from the test’s format to misalignments with instruction.” However, the Park Falls teachers ran out of time and didn’t focus on next instruction steps. After a discussion about students’ confusion about the word “dimension,” for example, one teacher said, “Maybe we should hit that word.” [Creekside and Park Falls meetings each had their strong points, and an ideal team data-analysis process would combine elements from both: the principal providing overall leadership and direction but deferring to expert guidance from a math coach; facilitation to focus the team on a more-thorough analysis of a few items; and follow-up classroom observations and ongoing discussions of effective and less-effective instructional practices. In addition, it would be helpful to have higher-quality interim assessments and longer meetings to allow for fuller discussion. K.M.] “Making Sense of Student Performance Data: Data Use Logics and Mathematics Teachers’ Learning Opportunities” by Ilana Seidel Horn, Britnie Delinger Kane, and Jonee Wilson in American Educational Research Journal, April 2015 (Vol. 52, #2, p. 208-242

How do teachers teach responsively?

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

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

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

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

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

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

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

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

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

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

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

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!

Math Departments that Support All Students’ Learning

Awhile back, I wrote an article comparing two mathematics departments that managed to successfully support students’ learning, even among students with histories of low achievement. One department, at “Phoenix Park” school, was in a working class community in England and documented in Jo Boaler’s book, Experiencing School Mathematics. A second department was in a working class community in California. I studied and taught at “Railside School.” A book about that school is about to be released.

Here are the common threads I found across these two groups of teachers’ approach to supporting students in heterogeneous classrooms.

  1. Teachers presented a connected and meaningful view of mathematics.

Both Phoenix Park and Railside teachers managed to present a version of the subject that students found both meaningful and engaging. At Phoenix Park, 75% of students interviewed reported using school mathematics in their daily lives, compared to none of the students taught in comparative group in traditional classrooms. Likewise, Railside students frequently referred to mathematics as a kind of language, as stated by this senior:

Math seems like a second language or another language that we’re learning—because it is something that you can use to communicate to others through math.

This student’s view of the usefulness of mathematics was common among students at both schools.

How are the Phoenix Park and Railside teachers imparting a perspective of mathematics to their students that so diverge from popular conceptions? In part, it stems from their own views of the subject, which differ from what we typically find in our schools. Many math teachers in the United States and England have what is referred to as a sequential view of the subject. That is, they regard mathematics as a well-defined body of knowledge that is somewhat static and beholden to a particular order of topics. This perspective has logical consequences for both instruction and student learning. First, in light of this view, the main goal of teaching is to cover the curriculum in sequence to achieve content goals. Second, students must master prior topics in the sequence in order to move forward in the curriculum successfully.

The sequential view has strong consequences in instructional decision making. Gaps in students’ prior learning are seen as obstacles to their present learning, making divisions between low-achieving and high-achieving students a necessity.

Making sense of mathematics at Phoenix Park.
At Phoenix Park, the teachers directed students’ mathematical investigations in a deliberate way. As Boaler reports, they:

did not subscribe to the common belief that lower attaining students needed more structure. They merely asked different questions of the students to help them make the connections they needed to make.
(p. 168)

In this description, the teachers’ conception of mathematics appears different than the image of hierarchically organized topics; instead, mathematics is a network of interrelated ideas whose connections can be understood by students with different levels of attainment, given appropriate and differentiated scaffolding. These problems required students to make meaning of the mathematics they were using, as they had to clarify assumptions and explore and defend their choices in problem posing and problem solving. Boaler found that Phoenix Park students performed both more sensibly and creatively on an open-ended design task (designing an apartment that fits certain mathematical criteria) than students who had received traditional instruction. For the Phoenix Park students, mathematics was a tool they brought to bear on problems in the world, not just a set of procedures whose meaning was bound up in school.

Valuing Careful Thinking over Speed at Railside School.
At Railside, the teachers shared a similar conception of mathematics. In the following excerpt from a department meeting, Railside math department co-chair Guillermo Reyes advised a new teacher who was struggling with a perceived gap between the fast and slow students in her classroom:

“The [students] that are moving through things really quickly, often they’re not stopping to think about what they’re doing, what there is to learn from this activity. […]
“A kid knowing, ‘Okay, I can get through this quickly but I’m working on X –– being a better group member because it’s going to help me in my future classes. Showing off math tools because I know how to do it with a t-table[i] but I don’t know how that relates to a graph yet.

“But like think of the ones that you think of as fast learners and figure out what they’re slow at.”

Although mathematics was not discussed at length, a distinctly non-sequential view of mathematics undergirded Guillermo’s statements. In Guillermo’s talk, mathematics was a subject with connections: he imagined a student needing to connect “t-tables” and graphs. More subtly, Guillermo’s reworking of the novice teacher’s categories of “fast” and “slow” students ties in with notions of mathematical competence. Since students, in his terms, are not simply fast or slow learners of mathematics, the subject itself takes on more texture. Mathematics competence is not simply the mastery of procedures –– something that students are more or less facile with. Instead, because mathematics is viewed as a connected web of ideas, knowing mathematics requires careful consideration of the various facets of any particular concept and the identification of the relationships among them. Guillermo revealed this last view of mathematical competence when he expressed concern about “the ones who move things really quickly […] not stopping to think about […] what there is to learn from this activity.” In order to learn mathematics, in other words, students must make sense of mathematics, not simply complete their work to get it done.

The Need for Sensemaking. The complex and connected view of mathematics shared by both groups of teachers was fundamental to their practice. It implicated the kind of professional knowledge they sought to develop, creating a need for deeper instead of simply more content. Additionally, it shaped their attitude toward their students’ learning and, as discussed in the next section, their implementation of curriculum that would support student sensemaking.

  1. A Curriculum Focused on Important Mathematical Ideas.

Both Phoenix Park and Railside math teachers designed their lessons to focus on important mathematical ideas. This approach stands in stark contrast to typical American math lessons, which have been found to be remarkably uniform in structure, often taking the form of “learning terms and practicing procedures.”  The US lesson structure, common in Britain as well, reflects the underlying sequential view of subject. If success in mathematics requires mastery of prior topics, then the curriculum needs to be carefully sequenced by teachers and then thoroughly rehearsed by students so that they may master the material.

In line with their non-hierarchical view of subject, the curriculum at Phoenix Park and Railside countered the typical US and British lesson structure. Instead of learning terms and practicing procedures, both schools’ math lessons were organized around big mathematical ideas. This was a deliberate strategy, designed to minimize the deleterious effects of low prior achievement.

Projects and Investigations at Phoenix Park.
A leaflet put out by the Phoenix Park mathematics department embodied this concept-driven curriculum and its connection to detracking:

We use a wide variety of activities; practical tasks, problems to solve, investigational work, cross-curricular projects, textbooks, classwork, and groupwork. Every task can be tackled by students with widely different backgrounds of knowledge but the direction and level of learning are decided by the student and the teacher.

At Phoenix Park, the yearly curriculum consisted of four to five topic areas, each of which were explored through various projects or investigations. A topic area might have a title like “Connections and Change” or “Squares and Cubes.” Boaler provides a detailed description of one teacher’s introduction to a fairly representative Phoenix Park math project called 36 pieces of fencing (pp. 51-54). In the task, students are asked to find all the shapes they can make with 36 pieces of fencing and to then find their area. This single open-ended problem took up approximately three weeks of class time. At Phoenix Park, the teachers used mathematically rich and open-ended curriculum to differentiate their instruction. Although the teachers strongly believed that all students should have access to challenging mathematics, their activities provided different access points for different students. Problems like 36 pieces of fencing supported a range of mathematical activity. Students could investigate the areas of different shapes, collect data on and construct graphs of the relationships between shape and area, explore combinatorial geometry, or use trigonometry. If students finished work or became bored, the teachers would extend the problems to support their continued engagement.

Group-worthy Problems at Railside.
Similarly, Railside’s math teachers organized their detracked curriculum around what they called “group-worthy problems.” In their meetings, the teachers consistently invoked group-worthiness as the gold standard by which classroom activities were evaluated. In one conversation, they collectively defined group-worthy problems as having four distinctive properties. Specifically, these problems: (1) illustrate important mathematical concepts; (2) include multiple tasks that draw effectively on the collective resources of a student group; (3) allow for multiple representations; and (4) have several possible solution paths.

Railside math teachers also organized their curriculum into large topical units. For example, one unit called y=mx + b focused on the connections between the various representations (tables, graphs, rules, patterns) of linear functions, connections that are essential to the development of conceptual understanding. Their units were subdivided into a collection of related activities, all linked back to an overarching theme.

A typical activity in an Railside Algebra class was The Vending Machine. In this problem, students were told about the daily consumption patterns of soda in a factory’s vending machine, including when breaks were, when the machine got refilled, and the work hours in the factory. Students were then asked to make a graph that represented the number of sodas in the vending machine as a function of the time of day.

The activity focused on one larger problem organized around a set of constraints. While these constraints limited the possible answers, students had an opportunity to discuss the different choices that would satisfy the constraints and look for common features of plausible solutions as a way of generalizing the mathematical ideas. Embedded in the activity are important mathematical ideas (graphing change, slope, rate) that are linked to a real-world context.

Interpreting the World through Mathematics.
The two curricula had in common an approach to teaching mathematics through activities that required students to use mathematics to model and interpret situations in the world. These curricular approaches are aligned with the view of mathematics as a tool for sensemaking: students need opportunities to understand mathematics through activities that allow them to make sense of things in the world. Although there were differences in the execution –– there was more latitude for curriculum differentiation in the Phoenix Park curriculum and more structured group work at Railside –– the conception of mathematics that they shared allowed the participation of students of varied prior preparation.

  1. A Balance of Professional Discretion and Coordination for Teaching Decisions.

Heterogeneous classrooms may make it harder for teachers to proceed through the curriculum in a lockstep fashion. Heterogeneity increases the urgency for teachers to respond to the particularities of the learners in their classrooms. At the same time, teachers need frameworks for decisions about what is important to teach in order to articulate to the larger curricular goals. Both groups of teachers organized their work to allow for individual adaptation and, simultaneously, a degree of coordination.

At both schools, the teachers collaborated on the development and implementation of their respective curricula. In addition, it is probably not a coincidence that both groups controlled the hiring of new mathematics teachers in their department –– a common practice in England but highly unusual in the US. As a result, both groups of teachers were working with like-minded colleagues. Their shared values surely facilitated the implementation of common frameworks and practices.

Looping through a Common Curriculum at Phoenix Park.
At Phoenix Park, the teachers balanced professional discretion and coordination by keeping a group of students with the same teacher for several years (a practice known as looping) while teaching from a common curriculum that they consulted about in an ongoing fashion. The looping structure changed the time that teachers had to work with their students from one to three academic years, allowing for more adaptations by individual teachers and a more in-depth knowledge of particular students. Looping also minimized the transitions between teachers that can challenge low-performing students.

At the same time, in their math department meetings, the teachers would discuss the activities they planned to use and any modifications they planned to make. These meetings allowed teachers to vet ideas past colleagues and consult on challenges that arose, instead of requiring them to work in isolation. While the teachers drew on each other’s knowledge and experience with their common curriculum, their classrooms reflected their individual teaching styles and managerial preferences.

Coordinating for Student Learning at Railside.
The Railside math teachers’ course structure required a greater degree of coordination. Students stayed with the same teacher for one term, with the school year consisting of two terms. This meant that students could encounter anywhere from three to seven math teachers during their four years of high school, a structure that increased the demand for coordination. As a result, the Railside teachers had more explicit structures to support this coordination.

At the start of each new academic term, the teachers gathered for what they called a roster check. Each teacher brought class lists to show to all the other teachers. In this way, they could alert each other to vulnerable students and share effective strategies for working with them. Additionally, the teachers met weekly in their subject groups (e.g., Algebra, Geometry) and discussed curriculum and its effective implementation. They worked collaboratively to develop and refine their curriculum, adapting published materials to make them more group-worthy. In addition, the teachers paid close attention to the ways they presented ideas, the kinds of questions asked, and employed language that might make mathematical ideas most meaningful to students. For instance, Railside’s teachers avoided commonly used terms like canceling out to describe the result of adding opposite integers such as 3 + +3. Instead, they preferred the phrase making zeroes, as it more accurately described the mathematics underlying the process.

At the same time, individual teachers commonly took their own paths through the common curriculum based on their own judgments about their particular classes’ strengths and needs. They did so in consultation with the colleagues who would be teaching students in their subsequent courses.

Common Vision, Adaptive Implementation.
Both groups of teachers had structures that supported the student-centered coordination of their teaching. At Phoenix Park, the common curriculum and the department meetings were the main vehicles for coordination. At Railside, where teachers’ interdependence was increased by their course schedule, a greater number of structures were required: roster checks, weekly subject-specific meetings, and attention to common language.

Although their contexts demanded different means for flexibly coordinating practice, both groups of teachers had one thing in common: they effectively used their colleagues as resources for their own ongoing improvement of practice. They had structures in their workweek that allowed them to consult with each other and learn from their collective experience, breaking through the privacy and isolation that often characterizes teachers’ work. This has been found to be true more generally of departments that support students’ participation in advanced mathematics courses.

  1. Clear distinctions between “doing math” and “doing school” for both students and teachers.

One of the effects of ability grouping is that, despite its name, students are placed according to their prior school achievement, not by their potential to learn. In this way, schooling savvy is conflated with mathematical competence. If students know how to turn in homework and study for tests, they will likely be placed in a higher track than equally capable students who have not mastered these school learning practices.

Within two very distinct school contexts, both the Phoenix Park and Railside mathematics teachers worked to make practices of schooling transparent to their students. Phoenix Park and Railside themselves afforded different kinds of teaching and learning, and therefore placed different demands on students’ schooling know-how.

Phoenix Park School, a comprehensive public school with no entry requirements or special charter, had about 600 students. Many of the departments used project-based curricula. The school’s progressive philosophy aimed to develop students’ independence. In contrast, Railside High School was a more traditionally configured comprehensive public school of 1500 students. The subject departments varied widely in their approaches to curriculum and instruction. Within the school, the math department was seen as a leader for many school-wide reforms, such as the shift to block scheduling and the creation of a peer-tutoring clinic. The two schools brought different resources and challenges to addressing heterogeneity.

Focusing on Student Thinking at Phoenix Park.
At Phoenix Park, the classrooms were minimally structured, with students electing to work independently or in groups, often socializing in between their pursuit of solutions to their open ended projects. This complemented the larger school goals of fostering students’ independence. Within this open setting, however, the teachers valued particular learning practices and made these standards clear to their students. For example, their teaching approach relied on students explaining their reasoning, thus teachers would frequently prompt students to do so. They paid particular attention to reluctant students, regarding students’ difficulties in communicating their thinking or interpreting their answers not as resistance but instead as a gap in the students’ understanding about classroom expectations. In addition, in their progressive setting, the teachers had the liberty to emphasize learning through assessments, commenting on the quality of student work without assigning it particular grades. This allowed both teachers and students to focus on individual students’ learning over their ranked school performance.

Teaching All Students How to Learn at Railside.
In the more traditional comprehensive high school setting of Railside, the math teachers conducted their classes in a more structured fashion. Although the curriculum was open-ended, the students were expected to work while in class, usually in small student groups. The teachers had received extensive training in a teaching method called Complex Instruction that allowed them to use groupwork as a vehicle for challenging students’ assumptions about who was smart at math. They aimed to broaden students’ notions of what it meant to be good at math, thereby generating greater student participation and success in the subject.

In line with their goal of increased participation, the teachers were explicit that learning to be a student was an important part of their curriculum, and they came up with structures to support that learning. At the front of each classroom was a homework chart laid out much like a teacher’s roll book, with students’ names in a column along the side and the number of each homework assignment across the top. Although actual grades were not posted, completion of homework was represented by a dot. The homework chart reminded students of the primacy of homework in their job as students. The teachers and the students could glance at it and see if the students were doing their job. If students did not complete their homework on a given day, they were assigned an automatic lunch or after-school detention. It was viewed as a major coup when the math teachers got the sports coaches to agree to not allow athletes to come to practice on days when they had missed their math homework.

At the same time that they emphasized traditional student skills like doing homework, they did not confuse failure in class with students’ intelligence or ability. In interviews, the Railside teachers frequently used the following phrase to qualify a student’s poor performance: “He was not ready to be a student yet.” They worked to convey this mindset to their students too: all Railside math teachers had a large sign with the word YET placed prominently in their classrooms. In this way, when a student claimed to not know something, the teachers could quickly point to the giant YET to emphasize the proper way to complete such a statement.

Focusing on Students’ Potential to Learn.
By making clear distinctions between doing school and doing mathematics, the teachers at both schools focused themselves –– and their students –– on the students’ potential to learn. Many of the examples given above come out of a shared emphasis on formative assessment, activities undertaken by teachers (and students) to provide information and feedback that modified their teaching and learning activities.

This distinction also allowed explicit conversations about the schooling practices that would help support students’ learning and academic success. Given that students at both schools often came from families whose parents had not succeeded in formal education, the teachers’ assumption of this responsibility helped to create more equitable classrooms.

What are “teaching disasters” and how do we talk about them?

My #AERA14 session was on professional language in teaching. Stanford graduate student Jamie O’Keeffe organized a panel with Pam Grossman, Deborah Ball and me. Judith Warren Little provided the commentary.

Why focus on professional language? Many agree that professional language in teaching is underspecified, opening the field to a host of difficulties, especially inefficiency and confusion in communicating about pedagogical issues and the inability to delineate for those both inside and outside the profession what the professional knowledge of teaching is. Researchers worry that, as a consequence of this under-specificity, teachers’ conversations often become what Deborah Ball and David Cohen once described as “an exchange of buzzwords and slogans more than specific descriptions and analyses with concrete referents.”

So Jamie challenged us to engage in issues about what it would take to professionalize language about teaching.

The discussion engaged many interesting issues. My research involves spending hours and hours of video watching practicing teachers talk together about their work. I study how teachers identify and make progress on what they perceive to be problems in their work. It helps me get a better handle on teacher thinking, the differences between teachers of different levels of accomplishment, and how these conversations contribute to classroom instruction.

The work I drew from was done in collaboration with my graduate student, Britnie Kane. Here are a few premises derived from our research:

  • Words in themselves are not inherently meaningful. Terms develop meaning in use in particular contexts. What one teacher means by “scaffolding” may not align at all with another teacher’s meaning. Meanings are dependent on larger perspectives and stances on the work.
  • Terms in teaching overlap with a number of everyday terms, leaving them open to common sense (rather than technical) meanings. “Think” is the 12th most frequently used verb in the language. We also say things like “learn” or “understand” all the time in everyday life.
  • Teaching contexts matter in meaning construction. David Cohen once described teaching as “the deliberate cultivation of learning in others.” We add “in particular teaching situations.” The details of teaching situations — who are the students, what is the context, who is the teacher, what resources and constraints are available –– matter enormously in what is possible and interact deeply with any notion of expertise. Our current vocabulary for teaching situations is clearly inadequate (e.g., “an urban school”).
  • Concepts in teaching evolve as teachers develop language and link them to particular teaching experiences. That means as teachers encounter new situations, their understanding of big teaching ideas changes too. For instance, the idea of status is never fully understood because status issues play out differently in different classrooms and schools.

In one study Britnie and I worked on, we compared the talk of teachers working in institutionally similar environments working toward similar mathematics instruction. The different groups were, on the whole, at different levels of accomplishment in this teaching practice. One important finding was that there was no significant difference in the number of technical terms used by teachers at different levels of instructional accomplishment. But there talk differed in other ways. Notably, there was a marked difference in the extent to which the most accomplished group focused on students and their thinking. They also consistently linked any talk of instruction or mathematics back to students.

So back to our AERA panel. What does this mean for the development of professional language for teaching? It is no doubt a challenge to try to coordinate meaning across one of the largest professions out there.

One idea really stood out to me in the course of the conversation.

Professions often develop the most precise vocabulary to avert potential disasters.

Think of pilots landing a plane. Think of doctors resuscitating a patient. There is a lot of extremely precise language to guide action in these events. So what is a disaster in teaching?

Listening to teachers talk, I often hear them debrief on the unexpected turns that lessons take. The post-mortem analysis reveals a lot about what they think are the critical aspects of keeping the classroom functioning, so I spend a lot of time listening to those parts of the conversation.

Deborah had a different take on teaching disasters. She told a story from her summer teaching, which she does with upper elementary students and makes public for observers. She talked about some wiggly boys who managed to stay engaged in her classroom. A principal who was observing said that he was sure that those boys would not have had the same opportunities to learn in his school because they would have been sent out of the classroom.

I agree that it is an educational disaster to have children left out of the classroom because they are being children. But since my work places primacy on how teachers are talking and thinking, I know that for many of them, those boys’ wiggliness would be the source of a potential disaster.

In looking at teachers’ workplace talk, I see a lot of language develop around these potential disasters. Students who “act out” or are “disruptive,” “unmotivated” or “unfocused.” These students interfere with the smooth and successful execution of the lesson, so teachers talk a lot about them, sometimes in ways that are not constructive.

The question I have been pondering in the wake of that discussion is how do we align teachers’ perspectives on what is and isn’t a disaster to the larger picture of access and equity? More fairly, how do we support teachers in effectively engaging all students when there is increased pressure to stay up with pacing guides in preparation for ever-more-consequential standardized tests?
It is an educational disaster if what feels like averting a crash at the classroom creates true disasters in our society.

Seeing Status in the Classroom

In my last post, I discussed the idea of social status and its consequences for classroom teaching and learning. I was introducing you to my way of thinking about a concept and making a case for its importance in teaching.

Some of the comments and questions I got in response involved specifics about how it plays out in the classroom. In response, I will specify further how status actually looks in mathematics classrooms.

Recall that status makes for hierarchies in the classroom. Students who are perceived as smarter or more socially valued get more opportunities to speak and be heard. Almost all kids catch on to the order of things.

Status hierarchies manifest in classroom conversations and participation patterns, often leading to status problems, or the breakdown of mathematical communication based on status rather than the substance of mathematical thinking. Before we talk about remediating status problems, let’s clarify how teachers can see status problems in their classrooms.

head on desk


One of the most important and tangible status assessments teachers can do is ask who speaks and who is silent. Some students might dominate a conversation, never soliciting or listening to others’ ideas. These are probably high-status students. Some students may make bids to speak that get steamrolled or ignored. Some students may seem to simply disappear when a classroom conversation gains momentum. These are probably low-status students.

If you want to get a better handle on the participation patterns in your classroom, give a colleague a copy of your seating chart and have this person sit in your classroom. He or she can check off who speaks during a class session. This simple counting of speaking turns (without worrying about content or length for the moment) can give you a sense of dominance and silence.

Surprisingly, teachers’ impressions of speaking turns are sometimes not accurate, so this exercise can help sort out participation patterns. I have seen this in my own work with teachers and in earlier research. Back in the early 1980s, researcher Dale Spender videotaped teachers in high school classrooms, many of whom were “consciously trying to combat sexism” by calling on girls and boys equally. Upon reviewing the tapes and tallying the distribution of participation, the teachers were surprised that their perceived “overcorrection” of the unequal attention had only amounted to calling on the girls 35 percent of the time. The teachers reported that “giving the girls 35 percent of our time can feel as if we are being unfair to the boys.” Although (we hope) the gender ratios in this research may be dated, the phenomenon of teacher misperception still holds.

Teachers attending to participation patterns can use certain moves to encourage silent students to speak. For example, teachers might introduce a question with “Let’s hear from somebody who hasn’t spoken today.” High-status students sometimes assert their standing by shooting their hands up when questions are posed, letting everybody know how quickly they know the answer. To get around this, teachers can pose a difficult question prefaced with the instructions, “No hands, just minds. I want all of you to think about this for the next minute. Look up at me when you think you know and I will call on somebody.” By allowing thinking time, teachers value thoughtfulness over speed and have more opportunity to broaden participation. Eye contact between students and teacher is a subtle cue and will not disrupt others’ thinking in the way that eagerly waving hands often do. Finally, teachers can make clear that they value partial answers as well as complete ones. When posing tough questions, they can say, “Even if you only have a little idea, tell us so we can have a starting place. It doesn’t need to be all worked out.”


Part of effective participation in classroom conversations requires listening and being heard. As a follow-up to an initial assessment of participation patterns, having an observer pay attention to failed bids for attention or to ideas that get dropped during a conversation might be useful.

Of course, part of the complexity of teaching is deciding which ideas to pursue and which ideas to table. But the choice of whether to entertain students’ thinking communicates something to them about the value of their ideas, which ties directly to status. Students whose ideas are consistently taken up will have one impression about the value of their ideas; students whose ideas are consistently put off will have another idea entirely.

Teachers can model listening practices during class discussions, directing students to listen to each other. By showing students that rough draft thinking— emergent, incompletely articulated ideas—is normal, teachers can help develop a set of clarifying questions that they ask students, and eventually, that students ask each other. For example, a teacher might say, “I’m not sure I follow. Could you please show me what you mean?” Saying this makes confusion a normal part of learning and communicates an expectation that students can demonstrate their thinking.

Body Language

During class, where are students focused? Are they looking at the clock or at the work on the table? Students who have their heads on the desk, hoodies pulled over their faces, or arms crossed while they gaze out a window are signaling nonparticipation. In small-group conversations, their chairs may be pulled back or their bodies turned away from the group. Body language can tell teachers a lot about students’ engagement in a conversation.

Teachers’ expectations for participation can include expectations about how students sit. “I want to see your eyes on your work, your bodies turned to your tables.”

Organization of Materials and Resources

If students cannot see a shared problem during group work or put their hands on manipulatives, they cannot participate. If fat binders or mountains of backpacks obstruct their views of shared materials, they cannot participate. As with body language, teachers can make their expectation for the organization of materials explicit. “No binders or backpacks on your desks. All hands on the manipulatives.”

Inflated Talk about Self and Others

Certain phrases or attitudes can be defeating and signal status problems. Adolescents often engage in teasing insults with each other, but such talk might become problematic in the classroom. Scrutinize judgments about other students’ intelligence or the worthiness of their contributions. The statement “You always say such dumb things!” signals a status problem. “Gah! Why do you always do that?” might be more ambiguous. Teachers need to listen carefully and send clear messages about the importance of students treating each other with respect. “We disagree with ideas, not people” might be a helpful way to communicate this value.

Negative self-talk can be just as harmful. It not only reinforces students’ impressions of themselves but also broadcasts these to others. “I’m so bad at math!” should be banned in the classroom. Give students other ways to express frustration: “I don’t get this yet.” The word yet is crucial because it communicates to students that their current level of understanding is not their endpoint. In fact, several teachers I know post YET on their walls so that any time a student makes a claim about not being able to do something, the teacher simply gestures to the word YET to reinforce the expectation that they will learn it eventually.

The converse of the negative self-talk issue also exists. If a student defends an idea only on the basis of his or her high status, this is a problem. Arguments should rest on mathematical justification, not social position. “Come on! Listen to me, I got an A on the last test” is not a valid warrant and should not be treated as one. By emphasizing the need for “becauses” or “statements and reasons” in mathematical discussions, teachers can winnow away arguments that rest on status.

I’d love to hear some of the ways you see and address status problems in your classroom. Please share freely below.

Once again, much of this text comes from my book Strength in Numbers.


More than Reflection: How Teachers Learn from Each Other

I have had a really interesting twitter conversation this morning with Luann Lee, a science teacher who is thinking hard about instructional coaching. Luann wrote a post suggesting that, instead of instructional coaching, it might make more sense to use resources to buy an assistant teacher for her and her colleagues to free them up to do more visits to each other’s classrooms.

While I agree that this may be a great idea for Luann and her colleagues, I do not think it would work in every circumstance. Recently, my graduate student Britnie Kane and I did an analysis of teachers’ collaborative conversations. They were all math teachers working in urban schools and involved in a professional development project. The key difference among the three groups of teachers was their level of accomplishment in what we call “ambitious teaching” — the kind of instruction that involves all students in high levels of content.

So we spent a year (I am not kidding) analyzing and coding 17 hours of video to make sense of the differences in how they talked about problems of teaching.

Using quantitative and qualitative analyses, here is what we found:

  1. Time spent on problems of practice increased with sophistication in ambitious teaching.
    While the average length of conversations about any one problem in the Sophisticated and Emergent Groups were relatively similar (11 min 26 s versus 9 min and 4 s, respectively), the average time spent in both of these groups was more than twice that of the Beginning Group (4 min 15 s). The differences between the Beginning Group and the other groups were significant (Sophisticated Group, p=0.0055; Emergent, p=0.0003).
  2. The Sophisticated Group consistently considered broad ideas of teaching in light of particular instances of practice.
    Their talk was neither overly vague (e.g., “we need to do more spiraling”) or overly specific (e.g., only telling stories). If they introduced a teaching idea like “spiraling”, it was always linked to examples from the past (what happened that makes them think that) and plans for the future (what will more spiraling look like in their classroom). The linking of general ideas to particulars was a hallmark of their talk, as was the constant pivoting between past and future classroom events.
  3. The Sophisticated Group typically linked discussions of students to issues of instructional decisions and content-specific learning. 

    The other groups might have a good debrief about why a lesson did not go as planned, but then the analysis would not be taken up in subsequent conversation. The Sophisticated group consistently linked any discussion of student learning to instructional decisions and content issues, while the other groups might reflect on these issues but not connect them back up.

In the end, we saw that all reflection is not created equal. The analogy we drew was to the differences between learning from a text when you are a strong versus a weak reader. Good readers can make inferences and extend their understanding, while weaker readers struggle to decode text and can’t see the larger implications. In other words, there is more to gain from reflective discussion once you have already learned quite a bit about teaching, making it an unequally valuable resource for different teachers.

When we looked at conversations with a strong facilitator, a lot of these differences disappeared. For this reason, we think that good facilitators and coaches can make for better conversations. (We know that they don’t always — but that is another post for another day.)

We also think our analysis sheds light on what it means to understand a concept in teaching. You can’t just have an abstract idea of student learning, scaffolding, cognitively demanding tasks, or status, and then know how to use it in your classroom. You need to see multiple examples, in different situations over time. By understanding the connections across these examples, you can really dig into what these things mean.

I guess this is why good teaching is so hard.