“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.