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.

Recognizing Smartness and Addressing Status in the Classroom

When status plays out in the complex world of the classroom, it takes many shapes. Although blatant dominance, insults, or non-participation are easy to spot, the more subtle manifestations take skill to identify and remedy. Effectively intervening with status problems first requires analysis of the situation. Figuring out the best strategy is often a trial-and-error process. Teachers get better at managing status in their classrooms over time, but even accomplished teachers run into challenges that force them to further sharpen their intervention tools.

The following strategies outline a starting point for status interventions. Unfortunately, this is not a recipe that will make status problems magically disappear. Status will always be part of our social world. The trick is to manage it such that students begin to reimagine themselves and their peers in the context of their competence and not their deficits. Every class you teach will have different personalities and dynamics, so these will play out differently in each circumstance. Nonetheless, here are some tested status interventions that can be adapted to any classroom.

Establishing and Maintaining Norms

Effective classroom norms support equal-status interactions. In the previous discussion of status problems, I suggested some structures teachers can use, such as “no hands, just minds,” that help curb status problems. These all communicate norms for participating and interacting. For our purposes, I will use the following definition of norms:

Classroom norms are agreed-upon ways of behaving.

Establishing norms requires a conversation with students. Some teachers do this interactively, asking students to contribute their answers to the question, “What makes you comfortable in a classroom?” Other teachers let students know that they have found certain behaviors helpful in making a positive classroom environment where students feel comfortable to learn. However they are arrived at, posting a list of norms on the wall as a reminder can help keep these at the forefront.

Norms can help curb status problems. For example, establishing the norm of no put-downs can minimize negative talk about oneself or others.  Examples of other norms that help support equal status interactions include the following:

  • Take turns.
  • Listen to others’ ideas.
  • Disagree with ideas, not people.
  • Be respectful.
  • Helping is not the same as giving answers.
  • Confusion is part of learning.
  • Say your “becauses.”

Since norms are associated with classroom behavior, they are often thought of as a classroom management tool. In a sense, they are, but they go beyond that. Classroom management is often understood as serving the important goal of managing the crowd in the classroom. Students may or may not value that goal. The use of norms as I describe them helps students learn.

To make norms more relevant to students, always link norms to your learning goals. For example, helping is not the same as giving answers values explanations and learning over the completion of work. Similarly, say your “becauses” values the mathematical work of justification over assertions of correct answers that may be based in status. This norm also helps alleviate the problem of nonmathematical assertion of an argument by helping a lower-status student demand that a higher-status student better explain an assertion. In classrooms where this norm is in use, I hear students say to one another, “Yeah, but why? You didn’t say your ‘because.’”

Telling students expectations for acceptable behavior does not, of course, ensure that they will always meet them. Norms require maintenance. New situations might create a need to reestablish them. Even new content—particularly content that highlights differences in prior achievement—can heighten status issues and therefore require a strong reminder about classroom norms.

Addressing Status through Norms

Over time, teachers get better at analyzing which norms might help shift negative status dynamics in their classrooms. Teachers pick one or two norms for a particular activity and tell students, “While you are working on this, I am going to watch how you do on these norms.” The teacher then reminds students of the expectation for acceptable behavior.

Sometimes the choice of norms comes from a teacher’s reading of the dynamics in prior class sessions. For example, if student conversations are coming too close to personal attacks, a teacher might highlight the norms be respectful and disagree with ideas, not people. If the teacher then circulates around the room and reminds students of these norms, he is not picking on problem students; rather, the teacher is stating a classroom goal that everybody is trying to work on.

Likewise, teachers can predict mathematical activities that might lead to status problems and use norms to head these off. Any topic that is confusing may make students vulnerable to status concerns. Reminding students that confusion is a part of learning can help. I have heard teachers say, “Now, I don’t expect you to get this problem quickly. It’s really hard and you will need each other’s help. If you get confused, that’s great because it means you are learning.”

Sometimes, specific topics expose students’ status concerns. Calculations with fractions commonly bring out insecurity in previously low-achieving students and impatience in students who are already fluent in these calculations: a recipe for a status collision. Anticipating this, a teacher can let the class know that she will be watching for the norms helping is not the same as giving answers and say your “becauses.” The first norm will send a clear message that students who can calculate quickly need to do more than show the other students their answers. The second norm offers less confident students a means to demand explanations from their peers (“Okay, but you didn’t say the ‘because’”).

Multiple-Ability Treatment

So far, this discussion of status has acknowledged the different status levels of students in any classroom and how it can undermine productive mathematical conversations. No doubt, addressing status through norms is crucial to creating equal-status interactions. By helping students interact more productively—listening respectfully, justifying their thinking—we help support meaningful mathematical conversations.

Norms, however, will take us only so far. Unless we address underlying conceptions of smartness, we risk reverting to the commonly held belief that group work benefits struggling students because smart students help them. As long as we have a simplistic view of some students as smart and others as struggling, we will have status problems in our classrooms. (Please see my previous post on different kinds of mathematical smartness.)  Students quickly pick up on assessments of their ability. For example, when teachers arrange collaborative groups to evenly distribute strong, weak, and average students, children will figure out that scheme and rapidly learn which slot they fill. No doubt, learners benefit from seeing more expert performance and should have opportunities to do so. But if we value only certain kinds of expertise, the same students will always play the role of experts. The question then becomes, What kinds of mathematical competence have a place in your classroom activities? If the mathematics is rich enough, the strengths of different students will come into play, rendering the common mixed-ability grouping strategy useless. Ordering the students by achievement and evenly distributing strong, weak, and average students across the groups will no longer be enough.

In fact, an essential practice for a multiple-ability classroom is random group assignment. If we believe that students can all learn from each other, then group assignments should have no underlying design based on assessments of ability. Teachers often do this by using a wall-hanging seating chart that has pockets for each student’s name. When it is time to rearrange groups, they will shuffle the cards and simply redistribute them in the pockets to make a transparent show of the randomness of group assignments. If a teacher judges a certain pairing of students to be unwise, she can publicly state the reason for this (e.g., “You two tend to get too silly together, so I think I will switch you out”). These reasons are not judgments about smartness but are instead social considerations. Random group assignment, however, is just one component of multiple-ability treatments.

As I said in my post on smartness, in schools, the most valued kind of mathematical competence is typically quick and accurate calculation. Evaluating people on one dimension of mathematical competence will rank students from most to least competent. This rank order usually relates to students’ academic status, and students tend to be aware of it. One way to interrupt status is to recognize multiple mathematical abilities. Instead of a one-dimensional rank order, we create a multidimensional competence space. Although some students may have multiple mathematical strengths, more places in which to get better surely exist. Likewise, a student who ranks low on the hierarchy produced when we focus on quick and accurate calculation may have a real strength at making astute connections, working systematically, or representing ideas clearly. We cannot address status hierarchies without emphasizing multiple mathematical competencies in the classroom.

A multiple-ability classroom represents a dramatic shift in the topography of mathematical ability. Instead of lining students up in a row in order of smartness, a multiple-ability classroom has students standing on different peaks and valleys of a hilly multidimensional terrain. No one student is always clearly above another. This structure may unsettle students who are used to being on top, as well as those whose vantage points and contributions have been presumed less valuable. In other words, challenging the status hierarchy by developing a multiple-ability view can provoke strong emotions from students, positive and negative. Teachers should not be surprised to see this response in their classrooms.

Multiple Ability Treatments

A multiple-ability treatment comes in the launch of a task. After presenting the directions and expectations, teachers list the specific mathematical abilities that students will need for the task and add the phrase, “No one of us has all of these abilities, so you will need each other to get this work done.” By publicly acknowledging the need for more than just quick and accurate calculation, teachers offer an in for a broader range of students. Multiple-ability treatments do other work too, particularly fostering interdependence.

Assigning Competence

The two status interventions described so far operate on the classroom level. Norms give clear expectations for behavior to push students toward more productive mathematical conversations. Multiple-ability treatments highlight teachers’ valuing of broader mathematical competencies.

The next step is to help students recognize where they and their classmates are located on the complex topography of mathematical competence to shift their self-concept and their ideas about others. Students need to recognize these other competencies for themselves so that they know their own strengths and can work confidently on hard problems. They need to recognize the strengths of their peers in order to interrupt assumptions based on a simplistic smartness hierarchy. If students believe their classmates have something to contribute, they have a mathematically motivated reason to listen to and learn from each other.

Teachers can communicate these messages to students through the practice of assigning competence.

Assigning competence is a form of praise where teachers catch students being smart. The praise is public, specific to the task, and intellectually meaningful.

The public part of assigning competence means that this praise is not an aside to an individual student or a communication with the parent. It takes place in the public realm of the classroom, whether in small-group activity or whole-class discussion. It needs to be specific to the task so that students make a connection between their behavior and their mathematical contribution. Simply saying, “Good job!” is not enough. Students need to know exactly what they did that is valued. The praise must be intellectually meaningful so that it contributes to students’ sense of smartness. Praising a student for a “beautiful poster” does not qualify as assigning competence, because making a beautiful poster does not display mathematical intellect. In contrast, if a teacher praises a student for a clear representation on a poster that helps explain an idea, that is intellectually meaningful because it is tied to mathematics.

I hope this post gives you some insight into how to address status and value smartness in your classroom. No doubt, this is challenging work, But I think the payoff in mathematical learning is well worth it.

What does it mean to be smart in mathematics?

In the last two posts, I discussed the idea of status. First, I talked about why status matters, then I talked about how teachers can see it in the classroom.

Sometimes, after I have explained how status plays out in the classroom, somebody will push back by saying, “Yeah, but status is going to happen. Some kids are just smarter than others.”

I am not naive: I do not believe that everybody is the same or has the same abilities. I do not even think this would be desirable. However, I do think that too many kids have gifts that are not recognized or valued in school — especially in mathematics class.

Let me elaborate. In schools, the most valued kind of mathematical competence is typically quick and accurate calculation. There is nothing wrong with being a fast and accurate calculator: a facility with numbers and algorithms no doubt reflects important mathematical proclivities. But if our goal is to address status issues and broaden classroom participation in an authentically mathematical way, we need to broaden our notions of what mathematical competence looks like.

Again, my naysayers roll their eyes and groan, assuming that I want to “soften” mathematics or dilute the curriculum. But I claim that broader notions of mathematical competence are actually more authentic to the subject.

Image

Let’s cherry pick some nice examples from the the history of mathematics. We see very quickly that mathematical competencies other than quick and accurate calculation have helped develop the field. For example, Fermat’s Last Theorem was posed as a question that seemed worth entertaining for more than three centuries because of its compelling intuitiveness. When Andrew Wiles’s solution came in the late twentieth century, it rested on the insightful connection he made between two seemingly disparate topics: number theory and elliptical curves. Hyperbolic geometry became a convincing alternative system for representing space because of Poincaré’s ingenious half-plane and disk models, which helped provide a means for constructions and visualizations in this non-Euclidean space. When the controversy over multiple geometries brewed, Klein’s Erlangen program developed an axiomatic system that helped explain the logic and relationships among these seemingly irreconcilable models. In the 1970s, Kenneth Appel and Wolfgang Haken’s proof of the Four Color Theorem was hotly debated because of its innovative use of computers to systematically consider every possible case. When aberrations have come up over the years, such as irrational or imaginary numbers, ingenious mathematicians have extended systems of calculation to encompass them so that they become number systems in their own right.

This glimpse into the history of mathematics shows that multiple competencies propel mathematical discovery:

  • posing interesting questions (Fermat);
  • making astute connections (Wiles);
  • representing ideas clearly (Poincaré);
  • developing logical explanations (Klein);
  • working systematically (Appel and Haken); and
  • extending ideas (irrational/complex number systems).

These are all vital mathematical competencies. Surprisingly, students have few opportunities to recognize these competencies in themselves or their peers while in school. Our system highlights the competence of calculating quickly and accurately, sometimes at the expense of other competencies that require a different pace of problem solving.

Evaluating people on one dimension of mathematical competence will rank students from most to least competent. This rank order usually relates to students’ academic status, and students tend to be aware of it. One way to interrupt status is to recognize multiple mathematical abilities. Instead of a one-dimensional rank order, we create a multidimensional competence space. Although some students may have multiple mathematical strengths, more places in which to get better surely exist. Likewise, a student who ranks low on the hierarchy produced when we focus on quick and accurate calculation may have a real strength at making astute connections, working systematically, or representing ideas clearly. We cannot address status hierarchies without emphasizing multiple mathematical competencies in the classroom.

A multiple-ability classroom represents a dramatic shift in the topography of mathematical ability. Instead of lining students up in a row in order of smartness, a multiple-ability classroom has students standing on different peaks and valleys of a hilly multidimensional terrain. No one student is always clearly above another. This structure may unsettle students who are used to being on top, as well as those whose vantage points and contributions have been presumed less valuable. In other words, challenging the status hierarchy by developing a multiple-ability view can provoke strong emotions from students, positive and negative. Teachers should not be surprised to see this response in their classrooms.

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

Participation

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.”

Listening

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.

 

Status: The Social Organization of “Smartness”

I wrote a book about a highly effective differentiation strategy for mathematics classrooms called complex instruction. Complex instruction is a research-based approach to teaching that treats the classroom like a social system. Using this idea, it helps teachers engineer the environment to make for rich learning opportunities for as many students as possible through collaborative student work.

I spend enough time in classrooms to be a realist about the various challenges teachers face day to day. I know that collaborative learning may be a far leap for some. Nonetheless, if there is one idea that would benefit any classroom learning environment, it is that of student status. In this post, I explain status and how it plays out in mathematics classrooms.

By the time students are in secondary school, children enter their mathematics classes with strong ideas about who they and their peers are as mathematics learners. They can tell you who is smart and who is not. They base these judgments on earlier school achievement, as well as on categories such as race, class, popularity, and gender. These assessments play out in the classroom. Some students’ contributions are sought out and heard, whereas others’ contributions are ignored. This imbalance obstructs productive mathematical conversations because an argument’s valuation comes from who is speaking and not what is being said.

Productive mathematical conversations are ones in which arguments are weighed on the basis of the mathematical validity of what is being said, not on who is speaking.

Judgments about who is smart based on prior achievement or social categories violate a fundamental principle of equity and are consequential: learning is not the same as achievement. Confounding this problem, American schools tend to be organized in ways that obscure distinctions between learning and achievement. In fact, they are often built around the idea that differences in student achievement are the natural consequence of differences in ability. The logic of tracking, particularly in the early grades, rests on notions of identifiable differences in ability that require different approaches in teaching.

In reality, tracking often only reinforces achievement differences by giving high-achieving students better teaching and more enriched learning environments. An important principle of equitable teaching is that achievement gaps often reflect opportunity gaps. We typically think of opportunity gaps as existing across schools, with schools serving upper middle-class populations having greater resources than schools serving poor students. Although this tragically remains the case in the United States, the resource differences within schools are often overlooked. Two students in the same school placed in different tracks—on the basis of their prior achievement—typically have radically different learning opportunities through the quality of their teachers, the time spent engaged in academic activities, and the rigor of the curriculum. Once you are behind, getting ahead is hard.

Status is not just a concern for low-achieving students: all students in the United States need the opportunity to learn mathematics more deeply.

The belief in ability as the root of different levels of achievement is so entrenched in the organization of curriculum and schooling that many people have a hard time imagining another model. Other conceptualizations are possible, however. Japanese education attributes differences in achievement to students’ different levels of effort instead of differences in ability. Classrooms are organized to see student differences as a resource for teaching, instead of viewing them as an obstacle to be accommodated. Tracking does not occur in the early grades.

Considering students’ robust views on who is smart along with schooling practices such as tracking, which naturalize differences, it is no wonder that most students’ mathematical self-concepts seem immutable by the time they arrive in secondary classrooms. Everything around them fixes their sense of their ability, be it low, high, or average.

If learning is not the same as achievement, and if achievement gaps often reflect opportunity gaps, what do we make of students’ prior achievement when they enter our classrooms? Who are the students who have succeeded in mathematics before entering our classrooms? How about those who have not? Disentangling achievement and ability may sound reasonable, but we need a new model for thinking about students we teach. Elizabeth Cohen’s work on complex instruction frames these issues around status, a concept that clarifies the conflation of achievement and ability. Status gives teachers room to analyze this problem and respond through their instruction.

In this context, we will use the following definition of status:

Status is the perception of students’ academic capability and social desirability.

The word perception is key to this definition. Perception drives the wedge between social realities and perhaps yet unrealized possibilities of what students can do mathematically. Perception involves our expectations of what people have to offer.

Where do these status perceptions come from? Typically, the perception of academic capability often comes from students’ earlier academic performance. It might come from their academic track, with honors students having higher status than that of regular students. Status judgments about ability might also draw on stereotypes based on class, race, ethnicity, language, or gender.

The perception of social desirability arises from students’ experiences with peers. For instance, students often see attractive peers as desirable friends—or perhaps just undesirable enemies. Likewise, whatever drives popularity in local teen culture will show up in the classroom as status. The team captain, the talented artist, or the cut-up rebel—whomever students clamor to befriend or win the approval of—will have higher social status.

Status plays out in classroom interactions. Students with high status have their ideas heard, have their questions answered, and are endowed with the social latitude to dominate a discussion. On the other side, students with low status often have their ideas ignored, have their questions disregarded, and often fall into patterns of nonparticipation or, worse, marginalization.

Recognizing the relationship between status and speaking rights highlights an important way for educators to uncover these issues in their classrooms. Status manifests through participation patterns. Who speaks, who stays silent, who is excluded, and who dominates class discussions are all indicators of status. Individually, this concept influences students’ learning. If some students’ ideas are continually ignored, their questions will go unanswered and their confusions will remain unaired. Over time, this system may reinforce negative ideas they have about themselves as mathematics learners, because they may conclude that their ideas are not valuable. Conversely, students whose ideas are consistently heard and worked with will have greater opportunities to engage and sort through them. Socially, if students’ dominance becomes unregulated, they may develop an overblown sense of their value in the social and intellectual world of the classroom. Thus, status-driven interactions not only influence learning but also reinforce existing status hierarchies.

Skeptics might protest linking participation and status. “Some students are just shy,” someone might say. That is true. Likewise, students learning English often go through a silent period or may be self-conscious of their accents. Our goal with reluctant speakers is to design ways for them to comfortably participate more than they are perhaps naturally inclined to do. Strategies such as small-group talk first or individual think time may help build the confidence of shy or nervous speakers. The emphasis on participation in classroom discussions comes from several research studies showing that such involvement is essential to developing conceptual understanding and academic language.

Socially, status plays out in participation patterns. Individually, status influences students’ mathematical self-concepts, or their ideas about what kind of math learners they are. As mathematics educators, we have all encountered students who claim that they are not “good at mathematics” before they even give a new idea a chance. Intuitively, we know that students’ mathematical self-concept influences their motivation and effort in mathematical learning. If students know they are not good at mathematics, why should they push past their confusion when problems become difficult? If students know they are smart, why should they bother to explain their thinking, let alone pay attention to a classmate’s? Students’ self-concept is deeply tied to their attitudes about learning mathematics, in and out of our classrooms. Societal biases predispose students to think of themselves and their peers as more or less competent in mathematics, playing into students’ choices to engage, persist, and take risks in the classroom.

Text adapted from my book, Strength in NumbersAs always, I invite your respectful and curious questions and comments.