Renegotiating Classroom Treaties

Many classrooms are governed by tacitly negotiated treaties. That is, students trade in their compliance and cooperation –– student behaviors that alleviate the challenges of crowded classrooms ––  for minimal demands for engagement by the teacher. When I have worked with teachers trying out open-ended tasks for the first time, I will often hear about “pushback” or “resistance” from the students: “I tried using this activity but the kids balked. They complained the whole time and refused to engage.”

These student responses indicate that teachers are violating their part of the treaty by going beyond minimal demands for engagement and increasing intellectual press. Put differently, by using an open-ended task, teachers raise the social risk, leaving students open to judgment since they can not rely on the usual rituals of math class to hide their uncertainty. Treaties may, as their name suggests, keep the peace, but they reflect norms of minimal engagement that interfere with deeper learning.

In my own observations, I see teachers struggle to move students past their initial reluctance to participate and make it clear that active involvement is required in their classrooms. Renegotiating classroom treaties requires a clear vision for what student participation can look like, structures to support that vision, along with the determination to see it through. The teachers I interviewed for my forthcoming book all emphasize how critical the first days are for setting these expectations for their students, particularly since their expectations may differ from what students are used to in math class. “It’s entirely intentional that I begin setting norms and structures on the first day of school,” Fawn explains. By launching the new school year by showing students what it means to do math in her class, Fawn renegotiates the classroom treaty through norms and structures, introducing the Visual Pattern and other discourse routines from the start. She says, “I need to provide students with ample opportunities to experience the culture that we have set up. We need to establish and maintain a culture that’s safe for sharing and discussing mathematics, safe for making mistakes, and a culture that honors each person’s right to contribute. There needs to be a firm belief among everyone that mathematics is a vital social endeavor. Building this culture takes time.”
Starting the school year with clear expectations is important, but guiding individual students’ participation is an ongoing project. The teachers I interviewed have numerous strategies for monitoring and building positive participation throughout the year. Students students who hide or students who dominate make for uneven participation. The teachers describe how they contend with these inevitable situations.
When figuring out how to respond to quiet students, the teachers try to understand the nature of students’ limited participation. Not all quiet students are quiet for the same reasons. At times, quietness is rooted in temperament: some students inclined to hang back until they feel confident about what is going on, but they are tracking everything in class. These students do not contribute frequently, but, when they do, their contributions add a lot to conversations. This kind of quiet is less of a concern and can even be acknowledged: “Raymond, you don’t talk a lot, but when you do, I always love hearing what you have to say.”
Other times, quietness signals students’ lack confidence. That is, students indicate some understanding in their work or small group conversations, but they do not have the confidence to participate in public conversations. With these students, the teachers seek out individual conversations. Chris calls these doorway talks, while Peg calls them sidebars. (“Trying to deal with calculators and rulers at the end of class, I couldn’t make it to the doorway!” Peg tells me when I note the different names.) “I might say to a kid, ‘You know, you had really good ideas today, and I would have loved to have heard more of them in the conversation we had a the end. I think you have a lot more to contribute than you give yourself credit for.’” Sometimes, there are ways of encouraging good ideas to become public that do not directly address the student. Chris explains that he might say something like, “I haven’t heard from this corner of the room.” He then asks other students to hold their ideas while waiting for a contribution from the quiet group.

Of course, some students are quiet because they really do not know what is going on. This could be due to a language issue, in which case, the teacher needs to modify instruction to give them more access to the ideas. If there are other learning issues going on, this might suggest the need to check in with colleagues about the students performance in previous years or in other subjects.

eager-students
Talkative students pose another kind of challenge to the expectation that everyone participates.  On the one hand, they can provide wonderful models of sharing their thinking. They can be the “brave volunteers” who explore their thinking publicly, and teachers can lean on them to get conversations started. On the other hand, they can be domineering, making it difficult for other students to get a word in. The quiet students who temperamentally need to think before they speak have their counterparts in some talkative students: these are the students who think by talking. Asking for their silence sometimes gets heard as asking them not to think. When I have had students like that in my own classes, I make sure to assure them that I value their engagement but that I need them to find other strategies for processing so that other students can be heard. Sometimes, students with impaired executive functioning, like those with ADD, have a hard time with the turn-taking aspect of classroom dialogue, so not only do they talk a lot sometimes, they struggle to take turns. Again, teachers can respond by valuing students’ ideas while helping them participate more effectively: “I know you get excited, but we need to take turns so that we can hear each other.” Finally, domineering behavior can get expressed through a lot of talking: students who are highly confident in their understanding and want to explain to others. Teachers need to judge the extent to which this is altruistic, a sense of trying to share knowledge, and the extent to which it shuts conversations down. In the first case, students can be coached towards asking questions of their classmates, channeling their impulse to talk into something constructive. In the second case, the dominance can be corrosive to the classroom culture and the students might need stronger redirection. For all of this feedback, similar strategies of direct address (via sidebars or doorway talks) and indirect address (“Let’s hear from somebody else”) can help teachers manage participation.

Who Belongs in our Math Classrooms?

Many students enter mathematics classrooms with a sense of trepidation.  For some, their discomfort reflects a larger sense of detachment from school. They may have not felt welcomed because of the gaps they experience navigating between their home language or culture and the expectations at school. The social milieu of school may make them feel like an outcast, as they see peers who seamlessly “fit in” while they remain on the outside. Unlike the sports field, their community center, or the stage, academics may make them feel untalented and incompetent. For other students, school itself is fine, but there is a distinct dread upon entering math class. Math has never made sense –– or perhaps it used to make sense when it was whole numbers and counting, but as soon as the variables showed up, all hope was lost. They may have been demoralized by a standardized test score that deemed them below grade level. They may get messages at home that “we’re not good at math.” For still other students, they love the subject, but must contend with people who do not see them as fitting their ideas of a person who is good at math. They have to combat stereotypes to be seen as legitimate participant in the classroom, as they defy expectations by holding forth with their smartness even as others look on in dismay.

For most students, alienation can be overcome by teachers who create a sense of belongingness. Belongingness comes about when students experience frequent, pleasant interactions with their peers and teacher. It also comes about with the sense that others are concerned for who they are and for their wellbeing.

Why Belongingness Matters
When I go and observe in mathematics classrooms, I can usually ascertain students’ general sense of belongingness. What is their affect as they walk through the door? How warmly and personally do they greet the teacher and each other? Are they represented –– through math work or other means –– on what is posted on the walls?
All too often, I see students enter their math classrooms with a sense of gloom. Smiles disappear as they cross the threshold of the doorway. Their posture slumps. They sit at the back of the room or put their heads on their desks.They may even groan or launch into a litany of complaints. When I observe these student behaviors as a teacher, it signals that I have work to do to make children feel more welcomed and excited about spending their time with me learning mathematics.

Teachers’ relationships with students are an important source of of belongingness, but peers are equally (if not more) important. Even if a teacher welcomes each student with a smile and takes an interest in who they are, frequent insults or intimidation from other students can create a negative classroom climate. To support belongingness, then, teachers need to do more than create strong relationships. In addition, they need to create norms and expectations about how students treat each other.

During adolescence, children face the enormous task of developing a strong and stable sense of themselves. Although this identity development happens over the course of a lifetime, adolescence is distinct because it is when children are first able to think abstractly enough to grapple with both their own emerging self-understandings as well as how society views them. This leads to both a delightful self-awareness as well as a sometimes painful sense of self-consciousness for many students, as they are more sensitive to others’ perspectives and feedback. Necessarily, then, inclusive and inviting classrooms provide a place for this crucial developmental work, particularly in relationship to school in general and mathematics in particular.

What Gets in the Way of Belongingness
Although I generally avoid absolutes when it comes to describing good teaching, I will highlight a few common instructional practices that feed a negative classroom climate, thus working against belongingness. First, many math classrooms emphasize competition. Whether this comes from formal races, timed tests, or just students’ constant comparison of grades, competition sends a strong message that some people are more mathematically able than others. This is problematic because there is typically one kind of smartness that leads students to “win” these competitions: quick and accurate calculation. To paraphrase mathematician John Allen Paulos, nobody tells you that you cannot be a writer because you are not a fast typist; yet we regularly communicate to students that they cannot be mathematicians because they do not compute quickly. While a competitive dynamic may be at play in other school subjects, it is especially toxic in math classrooms because students do not have other venues to explore and affirm their diverse mathematical talents.

Another contributor to negative classroom climate comes from devaluing who students are. This may come in many forms, some of which teachers may not realize. For instance, some teachers avoid using what is for them an unfamiliar (thus difficult-to-pronounce) name. Not only does this lead to fewer invitations to participate, it communicates to students that we are not comfortable with something that might make them different than us. Names are deeply personal, one of the first words students identify with: They often reflect home cultures and personal history. When teachers avoid them or change them without consent, they devalue something of who students are.

Likewise, when teachers problematically differentiate their treatment of students based on cultural styles, they can devalue who students are. For instance, educational researcher Ebony McGee studies successful students of color in STEM fields. She interviewed a Black chemistry major at a primarily White institution who reported that a White instructor avoided her when she dressed in a way often perceived by middle class teachers as “ghetto.” When she changed her clothing and hair style, he told her, “Now you actually look presentable. I bet you are making better grades too.” Similarly, in a research project I conducted, a female high school student concluded that her math teacher “didn’t like” her after the teacher emailed her mother that her skirts were “too short.” Adolescents use clothing to express themselves and their culture as a part of the identity work they engage in. Avoiding or rejecting them because of these forms of self-expression can further estrange them from the classroom or school. If concerns need to be raised, they should be done in a way that respects students’ self-expression.

Finally, teachers may alienate students by correcting the inconsequential. Although our job is to help students become educated people, when we correct the inconsequential, we may work against other goals of engagement and inclusion. Deciding what is inconsequential is, of course, a judgment call: context is everything. For instance, our standards for speech and language differ when students try to explain an idea they are in the midst of grappling with versus when they are preparing for a job interview. In the former situation, correct grammar is not the point, while in the second, it may matter a lot. If our students are learning English as a second language, speaking a pidgin or African American Vernacular English (AAVE), our focus on correct grammar in situations where it is inconsequential may disinvite their participation.

Faking Excellence: The Art of Milking Mediocrity for all its Worth

(Note: This is a guest post by my high schooler, an excellent student. It came out of a chat with some of her high-achieving friends.)

An Informative Guide

Part I: In the classroom

In order to uphold the image of “dedicated student” in the eyes of one’s educators, it is important to maintain a certain level of pseudoattentive behavior. Always have a notebook and a writing utensil out on your desk. Try to sit towards the front of the classroom, and make eye contact when teachers are lecturing. Take notes. Ask questions when you have them. (This practice both elevates the teacher’s opinion of you and helps to further the image of you as caring.) Greet your teacher upon entrance to the classroom. Converse with him or her on the finer points of their subject that you have diligently researched (see below). Bid them farewell upon your leave. Have books and supplies with you at all times. Participate in classroom activities. Make yourself feel actively present, and your teacher will take note.

Part II: Active Procrastination

In a survey of 2 high school students, both agreed that most to all of their homework is “boring.” As a result, we can conclude that one’s homework may not always come first in their lives. So if you don’t want to do it, does that mean you should binge watch Parks and Rec on the floor of your bedroom in shame? No! Procrastinate actively. Use homework time to expand your mind in more interesting ways. Read articles that are somewhat vaguely related to classroom materials (see above). Talk to your friends about how much you would prefer to do nearly anything but said assignment. Live the life of an overworked student while only spending a fraction of your time acting like one.

Part III: Completing Work

Close your eyes. Take yourself back to the last time you put off an assignment until 11:30pm the night before it was due. Get a good, long look at this mental image of last night, and open your eyes. Sure, you know how to put off work. But do you know how to cram it? The first lesson to be learned when attempting to do three weeks of work in one night is that you never outright admit this weakness. When dating the paper, always think back to when it was originally assigned. Then, count forward to the due date. Take one third of that number. Count that many days ahead from the original assignment date. There you have it: a believable but still respectable starting date. Exceptions may apply, but this is a good rule of thumb until you are a more seasoned procrastinator. The next lesson to be learned is the art of rephrasing. Many, many teachers steal each other’s work sheets. It is in their nature. So many, many foolish students at schools without honor codes (or with flagrant disregards to them) post the exact wording of these questions onto Yahoo Answers. And many, many Good Samaritans spend their time answering these questions. Learn to rephrase the work of these kind souls and make it sound like your own. Chop up sentences. Rearrange. Use synonyms. Expand on ideas. Cut down ideas. The Best Answer on Yahoo Answers is your marble, and you are Michelangelo. Now get on it, before you switch into complete sleep deprivation mode.

Part IV: Emulating Those More Well Rested than Yourself

The ideal model of a student is one who is not only well educated, but bright eyed and bushy tailed each and every morning. Now, on days when your eyes are more shadowed than bright and your tail is a deflated balloon, what is there to do? Worry not. The first step is hydration. Cold, cold water can jerk anyone out of dreamland, as can some nice old fashioned caffeine. Another tip is to remember the saying “dress for the sleep you wished for, not the sleep you got.” Wear clothes that make you look alive. Dead zombie clothes will turn you into a dead zombie. It’s science. Smile at things so that you do not appear to be a sleepy lump. And heaven forbid you fall asleep in class.

Part V: Eloquent BS

When completing various forms of free response questions, it is important to master the art of key term dropping. Sometimes, a question or prompt will only evoke a 404. message from your brain, and in that moment, do not be afraid. Recall the overall subject matter being assessed. Bring to mind the key terms of the section (often found alongside textbook passages) and think about whether you have any recollection whatsoever of how to use them. If so, you’re in luck! Teachers do not always read all 120 essays they have to grade, (and so especially for a class that isn’t a language class) they sometimes just skim to make sure that you have captured the general essence of the subject matter. Term dropping will not hurt, especially if you can bulk it up with any other somewhat related content. The author has personal experience of herself and very close friends getting 100s on answers for simply using the phrases “Christian-based society,” “complex gender issues,” “King John,” and “high death rates” in a paragraph with hardly any other coherence. Miracles do happen, but sometimes you have to help them along.

Part VI: Tying it All Together

In our short time together, you have learned how to become a more deceptively talented student. This skill, however, can only take you so far. Without a deep commitment to maintaining your facade of greatness, it will collapse like the Berlin Wall in 1989 and your lies will become obvious. Treat your mediocrity like a channel for something greater. Believe.

 

“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.”

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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.
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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.

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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…

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Then I talked about how status problems play out in classroom conversations.

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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!

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.

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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.

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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.