Informal Poll on on Math Edu Tweeters

In preparation for a discussion some of us are going to have at NCTM, I conducted a totally unscientific poll of the #MTBoS and others in the general math edu constellation. I wanted to get a snapshot of how math educators engaged with others through social media so I could feel more confident sharing my impressions with the wider world.

I asked three questions.

1. How often do you engage with educators on twitter?

2. Which of the following ways do you engage with other teachers online?

3. Please describe the most useful learning experience you have had online. You can provide links to specific posts or tweets.

Question 1 revealed how totally unrandom my sample was. I would say it characterized my sample (n =52) as highly involved in social media. Most respondents tweet almost every day.

This means that the casual users, the lurkers, the toe-dippers are not well represented in this poll. That’s fine. This gives us a good picture of why people would feel like engaging heavily in social media. It’s a good group to hear from.

For Question 2, I let folks select as many answers as they needed. I didn’t ask for “top three ways” or anything, so some categories were frequently selected.

Tweeting, reading blogs, and writing blogs were the most frequently selected uses of social media. Less than half of our heavy users reported participating in real-time exchanges like #edu chats or the Global Math Department. Slightly more (but still less than half) talked about collaborating on specific projects using google docs or other collaborative environments.

Question 3, what was the most useful learning experience you have had, let me get my qualitative analysis game on. Aside from the frequent response of that it was hard to pick one, people pointed to the following kinds of learning via social media:

• Idea exchange: This was one of the most frequent responses. Twitter was especially praised for the access it provides to others’ ideas: “I post a question. People come along and make me smarter.” Reading and writing blogs and tweeting let people share ideas and comment on others’ ideas about their math teaching. As one respondent described, “Whenever I’m interested in a new approach to teaching something, I can read many different implementations and see how it actually looks in the classroom. Helps make my abstract ideas more concrete.” A number of teachers pointed to the rich resources as providing more opportunities for personal development, as new ideas become more immediately accessible: “Being able to find lesson ideas, extension materials, and intriguing pics and videos has brought a whole new dimension to my classroom.”
• Sharing resources: Teachers have to plan lessons everyday. Social media is a great way for sharing resources. By turning to an online community, teachers know something of the values and practices of their sources. As one teacher described, “The idea/lesson exchange is better than a huge google death for activities.” Sometimes folks get new preps or new groups of students who require different kinds of materials, or sometimes teachers realize that their old lessons aren’t quite doing all they could. “A few years ago I was informed I was teaching a brand new AP course three weeks before school started. I was scrambling for resources so I took to twitter. The amount of support and resources I gathered from teacher on twitter was a life saver.” Teachers will often document their new work as they build new classes, only adding to the accumulation of resources in the community.
• Connecting with like-minded educators: A number of responses indicated that many educators who develop professional learning networks online do so to break the isolation they feel in their own schools or departments. “Just the several years I’ve had building my PLN (and making friends!) has been invaluable. I’m not sure I could point to anything specific. It changed my career.”
• Constructing resources together: Question 2 shows that this is not as common of a practice, but those who have done it have reported its value to their professional learning.”I think the biggest learning experiences for me have been the times when I’ve constructed a resource with others online and we’ve learned as we’ve gone through the act of co-creating.” One example of a collaboratively developed resource is Nix the Tricks, whose curator Tina Cardone explained, “There’s no way I would ever have taken on this project without the crowd sourcing and the encouragement that the MTBoS provides.”
• Developing shared critiques of educational tools and practices: Teachers, especially those committed to developing student understanding, are facing challenges on numerous professional fronts. A few teachers mentioned the support they find for particular visions of teaching. Additionally, the online community has developed numerous critiques of popular teaching tools like Khan Academy. “The mtt2k (Mystery Teacher Theatre) initiative [see an example here] encouraged me to learn more about Khan as well as doing my own video editing, plus sparked new connections.” There is a lot to keep up on the educational landscape. As one teacher described, “Twitter is my education newsfeed!!”
• Getting emotional and moral support: Especially when teachers are working against the institutional grain, pursuing more ambitious forms of instruction can get discouraging at times. A number of teachers mentioned the emotional support they get from colleagues online. “The most valuable past of this for me is knowing there are other teachers out there that are working towards the same goals as me. That there are other teachers that will support me in my journey to those goals.” Teachers also talked about getting more specific images of the kinds of classrooms they aspired to and having people to vent with on hard days.
• Learning about a specific practice, tool, or idea: A number of responses pointed to teachers who developed specific interests and pursued them in online communities. Some examples were practices like standards based grading, tools like desmos or GeoGebra, or unexpected insights into mathematical topics.

How did our enthusiasts do in capturing the learning potential of online professional communities? How did our friendly neighborhood educational researcher do in summarizing the responses? Is there anything that particularly resonates or that you think I left out?

Please share! I am going to be telling the world –– okay, at least the people coming to our session –– what this whole thing is about, and I am committed to getting it as right as possible.

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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.

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.

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.

retrieved from http://flic.kr/p/edWC5r

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 Numbers. As always, I invite your respectful and curious questions and comments.

What do we get with the “highly qualified teacher” clause of NCLB?

The conception of teacher competence animating NCLB can be traced to the landmark 1966 Coleman Report. As a part of America’s War on Poverty, Coleman, a sociologist, examined educational opportunity in the US, finding that school funding did not impact educational outcomes as much as teacher quality did. Part of what Coleman identified is what has come to be known as the maldistribution of qualified teachers. In the United States, poor children and children from historically underrepresented groups are disproportionately assigned to the weakest teachers –– a situation that persists to this day. International comparisons reveal that the United States stands out in this maldistribution problem: our country’s disparities in students’ access to qualified teachers is among the largest in the world.

Coleman’s formulation has had a continuous impact on notions of teacher quality. Numerous subsequent studies have verified positive associations between student achievement and teachers’ academic qualifications. This relationship has been confirmed through correlations between student outcomes with teachers’ scores on various standardized exams; and level of teachers’ content knowledge, usually proxied through course-taking counts; and years of teaching experience.

NCLB is animated by this legacy. The law sought to address the maldistribution problem by legislating a definition of teacher quality by mandating highly qualified teachers in every classroom. True to the Coleman logic, highly qualified teachers, according to NCLB, are those with full certification, a college degree, and demonstrated content knowledge in the subject being taught.

However, the legal definition of highly qualified has left much open to debate. For instance, in the description of highly qualified teachers, the word “student” only appears once, and then only as a moderating adjective.

In other words, the relational work of teaching is completely ignored.  Social, emotional, and cultural knowledge does not come through the policy text, nor does the specialized content knowledge teachers require to effectively represent and cultivate students’ understanding –– what Shulman called pedagogical content knowledge. In light of these omissions, the definition of highly qualified teachers may not reach deeply enough into the kinds of knowledge and qualities teachers need to best serve students living in poverty, thus potentially undermining part of the law’s intent.

Research on teachers of historically underserved students emphasizes that effective teachers engage particular forms of knowledge as well as the moral and political qualities, emphasizing cultural knowledge and a commitment for social justice and change, as well as pointing to organizational structures and collegial practices that support more equitable outcomes. Furthermore, the law’s emphasis on teachers and not teaching conflates teacher qualifications with quality teaching, which are not interchangeable. Organizational supports, colleagues and other resources contribute to what is possible instructionally, aside from an individual’s training.

In the end, the highly qualified teacher clause sets a minimum standard, but given the pragmatic demands of legislation, it falls short of more professionalizing notions of teacher quality. There is a huge difference between competence in rote instruction and what has been termed “ambitious instruction”  ––  teaching that strives to include as many children as possible in rich forms of content.

There is not an obvious mapping between the qualifications outlined in NCLB and ambitious instruction. Although the research on teacher quality following the Coleman report related teacher characteristics to student outcomes, ambitious teaching aims for a higher bar. Even under the best versions of traditional instruction, not enough children learned in ways that would support them in developing robust understandings of content. In one study investigating links between “highly qualified” teachers and more effective instruction, neither certification nor formal education within a subject predicted the use of ambitious methods; however, the combination of subject training and subject-specific pedagogical training did.

More than Reflection: How Teachers Learn from Each Other

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

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

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

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

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

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

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

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

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

I guess this is why good teaching is so hard.