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.

Why Meaningful Math Learning Matters

What Meaningfulness Means

Learning and schooling are not the same thing. There are children who are great learners but terrible students. These young people are full of ideas and questions, but they have not managed to connect their innate curiosity with their experiences in school. There are many possible reasons for this. Children may find school to be a hard place to inhabit, due to invisible expectations that leave them feeling alienated. Sometimes, school curriculum just seems irrelevant: their personal questions about the world do not find inroads in the work they are asked to do.
Although many parenting books extol children’s natural curiosity and emphasize its importance in their learning and development, schooling too often emphasizes compliance over curiosity. Thus, it is not surprising that children who are great learners and weak students have their antithesis: children who are great students but who are less invested in learning and sense making. Make no mistake: these students hit every mark of good organization, compliance, diligence, and timely work production, but they do not seek deep engagement with ideas. Given the freedom to develop a question or explore an idea, they balk and ask for more explicit directions. I have heard teachers refer to these children as “teacher-dependent.”
Too often, meaningfulness falls through this gap between learning and schooling. There is a fundamental contradiction at play: meaningfulness arises from and connects to children’s curiosity, yet “curious children” is not entirely synonymous with “successful students.” Meaningfulness comes about when students develop an appreciation for mathematical ideas. Rich and meaningful learning happens when students draw on prior knowledge and experiences to make sense of ideas and explore problems, invoke their own strategies, get to ask “what if…?”  In short, meaningful learning happens when students’ activity connects to their own curiosity. To make meaningfulness central to math teaching, then, teachers need to narrow the gap between being curious and being a good student.

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Meaningfulness: When students connect their own curiosity and experience to ideas, thereby developing an interest in and appreciation for mathematical content.
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Why Meaningfulness Matters

Every math teacher, at one one time or another, has been asked the question, “When are we going to use this?” While this question often gets cast as students’ resistance to learning, it can be productively reinterpreted as a plea for meaningfulness. When the hidden curriculum of math class –– the messages that are inadvertently relayed through classroom organization and activity –– consistently communicates that meaning does not matter, we end up with hordes of students who no longer reason when they are doing math. They instead focus on rituals, such as following the worked example, and cues, such as applying the last learned procedure to the current problem.
As researcher Sheila Tobias explained in her classic exploration of math anxiety, a lack of meaning exacerbates many students’ negative experiences learning mathematics. When math class emphasizes rituals and cues that rely on memorization over sense making, students’ own interpretations become worthless.

For instance, they memorize multiplication facts, and, in a search for meaning, they decide that multiplication makes things bigger. Then, they learn how to multiply numbers between 0 and 1. Their prior understanding of multiplication no longer works, so they might settle on the idea that mulitiplication intensifies numbers since it makes these fractional quantities even smaller. Finally, when they learn how to multiply negative numbers, all their ideas about multiplication become meaningless, leaving them completely at sea in their sense making. The inability to make meaning out of procedures leaves students grasping and anxious, as the procedures seem ever more arbitrary.
In contrast, when classrooms are geared toward supporting mathematical sense making, they reap multiple motivational benefits. First, students’ sense of ownership over their learning increases. Students see that multiplication can be thought of as repeated addition, the dimensions of a rectangle as related to its area, or the inverse of division. When they learn new types of multiplication, the procedures have a conceptual basis to expand on. Relatedly, their learning is more durable. Because they understand the meaning behind the mathematics they are learning, they are more likely to connect it to their own experiences. This, in turn, provides openings for their curiosity and questions. Beyond giving students opportunities for sense making, meaningful mathematics classrooms provide students chances to identify and explore their own problems. Indeed, in a systematic comparison of teacher-guided and student-driven problem solving, educational researchers Tesha Sengupta-Irving and Noel Enyedy found that the ownership, relevance, and opportunities to engage curiosity in student-driven problem solving supported stronger outcomes in student affect and engagement.[1]

The challenge, then, for teachers is how to help students engage in meaningful mathematical learning within the structures of schooling. I would love to hear your ideas about how to achieve this.


[1] Tesha Sengupta-Irving & Noel Enyedy (2014): Why Engaging in Mathematical Practices May Explain Stronger Outcomes in Affect and Engagement: Comparing Student-Driven With Highly Guided Inquiry, Journal of the Learning Sciences, DOI: 10.1080/10508406.2014.928214

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.

 

How Does School Culture Reflect Middle Class Culture?

Class is rarely talked about in the United States; nowhere is there a more intense silence about the reality of class differences than in educational settings.

bell hooks

One of the things teachers often hear in the course of teacher education is that school culture typically reflects middle class culture. For teachers who grew up middle class, this statement can be perplexing. It’s like trying to alert fishes to the unique presence of water: they are so immersed in it that alternatives cannot be fully imagined.

Yet class shapes everything from interactional styles to the kinds of competencies valued in the home. In her famous ethnography of class and American childhoods, Annette Lareau characterized working class and poor families as tending to promote natural growth in children. These parents tend to let children determine their leisure activities. When they interject authority, they tend to do so with directives.

lareau cover

In contrast, middle-class families tended to practice a form of parenting Lareau calls concerted cultivation. These parents tended to equate good parenting with deliberate development of their children’s talents, especially through organized leisure activities. They also used fewer directives, instead reasoning with their children when seeking to change their behavior.

(There are other contrasts between these approaches to parenting, as summarized in this table.)

Lareau’s point is not that one style is better than the other, but instead to point out that school often assumes middle class parenting, leaving poor and working class families with less of an institutional fit. In fact, as somebody who was raised in this manner, I personally see many strengths that come out of the accomplishment of natural growth. Children have more opportunities to develop autonomy and engage in more social problem solving than children whose leisure activities are organized and led by adults.

How do these middle class assumptions play out in school? Classrooms are crowded places, and teachers frequently need to direct children’s attention and activities. Many teachers tend toward the middle class style of suggesting a transition (“Would you like to join us on the rug?”) rather than directing it (“Please come to the rug now”). If you are used to the latter, the former can be understandably ambiguous and confusing.

What is more, middle class children, through their greater experience with formally organized leisure activities, usually come to school with tacit understandings about how to participate. They have more experience responding to the authority of a non-kin adult with whom they will likely form a superficial and transitory relationship. In contrast, if your early socialization has been primarily with family, taking directions from a stranger may seem like a strange and maybe not entirely wise endeavor.

There are also subject-specific ways that social class makes school more or less a fit with children. Valerie Walkerdine has documented the ways class can interact with mathematics education in particular. She points to the quantitative fictions common to math class, describing, for example, an elementary number game requiring the “purchase” of various items for 1 to 10 pence and then making change. The working class children she observed, whose lives were much more consequentially tied to actual prices of things, found the premise of the game absurd. As I often tell my pre-service teachers, which of your students knows where to find the best price on a gallon of milk, and which simply look to make sure it’s organic? How does that change your job in making sure the cost in your word problem is realistic?

To feel comfortable participating in classrooms, children need to have a reason to be there. They need to see a connection to their lives and experience a sense of belonging. Social class differences are sometimes the source of cultural barriers to feeling like you belong in school, that school is a place that matters, that things make sense. Teachers need to be thoughtful in how they bridge these differences with their students.

First, Do No Harm

I have often wondered if teachers should have some form of a Hippocratic Oath, reminding themselves each day to first, do no harm.

Since the network of relationships in classrooms is so complex, it is often difficult to discern what we may do that causes children harm. Most of us have experienced the uncertainty of teaching, those dilemmas endemic to the classroom. Was it the right decision to stay firm on an assignment deadline for the child who always seems to misplace things, after giving several extensions? Or was there something more going on outside of the classroom that would alter that decision? Why did a student, who is usually amendable to playful teasing, suddenly storm out of the room today in the wake of such an interaction?

What I have arrived at is that there are levels of harm. The harm I describe in the previous examples can be recovered if teachers have relational competence — that is, the lines of communication are open with their students so that children can share and speak up if a teacher missteps.

What I am coming to realize is that mathematics teachers have a particular responsibility when it comes to doing no harm. Mathematics, for better or worse, is our culture’s stand-in subject for being smart. That is, if you are good at math, you must be smart. If you are not good at math, you are not truly smart.

I am not saying I believe that, but it is a popular message. I meet accomplished adults all the time who confess their insecurities stemming from their poor performance in mathematics classes.

Here is an incomplete list of common instructional practices that, in my view, do harm to students’ sense of competence:

1. Timed math tests

Our assessments communicate to students what we value. Jo Boaler recently wrote about the problems with these in terms of mathematical learning. Students who do well on these tend to see connections across the facts, while students who struggle tend not to. But if timed tests are the primary mode of assessment, then the students who struggle do not get many opportunities to develop those connections.

2. Not giving partial credit

Silly mistakes are par for the course in the course of demanding problem solving. Teachers who only use multiple choice tests or auto-grading do not get an opportunity to see students’ thinking. A wrong answer does not always indicate entirely wrong thinking. Students who are prone to getting the big idea and missing the details are regularly demoralized in mathematics classes.

Even worse, however, is …

3. Arbitrary grading that discounts sensemaking

Recently, a student I know had a construction quiz in a geometry class. The teacher marked her construction as “wrong” because she made her arcs below the line instead of above it, as the teacher had demonstrated. This teacher also counts answers as incorrect if the SAS Theorem is written as the SAS Postulate in proofs. Since different textbooks often name triangle congruence properties differently, this is an arbitrary distinction. This practice harms students by valuing imitation over sensemaking.

4. Moving the lesson along the path of “right answers”

Picture the following interaction:

Teacher:    “Can anyone tell me which is the vertical angle here?”

Layla:        “Angle C?”

Teacher:     “No. Robbie?”
Robbie:       “Angle D?”

Teacher: Yes. So now we know that Angle D also equals 38˚…

That type of interaction, called initiation-response-evaluation, is the most common format of mathematical talk in classrooms. Why is it potentially harmful? Let’s think about what Layla learned. She learned that she was wrong and, if she was listening, she learned that Angle D was the correct answer. However, she never got explicit instruction on why Angle C was incorrect. Over time, students like Layla often withdraw their participation from classroom discussions.

On the other hand, teachers who work with Layla’s incorrect answer –– or even better yet, value it as a good “non-example” to develop the class’s understanding of vertical angles –– increase student participation and mathematical confidence. And, they are doing more to grow everybody’s understanding.

What are other kinds of teaching practices that stand to “harm” students?

Relational Density in the Classroom

Recently, Michael Pershan has been thinking about why it’s so hard for teachers to share knowledge and ideas. He has been playing with building cases to discuss as teachers, wondering about what counts as sufficient description to invite consultation.

In my work, I find that one of the challenges to building shared professional knowledge comes from the irreducible situativity of teaching. If that sounds like an academic mouthful, my apologies. But what I mean is that we can’t escape how much of what works in teaching comes out of nuances of our practice and resources in our context that we may not even be aware of. Just as fish don’t see the water they swim in, so too teachers often miss things like community norms or material resources that shape what is possible in the classroom.

In addition, I think the relational part of teaching has been understudied –– especially in mathematics education. As I have said before, asking students to share their thinking is a socially risky proposition and depends on the relationships in the classroom and the norms for participation.

Here is Courtney Cazden on this:

“In more traditional classrooms, social relationships are extracurricular, potential noise in the instructional system and interference with ‘real’ schoolwork. What counts are relationships between the teacher and each student as an individual, both in whole-class lessons and in individual seat-work assignments. In nontraditional classrooms, the situation has fundamentally changed. Now each student becomes a significant part of the official learning environment for all the others” (2001, p. 131)

So to get students to share their ideas, teachers have to attend not only to their individual relationships with students, but to students’ relationships with each other.

This is decidedly challenging work. Most classroom teaching situations exhibit tremendous relational density. As Philip Jackson observed decades ago, classrooms are among the most crowded of institutional settings. In order to function, they require some degree of cooperation from the students. Teachers often achieve that through setting up systems of compliance, by building relationships with students, or some combination of the two.

Although students who have an instrumental view of schooling are less dependent on a teacher’s relational skills, a teacher’s success often depends on engaging and shaping students’ sense of purpose.

But the relationships in the classroom do not simply exist between the teacher and students; they exist among the students themselves. Once we take this into account, the social complexity of the classroom is stunning. Instead of just seeing the relationship one teacher builds with each student, we must account for the combinations of relationships among the students themselves. As a consequence, the difference between having 16 students or 32 students in a classroom does not simply double the relational density of a classroom: each set of students has potential for harmony or conflict. Just considering the smaller class of 16, there are 120 possible pairings between students. In the larger class of 32, there are 496. The number of students only doubled, but the relational complexity has more than tripled.

CocktailPartyGraph_700

Figure 1. Student pair-wise relationships grow quadratically while the class size grows linearly. The red dots represent students, and the connecting lines represent potential relationships.The last diagram represents the relationships among 5 students and a teacher, illustrating the fast growing relational density with every added student.

Relational density serves as a backdrop of potentialities in classrooms: not all relationships are actively engaged. When I talk to experienced teachers, however, I notice that they are alert to the relational potentials across the classroom social network, usually framing them as classroom dynamics.

Returning to Michael Pershan’s question, how do we adequately capture these dynamics when we describe our teaching situations? Some teachers talk about the kids with “strong personalities” or “the quiet kids.” I have heard teachers talk about students who are hot spots in the classroom relational network: most other students have an active experience of liking or disliking them. These experienced teachers respond by building lessons with their hotspot students in mind, anticipating possibly corrosive behavior or harnessing potential leadership.

Obviously, not all teachers attend to classroom dynamics in this way. Whether or not these dynamics are  on a teacher’s radar, they contribute to the situativity of teaching. That is, we can’t really talk about teaching without addressing some of these particulars. Inattention to details of a teaching situation leads to invisibility in critical aspects of the work. This makes knowledge sharing hard.

So the question is: what sufficiently describes the character and dynamics of one situation to help teachers productively compare it to another? Often, teachers fall into language that relies on stereotyped understandings: an urban school, an honors class, an ADHD kid. These everyday categories stand in for broader dynamics but, in my view, do not adequately describe teaching situations.

Yet leaving critical dimensions of teaching situations underspecified contributes to the lack of consensus around expertise. What constitutes successful teaching remains hotly contested, evidenced by policy debates around standardized testing and value-added models of teaching. Grossly underdescribing teaching situations has led to an overdetermination of desirable, visible outcomes like test scores. In this way, invisibility creates a reliance on other kinds of representations of the work when communicating about instruction.

“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!