The Best of the #MTBoS is Here!

Well, Tina and I didn’t plan it this way, but the publishing gods have given our little project a Pi Day release! We told you about it here, and now you can get your own copy for a good cause. 

Last year at Twitter Math Camp (TMC), participants were asked whether they paid their own way. Here was the show of hands:

(📷: @_levi_)

We know that there are some fantastic math teachers out there for whom the cost of TMC is prohibitive. We would love to have you join the fun and learning!

So please purchase a copy, give them as gifts to your colleagues, and re-tweet, re-post, and share!

Online preview (intro, table of contents with titles only, index, glossary):

Direct purchase (more of the money goes to the scholarship if you buy from here):

Amazon Paperback:

Amazon Kindle:


The Best of the Math Teacher Blogs 2015

It’s never been easier to miss a great math blog post. The MathTwitterBlogoSphere –– known as #MTBoS around social media –– was once a small group of math teachers willing to make themselves vulnerable, putting their practice online. As the community has expanded, even the most dedicated readers struggle to keep up with the deluge of thoughtful commentary, engaging and interesting tasks, and stories that we can all learn from.

To help keep you from missing out, we have compiled some favorite posts from this past year, as nominated by #MTBoS folks on Twitter, into a book. These posts are as rich and varied as the educators who wrote them. Some delve into specific content. Some tell stories of change and growth. Others explore teaching practices, new or well established. We hope that you find some that provoke and push you, and others that make you smile. Most of all, we hope you make some new connections in the MTBoS community.

This book has another purpose as well. Since 2012, folks from the MTBoS have participated in an annual “tweet up,” a two-day math extravaganza called Twitter Math Camp (TMC). Unlike regular conferences, teachers come knowing who they want to meet. They come to continue conversations that have been taking place online, through blogs and twitter. TMC is a rich and personal learning environment. The grassroots nature of TMC means it is lively, personal, tailor-made, and unpredictable. However, most teachers have to pay their own way. We will use the money raised through sales of this book to start a fund to bring along some of the teachers who would not otherwise be able to participate. We think that TMC is a unique professional learning experience, and we hope to share it while we grow our community.

The book is nearly ready for publication, but we need assistance with a few tasks (we’d like to add an index and list embedded links at the bottom of each post so they’re accessible to anyone reading a paper copy). If you’re interested in assisting please email Tina (tina.cardone1 on gmail) and she’ll get you set up with a task.

Thank you for reading, and thank you for your support.

— Lani Horn & Tina Cardone
P.S. Sorry that we were super secret on this project! We didn’t decide to do this until after the #MTBoS2015 conversation started. We were so impressed by the quality of the nominated posts, it seemed like a great opportunity to do something for this amazing community. As long as we are confessing, we also didn’t announce it until now because we weren’t sure we’d be able to finish it! If people like the idea then we’ll have a more public and organized process for 2016.

#TMC15 Reflection: Gratitude

Last week, I had the pleasure of joining 200 math educators for Twitter Math Camp (#TMC15) at Harvey Mudd.

TMC is a place with a lot of heart: part reunion, part meet up, and a whole lot of hugs and mathy goodness. Most everybody travels on their own dime. They come because they want to connect to people who have sustained them and helped them grow as teachers. They want to deepen their mathematical knowledge and expand their teaching toolkit, alongside people of goodwill.

Heart. Many of us connected to Christopher Danielson‘s admonition:

Find what you love. Do more of that.

TMC was like a re-set on me for connecting to my purpose.

And I realize that what I love is being with really thoughtful and passionate teachers. So I am grateful for that. I felt recharged after having the chance to attend workshops and learn alongside everyone. I also made some great connections to thoughtful research colleagues. We are already scheming.

Heart. Like when Fawn Nguyen made us both laugh and cry, describing what she has learned after 25 years of teaching.

I also had a chance to give a keynote. It was about how teachers can use social media to grow their own practice. I have studied math teachers’ learning extensively, mostly by listening to them talk with colleagues. I challenged myself to think about how to apply what I have learned in real life professional communities to the online space known as #MTBoS (which, I learned, we can say aloud as “mit-boss”).

Here is a link to my slides. I don’t know how much it will make sense as a slideshow. I am trying to track down the guy with the video camera in the third row so you can hear me.

So thank you to everyone who organized #TMC15, especially Lisa Henry, who knows how to build community like nobody’s business. Thank you to everybody who participated, both IRL and virtually. I look forward to continuing to learn with you.

UPDATE 1: Here is the YouTube of my talk (Part 1 and Part 2 — thanks Richard Villanueva! You can also see Fawn and Christopher’s talks on the same playlist.)

UPDATE 2: Here is a googledoc started by Jonathan Newman for us to put in common teaching problems, along with unproductive framings vs. actionable framings of those problems.

What are the Grand Challenges in Mathematics Education

Back in March, the National Council of Teachers of Mathematics put out a call for Grand Challenges in Mathematics Education.

A Grand Challenge is supposed to spur the field by providing a focus for research. NCTM came up with the following criteria for a Grand Challenge in math education:

Research Commentary-Grand Challenges_1

So I ask you to help the brainstorm. What are the complex yet solvable problems we face in mathematics education that can have a great impact on people’s lives?

Add your thoughts in the comments below or through Twitter (@tchmathculture). Use the hashtag #NCTMGrandChallenge.

Facilitating Conversations about Student Data

Sometimes, you ask and the internet answers. My research team and I (doctoral students Britnie Kane, Jonee Wilson and Jason Brasel and postdoctoral fellow Mollie Appelgate) wrote this a couple of years ago at the request of one of our district partners. We have been studying how teachers learn through collaborative time.

This memo focuses on research-based ideas on how to support common planning time so that it has the greatest potential for teacher learning about ambitious mathematics teaching. To that end, we provide a framework for effective conversations about mathematics teaching and learning. We develop the framework by using vignettes that show examples of stronger and weaker teacher collaboration.

We use vignettes for a couple of reasons. Primarily, we are concerned about the confidentiality agreements we have with study participants, which protect their right to remain unidentified. Additionally, sometimes raw conversational data takes ramping up to understand. Details about particular group histories or the nature of the problems they are looking at that do not communicate well in brief excerpts. The vignettes are clear illustrations of key ideas that also protect our participants’ confidentiality.

Part I: A Brief Conceptual Framework for Understanding Teacher Collaboration

Our research centers on how teachers learn about important instructional issues through their collaborative time. Based on our work in MIST as well as previous research, we have found that teacher workgroups’ discussions are richest when they include rich depictions of connections across students’ thinking, teaching, and mathematics.


The Instructional Triangle

These are three critical aspects of teaching that are frequently represented as Schwab’s Instructional Triangle (see above). Rich collaborative discussions draw upon and make connections among the three elements of the instructional triangle. For instance, teachers can consider how students’ understandings of particular mathematical ideas can be drawn out and developed through the design of a particular lesson. Notice how this example accounts for relationships at each point of the triangle. As we will elaborate, the consideration of multiple dimensions of classroom teaching makes such a conversation richer for learning than, say, one that solely focuses on what mathematics comes next in the curriculum, without accounting for the particulars of students’ thinking or other lesson details.

Sometimes, it is assumed that doing certain types of activities will lead to better learning opportunities for teachers. For example, the very name “common planning time” implies that planning should be a central activity, perhaps with an assumption that co-planning is more important than looking at student work. One important finding in our research is that activities in themselves are not richer in learning opportunities. In other words, there are versions of “planning” that are replete with teacher learning opportunities and there are versions that have few of them. Likewise, there are strong and weak versions of “looking at student data” or “looking at student work.” In the following sections, we provide examples of strong and less-developed collaborative conversations, along with commentary to help you all make sense of these. It is our aim to both illustrate this point and fill out our notion of rich teacher conversations.

Part II: What does rich teacher collaborative talk sound like?

In our data, we often see teacher workgroups participating in three different activities: planning (which teachers seem to find most useful, since they have to plan anyway), looking at student data (which administrators often encourage because of accountability pressures), and looking at student work. We will provide stronger and weaker examples of “looking at student data.”

Vignette 1: Rich Talk about Student Data

In this vignette, three teachers are discussing their students’ interim assessment results. In front of them, they have the test booklet and their school’s distribution of student responses. Using these materials, they have been looking at which questions were frequently missed and then looking at the items to make sense of what their students struggled with. Their conversation included the following discussion about a problem involving supplementary angles.

Maricela: On this one I think our kids are having a hard time with this because it asks for supplementary angles but the angles are next to one another. And that is not what the kids are used to seeing.

Diane: Yeah, that’s how I showed ‘em too.

Marcus: Exactly. I think it was confusing to them because they were looking for angles that butted right up next to each other and obviously, on this one, there is not a straight line at the bottom which would say supplementary to them.

Diane: I don’t think this was so much about not understanding what supplementary meant as…

Maricela: No, I agree… which is frustrating that they would understand what it meant but still miss it just because of the picture.

Diane:  So how can we teach this differently next time?

Maricela: We should probably use different ways to represent the supplementary angles.

Diane: Yeah, not always using the straight line and asking, “What is the supplementary angle?” but also just drawing an angle.

Maricela: Or even stressing that the definition is really adding up to 180 degrees, that the angles don’t all have to be together to be supplementary.

Commentary: This conversation provides teachers with a rich opportunity to learn from the assessment data. Their conversation integrates student understanding, the mathematics, and the implications for teaching. Their discussion of the student understanding of supplementary takes the reason for the error into account. Specifically, the data push them to think about how they have been teaching about supplementary angles in their classrooms. When Diana asks about how they could teach this differently next time, all three suggest ways they could be more versatile in both their representations of supplementary and their use of the definition. In this exchange, they integrate students’ thinking, the mathematics, and how they should adjust their teaching to help the two come together more effectively. This is similar to rich talk about planning but the teachers are making connections between the test results and making sense of what it tells them about these critical aspects of teaching.

Vignette 2: Weak Talk about Student Data

In this vignette a group of teachers are looking at interim assessment scores and they are asked to label each student as commended, passing, bubble or growth based on what the teachers feel the students have the potential for earning on the state-wide test.





After the teachers have labeled each student they review their numbers.


Okay, looking at “commended,” I have 0%. “Passing” I believe I only have about 20%. Bubble kids need that extra little help – that’s 50% And 30% on “growth.” Of those, that 30%, a fifth failed it last year.

I have about 33% “commended,” 17% should pass, 30% that are borderline with a little help could probably be passing, and then one or two students not.

Commentary: The majority of the 45-minute meeting was devoted to this activity and the conversation that followed. While this may be a useful administrative activity, there are few opportunities for the teachers to consider the relationships among student thinking, mathematics, and teaching. Instead, the teachers focus on the distribution of students in the different NCLB categories.

To make this activity richer, it would help to connect the data to the particulars of instruction, student thinking and mathematics. While this may help teachers recognize their students’ progress and which students might need extra support, there is little in this conversation that would help teachers to think more deeply about their teaching or change their instruction. Even looking for patterns in what topics are frequently missed, as the teachers did in Vignette 1, would get closer to this goal.

Summary of Vignettes 1 and 2: What makes for more productive discussions about student data

To move discussions about student data from a weak to a strong activity, with richer opportunities for teacher learning, ensure that data conversations discuss and make explicit connections between student thinking, the mathematics of the questions and the implications for instruction.

Below are some questions that may help to productively discuss data and more clearly make connections between student understandings, mathematics and instruction:

  • Making sense of the data: What did we learn about student understanding on a particular math topic from looking at this data? What trends in student understanding do we notice?
  • Thinking back on previous instruction: What are students thinking about the math to have answered in this way? How might our instruction led them to think this way?
  • Thinking ahead to subsequent instruction: How should we consider adjusting our instruction to address what we found for this group of students? Why would that work? How can we address these issues in student thinking when we teach it next time?

When teachers look at student performance data, learning opportunities will be richer if the teachers have to resources for looking at overall trends alongside the details about mathematical topics, individual students, or both. These details allow teachers to delve more deeply into the connections among what they know about student thinking, the mathematics and their instruction.

Sometimes administrative activities, such as those in Vignette 4 must happen. However, it is important that these take up a minimum amount of time or that the information garnered from such analysis gets taken up later to develop connections across mathematics, student thinking, and instruction.

Part III: Facilitation

As the examples in Part 2 illustrate, conversations that are richer for teachers’ learning build connections across teaching, students’ thinking, and mathematics. Sometimes, we have found that facilitators can help support these kinds of conversations. In other words, the facilitator’s job is to support teachers in connecting the three elements of the instructional triangle –– and to do so as specifically as possible.

One challenge of teacher collaboration is that some critical aspects of teaching –– students and the classroom interaction –– are not available to examine together. Good facilitators come up with strategies to help teacher groups get “on the same page” about some issue in teaching. Sometimes they do this by re-enacting the voices of students and teachers in the classroom to creating shared images of actual classroom talk. Once the teachers have some shared image of the issue, they have to work together to make sense of it together. To this end, good facilitators ask teachers to provide rationales for instructional decisions that they make (e.g. “so, what are students learning in doing this for homework?” “How does this activity help students in thinking about and understanding the idea of what volume is, beyond memorizing the formula?”)

Good facilitators also support teacher engagement. They do this in several ways. First, they build supportive relationships with individual teachers, identifying their strengths and coming up with reasonable next steps for their professional growth. When teachers are engaged, they participate more readily in conversations. Of course, when teachers share their ideas honestly, there is greater potential for conflict. Good facilitators make a safe space for learning, respectfully listening to different ideas while continuing to press for deeper understandings about teaching, students, and mathematics.

In summary, good facilitators:

  • Get teachers on the same page about some important questions in teaching.
  • Press teachers to explain their pedagogical reasoning.
  • Link instructional issues to clear statements that connect teaching, students, and mathematics.
  • Support individual teacher engagement and development.
  • Develop norms for honest but respectful conversations.

As we did with our framework for teacher conversations, we will develop our notion of good facilitation through vignettes that show facilitators of different skills. As with the other vignettes, these are based on our data but have been cleaned up for reasons of clarity and confidentiality.

Vignette 3: Sophisticated facilitation

In this vignette, two teachers, Jack and Soledad, work with and Coach Rachel. The team works to plan a launch for the following day’s lesson. Coach Rachel begins by asking teachers to make connections among instruction, mathematics, and student thinking:

Coach Rachel:          Ok. So, would you look at the book’s lesson on place value, and decide what you think the kids have to know in order to be able to do it?

Soledad:                    They definitely have to know exponents, which is scary, because we haven’t done exponents yet. See how it says “10 to the—”

Jack:                         Yeah. Neither have we. I hate how this book skips around. Like, my kids don’t get exponents yet. Why can’t we stick to place value if 2.3 is about place value?

Coach Rachel:          Ok. I hear ya—you guys are worried about the exponents. Let’s pretend we’re students, and we have a shaky understanding of exponents. How else could we approach this problem?

Jack:                         They could. Um. They could use the idea of multiples of ten—or, you know, like what an exponent actually means. Like ten to the first is ten times one, ten squared is ten times ten, you know…

Soledad:                    Oh! I see what you’re getting at, you sneaky thing. ((laughs)). You’re saying it has to do with the, like, base ten?

Jack:                         Like, 10 times 1 is 10 and 10 times 10 is 100? Ok, so how can we connect that to place value for them? Because that’s tough.

Coach Rachel:          Yeah. You’re right, it is. Um. So, the idea is that place value stands for an order of ten, right. That’s what we need kids to understand in order to be able to do this problem.

Soledad:                    Yep. Especially when we get into decimal numbers. My kids get really freaked out by decimal numbers.

Jack:                         Right, so how can we launch this so kids get that?

Soledad:                    Ok. So, what if we use money. Like, kids get money. Right? Like pennies, dimes, dollars, you know…

Commentary: Coach Rachel’s work in this vignette illustrates some of the important qualities of effective facilitation. To get the teachers on the same page about their lesson planning, the group works together from the textbook and teacher guide. There is a positive, supportive, and honest tone in the conversation. Jack does not hesitate to share his frustration with the curriculum ( “Why can’t we stick to place value if 2.3 is about place value”), and Coach Rachel uses it as a way to connect the topic of place value to their concerns about the exponents. She is respectful of this concern (“I hear ya—you guys are worried about the exponents”) but manages to redirect the group so that they consider students’ perspectives and new thinking about mathematics. This is a critical move: the conversation could easily devolve into a gripe session about the curriculum, but she brings it back to the territory of the Instructional Triangle we introduced in the introduction of this memo. She does this by building an additional representation of the classroom, getting them further on the same page, asking the teachers to pretend like they are students and to think of alternate mathematical approaches to the work. Once teachers begin to reconceptualize the task from students’ perspective, Coach Rachel then marks exactly what students need to be able to see. The summary statement she provides links teaching, students, and mathematics. In the end, Coach Rachel helps teachers arrive at a specific instructional goal, based on a (re)consideration of student thinking and mathematics.

Vignette 4: Weak facilitation

In the following vignette, a group of teachers plans a launch on place value, using money as a jumping off point. They are using the same unit we heard about in Vignette 5, but it is a different teacher team and facilitator.

Coach Melissa:








Coach Melissa:



Coach Melissa:



Coach Melissa:

Ok. We’re going to role-play this launch. Tamara, will you play the teacher, and then, Trent—you’ll be the student.

Ms. White, I’m tired. ((laughs))

((laughs)) Ok. Um. So, what is this ((holds up a dime))?

A dime.

Right. So, how do we write that down?

Like this, “10¢.”

Nooo. Write it down the right way.

Well, that IS the right way ((crosses arms in front of chest)).

How about, “Like you’d see it on a price tag?”

I’ve seen price tags like that.

Ack! You’re just as frustrating as real students ((laughs)).

Ok. Go back to the role-play. Tamara, you’re the teacher.

Um. Ok. Are there any other ways to write it? ((Trent writes 0.10)). What happens if you multiply that by ten?

I don’t know, the decimal moved.

Why do you think the decimal moved?

Commentary: Coach Melissa does a number of things well in this interaction. Clearly, she has strong rapport with the teachers, who joke around and eagerly participate in the activity she has designed. The idea of the role-play has some potential to get the teachers on the same page about some issue in teaching. Nonetheless, we see this facilitation (and the meeting that surrounds it) as providing very few opportunities for teachers to learn about ambitious instruction. Drawing on our framework for rich teacher conversations, we see that few connections are made across teaching, students, and mathematics, and Coach Melissa does very little to press it in that direction.

Weak facilitation may result from focusing on any one of the three points of the instructional triangle, to the exclusion of the other two. This is an example of an over-emphasis on teaching with little consideration for mathematics or students. Strong facilitators often use role-plays, but effective role-plays allow inquiry into the connections between student thinking, teaching, and mathematics. Although Coach Melissa asks Trent to enact a student, the “student” gets very little air time, and “student” contributions are not taken up as meaningful: Coach Melissa revises Tamara’s question to “[Write it down] like you’d see it on a price tag,” but ignores the “student” objection to using decimal numbers or, as the following section points out, the mathematical import of that objection. Tamara playfully tells Trent that he is “as frustrating as real students,” and Coach Melissa urges a return to the role-play.

Less Attention to Student Thinking and Math: Coach Melissa overlooked an important moment for considering student thinking and mathematics in conjunction with teaching. We see this as a missed opportunity for teacher learning. For example, when Trent writes “10¢” and argues it IS the right way to write down the value of a dime (line 19), Coach Melissa could have led a discussion about student thinking: Why do students prefer to think of a dime as “10¢,” rather than 0.10? Perhaps they prefer whole numbers to decimal numbers? What is the relationship between the two ways of writing ten cents, and how does this connect back up to the unit topic of place value? Such a discussion would be relevant to a planning the lesson and would involve a richer consideration of student thinking and of math.  


Strong facilitation involves effectively working to build trust among group members so that teachers feel secure airing their confusions and struggles. It also involves connecting the three elements of the instructional triangle. In order to have rich opportunities to talk about connections between student thinking, teaching, and mathematics, the facilitator needs to help teachers get on the same page about important questions in teaching. To do this, the facilitator can press for pedagogical reasoning and ask teachers to re-enact student and teacher voices in order to create rich representations of students’ thinking. Because facilitating teachers’ opportunities to learn is complex work, it is also important for facilitators to make clear statements that tie together teachers’ representations of student thinking (which are often “impersonations” of students’ voices or examples of student work) with teachers’ understandings about the mathematics involved in particular lessons and, ultimately, the reasons for their instructional decisions.

Suggestions of ways for facilitators to press on teacher learning:

Here are some strong questions facilitators might ask:

  • What do you think students will need to understand in order to do this task?
  • How does (this activity) help students develop their understanding of (a key mathematical idea)?
  • What do you hope students will learn by (doing this activity/worksheet)?
  • What is the big idea that you want students to come away with from this lesson?
  • What happened in your classroom when you tried to do (a new teaching technique, for instance)?
  • What did students say in response?
  • What were students’ misconceptions?
  • Why do you think students had that misconception?
  • What led to students’ misconceptions? (Help teachers to focus on things over which they have control)
  • How can we address that misconception in our class next time? (“Re-teach it” is not a clear enough response—it doesn’t help teachers think about what they did last time or what they need to do differently next time.)

Concluding Thoughts

Over the years, research has repeatedly shown that teachers can benefit from professional interactions with other teachers. At its best, working with other teachers supports teachers to more deeply understand their work and prevents isolation. This is particularly important when teachers are attempting to try new practices, including moving toward more ambitious mathematics teaching. Our research has found that in collaborative conversations that allow teachers to make connections across student thinking, the mathematics being taught, and instruction have a greater potential to move their teaching closer to the ambitious instruction. Given the rigor of the current set of state assessments, students will need to access to this instruction to increase their chances of success.


Smith, Margaret, Victoria Bill, and Elizabeth K. Hughes. “Thinking Through a Lesson: Successfully Implementing High-Level Tasks.” Mathematics Teaching in the Middle School 14 (October 2008): 132–38

What I Notice and Wonder about Teaching Like a Champion

Last night, Chris Robinson shared an experience with an administrator who observed his math classroom. He had been doing an activity called Noticing and Wondering with his students, something that Max Ray of the Math Forum has written about extensively. Noticing and wondering is a great discussion starter. You share a mathematical object or situation with children and open up the floor to their curiosity. They can connect the mathematical thing with their own ideas, then a teacher can shape the conversation by building connections to formal math.

Here is the administrator’s feedback:

Now, I am not naive. I understand that our lack of consensus about good teaching leaves a lot of room for interpretation about what is working and what is not. The administrator was obviously perplexed by the wide berth Chris gave to his students to wonder about the math. Kids do say and think goofy things, as do all people. But sometimes our odd ideas need a good airing to connect to what we are learning.

Normally, seeing Chris’s tweet would frustrate me. What do we need to do to drive a wedge between people’s confusion about students being compliant and being engaged? What do we need to do to help educators understand that the path to deep understanding is often not a straight line, and that to connect ideas to our lives, our own thinking –– goofy or not –– needs a chance to come out?

Yesterday, however, the administrator’s problematic response did more than frustrate me. As I told Chris (and the others on the thread):

In my class Teaching as a Social Practice, we have been discussing the consequences of our lack of consensus on the nature of good teaching. We often examine what gets put out and circulated as good teaching and hold it against various research on things like  how kids learn or how teachers can teach responsively.

I showed this Doug Lemov video related to his best-selling book, Teach Like a Champion, with the intent to dissect the underlying assumptions about teaching and learning. The 100% technique is a way of managing students’ attention during instruction. Take two minutes to watch it.

What do you notice? What do you wonder?

I notice that these are all White teachers and that the students are nearly all Black.

fold hands

I wonder why the teacher (above) is signalling this boy to have his hands folded. I wonder if there is any research anywhere showing that folded hands will help with his learning.

Whisper to Jasmin

I notice that when this teacher reprimands this student for not having the answer to a question (1:11 on the video), she jumps immediately to the assumption that the girl needs to work harder. I wonder why the teacher doesn’t ask her if she has any questions about what was being asked or if everything is okay today.

Giving you a gift

I notice that this teacher says the following to his class as a motivational speech (1:44):

I can bring it to you but I can’t give it to you. You’ve got to reach for it. If they were free at Toys R Us you would reach. I’m giving you the same kind of gift, just not wrapped up. The gift of knowledge.

I wonder what is going on in this metaphor. I am wondering if I ever have seen wrapped up gifts at Toys R Us. I wonder if other overly analytical kids in this class also got lost down this rabbit hole of wondering.

I wonder if the kids would like the gift of being able to keep their hands unfolded and moving their bodies more freely more than the gift of repeating after the teacher in the name of “knowledge.”


What does all this have to do with Chris and his interaction with his administrator?

Teach Like a Champion has been a huge seller, especially in urban schools. It’s highly rated and ranked on Amazon and I have talked to numerous new teachers who report getting handed a copy by administrators. There is even a new edition Champion 2.0.

Activities like noticing and wondering open up classroom discussions and invite kids (goofy ideas and all) to think. Techniques like 100% in Teach Like a Champion limit permissible activity and thinking by students.  Contrasting the two is a productive microcosm on current debates about teaching. The issue is particularly urgent in urban classrooms, where methods like those promoted in Champion emphasize the control of Black and Brown bodies by White teachers instead of the celebration of children’s own ideas. This is especially troubling given what we know about disproportionate discipline of these children.

With this vision of teaching dominating the landscape, it becomes increasingly difficult for teachers like Chris Robinson to invite their children to think with him in the classroom without the risk of being reprimanded.

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.


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.

Global Math Department Talk

Last night, I had the great honor of chatting up some #MTBoS teachers via the Global Math Department. I was asked to discuss mathematics education research. (You can watch the hour long discussion here.)

Part of what I get from the #MTBoS is the chance to engage with people who are personally and professionally invested in the things I care about. What a gift.


And no one blinked when this guy burst in unexpectedly to say goodnight.
Another reason to love teachers.

I promised folks I would link to some of my work that I mentioned over the course of the talk, so here goes. I am just linking to one of what are usually several papers from the different studies. In some cases, I link you to blog posts or, in the last case, the project website.

If you want to know more citations for any of these, just leave me a comment and I will provide further links. Or you can poke around on my page if you are not comment-inclined.

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.