Laying the Groundwork for Logarithms


Strangely, I have had occasion to do a few tutoring sessions with different kids recently around exponential and logarithmic functions.

This particular mistake set off a few alarm bells:



Do you see what the student is doing here? She is treating

log a

like a variable that is being divided instead of a function.

I looked at the student’s notes, and all the usual log laws were there. But she did not yet have the unshakable understanding that logs are functions. I realized that there are some foundational ideas that she needed before we could really make sense of all of this.

Here are a couple of essential ideas I want to communicate to students about logarithm functions.

First, functions can be described as actions, so I always make students explain what a function is doing.

The question you should ask about every function is: what are we doing to the input to get to the output? I call it “saying the function in English.”

Since we usually teach logarithms after exponential functions, let me start with them.

I ask, What do exponential functions do? They provide rules based on repeated multiplication. So the function


tells us that “some number (y) equals 2 multiplied by itself any number of (x)  times” to get y. We can do this with different examples, talk about how the function grows, look at the graph, look at tables, compare the growth of exponential functions to linear and quadratic ones. My goal is to get kids to have a feel for what is happening with exponential growth so well that when somebody says, “It was growing exponentially!” they can decide whether that is an accurate statement or not.

This is the first part of the groundwork for understanding logarithms.

Second, remember that anything we do in mathematics, we always find ways to undo.

This is thematic in all of mathematics. It becomes a chant when I teach math.
I say to students:
“Since this is math, anything we learn to do, we need to ….?”
They soon learn to respond with:

Doing-and-undoing is a good mathematical habit of mind to emphasize, because students start to anticipate that when we learn some new funky function or operation, an inverse is coming down the pike. They are not at all surprised to learn that trig functions have an inverse and so on.

In this case, since we have learned to exponentiate, they can guess we need to un-exponentiate.


That’s just how math works!

I like to show inverses of functions in all of the representations. The idea is the same in tables, graphs and equations: the x’s and y’s switch places.

For tables and graphs, it’s fairly easy for students to figure it out. But the algebra gets tricky. To find the inverse of the previous exponential, for example, we need to derive it from:

inverse 1

This immediately creates a mathematical need to “un-exponentiate.”

So when we want to solve that equation for y, let’s undo exponentiation with a function we call a logarithm. Logarithms undo exponentiation.

logging the inverse
Since the log undoes the exponentiation, we end up isolating the y.

this one!

I also tell them we read this as “log base two of x equals y.”

So when you see an equation like:


you are asking “2 to what power equals 8?” I have them practice explaining what different equations mean.

Now your students are ready to learn all the details of working with logs!

Tell me your ideas in the comments.

[Before I close, vaguely related Arrested Development reference:

bob loblaw

Because this is a log law blog. But I guess I don’t really want to talk about log laws. Anyways…]


Building Teaching as a Responsive Profession

Those of you who spend real or virtual time with me have heard me talk about how hard it is to talk about teaching.

One frequently mentioned issue is that, unlike other professions, teaching does not have its own technical language. Professions like aviation and medicine have common professional terms that highlight important features of critical situations and guide practice. In aviation, for instance, pilots identify wind patterns to aid in landing planes. Likewise, surgeons have cataloged human anatomy and surgical procedures so the protocol for appendectomies can be named and routinized, with appropriate modifications for anatomical variations such as hemophilia or obesity. But a strong headwind in China is similar to a strong headwind in Denmark; a hemophiliac in Brazil will require more or less the same modifications as a hemophiliac in Egypt.

In contrast, an urban school may not be the same as an urban school a few blocks away, nor an ADHD kid the same as an ADHD kid in the same classroom. Although such terms attempt to invite descriptions about particular teaching situations, the language often relies on stereotyped understandings. Everyday categories like an urban school, an honors class, or an ADHD kid seldom work to describe teaching situations adequately to help teachers address the challenges they face. Words characterizing social spaces and human traits are inherently ambiguous and situated in particular social, cultural and historical arrangements.

The variation teachers encounter cannot always be codified, as they often are in aviation and surgery. In fact, in the United States, when educational situations are codified, they often presume the “neutral” of White, English-speaking, and middle class culture. However, the widespread practice of glossing cultural particulars, or only seeing them as deviants from a norm, reduces teachers’ ability to teach well. From Shirley Brice Heath’s  seminal work comparing home literacy practices in White and African American communities to Annette Lareau’s identification of social class-specific parenting patterns, we see time and again that children from non-dominant groups frequently encounter schooling expectations that are incongruous with their home cultures, often to the detriment of their learning. Conversely, when instructional practices align with children’s home cultures, teachers more are more effective at cultivating students’ learning. (See, for a few well documented examples, this work by Kathryn Au and Alice Kawakami, Gloria Ladson-Billings, and Teresa McCarty.)

Culturally responsive pedagogies are, by definition, highly particular and have been documented to yield better student learning. To communicate sufficiently, professional language for teaching would need to encompass this complexity, avoiding simplistic –– perhaps common sense –– stereotypes about children, classrooms, schools, or communities.

How, then, can we develop shared professional language for teaching and build professionals responsive to the children they serve? I have some ideas I will share in another post.

Reinventing Mathematics Symposium at The Willows School

I am honored to be presenting tomorrow at the Reinventing Mathematics Symposium at the Willows School in Culver City, CA.

My workshop is on Playing with Mathematical Ideas: Strategies for Building a Positive Classroom Climate. Students often enter math class with fear and trepidation. Yet we know that effective teaching engages their ideas. How do we lower the social risk of getting students to share to help them understand mathematics more deeply? I will share what I have learned from accomplished mathematics teachers who regularly succeed at getting students to play with mathematical ideas as a way of making sense.

In my workshop, I will develop the concepts of status and smartness, as well as share an example of “playful problem solving.” Here is the Tony De Rose video we watched, with the question: How is Tony De Rose mathematically smart? If he were a 7th grader in your classroom, what chances would he have to show it?

Usually teachers like  resources, so I have compiled some here.


Bellos, A. & Harriss, E. (2015). Snowflake, Seashell, Star: Colouring Adventures in Wonderland. Canongate Books Ltd; Main edition

Childcraft Encyclopedia (1987). Mathemagic. World Book Incorporated.

Jacobs, H. (1982). Mathematics: A Human Endeavor. W.H. Freeman & Co Publishers.

Pappas, T. (1993). The Joy of Mathematics (2nd Edition). World Wide Publishing.

Van Hattum, S. (2015). Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. Natural Math

Weltman, A. (2015). This is Not a Maths Book: A Smart Art Activity Book. Ivy Press.

Blogs that Feature Playful Mathematics

Math in Your Feet Blog

Talking Math With Your Kids

Visual Patterns

Math Munch

Some Inspiring Ignite* Talks that Give Ideas about Teaching Playfully

*Ignite talks are 5 minute long presentation with 20 slides and with the slides advancing automatically every 15 seconds. It’s the presentation equivalent of a haiku or sonnet.

Peg Cagle, What Architecture Taught Me About Teaching

Justin Lanier, The Space Around the Bar

Jasmine Ma, Mathematics on the Move: Re-Placing Bodies in Mathematics

Max Ray, Look Mom! I’m a Mathematician

There are tons more. The Math Forum does a great job of getting outstanding math educators to share their work in this series of talks.

Please feel free to add other good resources in the comments section!


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.


Teachers’ Work Conditions

Today I was feeling chatty on twitter, so I wished everybody a good morning. It’s nice to hear about what is going on with folks, so it’s a pleasant way to start a day. I got several responses from people I was happy to hear from.

One exchange in particular got me thinking. At an early hour, where I still had one last child to bring to school, Tina Cardone had already attended an intense IEP meeting and faced off with complaining students.

In just a few tweets, Tina reminded me of some challenges of teaching, ones that are beyond the reach of teacher preparation or most education reforms: teachers’ work conditions. Most of the public debate about the profession skips the work conditions part (although there certainly are many discussions of teacher compensation).

An IEP meeting is usually an add-on to a teachers’ day. Teachers need to attend, both because they are legally beholden to IEPs but also to provide a team feedback on student. However, this time is not typically compensated. The teacher comes early, gives up a preparation period, or stays after school to attend an IEP meeitng.

Aggrieved students can be an emotional drain, as a teacher can find herself defending her professional judgement about something  — a grade, an assignment, a grouping arrangement — to a group of young people who may not see the big picture of her work.

Finally, Tina threw in the bit about her “lunch” time being scheduled for 10:30 AM. It brought me back to my last teaching job, when I was pregnant and hungry at odd times throughout the day. I have talked to other pregnant teachers who commiserate about that physical struggle. The half hour teachers typically get for lunch is seldom enough to eat properly in the best of circumstances. Throw in an early time slot or a physical condition that requires extra nourishment, it becomes difficult to keep the energy and mood up.

I am not singling Tina out here. To be sure, Tina knows how to hit the re-set button better than most folks. She is a frequent tweeter on the #onegoodthing hashtag (some of her #MTBoS pals even have a blog dedicated to this). Even in telling me about what was going on, she took these conditions as a part of the deal, focusing on what she could do: take her preparation time to get her emotions together (“re-centering”) so she can be in a good space for the rest of her classes.

When I think about conversations about teacher turnover, I notice how little we attend to these very basic conditions. Even when talk about making schools welcoming and comfortable places for students, we too often skip the part about making schools welcoming and comfortable places for teachers. We pay attention to school climate for kids so they can do their best work. What would happen if we did the same for teachers?

Here is one idea that could alleviate some of the time intensity of teachers’ work: What if schools staffed one or two adults as permanent in-house substitutes, whose primary job it is to know the students, teachers, and classrooms, so they can step in seamlessly when somebody needs a moment for re-centering after a difficult meeting, to compensate teachers’ time taken for additional meetings, or to allow a pregnant teacher to step out and use the bathroom during class?

In the years since NCLB, I have seen schools find funding for “data managers” so they can generate the tables and spreadsheets needed for evidence-based practice. Why not support teachers in bringing their best selves to each class by giving them an additional resource through by funding the floating support person?

What other ideas do you have for improving teachers’ work conditions?

The Moral Qualities of Teaching

A few years ago, my colleague Rogers Hall and I looked at how biostatisticians and epidemiologists’ workplace conversations compared with those of instructional coaches and teachers. (We both study how people learn at work.)

As we compared our methods for analyzing workplace learning, we had a few great a-ha! moments. Rogers focuses a lot on epistemic communities in his analysis — that is, how different professions collectively agree about what qualifies as knowledge. The architects, etymologists and epidemiologists he has studied all have different standards for saying that something is “known.” Sharing analytic methods benefited me: the idea of epistemic community helped me describe how different teachers take different tacks on what counts as knowing in teaching.

My work informed his in a different way. In my studies, I examine how teachers justify instructional decisions. Oftentimes, they provide affective reasons for what they do  (“I am skipping this lesson because I don’t like it.” “I am going to do this activity because the kids love it.”)  Sometimes, they ground their choices in technical knowledge (“We need to give kids more time on subtracting integers. Those are hard ideas, and they need to see them lots of different examples.”) In addition, teachers will invoke moral reasons (“I am doing re-takes because every kid needs a chance to learn this. I don’t care who your 8th grade teacher was, you are going to learn in my class.”)

Through the comparison, Rogers saw that morality played in epidemiologists’ decisions too. For instance, in one observation, a scientist and a biostatistician debated how to sample a population to look for relationships between HIV and HPV –– whether to do fewer numbers of a better HPV screening or to get more statistical power by using a less expensive HPV test. If quality data were the only consideration, the need for statistical power would prevail. However, the epidemiologist had a had a strong moral commitment to improving the lives of poor women being recruited in the study and wanted to make sure they got the best screening available. This consideration played into his research design. Even supposedly “objective” scientists have reasons to weigh moral and ethical issues in their research.

Why do I bring up the role of morality in teaching? At the moment, I have intellectual and personal reasons.

Intellectually, I need to push back on how the cognitive revolution impacts how we think about teacher knowledge. Lee Shulman had a critical insight: good teachers have a special kind of content knowledge — what he called “pedagogical content knowledge”:

Pedagogical content knowledge (or PCK) includes: (a) knowledge of how to structure and represent academic content for direct teaching to students; (b) knowledge of the common conceptions, misconceptions, and difficulties that students encounter when learning particular content; and (c) knowledge of the specific teaching strategies that can be used to address students’ learning needs in particular classroom circumstances.

By acknowledging the specialized kinds of understanding that good teaching demands, Shulman did his part to elevate the teaching profession, opening entire programs of research that specify different facets of PCK.

Yet, somewhere in the years that followed, the moral element of teaching has too often been devalued. In our quest to professionalize teaching by defining its specialized knowledge, we have downplayed that teaching, at its best, is a deeply moral act.

For example, the PCK construct says nothing of what Rochelle Gutierréz calls “the political knowledge” teachers need to have truly equitable and inclusive classrooms. For instance, teachers need to understand the often biased structures of schooling and work deliberately against them. Recognizing bias and working against it is inherently moral: it acknowledges the inequities built into schooling, from unequal resources to cultural bias to curricular marginalization.

On the personal level, I have a child who has struggled in school. This child’s school experience has vastly improved when teachers are morally invested, sometimes beyond what would be sensible. I am fortunate because this year, my child’s teacher deeply understands the nature of these struggles.

When we first met, we discussed the history and nature of what has gone on. She shared that she had a child with similar challenges. Then she looked me straight in the eye and said, “So when I say I get your child” –– she tapped her hand to her heart –– “I get your child.

Since then, she has told me that she finds my kid an “interesting challenge” and a “delight.” I have heard her talk to other parents as well and can attest that this teacher has a strong commitment to find a way to connect with and reach every student in her classroom.

Calling her commitment a form of knowledge does not do justice to the deep place it comes from: from her heart, from her very purpose as a teacher. And I know that has made all the difference.

A Fallacy about Teacher Learning

In schools across the United States, professional development (PD) season is coming to its grand finale. Summer workshops end and district-mandated in-services begin.

My #MTBoS Twitter pals know this is a season of schadenfreude for me. They tweet me the ironic misfires, like when a teacher who develops sophisticated lessons around technology was obliged to attend an all day workshop on Google docs. Or when another teacher who travels the country leading sessions on classroom math talk is made to sit through a full day on classroom norm setting.

These examples of bad PD stem from a total lack of differentiation. Those teachers had expertise that did not matter in the one-size-fits-all mandates of their schools or districts. The workshops were not responsive to their needs or respectful of what they had already accomplished.

Even when PD is matched to teachers’ needs, it still often falls short. Anyone who has eagerly signed up for a workshop based on a title and description and left unsatisfied is familiar with this. These workshops are often full of activities, handouts, and tips and tricks, but they do not help teachers make sense of how to get these ideas going in their own schools.

In my view, centering descriptions of what to do in PD stems from a fallacy about teacher learning: to get teachers to do better, we need to change their behavior. 

To be clear, of course it matters what teachers do in the classroom. But actions are not the same as behavior.

Behavior involves a description of a sequence of events, such as:

 A woman was tied to a stake and set aflame. She died.

Action considers the meaning involved, which is derived from who people are and where they socially and historically situated, like:

Joan of Arc, who resisted the English because she heard the voice of God,
was tied to a stake and burned. She died as a martyr.

Teaching involves creating meaning. To develop teachers, we need to make them more effective actors in the complex social world of the classroom. If we only focus on providing activities or developing sequences of behaviors, we miss out the opportunity to grow their ability to interpret situations, make judgments and take the purposeful action that shapes meanings for and with their students.

In order to make teacher professional development more effective, then, we need to take seriously what it means for teachers to learn –– and not just learn what to do, but also how and why as they respond and adapt to the myriad and complex situations they face in their classrooms everyday.

#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