TRU, Checklists, and Getting Better

I’ve spent some time recently reading the materials in the TRU (Teaching for Robust Understanding) framework. From the website:

TRU is a framework for characterizing powerful learning environments in crisp and actionable ways

TRU focuses on what they call the five dimensions of powerful classrooms:


Buried in the framework is a thought-provoking footnote:

Making a practice of reviewing what counts can result in significant improvements. For example, Gawande (2007, 2009)  has shown that checklists that remind doctors and nurses of things they know they should be doing result in significant  improvements in hospital recovery and mortality rates. If reminders to wash one’s hands before interacting with patients  can improve medical results, then it stands to reason that instruction can be enhanced by routinely asking (for example) where in a lesson students have opportunities to engage in sense making at an appropriate level of cognitive demand.

TRU advocates for looking at lessons from a student’s perspective and asking targeted questions, connected to their five dimensions of powerful classrooms, about what instruction looks like:

TRU obs

I find these questions really fascinating. I’m often struck by the interest in the education world on ideas that are “new” and “innovative”, but these questions focus instead on some fundamental elements of classrooms and lessons that are conducive to learning. It brings up an interesting question for me. Am I more likely to improve my teaching by trying to learn the newest and most innovative strategies, or by asking myself some simple questions about my pedagogy to better practice the foundational elements of teaching?

I think I’d argue for the latter. I just pinned the observation questions above my desk, and I wonder if I will be able to consistently put in the effort to ask myself these questions each day.

Mathematics as Ambiguity

Here’s something I’ve changed in my teaching:

I came into teaching in part to help students become curious about mathematics and see learning math as a process of exploring and discovering beautiful ideas in the world of abstraction.

I still want to create opportunities for students to feel that curiosity and wonder. But as I get better at seeing what is actually happening in my classroom (not only what I wish was happening), I see students who feel like learning math is a process of following rules and jumping through hoops that were created to make them feel stupid and unsuccessful. To help these students feel a sense of agency and ownership over their learning, I have tried to slow down and unpack some of the ways that mathematics has been socially constructed — the choices of content to include or not include in the curriculum, the emphasis on certain types of skills and practices, the notation that is common in textbooks, the conventions of mathematical communication, and much more, are full of arbitrary rules and ambiguities.

I’ve learned that seeking out those ambiguities as learning opportunities can actually help students to better understand content, and also create a space where students feel like mathematics is a human enterprise that is, like humans, imperfect and always changing.



Sometimes it seems to me like teachers spend 80% of their time and energy dreaming up and implementing elaborate systems of sticks and carrots to try to motivate students to learn, or at least to sit in class and act like they are learning for the better part of a day. The word “coercion” sounds ugly, but that’s what it is. Points, prizes, incentive systems, grades, transcripts, recommendations, rules, routines, structures, and all of the little things teachers do to manage class on a daily basis.

I wonder what education could look like without coercion. I wonder how much more energy humans in schools could put toward learning instead of being distracted by the systems we build to try to get young people to learn. I wonder what those young people would act like in a world where they experienced genuine agency in their learning.

I don’t think we should blow up all of education. I think there’s too much we don’t understand about motivation, the stakes are too high for the students who could slip through the cracks, and we don’t have the adults to build a new education when we can barely staff the current one. I know I would struggle to teach in a totally different paradigm. But it’s fun to dream.

I think the toughest thing for me is that every way I have tried to hack my way to a place where I rely less on coercion to help students learn, I feel like I’m putting lipstick on a pig. Every new system feels nice on the surface but it’s the same nastiness underneath. I can do the best I can within the schools we have, and I can focus on all the great things happening, but I can’t help thinking there’s potential for so much more.

Expectations and Paying Attention


I live in Leadville, a little town in the Colorado mountains. We have two claims to fame. First, we are the highest incorporated municipality in the country at just over 10,000 feet in altitude. Second, we host a bunch of high-profile endurance events, culminating in the Leadville 100 Trail Run, a 100-mile ultramarathon in the mountains around Leadville each summer.

There’s a pretty absurd culture of athleticism here. Many of my coworkers have run the Leadville 100 or completed the 100-mile mountain bike race, and many who haven’t regularly run local  trail marathons or participate in 25-mile backcountry ski races. It’s a funny place. I might casually talk to the barista at the coffee shop in town about how the snow finally melted off the Mt. Elbert trail so he could run to the top (Mt. Elbert is the tallest peak in Colorado). Or I might go for what I feel is a hard morning mountain bike ride with a friend, to later learn that he ran 20 miles later that afternoon. I’ve definitely pushed myself further here than I would have living somewhere else, whether through implicit comparison, jealousy, or just the practical goal of keeping up with friends.

There’s also a negative side to this culture. Many folks in town are intimidated by the athletes, thinking that matching the serious racers is impossible and that it seems like a waste of time to try anyway, because they’re endowed with some special abilities the rest of us mortals aren’t. There’s a particularly tough gender divide — the mountain bike races are typically 80-90% men and the other races aren’t far behind. The dynamics of the gender divide are self-perpetuating, and while many folks feel motivated to push themselves many more feel left out of the dominant culture.

To summarize, seeing people working hard and achieving at high levels motivates some people, but alienates others, particularly those who are already in the “out-group”.

Math Class

This isn’t so different from my math class.

Some students share much more often than others. This might motivate some folks to work harder, but also alienates others as they believe working hard and engaging in class is for someone else. Every interaction they have either reinforces these paradigms or works against them.

Here’s something I want to work on to try and mitigate the inequitable outcomes of this cycle:

I want to work on paying attention to students. Sounds easy. But when a student is sharing, either to the whole class or in a small group, I want to watch the other students. Who is listening? Who is staring off into space? Who looks frustrated or hopeless? There’s a lot to see in young people’s faces when I look.

My instinct is always to watch the person who is speaking. It always has been, unless I’m scanning the room specifically for misbehavior. But the more I pay attention to all of my students — how they listen to or disengage from the conversation when certain peers share — the more I learn about the social dynamics of the class and the ways that students experience learning in my room.

It’s not surprising that I’ve pushed myself to keep up with the Leadville endurance scene. I’m a tall white guy from an upper-class background. I saw lots of people who looked like me, so it seemed natural to join in. In the same way, I need to look at my class and ask myself: which of my students see pictures of academic success that they feel like they can strive toward? And which of my students interpret learning as something for someone else, something risky, and something not worth their effort?


One conversation with a colleague this year has stuck with me. We were chatting after the first class of the year, and he made an offhand comment about how math was unambiguous and logical, with only one right answer.

I’m fascinated by this perception, and I need to remember that while I think of mathematics as a discipline grounded in struggling with ambiguity, resolving complexity, and working through confusion and uncertainty, most humans do not. I want to create structures in my class to communicate to students that math can be ambiguous, and that seeking out ambiguity can be an important way to learn about mathematics. I realize that many students say that what they like about math class is being able to find the one right answer and I want to value that aspect of mathematics — I definitely feel satisfied after finding and confirming the answer to a hard problem. At the same time, I want to expand student conceptions of what mathematics can be. They’ll have lots of experiences valuing right answers in my class and beyond; I want to make sure I also value ambiguity.

One way I try to value ambiguity and use ambiguity as a teaching tool is in my warmups. Which one doesn’t belong, visual patterns, number talks, and between two numbers are four great low-prep resources for tasks that students can look at in lots of different ways. Other folks have written about the finer points of each of these tools. I want to think for a minute about three teaching moves common to all of them that I try to use to communicate my values to students.

Who Thought About It Differently? 

This is my favorite question, and I get to ask it every time I do one of these warmups. I phrase it purposefully to assume that students approached the task in different ways, rather than saying “did anyone think about it differently?” I want students to understand that looking at a problem from a unique perspective is valuable for everyone’s learning, and to highlight those perspectives each day.

Rough Draft Thinking 

I think the most valuable part of these warmups is rough draft thinking — hearing students reason through a problem out loud and share ideas that might be wrong in front of the class. In all four structures I’m likely to start with individual think time and a partner share. Students are unlikely to offer rough draft thinking on their own, but I can listen in and ask students who have valuable but unfinished ideas to share. I’m not trying to find students making mistakes for the purpose of mistakes, instead seeking out partially-formed strategies that offer a new avenue of approaching the problem and creating an opportunity for the class to help. Creating a space where students feel comfortable sharing this type of thinking is hard, and involves celebrating mistakes every time as useful ways for the class to learn. But it’s worth all the effort to allow students to share ideas more freely and feel more comfortable taking risks.

Value Divergent Ideas 

Students often say things that, strictly speaking, are wrong. Highlighting rough draft thinking is one example of this. But ideas that students share often have important grains of truth. They might think about the step number of a visual pattern differently than the rest of the class, make a computational error despite sharing a unique strategy for a number talk, or misuse vocabulary while describing a new idea for why a certain graph doesn’t belong. I have tried to actively cultivate the habit of looking for the valuable ideas in everything students share. This has been hard. For the first few years of my teaching, I spent a lot of time asking the class questions, calling on a student, and then telling them, implicitly or explicitly, whether they were right or wrong. I have had to practice slowing down, unpacking what a student has to share, valuing their contribution, and building off of it to create an opportunity for the class to learn.

These three practices build off of each other. First I need to create a space where students see a task as an opportunity to compare and contrast approaches rather than to guess the right answer hiding in the teacher’s head. Then I need to help students see rough draft thinking as worth sharing and valuable for learning. Finally, I need to approach student ideas with a disposition to build off of strengths rather than point out mistakes. It’s an iterative cycle that, hopefully, over time, creates an environment where students see math as a discipline grounded in communicating, working with ambiguity, and connecting ideas. And all for five or ten minutes for each warmup.


I was super lucky to spend last week in the Bay Area, visiting schools and observing awesome teachers I’ve met around the twitterblogosphere and at conferences. My school takes students on backpacking trips a number of times each year, and each teacher gets one trip off to do something professional development-related. I got my school to agree to pay for a plane ticket to San Francisco if I spent the week observing teachers and staying with friends. I’m incredibly grateful to the awesome awesome folks who played host for me and opened their doors. All in all I visited five very different schools and observed 14 teachers. I don’t want to call anyone out individually on here, but I do have a few reflections on the experience.

First, it’s been some work to figure out what I want to take away from the visits. I learned a ton, but I learned a ton of different things that, at first, didn’t fit together neatly into some lesson I can put into practice right away. Instead, a lot of what I learned was a gradual process of watching really thoughtful, passionate teachers do their work, expanding my mental model of what great teaching can look like in all sorts of directions, and soaking up the hundreds of little teacher moves I got to see in each class.

Going through my notes, I realized that my biggest takeaway was seeing so many teachers show really remarkable patience with student thinking. This is tricky — it’s easy as a teacher to generalize about a class, to say “they understand this,” or “and then I got them to figure that out.” It’s a lot harder to navigate the complexities of all the individuals in the room. The awesome teachers I saw had a ton of tools to give every student in the room a chance to engage with the big ideas of the lesson, to check in on them, to create opportunities for students to think together about worthwhile problems, and to structure content in a way that helped each student make sense of new math.

This played out in lots of ways. Wait time, purposeful scaffolding, carefully chosen tasks, questioning to help students think metacognitively about their learning, and structured places to see where students were and where to head next. These aren’t remarkable teaching moves on their own. They became remarkable paired with this sense of patience, giving students time and space to think and work through ideas together, having the restraint to avoid jumping to the end or short-circuiting where students were heading, and creating a space where every kid in the room could wrestle with the math in ways that made them feel competent and valued in the class.

In writing about this, I’m realizing this type of patience with student thinking is a bit of a fuzzy idea. It’s not sexy, it’s not the flash that students might remember months or years later, it doesn’t even happen in a particular moment or teacher move. Instead, it was a sense that seemed present in very decision teachers made, that was the means to the end and the atmosphere in the room underneath and around everything else that was happening. It was that atmosphere that shifted from a class where the teacher was headed somewhere with a bunch of students on board and the rest trying to keep up, to a class that really felt like it was working collaboratively toward a shared goal on a level playing field.

So there’s my goal. I feel incredibly lucky to have received this masterclass in patience from so many awesome teachers. Now I need to figure out what, exactly, it is I’m talking about, and how I can put it to work in my teaching.

A Book I’d Like to Read

I often feel like books about education offer too many easy solutions to hard problems. Here is a summary of a book about teaching that I would like to read. It doesn’t exist, but if someone wants to write it that would be great.

Teachers everywhere disagree about what good teaching should look like. Whether we are talking about cold-calling, engaging students in inquiry-based learning, personalizing instruction, selecting resources into a curriculum, or implementing classroom management behavior systems, one teacher’s passion is another teacher’s malpractice. In this book find classroom teachers cutting through the hype and the jargon, talking about how their ideas of teaching play out in actual classrooms, sharing how teaching strategies fit their specific goals and context, and engaging in dialogue with other teachers they disagree with to find common ground and share differing perspectives.  Spoiler: everything is much grayer in practice than in theory. You’ll be reminded that teaching depends enormously on context, that practices which are invaluable in one school may be verboten in the next, and that the judgment of teachers trying to do right by students is at the heart of making schools work for all children. Most of all, you’ll come away with a few new tools, perspective on implementing them with fidelity, and an understanding of how they are likely to play out in your school, and with your students.