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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.

Tensions in Motivating Students

Participating in math class feels socially risky to students. Staying silent often feels safer.

This is the challenge that Lani Horn takes on in Motivated: Designing Math Classrooms Where Students Want to Join In. I’ve learned a ton from this book, and while there are plenty of concrete strategies to try, it also gives me new perspectives to understand my students and their engagement in class. I’ve made some progress on some elements of what Lani talks about, but more than anything this book has helped me to understand how far I have to go to create a truly engaging classroom.

In reflecting on that progress, I’ve come to think of engaging students in terms of the inevitable tensions that come with trying to create a motivational math classroom. I’ve thought more and more about tensions since reading Rochelle Gutierrez’s paper, Embracing the Inherent Tensions in Teaching Mathematics From an Equity Stance. Naming some of these tensions helps me to better navigate and learn from them in my classroom each day, and acknowledge that there are no easy answers. Lani names five components of motivational math classrooms, and here are five corresponding tensions that I now think about in my day-to-day practice:

Belongingness: Students are motivated when they feel like they are part of a classroom community that values them. I want students to feel like they are all working together toward a common goal, yet teenagers naturally and inevitably compare themselves to each other, highlighting their differences rather than their common goals. I need to help students see themselves as collaborators in the enterprise of learning mathematics, while also highlighting their commonalities and making differences in knowledge and skills feel like productive and useful parts of the classroom community.

Meaningfulness: Students are motivated when the math they are learning feels meaningful to them. Students want to feel like they can use math in their lives outside of school, yet I am responsible for teaching students both application and abstraction, both math that is likely to be used in the world and math that is detached from any obvious real-world meaning. I need to create a classroom that articulates the ways that math can be practical and empowering outside of schools, and argues for the value of mathematical knowledge in and of itself.

Competence: Students are motivated when they feel like they can be successful at learning and practicing mathematics. I want students to be able to experience math class as a place where they can be smart, and at the same time give students cognitively demanding tasks. Either of these alone is relatively simple; I could help students feel smart by dumbing down content, and I could give students demanding tasks without support. Doing both involves navigating the complexities of task design and facilitation to give students entry points into rigorous work, while also constantly learning about student thinking to better scaffold them from where they are to where I want them to be.

Accountability: Students are motivated when they feel like others care about their learning and their thinking. I want students to know that I care about them, and that I will follow through with my expectations because I care, yet the pressure of high expectations can increase stress and anxiety in already tense experiences in math class. Every interaction runs the risk of publicly exposing a student as less intelligent, and every interaction also has the potential to reinforce my expectations and communicate to students that their learning matters. Holding students accountable doesn’t have to exacerbate the social risk of math class, but it involves walking a tightrope that involves knowing my students and understanding how to respond when engagement isn’t where I would like it to be.

Autonomy: Students are motivated when they feel like they have choice and agency in a math classroom. I want to provide students opportunities for that choice and agency, and also acknowledge that teenagers’ natural state is not one of learning mathematics. Left to their own devices, most teenagers wouldn’t learn very much math, but a class with a very rigid structure to ensure students learn also erases any autonomy they might feel. I have to find opportunities for students to feel like they have a say in their experiences in math class, while also creating structures that drive learning forward.

In writing this post, I noticed that I used the word “feel” often. One challenge in these tensions is that belongingness, meaningfulness, competence, accountability, and autonomy are not static concepts. They depend on the perspective and perception of the student, and each student experiences my class differently. Maybe the most challenging tension in creating a motivational classroom is navigating the complexities of creating these conditions for all of the student in each class, each day.

Student Noticing

Really interesting quote, especially given how popular noticing & wondering has become in the twittersphere. I think Henri makes an excellent point.

I’d like to offer another way of thinking about noticing. One perspective is that noticing activities are designed to help students notice what is mathematically important about a certain problem, and teachers need to ask questions to help students get there. This seems to be what Henri is describing, and can devolve into a game of “guess what’s in the teacher’s head.”

Another perspective is that noticing mathematically important features of a problem is where we want students to get, and asking students to write, discuss, and share their noticings is a way for teachers to probe student understanding and see where they are at a particular moment. If I am teaching a lesson on slope and I want students to notice that larger coefficients correspond to a certain notion of “steepness”, I could ask them what they notice about a set of functions, circulate, and see what students write about and discuss with each other. Then, I can make a decision about what to do next based on what they seem to be noticing. Maybe none of the students will notice what I want them to notice. If that is the case, they will probably learn more if I step back and offer some more explicit guidance, rather than following me on a wild goose chase to try to get them to say what I want them to say. And if many students are noticing and discussing my goals for the task, I can select and sequence several to share out and then offer a new task to dive deeper into that concept.

I think noticing is a valuable tool for diving into student thinking, but it may be a better tool for formative assessment than for unguided discovery.

Learning the Standards for Mathematical Practice

Really interesting tweet and replies today:

For passers-by, the SMPs are the Standards for Mathematical Practice in the Common Core math standards. From the standards themselves:

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

Seeing all this got me thinking. Here’s an opinion I have that other math teachers might disagree with:

I don’t think that the SMPs are describing something that a student, or any human, can become “proficient” in. I don’t think they are skills that can be developed and employed regardless of content.

Instead, I think of the SMPs as ways of learning content more deeply, and learning content in ways that help students to transfer that knowledge in the future. When I teach my students about conic sections, I want them to construct viable arguments and critique the reasoning of others while trying to better understand the multiple meanings of eccentricity. I want them to model with mathematics to better understand how conic sections are related to planetary motion. I want them to use appropriate tools strategically to visualize and better understand the relationships between different types of conic sections. I want them to look for and make use of structure to better understand the big ideas of conic sections, and see them as one interconnected whole rather than a set of procedures to memorize.

I think that, if my students successfully use the SMPs to engage more deeply with the content they are learning, they learn that content in a way that helps them to apply what they know more flexibly in the future. And the more math students know, and the deeper they understand that math, the better position they are in to be quantitatively literate in the world, and be able to solve new problems by connecting and applying what they already know.

On a fundamental level, I teach math because I hope that what I teach helps students to solve new problems and reason in the world beyond my classroom. But I try to do that by teaching content first, and I use the mathematical practices as a lens through which to help my students learn that content. My goals on a daily basis are for students to learn math and to come to believe that they are people who can be good at math and use math in their lives. The practice standards are a means of meeting those goals, not goals in and of themselves.