Pedagogical Responsibility

I feel frustrated with a lot of the conversations I’m seeing about remote learning during a pandemic, on Twitter and at my school. It’s often “here are five great resources” or “have you tried this website?” or “the six keys to a great online lesson.” If I want to try a new technology tool or checklist for my lessons I’ve got plenty to choose from. But I find these conversations speed past core questions of what we’re trying to accomplish in remote learning. What are our goals? What are our responsibilities? Which students are succeeding, which students aren’t, and what can we do about that?

A great recent article by Grace Chen, Samantha Marshall, and Ilana Horn helped me to better understand these questions. When teachers talk about teaching, we often focus on pedagogical actions. Pedagogical actions are the surface-level observable behaviors in our classrooms our online lessons. How do we call on students? What do our handouts look like? What websites are we using? How will we give students feedback? How do we assess learning? These are important questions. But they’re also expressions of our beliefs, our values, and our contexts. It’s hard to communicate when so much is unsaid.

Horn encourages a focus instead on what she calls pedagogical responsibility. Pedagogical responsibility is who or what I feel beholden to as a teacher. What am I trying to accomplish? Why? My responsibilities are my starting point. The most helpful teachers I’ve learned from as I try to figure out this online teaching thing have begun by articulating their responsibilities and their ethical obligations in this challenging moment.

I’ve decided my core responsibility right now is connection. We are all disconnected from each other, and I want math class to be a chance for students to connect with me and with each other. I keep coming back to that responsibility in my planning as I figure out this online teaching thing. I care about other things — I want learning to feel meaningful for my students, and for students to enjoy class as much as possible. But I care most about connection, and I want to make decisions about what to do each day through that lens.

I’m not saying every conversation about teaching needs to start by stating our pedagogical responsibilities. I have plenty of practical concerns right now. My Google Classroom is an overhwelming and disorganized mess, and today I need to deal with that. My pedagogical responsibilities won’t be the first thing I think about. But I often see teachers talk past each other because they are starting from different places. We often jump right to the teacher actions in a situation. Articulating pedagogical responsibilities and reasoning is a way to bridge that divide and better understand each other. And articulating my own responsibilities helps me to stay grounded in what is most important in my teaching. Chen, Marshall, and Horn write in their article that pedagogical responsibility is often implicit. It’s always a factor in how teachers teach, but often in a way that goes unstated and unexamined. By making my responsibilities explicit, I hope to both make better decisions and to interrogate why I make certain decisions in the first place. Teaching today is an environment of uncertainty and experimentation. For me, pedagogical responsibility provides a grounding force to make better decisions under challenging condtiions.

Desmos + Feedback

I tried the new written feedback feature in Desmos activities yesterday. It was fun! It was also a good reminder about what feedback is good for, and what it isn’t.

I was teaching Burning Daylight. The lesson asks students to write and interpret functions for the hours of daylight over a year in different places using sine or cosine functions. Class was synchronous, so students were also on Zoom. The written feedback feature allows me to write feedback for a student on a specific screen. A notification shows up for the student that also links back to the screen in case they’ve moved ahead. I found written feedback especially useful for small obstacles students encountered. For instance, one student was using 14 as the period for a function modeling months in a year. They kept tinkering with different things but couldn’t get their function to fit the data. I prompted them to think about the connection between the period and months in a year, which helped the student to move forward in the activity. The ability to write equations is nifty too. I can cut and paste a student’s equation and change something to make a suggestion for how they might approach a certain problem. These small pieces of feedback can help to clarify student thinking in the moment or help them through a spot where they feel stuck but only need a quick hint. I could see myself using written feedback in an actual classroom as well. Walking around from student to student can feel cumbersome for small prompts. Giving digital feedback doesn’t carry the social stigma of the teacher coming over to a student or pair of students and asking them to try something again. Feedback is also great for students who rush through the activity, to prompt them to return to a previous screen and expand on their thinking.

One issue I encountered is when students are more than a little stuck. In those situations, a quick question or hint often isn’t enough to get them unstuck. Occasionally students tried to get into a conversation with me, typing questions in different places in the activity. But the interface isn’t designed to communicate from student to teacher, leaving students floundering. When students were really stuck, I had a hard time figuring out where the trouble was coming from and communicating a path forward through the written feedback feature. Feedback is great, but it’s not the solution to everything.

Those moments when students are really stuck are ones I usually watch out for in Desmos lessons. I often pause the class to address common issues, or offer a task separate from the activity to help move students forward or get at something they’re missing. Written feedback isn’t a substitute for those moments. Feedback can create a spiral where I’m so focused on getting the student through the task in front of them that I lose the bigger picture and miss better opportunities for helping that student. There’s also the challenge that, once I start giving feedback, some students become reliant on it rather than trying their own ideas and seeing how far they can get without help.

All in all, it’s a nifty feature. It’s great for those small pieces of feedback that can move a student forward. But when a student is really stuck, what they often need isn’t feedback, it’s a bit of instruction and a new task to act as a bridge from where they are to where they’re going.

Math in the Real World

Math teachers often tell students they need to know math so they can use it in the “real world,” whatever that means. Well, the real world is here, and math isn’t offering me much help.

I started teaching online this week and it’s a mess. I’m stressed and anxious about the pandemic. I keep talking with mute on. I screw up breakout rooms in Zoom. I have no idea how to help students visualize volumes of solids of rotation over the internet without resorting to a Youtube video. I hate assigning Youtube videos. The little things are getting a little better, but I’m not sure graphing trig functions is what students need right now. The more I read about the pandemic the more pessimistic I feel. And I have a stable job, financial security, and I live in a rural area where social distancing can involve lots of fresh air. Lots of folks are doing worse.

I asked my students earlier this week what questions they had about COVID-19, and whether they’d like to learn a bit about it in math class. Their questions were excellent and the answer was almost unanimously yes. I felt weird teaching about the virus, like a politician using a crisis to score points for a pet cause. “Math in the real world, kids. This is why you should pay attention in class.” It’s affecting real people, it’s not some word problem about ferris wheels or watermelons.

Here’s the thing. The data is incomplete and there’s a ton of uncertainty. Math can’t tell us what’s going to happen. The first thing I tried to do — and I suck at this, but I’m trying to get better — was to be transparent about how I’m feeling. I feel scared and anxious, and that feels like a normal response to the crisis. I’m not sure it was the right thing to say, but being stoic feels dishonest and teaches students that emotions don’t have a place in confronting the virus. Scared doesn’t mean hopeless.

A lot of students asked about the meaning of “flattening the curve.” I used these visualizations from the Washington Post to help students understand different possibilities, and this data from Our World in Data to look at the difference between our growth rate and, for instance, South Korea’s. The data might not be optimistic, but I want students to feel a sense of agency in understanding how their actions matter. I also wanted them to know what to look for if our response is successful. I emphasized that the data is incomplete and the models are just guesses; we don’t know.

Math isn’t the hero of this story. But I do believe that knowledge is power, and I hope that knowing a bit more about what is happening and how their actions matter will help students find their way through the pandemic. And I hope that being transparent in how I feel can help to model the type of emotional vulnerability that I think is essential to taking care of each other in hard times.

I did this with a lot of trepidation. It was especially hard online, without a good way to read students’ responses. But I’m happy that my students are a bit more informed, and had some of their questions answered.

Online Learning

We’re closed. I start teaching online tomorrow, and I’m nervous. Nervous about the teaching, but also unsure how to navigate this moment and do right by students.

It’s easy to fall down rabbit holes right now. I’m exploring new digital learning tools and wondering whether I should try Flipgrid or Slido or Formative. I’m playing with Zoom’s different capabilities. I’m wondering whether to keep going with my regular curriculum, or find some fun accessible problems and try to get students enjoying math. I’m thinking about the exponential growth of COVID-19 and the game theory of toilet paper hoarding and the probabilities of social distancing and wondering if those are teachable moments or too close to home.

It’s a great moment to step back and think about my responsibilities as a teacher. I would love to just take a break and let people take care of themselves right now. But if I have to teach, starting with my responsibilities can help me make priorities and consider the big picture. I think my first responsibility right now is to make sure online learning doesn’t suck. Life sucks for a lot of people right now, and I don’t want the requirements of school to make things worse. Second, I want to maintain relationships with students so they feel like they have a teacher and not a robot behind the computer. Third, I want students feel like online learning is meaningful to them and not busywork or going through the motions.

Those three responsibilities are my starting point. When life gets messy it’s easy to lose the forest for the trees. For me, teaching online means focusing on what’s important, doing less, and trying to do it well.

Student-Centered or Pedagogy-Centered?

In my second year teaching I received a piece of feedback I still think about. I was using number talks as a daily warmup, asking students to use mental math to make estimations, compare arithmetic strategies, and analyze visual patterns. My principal had stopped by a few times to see what was going on. And in a meeting after one of those visits, he shared the observation that my students hadn’t seemed to get any better at number talks over the last several weeks.

He was right. I can give lots of reasons now why my number talks didn’t make a difference. I didn’t create a culture where students listened to each others’ strategies. I chose problems haphazardly, without sequencing them toward coherent goals. I made the number talks too difficult too fast.

But those problems masked a deeper issue. I wasn’t looking for evidence of student learning. I was convinced that number talks were a great routine, that students had so much to benefit by improving their number sense. But I wasn’t willing to be wrong. I was so focused on the routines and structures I created as a teacher that I ignored the actual goal of learning.

In the online-math-education-writing space, it’s much easier to focus on the inputs than the outputs. It’s easier to argue that teachers should do this or not do that than to think about how students experience the math classroom. But I’m a better teacher when I focus on what students experience first, and my actions as a teacher second.

I’m not sure, but I think these are my values for the student experience, more or less in order:

  • I want students to see that they have agency in their learning; that when they work hard they are able to learn math.
  • I want students to consistently recognize their own successes.
  • I want students to recognize that I care about their success.
  • I want students to enjoy learning math.

I care about these experiences because I find them self-perpetuating. When students feel agency they are more likely to persevere through challenges in the future. When students feel successful they develop a positive self-concept as mathematicians and see themselves continuing their math education. When students see that I care about their success they are willing to ask for help when they need it and let me know when things aren’t working. When students enjoy learning math they show up to class ready to think hard and engage.

But it’s easy to focus on the inputs. To assume that if I teach a certain way, students will have a positive experience and everything will be great. That type of thinking encourages me not to notice what’s actually happening in my classroom. I become an ideologue, spouting truisms about the way math should be taught and blaming students if they don’t learn. I’ve done this in two directions. I’ve practiced “I do, we do, you do,” telling students, “I explained it,” confused when they didn’t understand. And I’ve tried to lead students to their own understanding, telling students, “you’ll understand it if you figure it out yourself,” and griped afterward about how my students aren’t independent enough.

In both of these cases, the students who succeed are the students who have been successful in the past. They’re the students with more privilege and more social capital, who are more likely to receive affirmations from teachers and to have a positive self-concept as a learner. If I’m not willing to ask myself hard questions about how students experience my class, I’m teaching for the students who need it the least. In both cases, I’m treating students as a variable and my pedagogical choices as a constant. Instead, I want to see my pedagogy as the variable, and to measure the success of pedagogy by the experiences of the students in the room.


I entered college without much direction. I was interested in philosophy, history, computer science, creative writing. I could have gone a few different directions. I’m very grateful I ended up studying math, but I made that choice as a result of one individual.

I had done well in math in high school but hadn’t enjoyed it very much. Calculus especially was a slog. Entering college I would’ve had to finish the calculus sequence before moving on to any other math classes, along with most other freshmen. But by coincidence, my “pre-major advisor” was the math professor teaching an introduction to proof class that first semester. She told me she thought I would like it, and offered to waive the requirement of completing multivariable calculus. I took the class, loved it, kept taking proof-oriented math classes, became a math major, and then a math teacher. My professor’s decision to waive me into that class was probably one of the more influential moments of my life.

I often think about the study outlined in this Atlantic article. Here’s the key result:

Then the researchers randomly attached one of two sticky notes to each essay. None of the students were aware that they were part of a study and thought their teachers had written the notes. Half of them received a bland message saying, “I’m giving you these comments so that you’ll have feedback on your paper.” The other half received a note saying, “I’m giving you these comments because I have very high expectations and I know you can reach them”—a comment intended to signal teachers’ investment in their students’ success.

Then teachers offered the students an opportunity to revise their essays.

The results were striking. Among white students, 87 percent of those who received the encouraging teacher message turned in new essays, compared to 62 percent of those who got the bland note. Among African American students, the effect was even greater, with 72 percent in the encouraged group doing the revision, compared to only 17 percent of those randomly chosen to get the bland message. And the revised essays received higher scores from both the students’ teachers and outside graders hired for the study.

The article connects feedback to the idea of belonging. It’s not that Black students are uniquely receptive to a certain type of feedback. Black students are less likely to receive cues that they belong in academic environments, because Black students have fewer role models who look like them, and also racism. So even one strong cue around belonging can make a big difference.

I became a math person in part because I got the message in my first days at college that I belonged. How many people don’t get those same opportunities, those same cues that they are capable?

One message I might take away from the study above is that I should tell Black students I have high expectations. But that would oversimplify the message. I’m a cis straight well-off white dude. It’s not surprising I received belonging cues in a college math department. Existing narratives of power and privilege serve to elevate some students at the expense of others. Part of my role as a teacher is to undermine these narratives. It’s my job to seek out brilliance, especially among those who are least likely to be recognized, and to help those students see their own potential.

More Explicit

I’ve taught at the same school for almost five years now. Not very long in the scheme of things. But long enough to teach the same lessons over a few times, and to notice the little tweaks I make each year based on the last.

One common change is that, in most of my lessons, my teaching becomes a little more explicit every time. Each time, I’m more likely to say exactly what I want students to know, and to say it myself, early in a lesson.

It feels weird to write that. There’s a bit of orthodoxy in parts of the online-math-education-world I hang out in where students should be figuring things out themselves and playing with mathematics. I see lots of people take as a given that good math teaching means letting kids figure things out themselves.

Thinking about why I’ve shifted toward more explicit teaching, I need to start with my values. My most important value is that I want my each student to develop a positive self-concept as a math learner. I want students to feel like math is a place where they can thrive, that math is worth learning, and that with effort they can learn it. My second value is that students learn some math. Exactly what math they will learn is mostly prescribed to me, but I can find a reason why almost anything is worthwhile. I enjoy the challenge of trying to make every concept engaging.

Back to the changes in my teaching. One motivation is that students learn content. It’s not flashy, but it’s a real responsibility I have. Lots of things are hard for kids to figure out on their own. I find that students who spend too much time struggling often end the class confused. Even if some students figure out what I’d like them to figure out, this can just widen the gaps between students who are doing well and students who are struggling. Through their learning, I want each of my students to develop a positive self-concept as a math learner. A big part of that self-concept is whether they experience regular success in class each day. I hear lots of messages that mistakes are important learning opportunities, but that ignores the reality that some students are told they are wrong more often than others. They’ve been wrong more often than they’ve been right for years. A few nice messages about brain growth aren’t enough to undermine their negative perceptions of themselves as learners. If I can do a few small things to set them up for success, to help them feel successful in class each day, I can help all students start to build more positive beliefs by acting their way into a self that is more confident in math class. That means knowing the places where they’re likely to get tripped up and smoothing the path a bit.

I don’t mean that I never have students explore and conjecture and play with math. I do, and I believe in these experiences as critical parts of math education. But while they are critical experiences, they don’t need to happen every day. I’d rather put my energy into doing an occasional exploration well than facilitating mediocre discovery every day.