Author Archives: dkane47

Coronavirus and Mathematical Structure

I think math is worth learning because mathematical thinking can help humans understand the world. As I’ve been inundated with news about COVID-19, I’ve tried to practice habits of mathematical thinking to better understand my place in the pandemic.

One of the toughest yet most gratifying things about teaching math is helping students to see the deep structure of a problem and not only the surface structure. The human mind has a built-in bias toward surface structure. We first notice what’s most visible and salient about a situation, even if it’s not the most meaningful. Students might look at three word problems, see cheeseburgers, drag racing, and cell phone plans, and assume they are unrelated concepts. These situations look different on the surface, but have the same deep structure of linear functions. My job is to help students recognize deep structure. Structure is everywhere in math. I want students to connect unit circle and function representations of trig functions, choose fractions or decimals strategically depending on the context, or recognize that solving a system of equations is the same as finding the intersection point of graphs of functions. It’s all structure.

I’ve been trying to practice understanding the pandemic with a similar understanding of structure. What are the surface features that can distract from the deeper structure that I should be paying attention to?

One piece of surface structure I see everywhere is the six foot rule. We’re constantly told to stay six feet away from humans not in our household. We’re redesigning schools to keep students six feet apart, lining up customers checking out at six foot intervals, and rethinking public spaces like subways and planes where that level of physical distancing is impossible. Six feet of separation is clearly a good idea! The evidence seems clear that if we had begun physical distancing sooner, thousands of lives would have been saved. At the same time, staying six feet away from people around me does not make me immune from the virus. I can walk right by someone without getting infected, but catch the virus having a sustained conversation from ten feet away.

The six foot rule is surface structure; the deep structure is that infected people exhale virus particles. If a healthy person inhales too many of those particles, they are likely to get sick. Erin Bromage has a great article that helped me to understand the subtleties of virus transmission. Walking past someone outdoors, away from large crowds, presents almost no risk. Yet if a pre-symptomatic infected person were to spend an hour in an enclosed space with a dozen other humans talking about rational functions, even if everyone wore a mask and stayed six feet apart, infection would be pretty likely. If a pre-symptomatic infected person sneezed in a bathroom and a healthy person walked in a few minutes later, they would have a decent risk of infection without any direct human contact. I don’t think we should get rid of the six foot rule. But we should recognize that, while it’s a useful rule of thumb, staying six feet away from others does not reduce risk to zero. If we can use the six foot rule but also be mindful of risk that is not mitigated by six feet of distance, we can make better decisions about reopening spaces while minimizing risk. If we reopen spaces, for instance schools, with a laser focus on keeping everyone six feet apart but a lack of understanding of the deeper structure of virus transmission, we’re likely to end up in trouble.

I find the habits of mathematical thinking that I try to teach helpful in understanding the pandemic. I’m not arguing that math is the only thing we need or that math is the solution to everything. But I believe mathematical thinking is valuable. Mathematical thinking also really hard to teach. What would math class look like if our we put more emphasis on helping students understand the world around them, and less emphasis on the standards and concepts that we’re used to?


When I started teaching I thought that accountability meant having consequences. Students need to pay attention and write certain things down and only talk to their seat partner and ask to go to the bathroom and not throw small objects across the room every time I turn my back. If they don’t then I punish them because behaviorism or something.

I think about accountability differently now. Accountability is how I let students know that their learning matters. My goal isn’t to catch them doing things wrong, it’s to make sure they have what they need to do things well.

This shift in perspective is particularly important in pandemic distance teaching. Do I assume that students are trying to avoid learning and seeing what they can get away with? Or do I assume that students are doing their best in a tough situation? Those assumptions lead to different actions for me as a teacher. Who do I follow up with? Do I ask questions to learn more? How flexible am I willing to be?

Sometimes consequences are important. But young people are complicated, and it’s easy to build a narrative in my head that oversimplifies a tough situation and assumes the worst. We all work in an education system that has conditioned us to use carrots and sticks to coerce certain behaviors. I want to practice accountability in a way that looks for the good first and offers support. This doesn’t mean that I look the other way or lower expectations. I still want to hold students accountable, but my first instinct isn’t to exclude or punish students.

Turtles All The Way Down

A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You’re very clever, young man, very clever”, said the old lady. “But it’s turtles all the way down!”

-Stephen Hawking, A Brief History of Time

There’s a lot of rhetoric flying around of the fallout from distance learning. It’s real, and I’m concerned about how schools will adapt when we can all return to our classrooms. At the same time, the rhetoric feeds into the popular perception of math as a giant ladder. If students don’t learn this then they’ll be confused next week and next year and they’ll fall behind forever and the STEM pipeline and college remediation and whatever. In some ways math is cumulative, but in others it’s not. Some ideas do come up again and again. But in any year of math we could leave out or cut short big chunks and students would be fine. When we assume everything builds inexorably toward wherever math education ends it becomes a self-fulfilling prophecy. We stuff more content in because “they’ll need it next year” and math becomes nothing more than a tool for learning more math. It’s turtles all the way down.

I’m making those types of decisions right now in precalculus. You would think, given the title, that most of what students learn they will need in calculus. But I’m cutting lots of content, and students are going to be fine. Out go most of the weird algebra we do with inverse trigonometry, most of the trigonometric identities, and a lot of the more obtuse trig equation solving. That’s not to say it’s all worthless or that it will never come up again. But no student who would have been successful in a calculus class will fail because they can’t solve some weird equations with compositions of inverse trig functions.

But now I’m following the turtles down. I’m worried more about getting students ready for next year than teaching meaningful mathematics. There are all sorts of dead ends and corners of math education, but they can be corners full of curiosity and low-stakes exploration. Those inverse trig functions are beautiful windows into symmetry. Identities can turn complexity into elegance with a single insight. And trig equations connect algebraic and graphical representations and bring us back to the unit circle, connecting the web of trigonometry.

I don’t need to do everything. Students will be fine without most of it. But if distance learning reduces the curriculum to a hollow shell of math that’s only designed to get everyone to next year, I’m doing students a disservice. I want to offer opportunities to engage with the essential math of my classes, while also finding as many moments as I can to capture the beauty and depth of mathematical exploration.

Doing Math

I’ve had a bit of time recently to do math and explore some problems. Here are a few problems I’ve been thinking about.

Play With Your Math keeps coming out with great problems. I’ve enjoyed their most recent ones, but this week I returned to an oldie with new eyes:

I had solved the specific problem about 72 before, but I had trouble coming up with a general rule. I played with the problem a bit more this week, and started to find some fascinating patterns. I still have a ways to go until I have a general rule, but it’s been a ton of fun.

Here is another fun one. Deceptively simple, but lends itself to exploration:

This problem reminded me of an oldie from the now defunct Five Triangles blog. The three triangles are congruent and equilateral. What fraction of the total area is shaded?

Annie Perkins’ math art challenges have been a fantastic daily exploration. I’ve especially enjoyed playing with Mondrian puzzles and alternating knots:

Finally, this tweet from Marilyn Burns led me to a great puzzle.

Start with an even number less than 50. Then, pick a number that is a factor or multiple of the number you picked. Then again, and again, without repeating numbers, until you get stuck. How far can you go? I tried it with smaller numbers first, limiting myself to 1-10, and then 1-20.

I notice looking back at these problems that I can approach each one with some form of trial and error. So much of the math I do for fun has this feel. I might have a hard time finding the best answer, but I can find a few initial ideas, and then work to improve on those. I don’t think that’s an experience I give students very often in math class. I wonder how I can create a similar environment in the ways I ask students to learn math.

More On Distance Learning

I value having students do math. The core structure of my class is students do a bit of math –> I look at some evidence of student thinking –> I figure out how to discuss and summarize what they worked on, give a bit of explicit instruction, and figure out where to go next –> I give students more math to do.

Distance teaching, I’m having a hard time getting a window into student thinking. I usually have lots of tools. Looking at written work during class, listening in on partner conversations, in-class questioning, looking at exit tickets, and more. When student thinking has to pass through the internet it’s all a little less effective. I’m making inferences based on little snippets, trying to figure out what students know and what they don’t know. I’m also leaning heavily on prior experience teaching a topic. Where do students usually get tripped up? What are some common ways of looking at this idea? Where do we usually head next?

I’ve been thinking a lot about the question of synchronous vs asynchronous online teaching. I’ve been doing synchronous classes with some time set aside for one-on-one checkins. But when I’m teaching synchronously, I’m making those decisions about how to respond to student thinking on the fly, and not very well. Classes are short, so the whole thing feels rushed and students often don’t have the time they need to really think. Taking away the time pressure of short full-class sessions could give students more time to think and articulate that thinking, and more time for me to understand where they are and figure out where to go next.

I’m not sure where I’ll end up. So far I haven’t made any massive changes to my pedagogy teaching online. I go a lot slower, I’ve changed assessment to be low stress and low stakes, and I’m setting aside more purposeful time to check in with students one-on-one. But those are just tweaks, and I’m fundamentally trying to do the same things I was before. I’m not convinced it’s working, and I’m wondering what to try next. What are other assumptions I’ve brought from my normal classroom to my online one?

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 or 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.

The authors encourages a focus instead on what they call 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.