What Cognitive Science Misses

Architects, like teachers, usually have multiple goals they try to satisfy simultaneously. Safety is nonnegotiable, but architects may also be thinking to a greater or lesser extent about energy efficiency, aesthetics, functionality, and so on. In the same way, some goals for teachers are nonnegotiable — teaching kids to read, for example — but after that, the goals are likely to vary with the context. In addition, architects make use of scientific knowledge, notably principles of physics, and materials science. But this knowledge is certainly not prescriptive. It doesn’t tell the architect what a building must look like. Rather, it sets boundary conditions for construction to ensure that the building will not fall down, that the floors can support sufficient weight, and so on.

In the same way, basic scientific knowledge about how kids learn, about how they interact, about how they respond to discipline — this knowledge ought to be seen as a boundary condition for teachers and parents, meaning that this knowledge sets boundaries that, if crossed, increase the probability of bad outcomes. Within these broad boundaries, parents and teachers pursue their goals.

-Daniel Willingham, When Can You Trust the Experts? (p. 221)

I think about that quote often. I love learning about cognitive science and better understanding how the human mind learns. But I also try to remember that cognitive science can’t tell me how to teach. Even at its best, cognitive science can often only help me understand when I’ve gone wrong, and give me some hints as to where and what I might try differently.

Along those lines, this tweet caught my eye:

I think the slide is a great distillation of some ways that cognitive science can help teachers. These are simple questions I can ask myself about my teaching. Are students thinking right now? What are they thinking about? Do they have the knowledge they need for that thinking? These are important questions, and questions I could ask myself every day and continue learning about teaching.

But what cognitive science often misses is an equity perspective. Equity is hard to measure, so cognitive scientists don’t spend too much time thinking about it. At each step I can ask questions about why a student isn’t learning. A student isn’t engaged? Why not? A student is compliant but not truly thinking? Why not? A student isn’t thinking about the right stuff? Why not? A student is cognitively overloaded? Why?

These questions often lead to important realizations about learning, classrooms, schools, and people. Students don’t learn for all kinds of reasons. Uncovering those reasons is the part of teaching that tries to be responsive to every student.

False Dichotomies

I am trying to make antiracist teaching a priority in my classroom. Something I’ve noticed in myself is a tendency to silo antiracist and equity work. I tell myself that work happens when I do my mathematician presentations, or when I get to that one social-justice-y lesson plan, and then I focus on teaching math and tuck it away until next time. I need to get better at looking through an equity lens each class, each day.

One tool I’ve found useful for breaking down the false dichotomy between equity work and math is the TRU framework. I have this pinned above my desk:

I am trying to ask myself these questions each day. And when I observe the lesson through a student’s eyes, I ask: which student? Are some students invited to explain things, and others not? Can some students hide or be ignored? Are some students recognized as more capable? Which students? Why?

These are questions about math teaching, but they are also questions about which students are invited to contribute fully in math class, and which students are not. If I put my equity lens away because I’m focused on “just teaching math,” I miss opportunities to recognize where I’m falling short, and how I can do better. It’s something small, but it’s something.

“Ignorance is no protection from the consequences of inaction.”

America is an old house. We can never declare the work over. Wind, flood, drought, and human upheavals batter a structure that is already fighting whatever flaws were left unattended in the original foundation. When you live in an old house, you may not want to go into the basement after a storm to see what the rains have wrought. Choose not to look, however, at your own peril. The owner of an old house knows that whatever you are ignoring will never go away. Whatever is lurking will fester whether you choose to look or not. Ignorance is no protection from the consequences of inaction. Whatever you are wishing away will gnaw at you until you gather the courage to face what you would rather not see.

We in the developed world are like homeowners who inherited a house on a piece of land that is beautiful on the outside, but whose soil is unstable loam and rock, heaving and contracting over generations, cracks patched but the deeper ruptures waved away for decades, centuries even. Many people rightly say, “I had nothing to do with how this all started. I have nothing to do with the sins of the past. My ancestors never attacked indigenous people, never owned slaves.” And, yes. Not one of us was here when this house was built. Our immediate ancestors may have had nothing to do with it, but here we are, the current occupants of a property with stress cracks and bowed walls and fissures built into the foundation. We are the heirs to whatever is right or wrong with it. We did not erect the uneven pillars or joists, but they are ours to deal with now.

And any further deterioration is, in fact, on our hands.

Unaddressed, the ruptures and diagonal cracks will not fix themselves. The toxins will not go away but, rather, will spread, leach, and mutate, as they already have. When people live in an old house, they come to adjust to the idiosyncrasies and outright dangers skulking in an old structure. They put buckets under a wet ceiling, prop up groaning floors, learn to step over that rotting wood tread in the staircase. The awkward becomes acceptable, and the unacceptable becomes merely inconvenient. Live with it long enough, and the unthinkable becomes normal. Exposed over the generations, we learn to believe that the incomprehensible is the way that life is supposed to be.

Isabel Wilkerson, Caste: The Origins of Our Discontents

I keep coming back to the phrase “learn to step over that rotting wood tread in the staircase.” It’s easier to look away than confront what’s broken. It’s a learned behavior, but one that fades into the background over time. “Ignorance is no protection from the consequences of inaction.”

I’m approaching this school year attempting to look with new eyes. Where is the rotting wood I’ve learned to avoid? What injustice has become normal? I’m also trying to recognize my own defensiveness. I might not have built the house, but I live here now. That’s what matters.

There is no neutral. Being neutral or unbiased is a pleasant fiction we tell ourselves, rather than recognizing we’re biased toward the status quo. Instead, I want to cultivate values of justice and equity, and a bias toward action. I don’t mind being labeled an “activist” or an “instigator” if that is what living out my values looks like.

What does all that actually look like? Too much for one blog post. Seems like I’ve got some writing to do this year.

But though I was initially disappointed at being categorized as an extremist, as I continued to think about the matter I gradually gained a measure of satisfaction from the label. Was not Jesus an extremist for love: “Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you.” Was not Amos an extremist for justice: “Let justice roll down like waters and righteousness like an ever flowing stream.” 

So the question is not whether we will be extremists, but what kind of extremists we will be. Will we be extremists for hate or for love? Will we be extremists for the preservation of injustice or for the extension of justice?

-Rev. Dr. Martin Luther King Jr, Letter from a Birmingham Jail

Opening School, Lies, and Fear

I often hesitate to write about my school. I work at a small private school that is unusual in a lot of ways. I won’t go into all that detail here, it doesn’t matter much. This week we’ve had meetings to prepare for students coming in person on Monday. And my experiences echo what I’ve heard from teachers around the country.

The first few days were fine. We had a mix of distanced outdoor meetings and work time. Administrators struggled to articulate our plan clearly and qualified literally every protocol with “when possible,” but the plan wasn’t totally terrible. In the middle of the week they changed “possible” to “practical.” Then, on Thursday, we had our first “all-staff” meeting, with both teachers and non-teaching staff. The meeting was in a large multi-bay garage. The first red flag was that tables were set up using less than half of the space, with many of us spread only about three feet apart. There was room to spread us out, and no reason not to besides carelessness or indifference. I chose to sit in a camp chair on the floor distanced from the tables. An administrator asked me whether I wanted to sit at a table, seemingly unaware that a reasonable person might choose to distance themselves from other humans during a pandemic. Then, that same administrator began the meeting by pulling his mask below his chin and telling us that the school was committed to our health and safety, and that they had worked hard all summer to prepare for student arrival. The cognitive dissonance nearly made my brain explode. Then things got worse. An outside presenter began by asking us to do introductions. He also had a mask pulled down below his chin. Every other adult kept a mask on while sharing introductions except for the same administrator, who pulled his mask down to speak, now only about three feet away from one teacher. Next, the outside presenter asked one teacher to come up to preview a demonstration of a helium stick. Thankfully, a different administrator asked him to use his mask. Then he asked for a group of volunteers who felt comfortable to arrange themselves in this formation for the demonstration:

That’s when I spoke up. I said there was no need to do this activity, and it didn’t matter if individuals felt comfortable because they were breaking protocol and that decision affects everyone in the room, and why weren’t we physically distanced when there was plenty of space, and why are administrators sending the message that it’s ok to take your mask off whenever it’s inconvenient?

I was terrified. It’s really scary to speak up in a room like that, and to call out literally everyone I work with. And I regret that I took half an hour to say something. I still can’t understand why no one else spoke up. I felt like I was going to start yelling something much less polite so I left to go for a walk, cry a bit, and blow off some steam. The message in that meeting was crystal clear. My school’s commitment to health and safety is a lie.

There are two ways to decide to open a school. The first is to start with safety. You decide what conditions allow schools to open. These might include minimal local and statewide community spread of the virus, ability to distance students, ventilation to circulate fresh air, protective barriers, robust mask and behavior policies, confidence that students can learn in the new classroom environment, and more. Then schools decide whether they can meet those standards, and the standards determine whether the school opens. The second way to make the decision is to start with the decision itself. Many schools are opening for political reasons whether they are prepared or not. Then schools make some efforts toward safety, but they’re all optional because the school is opening no matter what.

That’s what we’re doing. And opening without clear protocols will cause all kinds of problems. Strictly speaking my school could argue that they didn’t break protocol in that meeting. They can just say that it wasn’t “practical.” They’d be lying, but that’s how our protocols work. And there’s a lot of lying going on right now in schools around the country. When every protocol is subjective every teacher will interpret it differently. We’ll criticize plans only to get lectured about how hard administrators worked all summer on the plan that doesn’t make sense. We’ll spend our time bickering with students about how the teacher down the hall told them they can do this or can’t do that. We’ll speak up about inconsistencies and watch administrators retaliate against us. We’ll get lucky without an outbreak for a few weeks and watch norms erode just in time for a superspreader event. We’ll raise concerns about our health and safety only to be dismissed with “we’re all doing the best we can.” All because we’re telling ourselves some pleasant fictions about the value of schools and consoling ourselves with token gestures, rather than making real commitments to health and safety.

I read some drivel a few weeks ago about how teachers are essential workers, and we need to show up the same way nurses and doctors have been doing. I won’t link to it here. But go talk to some medical professionals. It’s become incredibly rare for nurses and doctors to get sick in hospitals. We have invested in high-quality PPE, and health-care settings are diligent about protocols and provide every resource they can to their employees. Now go down the street to the grocery store in my town. I don’t think a single cashier from six months ago is still working there. Because we, as a society, chose to value efficiency over safety in grocery stores and many other front-line workplaces. Now we’re doing the same with schools. It’s the same story we’ve seen as long as the teaching profession has existed. There are lots of people willing to stand up front and say nice things about how much we value teachers. But actions speak louder than words. Where the rubber meets the road we didn’t give a shit this week about safety. I don’t see any evidence that will change when students show up. Our school has sent a clear message. Teachers are disposable. We might be useful today but they can find someone else tomorrow.

The story is different at every school. The decisions are made at different levels, administrators are varying levels of incompetent, buildings range from decrepit to brand new. I’m lucky. I work with incredible teachers, with more resources than most. We have several compassionate and competent administrators. I think we will get through this. But that’s not the case for everyone. It’s because we as a society choose not to value teachers. That’s a choice. That choice can change. It can change when we decide that teachers are professionals, and that every professional — teachers, other frontline workers, and everyone who shows up each day ready to work hard and contribute to their community — deserves a healthy wage, a safe workplace, and the dignity of respect for their efforts.

I feel anxious, and scared, and angry. I can’t sleep. I can’t think about what I experienced this week without feeling rage. I don’t know what the next week, or month, or year will bring. I’m trying to practice gratitude. I am excited to see students on Monday. I’m excited to teach math in person again. But I can’t shake the fear. I feel like a pawn in a game run by people that don’t care about my well-being. It’s a fear I will carry with me this year. Over time it might fade into the background, like all of the other indignities teachers suffer. But right now the fear is suffocating. And I know teachers across the country feel different flavors of that same fear right now.

Black Lives Matter

Black lives matter.

I recently finished reading The Nickel Boys by Colson Whitehead. It is a haunting and beautiful novel based on real events in the 1960s at the Dozier School for Boys, a reform school in Florida. Whitehead writes in a note at the beginning, “As I contemplate how to prevent tragedies such as the one in these pages, I tumble into another, equally maddening netherworld: the one between action and de facto complicity.” In short, the book is about the Black boys at Nickel Academy. Nickel is ostensibly a reform school for boys too young to enter the adult criminal justice system. Instead, it is a hellish place of violence and abuse of power. I won’t say too much because it is worth reading in full without spoilers. But the thing that struck me most as I read is how violence can become mundane. The staff are cruel and inhuman toward the Black boys in their care. At first it is horrifying and tragic. Then it becomes normal. Eventually violence fades into the background, just another fact of life. It is shocking how humans can become inured to everyday cruelty. We see it, but it is easier to stay silent. Soon that silence becomes a habit. Then that silence becomes another piece of the structure that props up racism. Many of the boys who lived at Dozier are still alive today. I’m sure many are marching. That violence is our history. And, like much of our history, that story is often untold because the narrative is not convenient for white America.

Black lives matter.

I’m writing about The Nickel Boys today because our country has risen in protest of that same type of violence. I hope that justice will come for the killers of George Floyd, Breonna Taylor, and Ahmaud Arbery. I also know that white people need to recognize the violence that has become mundane in our country. Millions of voices are sharing their experiences as the victims of state-sanctioned violence. In this moment, protest is fashionable. I hope the protests will create change. But at some point we will go back to our lives, go back to “normal.” And when we do we risk allowing that same everyday violence that has plagued Black people in America for four hundred years to become normal again.

Black lives matter.

I am a white teacher. I see echoes of that same violence play out in our classrooms. The details vary from school to school, but the story is the same. The Black students who we label as “scary” or “aggressive” or “violent.” The teachers who aren’t willing to say “Black” or “racism” out loud. The silence as another Black student falls through the cracks. The excuses or euphemisms, blaming anything but race. The pressure on teachers of color to carry the weight of equity work. We’re often willing to accept that racism exists in the abstract, yet do mental gymnastics to convince ourselves that it’s not right in front of our eyes. George Floyd’s death was not unique. He just happened to be caught on camera. In the same way, racism in schools does not present itself on a silver platter with clear-cut solutions. Racism is pernicious and pervasive, and it will take more from white people than holding a sign or chanting a slogan to dismantle.

Black lives matter.

Now is a time to speak up and speak out. But every time is a time to speak up and speak out. White folks have a responsibility to build capacity in themselves, to build capacity in others, to recognize inequities, and to use our power and privilege to do something. I’m writing today to recognize the importance of this moment, and also to remind myself of the importance of every moment. I remember the moments when I was silent. I remember my missteps and mistakes. I will remember that my privilege compels me to use my power in every moment, not only when it is convenient. Today I am challenging myself, and challenging white folks who are reading this. What will I do now, in a moment when I can contribute to a movement for long-overdue change? And what will I do when I return to school in August? How will I continue the work of speaking up and speaking out to create a world where everyday racism is no longer mundane or normal, where normal is freedom and flourishing, and white people embrace the burden of lifting up those who have been oppressed?

Black lives matter.

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?