Category Archives: Uncategorized

Some Math I’m Playing With

I’m trying to spend more time playing with math. Because it’s fun, because I learn about learning, and because I want to practice the discipline I teach. Here are a few problems I’m playing with right now. No spoilers please, but feel free to play along!

The two diagrams above are different pictures of “Borromean Rings.” They are each a link of three components, which basically means three loops of string arranged together. The Borromean Rings have the property that the components cannot be separated as pictured, but removing just one of the components means that the other two can be separated. A link with this property is called “Brunnian.” Can you find a Brunnian link of four components? Of more components? (problem from Colin Adams’ The Knot Book)

This is the most recent problem from Play With Your Math and I have already had a ton of fun with it. It’s neat because the number of possible mountain ranges grows quickly, so it is very hard to count each possible mountain range individually for large numbers. I think it is possible to organize my work to find patterns that help me to make better predictions. I thought I had a breakthrough yesterday and then realized it didn’t work. So fun!

The Fibonacci numbers are really cool. They go like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,… where each number is the sum of the two numbers before it. They seem to have a neat property. 21 is the 8th Fibonacci number. 8 is divisible by 2 and 4, and 21 is divisible by the 2nd and 4th Fibonacci numbers, 1 and 3. Is this property true in general? Why? What other divisibility properties do Fibonacci numbers have? (problem inspired by this book, which was based on the PCMI 2012 math course)

I played a neat game recently. In a group of at least three people, each person randomly chooses two other people. Your goal is to stay equidistant from (though not necessarily at the midpoint of) your two people. Some positions act as an “equilibrium.” For instance, with three people, an equilateral triangle is an equilibrium position. What do some equilibrium positions look like for larger numbers of players? What is the probability that equilibrium is possible with randomly chosen people? What about a different version where each person chooses a “hero” and a “villain” and tries to keep their hero between them and their villain?


Precalculus, and Making Sense

I’m getting sick of teaching precalculus. Don’t get me wrong, there are some gems in there. I love teaching probability and bringing in Ben Orlin’s excellent The Bear in the Moonlight series of fables. Teaching sequences and series is a ton of fun; I get introduce students to some fascinating and thought-provoking perspectives on infinity.

But too much of the curriculum is there for the same reason: “you’ll need this in calculus.” Nothing deadens my soul like hearing that students needs to learn something for the sake of learning something else in the future, on and on forever.

Juxtapose with the big event in the mathematical news this week:

My favorite response:

The first has become a bit of a trope. Problems like it come along every few months, with the apparent goal of making people feel dumb and reinforcing the idea that math is this inscrutable language with arbitrary rules that don’t make sense. The second isn’t a thing; while language can be ambiguous, we understand that meaning comes from context and don’t feel stupid if we aren’t sure what the writer is saying.

Back to precalculus. One negative experience I have had more times than I can count teaching this course for the last four years goes something like this. Let’s say I’m teaching logarithms. Most students have seen them before, and most students have forgotten what they are and why they might be important. We take our time making sense of the idea of a logarithm, first informally (credit to Kate Nowak) and then with more precision. It seems a bit arbitrary that students have to learn this, but it makes sense. Then we move into the other log rules and it all goes to shit.

Sure, if I was a great teacher we could take our time and do this all right — there’s plenty to make sense of here. But precalculus is a race. There’s so much to cram in that spending the time to do log rules right gives short shrift to something else. More likely I skim a few key ideas and move onto the next thing.

The problem is the curriculum. It’s not designed for students to understand a coherent body of mathematics. The goal is to be able to push symbols around and remember some disembodied rules that might be useful in the future. And students often come out the other end with their worst ideas about math confirmed. Math is about manipulating letters and numbers in weird ways, doing what you’re told, and getting on with your life.

Much of high school math works like this. It’s a race to calculus, with everyone’s pet topic shoved in just in case. And it leaves teachers with a choice. Take the time to do things right and skip out on some topics, or sprint through everything and pray.

Right now, the central goal of high school math is to prepare students for calculus. What if, instead, our goal was for students to believe that math makes sense? What wouldn’t we teach? What would we add? What would math class look and feel like?

Whiteness and Racism

To any citizen of this country who figures himself as responsible—and particularly those of you who deal with the minds and hearts of young people—must be prepared to “go for broke.” Or to put it another way, you must understand that in the attempt to correct so many generations of bad faith and cruelty, when it is operating not only in the classroom but in society, you will meet the most fantastic, the most brutal, and the most determined resistance. There is no point pretending that this won’t happen.

James Baldwin

[W]e as dark people see—which White Americans cannot—a country with enough promise to capture and hold four hundred years of freedom dreams while systematically attacking, reducing, and/or destroying each and every aspiration…We who are dark want to matter and live, not just to survive but to thrive.

Bettina Love

I have a lot to learn about fighting racism. It’s June and I’m reflecting on the moments I came up short this school year. I’d like to share a bit of what I’m thinking about, from reading broadly and trying to better understand racism at my school. I’m writing in the hope that reflecting on my experiences might help me and other white educators to see in new ways and develop new tools to address racism in our schools.

For four hundred years, people of color have been locked out of the promise of the United States. Whiteness is the system that perpetuates this racism; it ensures that white people remain at the top of the racial hierarchy, and prevents change that might provide more opportunity to people of color. In Making Meaning of Whiteness, Alice McIntyre defines whiteness as “[A] system and ideology of white dominance that marginalizes and oppresses people of color, ensuring existing privileges for white people in this country.” In its modern form, whiteness is rarely explicit; it isn’t acceptable to advocate openly for white superiority. But even in the absence of explicit advocacy, whiteness perpetuates and protects white hegemony.

What Is Whiteness?

Whiteness is a system, and I mean something very specific by the word “system.” Racism didn’t happen by accident, and it can’t be eliminated overnight. It was built over centuries and permeates modern culture. Whiteness is a system that, when the racial hierarchy is challenged, responds by undermining the voices working against it and reinforcing the existing hierarchy in a self-perpetuating cycle. Any time the status quo is threatened, whiteness acts to maintain control of the narrative and distract and deflect from the truth of racism, preventing progress.

Whiteness is a central narrative in American history. Here’s just one example. For centuries, white people have plundered wealth from Black communities. First through slavery, but in the century and a half since slavery plunder has continued through discriminatory housing practices, underresourced schools, mass incarceration, and more. Any time an element of the system of plunder is threatened, the Black community is given a small fraction more opportunity. Immediately, white people say “Well we solved racism. Let’s move on.” and “Why haven’t Black families caught up to white people on [insert measure]?” Why aren’t Black families buying homes immediately after they’ve first been allowed to take out mortgages? Because mortgages were just one of hundreds of obstacles to building wealth for the Black community. The fact that Black families don’t instantly recover and match the economic success of whites is taken as justification for reversing progress and reinscribing the same systems that have kept the Black community from accumulating wealth. In the same way, any gains from affirmative action, decriminalizing drugs, and more are immediately eroded as whiteness reframes the problem as a deficit of people of color, ignoring the impacts of centuries of plunder. It’s a self-perpetuating cycle of incremental progress and blame undoing that progress, as each step reinforces the other.

I see these same dynamics at play in schools. For instance, racism is likely to prompt an emotional response from those most affected by it. But when someone speaking up about racism expresses emotions, they are characterized as letting their emotions prevent them from thinking rationally, minimizing their perspective and experience. Racism is allowed to continue, and the frustration of watching oppression perpetuated engenders more emotions. The cycle continues.

Intention is another weapon in perpetuating racism in schools. An individual in power does something that perpetuates racism. If someone speaks up, those in power claim good intentions, and others defend those “good intentions.” This acts as a distraction, deflecting attention, preventing further conversation, and slowing change. Robin DiAngelo writes:

We’re never going to be able to come to an agreement on intentions. You cannot prove somebody’s intentions. They might not even know their intentions. And if they weren’t good, they’re probably not going to admit that. The question I ask is, “How does this function?” The impact of the action is what is relevant.

Self-perpetuating systems also function because of the lack of racial awareness in white people. White people rarely need to monitor their language or social context. Meanwhile, people of color grow up knowing they have to act in specific and arbitrary ways to ensure they survive encounters with the police, aren’t accused of shoplifting, and are perceived as competent in the workplace. Then, when white people are asked to think critically about the impact of their language on those around them, they bristle with defensiveness because they’ve never had to check their tone before. The existing hierarchy is recreated because of a lack of racial awareness. Similarly, white people are socialized to believe that if they work hard, they deserve to be successful. If something doesn’t go their way, they fight and try to bend the rules. Meanwhile, people of color know the system is rigged and might feel a sense of futility, knowing that those in charge don’t look like them or understand them. White people continue to hoard opportunity and control the levers of power. Finally, white people rarely talk about race, yet race is always impacting the lives of people of color. So when race comes up, white people are likely to be uncomfortable and shut down the conversation, or accuse someone of “just making it about race, when there are so many other factors.” Of course it’s never just about race; humans are complex, and no one can be essentialized to one element of their identity. But white reluctance to talk about race means that the ways that racism is never surfaced and opportunities for change pass by.

So I’m a white teacher. I work in a school where I see whiteness perpetuating racism. What is my role?

How Is Racism Operating Here?

First, without understanding the way that whiteness and racism work, and realizing where they manifest themselves, they cannot be undone. Paul Gorski writes:

The path to racial equity requires direct confrontations with racial inequity—with racism. We start… by asking, “How is racism operating here?”

I need to be able to recognize how racism operates, and how whiteness works to entrench the existing racial hierarchy, slowing progress toward racial equity. This takes practice; I lean heavily on Gorski’s work identifying common racial equity detours. When those in power blame people of color for their challenges or insist that “we’re working on it, but this takes time,” I know that I’m witnessing whiteness at work.

One central part of doing this work as a white person is recognizing that I will never see dynamics of marginalization in the same way as those most affected by oppression. As a white man, people are likely to listen when I speak. I can use my position to elevate voices people of color, and members of marginalized communities more broadly, to help surface problems of equity and explore potential solutions. I will never have all the answers, but by learning from others I can help recognize the problems that need to be solved, problems that might be ignored otherwise.

Tell the Truth Harder

As I get better at seeing the truth, I have a responsibility to speak up. Naming racism makes those in power uncomfortable, and I should expect to see backlash. But as a white person, I have an opportunity to break the cycle of racism and whiteness that is often denied to people of color who speak up. I can amplify those whose experiences and perspectives allow them to notice dynamics that I don’t, and practice telling the truth harder when racism comes to light.

Why do emphatic equity advocates often face harsher repercussions for their advocacy than equity heeldraggers face for their inaction? Why is taking a strong, impassioned stand on racism interpreted as deviant while refusing to take a stand on racism is interpreted as in a developmental process (Mayorga & Picower, 2018)?

Paul Gorski

In Dare to Lead, Brene Brown describes the emotional responses people are likely to experience in these situations:

The majority of shame researchers and clinicians agree that the difference between shame and guilt is best understood as the difference between “I am bad” and “I did something bad.”

Guilt = I did something bad

Shame = I am bad

While shame is highly correlated with addiction violence, aggression, depression, eating disorders, and bullying, guilt is negatively correlated with these outcomes. Empathy and values live in the contours of guilt, which is why it’s a powerful and socially adaptive emotion. When we apologize for something we’ve done, make amends, or change a behavior that doesn’t align with our values, guilt—not shame—is most often the driving force.

As I tell the truth harder, I also need to recognize the emotions at play when white people hear hard truths. I need to put my defensiveness aside and find opportunities for action within my sphere of influence, while naming unproductive emotions in others and using my position to help other white people see the ways that their emotional responses function to perpetuate racism.

Creative Insubordination

Whiteness is a self-perpetuating system, and working within the system inevitably leaves racist structures intact. Rochelle Gutiérrez offers strategies for undermining power dynamics “creative insubordination.” I can press for explanation, uncovering the reasoning behind racist behavior while I formulate arguments. I can counter with evidence, drawing on broader research to undermine faulty logic. I can use the master’s tools, flipping racist structures upside down and use them to dismantle oppressive systems. I can seek allies, building a network of support to better advocate for change. I can turn a rational issue into a moral one, shifting the discussion from logic that minimizes the feelings and experiences of marginalized communities to ethical questions that require action. And I can fly under the radar, making change within my sphere of influence without drawing attention to equity work that might otherwise cause a backlash.

A sentence each doesn’t do justice to these strategies; read Gutiérrez’s paper for more context. The most important strategy for me is to seek allies. I learned this year that when I go it alone, I am easy to dismiss as “radical” or “biased.” When I build relationships and seek allies, I can help others hear hard truths. And I’ve found that people I don’t think will be interested in speaking up can become powerful allies and sources of learning; I often position myself as a lone crusader, letting my frustration convince me that I’m fighting the battle alone when I can be much more effective and learn much more by leaning on others.

Action Steps

I’ve experienced a lot of emotional responses as I’ve come to see racism in schools in ways I didn’t before. As a white person, shame can paralyze me and prevent action. At the same time, I can fall into the trap of easy fixes and token change that don’t actually address racism. I’ve often come up short at speaking up and working to redress inequity when I see it. I want to practice using my understanding of whiteness and my identity as a white man to feel a sense of agency, and to search for the structures and decisions I can influence and moments when I can disrupt racism. I have three commitments to make to better practice equity work in my sphere of influence:

  • Use my privilege to elevate voices of people of color, creating space to hear and see in new ways for myself and others.
  • Tell the truth harder, speaking up when I see whiteness in action and articulating the ways that systems perpetuate racism.
  • Seek allies, finding ways to collaborate with others to surface inequity and create change.

Anyone who has been transformed through a struggle can attest to its power to open up more capacities for resistance, creativity, action, and vision.

Nick Montgomery & carla bergman

I’ve experienced sadness and frustration in attempting to do equity work in the last year. At the same time, I notice and understand in ways I did not before, and I know I have made change, even if that change is smaller than I would like. I want to focus my energy on my capacity for action and remember that, while fighting racism seems like an endless uphill battle, my role is not to center my frustration, but to support those most affected and continue to develop my ability to create schools where every student is truly seen and valued.

The “Compulsion to Calculate”

I just read this great paper by Karen Bogard Givvin and James Stigler, “Removing Opportunities to Calculate Improve Students’ Performance on Subsequent Word Problems.”They ask one group of students to explain typical word problems, and ask another group to explain similar word problems but with the numbers removed so that calculating an answer is impossible. Students who solved problems with numbers removed both solved similar problems more accurately, and transferred their knowledge to new and different problems more successfully than the control group. Cool!

Brian Bushart calls these numberless word problems and has done awesome work writing, collecting, and thinking about them on his blog. Both Brian and the authors of the paper describe the strategy as a way to help students slow down rather than immediately applying procedures without thinking about the relationships in a problem. Givvin and Stigler reference this as the “compulsion to calculate.” They write:

They found that when students are presented with a mathematics word problem, their first response often is to try to compute an answer, even before they have tried to understand the problem. The description offered by Stacey and McGregor (1999) reminds us of the community college students we interviewed, who appeared not to think long about the problem posed, but instead to search their memory for a procedure that some teacher, at some point, had told them to use.

Something really interesting: almost half of the students in the study made up numbers to calculate with, seemingly feeling that number are always necessary to solve a math problem.

I’ve never used numberless word problems with my students (though now I want to). But the strategy reminds me of other activities I’ve come to find useful. I’m interested in better understanding this idea of the “compulsion to calculate” in this context. I worry that, as I experiment in my teaching, I’m drawn to activities that feel fun and new and clever, but might not actually have much value for student learning. The online math education world can lend itself to style over substance. I think that the “compulsion to calculate” articulates something useful that goes beyond clever ideas. If students are prevented from jumping into a procedure, but instead think about the relationships in a problem, focusing on where a certain procedure is useful and why, they are making important connections that are often lost in the race to calculate.

I think calculation is important! Students should calculate things in math class. But if students are only calculating, they are missing opportunities to make broader connections. Is a problem about multiplying numbers, or knowing when a problem requires multiplication? Is a problem about solving a logarithm, or considering what a logarithm represents? Is a problem about graphing a function, or thinking about the structure of the expression that determines the graph? As the study demonstrates, students are often predisposed to lean on calculation as the central component of what it means to “do math.” Finding ways to mitigate the compulsion to calculate acts as a corrective, balancing calculation with relational thinking that is otherwise lost.

Having this lens helps me to think about which new activities I want to find ways to bring more regularly into my class. I don’t want to try new things for the sake of trying new things, but ideas like menu math, connecting representations, and which one doesn’t belong? provide opportunities to facilitate this type of thinking. I also appreciate having a new lens through which to understand student thinking and diagnose when things are going wrong. When students all struggle with a certain type of problem, maybe the issue is over-reliance on a procedure without understanding where that procedure applies. Strategies to mitigate the compulsion to calculate are particularly useful in those moments to create opportunities for thinking that is missing.

Mathematical Magic

(via Robert Kaplinsky)

I notice that I’ve seen this gif on social media a few times recently.

I notice that the exterior angles seem to add up to 360 degrees.

I notice that I’m drawing on a lot of prior knowledge to make that conclusion. There are lots of angles one could highlight in the original polygons; I understand which ones are exterior. I understand the idea of angles greater than 180 degrees, so that a full rotation makes sense as 360 degrees. There are probably other things I don’t realize I know because I’ve spent lots of time solving angle problems with polygons.

I wonder if this would make any sense to a typical 8th grader.

I wonder when in an instructional sequence this gif would make the most sense.

I wonder if this gif makes a substantive mathematical idea seem like “magic.” I wonder what message that sends about the nature of mathematics.

I wonder how often I explain an idea in a way that connects with students who already understand an idea, but not with students who feel confused.

Prerequisite Knowledge

I had an odd extra day in one of my precalc classes last week and picked up my college combinatorics and graph theory textbook for some math to explore. We spent the period playing with graph theory. This fun problem from Play With Your Math was an awesome place to start:

We played with this for a bit, and then we looked at colorings of graphs. How many colors do you need to color a graph of the United States? What about Canada? (Hint: they’re different!) Then we played with the Bridges of Königsberg. The map below is of the Prussian city of Königsberg, with bridges highlighted in green. Leonhard Euler wanted to go for a walk crossing each bridge exactly once, without repeating any bridges. This led to explorations of Euler and Hamiltonian paths.

It was a pretty relaxed class. I’m not sure how much students learned, but it was a fun one-day excursion into some neat math. The best thing about the class was that it relied on so little prerequisite knowledge; students didn’t need to know about fractions or factoring or functions, but could play with new and challenging math on their own terms. I’ve taught Algebra II and Precalculus this year, and I hear from so many disaffected students that geometry is the only math class they ever enjoyed because they didn’t feel behind from the beginning and the math made sense to them. But geometry is an outlier. Most of our math curriculum is designed sequentially in the race to calculus, building on ever more complicated layers of algebraic manipulation.

Is mathematics fundamentally sequential, or do we just choose to make it so? I wonder what a school math curriculum would look like if it were designed to minimize the impact of prerequisite knowledge, to help every concept feel accessible to every student. Which topics would we eliminate? Which topics would we add? Which topics would we teach differently? My current teaching load feels like it is designed to do the opposite; there are so many places a student is likely to feel confused because of something they missed or forgot from a year or several years before. How does that make students feel? Which students who aren’t invited into the math conversation now might be if we approached math in a new way?

When to Nix the Tricks?

I love Nix the Tricks. I try to teach in ways that encourage students to make sense of mathematics, to believe that mathematics makes sense, and to understand what they’re doing rather than following a recipe provided for them.

But today, I told a student to flip and multiply when dividing fractions.

I’m teaching trig identities with my precalc students. I love this part of the unit; it’s abstract and challenging at first, but over time students start to see problems as little puzzles to figure out. I don’t mess with the product-to-sum and other more obscure identities, focusing more on reciprocal, quotient, and Pythagorean identities. I love questions like this one:

It seems so counter-intuitive that these expressions could be equal, yet they are. Students need to know their basic identities, need to be comfortable with algebraic manipulation, and need to remember some things about fraction operations. I like this unit because I find students can transition from seeing these problems as inscrutable and pointless symbol-mashing to seeing them as satisfying and logical puzzles.

But this isn’t a unit on fraction operations. Students were working on this problem:

Those divisions are hard! How do they work again? And we’re early in the unit, so there are some feelings of frustration that what we’re doing doesn’t make sense and students feel dumb. Frustration with fraction division is layered on top of all that. I could dive into an explanation of why dividing by a fraction is the same as multiplying by the reciprocal, but what I really want right now is for my students to feel successful working with trig identities and recognize the ways that they already know most of the math they need to solve these problems. Digging into fraction division feels like a distraction from the key ideas of the lesson, when students are happy to be quickly reminded of a procedure that will help them solve the problem in front of them.

Does this make me a bad teacher? Maybe I missed an opportunity to anticipate the difficulty with fractions and preteach some of those ideas before the unit. Maybe slowing down to spend time on the conceptual basis of fraction division is the right move. Maybe I should revisit fractions tomorrow. But these moments come up all the time, especially with precalc students. Are these moments distractions from the heart of the math we’re working on, or opportunities to circle back to math students have seen before?