The Mathematicians Project

I first learned about The Mathematicians Project from Annie Perkins at Twitter Math Camp in 2016. I finally got around to doing it with my students this year. I start class once a week by talking about a mathematician from an underrepresented background. I find an interesting mathematician, throw 2-3 slides together with some fun images, share for a few minutes, and take questions. It’s been a ton of fun! I recommend giving it a try. Annie’s page has a long list of mathematicians. The tagline is “not just white dudes” and the list is sorted by identity. I found Wikipedia and Google Image searches work well to find some useful info and pictures. Doing an image search for “moduli spaces of Riemann surfaces” while learning about Maryam Mirzakhani was a ton of fun.

I think it took me so long to start doing this with my students because I’m very protective of class time, and I didn’t see this as “doing math.” It’s still worth doing because it impacts my students’ perceptions of who does (and doesn’t) do math. Those perceptions influence their learning as much as their grasp of logarithms. And the conversations led to more mathematical curiosity and rabbit holes to explore.

I found it useful to think through why I am choosing to emphasize mathematicians from underrepresented backgrounds. Why not include some old dead white dudes? They did plenty of interesting math. My goal is to undermine deficit narratives about who does math. If a cis straight white male student in my class doesn’t see a mathematician who looks like him in my class, there are plenty of other messages in math classes and in the world telling that student that he can be a mathematician if he wants to. But other students get the opposite message, that math isn’t for people who look like them. My goal is to undermine that narrative by giving counterexamples to societal messages. There are plenty of mathematicians to go around, but some don’t often have their stories told. Those are the stories I want to share.

Here are three things I’ve learned:

  • I better understand the biases that have shaped mathematics. There are lots of white dude mathematicians; I knew that. But I underestimated how hard it would be to find information about women of color mathematicians. I’ve found plenty of white female mathematicians, plenty of men of color mathematicians, but women of color have required more effort to learn about. This both reflects the biases that prevent people from becoming mathematicians, and the biases in whose stories get told.
  • The Mathematicians Project is about expanding student conceptions of who does mathematics, but it can also expand student conceptions of what it means to do mathematics. I’ve shared about Vi Hart, who makes playful videos about mathematics and creates mathematical artwork, and Artur Avila, who won a bet for proving something no one else could figure out. Sharing less conventional mathematics can help students to see more different ways someone can be a mathematician.
  • Mathematicians don’t have to be famous. Becky recently shared this paper by Federico Ardila-Mantilla with me. In it, Federico shares new ideas in optimizing object movement that could make robot arms more efficient. But he also discusses the ways that math can act as a tool for good or evil, unpacking a choice by Dallas Police to use a robot to kill a suspect. Federico asks moral questions and puts math research and Black Lives Matter in tension. These types of stories are worth telling as much as pioneering mathematicians from decades or centuries ago.

What should students learn in math class? They should learn math, of course. But the discipline of mathematics is not static. By making conscious choices to push at the boundaries of our discipline, I have an opportunity to help students see not just what is, but the potential of what could be. What could be is a discipline of mathematics that is more inclusive and broader than what we have now. That might not mean eliminating what exists, but it definitely means heading in new directions. The Mathematicians Project is one tiny way I can model what a search for new directions looks like in my class.

Who Does Conventional Wisdom Work For?

I’ve spent more time recently questioning the received wisdom of the teaching profession. Things that I did without thinking about them because everyone else did, like assigning homework every night or modeling examples on the board while students take notes. There’s usually a useful idea under the surface that’s worth keeping. I don’t think assigning homework helps students learn, but students do need to practice applying mathematical ideas. I’m starting to avoid modeling examples on the board, but analyzing examples is helpful in building flexible understanding. I’ve noticed that these ingrained teaching practices work well for students who have been successful in math class in the past. I wonder if there’s some confirmation bias at play. Which students end up becoming teachers? Do teachers just replicate the practices that worked for them in their own schooling?

It’s easy to caricature “traditional education” as this outdated factory model preparing kids for jobs that don’t exist. A teaching practice isn’t bad because it’s been around for a while. There are plenty of “progressive” teaching ideas I don’t find useful. I hear that students will remember concepts better if they discover the math themselves, or that not answering questions helps students think for themselves, or that students shouldn’t have to memorize anything because it’s the 21st century or something. I don’t find those practices valuable either. And I think they follow the same pattern. They work well for students who have strong background knowledge and more academic supports, and leave behind those who are already struggling.

I don’t want to throw away anything because it’s conventional wisdom or because too many other teachers do it. I don’t want to be contrarian for the sake of being contrarian. I do want to ask why these practices persisted. Is this something that has stuck around because it works for everyone, or just for those who math class has worked for in the past? The status quo is a hell of a drug.

Notes

For a few years I was an “I write you write” teacher. Every class there were things I would write on the board and ask students to copy into their notes. Notes like this are a staple of math classrooms all over. But now I avoid it like the plague.

The biggest reason: students didn’t learn anything from taking “I write you write” notes. Many students fall behind and copy blindly, and those are the students who most need to process what’s on the board. Plenty of students take great notes, but most of them would’ve been fine without anything written down. Too many kids are just miming without meaning, without actually doing any thinking, and never looking at their notes again.

I’ve found teachers are quick to defend notetaking, and they often talk about how much they used their own notes in school. But teachers are a biased sample — kids who take good notes are more likely to become teachers. Not to say notes are useless, but teachers’ impressions might not be representative.

I wrote this summer about a notetaking routine I’ve been playing with. I ask students to summarize the key ideas of the lesson in their own words, share that summary with partners, and add to their notes based on what their partners shared. But I’m not happy with how the routine has developed, and I want to tweak or redesign it to better meet my students’ needs. Here’s the back and forth I’ve been going through:

I don’t want students copying notes off of the board, because the students who most need to be thinking aren’t.

Having students summarize key ideas in their own words and share what they wrote with a partner causes more thinking, but students often feel unsure they’re writing down the “right ideas” and feel like they aren’t learning without formal notes.

Giving students a reference sheet with key formulas and terms saves time from notetaking and gives students a useful and reliable resource, but also means there’s less of an incentive to write notes in their own words and consolidate their understanding if they know they’ll get a sheet later.

Summarizing notes in their own words often means students don’t feel like they’re learning, because they’re used to taking notes verbatim and the extra effort means they feel less successful. I do think there’s as much learning happening, but I want students to recognize that too.

Examples are an important part of notes, because they give context to formulas and terms. But giving students an example doesn’t always elicit thinking, and students struggle to come up with their own examples.

Giving students worked examples can be a good way to avoid blindly copying and have students analyze and annotate an example, but if future problems don’t look just like the example students often struggle.

I’m not sure what the answer is, but I’ve learned a lot this fall about the different pushes and pulls in trying to create a notetaking routine that causes thinking for all students, creates a useful resource, and helps students feel like they are learning.

How Helpful?

How helpful should math teachers be?

More and more, other questions I have about teaching seem premised on my answer to this one.

How much time should students spend doing math? How much time analyzing math someone else has done, and how much time watching someone else do math? When a student asks for help, how should I respond? When am I contributing to learned helplessness by helping too soon? When am I sending a message that I don’t care by helping too little? When can a student learn as well by figuring something out for themselves? When do students need guidance to consolidate and connect ideas? When is a student so overwhelmed that even figuring something out on their own doesn’t lead to any real learning? When is frustration productive, and when does it reinforce negative ideas about what it means to learn math?

It’s easy to find speakers at conferences with what seem like clear and confident answers to questions like these. I find that those answers tend to break down in the complexities of my classroom. If I’m confident about one thing, it’s that these questions depend a lot on the particular student in front of me. How well I know them, how one experience fits with the rest of their time learning math.

The toughest part is the uncertainty. It’s hard to know what’s going on in a student’s head, and it’s hard to know whether I made the right decision last period or the period before.

Lesson Plans

I have always found that in preparing for class, lesson plans are useless, but lesson planning is invaluable.

-Not Dwight Eisenhower

Some knowledge is what cognitive scientists call “descriptive knowledge.” This is knowledge that one person can quickly transmite to another. Like “adding sugar keeps the vinegar from overwhelming the dish” or “be careful at the intersection on 8th street, it can get icy” or “if the freezer isn’t staying cold try cleaning the condenser coil first.” Other knowledge can’t, like riding a bicycle or reading an MRI. For me, lesson plans fall in that second category. They make sense to me, but I’m sure they wouldn’t make sense to anyone else. More and more I adjust as I go, adding or subtracting problems, finding different ways to get students talking, or responding to the energy and engagement in the room. I would have a hard time explaining where those decisions come from. Which is cool! But it also leads to a different type of planning. Rather than scripting questions or discourse moves, I find it more and more helpful to dive into the content I’m teaching. Which examples will be useful? Which might trip a student up later? Where are students likely to get stuck, and why? This isn’t lesson planning, at least not the way I would describe it. It’s not something I can write into a plan. It’s getting a sense of a concept, including all its ugly moments and awkward corners. And then getting a sense of how a learner might start to think about that concept, what leads to what. And it’s often counterintuitive to me how different a learner’s perspective is from my own.

To me, this is one of the cool things about teaching. It’s pretty different than I thought it would be when I started. Different mostly in that it’s way harder and richer intellectually. It’s also one of the reasons why I love writing about teaching. It’s a chance to try and take all of that implicit knowledge, make it explicit, and learn something in the process.

Curb Cuts

I’ve written often in the past about low-floor high-ceiling tasks. This week I’m thinking about what exactly I mean when I use those words. I started thinking because of this problem from Play With Your Math:

I’ve had so much fun playing with it. No spoilers here. But there’s something about the problem that invites exploration and curiosity.

Second, this recent tweet thread from David Wees:

The words “low” and “high” have the connotation of ability. One might imagine an elevator. A low-floor high-ceiling task is then one where the elevator can go “lower” to pick up certain low students, and also go “higher” to accommodate certain high students. This metaphor seems likely to reinforce fixed ideas about ability. I also worry about any task where different students work with and learn different math. The more time students spend doing different math, the harder it is to bring the class together for a productive discussion.

A second metaphor might be a spider web. Rather than presenting a task as a linear sequence of strategies, a task might have several different entry points and exit points. Here, one strand is not “higher” or “lower” than another. Instead, they might illuminate different representations of a problem or connect a problem with different mathematical ideas. There is still the challenge of bringing together different perspectives, but if those perspectives are all connected to the same central ideas they support different student approaches, rather than positioning some as better than others.

In David’s thread, he argues that teachers should emphasize student knowledge, rather than student ability. Ability is often seen as fixed, while knowledge can change. On the Play With Your Math blog, Joey Kelly describes three principles they use to design problems:

  • To make success attainable
  • To make space for curiosity
  • To shelter from inaccessible questions

I think these principles are particularly useful through the lens of knowledge. If I give students a task because I like the problem and think it’s interesting, that task probably isn’t going to go very well. If, instead, I focus on finding problems where success is attainable, I frame my planning through the lens of what my students know and can do. When students feel successful in class, they’re more likely to take risks, share ideas, and enjoy math. Discussions of low-floor high-ceiling tasks often focus more on the task than the students. Which is inevitable, because it’s much easier to talk about tasks than students, but also probably unproductive.

I have a third metaphor that might be more useful. I first saw it in a talk by Andy Gael.

This is a curb cut. Curb cuts were originally designed for wheelchair users, but they benefit everyone. People pushing strollers, transporting large objects, walking with other mobility issues, or even walking home intoxicated have an easier time getting around with curb cuts. And curb cuts aren’t seen as some niche accommodation for people with wheelchairs; they are ubiquitous, and they’re just seen as normal.

Returning to math problems, curb cuts provide an on-ramp to a problem for all students. The Play With Your Math problem above has a lot to do with prime factorization. But rather than asking a question about prime factorization, it provides an on-ramp by giving students a chance to play and experiment first. We can formalize their understanding of prime factorization later, but that won’t be an obstacle to entering the problem. The problem avoids making assumptions about what students already know. Know nothing about prime factorization? No problem. At the same time, the opportunity to explore and extend the problem offers space for curiosity. I can provide access without reducing the interest of students who already know some things about prime factorization.

I like the metaphor of a curb cut because it focuses on getting someone from where they are to where they are going. Curb cuts work for everyone, and they provide access rather than transforming the destination. They also don’t make assumptions or separate people based on their mobility. Most people don’t even notice them. Similarly, an effective task doesn’t need to be some crazy complex production that offers different options for different students. Instead, tasks often need simple changes that provide access to more students. I’d love to spend more effort finding curb cuts, and using this metaphor to guide how I design tasks that engage all students.

Explain Less

I’ve developed an instinct against explaining things to students. It’s an instinct I’ve developed for practical reasons. The more I talk, the less students listen. The longer I talk without a break, the more tenuous my hold on what students understand. Students get confused when I spend too much time sharing how I think, and too little time understanding how they think.

I have no philosophical issue with telling students things. In short bursts, when students feel a need, there’s nothing better than a concise explanation that lets them solve a new problem. Then, I ask students to solve the problem, see how it goes, and consider whether there’s something else I need to address or if they are ready to extend their thinking a step further.

Something I’ve found myself doing is finding more and more ways to explain less. Asking students to figure things out for themselves often doesn’t work, and when it does it tends to privilege students with strong backgrounds and positive past experiences in math class. Explaining less might mean breaking a lesson into little chunks, so that I’m only talking for a minute at a time in between having students solve problems, warm calling students after each problem to summarize a key idea for the class. It might mean using worked examples to have students generate an explanation themselves, and listening in to notice their thinking and build from it. It might mean presenting a problem as a puzzle drawing on prior knowledge, figuring out where students are with that prior knowledge, and building my explanation from there.

I read this great post about teaching Computer Science yesterday and couldn’t stop thinking about the elegance of the ways this teacher found to explain less. My favorite:

Here’s an example — the second thing I ever show students, right after print("hello world") , is this right here:

name = "Tamara"
print("Hello" + name)

And then I ask one simple question:

Don’t answer out loud — just think. What will happen when you run this program?

Don’t answer. Just think. What will happen?

Turn and ask your neighbor for their prediction.

Literally every student intuits that this program is going to greet Tamara.

And then after that, we run the program, find out that it prints  HelloTamara  without a space, and we also do our first round of debugging. High fives all around!

I can imagine myself as a novice computer science teacher trying to explain what this example will do. I imagine it as a complete mess. There will be a time, later, to formalize student knowledge of variables, strings, and more. In the meantime, students learned something , and they’re primed to learn more. This is a beautiful example. It’s not too complex, builds from intuition, and packs in surprise. I’d love to find more ways to do this in a math classroom.

Here are three things that I think matter in finding moments to explain less. First, you can’t force it. I never want to become dogmatic and refuse to explain things. Explanations are valuable, and they’re especially valuable when they’re used at the right time and place. Second, I don’t want to let perfect be the enemy of good. My worst explanations are when I try to explain something with mathematical precision and address every possible case. But that precision can create confusion. Sometimes students are ready for an informal understanding of a concept, but struggle to engage with too much complexity at once. Finally, finding ways to explain less trusts students more, and sends a message about what they can do. If every lesson begins with explanation, students learn that knowledge always moves from the teacher to the student. I don’t expect students to derive centuries of math on their own, but there are plenty of opportunities for students to extrapolate form their knowledge to something new. And every time we do that, students have the opportunity to trust themselves and their ideas a little more.