Monthly Archives: August 2012

Bill Thurston, Math Hero

William Thurston passed away yesterday. I’m feeling really torn up about this, and I thought that writing about it might help. William Thurston has been a hero and inspiration of mine for a number of years now. I’ve often daydreamed of getting to study with him, or even just getting to hear him speak. And now that will never happen.

In my head, he’s “Thurston”—I would get excited about “reading some Thurston” or running across some new way that he was involved in mathematics that I’m interested in. And maybe he’ll feel like “Thurston” again to me someday. But right now—even though I don’t have any right to speak of him familiarly—I can’t help but want to call him Bill. I know intellectually that the feeling of closeness is illusory, but I can’t help myself.

I came to know Bill’s work through my ongoing interest in non-Euclidean geometry. It’s an amazing subject, full of beautiful objects, ideas, and theorems. It connects with compelling questions of philosophy and psychology, and its historical arc is an epic human story. And maybe most of all, it feels like a world unto itself, one that I’ve inhabited for a while and feel kind of at home in.

Bill’s ways of thinking about three-dimensional manifolds revolutionized their study.  I’m a rank amateur when it comes to this stuff, but I’ve applied myself to it the best that I can. Reading and re-reading parts of his The Geometry and Topology of Three-Manifolds—first the lecture notes, and then in book form—has been an ongoing love affair. Reading papers that referenced Bill’s work made me feel like I was circling around something important, something central. It’s just the most beautiful and compelling and exciting stuff–at once dizzying and serene.

And then one day I saw “Bill Thurston” pop up in a thread on MathOverflow, and I was just floored. Could that be the William Thurston? It was. And there he was, just mingling and sharing, probing and wondering. I quickly came to relish the thoughts he would share on the site. Here’s his profile page, which contains links to his MathOverflow contributions. It includes the following text:

Mathematics is a process of staring hard enough with enough perseverance at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.

That is not the tone that is associated with mathematicians. I remember reading the above and feeling so struck and moved—that here was this utterly brilliant person, a living legend, who was expressing forthrightly feelings of confusion—trying to negotiate his own mind and social dynamics and, simply, living. It felt like a revelation. Here’s another of his wonderful contributions to the site.

So often the world of mathematics—the content, the texts, the institutions, the social dynamics—can seem so daunting. Isolating. An enormous edifice. It’s easy to feel small and insignificant.

Here’s an entirely different feeling: the excitement and energy and most of all integrity of living in a world where your heroes are alive. Integrity in the sense that the world feels whole, like something that fits together and whose parts are in harmony. And that you just might have a place in it.

And now the world feels a little diminished to me.

I remember feeling similarly to this when Kurt Vonnegut died. Reading Vonnegut’s novels has been a thread in my life since high school. There are just these moments when I feel an urge toward his humor and outlandishness and compassion and moral depth. I admire and enjoy his work, and his turn of mind often thrills me. I always thought it would be cool to meet him. And then he passed away.

But I never wanted to be Kurt Vonnegut, or even be like him. There were some ways that I felt kind of like him, and that’s why what he wrote resonated with me. And even: reading what he wrote helped to shape who I am.

But I never wanted to see through Vonnegut’s eyes.

With Bill, though, I always really wanted to see the way that he saw. I wanted and want to be able to see and see into geometric structures the way he did—in a way that feels intimate and intuitive. And I wanted and want to see mathematics as a vibrant, interconnected, human pursuit in a way that I feel he had special insight into.

A few months ago I ran across this video of a lecture Bill gave at the Clay Research Conference in 2010. The introduction Bill is given is a great overview of his contributions to mathematics, and the way that Bill talks is marvelous to listen to. But I bring it up most of all to say—just look at the way he moves. There’s something in it that’s just breathtakingly amazing.

Bill both represented something to me and at the same time was the incarnation of the thing he represented. That’s a little poetic and high-falutin, but it’s the best way that I can say it. Bill shaped my idea of what a mathematician and a person can be, and then also satisfied and realized that idea in the concrete. He was at once the “great man” who I admired and a person who made efforts to share himself with other people—including me, in a very small way, without his knowing—so simply and so humbly.

Bill, I only knew you from afar, but I still feel close to you. Your book sits in a special place on my shelf, and your work holds a special place in my heart. You told me things that blew my mind—on the page, yes, but it was like you were talking to me and only me. That first time I really saw what it would be like to live inside of a 3-sphere, it was electric. And in my eyes, at least, you moved through this world with grace and humility and curiosity and unmixed genuineness. I’ll never be able to tell you so in this life, but I’ll say it here anyway:

Thank you, Bill. You are my hero.

Post-script: If you’re interested in reading some of Bill Thurston’s writings, this blog post at Secret Blogging Seminar has a list that’s a great jumping-off point.

Also, here is the celebration of Bill’s life put together by Cornell’s math department, where Bill taught.


Math Heroes

I want my students to have math heroes. This is another aspect of my commitment to consciously cultivating values in my classroom this year.

Lots of kids have sports heroes and music heroes. It’s not uncommon for them to have literary heroes or science heroes. But math heroes are something of a rarity.

Why? I’m not really sure. Is it because kids have little exposure to contemporary mathematicians? That the math that “famous” mathematicians do can be opaque, inaccessible, or unrelatable? Or because it can be hard to tell a good math “story” because mathematicians’ achievements are so abstract?

Whatever the reason, I know that if someone is going to have a personal connection to a subject or pursuit, it helps a lot if they have heroes—people to look up to, people to cheer for, people whose work and achievements they just can’t get enough of. People who are both very human and larger-than-life.

Sharing the work and human stories of mathematicians—present and past—is one part of what thrills me about co-writing Math Munch for my students and others. But in addition to this online locale, I find myself with the desire to share my own math heroes with my student in concrete, present-in-the-classroom kinds of way. Even further perhaps, I want to model how enthusiasm for a mathematician can be shown and shared.

For a while I’ve had the thought of setting aside a space on the wall of my classroom for the celebration of math heroes. I’ve tinkered with this thought just a little. Year before last I had a student who made a nice drawing of Augustus de Morgan during our 7th grade logic unit. But I realized that if this “math hero wall” was really going to take off, I’d have to put my money where my mouth is. So last summer, I sketched out this idea for a portrait of Felix Klein, who is one of my math heroes.

And then, of course, the school year started up.

Then two weeks ago, I was feeling like I needed a really concrete project to take on, and my mind went to my Klein sketch. After an initial couple of hours, and then some further detail work a few days ago, things now stand here.

I just need to add his name toward the bottom, and the names of the four objects that surround him. They’re all named after Klein. Clockwise from the bottom-left, they’re his quartic, his bottle, his model, and his four-group.

Hopefully my painting will serve as a good anchor for my room’s math hero wall. I’m looking forward to seeing what it helps to inspire my students to do this year. Drawings? Collages? Sculptures? And most of all, to have math heroes of their own?

Plus, I’m pretty pumped to have made a painting myself and to get to share it with you!

Read more of this week’s #made4math posts here!


This is my first attempt at a “Made 4 Math Monday” post. What follows is certainly making, and definitely math.   I hope you enjoy!

The other day I tagged along with Paul when he went to NYC Resistor to laser cut a piece of mathematical art.  For a while now, Paul’s been creating all kinds of amazing art that involves stars.  That’s STARt, for short.  He should really put up a gallery of his artwork on his blog. nudge, nudge.

Anyway, for this piece he was cutting all of the twelve-pointed stars out of plexiglass, planning to stack and glue them into a sculpture.  There were of course some leftover scraps from the cutting, including 12 nice isosceles triangles that he passed on to me.  They’re just the shape of triangle that you get when you pizza-slice a dodecagon.

After our outing, I was in an artsy mood and found myself playing with the triangles.  I was amazed by how many ways of arranging them presented themselves to me, just by futzing around with them.  It was a pleasant and exciting activity.  Here are a couple of pictures I took of my creations:

Playing with the triangles got me thinking about some further patterns that I wanted to try out. Like: what would happen if I stuck the triangles together along their long sides in a chain, pointing up and down at random? What would the long-term behavior look like?

So I did some programming, first in Scratch and then in Processing. Here’s a short clip of what I’ve gotten to see so far. It’s given me food for thought about connections between this pattern and David Chappell’s meander patterns.

And if you tweak a few parameters–including weighting the probabilities as a function of the number of triangles that have been laid down–you can get something like this instead. Nice!

And so?

What I took away from this was how having unfamiliar and tangible mathematical objects around led to play and product and inquiry. This felt real and striking to me.

What does this mean for my classroom? I want my students to have experiences like these—open, creative, productive experiences with mathematical objects that they feel connected to. Having lots of accessible and flexible mathematical objects in the room is a good direction to head in.

Writing Math Munch is one way I try to expose my students to new objects, patterns, and structures, but now I’m vicariously craving really tangible math experiences for my students. This gave me the thought (a recurring one) of how I want there to be a superabundance of supplies in my room.  I was in an art classroom at school earlier this week working on another project, and I was just so struck by how much stuff there is in that room.  I want there to be stuff in my room, too—both clearly mathematical stuff and stuff that has the potential to be.

What do you make sure to keep around in your classroom for kids to have access to?  Or would like to?  There are some things I keep around, like pipe cleaners and twisty balloons and poster board; markers and protractors and geometry building tools (Zome, Geofix, pattern blocks). But I want more. Because I want my students to futz. Thoughts?

Hooray for making and math! #made4math

P.S.  One final product of futzing: a further riff on STARt, inspired by Paul’s.  Made of thread and a water bottle.

In what ways can math be a part of your life?

A goal I have for this coming year—or rather a commitment I’ve made, as John Spencer reminds me—is to more consciously cultivate values in my classroom.  Specifically, I want to help my students to think about learning and about mathematics.  Thoughts have been kicking around in my head for a while, and so I’d like to make a first stab at sharing some of them.  Hopefully you’ll be able to give me some feedback and/or find something here useful to your own thinking.

I don’t often get asked “why are we learning this?” or “when are we going to use this?” or other “what’s the point?” questions.  Maybe it’s the kids, or me, or the stuff we do, or who knows.  I do know that this is a live question, whether it comes out or not, and that it’s a question where math teachers usually find themselves on the defensive.

So regardless of whether the questions comes up, I want to address them and teach about them.  Especially:

  • In what ways can math be a part of your life?
  • In what ways will math be a part of your life?

Getting kids asking these question in personal ways is most important—of themselves, their peers, their parents, their parents’ friends, other teachers, and so on.  I’m planning to put together some assignments, journal reflections, and/or free-choice projects to help them to seek out answers.

But I also have my own answers that I want to share with them, and with you.  It’s in four parts.  It’s not meant to be canonical—just the way I’m thinking right now.

  • Math can be a useful tool in everyday activities. When shopping, managing money, estimating, or reading about and discussing issues, some mathematical ideas and skills come up time and again.  These might be lumped under the generic heading “numeracy”.
  • Math can be a social token.  The classes you take, what you score on the SAT, whether you consider yourself a “math person” or not.  Your relationship and experiences with mathematics can be and often are a status symbol, a tattoo, for your whole life.  It can be a burden, a weapon, a bridge, a currency, or a welcome mat.
  • Math can be a tool to investigate the world.  There are parts of mathematics that are essential tools in any specific human pursuit—especially the physical and social sciences.  If you want to do physics or economics, play poker or create mathematically-inspired art, there’s some specific parts of math that you’re going to need to know for each one.
  • Math can be a constant companion and source of enjoyment.  You can do math for fun.  You can explore parts of math that excite you, work on problems and puzzles that are tough but gratifying, let math concepts become your playthings and color your life.

A chart came to mind as I pondered these different roles that math can play.

I especially want to point to the distinction between required and opt-in categories.  There are mathematical situations we’ll all encounter, regardless of the lifepaths we take.  At some point we’ll all do, for instance, some money-related calculations, and at some point we’ll find ourselves in a social-token situation.

Other math experiences are opt-in.  There is math that will be very important to understand if you want to be a physicist or a juggler, but that you can easily enough ignore if you don’t.  There are of course questions about what kinds of foundations students need in various mathematical disciplines so that they aren’t later shut out of opportunities.  Drawing the line between a math education for a generalist and a specialist is a live question for me (and maybe you), but I feel like the governing math ed consensus wrongly forces kids to specialize in ways they’d rather not.  I think this has to do with how unclear the meaning of “math is useful in life” usually is.

It also has to do with the fact that math as a social token is such an enormous but often understated or unseen force in our world.

I’m particularly excited about sharing this idea of math as social token with my students, especially in the context of the other three roles.  I feel like it’ll help them take greater command of their own mathematical identities.

Students have a fair amount of control over these four aspects of their mathematical lives, and they certainly have control over their attitudes towards these aspects. I think that pointing them out could be wonderfully empowering for students.

Classroom experiences and institutional structures that support these different values of course look different from each other.  I feel like just articulating them to myself helps me to think about how I’d like to talk about math with my students.

What do you think of my answers?  What would you add, subtract, merge, or reframe?  Or do you have a different way of thinking about this altogether?

And if you can point to resources for having kids think about these issues, I’m all…clicks?