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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.

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.

Over the weekend, I gave a talk at ATMNYC‘s fall conference.  I talked about formal systems problems, which I use both with my middle schoolers and my geometry students.  If you attended my talk, thanks for coming and for dropping my blog! For everyone else, I hope you find something here worth using or thinking about.

In brief, by a formal system I mean a set of objects and a set of transformation rules.  A formal system that I use in my classes goes like this:

Briefly, in middle school classes I generally have students work through some sheets like this, create some puzzles of their own, and share these with each other. We talk about “skipping steps” and impossible problems. I have them add rules to this first system that they think might make it more flexible, interesting, or fun. Sometimes we come back to the idea of formal systems later on during discussions of vectors or symmetry groups or modular arithmetic or permutations. Once you have this formal-system model in your head, it’s really easy to put many other topics under its umbrella.

Shadow puppets were clearly a key element to my presentation.

In my geometry class, formal systems serve a couple of functions. They’re a nice introduction to proofs, as well as to the notion of a mathematical system of that hangs together as a logical whole. Discussing these without having to juggle geometric content at the same time allow students to build up a model of proofs and systems that they can later connect back to. Proofs with justifications for each step, independence of axioms–these and more can arise from discussions of really basic formal systems.

A second reason I use formal systems with my geometry students is that they provide a nice parallel to geometric transformations. When taking Felix Klein’s approach to geometry–that geometry is about transformations that leave certain properties of geometric objects invariant–it’s nice to have formal systems in the background. Invariants, shortest-paths, and just the basic notion of a step-by-step transformation all can arise from discussions of simple formal systems.

So for my geometry class, it’s like killing two birds with one stone.

One thing to have in mind is that formal systems activities are very modular. It’s an idea more than a set of lessons, and a flexible one at that. You can drop it into a class for a day or for a week or scattered throughout the year. Bringing them into a classroom can have an impact regardless of the duration or what your particular goals are.

Here are the little charts that I had in my slides about my reasons for doing these kinds of activities.

Here are links where you can download some worksheets that illustrate this sequence.

WORDs sheets (PDF)

WORDs sheets (DOCX)

Finally, should anyone be interested, here are my slides from the presentation.

ATMNYC slides

As I said at the end of my presentation, I’d love to have your questions and thoughts, and I’d be crazy interested should you try this or something like it with your students!

What follows are some of my more personal thoughts about a project that I’ve worked on with several other teachers.  We wrote a letter to the College Board to express our hope that they will update their calculator policies to better reflect current technologies and the classroom practices that they make possible.  You can read and also sign the letter we’ve composed here:

http://calculatorpetition.wordpress.com

*****

In 1997, the College Board drew a line in the sand that was a good but temporary solution for a novel situation: they created a policy that embraced the advent of the graphing calculator.  At the time, it was a new tool that was rapidly changing the way that calculus and all of high-school mathematics was being taught.  At the same time, the College Board tried to distinguish between a graphing calculator and a full-fledged computer, believing one to be an appropriate and fair tool to be used on their tests and the other not.  The way they drew that defining line was the commonsense idea of banning devices with QWERTY keyboards.  At the time, this token gave them the distinction they were after.

Over the past fifteen years, mathematical technology has grown both more powerful and more ubiquitous.  One way that this has played out is that the divide between graphing calculator and computer has greatly narrowed.  Anyone who has seen the latest Texas Instruments device—the TI-Nspire CX CAS—would recognize it to be a souped-down computer with a ban-evading ABCDEF keyboard.  With the radical changes to technology in the past fifteen years—not to mention how these have begun to shape classroom practice—make it time for the College Board to revisit their calculator policies.

It’s most compelling feature? Permitted on the SAT and APs.

Bottom line: we should not be sharing inferior tools with our students because of an outdated policy.

My own connection to this story began about the time of the policy was put in place, when I was in high school.  It’s pretty wild to think that in that bygone era, I had a box that I could carry around with me that contained such powerful mathematical tools.  It was truly a marvel.  Not that I thought so much of it, of course.  After all, I never had the experience of learning much of high school mathematics without a graphing calculator.  I loved my      TI-89.  It could multiply out .  It could give exact answers like .  I could explore all kinds of crazy functions by graphing them.  And I used it for years to great effect.

Still, when I lost my graphing calculator a few years ago—saddened though I was—I did not buy another one.  By that point, I had begun to carry it around more out of nostalgia than for use’s sake.

These days, when I need to do some arithmetic quickly, I’ll often type the problem into a Google search bar or into the calculator on my phone.  When I want to graph something, I use Grapher or Winplot or GeoGebra—all of which are on my laptop.  When I need to crunch some big computation, I use Wolfram Alpha or, more recently, Mathematica.

I never yearn for a graphing calculator.  I’m convinced that the tools that I use are more powerful, more accessible, and more useful than stand-alone graphing calculators.

Because of this, I want to share these awesome tools with my students.

When I give my students the opportunity to create awesome GeoGebra sketches like this one, there is a trade-off.  They aren’t getting as much practice using a TI-84 or what-have-you.  This year it’s not so much an issue for me, since just one of my students will take the AP Calculus exam.  But I know that it will be more of a consideration in future years, and I know that it already is and has been an issue for teachers around the country.

We live and teach in a world where there are more and more one-to-one laptop programs and where a significant minority of students walk around with cell phones that can run cheap apps that outstrip the capabilities of top-of-the-line graphing calculators.  In such situations, graphing calculators are at best redundant and at worst an impediment.

Texas Instruments—to pick an example at random—is in a hard place.  The new bells and whistles that they continue to add to their single-purpose, stand-alone devices are weak attempts at staying relevant.  At a math teaching conference over the summer, a TI representative was practically apologetic in pitching us the new version of the TI-Nspire.  And yet there is a real sense in which they are in an enviable position.  Even if their sales numbers are low for their newer models–because folks are slow to move to technologies that either seem like not much of an improvement or already surpassed by other products–they have a captive audience.  As long as they are suggested and required for the SATs and APs, TI can sell us 84s and 89s ad infinitum–that is, as long as the College Board calculator policy stands.

It seems to me that two forces are keeping the graphing calculator industry in business.  One is understandable inertia.  Many teachers and schools and districts have invested a lot of time, effort, and money into incorporating graphing calculators into their classrooms.  Teachers understand how to use TI-84s and 89s in their classrooms and do a great job of it.  That’s awesome.  I have no frustration about that.  New tools and new technologies take time to become the mainstream.  I’m not looking for any kind of bloody revolution.  May these calculators be used in great ways for years to come!

Inertia is natural and fine, but I cannot abide impediment.  That is what the College Board calculator policy is—an impediment.  It stands in the way of the education community even imagining real technological progress.  It’s keeping graphing calculators on indefinite life support–pull the standardized-test plug, and they would die a slow but natural death.  That is why I’m so passionate about raising this issue and working for change—not because I think that old technology needs to be replaced by new technology, but because the College Board policy is preventing us from collectively considering the full space of the mathematical experiences we can share with our students.

As educators, I feel like we too often feel beholden to the College Board and standardized tests and take them as monolithic givens.  We betray this in the way that we talk: “But they have to take the SATs,” and “What about the AP scores?”  But the College Board is just a human institution that can change over time, and we can help that change to happen.  In fact, we are best positioned to do so.

Issues about standardized tests range far beyond these questions about calculator policy.  One thing I like about the issue of calculator policy is that it’s circumscribed and a manageable size.  If some teachers can pull together in a unified voice to say that we believe that the College Board’s policy about technology needs updating–for the good of our students–then we may both achieve a small goal and feel the power of our collective say.

The College Board provides us and our students with a service.  They certify that our students can perform certain tasks and think in certain ways.  As such, they should cater to our needs, and not the other way around.  We should call the tune.

Please read the open letter.  If you find yourself nodding in agreement, then please consider signing it and sharing it with others:

http://calculatorpetition.wordpress.com

If you’ve come here looking for Math Munch, click here!

***

If you’re not here looking for Math Munch, let me explain.  This past weekend, Paul and Anna and I created a new blog called Math Munch.  It’s an idea that I’ve been kicking around for a while now–a venue for curating some of the fabulous mathematical content that exists on the internet for my students.

I’ve learned so much math and found so much math to enjoy through the internet that it’s just crazy.  I mean, I love good math books as much as anyone, and person-to-person mathematical discourse is irreplaceable.  But the math content that exists on the internet is astoundingly deep and varied and gorgeous. I want my students to have these gems as part of their relationship with mathematics, and I want them to have an empowered and seeking relationship with learning where they know that they can reach out for knowledge that will delight and inspire them.

This kind of independent and personal-growth mindset has to come from somewhere.  The usual classroom techniques of chalk and talk don’t go far enough, at least in my experience.  It has taken me years to build for myself the habits and touchstones that make for a fruitful experience of learning with the internet as a tool.  Connecting students with the wealth of resources that the world has to offer is a first step in realizing that this is even possible.

Math isn’t just in me the teacher.  It’s not just in my classroom, the worksheets that I share and the problems we put up on the board–as good as I try to make these.  The truth is that there is an enormous, multifarious, unbelievably rich ecosystem of mathy people and stuff that is out there.  I have some significant experience with it and want my students to have an experience of it, too.  Math Munch is an attempt to make that happen.  Like it says in the inaugural post, “Here you will find links to lots of cool mathy things on the internet.  We’ll post some new items each week for you to enjoy.  We hope you are as inspired and excited by these creations as we are!”

Each week I’ll ask my fifth graders to do a journal entry about something they’re jazzed about from the mathematical internet.  I’m hoping these researches will inform and guide the way they choose to spend their free-choice time in class and the way they spend their time outside of school.  We’ll see where things lead!

So finally, the proximate cause of this post is that I got several search hits on I Choose Math that were clearly trying to find Math Munch.  Hence the redirect. But I’m also so glad to share this new venture with you!  You’re of course welcome to read Math Munch yourself, as well as to share it with your students as you find useful.  And if you have some cool mathy links that more people should know about, please send them my way!

“Bon appetit!”

In what I’m pretty sure was the first-ever instance of a live-tweeting of our weekly math department meeting, Paul sent out a tweet about a presentation I was giving in which he called me out to blog about its contents.  So here it goes.

First, a backdrop: this year for the first time, our department is reserving a chunk of our weekly meeting to share curriculum.   Nice.   I signed up to be the sharer for the second week.  I wasn’t sure what I was going to present.  Many of the things that I do in my classes I’ve shared with my colleagues over the course of time in less formal ways, and I didn’t want to be redundant.  In the end, I decided to share two different takes on mathematical “modeling” that I’ve done with my middle schoolers.  This post is about the first one.

Here is a first worksheet of an investigation of something that’s called a “measure task”:

The idea is to take some mathematical property that has some intuitive meaning and to attempt to formalize it—in particular, to try to come up with a way of measuring it numerically.  I think this “squareness” measure task is an awesome one because it has a very low threshold and there are many ways that students can approach it, regardless of their prior knowledge.

For example, last year I did this “squareness” worksheet with a faster-paced seventh grade class and a slower-paced sixth grade class.  The seventh graders had a fair amount of experience with expressing quantities algebraically, both from prior years and from our previous units.  They went straight for comparisons of length and width, coming up with formulas like  and .  This brought up interesting questions—like whether these different expressions would give the same ordering of the rectangles, and whether they were really capturing our intuitions.  On the latter point, notice how the formula  results in widely different measures for the squareness of 1×5 and 5×1 rectangles.

My sixth graders approached this problem through the lens of fractional parts.  They decided to measure squareness by the (smallest) fraction of the rectangle that needs to be chopped off in order to make a square.  Lacking algebraic sophistication, they landed on a simpler, “uncoordinatized” method for measuring squareness that was visceral and re-upped a core concept that we had been discussing.

I’m pretty sure that I first heard about this kind of task from Avery.  He’s even blogged about it here.  Some fabulous resources for finding out more—both in terms of theory and practice—are these two articles:

‘Creating Measures’ Tasks by Swan and Ridgway

Formulating Measures by Schwartz

In my presentation, I said that the measure task paradigm is both flexible and mathematically potent–in terms of content, lesson structure, and aims.  Measure tasks can be used to tackle mathematical concepts from the most elementary to the most advanced.  Like I said, I’ve used the “squareness” task with 6th and 7th graders, and I’m sure that this and similar tasks could be done with elementary schoolers.  On the other hand, some of the tasks that my colleagues brainstormed—like measuring how curvy a curve is—would be great tasks for a calculus class.  Once you get rolling, there are so many mathematical concepts that can be approached in this manner.

The lesson structure of measure tasks is also flexible.  They can serve as brief intros to a new concept at the beginning of a period that then swiftly moves on to other pedagogical modes.  They can be one-off lessons.  They can be used as review or extension activities, or even as luxurious several-day explorations.  As with content, the time-frame and commitment needed to bring measure tasks into a classroom are both as flexible as could be.

Finally, the goals that one might have for introducing measure tasks to students are varied.  The focus could be on particular content (rectangles or squares, in my example), applying concepts in new contexts (fractions or algebraic translation), or just on the mathematical process of creating, testing, modifying, and comparing.

I found talking (and now writing) about measure tasks to be exciting–I can’t wait to use them more often in my classes!  My colleagues really got into the idea as well, making great connections and brainstorming task ideas of their own.

Brainstorm in the comments?

Next up: Part 2.