Category Archives: Uncategorized

I Choose Math (self-titled, part one)

tl;dr — In the fall I’m moving to Atlanta and starting a math PhD program at Georgia Tech. I’m really excited!

The long version: For the past two years, I’ve worked at Princeton Learning Cooperative. I’ve had the opportunity to help a diverse collection of teens to shape their lives and to pursue their interests and goals. In this time, I’ve had the chance to witness some incredible transformations. Kids who were pretty shut down when they came to us—avoidant, angry, listless, or painfully shy—have become happy, engaged, and funny young people who are excited about their futures. I’ve seen kids and families open up, unclench their fists, move past what their school experiences had led them to believe about themselves and the world, and start to build new and more joyful lives.

It’s a good gig.

More places like PLC need to exist, and more are popping up each year. If our mission and vision resonates with you, be in touch—with me or with Liberated Learners, our umbrella organization. I can speak from first-hand experience that operating an unschool can be a great life—immeasurably complex and immeasurably rewarding.

I knew from the start that I would only be at PLC for the short-term. I was interested in encountering a certain kind of educational environment, but I knew that central New Jersey and PLC as an organization wouldn’t be long-term homes. One idea for my next step was to move somewhere and to start an unschool of my own, modeled on the thoughtful and well-executed systems that PLC has in place, and done up with all of my own little personal touches.

But life is long, you know. Starting an unschool means putting down significant roots, and it’s just not where I’m at yet, both in personal and in life-arc terms. I won’t be surprised if starting an unschool is still in the cards for me, but I don’t think it’s a calling I have for the next fifty years of my life. I have another chapter or two to write first.

So what, then? Well, in the past two years there’s been another storyline, too, which I didn’t anticipate.

Pursuing a PhD in mathematics was never a part of my picture of the world when I was growing up, or even in college. I really only met PhD mathematicians once I started teaching, and even then it felt like a whole different world—something some of my colleagues had once done. When I was at Saint Ann’s, I took some graduate coursework in math, but stopped after a few classes. I couldn’t give the classes the attention required to get out of them what I wanted to. Also, I just didn’t just enjoy sitting in lectures so very much—#teachingsnob—and that plus cramming to get homework done was my whole experience. No interactions with classmates or professors. At a certain point, I just stopped. I figured if I were ever going to study more math, I would have to do it whole-hog—not just in my “free time”—and I’d have to figure out how to do it on my own terms. It didn’t feel particularly likely, because I couldn’t imagine leaving classroom teaching. I’m sort of a teaching addict. :)

Even two years ago, pursuing a PhD in math didn’t feel like a possible reality to me. But since then, I’ve had some experiences that have opened me up. I got to attend a couple of math conferences. I got to meet a number of mathematicians, including through Twitter and through writing Math Munch. I started taking stabs at reading math research papers and following chains of references and attending seminars. All of this came to a head last summer when I made a list of goals for this year and wrote “Get accepted to grad school(?)”

And so as it turns out, over the past year I’ve studied for the GRE math subject test, researched different math PhD programs, opened myself to the possibility of falling in love with this path forward, contacted potential advisors, got support from family and friends and colleagues, put in some applications, got some acceptances (yay!), made visits, and made a final decision. And I’m really glad to say that I’ll be starting a PhD program in the School of Math at Georgia Tech in the fall. My focus will be in geometry and topology, and particularly in geometric group theory and geometric topology. Both of these fields were—no accident here—shaped by Bill Thurston.

There are a lot of reasons I’ve decided to go to grad school. I’ve got some of this typed up, but this is getting long as it is—so stay tuned for part two.

Thanks for indulging me this mind dump. These and connected thoughts have been bouncing around in my head for many months now. I’m really glad to be sharing them now!

Helping with Explore MTBoS

(This is a reiteration of Tina’s post here.)

Last Saturday, as NCTM Boston was wrapping up, Tina and I kicked off a round of Explore MTBoS. It’s an introduction to reading blogs and tweeting as professional learning, as well as an exploration of the many resources and projects that the fine folks of the MTBoS have created and shared.

Participating in the activities might be a great fit for you. If so, join in!

Or maybe you have a colleague who is ready to take their professional development into a ongoing, online space. Please recommend Explore MTBoS to them!

If you have some experience with blogs and Twitter, here are some ways that you can help newcomers during this Explore event:

  • There will be several #MTBoS chats this week as newcomers start up their Twitter accounts. Each chat will last for 30 minutes. Please stop by and say hello!Monday (April 27): 4pm ET / 1pm PT and 7pm ET / 4pm PT

    Thursday (April 29): 9pm ET / 6pm PT

    Next Saturday morning (May 2): 9am ET and noon ET / 9am PT

  • Even if you can’t come to those specific chat times, maybe you’ll try and keep an eye on the #MTBoS hashtag this week. Say hello and answer people’s questions!
  • Week 3 of Explore is about Organization—keeping track of everything one runs across online. Do you have a blog post where you discuss how you stay on top of things in a way that works for you? Please let me know about it! Or if you don’t have such a post, take this as encouragement to write one and to let me know. It’d be great to share the fruits of your experience!


#MTBoS Booth Opens Tomorrow!

I don’t know how to reblog from Blogger, so I’ll just point you over to Tina’s post about the #MTBoS Booth manually. If you’re attending the NCTM Annual Meeting & Expo in Boston this week, please stop by Booth 841 to say hi!

I do want to highlight one of Tina’s paragraphs, though:

An even better way to deal with your #BostonJealousyCamp woes? Check out when the next math conference near you will be. See if they have an exhibit hall. Ask if you can get a deal for MTBoS – for NCTM, a booth comes with 6 free registrations, for the price of 6 registrations. So we gathered 6 people, NCTM helped us pool our funding internally and ended up with a booth! Then they helped us out with the carpet and table. Justin and I have done all the work already, and we did it really strategically. We saved our to do lists and descriptions and emails. There’s a google drive folder just waiting for someone to say they want to run a #MTBoS booth at their next conference. We’ll ship you some supplies. The rest you can print from all the stuff we’ve saved in organized folders. You could have your own scavenger hunt! So don’t be sad, get planning. Make this the first of many opportunities for the math teachers of the world to learn there’s an amazing online community just waiting to talk to them about content and pedagogy and more.

I hope to see you at a MTBoS booth soon—either during my time in Boston on Saturday, or at the booth you help to coordinate at a future conference! Happy scheming!

Boothing It Up at NCTM Boston

For full details, hop on over to the Explore MTBoS site, but tldr: There’s going to be a MTBoS booth at NCTM Boston! We could use your help in the following ways:

  • Sometime soon, you can tweet on the hashtag #WhyMTBoS a reason why the MTBoS is great.
  • If you’re attending NCTM Boston, you can sign up to spend time staffing the booth.
  • If there’s an MTBoS project or endeavor that would be great to highlight at the booth, let us know about it!
  • Let us borrow your internet browsing device for NCTM— iPads would be excellent.

And there will be a new Explore MTBoS online excursion after NCTM!

Again, for more details, hop on over to the Explore MTBoS site. Yay!


Last summer I ran a smOOC (small, open, online course) designed for teachers called Math is Personal. I’m running it again this month. If you think you might be interested in unpacking your relationship with math—and also building it up—you can find all the details on this page. The course starts on August 10 and runs through August 31. Registration opens on Monday at noon. Please share this opportunity with folks who might be interested!

Math Circle – Halving Fun

I recently led a recreational session at (MC)^2—that’s the Mercer County Math Circle. I called the session “Halving Fun”. I don’t remember exactly how I hit upon halving—maybe because of a partition puzzle I’ve been contemplating—but I knew I was excited to share how much great mathematics can come from such a seemingly simple concept. Here’s the session description:

Join us as we “halve” some fun while creating some art and solving some puzzles that involve breaking shapes in half. What’s the most beautiful way you could cut a square into two equal pieces? Is it always possible to cut a funky shape in half through a specified point? Whether on grids or with free-form shapes, halving tasks provide many fruitful mathematical opportunities. Come explore and these questions and more!

I kicked off the session by showing off some examples of halving art, and giving a little spiel about each.

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Here’s the slide deck if it’s useful to you. I also shared the Yoshimoto Friends video that Paul and I made.

Then I set everyone loose to either make some art or try out some of the problem sheets I’d brought. I also had some physical “halving” puzzles and other models for attendees to play with.

Here are the problem sheets:

  • It’s Half – I’ve gotten a lot of mileage out of geometric fraction problems like these. On this sheet, you already know that each picture represents a half. The goal is to make clear just why this is the case.
  • Half Frames – These are two problems that anyone can work on, but that have rich connections to algebra and geometry.
  • Halving – I always share this sheet with my Geometry classes, ever since I found it in Girls’ Angle Bulletin a number of years ago. It’s also a great jumping-off point for students to make problems of their own!
  • Half Tetra – This is a template for a 3D puzzle to cut out and fold up. Two identical pieces combine to make a regular tetrahedron. Cardstock recommended. Via Futility Closet.
Can you say why the hexagon is half shaded?

Can you say why the hexagon is half shaded?

Attendees made some lovely Truchet-inspired drawings, and one attendee especially dug into the half frames problem—coming up with an interesting conjecture in the process. :) A middle school math teacher who attended even planned to share some of the problems with her students. Maybe you’ll find fit to do so, too. Let me know if you do!

Math Circle – Celtic Knots

On Saturday at (MC)^2, Harini and I led a session on Celtic knots. Many of the ideas for it were taken from these great articles on the NRICH website. Here’s the description we sent out ahead of time:

It’s St. Patrick’s Day this week! We’ll look at some ornate Celtic knots, observe and explain some patterns, and make some knot drawings of our own. We’ll also discuss the difference between knots and links, ways to identify each, and some other basic topological ideas. Come shamrock some great math with us!

Here is the warm-up sheet we gave to people on Saturday as everyone arrived and got settled.

How many bands are each of these designs made up of?

How many bands are each of these designs made up of?

We started with some introductory remarks and showing a few examples of Celtic knots:


Here are all of the examples we showed.

Then we played Vi Hart’s video “Snakes + Graphs”:

examples curveAfterwards we focused in on the first doodle game Vi shares. We drew and talked through a simple example on the board, then game folks time to try some of their own or to construct the over and unders on this example curve.

Next we shared a second, much more restrictive method of drawing knots. It produces images like the A and B weavings pictured above. We showed Alison’s video on this NRICH page twice and let folks follow along with this sheet (or work on whatever they liked).

Then I shared a third method for drawing knots that I learned through these lecture notes. Here was my first try, which I made in preparation for the session:


Here’s the handout of dot paper that we passed out for people to try on. You’ll need to tilt the page at a 45° angle.

We showed this photo of a project Harini did at a Bridges math art workshop a couple of summers ago, as inspiration for them to take next steps on their own at home.


Much of the 45 minutes of the session was giving attendees time to try out these methods on their own and giving them support and answering their questions as they arose. As a result, it was a much quieter math circle than what I’ve been used to, but still a lot of fun. People seemed pretty happy, and lots of great knot drawings got made.

Dan’s Circle-Square Challenge

Dan Meyer has posed some questions in a recent post about the merits and potential of the following problem:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

Dan asks:

What can you do with this? How can you improve the task?

Here are the thoughts I had (comment-turned-post) and an attempt at reworking the task.

Experiences with problems often become more powerful when they’re not just one-offs—when they’re surrounded by some kind of context. One kind of context that I find fruitful is the notion of a problem space. Most of the interesting aspects of the circle-square problem that Dan highlighted—like “assigning a variable” and “constructing a model” and “using operations on it”—aren’t specific to the geometric figures of circle and square. They equally well apply to similar questions (some easier and some harder) involving many different kinds of shapes. The circle-square combo does have a nice “classic” feel to it, but if we care most about the mathematical practices and methods involved in solving it, then we might do well to also explore variations on it.

In addition, the circle-square task as it stands puts the bar at one specific level of difficulty and requires certain algebraic and geometric skills to be firmly in place for a successful solution—and really even to make any significant headway. Whenever possibly I prefer to lower the threshold for success in a task while also raising the ceiling—an easy way to let students self-differentiate a task.

Here’s a first shot at a warm-up worksheet to introduce the problem space:


I might have string and scissors on hand for students who were interested in using them, but my sense is that the problems as I’ve stated it—and the solution methods students are likely to take—is sufficiently concrete to make manipulatives perhaps useful and fun, but not necessary.

What would happen after student had a chance to work on this sheet on their own for a bit? Well, they’d have time to share what they came up with in pairs or small groups. Then we’d do some sharing as a whole class and discuss solutions and conjectures. That’s another thing—there are so many equally and perhaps even more interesting mathematical questions that can arise in this particular problem-space than the circle-square one that we’re (likely) headed to and that prompted this whole adventure. I’m sure the directions that students would take this in and the questions and problems they would pose would be fascinating and worth pursuing. They’d also likely provide richer and more powerful mathematical experiences than the circle-square problem on its own would, and solving this problem would carry a new significance in light of the other problems that had been posed and solved prior to it.

I can imagine using this warm-up worksheet with middle schoolers, but I also think it would be an appropriate onramp to the circle-square problem for sophisticated high school students. For both audiences it will clarify the circle-square problem and build context and experience for the students. If the circle-square problem is the main course, it deserves both to have appetizers to whet everyone’s appetites and to make sure everyone has a seat at the table before the meal is over. Because let’s face it: mathematical meals in school are all too often hurried affairs.

Those are my thoughts so far. What do you make of them?

I Choose Fractions


On Monday I was walking with one my colleagues (Paul Scutt) as we tromped around Cranbury, NJ, on an excursion to the local historical museum and the store fronts of the town’s main street. We’d just had a bit to eat at Teddy’s, a local diner. Paul was wondering about a possible discrepancy in the bill and was calculating what the price of one drink must be if the total we’d been charged for three drinks was $5.25.

Paul was working out the quotient aloud in the style of long division. “3 into 5 is one with 2 left over…” Going through the steps was tricky—there was some retracing steps and checking involved—but things carried through and the answer came out as $1.75. This squared with the prices on the menu. All was well.

Through some of this I was only half listening. My thoughts were distracted by something or other; perhaps I had already begun my alternate calculation. Soon enough I was sharing my thoughts with Paul: “$5.25 is 5 and 1/4 or 21/4. 21/4 divided by 3 is 7/4, and that’s $1.75.”

It was an occasion to recall that over the course of my years of teaching math—and middle school math, in particular—I’ve developed a preference for fractions over decimals. Decimals, I’ve been inclined to argue, are just a special case of fractions—fractions with powers of ten in the denominator that we can suppress with a bit of ingenious and pragmatic notation. And decimals are a very slick trick—calculating with them is practically verbatim from whole-number arithmetic.

But then again, seen in a certain light, so is calculating with fractions—whole numbers that we calculate with while carrying them around in separate buckets. And since calculations with fractions have their own shape to them—their own kind of mental juggling—their meaning isn’t sedimented and murked up with their relation to ordinary place-value arithmetic. What is gained in ease is often purchased with opacity. This is true throughout math, and I’ve certainly found it to occur with decimals in particular. From these experiences, I’ll claim that it’s easier to help kids to think about fractions than about decimals, and that a teacher with a particular bent of mind can safely leave decimals as an afterthought. (Note: not “should” be.)

The ability to think flexibly about both decimals and fractions and to move between them comfortably is certainly the ideal. But how do we help students get there? What are the paths that we carve?

This post was prompted by an exchange on Twitter:

Screen Shot 2014-02-19 at 9.32.21 PM

What thoughts do you have to add? How do you think about the relationship between fractions and decimals? How do you talk about this with your students, or how might it affect your classroom silently? What is your own relationship with computations involving fractions and decimals—any preferences, for any reason? And what are good examples of their relative strengths and weaknesses?

Screen Shot 2014-02-19 at 9.45.18 PMScreen Shot 2014-02-19 at 9.45.24 PMWhat lines do you draw?

Mission #8: Sharing is Caring in the MTBoS

We’re wrapping up “Exploring the MathTwitterBlogosphere” this week, which makes it a great time to share your favorite corners of the MTBoS with colleagues, students, and others!
Thanks, everyone who participated in Explore MTBoS! :D

Exploring the MathTwitterBlogosphere

It’s amazing. You’re amazing. You joined in the Explore the MathTwitterBlogosphere set of missions, and you’ve made it to the eighth week. It’s Sam Shah here, and whether you only did one or two missions, or you were able to carve out the time and energy to do all seven so far, I am proud of you.

I’ve seen so many of you find things you didn’t know were out there, and you tried them out. Not all of them worked for you. Maybe the twitter chats fell flat, or maybe the whole twitter thing wasn’t your thang. But I think I can be pretty confident in saying that you very likely found at least one thing that you found useful, interesting, and usable.

With that in mind, we have our last mission, and it is (in my opinion) the best mission. Why? Because you get to do something…

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