Bridge to Enter Advanced Mathematics (BEAM)

“Fundamentally, this is a question about power in society,” said Daniel Zaharopol, BEAM’s director. “Not just financial power, but who is respected, whose views are listened to, who is assumed to be what kind of person.”


My friend Dan said this to a New York Times reporter, in the context of an organization that he started called BEAM. It is a pathway for underserved middle school students to gain exposure, interest, and opportunities to see how amazing the world of mathematics can be. This is important. Why?

Guess how many math and statistics Ph.D.’s were awarded in 2015 to black students?



So how do we change that? Dan and I were both at a lecture at Teachers College at Columbia yesterday given by Erica Walker. Her thesis? That you need mathematical socialization, spaces, and sponsoring (mentoring) to build positive, strong math communities.

Dan’s organization is doing that.

I highly recommend checking out the BEAM website to read about the ways it is trying to change the status quo.

But more than that, I recommend reading the article the New York Times reporter wrote about BEAM. She took an intimate, in-depth look at the program through the eyes of its participants — from riding the subway together to their discussion of the Black Lives Matter movement to their families. You may get welled up, as I did.



Teaching is hard work. Election aftermath.

Yesterday, I told one of my precalculus classes how it was an exciting day. I was setting them up because it was election day, and kids at my school are heavily interested in politics, so I thought they’d say “yes! Election!” And I would say: “Actually, it’s because one of my best friends from college is having a baby.”

Of course that setup didn’t work, because of course a kid asked “why is today exciting?” Thanks, kid. But I told the class about my friend’s baby.

Yesterday evening, as the election results came in, I got more and more anxious. And when it was clear that Trump won, I was destroyed. I am not going to use this blogpost to explain my love for Clinton, or why Trump makes my blood boil. Instead, I want to just share how my day has gone.

I teach at an independent school in Brooklyn, and the population of kids and parents we serve are (for the most part) liberal. The kids are politically active and aware and interested. Today, I came to school and kids were destroyed.

In my first class, I talked to kids a bit, and then asked them what they wanted to do. After hearing them, I came up with the following plan. The kids who woke up to the news and wanted to learn more and get informed could read articles online. (There were about 4 of those kids.) I just asked that before they started reading, they take 3 minutes to type out all the questions that they have — so help them start processing. (Like “How could this happen? What was wrong with the polling? Who was voting for Trump? What does this mean for issue X?”). For the others, we formed a circle with the desks and I let kids talk. At points, kids cried. I didn’t join in — I wanted this to be a space for them. They expressed real sadness, hopelessness, optimism, anger, frustration, embarrassment, terror, empathy. I really heard my kids, and when talking about this election, they were speaking their truth, about their hopes and dreams (and how those hopes and dreams were altering). It destroyed me inside to hear them. To see how much this election has affected them. I guess I hated the fact that my kids are feeling what I’m feeling. I don’t want that for them.

I went to my second class, that precalculus class that I told about my friend’s baby. The first thing a kid said to me was inquiring about my friend’s baby. That small gesture — that this student would remember that — lifted my spirits. In this class, more wanted to read the news, and a handful of us talked. This discussion tended to a bit more political punditry — about the what’s and the how’s and less about their emotional state. I suspect they got many of their feelings out in their previous classes.

In my third class, we watched Hillary’s concession speech.I teared up twice during the speech. One kid left to gather themselves for a few minutes after the speech. I didn’t know what to do after. Kids said they didn’t feel like discussing things anymore — they were discussed out — but they also didn’t see how they could focus on work. I made the executive decision to spend the last 20 minutes of class having my kids watch the pilot of the West Wing. I hoped that some optimism in politics might help.

I have one more class to go. It’s a 90-minute block. I’m drained, right now. I don’t have much more in me. I suspect kids are also drained, but I don’t know. I’ll suss out how things are, and try to get through it.

I’m exhausted. Yesterday I woke up at 5:30am to vote. Yesterday I didn’t get to bed until very late (maybe 1pm), and then woke up at 3am to watch Trump’s victory speech. I then read articles until I forced myself to sleep from 4-6am.

Teaching is hard work. Yes, there are lesson plans and grading and meetings and a zillion other things. But days like today, days like today keep me in check. And reminds me how hard the hard work can really be. Because the hard work is being an emotional support. To let kids cry. To let kids know you cry. And to get through the hard times together.

Update: My last class came in with bags under their eyes. I was also tired. I asked them what they wanted to do. A few wanted to continue talking, a couple wanted to do some math and do some talking about the election (a mix), and one just wanted to do math. I decided we would go over the nightly work first, and then talk about the election.

When going over the nightly work, kids were actually focusing better than expected. They asked questions. They were able to answer questions. It was going well. I then ended up going on a fascinating tangent about fractals (related to one of the questions we talked about). And when I realized kids had never heard of fractals before, I showed them a youtube fractal video. Then they wanted to know how it was made. So I gave a short 10-minute lecture on the complex plane, and how the Mandlebrot set is formed. Kids were entranced by the video. I gave a 5-minute break before we sat down to talk about the election. (During the break, kids were in the hall watching more of the fractal video on one of their phones!) When we returned, everyone was silent. No one spoke. I just let it hang there. Eventually one voice. Then another. It wasn’t a rowdy discussion. Not everyone was in it. But most kids had something to say. And then when the day was close to ending, and there was a natural lull, I used a comment about “voting systems” to show a video about alternative voting systems. And then I let kids go home.

I just made the first four slides for class tomorrow. They’re not fancy. I’m tired. But I think they encapsulate what I’ve taken away from today.



Now I must end. I now have to change all my lesson plans for the upcoming days, prepare for parent visiting day tomorrow, and write narrative comments. This feels impossible. But I needed to process today.


UPDATE: A student gave me a paper flower she made today, to thank me for facilitating a conversation about the election in our class on Wednesday. And that flower is going to stay on my desk all year to remind me of the other things we do as teachers that can be meaningful.


Technically Beautiful

Last week, Technically Beautiful opened. It is a math-art show that I helped put together (with another math teacher and a museum educator) at our school. We have a small teaching gallery at my school, and we wanted to do something special for this space which capitalized on our expertise. That is how Technically Beautiful came into being.

Technically Beautiful Card Art Draft 4 - 640px.jpgThe poster for the show. The name, by the way, came from a #MTBoS tweep!

In this post, I wanted to share with you the gallery virtually. (In a future post, whenever I get a chance to breathe, I’d love to talk about the programming we made around the show and possibly a bit about how the show actually came into being.)

Here’s a walkthrough video of the gallery:

Here’s our vision statement for the show:

The website for the show is here. The five artists featured in the show are: George Hart, Edmund Harriss, Veronika Irvine, the Oakes Twins, and Paul Salomon.

And lastly, here are photos of many of the pieces:



Taking Stock

I spent 5 hours today cleaning out my desk, going through files, recycling mountains of paper. In concrete terms, it means school is over. Graduation is tomorrow. And then: I’m on summer vacation [1]. So now a bit of a brain dump as I take stock.

I’ve found this year to be an important transition point:

For the first time, I taught ninth graders, and for the first time, I taught geometry. And in order to do that, I worked an insane number of hours with my partner-in-crime and co-teacher BK in order to write an entire curriculum from scratch, from head to toe. Yup, you read that right. We — in essence — wrote a textbook. We sequenced the course, we wrote materials and designed activities for the course, and we had kids do all the heavy lifting. There are particular moments as a teacher which standout as “big moments.” Moments where we know we’ve developed immensely as a teacher. Transitioning from individual and partner work into total groupwork was one of those moments. Converting my non-AP calculus course into a standards based grading course was one of those moments. And writing a curriculum from scratch, in a single year, with an insanely thoughtful collaborator was the most recent of those moments [2].

The previous two years (before this school year) were two of the hardest years I’ve had as a teacher. We teachers were called on to do a lot in the wake of our school’s five year strategic plan — and it became overwhelming. I had no work-life balance. And  I became a bit curmudgeonly because of those tough years. But this year, things have been better. I still have no work-life balance, but the overwhelming onslaught of initiatives have subsided. One of the things I did to actively try to stay positive this year was to write down every single day one good thing that happened to me — big or small. From the first day of classes to the last. And those things are archived here. This was especially important because at the start of the school year, my mom was diagnosed with cancer (she is doing very well, fyi, no worries).

That being said, I am going to make a goal: that next year, I am going to just let the things that I can’t control go… There’s no point in getting worked up over something that you can’t do anything about. Instead, I’m going to stay loose, and bring back my frivolity and humor, and go off the beaten path in class more. While organizing today, I was looking through a number of old emails and cards from students, and saw so many inside jokes and fun times that they references… and then I thought about this year… and I came up blank. I couldn’t think of a time that I doubled over laughing in class. I couldn’t think of an ongoing joke that I had with a student. I could think of great lessons and a ha moments, but nothing frivolous and fun. So my vow is to make sure that next year involves more joy and laughter. For me, and for my studentsEvery day.

Wow, yes, this braindump led me to something big. With that, I’m out.

[1] That doesn’t mean I’m done with school. I have lucky 13 college recommendations to write. And two summer projects that each will take 25 hours each to complete (revise my multivariable calculus curriculum; plan for our new schedule next year with longer blocks).

[2] I’ve written entire course curricula before. Calculus, for example. But that took a few years to write and get added to. And Adv. Precalculus, which I did in a single year, but lacked the collaboration and innovation that I was able to do this year with BK.

Some Geogebra Fun

I have an awesome friend and colleague at my school who is a geogebra master. He has started keeping a blog — Geogebrart — posting fairly frequently some stunning, jaw-drapping mathematical art he created using this powerful program.  Check this recent one out — which happens to be one of my favorites! Dualities!


Although I know most of the basics of Geogebra, I have not yet progressed to the stage passed “novice.” However I really want to get there, because this program is so freaking awesome.

When I was at TMC14 this summer, there was a sesh run by John Golden, Audrey McLaren, and Jedidiah Butler. They are like Jedi masters of Geogebra (though I know Audrey will play coy and say she isn’t…). When I was there, I learned about conditional objects, and it was awesome. (The google doc they used to help people out is here.) In about 30 minutes, with the help of John Golden and some kind people near me, I was able to make a rinky-dinky geogebra file which has a triangle on it, and has three points on the three different sides. When you drag each point close to where an altitude of the triangle would hit that side, I had something like “WOW!” or “YOU DID IT!” pop up! And if you got all three points close, something like “ALL THREE?! YOU’RE A SUPERSTAR!” show up.

Okay, okay, I wasn’t going to show you it because it’s sooooo dumb. But heck, whatever, here it is. Click on the image to check it out.


Okay, you and me, we both know that file is totally useless a teaching tool. And it is gross looking. By all accounts, I should not be excited by it. But the weird thing was: I was really proud of it, and I wanted to show everyone around me what I created. Even though I know it was simplistic and useless, I wanted to create a file that did X and I was able to do it! Although it felt dumb to get psyched about it, I was so excited that I could create something that would do what I wanted it to — that I couldn’t do before!

Today I was again inspired by my colleague and friend’s geogebra art, so I wanted to create some of my own.

I was quickly able to make this in 10 minutes [click on the picture to go to the file and mess around with the parameters! cool things happen!]

zzz1My goal was to define a curve parametrically and then have — at a ton of points on the curve — a circle to be drawn so it would look like a tube. That ended up looking only moderately neat. So I changed it so that as one traveled on the parametric curve drawing the circles, the radius of the circles would change (based on some formula I fed it). The reason this wasn’t so hard for me? I knew all the commands to do this except for the parametric curve command, which was easy to figure out.

But then I wanted to try my hand at something that would take more than 10 minutes and that would challenge me. I wanted to have something “show” a sphere via the animated drawing of “slices” (ellipses). It was inspired by this beautiful gif, but I knew that was going to be too hard for me to start out with. So I decided I would start out with a simple sphere with slices going horizontally and vertically, with no rotation.

After somewhere between 90 and 120 minutes, I did it! (You can click on the gif to go to the file and play around with some of the parameters.)

zzzspheregifAlthough the image isn’t as cool as the one that took me 10 minutes to create, I’m way prouder of this. It is because it took a ton of learning and trial and error in order to figure out how to do this. The set of problems I encountered and somehow figured out:

  • I know how to create a single ellipse in the center of the circle, but how do I make another ellipse a certain distance away that still only touches the edge of the circle?
  • How do I make the ellipses “width” (minor axis) decrease so that it is fattest near the equator, and almost like a line near the poles of the sphere?
  • Without manually typing a zillion ellipses, how do I tell Geogebra to create all the vertical ellipses at once, and all the horizontal ellipses at once?
  • The way I was generating the ellipses resulted in a problem… once an ellipse “hit the pole”(became a point), it would turn into a hyperbola. So I needed to find a way to make sure that once an ellipse “hit the pole” it would disappear.

I figured all this stuff out! So even though the sphere doesn’t look nearly as cool as I’d like, I feel so much more accomplished for it than with the super-cool-looking circles of variable radii drawn on a parametrically-defined curve.


Note: it’s amazing how “simple” this sphere image is once you figure it out. Once you create three sliders:

t goes from -5 to 5 [incriments of 0.1]
StepSize1 goes from 0.05 to 2 [increments of 0.05]
StepSize2 goes from 0.05 to 2 [increments of 0.05]

and you enter the following two (that’s it!) geogebra commands:

Sequence[If[abs(t – n StepSize) < 5, x² / (25 – (t – n StepSize)²) + (y – t + n StepSize)² / (1 – sgn(t – n StepSize) (t – n StepSize) / 5)² = 1], n, -5 / StepSize 2, 5 / StepSize 2, 1]

Sequence[If[abs(t – k StepSize2) < 5, (x – t + k StepSize2)² / (1 – sgn(t – k StepSize2) (t – k StepSize2) / 5)² + y² / (25 – (t – k StepSize2)²) = 1], k, -5 / StepSize2 (2), 5 / StepSize2 (2), 1]

Then you’re done! Well, you should animate the t-slider to make it cycle through everything without you having to drag the slider!

Seriously, two commands, that’s all it takes. But hopefully from the commands themselves you can understand why it would take me so long to figure out…

Playing with Math

Sue VanHattum (of Math Mama Writes) is in the finishing stages of editing a rich collage of works that is aptly named Playing with Math: Stories from Math Circles, Homeschoolers & Passionate Teachers.

Truth be told, I tend to eschew reading about math education because most of what I’ve read feels dry and irrelevant to me. I tend to stick with who I trust when it comes to math education: my colleagues, whether they be in-person or virtual. And although I didn’t tell Sue this, because she was so kind to share an advance copy with me, I fretted about falling asleep while slogging to get through 67% of this book because of the subtitle. (I have never led or been to a math circle, nor do I work with homeschoolers.) I’m just an average joe teacher who keeps his sights on his classroom and his kids, and… well… that’s about it.

Now for the punchline: I couldn’t stop reading it. All 100% of it.

The book isn’t composed of traditional articles-as-chapters. Playing with Math is, rather, a collage. I was treated to bursts of math puzzles, activities, and games (the majority of which were completely new to me) wedged between short and medium-length vignettes from people who are working with kids on math. (There are almost 50 contributors to this book, some of whom I know!) I can see this book being a great present for one of my NYC colleagues, because as I was reading it on my laptop, I kept thinking how perfect this book would be for subway reading because each piece was only a handful of pages. A testament to the book is that as I was reading it, I wanted a zillion post-its and tabs to flag this or that.

Even though I haven’t been to a math circle nor am in any way involved with the homeschooling community, reading the pieces around those topics were interesting precisely because I know so little about them. But moreso, they got me thinking about ways I could differently think about my classroom and my kids. When it came to the math circles, it gave me ideas on how to let go and trust kids to take charge of their own mathematical learning more. And when it came to homeschooling (and unschooling), I wondered how much kids lose their love of learning precisely because of the structure of school. The author of the pieces did this by telling stories. Some were like video cameras, documenting and explaining the “teacher moves” in some particular math circle sessions. Some were powerful and wrenching first person narratives about mothers trying to help their children. And the teacher section was a curation of powerful stories of teachers like me, trying to be a little bit better each year. Some pulled lines to whet your appetite:

We began today’s math circle, the first of six sessions, sitting in an “ogre.” Not a circle, not an oval, but an ogre, the kids’ way of precisely describing the shape we made.

Peter Panov and David Plotkin can barely stay in their seats. They’re firing questions and comments and conjectures and quips at their instructor, Jim Tanton, as fast as he can respond. The whole class of thirteen-year-olds was giggling when I walked in. On the board is a list of some Pythagorean triples and a procedure for generating more. Tanton had just generated the triple (-1,0,1), and a general hilarity about the idea of a triangle with a negative side-length erupted. Now it’s as if he were dangling strings in front of a pack of puppies. They’re all worrying at the problem, tossing out ideas, wiggling in their seats.

Looking back now, I see how far off the mark we were. We should have advocated for our daughter to ensure she received an intellectually, socially, and emotionally appropriate education. But we were overwhelmed by the more-pressing problem of Ryan, so we missed her quiet desperation. I wish I had been more proactive and looked below the surface. I wish I had worked more closely with her teacher. I wish I had trusted my own instincts about my daughter’s needs and abilities.

I waited eagerly for him to arrive the next morning, looking forward to the moment when he would put AAAAAALLLLLL those tiles together in neat rows by category, and he would have to exchange several times (not to mention his surprise at seeing all the units disappear when multiplying by ten). Instead Roland came in, shook my hand, and said: “My dad told me that all I have to do is add a zero to 8,696 and I’ll have my answer, because when you multiply by ten you just add a zero.” My heart sank. Oh no, Dad! You robbed your son of such a cool experience!

Several years ago, my school experienced a shortage of geometry books. There was talk of teachers sharing class sets and photocopying pages for students. I decided to try a different strategy. I took this as a professional challenge to see how long I could teach without a textbook. I knew whatever happened would be a growing experience for me as well as my students. Through no fault of the school library, two or three weeks stretched to seven. By that time, I was well into my “textbook-free” strategy, so I just kept the ball rolling … for the rest of the year.

I like stories, and that’s what this book is. Not disquisitions or pronouncements or shallow research studies. Stories. The authors bring to life their experiences and interactions with kids and their insights and their frustrations, and I started care about these people, their children, their classrooms.

If there is one theme that stood out to me, it is this: we need to work at undermining the constraints that we are confronted with (whether it be textbooks for teachers, or the entire school experience for some parents) to allow us to do what we all know is best for kids… playing and engaging with math in a way that tugs at internal motivation (curiosity, the excitement of discovering something) rather than external motivations (praise, grades). We need to continue to find ways for doing math to be beautiful and creative acts of passion and wonderment and joy. The contributors of Playing with Math are working on this, and I am inspired by their stories.

Sue speaks about the origins of this book here:

And she is having a crowd-funding campaign. “The book has been written, edited, and illustrated. The money raised here will allow us to pay the artists, editors, and page layout folks, and it will pay for the print run.” I contributed so that I could get a paper copy of the book and finally mark it up with all the post-its and flags I want!

Teaching Award

About a month ago, I received a teaching award at my school. Technically, I suppose it isn’t an award, but a chair (“the William C. Stutt Chair for Math, Science, and Technology”). Fancy, right? I wasn’t going to blog about it, but it is something I want to archive and that’s the biggest (but not the only) reason I blog.



It’s given out every three years, and the last person to get it is one of my best friends at the school (who is also the person I look up to as a teacher).

When I was called up, there was a standing ovation from the faculty. Of course, let’s put the cards on the table here: there always is a standing ovation from the faculty when anyone gets an award. But I can’t help but admit I got a real glow-y feeling. I was overcome when I saw my parents there, a surprise! They popped out of the curtain and hugged me. I didn’t quite know what to say, so I babbled. All I remember saying is my teaching motto: “Try to suck a little bit less each day.” I posted this on facebook, me feeling babble-y, and a friend said: “You are amazing. Your comment to the faculty about trying to suck less everyday was perfect and came up again a number of times over the remainder of the meeting. I hope you and your parents had fun celebrating your awesomeness this afternoon. Also, please take that standing ovation personally. We could have gone on clapping forever. There was nothing perfunctory about it. Congratulations!” So yes, me all feeling warm and fuzzy.

I also posted this on facebook: “Although I’m not one who basks in honors and awards (I even skipped out on going to my college Phi Beta Kappa induction and a writing award in college), I do feel like teaching is a profession where you don’t get a lot of positive reinforcement for the emotional struggle that you carry with you every day. A few kind words from students occasionally, or a nice email from a parent, if that. 99% of what we do goes unseen and unacknowledged. It’s isolating and exhausting. So this award was a nice thing, something I can turn to when I feel like I’m emotionally drained and a failure. (Which is more often than not.) But more than that, it reminds me how important it is that we teachers give accolades and kudos to each other in a million unofficial ways, *everyday.* Because most all the teachers (especially the math and science teachers) at my school are pretty awesome. And every one of us are working to do right by our kids. And more than awards that get handed out once in a blue moon, we need to pay attention of the good that everyone else is doing around you, and acknowledging and huzzah!-ing those things. Yes, that’s what I see from this. Let’s prop each other up.”

The little news blurb on our school website is here. Archived.