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.

Multivariable Calculus Final Projects 2013-2014

Instead of doing traditional problem sets and tests during the fourth quarter, I have kids work on an individual project on something that relates to multivariable calculus that interests them. (During the year, I have them keep track of interesting tidbits or facts or something I go off on a tangent about [pun] that they find could be a possible final project. I also have this list of ideas I’ve culled to help them come up with a topic.)

I have them come up with a prospectus and I individually talk with kids about their proposed project and timeline for completion. Then when they get started and start envisioning a final product, they are asked to write a description of the final product out clearly, and come up with a rubric for grading that product. They are also asked to make a 20-25 minute presentation to their classmates, their parents (if they choose to invite them), math teachers, and administrators. This year, they wanted to give their presentations during senior thesis week, which means that lots of their friends could come to their talks.

And they have been! In the past week, students have given their talks and I have been way impressed by them. Honestly they’ve been more independent than in years’s past, so I was unsure of whether they were putting together a solid final project or not. They did.

Without further ado:

Title: Mathematical Change We Can Believe In
Description: This presentation shows how one region can be manipulated to form something more interesting, a process called Transformation of Axes. The 2D and 3D analogues, use of rectangular and rounded shapes, and proofs of the properties of transformations abound in this exciting journey through the wonders of the world of multiple (MANY) variables.


Title: Pursuit Curves: The Ultimate Game of Tag
Description: Pursuit curves are the paths formed when one point chases another point. In this program, we will be looking at the mathematical explanations of pursuit curves, and then using a computer program I have built to model a few.


Title: What’s Our Vector, Victor
Description: This will be an investigation into the history, origins, and evolution of vectors, their analysis, and notation.


Title: Economists working with Models: Understanding the Utility Function
Description: Firstly, we will gain a foundational understanding of economics as a discipline. Secondly we will discuss the utility function and the questions which it raises.


Title: From Chemistry to Calculus: a study of gas laws
Description: For my project I have constructed a “textbook” that analyzes the idael and real gas law through the lens of multivariable calculus. In my “textbook” I compare and constrast these two laws by means of graphical and derivative analysis.


Title: Knot Theory
Description: Knots are everywhere around us, from how we tie our shoes to how the proteins in our body wind themselves up. My presentation will give an overview of their place not only in the “real world,” but also the world of classroom math and calculus.


Intersections, 2013-2014

Today we had our launch party for Intersections, our school’s math-science journal. Last year a science teacher and I gathered interested students to produce this journal — and they worked tirelessly and did a spectacular job. This year, we have some new students and some old students who served as editors. Here they are giving their speech at the launch party (which was also a pizza-soda party).


More than anything, I have enjoyed watching the editors become independent leaders, organizing something involving so many people and moving parts, and presenting their creation to administrators, math teachers, science teachers, computer science teachers, and other students. I feel like I’m coming to understand the niche I play in my school: I find ways to make math exist outside of the formal curriculum for kids who want to get more involved. Intersections is one of those spaces — both for editors and for those students who submitted.

If you want to check out this year’s issue, please click on the cover photo (designed by a student) below and it will take you to the website.



(You can also click here.)

More than anything, if you have the time, just click around and see what cool things you discover!

Although it’s a lot of work, if you have any thoughts about starting something like this at your school, I highly recommend it.

Senior Letter 2013-2014

Every year I write a letter to my seniors. Each year the message is pretty much the same, though the way I deliver it may change a bit based on the class and what I’m feeling at the time. Each year I hate my letter when I’m done, but I decide I’m going to give it out because it’s a tradition and I don’t want to break it, and I convince myself it is not that awful. I hand it out. I’m grateful after I do, because… I suppose I need closure. I have worked with these kids closely for a year (sometimes more). And I have come to care about them all. And although it happens every year — they leave and I stay — and from this point on they slowly begin fading from my memory, right now they are in my life in saturated colors and I know I’m going to miss them and I want the best for them.

So even though I currently hate it, here is this year’s senior letter.

It came packaged with their “who I am” sheet that they wrote about themselves on the first day of class, and two cards I had printed.


Switching Things Up, Need Help: Geometry is on the Horizon

For the past seven years, since I started teaching, I have been teaching calculus. When I started, I had 8 students in my class. Now I have 36 students. I’ve had to shift how I’ve thought about the course tremendously, and I’ve undergone a dramatic transformation in the content I teach (it is non-AP) and in the style in which I teach it. Seven years with a class is both a blessing and a curse. And honestly right now, I’ve reached the end of my usefulness for the course. I’m spinning my wheels. The only way I would be able to do a better job with it is to leave it for a few years and come back to it with a fresh pair of eyes.

And luckily, I have the opportunity to try something new. Next year, I will be giving up Calculus to teach an Advanced Geometry course for the first time. In fact, it’s the first time I’ll have ever taught geometry at all.


When I first began teaching, I was scared of geometry. Partly because as a student in high school, I found geometry to be uninteresting. It certainly didn’t have the elegance of algebra, at least the way I was taught it. Partly because I realized in that course — more than any other course — you as a teacher really have to focus on hard things. If you want kids to be able to do a proof of any kind (two-column or not), you are really teaching intuition building and connection making. Which is tough, and daunting for any new teacher, and this is why I recoiled at the thought. Now, years later, I see this as such an exciting challenge.

Right now, I am not anywhere about how to teach this course. And in fact, I’m only teaching one section and the other teacher is teaching three sections. But he’s very open to really revitalizing the course. So now we’re in exciting territory. Before I go bananas on scouring everything out there, I thought I’d crowdsource.

For any of you geometry teachers out there, if you have time to answer one or two (or all!) of these questions in the comments, I’d be ever so grateful!

1) What are your favorite geometry teaching resources — both online and offline? I’m talking books, websites, applets, manipulatives, whaever?

2) What are your favorite math teacher blogs that focus on geometry?

3) Is there a lesson you absolutely could not imagine teaching Geometry without?

4) Do you teach the course with a connective thread? Like: We are studying space and the properties inherent in space as we build space? Or: We are studying exactitude –and in particular, how we define mathematical entities so they yield uniquely understandable creatures? Or: We are studying “measurement” (in the vein of Paul Lockhart’s book).

5) I’m concerned that our kids lose a lot of their Algebra I skills when they take geometry. The other teacher and I have talked about putting coordinate geometry front and center from the beginning to help with this. Do y’all do anything else that helps keep their algebraic skills sharp, and maybe even push them forward?

6) Anything else? Problem solving? Sangakus? Geogebra use? Things you throw out because you feel strongly it’s only taught because it’s always been taught? Incorporation of Euclid’s Elements or math history? Graphic-design-y projects? Math art?

UPDATE: WOW, everyone, thank you so much for your resources and advice and for taking the time to type out so much great stuff. Now I’m genuinely THRILLED and CHOMPING AT THE BIT to get started re-learning geometry (and then teaching it). I am going to sort through things this summer!

Experimentum Crucis: A Symposium Course


This January, for seven days, I taught a seven day course with a friend and fellow teacher. Our school eliminated midterms and instead instituted different programs for different grades. Juniors and seniors were given the opportunity to sign up for full-day courses designed and taught by faculty on topics of interest. Faculty were given the opportunity to design courses which got kids to think about topics in a different way.

My co-teacher and I developed a course that was designed to be interdisciplinary (we were working at the intersections of history, science, and philosophy), hands-on (students would be working in the laboratory), and rigorous (meaning kids would be expected to think and work at a high level).

Designing and teaching this class was one of the hardest things I’ve ever done as a teacher. And I don’t know — honestly, I don’t know — if we were successful or not. Even with the feedback we received. Thus even though it was challenging, I’m not sure I felt it was rewarding. In fact, the reason I’m writing this blogpost now, months after this, is because I was so exhausted with the whole thing I couldn’t bring myself to even think about it in a reflective or objective way.

The origins of the class go back to the previous year, when my co-teacher and I started trying to envision precisely what the big picture ideas were, and how we were going to get kids to go from point A to point B in their thinking. This also was coupled with the question: how the heck do you design seven days with the same group of kids, from 8:3o to 3:15. Seriously put yourself into our shoes for a second. Initially, it’s pretty exciting! All this time! Do what you want! But then you realize: you are going to have 12 to 16 kids in your charge, and you need to fill up that time with multiple activities! Quickly this went from exciting to daunting and anxiety-filling. For months, the co-teacher and I would have meetings, read books and articles, come up with ideas, refine our ideas, and throw out our ideas. Coming up with a lesson plan for a single day took weeks of work. The agony, the hours, the frustration… I don’t wish that upon my worst enemy. But we finished.

Our course abstract:

Can you imagine building a battery without the concept of electrons?  What would it be like to describe chemical reactions without discussing atoms?  Would you believe Einstein’s theory of relativity if no text book told you to and there were no way to test it?

In this course, you will have the opportunity to put yourself in the shoes of scientists who (in retrospect) revolutionized the way people viewed and understood the natural world.  By carrying out famous historic experiments, you will explore the process of creating “scientific models” and “scientific facts,” many of which we now take for granted as self evident. This course will be hands-on and interdisciplinary. In addition to lab work, we will read primary and secondary sources that will allow you to place science in historical context and understand scientific knowledge making as a process and a product of its time.

Our course objectives:

Through this course, students will explore:

  • science in historical context
  • how science is influenced by and a product of its time
  • that the process of science involves models changing over time
  • that what we take for granted is often messy, weird and sometimes illogical
  • that science is a human endeavor
  • that the making of science is a process
  • how scientific “facts” get accepted/discarded –  that ideas are nothing without the acceptance of many people

and ask the big questions:

  • What is an experiment?
  • What is a scientific fact?

Anchor Texts:

Thomas Kuhn’s The Structure of Scientific Revolutions
Original papers by Robert Boyle and Alessandro Volta
Secondary texts


Originally, we planned to have a number of experiments: Proust, Boyle, Volta, Oersted, Einstein. However because we had a snowday (there went Einstein and the discussion of thought experiments), and because some of the experimentation took much longer than expected, we had to eliminate more (Proust and Oersted). Thus, we only ended up working extensively on Boyle and Volta.


One day was spent on a field trip to the Chemical Heritage Foundation in Philadelphia, but the rest of the days were spent having deep class discussions and carrying out two in-depth experiments in the labs. We did Boyle’s Law experiment, and they had to bend glass to make their own J-tube, and play carefully with mercury. (We inducted all our kids into the Royal Society, after reading bits of the original charter, and administering the oath that the initial founders took.) Our kids saw that our modern instantiation of Boyle’s Law (PV=k) was nothing like the original formulation (they only were given Boyle’s original paper to guide their research and help them figure out how to reproduce the original experiment), and they started to get at the idea that Boyle was looking at his experiment through a totally different lens (“the springiness of air”). My favorite part was when kids saw how their little sidebar about Boyle in their chemistry textbooks was just a black box for so much! And how it wasn’t just “one crucial experiment” that suddenly worked and changed our understanding. Mwahaha, the title of our course is precisely the thing we aimed to get our kids to debunk. 

Our second experiment was building (well, improving upon) the first voltaic pile. Again they only had Volta’s original paper to work from, they were given many materials that Volta mentioned in his paper to play around with and test (e.g. lye, silver, zinc, tin, coins, leather, cardboard, salt water, etc.), and they were working to win le Prix Volta (a real prize Napoleon and the French Academy of Science offered for research in electricity, after Napoleon saw Volta’s original battery demonstrated). This contest was good to talk about collaboration and competition in science, but my favorite part was having kids read a challenging history of science article about what actually was behind the creation of the battery (a torpedo fish!) and what sorts of things had to have happen for there to be the physical and intellectual space for Volta to even have the conditions for him to come up with his Voltaic Pile. That the battery is historically situated, and tools, ideas, and people had to come together in a specific way for the battery to emerge and look the way it did. I also really liked that students could understand that there could be an explanation of electricity that didn’t center around electrons.

That dovetailed really nicely into how we were talking about Thomas Kuhn. We used Kuhn’s Structure of Scientific Revolutions as our core text that they were reading extensive bits here and there each night, and although I was worried it would be too abstract for them, they grappled with it and came out victors. And I think (hope) it was a real mind-blowing experience when they realized that “old” theories weren’t “bad” because those scientific practitioners who adhered to them were dumb (or at least, weren’t smart enough to see the Truth with a capital T). And listening to them discuss Kuhn, grapple with the idea of Normal Science, and start to see glimpses that (1) science isn’t accumulative in the simplistic way that textbooks tend to say it is, and that (2) we always are looking at data, theories, experiments, observations through specific eyes, and what we see is dictated by the paradigms we accept.

Images: Here are images from the Symposium, without student faces in them. (Hence, we don’t have the majority of my favorite pictures.)