Math Teachers’ Circle

Last Monday morning I was on a SuperShuttle from SFO to the Creekside Inn. But I was running a little late, so the driver took me straight to the American Institute of Mathematics (AIM). I was expecting a tall building with lots of glass windows — instead the driver stopped here.


I was at AIM to attend a one-week workshop on How to Run a Math Teachers’ Circle. I stumbled upon this opportunity simply by doing a search for a Circle that I could be a participant in. But there wasn’t one in my area that I knew of, hence my always-ambitious-but-never-know-what-the-hell-I’m-getting-myself-into evil twin said, Why don’t you start one?

I asked Erin, my next-door math colleague, to join me. She and I fulfilled the requirement for “two middle school math teachers” — we just needed to find “two mathematicians and one administrator or organizer” to make a team of five.

I contacted Brianna Donaldson, AIM’s Director of Special Projects, to help us round out the team. Within the week, she hooked us up with Nate and Hala, both are math professors at Cal Lutheran. We still needed a fifth member, so I asked Melissa, another math colleague (there are only 3 of us at our junior high) — she graciously got on board.

Needless to say I was thrilled to hear that our team was chosen among many (like hundreds of thousands of teams) to receive full funding to participate in this workshop.

Below is a summary of my week in beautiful Palo Alto, California. (For some reason it’s easier for me to write in present tense when I recount a story — maybe it’s an ELD or ESL thing.)


I meet the other two members of our team for the first time, Nate and Hala. I’ve seen their pictures online, and they look the same — thank God!

There are six teams: 2 from Texas, 1 Kansas, 1 New York (Rochester), and 2 California.

Josh Zucker appears first on the agenda, “Introduction to Problem Solving.” I’ve always wanted to meet Josh — his name appears on most of the cool math problems that I first encountered at the Julia Robinson Mathematics Festival in Los Angeles. Turns out he’s the director.

He facilitates this classic problem:

The numbers 1 through 100 are written on the board. You choose any two numbers x and yand erase them, writing xy + x + y in their place. You continue to do this until one number remains. What are the possible values for that remaining number?

I just want to post the problems here without further discussions on them so that you — the thinker, the mathematician, the teacher, the student, the problem solver — get to struggle with the problem and construct meaning for yourself.

There are three things that happen consistently each day, so I’ll just mention them once here:

  1. We get two hours (TWO HOURS!!) for lunch as the nearby restaurants are about a 15-minute walk away.
  2. After lunch, we work within our group on logistics and fundraising to run our own math circle.
  3. “Happy Hour” greets us at the end of each session, if we wish to stay.


Tatiana Shubin is a math professor at San Jose State University; she presents Grid Power. I’ve always required my students to use quadrille-ruled composition notebooks to take notes and such, but after hearing Tatiana speak, I want my kiddos to do ALL their math work on grid paper!

Tatiana gives us this delightful problem:

How many squares are there in a 7 x 7 square?

Paul Zeitz introduces us to “Mathematical Games.” He’s a math professor at the University of San Francisco and is taking a sabbatical this year. I’ve ALWAYS wanted to buy the book The Art and Craft of Problem Solving — lo and behold, Paul is the author! But Paul doesn’t once mention his book (I made the connection after the workshop was over), instead he recommends James Tanton’sSolve This. I need these books.

Two of the games we work on:

Takeaway. A set of 16 pennies is placed on the table. Two players take turns removing pennies. At each turn, a player must remove between 1 and 4 pennies (inclusive). The winner is the last player to make a legal move.

Puppies and Kittens, aka Wythoff’s Nim. We start with a pile of kittens and puppies. Two players take turns; a legal move is removing any number of puppies or any number of kittens or an equal number of both puppies and kittens. The winner is the last player to make a legal move.

I really like how Paul keeps track of winning/losing moves as oasis/desert points on a horizontal line. So with the two variables like in Puppies and Kittens, we use a coordinate plane instead to record.


Diana White is a math professor at the University of Colorado Denver. Diana facilitates us through the Exploding Dots problem. (James Tanton owns the dots.)

Say you have a machine that holds ping pong balls. If you put three balls in the far right slot, they’ll explode and two balls will move one space left into the next slot. Like this:

1This happens with any set of three balls in one slot, therefore the explosion continues until there are fewer than three balls per slot. Thus, starting with 9 ping pong balls, the result looks like this:


Tom Davis is a retired math professor; he walks us through Conway’s Rational Tangles. I don’t even know how to begin to explain and illustrate this problem. It requires four students and two long ropes — each student holding one end of a rope. The task is to do two (and only two) commands of “twist” and “rotate” to tangle up the ropes — the challenge then is to untangle the ropes in a systematic way that involves arithmetic with positive and negative fractions. Okay, my explanation is as clear as mud.

The point is it is a wonderful activity that kids and teachers will absolutely love to get their hands on, literally.

The other point is I’m better than you because I have the two ropes. See?



Paul Zeitz is back this morning with “How to Gamble If You Must.” We play a few dice games, then we work on Two Lottery Tickets:

It costs a consumer $1 to buy a Klopstockia lottery ticket. The buyer then scratches the ticket to see the prize. Compute, to the nearest penny, the expected profit that the state of Klopstockia makes per ticket sold, given the following scenarios for prizes awarded. (The state will make a profit if the expected value to the lottery ticket is less than $1.)



There is no math activity on our last day. Each team has 10 minutes to present the how/what/where/when of their own future Math Teachers’ Circle. We each get a T-shirt and our team pictures taken. We bid farewell at noon.

I’m grateful to everyone who made this workshop possible. And I love my team!! So I lied about us being chosen among hundreds of thousands.

I had never drank so much wine in one week. I took two wine corks home to practice a trick Josh had taught us —

Please consider joining a Math Teachers’ Circle in your area. Our team leads the Thousand Oaks MTC and would love to have you. (I’m working on our website.) We owe it to our students to make math engaging and accessible.

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