A Week of Play in Uppers Math

Over this summer, I was perusing a bookstore in Brattleboro. I have more fiction books than I have time to read, so I decided to take a look at the nonfiction section and see if anything spoke to me about the math and science I would be teaching this year. As soon as I picked up Ben Orlin’s book, I knew I had found my winner. His book is titled Math with Bad Drawings: Illuminating the Ideas that Shape Our Reality. As someone who, like many of our students, gravitates towards graphic novels, the idea of a math discussion with images really appealed to me. Once I got to reading, his humor and playfulness also sold me. It’s a book I would recommend to anyone, especially those people who say they don’t like math. 

Upon reading his book further, I found out that he had been a teacher, for students aged 11 to undergraduates. He also wrote other books, including Math Games with Bad Drawings: 75 ¼ Simple, Challenging, Go-Anywhere Games, and Why They Matter. I immediately bought that book and started forcing my husband to learn as many games with me as we could before school started. (To be clear, he wasn’t actually forced, he was a very willing and excited participant!) As we played, I started thinking about how much fun we were having and how I was excited to bring this joy into the beginning of the year math class. I also kept coming  back to a quote in the introduction of the book: “The secret to our brilliance is that we never stop learning, and the secret to our learning is that we never stop playing.” From this book, the formation of our first week of math together was born. 

As you read on, I’m not going to get too much into the games’ instructions, but if you are interested please look into these games or better yet, ask an Upper in 6th or 8th grade to teach you how to play! They are worth learning, but for the purpose of this bulletin piece, it’s more important to see how each game inspired our discussions in the classroom.

Classes for the Uppers begin the second week of school, easing our way from summer freedoms to classroom structures. On Tuesday, both the 6th and 8th graders were introduced to our first game: “Ultimate Tic-Tac-Toe”. Born from the minds of a group of math students at Berkeley, “Ultimate Tic-Tac-Toe" is played with the goal of getting 3 in a row, but this time, the larger board is made up of smaller boards in each square. (You can see the game board on our whiteboard table in the photo below). The game moves between the smaller tic-tac-toe boards depending on where you place your x or o, so you are rarely able to play in the same small board two turns in a row. This creates a chess-like quality where thinking 2 or 3 steps ahead is important.  

At the end of the game, we used a discussion on strategy to help us talk about classroom expectations. When playing or learning together, what is important to us? How do we include other classmates and other ways of thinking? The 8th grade discussion ended with Ruby reflecting on the 2 times we played Ultimate Tic-Tac-Toe. She said she changed how she played the second time based on other classmate’s strategies. When we went to add this to our classroom expectations for the year, it turned out to be two expectations. The first was one for ourselves, that we can “expect to focus on other people’s strategies”, and the second was expectations for groups, that we can “help think about other people’s ideas and strategies”. 

In Ben Orlin’s Math with Bad Drawings, he ends the first chapter, How to Think Like a Mathematician, talking about Ngô Bảo Châu. Châu is a winner of the Fields Medal for his work in algebraic geometry. He started as a super competitive student, but quickly realized in order to truly succeed in mathematics he needed to see his competitors as his collaborators. He needed to ask for advice and hear about other ways to approach a problem. Much like Châu, our classroom this year has now set the expectation not for competitive math thinking, but rather for the collaborative style of learning we all know is important for our success. Although the game we played was competitive, the way we discussed our strategies changed us into collaborators. 

On Wednesday, both groups played the game “Neighbors.” In this dice rolling game the goal is to score points by putting the same number next to each other. We used this game as another way to think about ourselves as a group. We noticed our differences (no one had the exact same board), but we also noticed some similarities. The 8th graders played the first round building rows of the same number (we rolled an 8 five times!), but when we looked back at how to score our points, many students noticed that putting four 8s in a row only got you 32 points, but if you made a square with the numbers you could score each 8 twice and it would actually double your points. This became a perfect example of our need for collaboration - once someone began pointing out one noticing, the group took that information and ran with it and everyone was rethinking their strategies for the second round. 

Thursday was a real teaching highlight for me. The game we learned was called “Medicority” and in the 8th grade it did bring out some good conversations about not always having to have the highest score (or even the lowest like in golf), but how do we stay in the middle? How do we change our thinking and occasionally allow for mediocrity to be something that is not only allowed, but in the case of the game, encouraged? While opening some interesting dialogue, the highlight for me took place before the game, during our math warm up in the 6th grade. They were shown a cylinder and a circle and asked to share what is the same, but different about these shapes. We went through the typical naming of the shapes, 2D vs. 3D, and which side of the object we could see. That’s when Sage said, “I wonder how you could find the area of the sides of the cylinder that aren’t circles.” From this simple wondering, our discussion evolved from a quick 5 minute warm up to a 25 minute group think. 

Gathered around our whiteboard table, we talked about a can of beans. What does a can have? A label. What shape is that label? A rectangle, so could we use our knowledge of the area of a rectangle to help us? Can we decompose our 3D object into shapes we know? Moving away from the sides to the top of the can came a wondering about the circle on top. Is there a formula for that? Why in the world does that formula have a weird symbol called “Pi” in it and what’s with the “r”? Students drew the conclusion that the “r” in r2 must function like the “L” and “W” in our area formula, making it possible for us to find the area for any size circle. It was some of the most curious and thoughtful conversation I’ve ever had with students, to see them using previous years experience and applying it to different objects to try and make sense of something that many of us still scratch our heads at. Only 4 days into the week and already these 6th graders are bringing their math teachers to tears with their brilliance. 

On Friday, our week ended with the 8s and 6s playing two different games. The 8th graders learned one of my new favorites called “Outrangeous”, the goal of which is to predict a value using the most narrow range you confidently can. So I would ask a question like, “What year was the first Nobel Prize given?” and students had to write a range that had the value in it, then order themselves from narrowest to widest ranges, gaining points for the number of ranges they were smaller then. It opened up a lot of discussions around reasonable ranges and the challenges of making guesses when the values of the answers were both specific and oftentimes very large (as was the case when I asked for a range for how much the Disney Corporation is worth. It's currently $209 billion, if you’re curious). The students ended the class coming up with their own questions and answers that we will use for our warm up on Monday.

The 6th graders were given Goldfish crackers (with the very challenging direction to not eat them!) and learned a game called “Turning points”. You take turns placing the fish on a square in a 4 by 4 grid until all the squares are full and get points for the number of fish facing your side of the board. If you place a fish that faces another fish, that fish gets turned 90 degrees. When there are enough fish on the board, doing this can set off a chain reaction of fish turning. In our group reflection, Max recognized that instead of playing the game to win, Sage was playing to try and create the longest chain reaction he could on the board. Max acknowledged that this actually made the game more interesting and it was fun to watch Sage work through this problem. This brings us back to the importance of collaboration; the game became more fun because someone was trying something new and different. 

One final thought from my new buddy, Ben Orlin: in his introduction to Math Game with Bad Drawings, he talks about a 1974 geneticist named Marsha Jean Falco. She was trying to create a research tool for dog DNA using different symbols on index cards. “But as she shuffled and rearranged the cards, all the specifics fell away.” From this shuffle of patterns, she developed one of my favorite games, “Set.” From scientific research to one of the most popular card games of the 20th century, Falco let play become the focus of her learning. The goal of this week in the Uppers returns to our quote that the “secret to learning is to never stop playing.” Our goal in math this year is to let play and collaboration to be the center of our work together in class. We will try to face our new learning challenges as puzzles to be worked through or games to be strategized. We will use the luxury of having a small group to really get into discussion and let that collaboration help us all become the best math learners we can.