How to Build Mathematical Thinking with Games

Games are a staple in so many maths classes and, unsurprisingly, they get used in different ways and for different reasons. At a recent Maths Teacher Circles session, which I facilitated, we explored games that have inherent mathematical value - they are fun and energising, but they also help students to be stronger mathematical thinkers. After the session, I noticed how each game we looked at provided a lesson on games more generally. Here's what I found:

Games can be simple, yet insightful.

A simple game is easy to get started. By not needing to concentrate on the rules, simple games enable students to start noticing what's going on and to identify helpful strategies. Take tic-tac-toe. Players take turns placing Os and Xs on a 3x3 grid, with the aim of getting three in a row, column or diagonal.

You can blindly play tic-tac-toe and hope for the best. Or, you can start to consider which moves are good to make. In the grid below, if you are Os and it's your turn next, where would you go? Is there anywhere you'd avoid going? Why?

Tic-tac-toe: the game that got us started at the Maths Teacher Circles session.

Think about: Thinking strategically about games like tic-tac-toe helps students to build transferable skills, including using logic and mathematical reasoning. These skills are relevant across the mathematics curriculum. How might you encourage strategic thinking amongst your students?

Games can lead to investigations.

At times, the game itself is just the starting point and provides the stimulus for an investigation into important mathematical ideas. Multiple Mysteries is a card game, which was designed and presented by Toby Russo and James Russo at the Maths Teacher Circles session.

During the session, we spent time playing the game (using virtual cards) before diving into two investigation questions that Toby and James posed:

1. Investigating game scenarios: When playing with three as their target multiple, John was particularly proud of himself for scoring a ‘4-pointer’ using the cards 2, 1, 4, 5. What number do you think John created? Prove that you are right. Are there any other possibilities?

2. Inquiring into the structure of the game (e.g., what are the best numbers to have): Is it possible to be dealt five cards that would potentially allow you to score a 5-pointer for any target multiple (for the target multiples 2 to 9)? What might these five cards be? Are there any other possibilities?

Think about: Investigations get students thinking about the deeper structures of the mathematics involved in a game. How might you plan for an investigation? Where might it fit in a topic or a term?

Games can be adapted.

At the start of the Maths Teacher Circles session, I said to the group, "If something isn't the right level of difficulty for you today, make it right for yourself," i.e. people could increase the challenge and complexity of the games or they could simplify them and focus in on some aspect. Adaptation can involve varying the numbers used, the rules, the number of players or other relevant parameters. Making such changes can increase excitement and, importantly, also give insight about what might be going on in the game.

David Butler ran a game he'd developed for university students called Digit Disguises. As he explained, however, by adapting aspects of the game such as reducing the number of letters or changing the goal (e.g. "Find the letter that is 0"), Digit Disguises can successfully be played even with primary-aged students.

The set-up for David Butler's Digit Diguises, which we played as a whole group at the Maths Teacher Circles session.

Think about: Take a game you are familiar with and have run with students. How could the game be adapted? What might students learn by making each change to the game?

Questions about games

Throughout the session, practical questions about running games in the classroom also came up:

We are often told to display explicit goals for a lesson. However I prefer to use games without making it explicit what they are for. What do you think?

If we consider the three examples above, we can tie in important learning goals that give nothing away about the games. That way students don't miss out on discovering what's great about the game, but can also see the overall purpose and value the game provides. Here are some suggested learning goals:

• Tic-tac-toe: To describe a winning strategy for the game

• Multiple Mysteries: To explain methods for finding multiples of the target number

• Digit Disguises: To adapt the game so it is the right level of difficulty for you

The above learning goals touch on the maths proficiency strands. One teacher at the session suggested, "by making proficiency-focused learning intentions the norm like being able to reason why etc., ...it can be applied to all levels that students are working at in a game , not a particular content focus."

How do we build time into the secondary classroom to play games? It seems like you would only just get the rules explained and the kids engaged with playing and then there goes your lesson. How do we make time for this but also to address the required curriculum?

First, let's look at the practical side to these questions. Secondary teacher, Samantha Hellessey, suggested using dedicated problem solving lessons each term that are planned in advance. Another teacher explained how they use short periods of class time on a regular basis. By coming back to a game multiple times, you build up understanding of what's going on.

Second, let's consider what the purpose is of the games. As primary teacher Kerri Smith pointed out, "it is important that teachers understand the value inherent in games, beyond just being fun." When games help students to become better mathematical thinkers, they provide a dual benefit: high learning and high engagement. So, they're not just being run for entertainment purposes, but to provide beneficial learning opportunities.

I always want to facilitate that moment when the students stop competing with each other to win the game and start cooperating to figure out best strategy for the game. Sometimes I can do this by having the challenge be to play against me later. Any other ideas for how to help that happen?

Challenging students to play against you is a great strategy! As you indicate, it shifts the goalpost for what students are aiming for - from just trying to beat their opponent, to understanding the game at a more sophisticated level (and potentially getting kudos from their classmates if they do succeed in taking you down).

A complementary approach, which I've learned from Dan Finkel and used with great success, is to say to students early on in the lesson: "Use every loss as an opportunity to learn something more about the game and about your opponent's strategy". This helps students go into the game thinking about strategy from the outset.

Do you have other questions about running games with your students? Post your question below.

To dive deep into mathematical ideas and get involved in conversations like these with other teachers, join an upcoming Maths Teacher Circles session.