# A numbers game

- Cambridge Mathematics
- Mathematical Salad
- A numbers game

## A numbers game

Games like 2048 – downloadable, slick apps that are fun to play – aren’t new. They can be highly addictive and many of the general public would classify them as ‘maths games’. Are they?

Now, I do find it frustrating when there is an assumption that anything involving numbers is blindly labelled ‘maths’, but it would also be a shame to dismiss something that has the potential to encourage curiosity, problem solving strategies, and resilience in children and adults alike. Oldfield (as cited in Way, 2011) says that mathematical games are ‘activities’ which: involve a challenge, usually against one or more opponents; are governed by a set of rules and have a clear underlying structure; normally have a distinct finishing point; and have specific mathematical cognitive objectives. How does this align with your definition?

As teachers, the main problem with ‘maths games’ is that we might feel that we can’t really justify the time spent on them in the classroom. Alternatively, too much pressure is placed on the game itself – a great game cannot guarantee a great lesson (if it could, then there would be no need for maths teachers!). It seems that using a game needs careful thought and facilitation in the lesson itself if students are to have the experiences and become aware of the learning opportunities that you intend.

I suspect that 2048, and many other ‘maths games’, might have been dismissed by some teachers as it was too much work to try to shoehorn them into a lesson. What if we took some time to think about how we could make it easier to craft a lesson that includes 2048?

Firstly, let’s not forget the benefits of stating the obvious out loud when thinking about or through a problem. Let’s assume that the game is visible as students arrive in the classroom, on iPads between pairs of students or on the interactive whiteboard. A selection of questions, such as the eight listed below, are also visible, perhaps printed out and placed on tables (Although I am focussing here on the students, I also urge you to take a minute to think about your own responses to the questions below).

- • What do you notice?
- • How do you make a move in the game?
- • What happens every time you make a move?
- • What do you notice about the numbers in the game?
- • What can you say about the numbers in the game?
- • What’s the aim of the game?
- • How do you lose?
- • What might the game board need to look like just before a winning move?

Students could be encouraged to explore the game and respond to the questions in the first 5 (or maybe 10) minutes of the lesson. They talk to each other, sharing prior knowledge of the game, making claims and predictions or simply trying to play the game. The teacher facilitates, verbally asking questions that appear on the list and in some cases probing for more detail: ‘Just say that again’; ‘Do you think that’s important?’; ‘So what numbers is it possible / impossible to make? Could you give me a list?’; and so on. They might have some screen-shots of the game board at particular points to encourage students to think slightly differently. For example,

- • The person playing this game has never scored higher than 64 in a single square. Can you suggest a series of moves to continue this game? Can you predict how the game board might look after 5 more moves?

Developing a bank of questions, and questioning strategies, such as this is an important process if we are going to feel able to tackle increasingly challenging and unfamiliar problems (in any subject, not just in mathematics). Forcing yourself to respond to these types of questions can help to expose the nature of the problem or puzzle and can direct your thinking down potentially fruitful avenues. If you would like to pursue these ideas further, Mason et al. extensively discuss questioning and ‘becoming your own questioner’ in their book *Thinking Mathematically*.

So: 5 (or maybe 10) minutes into the lesson, the teacher could collate responses to the questions from pairs of students, perhaps recording particularly interesting or seemingly important realisations. They then might guide the thinking to a particular focus for the lesson by posing further questions, perhaps asking students to consider the number of 2s that must be added to reach 2048, and then the number of times a 2 must be doubled to reach 2048. Perhaps index notation is introduced (or reintroduced) later on to aid with the recording process. Maybe the lesson ventures into the realms of binary information and digital imaging (this also features in some A-level Physics courses).

There are many lesson foci that could be initiated by this starting point, for students of many ages and prior attainments. The important point for me is that it isn’t necessary for the game to be the context for the lesson (it could be, but it doesn’t have to be): it’s more that it sets the scene and provides a concrete point to return to over time. It is likely to attract most students’ attention and, importantly, has the potential to motivate definitions and concepts to be developed in that lesson. Jenni Way briefly summarises some of the relevant research and implications for teaching here.

When searching for this game on the internet, I discovered that a variety of versions and similar games are available. Intrigued, I downloaded one called ‘Threes’ which, as I’m sure you can guess, is a variant that uses and combines 3s instead of 2s. This presents a nice opportunity for considering the ‘What’s the same and what’s different?’ question and perhaps inspires further ‘what if…’ questions such as:

- • What if the board was a different size?
- • How does the game change if the smallest tile is a 1? What about 4? What about 5?
- o What might you choose the target number to be in each case? Why?

I’ll leave you to think about that …