Guest author and maths mastery expert Dan Polak talks about teaching maths in context, and the impact it can have on students’ understanding.

Whenever I complain that every billionaire is a policy failure, I use a context to explain just how enormous a billion pounds is. So few people I speak to believe the following fact:

Imagine if you were to travel back in time to 1485 and the Battle of Bosworth, where Henry VIII was only a glint in Henry VII’s eye. You could select a soldier and give them £5000 that day. And the next day. And the next, and the next…until today.

You still wouldn’t have given them a billion pounds by the date of this article. That’s 535 years of enormous paydays. Last time I checked, Jeff Bezos had an estimate worth of £93 billion.

I don’t know why we did algebra at school, I never use it.

It’s a minefield, telling anyone you’re a maths specialist. But this is the most common refrain you are greeted by. Others I hear regularly are “I hated maths”, “I’m not a maths person” and “My brain just doesn’t work that way”. I try to suppress the eye roll and the urge to tell anyone who complains about algebra that they probably, unknowingly, used it today. They just didn’t connect the classroom with their lives.

Many of us were taught mathematics pretty poorly. Sludging through textbook questions and lining up to have our answers ticked off by the teacher sitting at the front. I used to negotiate my place in line with keener students who wanted to get back to their table and move on to the next page. I used to entreat them to step in front of me, lacking the motivation to repeat the process.

Coins spilling from jar

Some of these textbooks were designed brilliantly – the intelligent design of the questions diluted by the unthinking application of ‘turn to page 55 and continue from yesterday’.

The Power of Context

I remember one lesson – a post-SATs moment of complete indifference. The teacher distributed a number of Argos catalogues we had been using for collages. She told us we were overnight millionaires and we could buy what we want. Just work out what you’d like and show me the subtraction. Off you go.

Pie cut into slices

I was hooked.

Filling four pages of my maths book, I fell even further down the rabbit hole when the teacher told us that if our maths was accurate, we would get an extra budget in addition. I didn’t swap my place in line that day.

The increasing call for the use of Concrete, Pictorial and Abstract in our maths classrooms has directed teacher’s thinking towards the use of context. Many teachers consider context the icing on the cake; an unnecessary luxury which can obfuscate the ‘real maths’ of the lesson. Sometimes, they are absolutely right.

To my mind however, context serves two distinct purposes in the mathematics classroom:

1) Where have you seen this and where might you see it?

When teachers use beans into pots to teach division, they are using a concrete manipulative which creates the structure of grouping or sharing in a child’s mind. This can be recreated by the child in continuous provision and in play.

If they plant seeds, they will experience this structure again. If they distribute toys into their friends’ party bags, they are exposed to the same mathematical structure used in the classroom. These are ‘lived’ experiences that the child will encounter again and again.

2) What is the ‘feel’ of the mathematics?

When we ascribe a context to a number like a billion or a million, we are able to gain a sense of the magnitude and difference in the numbers. A famous way of describing how much greater a billion is than a million is to use seconds. A million seconds is just shy of twelve days. A billion seconds is 31.5 years. Now we have context, it helps us to clarify and understand. Many children can tell you that a billion is a thousand times a million, but this context gives gravity and worth to what we have learned.

Bizarre contexts have been a mainstay of mathematics education. I’m not sure who first decided that watermelons in wheelbarrows was the lens we should see multiplication through, but rather than work out the calculation, I was left wondering what plans this person with a surplus of watermelons might have.

There was a recent question in the KS2 assessments where a child was buying birdseed by weight. If context is to be best applied, it needs to resonate on either a ‘lived’ experience level i.e. the child might do it themselves and recognise the structure. Or on an inspirational level – it helps us understand more about the world around us. Birdseed was unlikely to stir too many of our children’s mathematical senses.

Context is regularly used to draw comparison in the media, but this is hit and miss. Wandering around the natural history museum recently, I read a fact about a blue whale’s tongue weighing roughly the same as an elephant. While I might not have an intuitive sense of the weight of an elephant, I understand it to be…really big. This sort of comparison helps us feel the scale of the blue whale and adds more clarity than the figure ‘3,600kg’.

On the same day I read that the Shard was as high as 78 London buses. While we get a sense this is high, I’m not certain that we have an intuitive sense of bus heights and this adds little to clarify the relative height of the building.

Context is the heartbeat of mathematics.

Dan PolakMaths mastery specialist

Using Context in the Classroom

Finding these contexts and applying them in the classroom isn’t always easy.

When I introduce most concepts, the context I choose is my first definition of what makes context useful. Where have you seen this and where might you see it?

Measuring cups

I don’t want to immediately open the mathematics to all the confounding factors of thinking about the height of Everest or the length of the Nile. However, as children become more secure we can move through lessons and sequences to the application periods of learning. The moments that Shanghai teachers would call the ‘dong nao jin’ (use your head) exercises, I open up the other types of contexts. We want to apply division, so why not tell me how high Everest is by comparing it to the highest building in the world, the Burj Khalifa? Will this help us ‘feel’ the mathematics?

To find easy-to-apply contexts, I normally search for whatever topic I’m interested in and the words ‘infographic’ or ‘data visualisation’. Often you find beautiful, grouped data which helps you step outside the abstract and apply the mathematics to help children see the point. Division has a purpose. Let’s find it.

Take this example: the top eight richest men in the world have the combined wealth of the poorest 50%. Another reason we need mathematically-literate populations – to get a sense of whether we think this status quo is desirable. Let’s take their wealth and distribute it. How much does each individual get? What would the societal effects of this and would it be fair?

My year six had a fierce debate on this point. Some argued that people deserved to hold on to money fairly earned. Others argued that the tiny proportion of women, non-white or non-American billionaires showed that this money wasn’t fairly earned at all. Advantages were evident in the percentages – another application of mathematics taught earlier in the year. If 50% of the world is female, why aren’t 50% of billionaires female?

When maths lessons dissolve into social justice debates, it shows that we can make connections between the classroom and the world. If we are more expansive in our use of context, we will create adults who recognise when they use structures like algebra in the ‘real world’.

When we stop thinking of context as window dressing for the ‘real’ mathematics, we will recognise it for what it really is.

Context is the heartbeat of mathematics.

Dan Polak Headshot

Author

Dan Polak

Dan is a Deputy Headteacher at Woolacombe Primary School, a National Support School in Devon. He works as a Mastery Specialist for the NCETM and as an SLE for the North Devon Teaching School Alliance.

Previously working as a secondary teacher, Dan works with a range of schools to make maths coherent and meaningful for all children.

Follow Dan on Twitter – @DanDoesMaths.