Using local context as a ‘hook’ to ignite interest and excitement
Y6 at Woolacombe School, Devon, have been asking:
"How many grains of sand on Woolacombe beach? … And how long would it take to pick them all up?"
Why it’s important to link maths to the environment around us
When a child learns to read, practice carries on outside the classroom as the child can’t help but read signs, notices and adverts around them. Perhaps if we can develop children’s ability to interpret the world around them mathematically, then we will see this effect with mathematics too.
When we ask mathematical questions about the world around us, maths becomes a tool for finding things out, rather than something that only happens in school.
Dabbling with maths outside the curriculum boundaries, particularly when stimulated by their desire to answer a question or find something out, allows children to taste that there is a whole exciting body of maths beyond the curriculum, waiting to be explored.
By provoking, developing and nurturing children’s natural excitement about learning we stand the best chance of helping the child to engage with education and to become a lifelong learner.
Consideration should be given to the context to make sure that the numbers are manageable by the children and that they have the skills, strategies and equipment to handle the calculations.
Woolacoombe School Y6 teacher, Dan Polak, tells their story
I love it when children become obsessed about an idea from one of our lessons. A child obsessed with finding answers and applying their understanding of mathematics to enhance their understanding of the world, is a child who knows why we learn maths in the first place.
We look at this beach every day so it gives us a great opportunity to build obsession by asking questions about something familiar. It is possible to ignite interest which builds to obsession when we present mathematics which tells us something more about the world.
We started speaking in class about the idea that there are more stars in the sky than grains of sand in the world. It started the discussion about how many grains of sand might be on Woolacombe beach.
To create a series of lessons to explore this, I needed to carefully consider what my year sixes knew which could help them, and what structures needed to be in place to aid their thinking.
The question I posed was:
“If I picked up one grain of sand per second, what year would it be when I finished?”
Of course, the caveat was that I never took breaks and lived forever! (And for the purposes of this exercise, leap years were excluded)
I decided to give them the first piece of information that the ‘sandy’ part of the beach was three metres deep so each square metre was the end of a cuboid (1m × 1m × 3m) comprised of around 163 billion grains of sand. I also told them that the area of Woolacombe Beach is 2.4 million square metres.
The maths Dan’s class did…
 Calculate the number of grains of sand on the beach using Area of beach × number of grains of sand in each cuboid = 163 billion × 2.4 million. This involved some pretty hefty understanding of the relative size of numbers and some complicated long multiplications, even with the use of equivalent calculations.
 Convert the number of grains (i.e. the number of seconds) into years by dividing by 31 536 000 (the number of seconds in a year). Plenty of extra maths can be generated here. How many seconds have you been alive? What about family members? Can you live to be a billion seconds old? Will anyone ever be three billion? Some ask me whether they can work on this for a bit, and since the general principle of the maths is the same, I encourage them to follow their questions.
 Add 2017 to the number of years, giving the year 12 404 872 641 as the answer.
Using Dan’s starting point, his class got involved in all sorts of complicated calculations as well as grappling with the concepts of very large numbers, and of time and the difficulties of converting seconds to years. Their knowledge of long multiplication and division was drawn upon and practised in a context that made sense to them. While the size of the numbers used goes way beyond the curriculum at Y6, and calculating with them is something that would normally suggest the use of a calculator, Dan was able to allow these children to be led by their enthusiasm to handle maths way beyond expectations. As he says:
“The brilliant thing about this task was that there was a huge buzz in the room. There is a real sense of curiosity so the difficulty of the task goes out the window, and the children are busy working out how they can solve this question. This is obsession  it might take a very long time but we will get there because we want to.”
In the words of one child:
“We spoke about one the most amazing concepts, which is how big these numbers really are. Most children know millions and billions at year six, but the fact that a billion is a thousand times bigger is illustrated dramatically when we think about time. A million seconds is just over 11 days, a billion seconds is 31 years.”
Dan used time on the carpet to support children in their methods. “I started them all off together and left them for ten minutes to identify for themselves whether they had a strategy to solve this. I then started the first multiplication on the carpet. I normally operate an ‘openinvite’ approach to guided maths. I give the class an opportunity to chat about how to solve it and then start solving it myself shortly into the lesson. I might say something like “I’m going to start this on the carpet, if you want to see my strategy, come on down.” I might have already identified some children who need support and ask them to join us if they don’t volunteer!”
So many children asked what the world would be like in the year 124 050 723. This gave us a huge opportunity for more maths, as humans have only been around for a tiny fraction of this time. This gave us more far questions about human history and time than I’d thought about when I first started thinking about this lesson sequence.
Curriculum areas that the children drew on, developed or learned about:
 multiplication
 multiplying and dividing by multiples of ten
 volume
 estimating
 relative size of large numbers
 conversion between different units of time.
Alison Hopper, one of the NCETM's Assistant Directors (Primary) comments on curriculum areas covered:
Once children are fluent in using formal calculation strategies, they need somewhere to apply their learning. In this case Dan looked beyond the classroom to find a context in which to apply learning of multiplication and division strategies to solve an exciting and challenging problem. The fact that the problem was rooted in the school’s environment provided an extra level of engagement for the children.
Using representation to break the problem down in to manageable stages made sure that it was accessible to all children in the class  in this case, breaking the beach into sections which the children could visualise easily.
Children were required to draw on other aspects of their mathematical understanding: to consider time, using their knowledge of the relationship between units of time. Fluency in calculations, both mental and written, was required to manage the conversions.
In this particular problem, the numbers went beyond the expectations of the primary classroom but children are often fascinated by really big numbers and the names (both mathematical and not) that we use to label seemingly unimaginable amounts. In this case, those amounts had a physical representation on the beach that the children can see from their school, helping the children to move their learning into the real world. Having said that, the scale of the problem needs to be considered to check that the task is a reasonable one for the children to tackle and that the numbers are appropriate for the strategies that you want them to apply.
Estimation, rounding, application of mental and written calculation strategies and finding the structure of calculations in the world around them are all skills that we want children to have and rooting problems in the children’s environment is a way of providing the opportunity to develop them.
Fermi Problems
The problem that Dan set is an example of a Fermi problem. Fermi problems, named after the physicist Enrico Fermi, are problems that involve estimation and reasoned guessing about values that are not known. A famous example is to estimate the number of piano tuners in Chicago. Ideas for using Fermi problems with primary school children can be found on the Fractus Learning and National Council of Teachers of Mathematics websites.
So, what about that beautiful tree by the school gate? How many ants might be living under the bark? Could you pick all the daisies on the school field in a day? Do a thousand buses pass the school gate in a week? Are there a million bricks in the building? How many times round the field/playground do we have to go to get to [nearest city]/Paris/New York/to climb Mount Everest? How many playtimes would it take to score as many goals as are scored in the Premiership in a year?
Image credits
Page header by Phil Whitehouse (adapted), some rights reserved
Woolacombe Beach by Watkins (adapted), some rights reserved
