Cicadas and prime numbers

Written by: Marcus du Sautoy | Published:
Photo: iStock

Mathematics expert Marcus du Sautoy explains why art is an ideal vehicle for teaching mathematics and introduces a simple lesson idea involving the cicada and its predators...

Mathematics expert Marcus du Sautoy explains why art is an ideal vehicle for teaching mathematics and introduces a simple lesson idea involving the cicada and its predators...

Mathematics is seen as a dry subject with a rigid structure; useful, important, but not much fun. At primary school, mathematics tends to be presented technically rather than creatively. But mathematics isn't just a useful science. It is itself a creative subject, and it can be taught just as playfully as the arts.

Mathematics is the language we use to navigate our physical environment in a scientific way. At its core, it is storytelling. The arts can be utilised to tell these stories, "rephysicalising" mathematics in the process. By exploring mathematical ideas through theatre, music and movement, children have a better chance of navigating the abstract side of the subject because they are able to visualise, play and make connections.

Play is related to experimentation, a strong element of maths and science. From birth, we are programmed to play, explore, and spot patterns useful for our survival. So much knowledge is discovered through play because it also allows us to make mistakes, go down blind alleys, and see patterns at work. Where would we be if our scientists and great innovators were not playful, creative types?

One of my favourite mathematical stories – a story that actually happens in nature – is of the cicadas in North America. One type only appears every 17 years, another every 13, but none at 12, 14, 15, 16 or 18 years. They have evolved prime number life cycles. Why? In doing so, they are less likely to come across predators that also appear periodically in the forest.

This story can be brought to life with children using drama and performing arts, by making numbers from one to 100 in the classroom and getting children to play the cicadas or the predator. As you go through the hundred years with the cicadas appearing, say, every nine years and the predator every six years, you will find that they coincide every 18 years. Each time they meet, the predator gets to choose one of the cicadas to eat.

But change the cycle so the predators are appearing every six years and the cicadas are appearing every seven years, and they don't coincide until year 42. Through this game children are exploring a real scientific example. It is basic mathematics, nothing beyond the multiplication table, but because it is a good story it stays in the memory.

It is important to introduce maths in this creative way at primary school and not wait until secondary school because it makes it more real and tangible. If children can understand why they are doing something, it gives them the inspiration to carry on the hard work.

Mathematics also hides behind many things across the curriculum but at school it tends to be compartmentalised as "numeracy hour". Children move from one subject to another without realising they are all intimately connected. But if you can make interesting connections between subjects, it is always going to help both sides. Mathematics is the glue between many ways of seeing the world, the perfect bridge between science and creativity.

If I was going to rewrite the curriculum I would say let's forget about maths being useful all the time and let us just tell great stories. The arts are the perfect vehicle to help tell them.

Cicadas and prime numbers in practice

The mathematical story of the cicadas was recently used with year 5 pupils at St Thomas of Canterbury RC Primary School in London. The session was led by specialist Stuart Barter with the aim of helping the pupils with their knowledge and understanding of prime numbers and timestables through performing arts. The pupils had a variety of abilities in maths and were not used to maths lessons in this practical, holistic way.

Stuart asked the pupils whether they had heard of cicadas, what they are and where they are from. A picture of a cicada was shown to the pupils. The pupils were then told about the cicadas' predators, including the wasp. The pupils were asked in pairs to mime a cicada popping up and being eaten by a wasp.

The pupils were told that the cicadas want to miss the life-cycle of their predators, the wasps, who pop up every six years. Stuart then asked the pupils "if the cicadas pop up every nine years, how many times will they meet their predators over 100 years?" To answer this the pupils wrote numbers one to 100 on individual pieces of paper and laid them out on the floor in a semi-circle.

Six of the children were invited to be cicadas and one to be a predator. The children then crawled around the number line with cicadas popping up every nine years and a predator every six years. Whenever the two coincide, the predator pretended to eat the cicadas.

This was then repeated with the cicadas popping up every seven years. This time the cicadas only met the predator twice, reinforcing the prime number theory in action.

The children were then split up into pairs to investigate other numbers up to 11. Some of the children worked with the cicadas popping up every eight years, some every 10 years, and some every 11 years. The children were asked to work out when the cicadas and predators would meet.

The children physically counted and walked on the spot, turning to face each other every six (in the case of the predator) and eight, 10 or 11 (in the case of cicadas), the children learned that the most effective number of years for the cicadas to wait was 11 – which is the next prime number.

Stuart explained: "The storytelling gave the maths a real purpose, as it became about the survival of the cicada, rather than just the abstract manipulation of numbers. I was struck by how invested the children were in the story and the dramatic re-enacting of it and they were able to participate in the lesson in a practical, kinaesthetic way, as well as cerebrally. They did some great 'death scenes' of the cicada being caught by his predator, with plenty of heart-clutching, gurgling, staggering and cries of 'why-me?'."

  • Marcus du Sautoy is a professor of mathematics at the University of Oxford. He has presented a number of landmark television series for the BBC including The Story of Maths and The Code. He is also a supporter of arts education specialists Artis, which ran the session described in the article, and a member of the judging panel for the Artis "What is Art?" competition. Visit

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