Can maths really be creative?

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Image: schoolatoz.nsw.edu.au

Carey Ann Dodah of Explore Learning thinks it can. In this week's guest post, she considers a few simple ways to promote the development of problem-solving skills and creativity in maths lessons, as expected by the new primary maths curriculum.

'Creativity in Maths!? Isn’t that an oxymoron?' was the response from many when we launched our creative maths programme last year, a weekly class for eight- to ten-year-olds. However, we are strong believers that creativity defined as ‘a process where something new and valuable is created’ is the very essence of maths and it’s those inquisitive mathematical minds that we want to develop in our future generations.

The new National Curriculum supports this - problem solving is implicit across all subjects in the maths Programme of Study and with the recent announcement from the government that from 2016, Key Stage 1 and 2 tests will now include more problem solving, we can see the emphasis being placed on this skill.

We love solving problems at Explore Learning – it’s a life skill that we want every child to enjoy and have the confidence to tackle, not just in our centres but in their school experience. However, first there are some hurdles to overcome.

What makes a good maths problem?
In classrooms, problem solving is typically taught in a formulaic manner. The problem is introduced, a technique is demonstrated, the process is practised and then used to apply to other scenarios until practice makes perfect. In life though, and in maths, there is not always a technique or set route to follow to reach a solution and remember if you know how to solve something then it isn’t really a ‘problem’. A problem has to be new and without an obvious route to solution to genuinely challenge us. In maths, as in life, we need to develop children’s confidence to find a route themselves. They own the problem and they can fix it. 

At Explore, we have been very fortunate to work with NRICH – a team of inspiring mathematicians, teachers and trainers based at the University of Cambridge. Together we have created the annual National Young Mathematicians Award competition. Its goal is to bring rich, creative problem solving into primary schools. Last year,1000 schools took part, entering a team of their top four mathematicians (typically two girls and two boys).

Unlike conventional maths competitions that are based on individuals attempting paper-based tests involving equations, computations and problem solving, the National Young Mathematicians Award allows young people to create maths. The problems are real, they are without an obvious solution, and they often involve making something. They provide opportunities for logical thinking, identifying patterns, looking for all possible solutions, hypothesising and perhaps most importantly communicating and working as a team. 

Image: getsurrey.co.uk

I am always inspired observing the final each year. I especially love seeing the children working together. It highlights that maths is not a solitary subject – its true power is seen in collaboration. All the tasks that children get involved with require collaboration and therefore good teamwork is essential. The tasks are also very open ended with limited instructions so the next ingredients to success are using logical thinking, working systematically, using trial and improvement and taking ownership of a problem. You can really tell the schools that have incorporated this type of problem solving into their lessons previously. Those children that haven’t find it very difficult to know how to start given so open parameters – so much freedom!

How can we teach problem solving?
So how can schools introduce these skills into existing maths teaching? How can we make maths more creative today? First, let’s adopt one important rule – maths is about more than a right answer. It’s about exploration and letting your imagination go!

Try this exercise to demonstrate the concept to students and colleagues. Ask for three volunteers and instruct them to tell you a two-digit number, it doesn’t matter which ones they give. Get students into pairs and ask ‘which is the odd one out?’. On the board, draw five bullet points. 

Let’s say you have been given 62, 27, 36. Give a couple of minutes to the group and ask each pair to tell you their ‘answer’ and the reasons why. Record all the reasons on the board and try to fill the five bullet points. 

You may have something like this:
• 27 – it’s an odd number
• 62 – it’s above 50
• 62 – not a factor of 9
• 27 – it’s below 30
• 62 – ones are smaller than the tens

These are all correct answers! Well done! 

Now add several more bullet points to the board and allow the group much longer this time to think up some other possibilities. Thinking time in maths is vital – if cut short then you are unable to uncover the rich, deep thinking that everyone is capable of. Too often this is missed because time is pressured.

Now you will have less obvious associations made:
• 27 – has a diagonal in the digit
• 62 – adding the two digits together does not equal nine
• 36 – square number
• 27 – the words have a different number of syllables to the other two
• 27 – has ‘open’ figures rather than ‘closed’ ones

And you can keep going and challenging your group to think of more reasons. It can be anything as long as they can explain it! You’ll be amazed at the amount of learning that can be gained from such a simple task and how many different directions this will lead your learners. Those without the confidence to share initially might suddenly find a gem. 

Your role is as a facilitator: little talking, lots of listening and then jumping on the great ideas that could pull out more maths from the activity (‘what is a square number? Can you think of any others?’) and in doing so help children clarify their ideas and share their knowledge.

This exercise demonstrates the power of peer learning. It tells students that you value different ideas, strategies and routes. It encourages clarity in explanation and opens opportunities to introduce new vocabulary in a subtle but powerful way. Last but not least it encourages creative, imaginative thinking.

Making maths accessible

Image: willowbrook.leicester.sch.uk

The task described above is a good example of the kind of activities that we cover with our children each week on in creative maths course.  Every week our centres fill with a group of excited mathematicians awaiting a new challenge. They are not all ‘strong’ mathematicians, in fact many of them lack confidence in class but this is maths with a difference!

All the activities apply the NRICH philosophy of ‘low threshold, high ceiling’. In essence, low threshold means that all the exercises are accessible to the whole group. To access the task, you may need a basic ability to add numbers or use a ruler – everyone can have a go. High ceiling refers to the task having the potential to result in very complex mathematics, nth terms and formulas, should the individual be able. 

One of our favourite examples in the creative maths course is buttons! This is an exercise that could run over a couple of classes.

To begin with, describe that you have a cardigan with three buttons. You like to do the cardigan up in different ways every day. Sometimes you might start at the top, sometimes from somewhere else. How many different possible ways can you do up the buttons on the cardigan? This can be a great practical task and you could have a pot of different coloured buttons that the children could use or they might prefer to use T M B (top middle and bottom). It’s about the children finding a method that makes sense to them, rather than the way you would do it. 

Quite quickly children will come back with answers – it’s great to ask at this point that you need to be convinced that they have found all the different ways. Once children have convinced you, ask if they can find out what happens when you have four, five or six buttons and the task becomes a greater challenge. Your able mathematicians may start identifying a pattern and they might use conjecture and develop a theory. They are reaching for a high ceiling but in gentle, structured steps that allow you to see them grow (intellectually and in confidence) within a matter of hours.

You can find more similar problem solving activity ideas here and here.

What other skills will creative maths develop?
This practical maths allows children to improve key mathematical skills which also translate into other subject areas such as science, technology and are used in later life: 

• logical thinking 
• developing a systematic approach 
• trial and improvement strategy 
• exploration and explanation skills 
• convincing and proving 
• collaborative learning.

The challenge was to prove that maths can be creative, collaborative and stimulating. I hope this post has given you some insight into how this can be the case and some inspiration to challenge your young students. Perhaps you can try some of these rich, low threshold–high ceiling exercises out before entering a team for the National Young Mathematicians Award this autumn. Let’s see how far they can go!

Carey Ann Dodah is Head of Curriculum at Explore Learning.

To enter a team into the National Young Mathematicians Award, visit www.explorelearning.co.uk/youngmathematicians. Registration closes 30th September so hurry!

For further activities to help students practise for the competition, visit nrich.maths.org. For more information on Explore Learning, visit www.explorelearning.co.uk. 

Image: explorelearning.co.uk