Why is mathematics so beautiful?

This blog is offered by Gareth Smith, Trust Champion. The blog is offered as a provocation to thought and discussion and represents the authors personal views which are not the views of Dartmoor Multi Academy Trust as a whole. We would love to hear your thoughts and questions. Do use the comment function below.

2 years ago, I was sat in a room with several heads of mathematics from across the county. We were discussing which aspect our respective mathematics curriculums we could align to save on teacher workload. One of the leaders called out with “if we’re all having to teach exactly the same lessons, where is the passion about maths going to come from?”. I was infuriated! For me, the excitement about mathematics comes from the teachers, and most importantly they should not be afraid to show it! 
Mathematics teachers are always frustrated when it gets to parents’ evenings and a dad will turn round to their child and say, “it doesn’t matter, I’m rubbish at maths and you’ve got a calculator on your phone.” Yet, they would not turn round to their child’s English teacher and complain about not being able to read. Being a good mathematician has not been ‘cool’ for a long time. Student’s need to see the example set by their mathematics teacher and see someone who is incredibly passionate about their subject. This way teachers are able to teach students far more mathematics than is in their textbooks. We want our learners to be inquisitive and ask why a piece of mathematics works. Teachers should not be afraid of the ‘when am I going to need to know how to solve a quadratic equation, when I’m going to be a car mechanic’ question. I have never been asked to quote a piece of Shakespeare, but I understand how learning about his works can help me to have a wider understanding of our culture and an appreciation of how the English language developed. 
This is about role modelling. If a teacher is unable to demonstrate how beautiful mathematics is then how are the students going to start to appreciate its wonder? One of my favourite areas of mathematics is prime numbers. They are incredible and their properties boundless. They are the building blocks of multiplication and the basis of internet security. Whenever I discuss proof with the students or want to show them how good they can be at mathematics I share with them the most elegant proof around. The proof that there is an infinite number of prime numbers in the universe. The reason why it is so elegant is that you can do it in your head and without doing any calculations. First, you must imagine a world where there is a finite number of prime numbers that you can write down in a list. 2, 3, 5, 7, 11, 13, etc. You then take this list of numbers and multiply them together, so 2x3x5x7x… and you then add 1 to it. Now consider this new number. Before we added 1, the number was a multiple of 2, 3, 5, 7 etc. as we used them in the multiplication calculation. Now we have added 1 to it though we have a new number which is not a multiple of 2, as it is 1 more than a number in the 2 times table. It is not a multiple of 3 as it is 1 more than a number in the 3 times table. It is not a multiple of 5 as it is one more than a number in the 5 times table. We could go on through each of the list of prime numbers. If this new number is not divisible by any prime number, then this number must be prime. In fact, a new prime! However, at the beginning we assumed. We assumed that there was a finite list of primes. This assumption must be false as we found a new prime. Therefore, there are an infinite number of prime numbers in the universe. This is an example of proof by contradiction. 
This proof is not on any KS3, GCSE or A-Level curriculum, however, it is an example of adding depth to the curriculum with expert subject knowledge. When I used this proof with my foundation year 11 group last year, I knew the some of them would understand, some would enjoy and probably most may not be interested. However, I feel that it is important for students to see their teacher’s passion for their subject, because if they are not passionate about mathematics who will be. It also gives students the opportunity to think wider and more like a mathematician. There was a stage last year where I felt that I had let down one of my students. I was helping one of my year 13 students prepare for his Cambridge University mathematics interview. He answered the questions about mathematics easily enough, but I then started to push him on areas outside the A-Level curriculum. Areas of interesting mathematics but ones which are not taught in the specification. He could not answer. He had not read wider around the subject or worse I had not encouraged students to read wider around mathematics. Fortunately, when I was head of mathematics, I had ordered a stash of books for the library so we went and took them out so he could do some quick reading! 
My point is that as educators (not necessarily just mathematics teachers) we are responsible for sharing with students the beauty of mathematics. We need to role model, enthuse, enquire and not be afraid to teach beyond the curriculum. If we do not do this, we will crush creativity in the mathematics classroom. I have a 5-year-old and I love seeing how is starting to make connections between numbers. His latest fascination is infinity. I haven’t gone into the different types of infinity yet, but we often have discussions about infinity plus 1 is also infinity and how nothing can be bigger! It is this level of enthusiasm we want in our classrooms. As a secondary teacher I always used to love the year 6 transition lessons as the enthusiasm was boundless. It does worry how we ‘put off’ some students from mathematics so by year 11 they cannot stand it anymore.  
Organisations like the National Centre for Excellence in Teaching Mathematics (NCETM www.ncetm.org.uk) and the Regional Mathematics Hubs are doing fantastic work publishing resources and developing pedagogy to encourage enquiry and understanding from students. The NCETM Mastery Specialist programs look like incredible opportunities to work wider that one’s own department and have impact on learners in other establishments. With teams of passionate mathematics teachers, teaching beyond the curriculum written down on paper we gone show students and wider audiences why mathematics is beautiful.