Lines of infinite beauty
"Mathematics possesses a beauty cold and austere, sublimely pure, and capable of a stern perfection such as only the greatest art can show."
Bertrand Russell (1872-1970)
The story of the proof of Fermat's last theorem is the most wonderful piece of mathematical television. For those who haven't seen it (you must), there is a moment in Simon Singh's joyous BBC documentary when mathematician Andrew Wiles tries to communicate the experience of finding the final piece of the jigsaw. This gentle, brilliant man sits for perhaps 10 seconds in complete silence as he grapples with the enormity of the vision granted to him. It's as if he has reached a mountaintop and is gazing out over the virgin valley beyond that he has somehow subdued through the power of his fallible human mind. "It was so indescribably beautiful," he says quietly, like Moses attempting to describe meeting God in the burning bush.
No subject creates such an extreme set of emotions in its students as mathematics. Some endure a phobia so distressing that they freeze when faced with anything labelled as maths. They fail to trust in the ability that they actually have and count the days until they can drop the subject. Others find maths so fulfilling and spend so much time in that mysterious world that observers accuse them of neglecting the roundedness of their personalities.
These feelings pursue us into adult life: a teacher friend of mine experiences palpitations when her health and social care syllabus asks her to cover a mathematical element. But when a different friend wins a day to himself (a birthday, perhaps) he always says, "I'll spend it doing maths."
The second person is in danger of being misunderstood. He does not want to do sums all day; he will not be ploughing through past exam papers. The fact is, he is profoundly aware of the transcendent beauty that lies behind the best mathematics. A day spent pursuing that beauty - tackling some delicious problem or reflecting on the extraordinary mental structures that mathematicians down the ages have conjured up - is, in his view, a day well spent. Such a day of wonders, he would argue, tends to leave your character more rounded and not less.
Some may doubt him. "What does this mathematical beauty look like?" I hear you ask. Let me pick an example, and a simple one at that, my point being that one can find beauty in the simplest of maths, not just the extraordinary efforts of Wiles or Stephen Hawking. For teachers, of course, this is key.
Bear with me. If you and a friend have a moment, you might try this quick game. Imagine that you have nine cards lying on the table between you, each displaying a different digit from 1 to 9.
You take it in turns to pick a card. The winner is the first person to have three cards in their hand that add up to 15.
A sample game might go like this. Player A picks 6 and Player B responds with 3. Player A then chooses 5 and Player B is forced to make 4 his second choice to block A from getting 6-5-4. In turn, however, Player A is forced to block B by picking 8. Player B is then in an impossible dilemma as he would have to pick both 1 and 2 to block A successfully. Player B picks 1 and A ends up with the winning combination of 2-5-8 in his hand.
A bagatelle to while away a minute, you might think, but a mathematician drawn into this might find herself asking questions. Is it better to go first or second? Does best strategy guarantee a win for one side or the other, or is a draw possible? How can I look at this in a different, more helpful way? Perhaps, even, in a more beautiful way?
Suppose our mathematician picks the cards up and rearranges them on the table. "The numbers from 1 to 9 and the number 15," she reflects. She places the nine cards like this:
"Each row adds up to 15, each column adds up to 15 and each of the main diagonals adds up to 15. The game asks me to pick three cards that add up to 15 - in other words, I am trying to get a line of three," she says.
Suddenly there is a wonderful moment of clarity: the game is noughts and crosses. All the questions about strategy are resolved, since we have all played noughts and crosses many times. With best play, the game (known by our American colleagues as tic-tac-toe) should always end in a draw, although if your opponent is an inexperienced player, it does help to go first.
There is one check left to make: is every way of making 15 with three cards represented in our magic square? A little research shows that the answer is yes, there are exactly eight different ways of making 15 with three different cards and exactly eight different ways are shown in our magic square. The analogy is perfect and the underlying structure for these two apparently different games is seen to be identical.
Mathematicians love this kind of thing. Indeed, they find beauty in the logic. It may be that this solution to finding the game's best strategy does nothing for you. This brings to mind a story about a woman who asked legendary jazz man Fats Waller what jazz was. "If you have to ask, you ain't never gonna know," was the reply.
Of course, this piece of mathematics may leave you cold. If that's the case, you could feel that I'm making a lot of fuss about nothing and ignoring true sources of beauty like poetry, art and music. My reply would be that no mathematician claims that their appreciation of beauty trumps anybody else's.
Every subject in the curriculum has the power to seduce every teacher, whether it's a chemist contemplating the benzene ring, a business studies teacher reflecting on the complex intertwining of inflation and interest rates or a PE teacher who has just witnessed an extraordinary goal. To set up a competition would be ugly in the extreme.
So for all of us the question is the same: how can I explain how beautiful my subject is? I gaze out each year at my new classes and wonder what their prior experience has been. How many will have chosen to pursue maths beyond the syllabus in their spare time? The percentage will be almost infinitesimal - how can I increase it?
I think back to my first glimpses of mathematical beauty - I definitely caught the love from an inspiring teacher. One day, Mr Russ came in and tried something completely different. He put a simple problem about points and lines on the board and I found myself gasping: "So this is what maths is!"
We teachers need first and foremost to nourish our own appreciation of our subject's allure. We need to find time in our impossibly busy lives to read around the bare necessities. We need to make sure our schemes of work contain more than just the effective and the expedient. We need to fight for space for the awe-inspiring, even if it doesn't immediately tally with a particular exam topic. Of course, the effective and the expedient will inevitably be in there, but even apparent drudgery can be approached in a multitude of ways, some of which acknowledge beauty more than others. As the great mathematician Carl Gauss once said: "You have no idea how much poetry there is in a table of logarithms."
Persistence and resistance
Which takes us back to Wiles attempting to verbalise the sublimity of his proof for Fermat. In the end, perceptions of beauty can only be discussed in spiritual terms. I am an atheist (I might be wrong), but there are plenty of atheists out there leading profoundly spiritual lives who I hope would agree. We live in a time when the spiritual is under threat, especially in our professional lives as teachers.
Bureaucracy that is thoughtful can be beautiful too, but when it becomes an end in itself, it leaches the wonder from teaching. Few teachers perceive the beauty in marking; increase our marking workload and the beautiful is likely to take a hit. Man cannot live by bread alone, Jesus taught us, but that doesn't stop lots of people from trying to make it so, including those who prescribe us a vast administrative workload and a utilitarian syllabus.
I have three raw goals in my teaching. The shallowest is to get decent exam results. Without these, I won't have a job and my students will be disadvantaged, which would be unfair. Second, I want us all to enjoy the process of tackling maths together - the social interaction, the battling alone with homework, the overcoming of failure, the discussion, the experience of concentrating hard in a group. If I achieve goals one and two, I keep my managers off my back. But have I even begun, really, the task of education? My third goal is the most profound and it is the reason why I am a teacher at all: I want to initiate my classes into the human conversation that is mathematics and that includes an appreciation of mathematical beauty. I want us to be able to look back at our hours in my classroom together as a time of wonder. I want my students to lose their fear of maths, even in the face of real difficulty, and I want them to learn that everyone on this Earth is a mathematician even if they don't know it yet.
Do I manage this? My colleagues will read this and smile. I think about my classroom and I know that my ideals do not make it into practice just yet. I still make the mistake of telling my students in advance that we have a beautiful piece of maths coming, only to be met afterwards by blank faces and "Was that it?" Some of my students have learned that I look for elegance in mathematics and will mischievously raise their hands after seeing a smart solution and ask, "Would you say that was `elegant', Jonny?"
But then, I've only been teaching and trying to reveal mathematical beauty for 25 years. Ask me how I'm doing in another 25 years' time.
Jonny Griffiths teaches at a sixth-form college in Norfolk
Bring a little beauty into your classroom with these resources showing off the true artistry of maths:
Explore Fermat's last theorem and the idea of proof. bit.lyFermatsLastTheorem
Brighten up classrooms with these posters displaying the beauty of maths. bit.lyMathsPosters
Learn about the mathematics of beauty with the golden ratio. bit.lyTheGoldenRatio
Introduce students to famous mathematicians, including Pythagoras and Einstein. bit.lyFamousMathematicians
Discover important female mathematicians in history. bit.lyFemaleMathematicians
Delve into number patterns to show the beauty of maths and motivation. bit.lyMathsandMotivation
Inspire students with these mathematical quotations. bit.lyMathsQuotes