It’s really a crying shame that more people don’t love maths. If you thinking I’m talking crazy right now, or are at all dubious, it’s because what you haven’t been shown what maths really is. What you were taught at school was almost certainly awful. There’s no better way to illustrate than Paul Lockhart’s analogy of what it would be like to teach art this way:
After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that. Of course we track our students by ability. The really excellent painters— the ones who know their colors and brushes backwards and forwards— they get to the actual painting a little sooner, and some of them even take the Advanced Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they get out there in the real world and paint their kitchen they don’t make a total mess of it.”
His point is that maths is art, not science, “dreamy and poetic … radical, subversive, and psychedelic … the purest of the arts, as well as the most misunderstood”. He has a musical analogy too – check out the full essay here. And before you read any further, forward it to any teachers you know 😉
It doesn’t have to be this way
I was lucky to have a few uniquely gifted maths teachers growing up. When I lived in Birmingham, I attended some after-school sessions with Tony Gardiner. I doubt that anyone can have done more for developing talent in young mathematicians in the UK than Tony. How many university lecturers talk about their subject this way?
Somewhere in all this I embraced the wild dream of trying to straddle the whole gamut of mathematics – from the young child learning about number, shape and logic to research. My wife and I started a Saturday morning maths club- which we used as a laboratory for trying out outrageous things on willing guinea pigs. I was also particularly fortunate to visit Freudenthal and his colleagues in the early days of their amazing research project in Utrecht. And the rest is history. I do not pretend to have fulfilled my wild dream, but I have tried to understand and to serve the needs of mathematics, teachers, and students as best I can.
I don’t think it was on Saturday mornings, but I did get into a club of his, where I really saw beauty in maths for the first time. I still have a certificate from a Russian maths contest he entered us into, totally incomprehensible to me!
When we moved to St Andrews, it was Tony who told us not to take the place I had planned at Dundee High School, but to go to Madras College instead – purely because of one man, Ken Nisbet. He was the best teacher I met at school, in any subject. Liked equally by the best and worst pupils in the class, he proved that great teaching doesn’t have to involve making a choice between the two. He gave me a push to take the maths olympiads a little more seriously, and organised for me to spend time with Nik Ruskuc after school, who was very generous in sparing his time to do old olympiad problems together on the blackboard.
Thanks to those great teachers, I made it to the UK team for the International Maths Olympiad (Taiwan 1998), a genuine once-in-a-lifetime experience. The IMO, and the pyramids of national qualifying contests leading to it, represents what maths is really about far better than the school education system. It’s not perfect – I think it’s possible to do well by preparing too intensively at a risk of removing some of the fun and creativity from the contests – but I think most contestants get it just about right. You need a little preparation to stand any kind of chance!
Our trip to Taiwan started badly: we missed a connecting flight and arrived a day late with no luggage. Add in jet lag, unpleasantly humid heat, and wooden boards for beds, and you don’t have ideal preparation for sitting two 4.5 hour exams! Despite all that, it was an enjoyable experience – particularly after the exams were over! While the papers were being marked, we got to do a little sightseeing and socialising with other countries (highlighted by an impromptu cricket game against the South Africans). Taiwan was amazing; it was a slight shame that the organised excursions focussed so much on museums and their history – we’d have liked to have seen some of their country’s natural beauty – but I guess you can’t please everyone.
In 2002, I had the rather odd experience of seeing the olympiad from another angle, as the UK hosted and ex-team members became coordinators, responsible for scoring the papers. Typically it’s unusual to get partial credit – 0 and 7 (yes, 7 is the maximum score for a question!) are by far the most common scores. The scoring system is surprising at first, but it actually makes a lot of sense: each team’s leader decides what score they believe their own contestants deserve, and have to justify this to the coordinators. When you think about it, it’s probably the only system that could work – as it saves so much coordinator time. For every time a team leader claims a score of 0 (which is quite a lot!), the coordinators don’t have to do any work wading through an incorrect solution. In fact in general, coordinators don’t have to wade through anything – the team leaders summarise and highlight the relevant parts, and coordinators just have to verify. This is particularly important when you consider that most of the scripts are in an unfamiliar language to the coordinators!
Without this time-saving device, I guess many host countries might not be able to find enough qualified volunteers to do coordination, or the scripts might take much longer to mark. Having a small number of coordinators is good for another reason: they can assure a high level of consistency and fairness across all contestants. The obvious downside to the system is that some serious debate can ensue when team leaders push for more than the coordinators feel they deserve!
I’ll never forget the Israeli team leader – I can’t remember his name but he was legendary for arguing with coordinators, and we had to call for assistance (from Tony Gardiner, as it happens). He wanted some partial credit for making partial progress, so I could see his point of view, but we’d already given a number of other contestants 0 for the same partial progress, and couldn’t go back. It didn’t stop him persisting! We were assigned to question 2, geometry, and it was fascinating seeing the same problem solved in so many languages. I’ll never forget Thailand, with the most ridiculously pretty handwriting you could imagine, Japan with their ugly but technically impressive brute-force solutions (assigning coordinates to point positions and using algebra, rather than the expected pure geometric approach), and Israel – not just for the haggling but for the only right-to-left maths I’ve seen (there’s something rather disturbing about having implications flow to the left!)
Anyway, I’ve digressed somewhat, but if you want to see what IMO problems are like, why not take a look at this year’s questions – it was just a few days ago. If you like interesting problems, they’re well worth a go; I think this year seems perhaps slightly easier than usual, and there are several questions which are highly accessible, in that they don’t need much knowledge. In particular, 2 and 4 need nothing more than basic high school algebra, and 5 just the ability to count (ok, so knowing the basics of binomial coefficients is recommended!). I wouldn’t recommend trying 3 or 6 (the last question on each paper is typically significantly harder), and although 1 isn’t too hard by IMO standards, the diagram is a bugger to draw and you really need to know a few theorems about triangles and circles to stand a chance. Give a question reasonable time – the exams are long for a reason – but if you haven’t done olympiads before, you probably won’t get far 😉 Once you’ve had enough, the answers can easily be found with Google – or drop a comment and I’ll provide a couple of hints …
Olympiad problems are a huge amount of fun, and you get to see the essence of what maths is all about, without years of study. There’s a full spectrum of difficulty levels to ease you in before IMO level. I strongly recommend this book:
It’s based on the British Olympiads (so it’s not too hard), contains a concise introduction to all the theory you need, references to further reading on specific topics, and some wonderful hints rather than just solutions – a much better way to enjoy the problems! 😉