Problem Solving, Andrew Wiles, and the Millennium Prize Problems
Could you teach the next Andrew Wiles?
In 1995 Andrew Wiles proved what is known as ‘Fermat’s Last Theorem’, one of the most difficult and famously unsolved problems in mathematics. In a recent interview Andrew Wiles talked about the importance of maths teaching and how his own maths teachers had introduced him to number theory during his time at school. While Andrew Wiles can take the credit for solving Fermat’s Last Theorem, his maths teachers from many years before also had a part to play. Their inspiring teaching set him on a path to becoming one of the most famous mathematicians of all time.
Not many teachers get the chance to teach someone like Andrew Wiles – but one of the exciting things about teaching is that you can’t imagine what your impact will be in the future. Andrew Wiles also believes strongly that great mathematicians aren’t just born that way – shaking off the popular idea that you are either born a maths genius or you aren’t. In order to become a research mathematician, you need to do a lot of hard work, have a lot of determination, but also have a lot of the right input from teachers around you.
Are mathematicians born or made?
Many pupils have soaked up the myths around mathematicians being born not made – for example lots of pupils in Set 3 will believe deeply that they are naturally meant to be in that group and that their mathematical ability is set in stone. This is simply not the case – undoubtedly there is natural variation in ability – but there is then huge scope for how that ability can be nurtured and developed. With the right teaching and attitude to learning, it is amazing how much pupils can improve.
There is also a feeling amongst a lot of pupils that the process of discovering new mathematics is finished. They clearly know that scientific discoveries are being made all the time, whereas when it comes to maths it looks as though everything has been completed and written down neatly in a textbook. One easy way to emphasise the fact that mathematics is a living breathing subject is to talk about what are known as the Millennium Prize Problems. It also helps that they have a huge cash reward attached to them.
The Millennium Prize Problems
To celebrate the millennium, the Clay Mathematics Institute collected seven unsolved mathematical problems and offered a prize of $1,000,000 for the solution to each of them. Only the first one has been solved so far (but not claimed).
Millennium Prize Problems
1. Poincaré Conjecture
2. P versus NP
3. Hodge Conjecture
4. Riemann Hypothesis
5. Yang-Mills existence and mass gap
6. Navier-Stokes existence and smoothness
7. Birch and Swinnerton-Dyer Conjecture
The Millennium Prize Problems are just the tip of the iceberg when it comes to unsolved mathematical problems. Perhaps you could make a wall display showing your pupils what the problems are. Only a few elite mathematicians will work on solving these problems – however it emphasises an important point:
It is ok to be stuck
One of the reasons that some pupils have a fear of mathematics is that it is a subject where you can get stuck. Teachers will also feel this fear acutely on training days where they have to do maths in the presence of other teachers. The fear of being stuck may well return, even to an experienced teacher! In English you will usually be able to add a few more words to a tricky essay, whereas in maths you can get completely stuck on a problem.
What the Millennium Prize Problems show is that even professional mathematicians get stuck on problems. Being stuck isn’t a nasty side effect of doing mathematics – it is what the subject is all about. It is one of the reasons employers love people who are good at maths – they aren’t afraid to take a difficult problem and work out a solution.
As a teacher one of your jobs is to teach pupils how to overcome the fear of being stuck and to move to on to developing their own problem solving strategies. It is obviously important to teach pupils how to solve the standard problems of mathematics such as factorising a basic quadratic equation. However, they are going to enjoy maths much more (and do much better in their exams and future careers) if they can become competent problem solvers.