 Thank you very much. Two weeks ago, I spent a wet Saturday afternoon at Bramol in Sheffield. I was watching the Sheffield United football team produce a kind of lackluster display I've come to expect after 40 years of following them. I spent the entire 90 minutes, along with 19,000 other people, bemoaning the lack of quality on the field in front of me. This issue of quality in relation to skill is, however, I think very interesting, whether it's in relation to football or even mathematics. Why do some people excel at certain things whereas others struggle? As a society, I think we tend to attribute this phenomenon to natural ability without ever questioning the nature of that ability. For example, we accept that great inventions, say, were dreamt up in an active genius without ever stopping to think what that act might have actually entailed. On the face of it, the third division footballers I was watching have a lot in common with top players. For example, they all know how to make an accurate pass. They all know how to win a header. They all know how to make a tackle. However, these observations don't reflect the golfing standards. The difference is, top players think differently. They think differently about the game. They perceive what's going on in the pitch differently. They make different decisions during the course of a game. The analogy with mathematics is actually surprisingly apt. As instructors, we teach students basic skills. We teach them procedures. We teach them the concepts of mathematics. But this does not make them a mathematician. Fluency with procedures and skills does not mean that they can think like a mathematician. I believe that the ultimate goal of education is to transform the thinking of students. Surely the ideal of any teacher is to be able to equip students with the know-how and skills to develop their own thinking. For me, the key question is this. Can we identify those aspects of cognitive skill which distinguish those who display quality in some area from those who don't? And if we can do that, can we learn that? And perhaps more importantly, can we teach it to others? The idea which lies at the heart of my approach to teaching is that metacognition that is thinking about thinking or consciously exerting control of one's thought processes is the key to improving thinking. Now, it's not enough to say to a student, be more aware of your own thought processes. This is akin to MTV's exhortation to free your mind. It sounds great in principle, but how do you actually do it? In fact, the mathematics education literature has been studying mathematical thinking for a long time, and there are things one can do. I myself have been heavily influenced by the writing of John Mason, and he was discovering a book authored by him in my teenage years which completely transformed my mathematical development and without which I wouldn't have become a mathematician. The need for cognitive transformation in mathematics under graduate education is nothing clear. For example, if one takes a school leaving certificate syllabus in mathematics and compares that with a typical first-year mathematics module at university, on paper, these things can look very similar. However, there is a difference, and the difference is in the way in which we expect students to think about that content. Actually, there are two transition points in undergraduate mathematics education. The kind of transition I've just described of students entering university, and if I need to deal with these issues head on, leads to a lot of the kind of problems that are very well documented. However, there is another, if anything, more serious transition problem in mathematics which students who elect to take a degree in mathematics encounter, and that's the boundary between computational procedural mathematics and rigorous abstract mathematics. I teach a module entitled Problem Solving and Proof. This is delivered to part of an evening MSc programme for mathematics teachers. The idea here is to try and improve mathematical thinking. What about mathematical content? I'm not interested in developing the procedural skills of the teachers so much as encouraging the kind of thinking which facilitates creativity and invention in mathematical problem solving. For example, we look at the role of attitude and beliefs in decision making during mathematical problem solving. We look at the role of emotion. We look at examples of good and poor metacognitive control. The module at its heart deals with problem solving, and a framework is presented to the students through which they can navigate the various options of possibilities that they encounter during mathematical problem solving. In broad terms, the problem solving process is broken down into several phases. Within each phase, key words are developed, and the use of these key words help students to punctuate their written thought processes. These key words either describe a state when reaching in the problem solving process, or offer a suggestion about how to proceed. In this way, these key words provide cognitive landmarks, and by writing them down, students are forced to take an overview where they're going and hopefully develop their own metacognitive control. Now, none of this is easy, either for the students or for the instructor. For example, last week I gave the students an example to work on in class, and the point of this example was to force them into trying things and experimenting, observing and refining. As I walked round the class, some students had written a couple of things down and were staring at a page waiting for inspiration. Other students were just reading and re-reading the question, waiting for inspiration. Other students were just staring at a blank piece of paper blankly. Unfortunately, this kind of observation is all too common in many mathematics classes. I was transported back to Bramol Lane, watching players waste good opportunities time and time again. A top player would take on the defender and try and make a shot. They may fail, but they would learn something about the defender in the process. They would keep trying, trying to make something happen. Not so the players I was watching. Inspiration and confidence wouldn't come to them. They passed the ball back, they passed on responsibility, and the opportunity was lost. Meanwhile, back in the classroom, after much encouragement to actually do something to not worry about being right to try and refine and observe and revise, the class eventually came up with an elegant solution to the problem, a solution which was actually quite unlike the one that I devised before the class. I think it's this kind of experience which lies at the heart of good education. It's the kind of experience which will stay with the students long after the precise details and facts have been forgotten. Thank you.