Paul is Professor of Mathematics and Mathematics Education at the University of Reading. He sits on many national bodies relating to mathematics education including the Council of the Institute of Mathematics and its Applications (IMA), is the Chair of The Joint Mathematical Council of the UK (JMC), a member of The Royal Society’s Advisory Committee on Mathematics Education (ACME), and an External Expert for Ofqual.
1. What’s your earliest memory of doing mathematics?
I guess I was ‘doing’ mathematics at primary school. But now I consider I am ‘doing’ mathematics when I am being creative, looking for generalisations, making connections, investigating new problems, and making conjectures to explore further; ultimately ‘doing’ mathematics for me is about being inquisitive and pursuing any and every possible avenue that results. Having been brought up on a diet of ‘closed’ problems set for me while at school and university, my earliest memory of this was a brief experience I had when solving problems on geodesics (a geodesic is a curve between two fixed points on a prescribed surface that gives the minimal path length between the points). To fully solve the problem in closed-form I needed to determine a pretty horrendous indefinite integral, and I tried every technique available to me to determine the primitive, and wasn’t satisfied by using one approach alone! I encountered three very different looking expressions as a result, which made me think I had made some mistakes. However, after much investigation, I managed to prove that these expressions differed by only a constant, and learned a great deal along the way, including some interesting and unexpected relationships between seemingly different expressions. I realised that I might be ‘doing’ some mathematics that no one else had done previously. Of course once I started doing research in mathematics for my PhD, and subsequently as an academic, I also had many opportunities for ‘doing’ mathematics that no one else had done before.
2. How has mathematics education changed in the time you have been involved in it?
I have been heavily involved in recent post-16 mathematics reforms, including AS/A levels in Mathematics and Further Mathematics, and Core Maths. The most noticeable changes in mathematics education I have witnessed at first hand are as a consequence of these new qualifications, and the associated teaching and learning that is, or will be taking place. For AS/A levels this will be as a result of the introduction of mathematical ‘Overarching Themes’ – argument, language, proof, problem solving, modelling – and overall a more holistic approach to learning mathematics that has been sadly lacking at this level for some years now. For Core Maths this will be as a result of the strong emphasis on contextualised problem solving, and on using and applying mathematics in meaningful contexts. Both qualifications build on the strong foundations provided by a pre-16 curriculum that emphasises fluency, reasoning and problem solving. Fundamental to these reforms is knowledge with deep understanding, and the ability to apply that knowledge with confidence .
3. Tell me about a time in your career when something totally flabbergasted you.
In mathematics it was when Fermat’s Last Theorem was finally proved – such a simple concept and yet so complex to prove. I will be even more flabbergasted if another equally simple problem is resolved: whether or not a perfect cuboid exists – a perfect cuboid is a box whose sides and face diagonals have integer length, and whose ‘space diagonal’ is also of integer length. How I would love to crack that one!
4. Do you practise mathematics differently in company?
In my experience most mathematics academics relish flexing their mathematical ‘muscles’ in full view of their more lowly peers. It’s not unlike the rankings in tennis where you know roughly where you sit compared with others. As such, I would never ‘practise’ mathematics in front of anyone unless it is the mathematics I happen to be teaching to a group of students. Otherwise, I ‘do/practise’ mathematics on my own, taking as long as I need, exploring every avenue I can find, without the fear of making mistakes in front of anyone and the resulting embarrassment that would cause me!
5. Do you think a brilliant maths teacher is born or made?
Both! It helps to have some innate ability, but much of it is down to sheer hard work. Teaching is still very much a ‘performance’, and for some this comes naturally, and for others it develops over time. The key is to be passionate about one’s subject. Of course it depends on what you mean by ‘a brilliant maths teacher’ – beyond the ‘performance’ and classroom management elements of teaching, good teachers need to be able to motivate students, help them enjoy their mathematics, and enable them to realise their full potential using all the skills that teachers have at their disposal, much of which is ‘made’.
6. What’s the most fun a mathematician can have?
Aside from listening to the cricket Test Match Special (TMS) broadcast, for this mathematician it is ‘doing’ any kind of mathematics where I am ‘in charge’, which I have had the fortune to do through my teaching career (mainly undergraduates). This includes all the many and varied mathematical challenges I have set myself, together with the resulting mathematics I have done, along with the diversions and surprises, and of course the mathematical cul-de-sacs one ends up in, culminating in my case in more than 400 publications. Both of these pursuits – teaching and ‘doing’ mathematics – have provided me with so much fun, and I feel very fortunate in that. Hopefully when I am too decrepit to do the former I will still be able to do the latter!
7. Do you have a favourite maths joke?
In a series of light-hearted letters published in The Times in 1990 from various correspondents on the topic of how different professions ‘add 1 and 2’, with some interesting suggestions for engineers, statisticians, pure and applied mathematicians, accountants, lawyers and so on, my contribution to the column – which I will now claim as my favourite ‘maths joke’ – was that: ‘A computer scientist will add 1 and 2 bit by bit.’