# My Mathematical (and Teaching) Journey

## In the beginning, there were Smarties

My earliest recollection of doing maths in school is Friday afternoon Smarties competitions in primary school, when I was aged 5. I think the questions must have been addition and subtraction – perhaps 40 of them written on the board. The first to finish would win some Smarties from the magical ‘giant box’ that our teacher kept in her desk. I have to admit that most of the time I was the winner, although I do remember some close competition. Strangely, though, I don’t remember any mathematics lessons as such. Perhaps my teacher was adept at not making maths stand out as a subject, and certainly not one to be feared, or perhaps I was just a bit distracted.

I think throughout primary school we had some workbooks (5-a-day? 6-a-day? If I remember rightly, they were a set of increasing difficulty) and a box of workcards which had different problems to solve. I still remember being stumped by one asking how to fold a strip of paper into thirds. Curiously, I also remember a hushed conversation between two teachers at the front of the class – clearly about something particularly difficult in mathematics – and at one point, a letter was written on the board. A letter? In a maths problem? I was so intrigued but they didn’t let me in on the secret.

At home around the same age, there were some supportive mathematical influences. I was tested on my times tables (up to 12) until I knew them thoroughly. We would play board games and card games. I remember a table cloth that was patterned with squares (maybe 1.5”) of different shades of red/purple. I would stare at it for patterns, trace out different blocks with my finger, and even start to discover simple ideas about area of rectangles. My dad had a financial calculator where you had to enter everything in the wrong order (apparently it’s called reverse Polish notation and works like a stack in computer science) and he taught me every geometrical construction he could remember – I think that was his idea of bonding as neither of us had any interest in playing football.

## Secondary school

Through secondary school I enjoyed many but certainly not all of my maths lessons. There was one about lighthouses flashing at different times (obviously an LCM problem now I look back at it), but I didn’t have a clue what was happening in that particular class. That was a bewildering feeling. I was steadily pushed up the rankings, ultimately destined to take GCSE early. I remember that matrix transformations was a topic we studied – I still have my STP 9(a) book and show it to my FP1 students when I’m annoyed at them: “A 13 year old could do this stuff!” I loved the ‘trick’ that the image of the unit vectors form the columns of the matrix representing the transformation, although I’m not sure it was quite presented in those terms. (To continue the mathematical thread, I loved learning about the concept of span and a basis in linear algebra and how it made sense of this ‘trick’ from year 9.)

Outside of school, the family influences also continued: my aunt taught me how to quickly multiply two-digit numbers by 11. That was amazing, but why did it work? (Three years later, our A level teacher taught us modulo arithmetic and the reasons this divisibility tests work. More on him later.) My brother, three years my senior, showed me the dark art of the integration sign and how it could be used to find areas of mystical shapes. I always wanted to know more, and I was fortunate to have people around me sharing these little ideas.

## Sixth Form

After GCSE in year 10, I took additional maths in year 11. My teacher, Mr Orton, was just as much a mathematician as a teacher and I was fortunate to have him all the way through my A level Maths and Further Maths, too. We didn’t know where Maths ended and Further Maths began. We didn’t know what was examinable and what was included at his discretion. We just knew that we were studying mathematics and, it turned out, we were more than amply prepared for the exams. Even the textbooks gave no clue as to the exam content either: his favourites included Caunt’s Elementary Calculus (1963), Backhouse and Houldsworth’s Pure Mathematics (1969) and a seletion from R. I. Porter’s oeuvre. (Don’t tell anyone, but I’ve still got copies of them all.) I remember a lesson where we were shown a video introducing the idea of fractals. I didn’t learn much from the video itself but, once again, I was captivated by the new concept. I started reading about them and learning to write computer programs that could recreate the images. Mandelbrot set on a Casio graphics calculator anyone?!

As for teaching mathematics, during year 11 I helped a fellow student out with his GCSE: teaching him a number of topics and practically writing a short book to help him. That was when I first thought that teaching might be for me.

The external influences continued, too. During my sixth form years, my brother was studying Electronic Engineering. I visited him at university and even went along to one of his maths lectures. The lecturer was explaining about a path in a plane, but the plane had to be cut so that the path could continue in another plane before passing back through the cut to the original plane. I had no idea about the details of the concept but was entranced! (With hindsight it was an introduction to the Riemann surface for z².)

## University

At Warwick I opted for the MMath course and got off to a flying start. Further Maths made some of the 1st year modules much easier (except probability which I barely passed – my whole A level was pure and mechanics) and I developed a liking for analysis. In my second year, I had the opportunity to be a ‘peer tutor’ for analysis which gave me another taste of teaching. I also discovered a significant distaste for algebra: PIDs, UFDs and who knows what else. I had loved group theory, but something went very wrong in that course. I’ve been allergic ever since. At least there were complex analysis and metric spaces. It seemed as though all the ‘fun’ options had been reserved for third years: topology, knot theory, complex function theory, functional analysis, fractal geometry and many others. I also had an additional opportunity to teach: as a peer tutor in metric spaces.

In my fourth year I honestly struggled: if you don’t like algebra or applications, then it’s difficult to choose enough courses. Riemann surfaces was top of my list but then there were PDEs, Waves, and Lie Groups. I’m ashamed to say I entered the Lie Groups exam still not understanding what a Lie group actually was (even now I have no clue). I memorised the definition symbol-for-symbol and managed to score a grand total of 8%. Fortunately, the other courses compensated for this shocker.

Teaching-wise, I was flying: more analysis teaching and supporting small groups of first years. After graduating, I even stayed on to do more teaching, including differential equations and some undergraduate courses for students on the primary maths BEd programme. Warwick also developed a programme of residential summer schools: NAGTY in the early years, becoming IGGY later in life. It was hugely enjoyable (and exhausting) to develop and run these maths enrichment courses for gifted kids from around the world.

## Dabbling in research (or ‘The Barcelona Years’)

Although teaching was what I enjoyed the most, in order to stay with the university I had to undertake research of some kind. I began a PhD with the idea of studying something in complex dynamics (fractals associated with complex numbers if that helps). There was a lot to enjoy about this (not least the people I’d meet at conferences – there’s something about complex dynamics that makes for an entertaining bunch!). I also had the opportunity to work at Barcelona University for a while, but even there, I was helping out with school liaison activities (and learning Catalan….)

Disappointingly, my research didn’t amount to much: a world-record for something that not many people understand, let alone care about (largest known ray equivalence class for the combinatorial mating of two quadratic polynomials, if you must know). However, it certainly wasn’t enough to claim the PhD. I returned to England, was incentivised by the ‘golden hello’ scheme to get into teaching and never looked back since. (Although I do miss Barcelona hugely!)