@DrBennison (Tom Bennison in real life) recently posted about his mathematical journey and wondered what other people's looked like. Happy to oblige.
I almost didn't study maths at university. Most subject teachers are thrilled when you suggest you might study their speciality, but I was extraordinarily fortunate that when I told Mr Smith - who spoke French with a distinct Brummie accent - that I was thinking of doing French for my degree, he looked at me with horror. I don't recall his exact words, but I paraphrase: "There's not really much you can do with a French degree. Maths is way more useful -- and you're much better at it." (This wasn't a put-down -- I was doing French A-level a year early. I'd caught up with the Upper Sixth group in a matter of months and would end up being the only person to get an A in it. I was good at French.) "Also," he said, "there's a scheme called ERASMUS where you can study French as a minor and spend a year in France..."
Well, I was sold1. I picked St Andrews over Cambridge partly because they had an ERASMUS agreement with Avignon, but mainly because the Cambridge admissions spiel was "If you're lucky enough to get onto the Maths course, you'll work really hard, struggle, and probably drop out"2, while St Andrews was more "Maths? Cool! Have a pie."
How did I get good at maths? I honestly don't remember. I don't think I had any special "maths for baby geniuses" books as a kid, my parents are competent with numbers but probably wouldn't call themselves mathematicians, we didn't have gifted and talented programmes in those days -- to the best of my memory, I was just left to get on with it. They encouraged me to read plenty. My parents loved (and still do) puzzles -- crosswords, logic puzzles, jigsaws. I never felt pushed in any direction -- the only parental directive I remember is an admonishment from my dad that I should never do anything but my best work, advice I still try to follow. One thing that did contribute to my maths education was having a ZX Spectrum in the house, a computer you could hardly help but program, especially if the tape deck was dicey, which it was.
I didn't plan to be a mathematician -- I wanted to be a journalist, maybe an ambassador3 I edited the sixth form newspaper, and when that didn't exactly work out4, I started a rival. (I have fond memories of sitting outside the Head of Sixth Form's office listening to the gales of laughter from inside, followed by a very stern-faced Mrs Minshall coming out and saying "you're not printing that.").
My late teens were a shock to my ego -- when I got to sixth form, I couldn't rely on generally being the smartest student in the room any more. At university, it was worse: I couldn't even rely on being the smartest mathematician in the room. I was good -- but I wasn't that good. A lot of the first year maths material I knew from A-level, so I was getting most of my enjoyment out of modules in Philosophy, Computer Science and French.
I have a hypothesis that everyone5 has a day in their life when they wake up and maths is suddenly difficult. For me, it was the day of my second year Analysis exam: I turned over the paper, read the first few questions and the best thing I could say was "I recognise many of those words." It was an acid test for me: up to that point, I'd been able to coast quite comfortably. In the end I scraped a decent grade by a combination of doing the bits I could do, bluffing the bits I could almost do, and inspiredly guessing the rest.
Third year, mathematically speaking, wasn't an awful lot better -- that was my year in France, where the maths is as dry as the Mistral wind and as pure as the snow it drives6. I, in the meantime, started getting panic attacks and depression, missing classes, not really understanding the classes I did go to, and -- in honesty -- phoning it in. My grades were nothing to speak of -- in one module, I got 1/20, and after looking at the paper I got back, I couldn't see where I'd been given the mark.
If this was a story, this would be the long night of the soul: I was at a crossroads -- I could either carry on phoning it in, or I could take my dad's advice and do my best work. I decided I wasn't a phone-it-in kind of person, and I was going to kick some serious backside in my final year.
Best decision ever. I persuaded professors to let me do courses without having the prerequisites. I did more modules than I needed to to get my degree. I got organised. I coerced fellow students into revising with me. And I kicked some serious backside -- enough for one of the professors I'd persuaded to stop me in the corridor and invite me to do a funded PhD.
The retrospectoscope is a marvellous thing: while doing a doctorate was, overall, positive (and a necessary step to getting to where I am today), I think I'd have been a bit more cautious about taking it on if I'd known how hard it would be on my mental health. I researched the magnetic fields around the Sun, and the topological changes it goes through en route to a solar flare. At least two very good things came of my PhD: I got to tutor small groups of first and second year students, with licence to experiment. Give them chocolate for handing in their work! Do classes in the pub! Come up with different methods from the one in the notes! I noticed I got good results if I put the responsibility on them to do the work, while making it clear that if they worked hard, I'd back them to the hilt.
The second good thing was spending a month in the USA working on a project at Montana State University. It went incredibly well, and they more or less offered me a post-doctoral position then and there. (Because it was a state university, they had to advertise for applicants; the job description didn't quite say "candidate's name must be an anagram of Bolin Ceveridge", but it wasn't far from it7.)
Once I got through my viva (and associated panic attacks), I carried on the same sort of work at MSU. I wrote a lot of code. I threw out a lot of code. I took some very general results (hi, Euler! Hi, Poincaré!) and hammered them down to one equation that worked just for my specific field, and then persuaded people that it ought to be called the Beveridge-Longcope equation -- which is how it's been referred to ever since. Equations, shmequations, though -- my main contributions were in coming up with ways of visualising and presenting results. (See the big bit of advice at the end.)
But I got to a point where my papers were taking longer and longer to write, and being read by fewer and fewer people, while the winters were getting longer and longer. I realised I wasn't really helping anyone, I wasn't enjoying myself, and I needed to go home.
Home I went. My mum's hairdresser's son needed help with his maths GCSE. I said "of course I'll help!". The realisation that people would pay me to show them how to think their way through sums was a revelation. I decided I'd do my best at it. Good decision.
I started writing up interesting problems. Somebody at Wiley found me on the Googles. They said "Would you write us a book?". I said sure, and decided I'd do my best at it. Good decision.
Big bit of advice: if you want to be influential, develop two orthogonal skills. @standupmaths is the mathematician who's funny. A mathematician who's good at graphic design could make a serious impact in research. I'm not where I am today because I'm a fantastic mathematician -- I'm really not, I was only ever a mediocre researcher with some good ideas and an eye for three dimensions, at best -- it's because the Venn diagram of 'decent mathematicians' and 'decent writers' has a relatively small intersection. By being a mathematical writer, I'm a biggish fish in a smallish pond which lots of people want to fish in.
Edited 2015-07-20 to give Tom his preferred name.