(On learning to love, and loving to learn, mathematics)
This article was written for purely selfish reasons. I wanted to reflect on and begin to make sense of my experiences in learning, teaching and doing maths in order to distil from them a clearer sense of purpose and direction going forward. I feel it’s been a worthwhile start. If it can prompt, or otherwise help, anyone else do the same for themselves and their learners – just as articles shared by other members of the online maths community have done for me – then its worth will increase.
I use maths every day. I need maths in my life.
I don’t mean that I have to manage my time, my money and all the other quantitative aspects of everyday life like finding clothes and shoes that will fit me and food that is of the right amount and nutritional value. Or getting around on public transport, or dealing with the statistics that help form the fabric of modern life, designed to either inform, persuade or even mislead me. I do need to do these things, of course, but doesn’t everyone?
We live in a world that simply wouldn’t exist in its current form without the advances in maths and technology that have accumulated and, more recently, accelerated throughout history. It goes without saying that this is testament to the importance of maths. But it’s not what I mean when I say I use and need maths on a daily basis.
I should point out that I work as a maths teacher in a secondary school. Of course, I hear you say, you use maths every day (or at least every school day). Think about this, though. What if I told you I travelled extensively, then went on to explain that I was a long-distance lorry driver or an airline pilot? The truth of my original statement would not change but its meaning would. What I want to explore here is the way maths has come to permeate not only my working day, but so many other aspects of my existence as well.
For a long time maths was not as central to my life as it is now. I was fairly successful in my maths studies at school. I was fortunate to have a great teacher for the last three years there and I still have fond memories of her class. Nevertheless, I can’t say in truth that I loved maths as a pupil. In fact, I preferred chemistry and it was this subject I chose to study at university after finishing S5. Again, this was down to an inspirational teacher: at that point I was clear in my mind I wanted to emulate him by becoming a chemistry teacher myself. The degree course involved more maths during the first two years. I remember very little of my university maths – suffice to say I managed somehow to get through my exams and was not unhappy when it was over. To be fair, I didn’t particularly enjoy my chemistry studies either and I gradually became disenchanted with academic life, including any thoughts of a career in teaching, leaving uni with a pass-degree after three years. (An honours degree here in Scotland involves four years of study.)
After graduating I was offered a full-time job in the betting industry, where I had worked Saturdays and summer holidays as a student. For almost 20 years the main part of my working day was spent calculating (or “settling”) the returns my customers were due from winning bets. This involved becoming skilled in working with fractions and decimals, with the occasional application of proportion or a percentage increase. While I could, in principle, carry out these calculations manually, the pressure of time often led me to use a basic pocket calculator – a skill in its own right – or, later, as the nascent electronic revolution gathered pace, a bespoke “settling machine”. These were not highly specialised skills: there was no requirement for any formal qualifications in maths – just good number skills, which were assessed as part of the recruitment and selection process. Most of my colleagues at the time hadn’t been to university (many had few or no formal qualifications at all), although I have worked with people who could perform these calculations without electronic assistance at impressive levels of speed and accuracy. While I took a pride in doing this part of my job to the best of my ability, I became intrigued by the underlying maths behind the permutations and combinations that told us, for example, what bets made up a “Yankee” – details that many staff and customers alike took for granted. (A Yankee, by the way, involves 4 selections and comprises 6 “doubles”, 4 “trebles” and 1 “fourfold”: 11 bets in all.) Even more fascinating to my enquiring mind were the amazing algorithms that allowed me to calculate the winnings from such “multiple bets” in seconds with just a red pen and paper or a few keystrokes on a calculator. I often wondered who came up with these algorithms. I never found out but secretly hoped it was someone with no special training in maths but who, nonetheless, was in possession of a restless, probing mind, a clear understanding of the problem at hand and a relish for finding a solution. Sadly, for me anyway, the job of the settler has become obsolete, now performed by software instead. All of this though, stimulating as I found it, was a mere drop in the metaphorical mathematical ocean.
Having mastered these aspects of my job, I felt my mind was stagnating somewhat. Seeing our three young children develop and, later, start school I became fascinated with the way they seemed predisposed to learn from experience and to instinctively ask succinct open questions; after reading a few library books on this broad topic I found myself registering for a course in psychology with the Open University. Once again, with my own and my children’s learning forming a key focus in my life, my ambition to teach gradually began to resurface. An Open University degree can be pretty flexible in its make-up and I ended up choosing some science and maths courses to supplement the psychology units. To be honest, all this study while working full-time and being a parent was not always enjoyable: had anyone asked me my feelings about maths at that point, I don’t think words like love would have featured in my answer. By the time I had finished my degree – with honours this time – I’d made up my mind that my future lay in teaching maths.
Since leaving the bookies around 12 years ago to begin my PGDE, my mathematical horizons have expanded both inside and outside of the classroom. (Now that I teach the binomial theorem I realise it was behind the settling algorithms I mentioned above – I hadn’t even heard of it back then!) However, when I say I use and need maths daily, I’m not really talking about the routine maths questions in the textbooks and worksheets I frequently issue to my pupils. In fact, these are the very things, devoid as they can seem of any inherent value or interest, that sometimes prompt frustrated learners to ask the question many maths teachers dread and struggle to find a meaningful answer to: “When will I ever need maths?”
Just as it was in my previous job, the ingenuity behind the rote processes – discovered and developed by people whose names have long since been forgotten – that form the bread and butter of maths is what appeals most to me. Through spending so much time thinking about maths I have come to see it as an expression of some of the highest human capacities: enquiry, sense-making and creativity to name just a few. I now use maths for pleasure and for thinking about how I can become a better teacher. I read books, journals and blogs on maths. I talk to colleagues and use social media (who’d have thought it?) to see what other maths teachers – and non-teachers – are talking about and to join them in both structured and informal discussions. I seek out puzzles that will stretch my problem-solving skills and, occasionally, try to come up with some of my own. I attend lectures and conferences and listen to podcasts to hear speakers share their experiences about innovation in the teaching and learning of maths. If there’s a programme or film about maths on TV, you can be sure I’ll be watching it; in fact, these are among my favourite TV programmes. Actually, I don’t watch that much TV. I’d rather spend time doing things like playing with shapes and patterns. More formally, I’ve learned how to make geometric drawings, including Islamic and Celtic patterns, and to construct 3D models using just paper or card, a compass, straight-edge, pencil, scissors or craft-knife and maybe some glue. (Note to self – I need to do this more often!) Maths is one of the most stimulating, enjoyable and, at times, exciting things in my life.
If you give me a maths puzzle or problem I can’t immediately solve, I don’t want anyone to tell me the answer. I might get it after a few minutes, an hour or several hours. Maybe I’ll put it aside and the answer will come to me later that day, the next day or even after a few days have passed. When you’re used to spending time thinking about maths problems, you often find your brain has been working away in the background outside of your conscious awareness. Or it may be the case I’ve figured out how to solve a problem, but it just takes a while to work through it. If I need to find a way to help time pass, for example during my bus journey to and from work, or when I’m travelling overseas on my annual holiday, a maths problem will always do the job nicely. Of course, I don’t solve every problem I come across. Some days, if I’m honest, I can’t be bothered. And some problems I just find too difficult and quickly give up on. However, I have invariably found that time spent on doing maths in this way (as opposed to just reading about it or practising a new process) – even when I don’t find the solution and have to, grudgingly, look it up or ask someone else – has been an investment that has paid dividends in terms of my mathematical development.
On the other hand, I might be doing a routine question that barely taxes my mind at all, when suddenly I see it in a new way. Even better, I might be teaching something I thought I knew everything about and one of my pupils will offer an observation or ask a question that has never occured to me before. Even professional mathematicians can find that study in one area of maths reveals unexpected links to others – who would have thought, for example, that pi, the ratio of a circle’s circumference to its diameter, would be involved in trying to find the sum of a series of fractions that get smaller and smaller, or in the answer to a probability question? Surprises like this are another of the reasons I love my subject.
I feel really fortunate to have a career doing one of the things I love most in life. If I didn’t need to earn a living, I like to think I’d still be teaching maths anyway; I’d certainly still be doing and learning maths. I have a “no regrets” philosophy in relation to events in my past. For one thing, I know I can’t change them; for another, I wouldn’t be at the happy place I am today without getting through the difficult times as well as the good times. Experience has also taught me that many of the important aspects of being human are context-dependent: the meaning and value of such things are not fixed, but fluctuate with time and with personal, social and societal circumstances. I do sometimes wonder, though, what would have happened had I properly discovered the intrigue, beauty and joy of maths at an earlier age than I did. For anyone reading this whose learners (or they, themselves) ask that dreaded “when will I ever …” question, don’t give up on the possibility that they, perhaps with your help, will find a way to learn to love maths.
In future, when l’m asked, “When will I ever need maths?”, I shall answer, “I hope the time comes when maths will enrich your life in the ways it now enriches mine”.
Through this website I hope to share this enriching intrigue, beauty and joy and, in doing so, to become better at learning, teaching and doing maths.
One of my other passions in life is the great outdoors. I took some time out on my customary Sunday afternoon pilgrimage in nature’s cathedral to mark the shifting seasons with this attempt at an ephemeral maths-art piece.