On Mathematics and its Teaching

And my journey from loving math to hating it and back to loving it again

01 Dec 2017

I love mathematics. It’s logical, universal and beautiful. I loved mathematics as a child and was naturally good at it in school. In fact it was one of the things that defined me as a kid, I was good at math and science. I wasn’t into sports or any other significant activities back then. Yes, I was one of those kids. Mathematics just came naturally to me and I wanted to learn more and more. I wanted to know everything.

I grew up in a nondescript family in a nondescript town and went to (I realize now, looking back) nondescript, mediocre schools. I wasn’t driven to achieve; there was no expectation that I should aspire to attend an elite university that would act as a stepping stone for me to then move on to a “career”. I just went to school and learnt what I was presented with.

I did have the benefit of having some great, fun math teachers though; in middle school (what, in the UK, we called high school - Redmoor High School in Hinckley to be precise) and then in high school (sixth form college in the UK, specifically the John Cleveland College, also located in the same Leicestershire idyll).

Now I described these teachers as “great” and at the time that’s what I thought they were. Looking back now though, with the benefit of hindsight, I can say that they didn’t do that great a job of preparing me to study mathematics at a higher level (more on this below). However, they were encouraging and they made the subject fun, and that is an essential aspect of mathematical pedagogy; and something that was noticeably absent from my experience of university-level teaching.

I was streamed ahead in middle school and then again in high school; and for the last two years there I was in a tiny class (with only five kids in total if I recall correctly) where everyone was motivated to learn and so the teachers could just focus on teaching without having to do general crowd control as well. There was also a competitive dynamic between three of us in the class that really drove us on. I recall it fondly.

One day my teachers suggested that my fellow math geeks and I should apply to Oxbridge (a portmanteau used to collectively refer to the two ancient elite universities in the UK, Oxford and Cambridge). This was an idea that would never have occurred to me or my family ourselves. As I said before, I was not raised to aspire to such things. The high school didn’t have a grand tradition of sending students to Oxbridge either but they clearly had an eye for it and we were considered worthy. From that point (towards the end of my penultimate year of school) our math class shifted to focus on the relevant entrance exams and preparing us for interview. We were successful too. Ultimately I and two of my fellow classmates were accepted, two of us to Oxford and one to Cambridge. And lo, I got to read mathematics at St. John’s College, Oxford. And boy was it different.

A few things changed when I got to Oxford. First, I went from being one of the brightest kids in school (without really having to try) to being middle of the pack at best. This forced me to reevaluate my sense of who I was and what defined me. Also, many of my fellow students studying mathematics, as well as the lecturers and tutors, were - how can I say this - really nerdy and not cool. Now God knows I was never cool myself in high school but I aspired to be, and I think I had the core ability to be social and funny, I just had to grow into it and that process really started once I was at university. I remember having to spend time together with some of my fellow mathematicians and tutors (one in particular) and feeling really awkward in their company. The tutor hardly spoke. I absolutely felt that these were “not my people” and thus began the process where I started to drift away from mathematics. I couldn’t love the subject anymore because to do maths at this level was to be like these people.

I gravitated more to some of the students who were studying engineering, physics and chemistry. They were still smart and geeky (geeky is NOT the same as nerdy) but more fun to be around. I also very much enjoyed the company of those who were studying a whole variety of completely different things (history, english, geography, psychology, philosophy, law, …) as I was exposed to these people on a daily basis. More on this below.

Now one of the great things about being at Oxford is the fact that you are a member of a college as well as a member of the university. There are 38 colleges that collectively make up the University of Oxford with each college being an independent institution with its own history. My college (St. John’s) was founded in 1555 and housed about 400 undergraduate students, 250 postgraduate students and 100 academic staff. The students and staff are drawn from all academic disciplines and so life in college is a wonderful mix of personalities, ideas and intellectual points of view. The college bar, when evaluated purely as a bar, was bloody aweful but it was the place where this wonderful mix of people hung out and where you could find people discussing and debating all aspects of intellectual pursuit as well as a variety of trivia and stupidity (which was just as awesome). It also served the absolute best cheese and ham toasties, and baked potatoes with cheese and beans. Then there was the Junior Common Room (or JCR) which would hold regular meetings and debates, well lubricated by booze provided by the JCR Pratt (an official position on the JCR Committee who’s responsibilities included being a pratt and getting the beer in for JCR meetings). I recall one debate in particular where great energy was expended in resolving the matter of whether Mr Nathan Byers (the JCR Pratt at the time) should change his name (by deed poll) to Nathan-Madonna Byers in honor of the “Queen of Pop”. We supported the motion and he did actually go ahead and change his name. What a stand-up fellow.

Anyway, back to the main point of this post. Just being at St. John’s was much more interesting than studying mathematics. Not to say that I didn’t want to study and learn, I did. Which brings me to my next point.

Undergraduate teaching at Oxford is done in two parts. I’ve already described the college system and the fact that when you are at Oxford you are a member of a college as well as a member of the university. The university as a whole still exists of course and there are university departments for each subject area. All tutors (professors) are a member of a college but also a member of a university department. The departments organize and teach the curriculum for a subject area and set exams. It’s your performance in these final exams that ultimately determine your class of degree. Teaching consits of group lectures at the department level augmented by small group tutorials (in my case it was two students to one tutor) with your local college professors within college. This all sounds great and it would be if all of the lecturers and tutors were good teachers. Unfortunately, in my experience and especially in Mathematics, often they were not.

An aspect of the teaching of mathematics at the undergraduate level is that it is presented in a fundamentally different way than in high school. A degree of rigor is introduced that represents a step change in the way things are taught. There is much less of a focus on the student developing an intuitive understanding of the concepts and instead things are presented in terms of a series of “definition, theorem, proof” cycles. This was something that I had not seen before and was quite hard for me to come to terms with. This is what I meant when I said that my high school teachers didn’t prepare me to study mathematics at a higher level. I learnt from several of my fellow students at Oxford that they had been introduced to some degree of rigor, and this style of presentation, in school and it really helped them. I guess that’s the difference between a state school in a nondescript provincial town and a fee-paying private school. Sigh.

I stuggled to grasp the new presentation of the subject. Often the rigor seemed pointless (“But that’s just obvious!”) and othertimes it obfuscated and hid the concept that was being introduced.

Given time it became more familiar and looking back now I absolutely understand and appreciate the focus on rigor and the logical construction of things. At the time though I was frustrated and didn’t feel like I could ask for help in getting up the learning curve since my tutors felt unapproachable. It was yet another factor that tainted my orignal love of the subject.

I ultimately graduated from Oxford and earned degree in mathematics with a good enough grade (a 2:1). The competitive high school kid I used to be would never have been satisfied with anything less than a first but I wasn’t that kid anymore. I just wanted to get on with the next phase of my life and that I did.

Some years later I read a book on mathematics written for the layman and I once again saw something in it that was beautiful. I read more, and especially read a lot more about the history of mathematical discovery and the iconic figures who developed much of what we take for granted today. The historical context was fascinating and the presentation, geared to the non-academic, emphasised the concepts and the consequences first and foremost. I ultimately came to the conclusion that you can’t study math in isolation, it’s too sterile and without meaning. By taking a leisurely tour through its history and landscape I rediscovered my love for it. I’ve since been motivated to try to re-learn much of what I was first exposed to back at university and also I now feel a personal mission to help others to learn math the “right way”, so that they can see it and grok it while they also learn it properly.

I commit to writing posts about math as I continue my journey of rediscovery through its gentle rolling hills.