How my supervisor saved my PhD — the Goldilocks principle for mathematics
Mathematics is all in the asking
I almost flunked my maths PhD.
I had been forewarned that the journey to graduation would be paved with uncertainty; nothing like the smooth, linear path I travelled as an undergraduate. Research is a messy enterprise, and pure mathematics takes no prisoners.
Almost two years in, I was caught in the jumbled mess that is the hallmark of research. To acquire a PhD, one must make an original contribution to their field’s body of knowledge (or similar; I can’t recall the exact wording). However subjective that criteria might seem, I was clearly not meeting it.
Trawling
I had spent most of the first year trawling through the literature, familiarising myself with the established body of knowledge in my field of Functional Analysis. Research is a fine line between plagiarism and creativity: you have to take inspiration from existing ideas and develop novel approaches that build on existing knowledge.
The PhD mathematician’s first charge is to identify a question worth pursuing — one that will invite creative solutions, and that can be answered within the 4-year timeframe of a doctorate.
By the first year’s end, I had my question pinned down. The second year would be all-out assault. That was the plan, anyway.
Toiling
Year two was the year of insignificance. I attacked my problem but met with resistance each time. I sensed breakthroughs were close, only to learn that my senses had betrayed me. My mathematical intuition — that holistic quality I had relied on in years past — had evaporated altogether.
When you perpetually fail to solve a maths problem, it can mean one of two things. Either the problem can’t be solved, or it is too damn difficult. With my research question, I had assumed the latter. Perhaps I just needed to take a deeper dive into the journals to uncover new lines of attack. A fleeting insight in the shower confirmed my gravest fears.
Archimedes in reverse
I had always dreamt of my own Archimedean triumph, where inspiration would strike me at the most inopportune moment. One morning, as I showered, that moment arrived. My intuitions finally awoke. But triumph turned to disaster as the mathematical breakthrough that came to me was not the solution to the problem, but the realisation that the problem could not be solved. For the past year, I had been chasing a wild goose; a maths problem with no solution. I had asked the wrong bloody question.
I stumbled into my weekly supervision a picture of dejection. Why waste my supervisor’s time, or even my own, with meek attempts at mathematical discovery? Impostor syndrome had taken firm hold of me; I was ready to quit. And cry.
My doctoral supervisor fits many of the stereotypes that mathematicians carry. Needless to say, I knew better than to lean on him for sympathy or emotional support. I shared the devastating discovery that my research to date was all for naught, my tone marked by desperation and a hint of shame. I honestly believed I would make unwanted history as his first failed graduate student.
Pivot
My supervisor then did what he does best. Paused, reflected, scratched his scalp. He drew a blank piece of paper and scribbled away, murmuring something inaudible. After another pause, he handed over the paper and declared in a neutral tone: ‘this problem might be worth your effort.’
His presentation may have been subdued — a bit awkward, even — but the message was riveting. He was not abandoning me! We talked through the intrigue of this latest problem, which had no resemblence to the one I had unwittingly pursued over the past year. In a single moment, with a single problem, my supervisor had pivoted my research.
Floodgates
It worked, too. My progress was not immediate. I continued to toil a while, but there was always a sense that the breakthrough was on the horizon. This time, my senses came good. I derived a partial solution to my supervisor’s problem, which opened the floodgates to further insights. Within a few months of that initial milestone, I had solved the problem in its entirety. Within a few more, I had extended the problem in several directions, asking new questions for myself. Smart, answerable questions too, the solution to one of which would later be labelled a ‘desideratum’ by my examiner (a mark of honour for mathematicians).
A noticeable shift occurred in my third year: I found myself driving the weekly meetings, educating my supervisor on new insights I had gathered. This transition marked the final stages of my PhD; I sprinted to graduation with renewed spirit and confidence in my mathematical abilities. My identity as a mathematician was secure.
I was under no illusion — I owed the turnaround to my supervisor. He possessed the knowledge and experience to pose the right kind of problem. He had no solution at the time, just a feeling that it fell within the scope of a doctoral research project.
Having devoted a year to an insoluble problem, I had only respect and admiration for my supervisor’s instincts. And gratitude.
Reflections
The PhD is defined by its struggle. In some respects, that ‘wasted’ second year was no waste at all; it engrained in me a resilience that serves me to this day. The achievement of my dissertation was not in the rush of breakthroughs late on, but in overcoming the doubt and uncertainty that preceded it.
My outlook as a maths educator has been shaped by my time as a PhD student. I know what it means to struggle in maths. We all hit brick walls along the way — I hit mine later than most but my struggle has connected me to the difficulties that students and teachers across the education system face.
The messiness of a PhD is emblematic of the learning journey that most students embark on throughout their time in school.
I know all too well how a conceptual blockage can be devastating for one’s confidence and enjoyment of maths; how it can destroy our sense of mathematical identity.
I also know the virtuous role a teacher can play in reviving that identity. My success is due to my supervisor’s deep expertise of the subject; in his ability to pitch a maths problem at just the right level; at the edge of my ability.
How many students flounder in school simply because maths problems are not tailored to their current needs?
Maths problems must be selected at the right level of challenge. Too easy and you lose any sense of challenge and engagement (worse still, you can induce complacency). Too difficult and you risk shattering a child’s confidence. It is futile to throw a child into a problem without first grounding them in the relevant background knowledge, or to leave them searching for a solution completely unaided.
But when the problem is pitched at just the right level — where it stretches the child’s knowledge, as well as their imagination — they will soar. It is the Goldilocks principle for maths educators.
Lastly, my time as a research student reminds me of the essence of mathematics itself: not just in solving problems, but in posing them. Maths gives us the freedom to question; it is shameful to strip students of this entitlement by reducing the subject to a set of pre-defined questions with right/wrong answers. The skill of a mathematician is in asking the right questions — a lifelong skill I contended with all the way through my PhD.
It is never too early to nurture the craft of asking questions, and probing what makes maths problems worth attempting at all.
