I no longer understand my PhD dissertation (and what this means for Mathematics Education)

Earlier this week I read through my PhD dissertation. My research was in an area of Pure Mathematics called Functional Analysis which, in short, meant it was self-motivated and void of tangible real-world application. I submitted the thesis in 2011 and after a successful ‘defense’ made a swift exit from research mathematics.

I was curious to see how much of the dissertation I can still grasp, five years after the fact. I figured it couldn’t hurt my ego if I refreshed my mind with past mathematical glories.

How wrong I was.

This was not the casual read I had in mind. The notation was alien. I even had to scour the examiner’s report to direct me to the key results. And while I could have sworn this was a well-written thesis, I repeatedly found myself bamboozled by my own prompts. “The result now follows easily…” may have made sense back when, but now the author-turned-confused reader can profess that it most certainly does not follow easily, at least in his own mind.

A snapshot of my dissertation. That implication is no longer trivial to me, the consequence of (b) is no longer immediate and I have no idea how (d) follows from Proposition 4.2.6.

A humbling experience such as this begs the question: what was the point? Given I have retained precious little of what I devoted myself to over four long years, was this a misdirection of my talents (whatever they are)? A prolonged delay to my career proper?

Beyond scarcely stretching the boundaries of obscure mathematical knowledge, what tangible difference has a PhD made to my life?

These are not merely the musings of a has-been mathematician. They are relevant to all of us working in Education as we probe the rationale behind existing models of mathematics curriculum and assessment.

So let us ask the question in more general terms: what is the purpose of studying maths? I offer you three reasons, each informed by a resolute belief that my doctorate was worthwhile after all.

  • Mathematics is an excellent proxy for problem-solving

My PhD trained me to be a better problem-solver. I cannot prove this except to say that research empowered me with unrivalled tools for problem-solving: scouring journal papers to draw insight on existing methods, brainstorming with peers, trying new approaches…the list is endless.

My choice of field was irrelevant; Functional Analysis was mere proxy for pushing my problem-solving skills to new levels.

Mathematics, by its concise and logical nature, lends itself to problem-solving (it is not unique in this regard).

Choices around content are far less significant than the experiences they afford students to develop the skills of reasoning and problem-solving.

This understanding of how to nurture critical thinking is lost on policymakers. Curriculum standards are based on notions of what students should know (with some movement towards performing discrete acts of reasoning, itself very limited). Assessment is dominated by an obsession with short-term knowledge gains. Yet knowledge without understanding carries no currency in the world students are being prepared for.

In the era of data-driven accountability, educational measurement must focus on the processes of learning. Problem-solving is a creative and even holistic endeavour; it can not be codified or captured in absolute terms. Nor should it be; mathematics is beyond rigid measurement and mathematical thinking can never be reduced to knowledge acquisition.

  • Mathematics embeds character in students

Rich mathematical experiences are steeped in so-called non-cognitive skills (you know the ones — grit, resilience, mindset et al).

A thesis represents the very small subset of ideas that came good. It does not include failed efforts, yet they are the ones that define much of the research experience. Those failed efforts often contain the key insights that inspired the final breakthroughs. They condition the mathematician with a mental toughness. My own results only materialised after three years of failure and frustration (which included several renewed commitments to quit the damn thing altogether).

Those failures are largely what made me a better problem-solver. They endowed me with a refusal to give up, which often trumps natural intuition.

The mark of a good mathematics problem is multiple solution paths that give students the opportunity to experiment with different approaches. And a good teacher will create a safe environment for students to take risks and fail, all the while emphasising the importance of positive beliefs and mindset.

  • Mathematics is fun

Nothing profound here. I had great fun studying mathematics and was modestly compensated for the privilege. There are worse reasons to sign up to a 4-year commitment, and few better ones.

Students deserve schooling experiences that bring them joy and happiness. This will even suffice as a utilitarian argument, for these traits will outlast narrow knowledge gains and make a more positive impact on the world. In its purest form, mathematics is the perfect expression of human thought that marries logic with creative expression. There is much beauty to seek in the learning and teaching of mathematics. Is that not reason enough?

QED.

I am a research mathematician turned educator. This piece was originally posted on my personal blog, www.fjmubeen.com