What is meant by dissertation?
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.