I’m going to tell you a story.
It is decidedly not mine. In fact, it comes from a book I didn’t write, stars two people I’ve never met, and describes an interaction the likes of which I’ve never had.
Still, once you read it, the story immediately feels like yours. Good stories are like that.
The book is King of Infinite Space by Siobhan Roberts; it’s a biography of the geometer Donald Coxeter. Like geometry itself, the book is full of strange and lovely moments, such as when young Donald is so distracted by mathematical fancies that his tutor forbids him from thinking in the 4th dimension, except on Sundays.
The story I want to tell lasts only a paragraph, yet it manages to capture all the peril and promise of the relationship between advisor and advisee. It concerns an older, wiser Donald and his final PhD student.
The tale begins:
Asia Ivić Weiss, now a professor of mathematics at York University, was… Coxeter’s seventeenth and final PhD student (and the only woman).
Academics describe their professional relationships with a charming, oddly intimate metaphor. Your advisor’s advisor is known as your “academic grandfather.” Your advisor’s other advisees are your “academic siblings.” Your advisor’s sibling is your “academic aunt.” And so on.
The metaphor is no accident. These relationships can rival the duration, intensity, and complexity of family. Strangely, the pillar at the center of the metaphor is often left unspoken: that your PhD advisor is your academic parent.
Anyway, back to Weiss…
Working on a problem, she couldn’t see Coxeter’s extrapolation from two dimensions to three, even after reams of calculations.
This is a typical PhD experience. The professor tosses you a challenge, casually, as if it were a capped pen or a fresh apple. Their nonchalance reassures you. It’ll be hard work, no doubt, but honest and straightforward, like digging a garden.
Then you sweat and strain, you huff and puff, and the boulder just won’t bulge. You become convinced you’re doing something wrong. After all, your advisor has years (or decades!) of expertise, while you’re still finding your way, still building the muscles. Surely he is right. Surely the mistake lies with you.
Teaching high school, I’ve found that some students assign a tremendous multiplier to the value of my time. They’ll spend hours toiling alone before they’ll trouble me for five minutes of help. I plead with them to value my time less, to come with questions before they’ve clocked days of frustration, but I understand why they’d rather struggle alone. It’s how I was as a student, too.
So it’s no surprise that Weiss chose to toil on her own.
[Weiss] persevered and, after days and days, successfully proved that the three-dimensional result was not a spiral on a cone, as Coxeter had said was so patently clear, but rather a spiral on a sphere. “I was afraid to go and tell him,” said Weiss.
PhD students in math are often in their early or mid-20s. They’re tiptoeing into the profession; most remain doubtful about pretty much everything. Where are my blind spots? Am I allocating my energies right? How can I know whether I’ll make it in this field? In this cloud of uncertainty, the advisor’s every word can resound like thunder, a divine answer to an unspoken question.
But the advisors are older, settled, and (most significantly) tenured. For them, a conversation is just a conversation, a comment only a comment. “No, that’s wrong” means “No, that’s wrong”—not “Please spend the next week wallowing in feelings of worthlessness.”
This is the age-old gap between student and teacher. A moment’s thoughtlessness for the authority figure can feel like a lifetime sentence for the pupil.
In that light, it’s easy to understand Weiss’s trepidation. She faced a steep power gradient. The pending conversation would be one of the highest-stakes encounters in her career to date. And for Coxeter, it would be a Thursday.
So how did the conversation go?
Coxeter’s response was pure delight: “Haaaaaaaa! Look at that!” he let out with glee.
Here comes the switch in perspectives. The duck is suddenly a rabbit.
If you’re an advisor, you’re not there to assert your superiority. You’re there to learn truth, to explore reality, and to help your advisees join the game. If you’re good, you look past bruised egos and smarter-than-thou competitions; if you’re really good, then you model curiosity and enthusiasm, helping remind your students why they got into academia to begin with.
For Coxeter, I’m sure this was the only natural reaction: gleeful pleasure at a new truth brought before his eyes. But for Weiss, the moment must have felt like a bonus check when she’d feared a pink slip.
And here we come to the closing credits scene, with the student-advisor relationship given a physical form:
Weiss married the following year and Coxeter gave her a beautiful glass ball with spirals winding around it as a wedding gift.
Your advisor is there to help you see things: patterns in the literature, paths of possible inquiry, norms in the profession. On rare and thrilling occasions, you can reverse the relationship, showing new things to your advisor—your findings, your progress, the fruits of your hard work.
Here, Coxeter transfigures that collaboration into a concrete, literal, lovely object. His gift to her honors hers to him. Advising, in this way, is an exchange of beautiful ideas, the giving and the receiving of visions.