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).

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.

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.

## 8 thoughts on “My Favorite Student-Advisor Moment”

Students are brought up to think asking help from anybody else is cheating (“paddle your own canoe”, although the fastest canoes have two paddlers), and I think that’s why they don’t ask for help explicitly.

2. Through the years, I eagerly awaited my students’ “Aha! moments”.

The existence of the mirror image moment in the teacher/advisor is significantly more profound.

Thank you for focusing on this story for us to see.

3. nmpdx says:

I didn’t go to grad school so never had a PhD advisor. But I did have the best ever prof in Chemistry who would have had this reaction too. He delighted in that moment when a student gets something, esp if the student is fully flabbergasted or overjoyed by the thing they just got. His answer to “um, er, hm, do you mean…(fill in the student’s timid question)?” was always either “Hey, yeah, you got it! Isn’t it great?” or possibly “Well, no, not quite that…here, let’s try it again together…” I’m sure there are many university corridors full of anger and resentment and confusion and trauma from inadequate, unbalanced, improper, harmful advisor-advisee interactions, but when they go right, it really does feel like the way the world should work.

4. aleshia says:

I love this . I recently completed my grad and didn’t have an advisor who gave much of a positive affirmation at any time. Anywho I completed but it wsaa a hell of an experiene as compared to the internship in which I was alloted a much better advisor. I truly believe what you wrote here and that bad drawing is not at all it is very accurate and relatable !

5. Matt says:

When I returned to my advisor just a couple days after receiving my problem and told him that “I think I’ve got it” and then proceeded to describe a strategy, my usually kind advisor boomed “THAT IS NOT A PROOF.” Many difficult months later, the last part of the “10,000 piece, all-white, no border jigsaw puzzle” fell into place, the sky opened and God spoke.

One can take graduate courses forever, but until one has this experience you do not feel this power of creation. A mountain climber once confirmed my metaphor that it takes an entirely different skill-set to make the final assent than what it takes to get to the base camp just below.

1. Joe Morin says:

Getting to the summit is optional. Getting back to base camp is mandatory.
Joe