lessons from year one as a college math professor

On day one of the fall semester, still in the heat of a mountain-pine perfumed August, I walked into the classroom and it fell silent. Twenty pairs of human eyes, full of light and elasticity, fixed on me, waiting.

That's the day I discovered the awesome responsibility of teaching: to monopolize human time and attention for one hour every other day, times forty, for ~twenty human beings. 280 hours per student is a lot to spend on one topic. I discovered that teaching is a sacred trust, a sacred responsibility. I can’t afford to squander human time or human awareness. It is a priceless treasure.

Professors can make a difference. We can kill any text, any mathematical theorem by making it as dry as the remainder of a biscuit after a Renaissance era crossing of the Atlantic. Or we can make mathematical ideas come alive for students, so that they can talk to their friends about the signal processing techniques Renaissance Technologies uses for their options trading strategy during a bear market.

I take it seriously. Teaching intelligent young minds is an enormous opportunity and a responsibility that I feel I rarely live up to.

The university at which I teach provided a month-long intensive orientation complete with a seminar in pedagogy and methodology, constant guidance from veteran tenured professors, observation and counseling, and new professors thinking through and practicing together every dimension of teaching, learning, testing, from the most practical to the most theoretical.

And then four sections full of college juniors majoring in physics, all the engineerings, and chemistry were set loose to be all mine. How did it go? I can’t explain mathematical concepts with perfect economy of time and words. Sometimes I am rushed. Hopefully my writing on the whiteboard was breathtakingly elegant, rapid, and precise. Here are ten lessons from my first year of teaching.

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1. There’s a myth that learning is meant to be serious business.

When you see someone learning it’s usually a person bent over a book. Rarely do you see someone in the throes of a fully belly laugh, eyes watering from laughing, but our brains love memorable moments. Humor is a learning juggernaut. The hippocampus determines whether something is worth filing away depending on if your amygdala had gotten worked up about it or not. So, the best learning is done in a safe and stimulating environment.

I swore I would not do to my students what had been done to me on occasion during my education. That is still fundamental in my approach: Whatever I do, don't bore them! Life is too short, too precious, for boredom. Boredom is a crime, a sin, a death to soul, reality, truth, life.

2. If you want to get a duck wet, you can't pour water on him only once. Three times is ideal.

Every important point gets cycled back three times in my lectures over a period of days or weeks, in different ways and forms, because of what I call the duck's back principle: if you want to get a duck wet, you can't pour water on him only once. Three times is ideal. That lesson lives in my teaching.

3. Think in bets.

I’ve written about this previously. Our tiny primate-evolved brains can’t comprehend the complexity of the world, so we simplify it. When A caused B it caused C is easy to understand. But the world is rarely this simple. Life is more in superposition—two states at one time in some percentage of both.

Thinking in terms of bets turns the linear reasoning above into a 40% chance B succeeds because of A when A has a 35% chance of happening, but there is also a 5% chance of something I didn't think of. This very quickly maxes out our cognitive ability to visualize our world model. But it is exactly this difficulty you must not shy away from, as tempting as the simpler model is, because this nuance is where objective truth lies.

Probabilistic thinking helps us learn when to ignore our senses and rebuild intuition and when relying on intuition closes the learning gap faster so we can stand on new ground. It’s a tricky balance to strike, and one an incomplete 21-year-old pre-frontal cortex would be keen to get exposure to now.

4. Education should not aim at putting knowledge into the soul, but at turning the soul toward right desires.

There’s a pivotal moment in Book VII of The Republic during an argument with Glaucon where Socrates asserts that education is presumably held to put knowledge into the soul that isn’t currently in it.

In his customary frustratingly circular logic, Socrates reminds us that there are two main types of disturbances to our eyes: one when transferring from light to darkness and the other from darkness to light. In both cases, for a brief moment we are unable to make anything out with clarity until our eyes re-adjust. The same thing happens to our intellect in periods of confusion. It’s natural to consider whether the confusion is because we are coming from a brighter life into darkness or moving from a lack of learning (darkness) to dazzling brightness. The first soul is the teacher turning the eyes of the second soul—the learner—towards the light.

There is an art of this turning around, concerned with the way in which this power can most easily and efficiently be turned around, not an art of producing sight in it. Rather, this art takes as given that sight is there, but not rightly turned nor looking at what ought to look at, and accomplishes this object.

The goal of education is to drag every man as far out of the cave as possible. Education should not aim at putting knowledge into the soul, but at turning the soul toward right desires.

5. I dig immediate feedback loops.

Many things in life and especially professional organizations have an inherent lag. In my time at SpaceX I once produced a training tool to better estimate the number of labor-hours spent on a specific type of maintenance performed on one of the propulsion systems. I didn't realize until months later that my supervisor had been using it weekly and actually relied on it for real-time status updates.

Students are kind of like the medieval Christian art children that are just depictions of little diminished adults with remarkable self-expressions—that is, they wear expressions on their face like it’s a plague. If it looks like the bubonic kind they are telling me class is heading towards a tumbleweed situation and I need to change course to explain it another way. Whatever I’m doing is not working. But their faces can also look like I just showed a union solider in the civil war a ten-pound bag of coffee—that is, everything in the world makes sense and is okay for the moment.

The best teachers are able to extract this immediate feedback out of students by creating an environment where it is freely given and then use it to dynamically change the heading of the course.

6. Use the lower register of my voice.

Probably too personal to generalize to many people, but I’ve found using this as an internal cue allows me to do a few things better than when I don’t focus on it. I speak slower and enunciate my words better. When responding to a question, this cue somehow forces me to take a longer pause before answering. That extra few seconds develops a clearer stream of consciousness (opening the class for questions, by the way, is scary because I am no longer in control of which direction we go in).

And best of all, because my brain seems to function with more clarity using the lower register of my voice, I sometimes come up with pretty witty responses.

7. If we're not different, if we have not changed, as a result of my class, let's all just go home.

I mentioned at the beginning not wasting class time. That is very important to me. The institution I teach at estimates a cost of $416K for a four-year degree. I calculated: that’s $52,000 per semester. With five courses of forty contact hours each in a semester students have 180 hours of class time per semester. That's $260 per hour of class time that students are paying to come and look at me. As they walk in the door, I imagine them dropping their payment in one of the USPS big blue drop boxes every time they walk in my class. Was that last hour of class worth $260?

8. Room 5D10 is a podcast studio.

On the good days my class feels like a conversation with a friend. Students are invited to witness this or be active participants, coming on a guided tour of a discovery of mathematics.

On the bad days it feels like I want to end the conversation and immediately switch topics and listen to an angsty playlist. I haven’t nailed all the variables that make class more like the former, but it starts with being great at cultivating stellar nursery environments.

I spend at least 15 minutes per lesson talking about a quote of the day, shower thought of the day, and interesting write-ups I have stumbled upon on my gate kept list of blogs I am loyal to.

We talk about the Red Queen Fallacy, about AlphaFold3 business ideas, about some guy who gets his daily news from a dot-matrix printer, about how love is responding to your partners bids, or Steve Ballmer’s favorite interview questions. During this time, I have never seen an iPhone out. I only see curious—skeptical sometimes—eyes. Stimulating nursery environments are the cure to Jonathan Haight’s Anxious Generation.

9. Set out on daring explorations from a secure base.

The Japanese concept of ikigai is the intersection of what you are good at, what the world needs, what you can be paid to do, and what you are interested in. Tapping into that calling arouses tremendous levels of energy. Do you think there are currently enough avenues in your 20s to find that purpose in life?

I beat into my students more than RFK would beat a dead horse that the key to success is not how smart you are at 21, but whether you keep growing in life. Most people are electric at 21. Fear is yield, not a stop sign. But if I met up with that same person for coffee a decade later, often that light has gone out.

I have given them responsibility for their own learning; I refuse to hold their hand. We can only keep the light bright by trying out a bunch of stuff and seeing what touches our soul.

10. The key to it all can't be taught. It is love. Love students.

I see that my students love me so much, because I love them, and give everything of myself. Not that I am not demanding, terrifying sometimes. I begin the semester with describing my core values, one of which is competency. I demand it. Competency should be as reliable as your car brakes—you never think if they will work or not. I can correct them in the sternest ways. I had four students work for months on a machine learning modeling project, and they had handed me their initial draft. A week later, I handed them a three-page document in a sealed envelope. That letter said lots of things, but the gist was that in my estimation, they had done nothing worthwhile. It was a complete demolition. A few weeks later, they began regularly coming to my office for lunch just to hangout. They produced their best work for me.

When I give love, I quickly discovered the generosity and love in their eyes too. I began to learn how much students will give, how much they will give of themselves, how they will respond, come towards you, if you step towards them.

Any success of any teacher depends on that light, love, generosity in our students. A shocking discovery over this entire experience so far is how 2-3 personalities in a class make impact on the flow and vibes and feel of the class than pretty much anything else. I need them too!

I learned then that the key to all teaching is: I must love my students with a deep, self-giving love. There is a famous phrase in Dante, about how no one loved can escape from loving in return. That is not actually true with sensual love. It is true with selfless, self-giving love. That love is the bond, the link of communication, through which real teaching and learning happen.

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On the final day of the semester, I walked into the classroom and it erupted into conversation: Captain Graham, you must see this blog about the longitudinal effects in Chile!—John Gruden should be your next quote of the day, did you see his speech to the Wisconsin football team!—I got a Melenzana hat for you!—I read Chekhov’s Act IV like you said and I totally disag—… I can’t help but smile inside and out. Twenty pairs of human eyes, full of light and elasticity, fixed on me, waiting. The Latin educo, educare means to lead out, to draw forth. Their curiosity spout is primed, bursting, and I know I’ve made a difference.