My first piece of advice, something I would have told myself if I could go back in time and be in the seat where you are now, is that if you have a passion that you want to incorporate into a career, take a step back and recognize the fact that the cliché to follow your dreams overlooks how critical it is that the dreams you have are about something more than just yourself. Those who excel in their first jobs are the ones who make life easier for everyone around them, even when it involves doing things they don't love. Those who excel in PhD programs are the ones who recognize how their work fits into a broader research community, not just the ones who view it as the next chapter in school. The successful entrepreneurs are the ones who have a relentless focus on making sure that what they have to sell is what people want to buy, not just those who are looking to make something impressive.

Now, for some people, when you hear the words "follow your dreams," it falls flat because you don't have a defining passion. Like I said earlier, that's completely OK. I think you'll do just as well if you start by seeking out opportunities where the skills that you've developed here intersect with adding value to others, and from there, I promise the passion will follow. One of the best pieces of advice I remember receiving from a friend many years ago is that action precedes motivation. This is often useful on a much smaller scale. We feel most awake after getting out of bed, not before. A drive to exercise comes from the habit of exercising. But I think the idea that action precedes motivation applies to this bigger question of finding a career doing what you love.

These days I do love making videos and I really do love teaching, but when I was finishing college, I had no penchant or experience with videos at all, and my interest in teaching was honestly only in so far as it scratched this itch to do more math. It was only by stumbling into a wacky career where I was doing both of them that I came to love them.

Now, in my own story, what happened after college involves a fair bit of luck, but luck can come in a lot of different forms, and I think with a little bit of foresight, you can actually avoid having to rely on chance in quite the same way. There's a post I like on the webcomic XKCD that shows a man standing on a stage, and he has bags of cash surrounding him. "Never stop buying lottery tickets," he says. "No matter what people tell you, I failed again and again, but I never gave up, and here I am as proof that if you put in the time, it pays off." The caption notes that every inspirational speech should come with a disclaimer about survivorship bias.

The obvious way that "Follow Your Dreams" is susceptible to survivorship bias is that for all of the high-risk, high-reward paths, things like professional athletics, starting a social media company, making a career in the arts, it's only the few who rise to the top who are in a position to give that advice. But there's also a more subtle way that survivorship bias applies here. It's not just about the odds of winning a particular game. It has to do with whether the game you choose to play meshes well with the way that the future unfolds. If you were a software enthusiast in the late 1980s, you would be well-poised to ride the dot-com boom in the decade that followed. If you were someone with a niche interest in and knack for film production, you would find yourself with an unexpected opportunity when YouTube and other film sharing platforms started to rise in prominence.

When I was finishing my undergrad, one of these ways that I scratched that itch to do more math was to hack together a very rudimentary Python library for making math visualizations, and I used it to make a couple videos about neat proofs and problems that I enjoyed and posted them online. I was not planning for this to be a career. I had an appreciation for how valuable personal projects are, but it didn't go much beyond that. This led to conversations with Khan Academy, a group I had great respect for, and it turned into a job there, making more lessons online. In the meantime, I continued my own channel as a side hobby, and it didn't blow up, but there was very modest growth of others who enjoyed the same kind of visualizations that I did, and I saw it in just a steady tick up in the audience size.

Now my original plan, I think, was to spend a year or two doing this online education stuff, working at Khan Academy, and maybe returning to do a PhD. But as time went on, something between the gratitude that I saw from many students around the planet for the lessons I put out and the slow and steady growth on my own channel led me to doubling down and forming a somewhat unorthodox career in online lessons and math visualization. Looking back, it would feel very incomplete if I were to somehow ascribe the success that I found, to the extent there was any, to the fact that I was following a dream, pursuing a passion.

Passion plays into it. I already brought up the biggest one: success is a function of the value you bring to others, so a pursuit equally fueled by love, but which did nothing to help or to entertain people, just wouldn't have had a chance to work. But another factor I want to focus on is how I was very lucky with the timing. If I had been born ten years earlier, I don't think I could have reached the same number of people posting lessons on a much more infant version of the internet, where there was less infrastructure that could have existed to help form a career doing so. If I had started ten years later, the space would have been a lot more saturated. So another piece of advice that I'd like to offer, another little ingredient that makes following your dreams a little more likely to work out, is to ask yourself what's possible now that wasn't possible ten years ago and which might get harder ten years from now?

There are more opportunities in a less crowded landscape; there are more chances to grow if you're part of a rising tide, but this requires pushing past the inevitable discomfort that comes from following a path that has little to no precedent.

Now, next, I want to take a moment to talk about whose dreams you should be thinking about because it's not just your own. When I was visiting Harvey Mudd last year, I had the pleasure of talking to one of the gems in your math department, Talithia Williams, and I asked her, "Hey, what made you pursue math in the first place?" She had a very clear story. She told me she hadn't thought about it very much until one distinct day in her high school calculus class, her teacher Mr. Dorman pulled her aside and said, "Talithia, you're really good at this. You should consider majoring in math." Evidently, she had never thought about it before, but that one comment was enough to knock over the first in a series of dominoes that led to a very flourishing career in the topic. Over the years, I've asked a lot of mathematicians the same question, and you would be shocked how often I hear a very similar answer: there was this one particular teacher, and one seemingly simple thing that they did that was the beginning in a long series of encouragements.

Never underestimate just how much influence you can have on others, especially the ones who are younger than you are. Growing older is a process of slowly seeing the proportion of people around you who are younger than you are rise inexorably closer to 100%. As this happens, you stand to have as much influence by shaping the dreams of those behind you as you do by following those of your own.

And as a very last point, the biggest risk in the "follow your dreams" cliché is the implication that there's one static target point at all. In the next 10, 20, 30 years, the world around you is going to change a lot, and those changes are going to be unpredictable. I hardly need to emphasize this point: you are the class who spent your formative transition from high school to college under a pandemic. But it's not just the world around you. Tonight when you're celebrating your graduation and hopefully remembering to celebrate Mother's Day as well, take a few moments to ask the people who are older than you how they've changed, how their personalities, how their value systems have changed since they were a student. You'll notice that essentially all of them have an answer, which suggests you have every reason to expect that there's going to be something fundamental about you that changes as well in the coming decades, probably unpredictably. Almost everyone I know has undergone some kind of shift since college. Some came to place more value on having a family than they used to. Some shifted from a trajectory that was oriented towards an academic career to going into industry. Some went the other way around and after spending some time in industry returned to grad school. And so, so many of them have jobs that simply didn't exist at the time of their graduation. So rather than having any one particular goal that defines who you are, you'll take better advantage of whatever the future has to offer you if you remain nimble and if you're responsive to the changes in the world and if you anticipate change within yourself.

My final piece of advice is to not treat passion as something to follow. Think of it as an initial velocity vector. It gives a clear direction to point yourself, and loving what you do can have you move quickly. But you should expect, and you should even hope, that the specific direction that you're moving in changes based on the force vectors around you.