When Richard Rusczyk became interested in math competitions as a middle schooler in the early 1980s, the contest problems looked nothing like the ones in his math classes. He couldn’t find any book to guide him — there were only the problems themselves.
In some of the more advanced competitions he participated in as he moved on to high school, he couldn’t solve a single problem. Gradually, though, he figured out how to “kind of connect the dots, and back out what was actually going on,” he said. He learned a lot of math, but also something he considers even more important: the art of problem-solving.
Later, as an undergraduate at Princeton University, he saw classmates struggling in math classes despite having gotten perfect scores in high school. Their earlier classroom experiences had taught them to memorize a grab bag of tricks, he said. “When you get to college, that doesn’t work anymore.”
So Rusczyk and a competition-loving classmate, Sandor Lehoczky, set out to write the book their 13-year-old selves would have devoured. The resulting two-volume series, The Art of Problem Solving, opens by addressing readers: “Unless you have been much more fortunate than we were, this book is unlike anything you have used before.” From the start, the books sold 2,000 copies per year — “enough to cover rent,” Rusczyk said. Word of mouth grew, and over the 30 years since, well over 100,000 math enthusiasts have bought copies.
Today, Rusczyk’s company, Art of Problem Solving (AoPS), offers not just a large array of textbooks but also online and in-person math classes for “ambitious problem solvers” that serve nearly 25,000 students each year. These courses include both contest prep classes and subject-matter courses, but they have a common goal of fostering a problem-solving mentality. The company is currently expanding its elementary school materials, called Beast Academy, into a full curriculum, with the goal of bringing the problem-solving mindset to more than just self-selected math lovers.
This mindset “should be baked into the curriculum,” Rusczyk said. “It shouldn’t be the thing you do on every third Friday.”
Quanta spoke with Rusczyk about how to turn math learners into problem solvers. (In the interest of full disclosure, our interviewer’s child has taken AoPS classes, and her sister taught AoPS summer camps online in the first year of the pandemic.) The interview has been condensed and edited for clarity.
Your Beast Academy textbooks are comics, and you introduce concepts through story. The characters are talking about their math homework on the school bus, or they’re in woodworking class, or they’re on a field trip. What made you choose that approach?
You can’t lecture a third grader. You need to have a back-and-forth. The comic book structure we use has little kid monsters in conversation with each other, parents, teachers, the different characters in the universe.
So you can model exploration, you can model overcoming challenges, you can model being OK with being wrong. You can create the environment for the child emotionally and intellectually. Every year we have parents sending in pictures of their kids dressing up as various characters for Halloween. They are putting themselves in these spaces.
We spent months trying to figure out: What is our delivery mechanism? We had 150 pages of worksheets, and we’re like, “No, this doesn’t work.” And then in one five-minute stretch, someone said comic books, and someone else said monsters. And we got a fantastic artist and started building out the books.
The lessons you’re trying to teach seem to go far beyond any specific math content, or even specific problem-solving techniques.
One of the main things we’re trying to get across is just the mindset of openness and willingness to engage with things we don’t understand at first. This is something kids are naturally inclined to do. But then something happens during elementary school, particularly in math classes, and we train that out of them.
We’re trying to encourage kids not to lose this curiosity or get into a mindset where the goal is to do everything perfectly. Because we have machines for that now. When we set kids up to compete with computers, we’re setting them up for failure, because anything a computer can do, it’s going to do better.
Within Beast Academy, the kids have different strengths. There’s one that’s wacky and does outlandish things that are sometimes not right, but sometimes really insightful. There are characters that are very precise and organized. And there’s a character who emerges over time as just plain brilliant. These are all different aspects of approaching different types of problems.
Your materials for older students don’t incorporate a storytelling framework. But one striking thing about them is how each new chapter or class session begins not by introducing concepts, but with a collection of problems. What made you choose that format?
This was how I learned math. It was a pretty powerful way to learn.
When I started experiencing high school math Olympiads, it was two years of getting zero right on every single test. That was really frustrating. But it was safe, because it was a math contest, and who really cares? It wasn’t the first-year math class in college, staring at four problems and thinking, “I am not going to be able to do this, I am not going to be a scientist, I’m not going to be an engineer.”
That’s the experience our educational system gives to a lot of students. They think they’re not good enough, because the first time they’ve had this experience is when they get to college. They’re good enough, they just haven’t been prepared.
So we show the problems first. If a student discovers math for themselves, it becomes their math, instead of just something that was told to them. They’re not always going to get there, and that’s fine. Or sometimes they’re going to do it very differently than we did. That’s great too.
Your classes tend to attract kids who are already excited about math, and that in turn attracts teachers with strong math backgrounds. It’s one thing to make a system that works well for such enthusiastic and experienced participants, and another to make something that will work in classrooms everywhere. What challenges do you anticipate in scaling up your Beast Academy materials to a full curriculum?
We are approaching it first as a learning experience for us. We have a strong perspective on a certain type of student, and a strong conviction about some of the approaches we think should be taught to students. As to how to best deliver those resources to teachers and students in different environments, that’s something we’re more than humble about.
I’ll step back further and say I believe a lot of the troubles in education right now are technology companies going to schools and saying, “This is how you should do things.” It has to be a partnership between the content providers and the most important delivery mechanism these kids will ever have, which is the teacher in the room and the other kids.
Two or three years ago, we started working with schools using Beast Academy as a supplement, and that’s been pretty successful. But to reach more students and have a deeper impact on them, you really want to be the entire experience.
When you say that Beast Academy has been successful as a classroom supplement, how do you measure that?
We just had a study completed in a school district in Minnesota. It was a little over 1,000 students in three groups: a “gifted” group, that passed some test; “Rising Scholars” students, who I think are defined as kids from diverse communities that didn’t pass this test but were close; and other students. They looked at the students’ performance on the Minnesota [standardized] test, and how that varied with the number of lessons they did on Beast Academy online. And they found a very strong relationship — the students who did more than, like, 150 or 200 lessons grew by a much larger margin than the kids who did 15 lessons, or no lessons. One really interesting thing is, the effect size was largest in the Rising Scholars group.
Who chose how many lessons kids did — the teachers, or the kids themselves?
It was during the pandemic, so my guess is a little of both. The outliers are almost certainly kids choosing it themselves. Whether this is revealing that the material teaches the kids or the material unlocks the kids, I’m not sure it matters, right? You have to give them material that’s going to make them want to do it. Getting the student to a place where they are interested in struggling with whatever you’re showing them, that for a lot of kids is the whole game.
There’s a lot of debate in educational circles about whether kids at both the high and the low end of performance are best served by being put on separate tracks or the same track. It sounds like you feel pretty strongly about giving extra challenge to kids who are ready for it.
We want to give students the materials that are most suited to help them realize their potential. If you give students material that is not speaking to them, you’re not giving them the opportunity to realize that potential.
When you remove advanced programs, you remove them for all students. So there’s going to be some kid who’s brilliant, but she will never know. And that’s a missed opportunity for her and for us, because these are the highest-leverage people in terms of making medical and technological advances.
Creating those experiences also helps the students find their people. Part of what we do with Art of Problem Solving is our online community. For some students, it’s the only place where they feel safe expressing a love of math and science, because it is not part of the culture of their schools.
When I went to math competitions for the first time, the thing that resonated with me was, not only were there other kids who liked the same geeky stuff I did, there were adults who were excited about me being good at math, and they weren’t my parents, they weren’t my teachers. They weren’t required by profession or relation to be happy that I could do math. I had never seen that before.
Math competitions can be great for kids who are naturally competitive, but that’s not all kids. What can we offer the other kids?
It’s one of the great failings of the math community that the primary way you can explore deep interest in math is through competitions. When I was a student, contests were the only game in town.
This has gotten less true in the last 10 to 15 years, which is great. Now there are summer camps that are not contest-focused, and math circles that came out of the Eastern European tradition where professors work with the top students in their city.
I started one of these math circles at UCSD here in San Diego before I started Art of Problem Solving. And we had Efim Zelmanov, a Fields medalist, come give a talk. This was joyous, beautiful math — he was just so magnetic and happy to be there. So I thanked him for coming, and his answer was, “Well, I’m here to do this because this is what people did for me growing up.” And I’m sitting here thinking, I have exactly the opposite answer. We’re building these things because we didn’t have this sort of stuff.
It seems like Beast Academy, the imaginary school in the comic books, is the kind of place you would have dreamed of attending as a kid. You’ve said that some kids dress up as their favorite Beast Academy monster for Halloween, but what about you? Is there a monster you especially identify with?
Bits and pieces of various characters. But I might have identified most with Fiona [the math team coach]. In her day, she was pretty strong. But her interest is in sharing beautiful, interesting things with students, and helping them become stronger than she was.
