Riki Rattner

Laurens Gunnarsen (center) works with his students during a Saturday math masterclass. Without grades or a set curriculum, the class drives to foster a “creative” approach to math.

The warm sun shines down on the neighborhood on a lazy Saturday afternoon as the cloudless blue sky engulfs the town in a sweet, relaxing atmosphere. Light illuminates every plant, flower, and leaf; the birds sing soft lullabies as if they want you to lay down and fall asleep. People all over the town take time to cherish theses artistic aspects of nature. However, somewhere in Palo Alto, a group of young geniuses appreciate a different type of art, tackling a difficult problem or grasping a challenging mathematical concept.

One of these mathematicians is former Palo Alto High School student John Boyle, who has been collaborating with a professor as well other mathematically talented youth regularly on Saturdays for almost two years.

After taking honors math Analysis under Math Department Supervisor Radu Toma at Paly freshman year and AP BC Calculus under Dave Deggeller at Gunn High School sophomore year, Boyle decided to take a different path to continue his study of mathematics. While many high school students enroll in Stanford University math classes after completing the AP BC Calculus course before senior year, Boyle chose not to continue his studies in mathematics at the university.

Instead, Boyle has been collaborating with Professor Laurens Gunnarsen, who was trained as a mathematical physicist with a Ph.D. in General Relativity from the University of Chicago. He currently teaches mathematically gifted youth like Boyle.

“For the last year or so, I've been meeting more or less weekly with a small group of math people and a mathematician,” Boyle said. “Usually we'll have something to talk about. The student will have a topic or a puzzle, [the] student will have some progress on a puzzle to present, or we'll talk about and work on a problem together.”

Gunnarsen founded the class in Dec. 2009 after teaching in Stanford Math Circles, a prestigious math program designed for students K-12 who are serious in math, and has been coordinating the classes almost every Saturday since. He likes to refer to the informal classes as “masterclasses” which are taught in the same style and structure as art classes are taught.

“Laurens had had [the idea] for a long time before he actually put it into practice,” freshman Aaron Slipper said, who regularly attends Gunnarsen’s masterclass. “He believes that mathematics is an art, and that it should be taught similarly to music, following a kind of Suzuki-method plan. For a long time he had entertained the notion that there should be private math lessons, math masterclasses and private math camps.”

According to Boyle, the masterclass does not have tests, grades or a set curriculum. Instead, it is a casual class with a very open and free environment, which allow students to relax and soak in information more effectively.

Saratoga High School sophomore Kevin Garbe, currently taking Trigonometry and Precalculus at his school, met Gunnarsen at Stanford Math Circle, a class for talented math students, around two years ago as well. At the time, Gunnarsen was one of the teachers there before he left to create the masterclass. When he did leave to pursue his own ways of teaching, Garbe continued under Gunnarsen’s instruction. According to him, the masterclass focuses on the art of math instead of viewing it as a duty.

“Compared to the school, the masterclass is more casual and into the derivations and beauties of mathematics rather than following a set curriculum with a set of problems,” Garbe said.

Gunnarsen believes that like arts, mathematics can only truly be learned by approaching it using one’s personal instincts instead of an imposed teaching and rote memory. By creating a learning atmosphere that is relaxing and without boundaries to retain students’ thinking, students learn to think outside the box and approach problems creatively.

“Mathematics is an art,” Gunnarsen said. “And like all arts you can only learn it by doing. But it's crucial to retain your own personal taste and instincts, and a teacher who tries to impose his own on you is doing you no favors.”

Before organizing the masterclass and becoming an active mentor of the students, Gunnarsen studied math and physics in college. He continued his studies in Japan, where he discovered his love for Japanese culture.

“I was an undergraduate math major at Berkeley and then I did my Ph.D. in General Relativity at the University of Chicago,” Professor Laurens Gunnarsen said. “After that I went to Japan to do a post-doctorate, and I loved the place. I stayed there for more than six years, and spent much of my time singing sentimental Japanese love songs on television.”

However, it was not until after Gunnarsen came back from Japan that he decided to mentor students. Through the inspiration of music and the arts, Gunnarsen explored different ways of teaching mathematics.

“I never planned to become a professional mentor to mathematically talented kids, but that's what gradually happened to me after I got back from Japan,” Gunnarsen said. “At first, I was pretty clumsy at the job; I used to lecture at my students. Now, though, I model my practice on that of the great music teachers, and the great game designers.”

After some time with the students, Gunnarsen devoted his time to working with students individually. He established close teacher-student connections to each of them, which he feels is beneficial to their achievements.

“I work with kids one-on-one, and I put a lot of energy into building the sort of relationship with my students that encourages them to relax and play, to be wild-minded and uninhibited and adventurous,” Gunnarsen said. “For my purposes, this is absolutely essential to success.”

Though Gunnarsen’s students learned a great deal throughout the years, Gunnarsen himself discovered his most preferred and effective way of teaching this particular group of talented youth that allows them to formulate their own ways of combatting challenging problems.

“Now I prefer to convey the things I want my students to understand through the medium of puzzles and games,” Gunnarsen said. “ I want my students to see for themselves what's lurking behind these devices. I want them to find their own path to the secrets implicit in them, and I rejoice if their approaches differ greatly from the one I had in mind.”

Gunnarsen believes that it is crucial for his students to be completely immersed and in love with math for them to feel like their work is important, especially with tedious problems that require lots of patience. However, Gunnarsen thinks that obsession for math, or any subject, can serve as an impetus for his students to continue to strive toward a deeper understanding of the subject as well as to maintain their enthusiasm and love for their work.

“I always strive to teach by tantalizing,” Gunnarsen said. “My most important goal is to cause my students to become obsessed by deep mathematical mysteries, because without obsession, you really haven't a hope of doing anything really worth doing. Real, serious math is just way too hard and way too exhausting to do out of a mere sense of duty. Obsession is the key thing for me, and I've noticed that the best game designers are really good at causing people to become obsessed, which is why I'm always trying to imitate their methods.”

Though these math students are taking classes at their schools that are relatively easier compared to the concepts that they explore in Gunnarsen’s masterclass, the weekly meetings have still helped them better understand the concepts that are taught in their regular math classes at school.

“I'd say that my understanding of any math taught in high school was already pretty high when the masterclass started, so it didn't have much effect on my high school math class,” Homestead High School senior Alex Mennen said. “Although, there are a few instances where I initially missed something in high school level math that Laurens pointed out.”

In addition, the way that the masterclass heavily emphasizes the creative approach to a problem as well as more specific topics that are not taught in school have helped students such as Slipper with his understanding of certain material covered in his math class.

“The math I do with Laurens has, to a certain extent, helped me in the Analysis classroom,” Slipper said. “Though [Laurens] stresses the creative side of math far more than he does the problem-solving side, there are some problems that I encounter in Analysis class that I am prepared for by concepts or theorems that I have learned under the provocation of Laurens.”

Though Gunnarsen’s masterclass has helped his students in the classroom in a number of ways, Garbe believes that both the traditional classroom setting and the the masterclass setting have their merits.

“For school it rigorously covers a set of curriculum, including both things I know and don't know,” Garbe said. “The masterclass includes a lot more understanding of the derivations and the methods rather than the implementation.”

According to Slipper, one of Gunnarsen’s goals as the professional mentor of the masterclass is to have a math summer camp. This goal was achieved last summer when the Gunnarsen and the students invited John Horton Conway, John von Neumann distinguished Professor of Mathematics at Princeton University, to stay with the masterclass for about two weeks.

According to Conway, he and Gunnarsen met each other through a series of math camps, where they both shared their love of teaching mathematics and meeting math professors like themselves.

"I used to go to these math camps that go on for five weeks," Conway said. "I would go to one week of each of them, and I met [Gunnarsen] at one of them."

Conway believes that the masterclass provides a great opportunity for mathematically talented youth to meet other students alike, since oftentimes the “math geniuses” are isolated from the rest of their math class at school.

"[They] meet other kids, and often in a math class there is one kid that is the best in math in his class with no one to talk to, and at these masterclasses they are able meet other kids in the same position,” Conway said.

Conway taught the summer math camp not only because Professor Gunnarsen asked him to, but also because he was interested in meeting new young mathematicians. The informal camp was not such a rigid responsibility for him because he mainly talked about what he felt like talking about, according to Conway.

"I taught the summer camp because Laurens asked me to, but also because I was interested in talking to young mathematicians,” Conway said. "[The camp] wasn't so much of a responsibility for me because I mostly just talked about the things I liked."

Not only did Conway receive the Nemmers Prize in Mathematics and the Leroy P. Steele Prize for Mathematical Exposition from the American Mathematical Society, but he is also the inventor of the famous cellular automaton "The Game of Life".

According to Slipper, masterclass students spent nearly the entire day discussing math with Conway. They listened to whichever topics Conway wanted to discuss about, including concepts that are not purely mathematics. At the end of the day, Conway would then stay at one of the students’ houses overnight.

“[It] was an intensive, home-tailored math camp--it was very free form,” Slipper said. “We allowed Professor Conway to discuss whatever he felt like speaking about, including physics, mathematical philosophy, projective geometry, partition theory, quadratic forms and lots more.”

In addition, masterclass students shared their own discoveries with him, and Conway would then review and discuss about their approaches.

“[We] also showed him our own investigations, and let him comment on our approaches and expositions,” Slipper said. “We learned a great deal from him, and enjoyed ourselves thoroughly.”

The summer camp that the students participated in with Professor John Conway was the highlight of their masterclass experience, according to Boyle and Slipper. Students explored mathematics at a deeper level of understanding and also formulated their own mathematical theories.

Boyle believes that the professor’s and students’ talent and passion for math is what brings them together as a team.

“We have fun, and it's nice to have a kind of community, a group of people who are all interested in math and whom you can see pretty reliably every week,” Boyle said. “The common denominator is that we're interested in math and pretty good at it.”