Many people associate mathematics with rote memorization of theorems and their application. This may be the reason that creativity is so often associated with the arts and humanities before math. However, mathematical research requires creativity, ingenuity and originality. While this may seem daunting for high school students, Pioneer has designed an innovative academic system to help Pioneer scholars reach their full potential. Supported by Pioneer’s academic system and faculty mentors, Pioneer scholars do creative, original mathematics research as high school students.
What does it take to do research in mathematics? Surprisingly, experience in advanced math courses isn’t the most reliable predictor of success. Professor Dresden, a Pioneer faculty mentor who teaches about Fibonacci numbers, explains, “The student I’m looking for hasn’t necessarily gotten straight A’s or taken all the advanced math classes… I’m looking for a student who enjoys puzzles, who likes working on brain teasers or riddles or card games or just figuring stuff out. Because aside from the technical aspects of some formulas and techniques, math is really about seeking patterns and using creativity to solve problems.” Michael (2019, mathematics), a Pioneer scholar from the US who conducted research in number theory, agrees. “You don’t know how to approach [a math problem] until you’ve found the solution. So it’s really just trying a bunch of things and seeing what works and what doesn’t…. It’s really just like a puzzle,” he says. Tobi (2020, mathematics), a Pioneer scholar from Nigeria, also worked with Fibonacci numbers. She chose to represent the sequences visually and filled a notebook with drawings of squares representing integers for inspiration. “Math can be very creative, and you can use it to create new things,” she says.
Dresden believes that the fresh outlook Pioneer scholars bring to mathematics is actually a powerful asset. “Let’s be frank here, I know a lot about mathematics,” he explains, “but because my students are just beginning in this area of research, they can sometimes come up with creative, out-of-left-field, crazy ideas. And every once in a while it works, and it just astounds me.” Ziyang (2019, mathematics), a Pioneer scholar from China, did research on coding and cryptography without any prior experience. Nevertheless, he came to Pioneer with a genuine love of mathematics and an interest in using creativity to solve math problems. “I think math is uniquely misunderstood,” Ziyang says. “In general mathematics is taught like you need to remember formulas… so it’s kind of a fixed way of thinking. But research in mathematics is not like that. You have to explore new things and activate an untouched area of the brain to think about those math problems.”
Attempting original research in mathematics comes with its own set of risks and rewards. For one, there is no guarantee at the outset that a given mathematical problem is solvable. “You may spend a couple weeks working on a project, and it’s too hard, or it’s simply too easy, or it just doesn’t go anywhere,” Dresden says, “But sometimes you’re working on a nice little problem, and you get some results, and it turns out nobody has ever thought of this result before. It’s kind of fun to know that for a brief moment of time, you and your students can see something that nobody else in the world can see or has ever seen.” Zijing (2018, mathematics), a Pioneer scholar who conducted research under Professor Dresden, testifies to this experience. “Sometimes with mathematical research you cannot get the answer. I didn’t have any breakthrough for two days, three days, or longer, because you get stuck, and you struggle with your proofs,” Zijing explains. However, this struggle makes “eureka” moments all the more satisfying. “Eventually, I got the formula, and I felt really excited. When you look at the sequence of a formula and feel satisfied with the result, you think ‘That’s it! That’s the answer!’”
In addition to being creative, mathematics research is a collaborative undertaking. In the Fibonacci Numbers and Visual Proofs concentration, Pioneer scholars contribute directly to the Online Encyclopedia of Integer Sequences. In so doing, they expand publicly accessible knowledge and join a community of mathematicians. Math has always been social for Ziyang, who enjoyed making friends in math clubs at school. At Pioneer, conversations with his faculty mentor and his peers boosted his creativity. “When you’re stuck on a problem or in research, you have to talk to a professor or a friend, because they will let you refresh your mind and come up with new ideas. I think that’s pretty important,” he says. Pioneer’s academic system operates on a cohort model, which ensures that students have chances for productive discussions with both peers and their professor. These cohorts are incredibly diverse, which adds to the learning experience for students. Professor Dresden keeps a map of the world on his wall with tags that show the location of each Pioneer scholar he mentors in a given cohort––California, Florida, Norway, India and Nigeria, to name a few. “One of the advantages of the international cohorts of students is that because they’re coming up through different educational systems, they lend their own unique perspective to approaching research issues,” Dresden explains. Because of the cohort system, students work together in small groups, giving them the opportunity to collaborate and learn from one another.
The experience of doing research in mathematics is unlike anything most students have done in high school. It has little to do with rote memorization or standardized testing. Instead, it is a puzzle to be solved creatively through trial and error. More than learning formulas, it requires collaboration, curiosity, grit, and a willingness to think outside the box. For those students who step up to the challenge, it provides an immense sense of accomplishment and intellectual satisfaction.