Terence Tao’s 5 Tips for Mathematical Problem-Solving

Terence Tao loves a tough problem—the kind that makes you sweat. He’s taken on some of the most baffling equations known to humankind, and he’s won. In the process, he’s been part of major advances in mathematics and sparked lifesaving innovation in the field of medicine. His work in number theory, prime numbers, and quantum mechanics has solidified his reputation as one of the most powerful minds on the planet, all while opening countless eyes to the beauty, artistry, and universal relevance of mathematical thinking.

Terence earned his Ph.D. from Princeton University and joined the faculty at the University of California, Los Angeles (UCLA). He has received the Fields Medal, the highest prize that mathematics has to offer; a MacArthur Fellowship, also known as a “genius grant”; and a Royal Medal from the U.K.’s Royal Society, the oldest continuously running scientific academy in the world. He’s also authored or coauthored more than three hundred research papers that have been cited more than eighty-three thousand times, and he’s published more than a dozen books.

“There is a certain way in which mathematicians approach problems,” says Terence Tao. “We abstract them. We break them up into pieces. We make analogies. We try to find connections with other problems.” You can begin to harness this creative approach to problem-solving using a few simple strategies.

1. 1. Choose wisely. Mathematicians can often select the problems on which they work. They’re also free to change the parameters of a problem to make it easier, harder, more generalized, or more specific. You can essentially play around with the dials to modify all sorts of variables. What should you look for in a potential problem? Like any creative field, math is most exciting and rewarding when you’re working at the cutting edge of feasibility, Terence says. You want problems that are just barely outside the range of all your tools.
2. 2. Trim the fat. When he’s working on a problem, Terence will often use mathematical tools of abstraction to simplify things. This could involve taking key components of a problem and representing them with mathematical “objects” like numbers or shapes. In order to do this in a way that makes sense, you first need to figure out which elements of the problem are most important. “We’re really stripping the problem down to its bare essentials,” Terence says. “By moving the inessential components of a problem, you can focus on what’s really going on.”
3. 3. Use analogies. Analogies can help you understand the mechanics of a problem or even its philosophical underpinnings. For example, the process of polling—gathering opinions from individuals to assess public viewpoints—often involves statistical estimations. Terence compares this to testing ocean waters for salt content: To understand the overall salinity, you need to collect results from multiple locations within the ocean. Similarly, you need to collect opinions from across a citizenry to measure a proposal’s political viability. (Mathematical theories, it’s worth noting, can also help correct for problems in polling.)
4. 4. Get attached. If you aren’t motivated to solve a problem, chances are you’re not truly invested in the process. Instead of solving for X or Y, Terence recommends thinking of the endeavor as a search for clues or a battle against a wily enemy. Because solving a problem can be labor-intensive and often involves failing again and again, finding (or ascribing) some semblance of meaning in the process is crucial.
5. 5. Question everything. As a young student, Terence was innately skeptical of the established methods for solving problems. He remembers trying to pass an exam without using the quadratic formula as recommended by his teacher; Terence tried his way first, only reverting to his teacher’s recommendation after spending an hour failing to solve the equation. He urges others to challenge the status quo, even if they turn out to be wrong.

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