"The ability to decode patterns is a key that opens many locks. Whether you're talking about data security, business, medicine, chemistry or even art, seeing patterns means seeing solutions to real world problems."

If you think mathematics is all about taking a problem and finding a solution, think again. As number theorist Sharon Garthwaite explains, the subject is far more involved than that. "The field involves problem solving, proof, and, at its most advanced level – patterns," says the professor of mathematics.

Garthwaite approaches teaching as an evolutionary process, beginning by having students solve complicated problems that have been solved before. Finding the solutions allows them to master known techniques and truly understand definitions and theorems. She says these are the skills they'll need to tackle the next stage: proof writing – showing how and why the solution is the only or most efficient answer.

"This is where language comes in," explains Garthwaite. "The answer isn't five anymore, it's multiple paragraphs explaining why the answer is five." By mastering proof writing, says Garthwaite, students become prepared to devise questions of their own. The exercise often turns out to be far more difficult than any of the students have imagined.

She also encourages them to tackle problems for which there are no known solutions. One classic example is the twin-prime conjecture, which states that an infinite number of primes appear as pairs with the second number two greater than the first, like 11 and 13. Mathematicians know the number of primes is infinite, but they don't know how to predict when large primes occur. Garthwaite explains that large primes are useful when it comes to cryptography and data security. "The challenge," she says, "is to produce a pattern that gives you every pair of primes or even the reason behind the pairs of primes. That way, we'd be able to efficiently access any prime we wanted to work with."

Many private and government agencies, such as the National Security Agency, are interested in employing graduates who have mastered Garthwaite's type of education. "The ability to decode patterns is a key that opens many locks," she says. "Whether you're talking about data security, business, medicine, chemistry or even art, seeing patterns means seeing solutions to real world problems."

*Posted October 3, 2013*

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