Discovering where light bulb flashes

April 01, 2004, vol. 29, no. 7
By Marianne Meadahl



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The world's top mathematicians do some of their best work in the shower, while cooking, or even while asleep.

That's how several internationally reputed math geniuses describe what happens when the proverbial light bulb goes on, otherwise known as the ah'ha experience, that pivotal moment when they think “eureka,” as the solution to a problem suddenly becomes clear. SFU education doctoral student Peter Liljedahl surveyed 25 mathematicians to determine how those moments are achieved.

Most said how the idea came about - and how it made them feel - held greater significance than the idea itself. “It was Monday morning in the shower, in conversation, while falling to sleep - these are how they remember their greatest ah'ha moments,” says Liljedahl. “Through their use of metaphors and visual imagery in describing these moments, it also became very clear that what mathematicians do is a highly creative process.”

Liljedahl wanted to study what prompted these moments to see if they could be manufactured in a controlled setting, like a classroom. He expected to find something more firmly rooted in their ideas. Instead, the mathematicians often conveyed the essence of these moments without telling him anything about such details.

“Usually the ideas were not that significant,” he says. “But the mathematicians painted these incredible pictures around them that were very personal and had deep pedagogical implications.

“They also showed an incredible respect and acceptance of the fact that a huge part of the mathematical process relies on chance, which has critical implications for problem solving.”

Among survey participants were five Fields medallists, including Italian mathematician Enrico Bombieri, a leading authority on number theory and professor of mathematics at Princeton University. The medal is comparable to a Nobel prize.

Liljedahl, who collaborated with SFU mathematics professor Peter Borwein, based his work on a survey created in the 1940s by Jacque Hadamard, who studied the psychology of mathematical invention.
Liljedahl didn't expect to draw such personal responses. Many mirrored the words of Dusa McDuff, a mathematics professor at the University of New York at Stony Brook and a major participant in calculus reform.

“In my principle discoveries I have always been thinking hard trying to understand some particular problem. Often, it is just a hard slog. I go round arguments time and again seeking for a hole in my reasoning, or for some way to formulate the problem. Gradually some insights build and I get to know how things function. But the main steps come in flashes of insight.

“This can occur while I am officially working. But it can also occur while I am doing something else, having a shower, doing the cooking. I remember the first time I felt creative in math as a student, trying to find an example to illustrate some type of behaviour. I'd worked on it all evening with no luck.

“The answer came in a flash, unexpectedly, while I was showering the next morning. I saw a picture of the solution, right there, waiting to be described.”

Participants also suggested dialogue and learning math through direct contact with people were key factors in how they processed problems.

Liljedahl's work could lead to new strategies for teaching math in the classroom. He has since restructured a math course on problem solving to monitor the occurrence of ah'ha moments. During a trial run, he gave a class of university students more time, more direct contact and encouraged them to keep journals documenting their feelings, and even their failed attempts, during the process. “It's clear that to be able to produce such moments they need room to move and time to incubate and talk things through,” surmises Liljedahl.

Not only is the ah'ha experience accompanied by an emotional response, Lijedahl says that response can be substantial enough to alter the negative belief structures and poor attitudes of resistant mathematics students.

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