Discovering New Ways to Solve Problems Can Be Glorious

Problem-solving. What a wondrous concept. One tribute to our rationality is our problem-solving skills in novel situations. Some problems we solve through trial and error. For other problems, we use algorithms, methodical, logical rules or procedures that guarantee solving a particular problem. It contrasts with the usually speedier-but also more error-prone-use of heuristics, the simple thinking strategies that often allow us to make judgments and solve problems efficiently. By using heuristics and then applying trial and error, we may hit upon the answer.

Sometimes, no problem-solving strategy seems to be at work at all, and we arrive at a solution to a problem with insight, or a sudden and often novel realization of the solution to a problem. It contrasts with strategy-based solutions. Teams of researchers have identified brain activity associated with sudden flashes of insight. When people were given a problem, researchers mapped the problem solver’s brain activity using fMRIs or EEGs. In the first experiment, about half the solutions were brought about by a sudden Aha! insight, which was typically preceded by a burst of activity in the right temporal lobe, just above the ear. When the “Eureka moment” hits us, we feel a sense of satisfaction, of happiness.

When Princeton mathematician Andrew Wiles was able to come up with the solution to Fermat’s last theorem, the moment demonstrates creativity, the ability to produce novel and valuable ideas. Studies suggest that a certain level of aptitude is necessary but not sufficient for creativity. Intelligence tests, which demand a single correct answer, require convergent thinking. Creativity tests require divergent thinking. Injury to the left parietal lobe damages the convergent thinking required by intelligence test scores and for school success, but injury to certain areas of the frontal lobe can leave reading, writing, and arithmetic skills intact, yet destroy imagination.

Robert Sternberg and his colleagues have identified five components of creativity. The first, expertise, is a well-developed base of knowledge and furnishes the ideas, images and phrases we use as mental building blocks. The more blocks we have, the more chances we have to combine them in novel ways. The second is imaginative thinking skills, which provide the ability to see things in novel ways, to recognize patterns, and to make connections. Having mastered a problem’s basic elements, we redefine or explore it in a new way. The third, a venturesome personality, seeks new experiences, tolerates ambiguity and risk, and perseveres in overcoming challenges. Venturing encounters with different cultures also fosters creativity. The fourth is intrinsic motivation, which is being driven more by interest, satisfaction, and challenge than by external pressures. Creative people focus less on extrinsic motivators than on the pleasure and stimulation of the work itself. The fifth, a creative environment, sparks, supports, and regulates creative ideas. Creativity-fostering environments often support contemplation.

We seek evidence confirming our ideas more eagerly than we seek evidence that might refute them. This tendency, known as confirmation bias, or searching for information that supports our preconceptions and ignoring or distorting contradictory evidence, can hinder problem solving. Once we incorrectly represent a problem, it’s hard to restructure how to approach it. Fixation is the inability to see a problem from a new perspective. Two examples of fixation are mental set, the tendency to approach a problem in one particular way, often a way that has been successful in the past, and functional fixedness, or the tendency to think of things only in terms of their usual functions. As a perceptual set predisposes what we perceive, a mental set predisposes how we think. Sometimes, a mental set based on what worked in the past prevents us from finding a new solution to the problem. Functional fixedness is demonstrated when a person may scour the house for a screwdriver when a coin would have turned the screw.

I encountered an interesting video on Ted-Ed that explains Vedic math, an ancient Indian method. The principle behind Vedic math is (10a+b)(10+1), which translates into 110a+10(a+b)+b (a is the coefficient). In the video, several ideas are introduced, such as the Base Method used to multiply numbers close to the powers of ten, which includes cross-subtraction and cross-addition. The other idea, mental squaring, is also very similar. The principle that underlies this is (x+a)(x+b)=x(x+a+b)+ab. X=base10,100. Another is the Vertically & Crosswise Pattern for 2 Digits.

The question posed at the end was also something to think about: Whether you want math to be dull and boring or fun and interesting, the choice is yours.

This was a very enlightening video that really requires you to think deeper than the usual problem-solving methods. It really shows how we can quite literally be "stuck" in solving a problem rather than realizing the world of possibilities that lie beyond our usual thinking strategies.

For your viewing pleasure, the video is below:

I encourage you to watch it. It might evoke an "ooh" or an "aah" at some point.