A problem, by definition, needs to be solved. Alas for many, effective solution methods do not include crying, complaining, or ignoring the situation. In fact, problem solving often depends on methods and approaches outside our ordinary routine.
Good old George Pólya knew a thing or two about problems and solutions. His masterpiece of mathematical heuristics, How to Solve It: A New Aspect of Mathematical Method, maps out the four essential steps to solving problems. We must always start, of course, by understanding what the problem really is. How else can you solve it? But the second and perhaps most critical step is not, as many believe, to jump right in and start making things happen. Instead, take a moment to come up with a plan about how best to solve the problem:
“We have a plan when we know, or know at least in outline, which calculations, computations, or constructions we have to perform in order to obtain the unknown. The way from understanding the problem to conceiving a plan may be long and, in fact, the main achievement in the solution to a problem is to conceive the idea of a plan.”
To clarify, any given problem probably has several solution methods. Some will be easier than others. Some will be faster than others. Some, depending on what kind of problems we’re talking about, will demand more time or money or even pain than others. The smart problem solver takes a moment to consider the different solution methods. weight the merits of each, and then commit to the best for that specific situation.
Think about a solution method as a means to get from what you have to what you need. For example, say you want to visit the beach. Depending on where you live, this may be an easy problem or a nearly insurmountable one. Once you’ve identified where you want to go (the problem), you can review the different air, land, and sea routes (the solutions) while factoring in complications like weather, traffic, and parking. Unless you prefer the scenic route, your optimal solution will entail the fastest and possibly cheapest journey from your doorstep to the coast. And if you’re smart, you’re going to commit to the best route before you start driving!
In academics and testing, we usually aim to solve every problem as quickly, easily, and accurately as possible. Since problems on tests like the SAT and ACT differ from question to question, it makes sense that the optimal solution methods for each one differs as well. Conventional problem solving methods like the ones learning in math class will serve in some instances but may be too cumbersome or time-consuming in others. Fortunately, savvy students can learn faster approaches to many math problems. Having a variety of problem solving strategies at your disposal opens up the opportunity for faster, easier, more successful problem solving… as long as you take the time to select the right one.
“Every solution to every problem is simple,” asserts author Derek Landy. “It’s the distance between the two where the mystery lies.” If you want to traverse that span between problem and solution as quickly, easily, and safely as possible, follow Pólya’s advice and devise a plan first.