At its roots, problem solving is exactly what it sounds like, the process of solving problems. However, problem solving methods permeate the studies of mathematics, science, and technology. The human processes involved in problem solving are often studied by cognitive scientists. Often, problem solving is more than rote memorization or raw calculation. It is an art, which Art of Problem Solving helps people recognize and use. Problem solving skills are often needed on mathematics competitions, such as the American Mathematics Competitions, Mathcounts, USAMTS, etc.
One of the major goals of the Art of Problem Solving website and company is to help develop problem solving skills. This means more than just formula sheets; it is not seeing a routine problem and plugging into the formulas, but seeing an extraordinary problem and attempting to bring together various skills to solve it. Without problem solving, progress is quite difficult; the old tricks won't work for new problems. But with problem solving, the new problem can be solved, too.