There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. Every new discovery in mathematics results from an attempt to solve some problem.
—Howard Eves (teacher, geometer, writer, editor, and historian of mathematics)
Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.
—George Polya (Hungarian who immigrated to the United States in 1940. His major contribution is for his work in problem solving)