For example, in the problem, "Write a persuasive essay in favor of year-round schools," the goal state is not clear because the criteria for what constitutes a "persuasive essay" are vague and the allowable operators, such as how to access sources of information, are not clear.
Only the given state is clear–a blank piece of paper.
When the goal of an educational activity is to promote all the aspects of problem solving (including devising a solution plan), then nonroutine problems (or exercises) are appropriate.
In a nonroutine problem, the problem solver does not initially know a method for solving the problem.
For example, for most adults the problem "589 × 45 = ___" is a routine problem if they know the procedure for multicolumn multiplication.
Routine problems are sometimes called exercises, and technically do not fit the definition of problem stated above.
The definition of problem solving covers a broad range of human cognitive activities, including educationally relevant cognition–figuring out how to manage one's time, writing an essay on a selected topic, summarizing the main point of a textbook section, solving an arithmetic word problem, or determining whether a scientific theory is valid by conducting experiments.
A problem occurs when a problem solver has a goal but initially does not know how to achieve the goal.
For example, "3 5 = ___" might be a problem for a six-year-old child who reasons, "Let's see. That makes 4 plus 4, and I know that 4 plus 4 is 8." However, this equation is not a problem for an adult who knows the correct answer. It is customary to distinguish between routine and nonroutine problems.
In a routine problem, the problem solver knows a solution method and only needs to carry it out.